phys.astr
Rapport De 
  
Soutenance De Thèse 
  
M Felipe Braga-Ribas 
  
Rodney Gomes 
  
Mme Antonella 
  
M Sylvio Ferraz-Mello 
  
M Rodney Gomes 
  
Jean-Marc Petit 
  
M Roberto 
  
Vieira Martins 
  
Antonella Mme 
  
M Barucci 
  
Ferraz-Mello M Sylvio 
  
Gomes M Rodney 
  
M Bruno 
  
Sicardy M Roberto 
  
Felipe Braga-Ribas 
  
Felipe Explorando 
  
O S Objetos 
  
Transnetunianos Pelo 
  
Método De Ocultaçoes 
  
Estelares : Prediçâo 
  
Quaoar Observaçào 
  
Roberto Vieira 
  
Bruno Sicardy 
  
Raoul Behrend 
  
Barucci S Fornasier 
  
M A De Bergh 
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
TRÈS HONORABLE) : TRÈS HONORABLE LE PRESIDENT ET LES MEMBRES DU JURY (SIGNATURES)

Keywords: quatorze ocultaçôes por TNOs jâ foram observadas: doze pelo grupo Ocultaçôes estelares, TNO, KBO occultations stellaires, OTN, KBO, 50000 Quaoar, objet de Maclaurin. 11

Le jury, dont la liste est donné au recto du présent document, a assisté ce jour (26 février 2013) à la soutenance de thèse de M. Felipe Braga-Ribas, sur le sujet suivant: "Exploration des objets trans-neptuniens par la méthode des occultations stellaires: prédictions, observations, Quaoar et les premiers résultats".

Le jury a apprécié la présentation orale du candidat, qui a répondu de manière tout à fait satisfaisante aux questions du jury.

Le manuscrit est également apprarue comme bien structuré et très com plet.

Le jury a souligné la quantité et la qualité du travail effectué. Le candidat a mi en place tout ma chaîne qui permet de prédire les événements, organiser les campagnes d'observation, et finalement analyser les donnés et en déduire les propriétés des objets étudié: taille, forme, densité, albédo, et contraints sur la présence d'atmosphère éventuelles.

également être calculées. Par ailleurs, des atmosphères éventuelles provoquent des baisses de signal juste avant et après l'occultation, permettant des détections à des niveaux de pression de quelques nanobars.

Cette thèse décrit (1) les méthodes de prédiction des occultations par une quarantaine d'OTNs, avec des précisions de l'ordre de 30 mas, (2) l'organisation de campagnes mon diales faisant intervenir des dizaines de sites professionnels et amateurs, (3) l'observation, l'analyse et l'interprétation de plusieurs occultations par de gros OTNs. 

Introduçâo

O estudo dos objetos transnetunianos (TNOs) esta intimamente ligado ao estudo da evoluçâo do Sistema Solar. A evoluçâo dinâmica pode ter deixado assinaturas na disposiçâo atual das ôrbitas (anexo A), densidade e distribuiçâo de massa dos TNOs. Os elementos primordiais da nuvem protoplanetâria também podem ter sido preservados por estes objetos, e entender sua abundância, distribuiçâo e processos quimicos aos quais estes corpos foram submetidos é fundamental para entender as condiçôes iniciais de formaçâo do conjunto. Dai a necessidade de conhecer as propriedades fisicas, como tamanho, densidade, albedo e composiçâo destes objetos. Uma ampla e detalhada visâo do que conhecemos sobre os objetos transnetunianos é dada no livro The Solar System Beyond Neptune [START_REF] Barucci | The Solar System Beyond Neptune[END_REF].

Para obter informaçôes sobre estas propriedades fisicas é preciso lançar mâo de diversas areas de estudo como: evoluçâo dinâmica, espectroscopia de superficie, fotometria, medidas térmicas, determinaçâo de massa, etc. Um dos parâmetros fîsicos mais fundamentais é o tamanho de cada objeto. No entanto, como esses pequenos corpos estâo a mais de 30 ua (unidades astronômicas) do Sol, nâo hâ telescôpio em solo que consiga medir estes objetos através de imagem direta.

Utilizando a técnica de ocultaçôes estelares, que consiste em observar o TNO enquanto este passa na frente de uma estrela para um dado observador, procura-se determinar os tamanhos e formas destes corpos com precisoes da ordem do quilômetro e, por consequência, seus albedos e densidades sâo obtidos com grande precisâo. Isto tem implicaçâo direta sobre sua composiçâo superficial e estrutura interna. Além disso, hoje em dia, a ocultaçâo estelar é a ûnica técnica que permite detectar atmosferas com pressâo superficial na ordem do nanobar em torno destes corpos, o que tem implicaçôes sobre os processos térmicos aos quais os objetos sâo submetidos.

Antes de observar uma ocultaçâo estelar por um grande objeto, é preciso realizar a prediçâo do evento (Capitulo 2). Para isto compara-se suas ôrbitas à posiçâo de estre-> las de catâlogos. No entanto, suas ôrbitas apresentam imprecisôes até dez vezes maiores que o necessârio para o câlculo destas ocultaçôes. Como uma das causas destes erros, tem-se a recente descoberta destes objetos em relaçâo com seus longos periodos orbitais (>200 anos), de forma que somente uma pequena porcentagem de sua ôrbita é conhecida.

Seu fraco brilho, em consequência de sua distancia ao Sol, também dificulta sua observaçâo. Para contornar este problema, realizam-se medidas astrométricas dos objetos para determinar quai é a diferença em relaçâo à posiçâo dada por sua efeméride. Esta diferença é aplicada como um desvio constante, para entâo se calcular os eventos de ocultaçâo.

Além disso, o lento deslocamento dos TNOs no céu faz com que eventos de ocultaçâo sejam raros, obrigando a recorrer a catâlogos estelares mais densos, com estrelas mais fracas. Isto traz outro problema, jâ que a precisâo da posiçâo astrométrica destas estrelas também é inferior à necessâria para uma boa prediçâo de ocultaçâo. Para contornar estas dificuldades, fez-se um programa junto ao telescôpio de 2,2m com do ESO/MPG1, equipado com a caméra WFI (Wied Field Imager), no quai se observou os caminhos que dez entre os maiores TNOs percorrerâo até 2015. Utilizando técnicas astrométricas adequadas, criou-se catâlogos estelares até magnitude R= 18, com precisâo melhor que 40 milissegundos de grau (mas) (anexos C e D). Com isto, prediçôes de ocultaçôes estelares por TNOs confiâveis forain obtidas ( Assafin et al. (2010) & Assafin et al. (2012)).

Centenas de eventos sâo previstos para cada ano, e como é impossivel observar todos, uma seleçâo é feita seguindo critérios como magnitude da estrela ocultada e localizaçâo da sombra, conforme a rede de colaboradores. Dias antes de um evento, quando o TNO e a estrela candidata podem ser observados numa mesma imagem, tomam-se medidas astrométricas relativas. Com isto evita-se erros sistemâticos de catâlogos. Assim, precisôes da ordem de 20 mas sâo atingidas, e o local por onde a sombra de ocultaçâo passarâ sobre a Terra é definido.

Campanhas internacionais sâo realizadas, envolvendo dezenas de pesquisadores e astrônomos amadores. Uma pâgina internet para cada evento é publicada2 com informaçôes aos colaboradores.

Neste trabalho, além metodologia de prediçâo exposta acima, discute-se em detalhes todo o procedimento de reduçâo dos dados provenientes das ocultaçôes de Quaoar em 2011 e 2012 (Capitulo 4). A ocultaçâo ocorrida em 04 de maio de 2011 contou com detecçâo em para se ajustar uina elipse que reflete o tamanho, forma, posiçâo e orientaçâo do corpo.

Pela disposiçâo das cordas, Quaoar parece ter uma forma bastante irregular. Neste caso apresentam-se duas soluçôes. A primeira caracteriza-se por ter uma cratera que, como é discutido, séria demasiadamente profunda. A outra soluçâo apresenta uma montanha, fisicamente improvâvel, considerando o tamanho e composiçâo de Quaoar.

Explora-se entâo uma terceira soluçâo, em que se considéra a possibilidade de haver problemas no registro do tempo absoluto em alguns observatôrios. Devido ao seu tamanho, rotaçâo e composiçâo superficial, espera-se que Quaoar esteja em equilfbrio hidrostâtico. Por isso, ele deve apresentar a forma de um objeto Maclaurin. Para se obter tal figura, as cordas sâo deslocadas no tempo, obtendo-se assim a superficie équivalente, que é comparada à curva de equilfbrio de um objeto Maclaurin. Obtém-se assim todos os parâmetros ffsicos pertinentes de Quaoar (anexo H, [START_REF] Apêndice | Size, shape, albedo, density and atmospheric limit of transneptunian object (50000) Quaoar from multi-chord stellar occultations F. Braga-Ribas 1 2 t[END_REF]).

Considerando todas as soluçôes, as seguintes faixas de resultados sâo obtidas: raio équivalente 504 -566 km, achatamento 0,0722 -0,33. albedo gcomctrico no visfvcl 0,102 -0,139 e densidade 1.40 -2,46 g cm-3. Ainda coloca-se um limite superior de 21 nanobar para a presença de uma atmosfera global de metano em Quaoar, mas nâo é possfvel descartar a presença de atmosferas locais.

A metodologia aplicada à Quaoar também foi utilizada em outras ocultaçôes por TNOs observadas pelo grupo, nominalmente Éris (anexo F, (Sicardy et al. 2011)), Makemake Capitulo 2

Prediçâo de Ocultaçôes

Predizer ocultaçôes estelares por pequenos corpos do Sistema Solar consiste essencialmente em comparar, sobre a estera celeste, a posiçâo do objeto num dado momento oriunda de sua efeméride, com a posiçâo de estrelas fornecidas por um catâlogo. Se as posiçôes forem coincidentes, uma ocultaçâo ira ocorrer. E évidente que diversos detalhes, e por vezes dificuldades, estâo envolvidos neste processo, e todos serâo apresentados nas seçôcs que seguem. Nas publicaçôes Assafin et al. (2010) e Assafin et al. (2012) os procedimentos descritos a seguir também sâo discutidos.

Para eventos observâveis a partir da Terra, compara-se a posiçâo de estrelas de um catâlogo com a efeméride geocêntrica de um dado objeto, calculada a cada minuto, por exemplo. Se a distancia estrela/objeto for menor que um dado valor, uma possfvel ocul taçâo é encontrada. Tal valor é definido como um raio de busca, que considéra o tamanho angular da Terra e do prôprio objeto. A uma distancia de 30 ua, a Terra tem raio de 293 masîncluindo-se o tamanho do prôprio objeto, nâo maior que 50 mas para os TNOs, estes valores definem o raio mmimo de busca de uma ocultaçâo sobre nosso planeta (dmin = 318 mas).

No que segue, serâo consideradas posiçôes aparentes no piano tangente à estera celeste com o observador no centro da Terra. A partir de duas posiçôes arbitrârias do corpo nos instantes t\ e l2, a uma distancia D, anterior e posterior ao encontro com a estrela S (ver V D2

(2.1)

(2.2)

Asin(D)

h -ti (2.3) LST = t0 + AR -MSTG = t0 + long (2.4) Para destacar fenômenos lentos, normaliza-se as magnitudes para velocidade tipica dos TNOs durante a oposiçâo v-20 km/s. Com isto, obtém-se a magnitude M* da estrela que sera menor que a magnitude real se a velocidade for menor que 20 km/s, ou maior que a magnitude real se a velocidade for maior que 20 km/s (Equaçâo 2.5). Eventos lentos podem permitir exposiçÔes mais longas, facilitando a observaçâo.

(2.5)

M* = M+ (,2Ôkm~s~^')

À exceçâo de Plutâo, que apresenta ~105 milissegundos de grau (mas) no piano do céu, os TNOs têm tamanhos angulares menor que 50 mas, portanto acurâcia da mesma ordem é necessâria para se fazer prediçâo de ocultaçôes estelares confiâveis. Infelizmente, nem a posiçâo do corpo, nem da estrela sâo conhecidos com tal exatidâo. A origem destes erros e as técnicas para minimiza-los serâo discutidas a seguir. A parte cinza escura représenta a noite no momento do C/A, a cinza clara o crepusculo civil com o Sol a uma altura <-18 graus. O ponto vermelho grande représenta o ponto de maior aproximaçâo (C/A), os pontos vermelhos estâo separados de minuto em minuto.

Na legenda acima, tem-se o nome do objeto e o offset aplicado à efeméride (ver seçâo 2.2). No rodapé sâo apresentadas as informaçôes, na sequencia: data e horârio do C/A, coordenadas geocêntricas J2000 da estrela, C/A em segundos de arco, ângulo de posiçâo em grau, velocida.de da sombra em km/s, a distancia do objeto ao observador em unidades astronômicas, as magnitudes R* e K* (ver Eq. 2.5) e a longitude do ponto sub-planetârio em graus. > 2.1

Catâlogos estelares

Encontram-se dispomveis em bases de dados, como o VizieR [START_REF] Ochsenbein | The VizieR database of astronomical catalogues[END_REF], dezenas de catâlogo estelares que fornecem posiçôes astrométricas para estrelas de todo o céu. Destacam-se catâlogos com estrelas brilhantes: HIPPARCOS [START_REF] Perryman | The HIPPARCOS Cata logue[END_REF] e TYCHO-2 (Hpg et al. 2000); ou catâlogos mais densos em nûmero de estrelas: 2MASS (Skrutskie et al. 2006) e USNO B1.0 (Monet et al. 2003); ou ainda aqueles atrometricamente mais precisos: UCAC2 (Zacharias et al. 2004a), recentemente atualizado para UCAC4 [START_REF] Zacharias | The Fourth US Naval Observatory CCD Astrograph Catalog (UCACf)[END_REF] Tabela 2.1 Alguns dos principais catâlogos astrométricos, com a quantidade de estrelas catalogadas, se fornecem movimento prôprio ou nâo, quai o limite em magnitude V e a precisâo astrométrica média. *0 catâlogo 2MASS refere-se à magnitudes no infravermelho e o limite dado é para a banda J (1.25 /rm).

Uma vez que uni evento é identificado, novas medidas astrométricas da estrela sâo necessârias para melhor determinar sua posiçâo, jâ que erros da prôpria medida do catâlogo e na determinaçâo do seu movimento prôprio sâo facilmente superiores à precisâo exigida para uma boa prediçâo de ocultaçâo por TNO (>50 mas).

Catâlogos poucos densos, somados ao pequeno tamanho angular e a baixa velocidade de deslocamento no céu dos TNOs, levarâo à poucas ou nenhuma ocultaçâo por ano.

Catâlogos densos mas com baixa precisâo astrométrica, levarâo à muitos falsos eventos, o que implica desperdfcio de esforço observacional, e de dinheiro de financiamentos para viagens e equipamentos.

Para superar este problema fizemos, durante um periodo de colaboraçâo entre o

Observatôrio Nacional e o ESO (European Southern Observatory), pedidos de tempo do telescôpio de 2,2m com do ESO/MPG1, equipado com a caméra WFI (Wied Field Imager). Esta caméra, composta por um mosaico de oito CCDs, tem um campo final no piano do céu de 30' x 30' com escala de plaça de 0",238. por estes objetos jâ estâo dispomveis (Camargo et al. 2013). No decorrer do texto os programas de prediçâo de ocultaçâo publicados em Assafin et al. (2010) & Assafin et al.

(2012) e (Camargo et al. 2013) serâo referenciados como caminhos.

Ôrbitas

Desde a descoberta do primeiro objeto transnetuniano (15760) 1992 QBi (Jewitt & Luii 1993) (excluindo Plutâo), pouco mais de 1260 corpos deste tipo jâ foram descobertos, sendo a maioria entre os anos 1999 a 2006. A recente descoberta, ligada ao periodo orbital maior do que 200 anos, faz com que arcos individuais nâo maiores do que 10% das ôrbitas sejam conhecidos. Além disso, devido ao seu tamanho e distancia do Sol, suas magnitudes V aparente sâo superiores a 18 dificultando sua observaçâo, salvo poucas exceçôes.

Por estes fatores, as ôrbitas dos TNOs apresentam erros que podem chegar a 500 mas, ou seja, mais de dez vezes superiores à precisâo necessâria para uma boa prediçâo de ocultaçâo. Para melhor compreensâo do que isto représenta, uma incerteza de 100 mas sobre um objeto a 45 ua, projetados sobre a Terra, représenta uma incerteza de 3300 km sobre a posiçâo da sombra.

O caso mais conhecido é a dériva de Plutâo em relaçâo à sua efeméride desde 2005 (Assafin et al. 2010). Quando um TNO oculta uma estrela, do ajuste das curvas de luz de ocultaçâo observadas é possivel obter a posiçâo do TNO relativa à estrela com precisâo abaixo de 1 mas. Assim, à medida que ocultaçôes estelares por Plutâo foram observadas com sucesso, foi possivel verificar que, a partir de 2005, sua posiçâo real se desvia numa taxa de aproximadamente 30 mas/ano em declinaçâo (DEC) da posiçâo prevista pela efeméride DE413 (disponfvel pelo sistema HORIZONS/NASA: (Giorgini et al. 1997)).

Hoje em dia, a diferença entre a efeméride e a posiçâo real em declinaçâo jâ chega a quase 200 mas. Com observaçôes de 2005 a 2012 a dériva linear em declinaçâo é muito clara (Figura 2.4). Esta dériva pode estar ligada à efeito de refraçâo cromâtica nâo corrigidas nas medidas astrométricas. Em ascensâo reta (AR) observa-se uma oscilaçâo conforme a distancia do evento à oposiçâo. Uma hipôteses é que um erro na distancia heliocêntrica poderia provocar tal efeito.

Com o exemplo acima, fica claro que, ao se fazer prediçôes de ocultaçôes estelares, é preciso considerar o erro da efeméride do objeto. Infelizmente, apenas Plutâo jâ foi observado em diversos eventos de ocultaçâo que permitem verificar tal dériva. Para os demais objetos, a estratégia usada para contornar o problema do erro de suas efemérides, Em declinaçâo a dériva é de aproximadamente 30 mas/ano. Em ascensâo reta o efeito é menos pronunciado (Assafin et al. 2010).

consiste em observa-los para medir suas posiçôes astrométricas e entâo determinar a diferença entre esta posiçâo medida e a prevista pela efeméride (do inglês, offset). Aplica-se o offset medido à sua efeméride para entâo obter as prediçôes de ocultaçôes.

Os offset utilizados em nossas prediçôes vêm principalmente de observaçôes feitas no

Observatôrio do Pico dos Dias (OPD) que pertence ao Laboratôrio Nacional de Astrofîsica (LNA) e no telescôpio Max-Planc 2,2 m do ESO.

Para os novos 39 TNOs e centauros para os quais observamos seus caminhos até 2014, obtivemos mais tempo no telescôpio Max-Planc 2,2 m para determinar seus offset (Camargo et al. 2013) (programas: 090.C-0118, 091.C-0454). Parte destes objetos jâ foi observada no segundo semestre de 2012 e o restante sera observado no primeiro semestre de 2013.

Capitulo 3

Seleçâo, Atualizaçôes e Campanhas

A prediçâo de ocultaçôes é apenas o passo inicial para se atingir o objetivo principal do trabalho, medir o tamanho e forma, e deles derivar diversos parâmetros ffsicos dos TNOs.

Todas as prediçôes dos programas dos caminhos sâo disponibilizadas na internet, acessfveis pela pagina de Bruno Sicardy: http://www.lesia.obspm.fr/perso/bruno-sicardy/.

Obviamente nâo é possfvel observar todos os eventos previstos, uma vez que cada observaçâo demanda um grande esforço logfstico na mobilizaçâo de pessoal, preparaçâo de observadores, telescôpios, instrumentos, que reflete também no uso de grandes montantes de recursos financeiros.

Seleçao

Das centenas de mapas de prediçâo de ocultaçôes gerados, é preciso selecionar quais os eventos mais favorâveis à serem efetivamente observados. Alguns parâmetros dispomveis nos mapas podem ser utilizados, mas ha ainda fatores como maior interesse cientffico, raridade do evento, rede de colaboradores e o clima.

Dos 10 objetos (mais Plutâo) para os quais observamos seu caminho no céu até 2015, alguns tem como fundo estelar o piano da galâxia, e, portanto, ha maior probabilidade de ocultarem uma estrela. No entanto, alguns objetos, mesmo estando em regiôes mais desprovidas de estrelas que significa uma baixa taxa de ocultaçôes por ano, foram selecionados por seu intéressé cientffico, à cxcmplo de Eris, Haumca, Makemakc e Sedna. Isto significa que sens eventos serâo raros mas bastante importantes. Portanto, mesmo que as condiçôes nâo sejam ideais, um grande esforço observacional sera feito na busca da detecçâo do evento.

A quantidade de colaboradores e a facilidade de acesso à telescôpios distribufdos pelo globo é bastante heterogênea (Figura 3.1). Por exemplo, eventos que passem sobre o

Chile, Europa e Estados Unidos, têm mais chace de serem observados do que eventos que cruzem a Africa Central ou a Rûssia. Entâo eventos que exijam telescôpios maiores e com maior incerteza sô serâo observados se passarem por uma regiâo favorâvel. No entanto, se o evento for muito raro ou envolver uma estrela brilhante (mag. V < 12), aventuramos a utilizaçâo de telescôpios em quase qualquer parte do globo.

Figura 3.1 Mapa com todos os sitios que colaboram com nosso grupo regularmente ou jâ colaboraram em uma ou mais campanhas observacionais de ocultaçôes estelares. A densidade de colaboradores pelo globo é bastante heterogênea e composta por profissionais e amadores. As diferentes cores representam diferentes campanhas, mas em gérai um mesmo si'tio jâ colaborou em diversas campanhas.

Nos mapas de prediçâo (Fig. 2.2) sâo fornecidas as magnitudes normalizadas R* e K* (ver equaçâo 2.5), estes dois valores nos dâo uma ideia dos tamanhos de telescôpios que podem ser utilizados numa possfvel campanha observacional. Quanto mais brilhante a estrela, maior o nûmero de telescôpios que podem ser utilizados, uma vez que telescôpios da ordem de 0,25 m de diâmetro de espelho sâo muito mais numerosos que telescôpios de 0,5 m ou maiores. A diferença entre as magnitudes R* e K* também podem nos dizer se uma estrela é bastante avermelhada. Isto costuma ser uma vantagem para muitos CCDs, jâ que sâo mais sensiveis na parte vermelha do espectro.

Eventos mais lentos permitem maiores tempos de exposiçâo. Por exemplo, uma ocultaçâo por um objeto de 900 km a uma velocidade tipica de 20 km/s pode durar até 45 segundos, mas hâ eventos que podem durar até mais de 150 segundos. Assim, CCDs que sejam menos sensiveis e que tenham o tempo de descarga da imagem maiores, que sâo mais comuns em telescôpios pequenos, podem ser utilizados para a observaçâo.

Outro fator importante é a possibilidade de atualizaçôes da prediçâo. Como serâ discutido a seguir, esta é uma etapa fundamental da prediçâo de ocultaçôes. Para isto verifica-se a elongaçâo do Sol. Se ela for pequena e a leste, significa que nâo sera possivel observar o corpo e a estrela a ser ocultada, nos meses que precedem o evento. Entâo, caso nâo exista boa determinaçâo do offset e/ou da posiçâo da estrela, nâo sera possivel lançar grandes campanhas observacionais.

E muito comum utilizar-se a queda de magnitude durante a ocultaçâo como um parâmetro de seleçâo. A queda em magnitude esta relacionada à precisâo fotométrica necessâria na observaçâo, isto é, à razâo sinal sobre ruido (S/R). Quanto menor a queda maior a precisâo fotométrica necessâria, e, por consequência, maior o tempo de exposiçâo necessârio para um dado equipamento. No entanto, no caso contrario, quanto maior a queda mais fâcil é a observaçâo do evento, jâ que um S/R baixo é suficiente para sua detecçâo. Como quase todos os TNOs têm magnitudes R superiores a 18, e as estrelas acessiveis à maioria dos telescôpios têm mag. R < 18, a queda de magnitude é, em gérai, superior a um. Por exemplo, para um baixo S/R= 5, uma queda de uma magnitude jâ séria facilmente detectâvel. Portanto, este nâo é um parâmetro muito relevante para a seleçâo de eventos envolvendo os TNOs. O mesmo nâo é verdade quando a ocultaçâo é por Plutâo, jâ que este tem magnitude R~14,5.

Para o câlculo da queda de magnitude Ama£M utiliza-se a magnitude do objeto ocultador mp e da estrela ocultada ms relacionados na equaçâo 3.1. Uma relaçâo aproximada entre a precisâo em magnitude crmag e o sinal sobre ruido é dada pela equaçâo 3.2. Nâo hâ dûvida que maiores valores de S/R trazem resultados mais precisos. Por exemplo, para detecçâo de atmosferas muito ténues, alta frequência de imagens e alto S/R sâo fundamentais.

Entretanto, para a determinaçâo do tamanho e forma dos objetos, precisamos do maior nûmero de detecçôes positivas em sitios diferentes. Entâo, nâo é incomum haver uma grande heterogeneidade de S/R entre os dados utilizados para a determinaçâo do perfil de um objeto.

quente, e disponibilizada em um site de internet com acesso restrito aos colaboradores para o pedido de tempo em telescôpios (http://devel2.linea.gov.br/~braga.ribas/ selec). Desde 2010 mais de 450 eventos jâ foram selecionados, dos quais cerca de 12%

foram efetivamente observados e uma parcela menor ainda foi detectada.

A mag = mp -171,, + (2,)) (3.1) (3.2)

A seleçâo dos eventos mais favorâveis é feita ao final de cada ano para o ano subse-

Campanhas

Uma vez com as novas medidas, novos mapas de prediçâo sào gerados (Fig. 2.2). Se as condiçôes da nova prediçâo forem favorâveis, lança-se um alerta a possiveis observadores.

Contactamos observatôrios profissionais e amadores pedindo para observarem o evento.

Em algumas oportunidades, caméras especiais sâo levadas aos telescôpios que nâo tenham equipamento adequado.

Como muitos eventos sô podem ser confirmados com alguns dias de antecedência, nem sempre é possivel obter telescôpios profissionais de grande porte. Uma alternativa é solicitar uma quantidade de tempo de alvo de oportunidade (ToO) durante o pen'odo Para cada campanha uma pagina internet é abcrta (veja figura 3.2 como cxcmplo), contendo informaçôes importantes para a observaçâo, cartas de campo para a loca lizaçâo do alvo e os mapas de prediçâo. Desde 2010 mais de 40 campanhas jâ foram realizadas e as informaçôes disponibilizadas nos sitios: http://www.lesia.obspm.

fr/perso/braga-ribas/list para os eventos de 2011, http://www.lesia.obspm.fr/ perso/braga-ribas/campaigns para os eventos de 2012 e http://dev.linea.gov.br/ "braga.ribas/campaigns para 2013.

Observaçâo

A ocultaçâo estelar é um evento fotométrico e temporal, isto é, mede-se a variaçâo da quantidade de luz recebida da estrela ao longo do tempo, o que chamamos de curva de luz de ocultaçâo, a exemplo da figura 5.4. Quanto maior a frequência de imagens, melhor a precisâo, em quilômetros, sobre o tamanho do objeto. Normalmente nâo se utiliza filtro, uma vez que precisa-se da maior quantidade de fôtons possiveis num tempo curto de exposiçâo. Em circunstâncias de Lua prôxima ou poluiçâo luminosa intensa, o emprego de filtro vermelho (R ou I) pode melhorar o contraste entre a estrela ocultada e o ruido do céu. Ou ainda, no caso da existência de brumas na atmosfera do objeto ocultador, a forma da curva de luz pode ser cliferenciada caso a observaçâo seja feita com filtros > Figura 3.2 Exemplo de pagina internet de uma campanha para a observaçâo de uma ocultaçâo. Varias informaçôes sâo dadas aos colaboradores como: o mapa de prediçâo, as coordenadas da estrela J2000 e "do dia", magnitudes R e K da estrela, queda de magnitude prevista, motivaçâo, dicas de observaçâo, cartas de campo para busca do alvo e referências e mapas de prediçâo atualizados. (Acessado em 26/01/2013: http: //www.lesia.obspm.fr/perso/braga-ribas/list/2011/2011_08_05_Quaoar.html) diferentes (Sicardy et al. 2006b). Até hoje, somente Plutâo e Tritâo1 sâo conhecidos por terem atmosfera no Cinturâo de Kuiper.

O objetivo é determinar o tamanho do objeto com precisâo de alguns quilômetros, utilizando para isto observaçôes de si'tios diferentes. Entâo, o registro do tempo absoluto por cada telescôpio é essencial. Para tanto, utiliza-se antenas GPS (Global Positioning System) que fornecem o tempo absoluto com precisâo melhor que o milésimo de segundo.

Caso uma antena GPS nâo esteja disponlvel, outras fontes de tempo absoluto sâo empregadas, como serviços NTP (Network Time Protocol), internet ou radio.

Em boas condiçôes de observaçâo, isto é, céu escuro e limpo, e uma caméra CCD sensfvel e râpida, estrelas de magnitude R<18 podem ser observadas por telescôpios a partir de 0,5 m de abertura; estrelas de magnitude R<16,5 sâo acessiveis a telescôpios de 0,3 m, estrelas de mag. R<15 podem ser registradas por espelhos de 0,25 m, pequenos telescôpios de 0,15 m podem detectar estrelas de mag. R<12 e, a partir de mag. R<11, os eventos podem ser observados visualmente com um telescôpio de ao inenos 0,2 m. No programa caminhos (Seçâo 2.1), entre 2009 e 2015, hâ apenas 6 ocultaçôes previstas para todo o globo que envolvam estrelas de magnitudes entre 11 e 13, para magnitudes entre 13 e 15 sâo 34 eventos, entre magnitudes 15 e 17 sâo 174 e acima de magnitude 17 sâo milhares (Figura 3.4).

1T'ritâo, satclite de Netuno, foi provavelmcnte um TNO capturado por Netuno no periodo da migraçâo planetâria [START_REF] Mckinnon | Gas drag and the orbital évolution of a captured Triton[END_REF] e [START_REF] Nogueira | Reassessing the origin of Triton[END_REF]). 

Reduçâo

Fotometria

A calibraçâo de imagens obtidas para ocultaçâo é feita de maneira tradicional, retiradose bias e corrente de escuro e dividindo pelo fiat -field. O fluxo da estrela + objeto é medido com fotometria de abertura. A abertura deve ser otimizada para o momento da ocultaçâo, jâ que minutos antes/depois os objetos podem nâo estar totalmente fundidos no piano do céu, havendo perda de fluxo medido. Devido às variaçôes de transparência do céu, utiliza-se estrelas prôximas como referência, obtendo fotometria diferencial de abertura.

Todas as curvas de luz, obtidas nos diferentes sftios, sâo posteriormente normalizadas ao nfvel do fluxo da estrela + objeto nos perfodos anterior e posterior à ocultaçâo, aplicandose um polinômio de primeiro ou segundo grau aos dados.

Devido à diferente resposta espectral dos sensores ou ao emprego de filtros, a queda de magnitude sera diferente para cada equipamento. Para calcular a razâo entre os fluxos dos dois objetos e entâo a queda de magnitude para aquele instrumento, imagens anteriores e/ou posteriores ao evento sâo obtidas, de tal maneira que o ocultador e a estrela ocultada possam ter scus fluxos mcdidos separadamcnte. Estas imagens devem ser tomadas preferencialmente à mesma altura do evento, jâ que o indice de cor dos objetos pode ser diferente. Neste caso, a absorçâo cromâtica diferencial pode dar origem a razôes de fluxo diferentes para alturas diferentes. Este proccdimcnto também é cssencial para os casos de objetos com atmosfera. Devido à refraçâo na atmosfera do corpo, recebe-se luz residual da estrela durante todo o evento, e este fluxo nos trarâ informaçâo sobre a pressâo da atmosfera e gradiente de temperatura prôximos à superficie do objeto (Sicardy et al. 2006b). Entâo é essencial conhecer precisamente a razâo de fluxo entre o corpo e a estrela. Alguns programas de aquisiçâo, como o comumente utilizado MaxImDL, truncam o horârio registrado, guardando apenas a parte inteira do segundo (Sicardy et al. 2011).

Recuperar a fraçâo de segundo é possfvel, e para isto ajusta-se uma reta aos instantes de tempo (z,G,i), sendo i o numéro da imagem e G,; 0 instante escrito no cabeçalho.

Os resi'duos deste ajuste mostram um padrâo de serra com dispersâo entre -0,5 e +0,5 Os instantes de imcio e fini da ocultaçâo sâo obtidos das curvas de luz, ajustando aos dados um modelo geométrico de um corpo sem atmosfera ocultando uma estrela pontual (Figura 3.7). O modelo é entâo convolvido com: o padrào de difraçâo de Fresnel de uma fonte pontual à distancia do objeto; o diâmetro da estrela; a banda em comprimento de onda utilizado; e o tempo de integraçâo (Widemann et al. 2009). Os programas utilizados para obter os resultados apresentados adiante, foram escritos por Bruno Sicardy.

A difraçâo de Fresnel é calculada para uma fonte pontual à distancia do objeto ocul- Propriedades fisicas de Quaoar a partir de ocultaçôes estelares A partir da detecçâo positiva de uma ocultaçâo, obtém-se o que chamamos de "corda".

Cada curva de luz é de fato a medida de uma latitude do objeto, e a duraçào da corda dependerâ da posiçâo relativa observador -ocultador -estrela e do tamanho do corpo.

Com a observaçâo simultânea de vârios locais, obtém-se o perfil instantâneo do objeto.

As cordas sâo combinadas corrigindo sua posiçâo do efeito paralâtico da posiçâo de cada sitio (Figura 4.1). A determinaçâo de sua ôrbita nos permite conhecer a massa de Quaoar que é utilizada para calcular sua densidade.

À época da descoberta, o diâmetro de Quaoar foi estimado em 1900 km, assumindo um albedo de 0,04. Com imagens do Hubble Space Telescope (HST), estimou-se seu albedo em p -0,1, entâo seu tamanho foi calculado em 1260 ± 190 km (Brown & Trujillo 2004). Telescope que permitiram a descoberta de Quaoar (Brown & Trujillo 2004).

Mais tarde, com medidas do Spüzer Infra-Red Telescope, seu tamanho foi estimado em 844lis9 km (Stansberry et al. 2008). Com imagens diretas do HST, Fraser & Brown Isto significa que Quaoar reteve volâteis cm sua superficie c podc, portanto, também ter retido uma atmosfera.

Prediçao

A descoberta da estrela candidata a ser ocultada foi feita pelo programa caminhos. A estrela é a UCAC2 26257103, com magnitudes B=17,P, V= 15.9 §, R= 15,8*, 1= 

Observaçao

Com as atualizaçôes favorâveis a uma ocultaçâo ocorrendo no sul da América do Sul, diversos observatôrios naquela regiâo foram contactados para realizarem a observaçao.

Uma pagina na internet foi criada com informaçôes e dicas aos observadores, como os mapas de prediçâo, cartas de campo para busca do alvo, coordenadas, magnitudes da es trela, duraçâo, queda de magnitude esperada, etc (http://www.lesia.obspm.fr/perso/ braga-ribas/list/2011/201l_05_04_Quaoar.html).

Uma grande diversidade de instrumentes foi utilizada, de telescôpios profissionais de até 2 m a telescôpios amadores de 0.254 m de diâmetro, formando uma rede de 

Fotometria

A qualidade das imagens obtidas foi bastante variada, como pode-se ver na figura 

Geometria

Como discutido, uma vez os instantes de ingresso e egresso determinados, obtém-se as cordas do objeto (Tabela 4.2), isto é, tem-se medidas do perfil do corpo no instante da ocultaçâo em diferentes latitudes. Como a ocultaçâo por Quaoar em 4 de maio de 2011 foi observada de cinco sftios diferentes, obteve-se cinco cordas (Fig. a base do V corresponde à e' = 0 e RequiV = L, sendo que L dépende do modelo ajustado.

A melhor soluçâo obtida ao se ajustar as cordas observadas, apresenta um Xmin = 9» 1, quando um valor prôximo a 1 séria esperado. Isto signihca que o modelo nâo é satisfatôrio.

Como pode-se ver na figura 4.6 a elipse nâo passa por varias extremidades das cordas.

Da figura 4.1 também pode-se pcrceber que o perhl dado pelas cordas realmente nâo corresponde à uma elipse, o que pode signihcar que Quaoar nâo apresenta a forma de um objeto em equilibrio hidrostâtico.

Espera-se que um objeto com cerca de 1000 km de diâmetro, como Quaoar, esteja em equilibrio hidrostâtico, isto é, que tenha a forma de um objeto Maclaurin ou Jacobi (Seçâo 4.8). O diâmetro crftico para isto pode ser determinado assumindo-se a composiçâo (gelo ou rocha) e sua temperatura (Tancredi &;Favre 2008) Nâo hâ nenhum indfcio do que poderia causar tais deslocamentos, jâ que todos os sitios estavam conectados à um servidor de tempo via internet (NTP). A ûnica exceçâo foi Rivera, que foi uma estaçâo môvel, que utilizou um GPS de mâo e manualmente acertou o horârio do computador que era usado para registrar as imagens.

Antes de deslocar as cordas e seguir para o ajuste Maclaurin, é preciso entender porque espera-se que Quaoar apresente tal figura e como pode-se utilizâ-la para restringir os i resultados.

Figuras de equilîbrio

Um corpo fluido isolado em rotaçâo ira assumir formas esfcroidais dcpendendo do scu momento angular. Sendo a equaçâo de um esferoide dada pela funçâo 4.4, pode-se définir très casos para a forma do objeto em revoluçâo (Chandrasekhar 1987):

1. Se nâo estiver em rotaçâo sua forma sera esférica, com todos os eixos cartesianos partilhando da mesma dimensâo (a = b -c).

2. Se seu momento angular for baixo, assumir a a forma de um esferoide oblato com apenas dois eixos com o mesmo tamanho (a = b > c), conhecido como esferoide Maclaurin (Fig. 4.12(a)).

3. Se seu momento angular for alto, ira assumir a forma de um elipsoide de Jacobi, onde (a > b > c), isto é, seus très eixos tem dimensôes diferentes (Fig. 4.12(b)). Um objeto Maclaurin nâo apresentarâ curva de luz de rotaçâo se tiver uma superficie uniforme. Jâ um objeto Jacobi terâ uma curva de luz com dois picos de mesma amplitude que podem ser calculados pela equaçâo 4.6, onde (3 = (b/a) e 7 = (c/a). A presença de diferenças locais de albedo, crateras e facetas irâo causar desvios à estas curvas [START_REF] Trilling | Light Curues of 20-100 km Kuiper Belt Objects Using the Hubble Space Telescope[END_REF]. .No caso do objeto ser visto pelo equador (C = 90°) a amplitude sera mâxima, c a equaçâo 4.6 se simplifica na relaçâo Ammax = 2, 5 log(/3).

A m = -1, 25 47rG/FyT (4.12) Verifica-se as condiçôes acima para 0 caso especffico de Quaoar. Ele apresenta curva de luz de rotaçâo de unico pico com amplitude A m = 0,112 ± 0,011 e perfodo P = 8,8394 ± 0, 0002 horas (Figura 4.15). A possibilidade de se tratar de uma curva de pico duplo com P = 17,6788 horas nâo é descartada, mas pouco provâvel jâ que a densidade de um objeto Jacobi com tal perfodo de rotaçâo séria proibitivamente baixa (Ortiz et al. 2003).

Considerando o perfodo de rotaçâo de pico unico e uma densidade tfpica entre os TNOs de 2 g/cm3 pode-se calcular fl para Quaoar. Usando a equaçâo 4.7, obtém-se Figura 4.15 Curva de luz de rotaçâo de Quaoar com P = 8,8394 ± 0,0002 horas e Am = 0,112 ± 0,011 (Ortiz et al. 2003).

Hq = 0,093, portanto encontra-se dentro da faixa de estabilidade de um corpo Maclaurin (Figura 4.13(a)).

A partir das soluçôes jâ apresentadas, calcula-se as densidades e considerando também do periodo de rotaçâo de pico ûnico, determina-se fl para Quaoar Oq = 0,08 -0,13.

Portanto ele se encontra dentro da faixa de estabilidade de um corpo Maclaurin (Figura gama de raios équivalentes obtidos, de 504 a 566 km, a gama de albedos geométricos no visi'vel para Quaoar é pVjQ = 0,102 a 0,139.

Densidade

A densidade global é um dos parâmetros fîsicos mais fundamentais de um objeto.

Ela é a primeira estimativa da composiçâo gérai do corpo. 

Ocultaçâo por Weywot

Como jâ foi dito, Quaoar tem uma lua, (50000/1) Weywot, descoberta em 2006 (Brown & Suer 2007). O satélite tem cerca de 5 magnitudes a menos que Quaoar (Figura 4.20(a)), entâo, assumindo que tenham o mesmo albedo, a razâo entre seus tamanhos é de 12:1.

Usando o tamanho de Quaoar, determinado do ajuste da ocultaçâo, Weywot teria um diâmetro équivalente de ~90 km e massa de 7,1 x 1017 kg, se a densidade for igual à de Quaoar (1,93 ± 0,5 g/cm3).

A ôrbita de Weywot tem semi-eixo maior de 14000 km, excentricidade 0,13, inclinaçâo em relaçâo ao piano J2000 de -38°e periodo orbital de 12,4 dias (Vachier et al. (2012)). os maiores e ricos em volâteis, e o menores e pobres em volâteis (Schaller & Brown 2007).

A presença de volâteis é um dos pré-requisitos para a presença de uma atmosfera. E preciso também uma fonte de energia para transformar os volâteis em gâs, e que a taxa de perda deste gâs, por mecanismo Jeans ou hidrodinâmico, nâo seja tâo grande a ponto de jâ terem perdido todo o gâs na idade do Sistema Solar [START_REF] Stern | The Solar System Beyond Neptune[END_REF]. Os pequenos TNOs nâo têm força gravitacional suficiente para terem retido uma atmosfera até a época atual, mas alguns objetos ainda podem possuir uma.

Très processos ffsicos podem dar origem à atmosferas nos TNOs: sublimaçâo, que dépende da temperatura de equilibrio do corpo e dos elementos présentes; perda de gâs do seu interior, devido a presença de gêiseres por exemplo; e impactos que podem ter energia suficiente para sublimar uma quantidade suficiente de gelos para originar uma atmosfera. Ainda, a temperatura no ponto subsolar pode ser suficientemente alta para gerar uma fina atmosfera local, que se recondensaria em outras regiôes do objeto, mudando as caracterlsticas da superficie, como pode ser o caso de Makemake (Ortiz et al. 2012).

Devido à excentricidade de suas ôrbitas, importantes mudanças sazonais sâo esperadas, como jâ se observou em Tritâo e Plutâo (Sicardy et al. 2003), e até mesmo recondensaçâo compléta da atmosfera, como deve ser o caso de Eris (Sicardy et al. 2011). Possui uma temperatura de equilibrio (Tabela 4.6) de 32 K, suficiente para sublimar parcialmente N2, CH4 e CO. Sua atmosfera tem pressâo de 2,3 fibar a 1400 km de altitude (seu raio é de 1352 km) (Elliot et al. 1998). Plutâo tem temperatura de equilibrio de 40 K, pressâo de 4,5 fibar a 1215 km do centro (raio de 1180 ± 22 km) [START_REF] Lellouch | Pluto's lower atmosphère structure and methane abundance from high-resolution spectroscopy and stellar occultations[END_REF].

Para investigar a estabilidade dos volâteis nos TNOs ao longo da idade do Sistema Solar, Schaller & Brown (2007) desenvolveram um modelo simples de perda de atmosfera pelo mecanismo Jeans, analisando os gelos de N2, CH4 e CO. Para Quaoar, levando em consideraçâo as propriedades fisicas, como diâmetro, temperatura média e distancia média ao Sol, eles concluiram que N2 é muito leve para ter ficado retido nesta escala de tempo, mas CH4 e CO estariam prôximos à curva de estabilidade (Figura 4.21). É importante frisar que o escape Jeans fornecc apenas um limite inferior para a taxa de escape. Outros mecanismos de escape, se atuantcs, serâo mais cficicntes cm rctirar os gases do objeto. A presença de gelo de âgua na sua fase cristalina em diversos TNOs, inclusive Quaoar, implica que suas superficies estiveram expostas à temperaturas maiores que ~100 K numa época relativamente recente, < 107 anos (Cooper et al. (2003) e Jewitt & Luu (2004)).

Alguns volâteis, ao atingirem esta temperatura, seriam rapidamente perdidos, podendo dar origem ao escape hidrodinâmico. A zona de transiçào entre uma atmosfera que perde lentamente seus gases pelo mecanismo Jeans e uma atmosfera que é rapidamente perdida pelo mecanismo hidrodinâmico é investigada (Levi & Podolak 2009). Na figura 4.22 é apresentado um grâfico entre a densidade do corpo e seu tamanho, com curvas de tempe ratura mâxima que poderiam atingir antes de submeter o gâs de metano ao mecanismo hidrodinâmico. Alguns TNOs sào mostrados, Quaoar encontra-se numa zona de transiçào.

Para oN2e CO, apenas o escape Jeans teria ocorrido.

No entanto, os régimes de escape dependem mais da razâo de mistura entre os gases do que de suas pressôes de vapor individuais, mesmo que um régime hidrodinâmico possa aumentar a perda de CH4 parcialmente. Além disso, a firme detecçâo de gelo de metano na superficie indica que estes cfcitos nâo foram suficientcmcnte cficazes a ponto de terem para Plutâo e Caronte, 0,64 e 0,30 m/s2 respectivamente (Tholen et al. 2008).

Muitos volâteis jâ foram detectados sobre a superficie de Quaoar (Figura 4.23): ~36%

de gelo de âgua em fase cristalina e amorfa (Jewitt h Luu 2004), ~13% de gelo de metano e possivelmente ~4% etano (Schaller &;Brown (2007b) e Guilbert et al. (2009)), e talvez ~20% de nitrogênio molecular (Dalle Ore et al. 2009). Segundo Schaller & Brown (2007b) seu espectro é de um objeto geologicamente morto e frio, perdendo seus volâteis.

Entâo, como Quaoar se encontra em zonas de transiçâo entre ter perdido ou nâo sua atmosfera pelos mecanismo Jeans ou hidrodinâmico, e possui volâteis em sua superficie, é possivel que possua uma atmosfera, assim como Plutâo. Como a ocultaçâo estelar é a técnica mais eficaz para se detectar (ou colocar limite superior à presença de) tenues atmosferas à pressôes tâo baixas quanto o nanobar, investiga-se sua assinatura nas curvas de luz de ocultaçâo obtidas em 2011.

A estrutura térmica da atmosfera de Quaoar obviamente nâo é conhecida, mas podese esperar que seja composta principalmente por metano ou diôxido de nitrogênio (se présente na superficie). A parte baixa da atmosfera deve estar em equilibrio térmico com a superficie, entâo calcula-se a temperatura de equilibrio da superficie correspondente à Figura 4.23 0 espectro de Quaoar mostra a presença de gelos de H20 em fase amorfa e cristalizada, de CH4 e possivelmente de CO e N2 (Brown & Suer 2007).

um corpo sem atmosfera usando as equaçôes da tabela 4.6.

A temperatura no ponto sub-solar de uma superficie lambertiana com emissividade 1 é T0, onde F©( 1 e da intégral de fase q que é a razâo entre o albedo de Bond A e o albedo geométrico p.

Para Quaoar q ~0,52, entào Tsubo = 59, 5 Kelvins.

A temperatura de equilibrio da superficie dépende da velocidade de rotaçâo do objeto.

As equaçôes dadas na tabela 4.6 correspondem: à esquerda, a temperatura sub-solar de equilibrio de um objeto com rotaçâo lenta Tss, e sua temperatura global Tssg; à direita, a temperatura de um corpo com rotaçâo râpida Tequ, e sua temperatura global Teqg. A emissividade s, que é a eficiência do corpo em emitir globalmente o fluxo térmico, e o fator relacionado à eficiência do corpo para emitir radiaçâo na direçâo do observador t/, também sâo considerados (Stansberry et al. 2008). Para os TNOs mede-se um valor tipico de e -0,9. Para Quaoar calcula-se p = 1,7 (Fornasier et al. (2013), comunicaçâo privada).

Para Quaoar calculam-se as temperaturas Tss = 54 K e Tequ = 40 K. No entanto, com um periodo de rotaçâo de 8,8394 h, Quaoar coloca-se com rotaçâo intermediâria, e por isto assume-se como temperatura superficial de Quaoar Tsup = 42 K, que é tomada como temperatura de sua atmosfera inferior.

A temperatura da parte superior de sua atmosfera pode ser calculada pelo "modelotermostato" do metano (Yelle & Lunine 1989). Nele o aquecimento pela absorçâo de radiaçâo no infra-vermelho é balanceada pela emissâo na banda do metano a 7,7\im. À Figura 4.24 Estrutura térmica da atmosfera de Quaoar, como gradiente de 5,7 k/km e um perhl isotérmico com temperatura de 102 K que se estende até a estratosfera.

da superficie [START_REF] Lellouch | Pluto's lower atmosphère structure and methane abundance from high-resolution spectroscopy and stellar occultations[END_REF]. À partir destas consideraçôes, a atmosfera de Quaoar é modelada com temperatura de 102 K, à partir de 10 km de altitude e com 42 K na troposfera (parte inferior). Isto résulta no perhl de temperatura mostrado na figura 4.24, onde o gradiente de temperatura, entre a superficie e 10 km de altitude, é de 5,7 K/km, seguido por uma parte isotérmica a 102 K até a estratosfera.

A pressâo de vapor de volâteis pvap, a uma dada temperatura, pode ser calculada (Figura 4.25). Usando relaçôes termodinâmicas propostas por Fray & Schmitt (2009), calculou-se a pressâo de vapor para CO, CH4 e N2 de 30 a 60 K (Tabela 4. modelada. Ao passar pela atmosfera do corpo, a luz da estrela sofrerâ refraçâo que dependerâ do seu elemento constituinte, sua temperatura e pressâo. A curva de luz observada ainda dependerâ da distancia do observador ao objeto De da distância y à Figura 4.26 Figura esquemâtica da refraçâo da estrela ao passar pela atmosfera do corpo.

A refraçâo dependerâ do elemento constituinte da atmosfera, sua temperatura e pressâo (Elliot & Olkin 1996).

Usando um côdigo de traçado de raios (ray tracing) como descrito em Sicardy et al.

(1999) & Sicardy et al. (2006b), modelou-se curvas de luz sintéticas para a atmosfera acima descrita. Estas curvas sintéticas sâo, entâo, comparadas com a curva de luz obtida durante a ocultaçâo por Quaoar, pelo telescôpio de 0,5 m, em San Pedro de Atacama.

Com tempo de integraçâo de 3 s e S/R-52, é a melhor curva de luz positiva do evento.

Como obteve-se diferentes soluçôes para a forma de Quaoar, a distância do centro do corpo ao limbo sondado nâo é conhecida. Para contornar este problema, modelou-se as curvas de luz sintéticas para uma gama de raios (de 550 a 560 km). Cada curva sintética foi comparada com a curva observada, e o x2 do ajuste para diferentes pressées obtido.

Todos os ajustes sâo entâo verificados obtendo-se uma curva gérai de x2 (Figura 4.27).

Desta curva pode-se determinar o Xmm> Xmin + R que corresponde ao erro a 1er, e o Xmin + 9, que correspondente ao erro a 3<r.

Obtém-se, entâo, um limite superior para a pressâo superficial da atmosfera de Quaoar de 21 nanobar a 1er, ou 56 nbar a 3cr. O perfil esperado para uma atmosfera de 21 nbar é mostrado na figura 4.28. O instante de cada exposiçâo é transformado em distância ao Mesmo nâo sendo compatfvel com "modelo-termostato" do metano, investigou-se o caso de uma atmosfera isotérmica. Neste caso haveria detecçâo de uma atmosfera a 42 K e pressâo superficial de 54 ± 25 nbar (1er) (Figura 4.30). As curvas de luz dos outros sltios nâo apresentam a qualidade suficiente para melhor interpretaçâo dos dados.

Importante notar que o metano tem pressâo de vapor pv = 32 nbar a 42 K, consistente com o limite superior obtido de 56 nanobar a 3a. O nitrogênio molecular é mais volatil que o metano (Tabela 4.7). À 42 K sua pressâo de vapor é de 176 nbar, muito acima dos limites de pressâo aqui estabelecidos. Portanto nâo espera-se que N2 seja um elemento abundante na atmosfera de Quaoar, mas nâo é possfvel descartar a presença de uma ténue atmosfera global de metano.

Como em Quaoar Tss = 54 K, e a pressâo de vapor do metano a esta temperatura é de 17 /ibar é possfvel que exista uma atmosfera local no ponto sub-solar que se condensaria ao nfvel de alguns nanobar no terminador, indetectâvel em nossos dados. o evento (Figura 4.8). No entanto, très deles, todos lia França, estavam separados por no mâximo 3,1 km, em termos de latitude de Quaoar. O quarto em Gnosca/Suiça, estava a cerca de 136 km mais ao norte. Portanto, na prâtica, obteve-se duas cordas do perfil de

Quaoar (Figura 4.34).

Para a reduçâo fotométrica das imagens de video obtidas, oriundas de Gnosca e Valensole, cada trama do video foi transformada em imagem FITS para posterior fotometria diferencial. Os instantes registrados com dispositivos "incrustadores de tempo em video" (ITV), onde a fonte de tempo oriunda de uma antena GPS é escrita diretamente sobre cada trama do video. A observaçâo de Tourrette-Levens foi feita com caméra CCD. A curva O centro fc e gc sâo calculados usando a efeméride geocêntrica de Quaoar JPL#21.

Eles podem ser utilizados para derivar a posiçâo de Quaoar no piano do céu no momento de C/A. Na tabela 4. Foram utilizados dois telescôpios no mesmo sltio (Long.: 30°10' 10",78 S, Lat.: 70°48'

23",49 O, ait.: 2.207 m), sem filtro, com exposiçâo de 8 segundos e tempo-morto de 1 segundo cada, e suas exposiçôes foram iniciadas com 4 segundos de defasagem. Assim, combinando as observaçôes, nâo houve perda de informaçâo por conta do tempo-morto.

A ocultaçâo teve im'cio a 00:45:30 ± 2,5 s e fim a 00:45:50 ± 2,5 s. Anâlise preliminar das imagens mostra a detecçâo positiva do evento com duraçâo de 20 s, isto é, uma corda de ~400 km (Figura 4.35).

Detecçôes positivas de corda unica, envolvendo objetos que jâ tenham seu tamanho determinado por alguma ocultaçâo multi-cordas (ou outro método), sâo importantes para determinaçâo da posiçâo astrométrica do corpo, jâ que neste caso, o erro limita-se à posiçâo da estrela (Tabela 4.11). Esta pode ser determinada com menos de 20 mas em relaçâo ao catâlogo UCAC4, por exemplo, o que équivale a cerca de um raio de Quaoar 2012)) e da implementaçâo de uma grande rede de colaboradores.

A Medidas térmicas feitas pelos telescôpios espaciais Spitzer e Hershel indicam que Makemake deve ter ao menos dois terrenos diferentes, um de maior area com um albedo superior a 0,7 e uma pequena regiâo com albedo prôximo a 0,1. Seu tamanho também foi estimado entre 1350 < D < 1510 km (Lim et al. 2010). Ele possui metano em sua superficie que deve estar em forma de grâos tào grandes quanto 1 cm. Hâ também evidências de N2 (Tegler et al. 2008). Ainda, devido ao seu tamanho, baixa amplitude da sua curva de luz (Am = 0,014) e rotaçâo (P = 7,65 h) [START_REF] Thirouin | Short-term variability of a sample of 29 trans-Neptunian objects and centaurs[END_REF], espera-se que ele seja um objeto Maclaurin com pequeno achatamento.

Por sua raridade e valor cientifico, a ocultaçâo por Makemake prevista para 23 de abril de 2011, envolvendo uma estrela de mag. R = 18,2, foi alvo de intensa campanha astrométrica para realizaçâo de atualizaçôes da prediçâo. Um total de 16 telescôpios foi utilizado para a observaçâo da ocultaçâo, dos quais 7 detectaram a ocultaçâo em cinco sitios diferentes (4 no Chile e 1 no Brasil) (Figura 5.6).

Das cinco cordas, quatro estavam dispostas a menos de 100 km uma da outra no piano do céu, prôximas à parte central de Makemake. É possivel ajustar às cordas uma elipse de Requit; = 732 ± 9 km e pequeno achatamento e = 0,05 ± 0,02 (Ortiz et al. 2012). Deste raio calcula-se seu albedo no visivel pv = 0,77 ± 0,03 que é intermediârio entre o albedo de Plutâo (pv = 0,52) e de Éris (jpv = 0,96). Como é de se esperar para os maiores e mais distantes TNOs, Sedna apresenta gelo de âgua, metano e possivelmente nitrogênio molecular em sua superficie. É um dos TNOs mais avermelhados, talvez por sua exposiçâo a radiaçâo galâtica [START_REF] Barucci | Sedna: In vestigation of Surface Compositional Variation[END_REF]).

Sua curva de luz de rotaçâo apresenta uma baixa amplitude Am = 0,011 ± 0,001 com periodo de revoluçâo P = -Frios, com baixa inclinaçôes i < 5,5°;

-Quentes, com inclinaçôes superiores a i > 5,5°.

2. Objetos espalhados, sendo aqueles com excentricidade maior que e>0,24, que no seu periélio chegam prôximos à influência gravitacional de Netuno (q~30), mas seu afélio se dâ a distancias muito superiores a isto. Photometry, spectroscopic, thermal (far infra-red) give information about their rotational period, surface composition, size and albedo estimations.

Nevertheless, due to their distance (greater than 30 Astronomical Units) and their sizes, we do not hâve the necessary technology to directly measure their size and detect possible atmosphères, as their angular sizes are smaller than a 50 miliarc seconds (mas).

The stellar occultation technique consists of observing the transit of a TNO in front of a star. The précisé duration of the event, observed from several different sites, allows to deduce their size and shape with kilométré accuracy. So, précisé détermination of their albedos can be derived, and for those bodies that hâve their mass deduced from its satellite orbit, précisé détermination of their densities is allowed. Besides, tenuous atmosphères at the nanobar level, can be detected as they promote an atténuation of the star light right before and after the occultation.

In this work it is described (1) the prédiction method applied to more then fifty TNOs, with accuracy of about 30 mas, ( 2 Ce travail a déjà permis l'observation de 12 événements, parmi les 14 occultations jamais détectées impliquant des OTNs (excluant Pluton et Charon). It is more advisable to select large TNOs as targets for stellar occultation campaigns. They usually hâve large apparent sizes (30 mas or more) and thus hâve a greater chance of a posi tive détection. In addition, theory indicates that larger TNOs are most likely to possess atmosphères (Elliot & Kern 2003).

In the case of atmosphères, even négative occultation chords ob served nearby locations with positive records are useful, giving valuable upper limits to the atmosphère size and profile. Larger UCAC2 star (Elliot et al. 2009). This was the first time ever 

Astrometry

We briefly describe the réduction procedures. Ail CCD images were corrected by overscan, zeromean, fiatfield, and bad pixels with IRAF (Tody 1993) using the esowfi (Jones & Valdes 2000) and msered [START_REF] Valdes | [END_REF] packages. Using the PRAIA pack age (Assafin 2006), the astrometric treatment for each TNO consisted of three distinct steps.

An assigned field distortion pattern was first applied to each CCD in the WFI mosaics for each run (see details in In the second step, astrometry was performed over the individual CCDs, where the (*, y) measurements were corrected by the pre-determined field distortions. Magnitudes were obtained from PSF photometry and calibrated with respect to the UCAC2.

The (a, y) measurements were performed with two-dimensional circularly symmetric Gaussian fits. The estimated (x,y) mea surement errors from Gaussian fits were about 5-10 mas for R < 16, then increased to 18 mas for R < 18, to 20 mas for 18 < R < 20, and to 30 mas at fainter magnitudes. Figure 2 shows the distribution of (a, y) errors as a function of R mag nitude for ail measurements of ail TNOs. The UCAC2 was the prime reference catalog, except for Eris. In this case, the 2MASS catalogue was used as the reference frame owing to the insufficient number of UCAC2 stars in the FOV. This is not an important change, as in the last step, ail frames were consistently corrected to the UCAC2 frame. In steps 1 and 2, we used the six constant polynomial model to relate the (a, y) mea surements to the (X, Y) tangent plane coordinates. The position mean errors in the (a, ô) individual CCD solutions are listed in Table 2; the average number of reference stars and number of CCD frames are also given for each TNO set of observations.

Detailed information on the procedures in this step are found in Sect. 4.2 of PI.

In the final step, ail positions coming from the differ ent CCDs and mosaics were combined to produce a global so lution for each sky path year, and final (a, ô) star positions were obtained in the UCAC2 System. In this step, ail positions were projected onto the tangent plane and a complété polynomial model of the third degree was used to fix ail the coordinates in the UCAC2 frame. from ail sky path years. Each final object position is assigned a flag that indicates its multiplicity status (see Table 4 for details).

In ail, more than 93% of the final positions corne with no multi ple entries (flag = 0).

Proper motions were extracted directly from the UCAC2 cat alog. For the other stars, proper motions were computed for each star using the 2MASS and USNO B1.0 catalogs as a first epoch.

Although these two catalogs may hâve random/systematic errors at the 100-200 mas level, they are still the most suitable choice for this purpose. The proportion of UCAC2, 2MASS-based, and

USNOBl-based proper motions of stars are given in Table fi.

Details about the computations can be found in Sect. 4.5 of PI.

The highest percentage of stars without proper motions was verified for Ixion, namely 62%, which is twice the average fraction (about 30%). Ixion is now Crossing the plane of the Milky Way Notes. UCAC2 stars with proper motions (directly extracted from UCAC2), stars with proper motions based on the 2MASS and USNO B1.0, and stars for which no proper motion could be computed.

Percentages are relative to the total number of stars for each TNO.

R magnitude

Fig. 3. Star distribution per R magnitude for ail catalog stars. It il lustrâtes the R magnitude completeness and limit of the catalogs.

Percentages were computed over 0.5 mag bins for ail stars in ail catalogs.

(see Table 1 ). One possible explanation is that the WFI magni tude limit is fainter than that of the USNO B 1.0 (see star dis tribution according to magnitude in Fig. 3 5.

For each TNO, Table 6 lists the total number of catalog stars per year, the R bandpass magnitude completeness, mag nitude limit, and average position errors for faint stars with R at the magnitude completness. Figure 3 shows the star distribution per R magnitude for ail catalogs. Figure 4 plots the position er ror of ail catalog stars as a function of R magnitude. Values were averaged over 0.5 mag bins.

Ephemeris offsets and search for candidate stellar occultations

To achieve the highest accuracy in predicting stellar occulta tions, it is necessary to pin down the actual paths of the TNOs in the sky. This can be done by correcting their orbits by offseting their ephemerides in both right ascension and déclina tion. We derived average ephemeris offsets for ail TNOs in the ES02p2/WFI program. France. Each TNO ephemeris version is displayed. They were extracted using the JPL Horizons service (Giorgini et al. 1996). The listed offsets are provisional in the sense that the parallel program is still ongoing and further updates are expected. For thrcc TNOs (Varuna, 2002TX300, 2003 AZ84), offsets came from fils to actual observed stellar occulta tions (see discussion in Sect. 7).

80 r-r I 1 Sedna, Varuna, 2002 TX1nn, and2003 AZ84 are available in electronic form at the CDS. In the electronic version, in addition to the star positions, we provide the TNO estimated coordinates (with ephemeris offsets applied) at occultation instant.

1 1 I 1 1 1 I 1 i 1 | i 1 1 | 1 I I | i 1 I | 1 1 1 1 1 I i | r right ascension x déclination a °x O x X x x% X , ' X xxxxxxxxxx I 1 i 1 I 1 1 1 I 1 i_i I i__i 1-! i_j 1 L R magnitude
Using the astrometric catalog of stars in the path of the sky through which the TNOs will pass up to 2015, we then searched for stars that could be occulted by the respective TNOs. To achieve this, the whole star catalogs were cross-correlated with the JPL ephemerides of the bodies generated each minute for the whole period between 2008/2009 to 2015. The JPL ephemerides were offset prior to the search using the quantifies given in Table 7. If the apparent distance in the sky plane between the star and the body is less than an Earth apparent radius plus 50 mas of safe margin, then a potential occultation was identified and ail astrometric and géométrie data relevant to the possible event were computed and stored. In addition to astrometric and photometric data, for each candidate star, we computed and stored the minimum apparent geocentric distance <7, the central instant of closest approach to, the shadow velocity v across the Earth, the position angle P/A of the shadow path, and the local solar time LST at the sub-planet point. These géométrie quantifies are defined and calculated as explained in Sect. The reference velocity of 20 km s"1 is typical of events around Pluto opposition. In slow events, M* may become one to two magnitudes brighter than the actual star magnitude, indicating that the observation is feasible after ail, because we can use longer intégration times, and consequently reach reasonably high S/N, without any losses of spatial resolution in diameter measurements. For the same reasoning, we may also probe the atmosphère altitudes in the light curves, despite the faintnesses of the stars. As mentioned in the introduction, in the case of TNOs, occultations involving stars as faint as R = 18.0 hâve been successfully recorded with 50 cm télescopes. instants and C/A values, or an extemal check for the accuracy and précision of our WFI prédictions.

The information in the prédiction tables allows us to déter mine the central instant and shadow path over the Earth. For ground-based observations, the sélection of events usually favors brighter stars and shadow paths that cross easily accessible places for the use of mobile télescopes. Faint stars may be selected in the case of observation at sites with larger observatories/telescopes. If the star is reddish, observations can be made in daylight with IR detectors. For TNOs with a low rate of oc cultations per year, such as Eris, opportunities are very rare, so broader nets of observers may be formed over larger areas, even if the event is not as promising in terms of star brightness or lo cation. We leave the choice of potential events among the many predicted occultations listed in the tables to the judgement of the reader. Our group publishes at our web page potential occulta tion campaigns on a yearly basis.

The errors in our TNO sky path prédictions are governed by the internai offset uncertainties listed in Table 7. These un certainties are based on optical observations made for a period of some years. The budget ephemeris error estimâtes given by JPL -which increase with each passing year -only ex press the random error features of the JPL orbit solutions, rather than represent a possible increase in systematic trends. As time goes by and more TNO observations and occultations are accumulated, more accurate JPL ephemeris solutions are being published, although in the new versions no significant shifts in the TNO orbital sky paths are predicted for the next few years.

For now, the adoption of a constant ephemeris offset in the pré dictions has the only side effect of absorbing some remaining systematic errors into the offset uncertainties. These uncertain ties, on the other hand, do not increase with time for the period considered and are realistic estimâtes for the overall sky path prédiction uncertainties. In the prédiction tables (available in eletronic form), in addition to the star position, we indicate the estimated TNO position (with ephemeris offset applied) at the predicted occultation instant. This should help us to evaluate the events in the future as updates become available at the star position and on the TNO ephemeris by follow-up astrometric programs or by the JPL.

A tentative way to estimate the probability of success in recording an occultation of a particular TNO is to consider the ratio of the apparent TNO diameter in mas to its ephemeris off set errors displayed in Table 7. We list TNO diameter values in Table 10 to support reliability estimâtes for the prédictions. 11.

Long-term prédictions are the only way of spotting potential stellar occultations in terms of favourable géographie access, observatories and instrumental facilities, etc. Once an event is chosen, then a complex international campaign is initiated, with astrometric follow-up of the TNO and star being mandatory. The only reliable long-term prédictions of stellar occultations that use solely all-sky star catalogs are those based on the UCAC2. 7). In some cases, offsets reached almost 200 mas (about one Earth radius at 30 AU) or more.

Given the time span of 2008-2015, we expect that the fixed corrections applied to the JPL ephemerides suffice for pinpointing the most interesting occultations among ail prédictions.

However, once an event is selected, the shadow path on Earth must be further refined. From the point of view of TNO orbits, improvements can still be made to the accuracy of the WFI prédictions. For instance, by observing TNOs at their op position and conjunctions to obtain the observed ephemeris off sets, one can model the geocentric parallax errors and correctly extrapolate the observed ephemeris offsets with time for the oc cultation dates. For a body at 30 AU in the Ecliptic, these errors can amount to 100 mas in right ascension for an ephemeris error of some few tens of thousand kilometers in heliocentric distance (Camargo et al. 2010). As successful occultations are recorded, the ephemeris offsets are more accurately determined.

For three TNOs (see Table 7), we could already improve the ephemeris offsets for this work using occultation results. As the new occultation results are Consolidated, we will update the ephemeris offsets and release new, higher quality prédictions.

An advancement that is beyond the scope of WFI prédic tions, for selected occultations, is an alternative procedure for improving the shadow path in the short term by direct fitting an orbital arc to the observed TNO positions. This can be done by numerically integrating orbits or, in principle, by using the current ephemeris as a template and then fitting it to the obser vations following a correction model. In the case of observations when both star and TNO are imaged in the same field of view, this procedure can yield very accurate prédictions, as high préci sion relative positions can be attained for the TNO with respect to the occultation star, once proper care is taken to correct for difîerential color refraction.

The astrometric follow-up of the selected stars and TNOs on a regular basis is therefore very important, regardless of the approach taken to improve the accuracy of the prédictions. In practice, however, the week or two before occultation date is still décisive in pinning down the shadow path on Earth. Updates on the TNO ephemeris offsets, star positions of selected occul tations, updated reports, and finding charts are available on a continuous basis from our group2. Among other similar projects conducted by other active groups in the USA, our efforts reflect a long-term collaboration with the international campaigns coordinated by the Observatoire de Paris for this purpose. tions such as those coming from albedo assumptions or from the modelling of optical, infrared (IR), and sub-millimetric observa tions (Cruikshank et al. 2005;Altenhoff et al. 2004;Stansberry et al. 2008). From these direct size measurements, one can dérivé more accurate albedos and place tighter constraints on the sur face composition of the TNOs. If the mass of the body can be estimated from the orbits of detected satellites or by other indirect means, the density can be estimated more accurately and thus the internai composition and structure of the body can be inferred far more reliably. The détection of atmosphères around TNOs will help to improve our current understanding of their dynamics and relationship with the surface. Détection of Chiron-like jets (Elliot et al. 1995) is also a possibility. A more robust characterization of companions/satellite orbits and rotational periods from stellar occultations and ground-based adaptive optics observa tions could also improve our models for binary formation and collision (Goldreich et al. 2002;Weidenschilling 2002;Sicardy et al. 201 la).

It is more advisable to select large TNOs as targets for stellar occultation campaigns. They usually hâve large apparent sizes (30 mas or more) and thus hâve a greater chance of a posi tive détection. In addition, theory indicates that larger TNOs are most likely to possess atmosphères (Elliot & Kern 2003).

In the case of atmosphères, even négative occultation chords ob served nearby locations with positive records are useful, giving valuable upper limits to the atmosphère size and profile. Larger bodies also hâve smaller relative size errors. The accumulation of positive and négative détections also resuit in significant improvements in the TNO ephemeris, thus refining the orbit and substantially increasing the accuracy of the subséquent oc cultation prédictions. Improving the chances of success rninimizes the money and time spent in the complex international campaigns, which usually involves large télescopes in wellequipped observatories, smaller mobile instruments, and specialized personnel.

Even as one of the most important and larger members of the TNO population, Pluto has only been probed in terms of oc cultations, prior to 2000, almost exclusively by the 1988 mutual events with its larger satellite Charon. The first observed stel lar occultation by Pluto took place only in 1985, when its at mosphère was first discovered (Brosch 1995). The atmosphère could only be more extensively observed in the next occulta tion in 1988 (Millis et al. 1993). In 2002, a new stellar occul tation by Pluto was recorded, but with higher time resolution (Sicardy et al. 2003;Elliot et al. 2003), revealing that there had been a dramatic expansion in the atmosphère pressure. Another UCAC2 star (Elliot et al. 2009). This was the first time ever 

The ES02p2/WFI TNO program and observations

The TNOs of the ES02p2/WFI program were selected according to their body size, which is usually related to their apparent sizes on the sky. Given the astrometric uncertainties in the star posi tions and the error in the TNO ephemeris, larger apparent diameters correspond to a higher probability of a successful prédic tion. Another sélection criteria was the star field. We preferred to select TNOs currently orbiting dense star régions close to the Galactic plane at Galactic longitudes of interest. The TNOs of intrinsically high science value were also selected, even if the star density along their sky path was low. In these cases, the advantage of using the large FOV of the WFI was even clearer.

Observations were made at the ESO 2p2 telescope 

Astrometry

We briefly describe the réduction procedures. Ail CCD images were corrected by overscan, zeromean, flatfield, and bad pixels with IRAF (Tody 1993) using the esowfi (Jones & Valdes 2000) and mscred [START_REF] Valdes | [END_REF] packages. Using the PRAIA pack age (Assafin 2006), the astrometric treatment for each TNO consisted of three distinct steps.

An assigned field distortion pattern was first applied to each CCD in the WFI mosaics for each run (see details in Sect. 4.1 ofPl).

In the second step, astrometry was performed over the individual CCDs, where the (x, y) measurements were corrected by (see text for details). The six constant polynomial modcl was used to relate the (x,y) measurements to the (X, Y) tangent plane coordinates.

The (a-, y) centers were pre-corrccted by the field distortion patterns (see details of these pre-corrections in Sect. 4.1 of PI). the pre-determined field distortions. Magnitudes were obtained from PSF photometry and calibrated with respect to the UCAC2.

The (a, y) measurements were performed with two-dimensional circularly symmetric Gaussian fits. The estimated (x, y) mea surement errors from Gaussian fits were about 5-10 mas for R < 16, then increased to 18 mas for R < 18, to 20 mas for 18 < R < 20, and to 30 mas at fainter magnitudes. Figure 2 shows the distribution of (x, y) errors as a function of R mag nitude for ail measurements of ail TNOs. The UCAC2 was the prime reference catalog, except for Eris. In this case, the 2MASS catalogue was used as the reference frame owing to the insufficient number of UCAC2 stars in the FOV. This is not an important change, as in the last step, ail frames were consistently corrected to the UCAC2 frame. In steps 1 and 2, we used the six constant polynomial model to relate the (x, y) mea surements to the (X, Y) tangent plane coordinates. The position mean errors in the (a, ô) individual CCD solutions are listed in Table 2; the average number of reference stars and number of CCD frames are also given for each TNO set of observations.

Detailed information on the procedures in this step are found in Sect. 4.2 of PI.

In the final step, ail positions coming from the differ ent CCDs and mosaics were combined to produce a global so lution for each sky path year, and final (o-, ô) star positions were obtained in the UCAC2 System. In this step, ail positions were projected onto the tangent plane and a complété polynomial model of the third degree was used to fix ail the coordinates in the UCAC2 frame. from ail sky path years. Each final object position is assigned a flag that indicates its multiplicity status (see Table 4 for details).

In ail, more than 93% of the final positions corne with no multi ple entries (flag = 0).

Proper motions were extracted directly from the UCAC2 cat alog. For the other stars, proper motions were computed for each star using the 2MASS and USNO B1.0 catalogs as a first epoch.

Although these two catalogs may hâve random/systematic errors at the 100-200 mas level, they are still the most suitable choice for this purpose. The proportion of UCAC2, 2MASS-based, and

USNOBl-based proper motions of stars are given in Table 5.

Details about the computations can be found in Sect. 4.5 of PI.

The highest percentage of stars without proper motions was verified for Ixion, namely 62%, which is twice the average fraction (about 30%). Ixion is now Crossing the plane of the Milky Way Notes. UCAC2 stars with proper motions (directly extracted from UCAC2), stars with proper motions based on the 2MASS and USNO B 1.0, and stars for which no proper motion could be computed.

Percentages are relative to the total number of stars for each TNO.

R magnitude

Fig. 3. Star distribution per R magnitude for ail catalog stars. It il lustrâtes the R magnitude completeness and limit of the catalogs.

Percentages were computed over 0.5 mag bins for ail stars in ail catalogs.

(see Table 1). One possible explanation is that the WFI magni tude limit is fainter than that of the USNO B 1.0 (see star dis tribution according to magnitude in Fig. 3 France. Each TNO ephemeris version is displayed. Thcy were cxtracted using the JPL Horizons service (Giorgini et al. 1996). The listed offsets are provisional in the sense that the parallel program is still ongoing and further updates are expccted. For thrce TNOs (Varuna, 2002TX300, 2003 AZ84), offsets came from fus to actual observed stellar occulta tions (see discussion in Sect. 7).

for stars up to magnitude R = 19, and 30 mas up to R = 17. The catalogs are available in electronic form at the CDS.

The catalogs are divided in terms of year. Stars that had mul tiple entries within 175 in the global mosaic solutions are flagged (see Table 4). The R magnitudes from PSF photometry were cal- 5.

For each TNO, Table 6 lists the total number of catalog stars per year, the R bandpass magnitude completeness, mag nitude limit, and average position errors for faint stars with R at the magnitude completness. Figure 3 shows the star distribution per R magnitude for ail catalogs. Figure 4 plots the position er ror of ail catalog stars as a function of R magnitude. Values were averaged over 0.5 mag bins.

Ephemeris offsets and search for candidate stellar occultations

To achieve the highest accuracy in predicting stellar occulta tions, it is necessary to pin down the actual paths of the TNOs in the sky. This can be done by correcting their orbits by offseting their ephemerides in both right ascension and déclina tion. We derived average ephemeris offsets for ail TNOs in the ES02p2/WFl program. Table 7 displays (UTC), the ICRS (J2000) star coordinates at occultation, the elosest apparent distance between star and TNO (C/A), the position angle (P/A) of the shadow across the Earth (counter-clockwise, zéro at north), the velocity in km s"1, the distance (D) to the Earth (AU), R*, J", H", and K* star magnitudes normalized to a référencé shadow velocity (v) of 20 km s"1 (50.0 means no magnitude available), longitude (d) of the sub-solar point, local solar time (LST), (Ae", Aeg) JPL ephemeris offsets in (cr, <5) for the central instant (see Sect. 5), catalog cross-identification (uc = UCAC2, 2m = 2MASS, fs = field star), proper motion existence and multiplicity flags (see Sect. Sedna, Varuna, 2002 TX300. and2003 AZ84 are available in elcctronic form at the CDS. In the electronic version, in addition to the star positions, we provide the TNO estimated coordinates (with ephemeris offsets applied) at occultation instant.

Using the astrometric catalog of stars in the path of the sky through which the TNOs will pass up to 2015, we then searched for stars that could be occulted by the respective TNOs. To achieve this, the whole star catalogs were cross-correlated with the JPL ephemerides of the bodies generated each minute for the whole period between 2008/2009 to 2015. The JPL ephemerides were offset prior to the search using the quantities given in Table 7. If the apparent distance in the sky plane between the star and the body is less than an Earth apparent radius plus 50 mas of safe margin, then a potential occultation was identified and ail astrometric and géométrie data relevant to the possible event were computed and stored. In addition to astrometric and photometric data, for each candidate star, we computed and stored the minimum apparent geocentric distance d, the central instant of elosest approach /q, the shadow velocity v across the Earth, the position angle P/A of the shadow path, and the local solar time LST at the sub-planet point. These géométrie quantities are defined and calculated as explained in Sect. 7 of PI. instants and C/A values, or an external check for the accuracy and précision of our WFI prédictions.

The information in the prédiction tables allows us to déter mine the central instant and shadow path over the Earth. For ground-based observations, the sélection of events usually favors brighter stars and shadow paths that cross easily accessible places for the use of mobile télescopes. Faint stars may be selected in the case of observation at sites with larger observatories/telescopes. If the star is reddish, observations can be made in daylight with IR detectors. For TNOs with a low rate of oc cultations per year, such as Eris, opportunities are very rare, so broader nets of observers may be formed over larger areas, even if the event is not as promising in terms of star brightness or lo cation. We leave the choice of potential events among the many predicted occultations listed in the tables to the judgement of the reader. Our group publishes at our web page potential occulta tion campaigns on a yearly basis.

The errors in our TNO sky path prédictions are governed by the internai offset uncertainties listed in Table 7. These un certainties are based on optical observations made for a period of some years. The budget ephemeris error estimâtes given by JPL -which increase with each passing year -only ex press the random error features of the JPL orbit solutions, rather than represent a possible increase in systematic trends. As time goes by and more TNO observations and occultations are accumulated, more accurate JPL ephemeris solutions are being published, although in the new versions no significant shifts in the TNO orbital sky paths are predicted for the next few years.

For now, the adoption of a constant ephemeris offset in the pré dictions has the only side effect of absorbing some remaining systematic errors into the offset uncertainties. These uncertain ties, on the other hand, do not increase with time for the period considered and are realistic estimâtes for the overall sky path prédiction uncertainties. In the prédiction tables (available in eletronic form), in addition to the star position, we indicate the estimated TNO position (with ephemeris offset applied) at the predicted occultation instant. This should help us to evaluate the events in the future as updates become available at the star position and on the TNO ephemeris by follow-up astrometric programs or by the JPL.

A tentative way to estimate the probability of success in recording an occultation of a particular TNO is to consider the ratio of the apparent TNO diameter in mas to its ephemeris off set errors displayed in Table 7. We list TNO diameter values in Table 10 to support reliability estimâtes for the prédictions. Notes. Comparison between central instants and C/As from WFI-based (Table S) and actual observations (fit to data) for 6 (out of seven) successfully recorded TNO stellar occultations to date for which data is available. Using P/A, distance and the relative specd, central instant différences are expressed in seconds and in mas, and C/A différences in mas and in km. Averages for the modulus of the différences are computed with and without Eris. 2003 AZX4 is excluded from the analysis as the fit to the actual occultation already uses the WFT star position, and the current WFI prédictions already uses the offsets derived from the occultation fitting.

They are given in km and as geocentric apparent sizes in mas for 2012.5. Values were obtained from previous stellar occul tations and by modeling visible and IR observations (see référ encés in Table 10).

The ES02p2/WFI prédictions can be directly compared to actually observed stellar occultations in order to estimate the external accuracy and précision of prédictions. Table 11 displays a direct comparison between prédictions (ES02p2/WFI-based)

and actually observed occultations, for which P/A, central in stant and C/A values were obtained from fits to the light curves of the chords. We used six out of the seven TNO stellar occul tations known to date, for which data is available (Elliot et al. 2009;Sicardy et al. 2011b;Braga-Ribas 2011, priv. comm.).

Using P/As, distances and relative speeds, we computed both the predicted minus fitted central instant and C/A différences.

Différences in central instants are expressed in seconds and in mas (in order to be comparable with C/A results), and C/A différences are given in both mas and kilometers. We give in Table 11 the averages for the modulus of the différences. We note that the WFI prédiction for the Eris' central instant was in strong disagreement with observations. This may indicate that much more accurate right ascension ephemeris offsets should be obtained in our follow-up program for Eris. A separate set of averages is provided in Table 11 without taking Eris into account. The TNO 2003 AZ84 is excluded from the analysis as the fit to the actual occultation is based on the WFI star po sition, and the current WFI prédictions already uses the off sets derived from the occultation fitting. In ail, the différences in C/A and central instant are consistent with the expected errors in the WFI star positions as a function of magnitude and ephemeris offset error (see Table 7). The somewhat larger différ ences found in Table 11 for the central instant may be associated with ephemeris offset uncertainties in right ascension (the di rection of the chords are usually east-west). One possible cause is geocentric parallax error (see discussion about the follow-up program and ephemeris offset détermination in the next para graphes). In ail, for R = 19 stars (catalog magnitude completeness) and 40 mas errors in the WFI positions, we may assume a bulk error of about 80 mas for C/A, that is dominated by the ephemeris offsets errors of about 70 mas. For about 40 AU, this implies a shadow path uncertainty over the Earth of the order of 2300 km. If the ephemeris offsets can be well-determined to a more accurate précision than 30 mas, then a bulk error of 50 mas in C/A can be achieved, leading to a précision of about 1400 km for the WFI occultation path prédictions, which is similar to the obtained values displayed in Table 11.

Long-term prédictions are the only way of spotting potential stellar occultations in terms of favourable géographie access, observatories and instrumental facilities, etc. Once an event is chosen, then a complex international campaign is initiated, with astrometric follow-up of the TNO and star being mandatory. The only reliable long-term prédictions of stellar occultations that use solely all-sky star catalogs are those based on the UCAC2. The sky paths covered by the star catalogs were determined by the chosen time span of our prédictions (2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015) in a compromise between the availability of WFI telescope time and the number of selected TNOs. The upper limit coincides with NASA's New Horizons spacecraft arrivai at Pluto in 2015.

Given this long time coverage of about eight years, the compu tation of proper motions for the stars was also of concem. That is why we used as first epoch data the best available astrometric resources at the time of completion of this work. In ail, the com parison made with actual events (Table 11 ) and the successful WFI prédictions specifically for the faint stars of Eris, 2003 AZ84 and Makemake occultations indicate that the WFI astrometric solution is sactisfatory in the context of long-term prédictions.

Owing to the lack of observations of TNOs for ephemerides, to the few positions sometimes derived without the appropriate astrometric care, and to the relatively short orbital arc length coverage, the actual position of TNOs can differ significantly from their ephemerides. To refine the orbits of large TNOs in the short and long runs, by regularly obtaining précisé astrometric positions, a parallel program involving many télescopes of 60 cm to 2 m class in Chile, France, and Brazil was started in 2007 (see further details in Sect. 5). Using the first results of this program for the prédictions published in this work, we applied average fixed offsets to the current JPL ephemerides of the TNOs in the list (Table 7). In some cases, offsets reached almost 200 mas (about one Earth radius at 30 AU) or more.

Given the time span of 2008-2015, we expect that the fixed corrections applied to the JPL ephemerides suffice for pinpointing the most interesting occultations among ail prédictions.

However, once an event is selected, the shadow path on Earth must be further refined. From the point of view of TNO orbits, improvements can still be made to the accuracy of the WFI prédictions. For instance, by observing TNOs at their op position and conjunctions to obtain the observed ephemeris off sets, one can model the geocentric parallax errors and correctly extrapolate the observed ephemeris offsets with time for the oc cultation dates. For a body at 30 AU in the Ecliptic, these errors can amount to 100 mas in right ascension for an ephemeris error of some few tens of thousand kilometers in heliocentric distance (Camargo et al. 2010). As successful occultations are recorded, the ephemeris offsets are more accurately determined.

For three TNOs (see Table 7), we could already improve the ephemeris offsets for this work using occultation results. As the new occultation results are Consolidated, we will update the ephemeris offsets and release new, higher quality prédictions.

An advancement that is beyond the scope of WFI prédic tions, for selected occultations, is an alternative procedure for improving the shadow path in the short term by direct fitting an orbital arc to the observed TNO positions. This can be done by numerically integrating orbits or, in principle, by using the current ephemeris as a template and then fitting it to the obser vations following a correction model. In the case of observations when both star and TNO are imaged in the same field of view, this procedure can yield very accurate prédictions, as high préci sion relative positions can be attained for the TNO with respect to the occultation star, once proper care is taken to correct for differential color refraction.

The astrometric follow-up of the selected stars and TNOs on a regular basis is therefore very important, regardless of the approach taken to improve the accuracy of the prédictions. In practice, however, the week or two before occultation date is still décisive in pinning down the shadow path on Earth. Updates on the TNO ephemeris offsets, star positions of selected occul tations, updated reports, and finding charts are available on a continuous basis frorn our group2. Among other similar projects conducted by other active groups in the USA, our efforts reflect a long-term collaboration with the international campaigns coordinated by the Observatoire de Paris for this purpose. The dwarf planet Eris is a trans-Neptunian object with an orbital eccentricity of 0.44, an inclination of 44 degrees and a surface composition very similar to that of Pluto1. It résides at présent at 95.7 astronomical units (Iau is the Earth-Sun distance) from Earth, near its aphelion and more than three times farther than Pluto. Owing to this great distance, measuring its size or detecting a putative atmosphère is difficult. Ilere we report the observation of a multi-chord stellar occultation by Eris on 6 November 2010 ut.

The event is consistent with a spherical shape for Eris, with radius 1,163 ± 6 kilométrés, density2.52 ± 0.05 grams per cm3 and a high visible géométrie albedo, pv = 0.96^oo9. No nitrogen, argon or methane atmosphères are detected with surface pressure larger than ~1 nanobar, about 10,000 times more tenuous than Pluto's présent atmosphère2'5. As Pluto's radius is estimated1'8 to be between 1,150 and 1,200 kilométrés, Eris appears as a Pluto twin, with a bright surface possibly caused by a collapsed atmosphère, owing to its cold environment. We anticipate that this atmosphère may periodically sublimate as Eris approaches its perihelion, at 37.8 astronomical units from the Sun.

The dwarf planet (136199) Eris was discovered9 in 2005. Its radius has been estimated to be 1,200 ± 100 km on the basis of direct imaging10, although détection of its thermal flux provided another estimate" of 1,500 ± 200 km, potentially making it larger even than Pluto, and the largest known dwarf planet. The motion of Dysnomia (Eris' satellite) provides Eris' mass, ME = (1.66 ± 0.02) X 1022 kg, 27% larger than Pluto's mass12. No short-term (day-scale) brightness variability has been detected for Eris at the 1% level13,14, suggesting either a spherical body with no albedo variegation, or-if elongated-a finelytuned, pole-on viewing geometry. The spectrum of Eris is very similar to that of Pluto and reveals a methane-ice-rich cover, and another dominant ice, presumably nitrogen, but not excluding argon1.

Stellar occultations by Eris are rare, as it subtends a minuscule angular diameter (~0.03 arcsec) while currently moving in severely depleted stellar fîelds at an angular rate of ~1.5 arcsec h"1 at most.

Using the techniques described in ref. 15, we predicted one Eris occultation in 2010, on November 6 ut. We attempted observations from 26 stations, and the occultation was detected from two sites in Chile, with two détections at San Pedro de Atacama (San Pedro for short) with the Harlingten and ASH2 télescopes, 20 m from each other, and one détection at La Silla, with the TRAPPIST telescope (for details, see Fig. 1, Supplementary Figs 1 and2, and Supplementary Tables 1 and2). Another station further south at Complejo Astronômico El Leoncito (CASLEO), Argentina, provided a light curve without occultation, but went close to Eris' shadow edge (~200 km; see Fig. 2).

The San Pedro and La Silla observations provide two occultation segments-or 'chords'-whose four extremities are used to constrain Eris' size (red segments in Fig. 2). When deriving the occultation times, it appeared that two equally satisfactory solutions for the star reappearance time at the Harlingten telescope in San Pedro are possible, yielding two different chord lengths. These two solutions are separated by 1.2 s, and are respectively called solution 1 and solution 2, in chronological order. This ambiguity is due to the fact that the star reappearance occurred during a gap between consecutive exposures, corresponding to a net loss of information. The ASH2 data collected next to Harlingten did not provide enough signal-to-noise ratio to discriminate between these two solutions, and are not used in the fît described below (see Supplementary Information). As a dwarf planet, Eris is expected to be in hydrostatic equilibrium under gravity and centrifugal forces. The most general apparent limb shape is then an ellipse with semi-axes a'>b' with effective radius Rp. = Va'b', defined as the radius of a disk that has the same apparent surface area as the actual body. This shape stems either from an oblate Maclaurin spheroid disappeared during the event (Supplementary Information, section 2). The red lines are the best square-well models fitted to the events. We show in b solution 2 for the light curve (solution 1 being very close at that scale, Supplementary Fig. 3).

The vertical arrow in d shows the time of closest approach (CA) to the shadow edge at CASLEO, at 8,368 s UT.

(small angular momentum régime) with assumed equator-on viewing, or an elongated triaxial Jacobi ellipsoid (large angular momentum régime) observed pole-on, as implied by the absence of brightness variations.

We hâve five free parameters to adjust: a', the apparent flattening (a1 -b1)/a', the ellipse position angle P in the sky plane, and the two coordinates of its centre,fc,gc (Supplementary Table 3). With four chord extremities, our observations allow for an infinity of limb solutions.

However, as the two chords hâve almost the same médian lines (Fig. 2), this strongly suggests that Eris' shape is indeed close to spherical, unless very spécial geometries occurred (see below). Using a circular model with three free parameters (RE = a',fc,gc) and adopting solution 2, we obtain RE = 1,163 ± 6 km (1er formai error). The minimum yj per degree of freedom, y}^=1.38, indicates a satisfactory Fit to the data (Supplementary Table 3). Moreover, the r.m.s. radial residual of 2.1 km is fully consistent with our formai timing errors. We may not exclude, however, the possibility that random topographie features with ampli tude approximately ±3 km exist along the limb, which would resuit in a slightly larger error bar for Eris' radius, RE = 4). However, as RE départs from 1.165 km, the flattening must rapidly increase, requiring fast rotations which are not supported by observations13,14. The extreme case of LETTER RESEARCH RE = 1,500 km previously found" can be ruled out, as it requires fine tunings in both Eris' limb and pôle orientations (Fig. 2). Thus, the most straightforward interprétation of our observations is that Eris is close to spherical, remembering that larger sizes are possible in a narrow région of the parameter space. Consequently, Eris is close in size to Pluto, whose radius3"8 is estimated between 1,150 and 1,200 km.

Our radius value implies a density of p = 2.52 ± 0.05 g cm-3, when combined with Eris' mass'2. This is comparable to Haumea's density'6.17 (~2.6gcm~3), for which a typical rock/ice ratio of 0.85/0.15 is derived18. This suggests that Haumea (and thus also Eris) is a large rocky body with a thin overlying ice shell. Note that the densities of trans-Neptunian objects (TNOs) span a large range, with p values of 1.0, 1.6 and 2.0gcm~3 for Varuna '7,Charon19 and Pluto19,tively, pointing to diverse origins and/or évolutions. Our radius value provides a géométrie albedo of pv = 0.96loQ4 in the visible range (Supplementary Information). This makes Eris almost as bright as a perfect isotropie Lambert surface (for whichpv = 1 by définition), and one of the intrinsically brightest objects of the Solar System. For comparison, Saturn's satellite Enceladus is even brighter, with a géométrie albedo of pv ~1-4, associated with its geologically active surface20. In contrast, Eris' brightness and lack of light-curve variations may stem from the collapse of a nitrogen atmosphère (see below). We find that Eris is brighter than the TNO 2002 TX30o, whose high albedo (0.88^q'Ôô) *s probably due to the exposure of fresh water-ice21.

We now reassess Eris' surface température in the light of our new results. Measurements by the Spitzer22 and IRAM" satellites imply disk-averaged brightness températures of Tb = 30 ±1.2 K and 7b = 38 ±7.5 K at 70 and 1,200 pm, respectively. As a completely absorbing surface at Eris' distance has a température T0 = 40 K, the second value is surprisingly high (and consistent with the fact that the previously found radius" of 1,500 km is about 30% higher than our value), but we note that a unique brightness température Tj.,^31 K matches both the Spitzer (at 70 pm) and IRAM (1,200 pm) measure ments at the 1.5a level (Supplementary Fig. 5). However, Eris' surface température is probably not uniform, because an atmosphère (if any) would be too tenuous to isothermalize the surface frosts, asoccurs for Triton and Pluto. We therefore consider two extreme standard tem pérature distribution models, corresponding to (1) a warmer slow rotator (or equivalently, pole-on orientation, or zéro thermal inertia, the standard thermal model, STM) with sub-solar température Tss, and

(2) a coder fast rotator with equator-on geometry (isothermal with latitude model, ILM), with équatorial température Teq.

In the STM, both Spitzer and IRAM fluxes are reproduced satisfactorily with Tss = 35 K (Supplementary Fig. 5, Supplementary Tables 4 and5). The thermal equilibrium équation Tss = T0[(l -pvq)/ (£//)]1/1 then provides a relationship between the beaming factor // (describing the effects of surface roughness), the phase intégral q and the surface emissivity e, where A -pvq is the Bond albedo, which measures the fraction of reflected solar energy. Using a standard value22 £ = 0.9 and a plausible range from q = 1 (no roughness) to 0.7 (large surface roughness), we obtain q = 0.49-0.66, fully consistent with the values for Saturn's brightest icy satellites20,23. The ILM in contrast leads to the extreme condition 0 <q< 0.24, which is an implausible range as bright objects also hâve large phase intégrais2'.

Essentially, the fast rotator model does not provide enough thermal flux given the new, smaller size of Eris. We therefore strongly favour the STM, implying either a pole-on orientation or a very small thermal inertia, as observed in other TNOs25,26.

The occultation puts an upper limit on a putative atmosphère around Eris. As discussed in Supplementary Information, our preferred model is an isothermal N2 atmosphère near 30 K, for which we can place an upper limit of about 1 nbar (1er level) at the surface (Fig. 3). Similar limits are obtained for hypothetical CH4 or Ar atmo sphères. Also discussed in Supplementary Information is the possibility that a Pluto-like atmosphère sublimâtes as Eris approaches its perihelion, at 37.8 au from the Sun. In that case, Eris would currently be a Offset^in right ascension (mas) with respect to DE405/JPL28 +212+50 +212+50

Offset^in déclination (mas) with respect to DE405/JPL28 -56+50 -56+50

^The timings from the San Pedro/0.4m-ASH2 telescope are not included in the fits, as their larger error bars do not bring further constraints on the length of the occultation at that site (Table 1S and Fig. 3S).

^The timings of Table 1S provide the star position relative to Eris expected centre, using the DE405/JPL28

Eris' ephemeris (http://ssd.jpl.nasa.gov). This position is projected in the plane of the sky, in km, where f is the relative position in right ascension, positive if the star is east of Eris' centre, and g is the relative position in déclination, positive if the star is north of Eris' centre. We use the ICRF/J2000 star position given in Eq. 1.

(c) Velocity of Eris' shadow at the considered site, perpendicular to the observer-Eris line (équivalent to Eris velocity relative to the star, projected in the plane of the sky). The radial velocities with respect to the shadow edge (assuming a circular shape) are 9.7 and 24.4 km s-1 at La Silla and San Pedro, respectively.

(d) This centre corresponds to the offset to be applied to Eris' position in order to match the occultation chords (see the cross in Fig. 2). This offset is relative to the DE405/JPL28 Eris ephemeris, but also dépends on the adopted star position given in Eq. 1.

^Value of minimum x2 Per degree of freedom, see text.

^^Defined as the r.m.s. of the four radial residuals.

TiTime at which Eris and the star are the closest in the sky plane, as seen from the geocentre. h)Distance between Eris' centre and the star in the sky plane at closest approach, as seen from the geocentre. Conversion to mas can be made using an Eris observer range of 95.738 AU during the occultation, corresponding to a scale of 69.436 km per mas. At closest approach, Eris was at a position angle of 167.57 degrees relative to the star, in the J2000 reference frame, as seen from the geocentre.

^These values are derived from the values of /c and gc given above. The error bars are dominated by the uncertainties on the star position (±50 mas), not by the error bars of the circular fit to the occultation chords.

semiaxes a'>b' and apparent flattening -b')/a'. In the small angular momentum régime, this shape results from the projection of an oblate Maclaurin spheroid with semiaxes a=b>c, where a and c are the équatorial and polar radii, respectively. The true flattening of the body is then e=(a -c)/a, and for small values of e, the latter is related to the apparent flattening through31 e'~e • sin2(C)

where ( is the angle between the rotation c-axis and the line of sight (C=0 corresponding to poleon geometry). In that case, we assume that the spheroid is observed equator-on (so that its true flattening is (.=('), remembering that more general viewing geometries will resuit in true flattenings e>e'. In the large angular momentum régime, the shape results from an elongated triaxial Jacobi ellipsoid with semiaxes a>b>c. In that case, the ellipsoid has to be observed almost pole-on because no brightness variations larger than 1% are observed for the dwarf planet13,14. We finally define Eris' effective radius as the radius of a disk that has the same apparent surface area as the actual

Topographie features

Even if globally spherical, a body composed of compact ice with Eris' size may sustain small reliefs with maximum amplitude of typically ^± 5 km (ref. 32). They may create randomly distributed local, radial limb distortions, thus altering the occultation chord lengths, and finally, the retrieved Eris' radius. However, ±5km corrugations are probably too high in the case of Eris. First, Eris'

interior has a density of ~2.5 g cm-3 (see main paper), more than twice the density of ice, resulting in a surface gravity larger than for a pure ice body. Second, the low thermal inertia of TNOs25,26 

Elliptical fit

As mentioned earlier, we can fit an infinity of elliptical limbs to the two occultation chords. If

Dysnomia orbits in Eris' équatorial plane, then the limb orientation is determined and its position angle P can be fixed, see the next subsection. If this is not the case, and since we are primarily interested in Eris' effective radius, we want at least to estimate a plausible range of values for R£.

To do so, we hâve fitted the two occultation chords with elliptical models where P and a' hâve been fixed to prescribed values. The free parameters of the fits are then e', fc and gCl each best fit returning a value for R.e and providing a radial residual which indicates the quality of that fit. We hâve varied P between 0 and 180 degrees by steps of 1 degree, and a' between 1,000 and 1,800 km by steps of 10 km, first using the chords derived from solution 1, and then, from solution 2.

Moreover, we hâve retained only the fits with radial residuals smaller than 4.2 km, which resuit from the quadratic addition of possible timing errors (corresponding to about ±3 km in radial direction)

and/or topographie features on Eris' surface (with amplitude of about ±3 km, see above).

The results are summarised in Fig. 4S. The histogram of derived R^'s reaches a maximum near 1,165 km, as expected because it is close to the value derived from the circular fit, 1,163 km. The distribution rapidly drops as 7?e deviates from that value. This is because elongated limb shapes require a fine tuning of P in order to go through the chord extremities, resulting in low probability of réalisation, see Fig. 2. The 68.3% probability of occurrence centered around the maximum of the distribution (roughly corresponding to the usual la level error bar for a normal distribution) corresponds to the interval i?E=l,165±90 km.

However, we note in Fig. 4S that the limb apparent flattening e' rapidly increases as R.£ deviates from 1,165 km, with a strong corrélation betwen Re and e'. At the boundaries of the interval i?E=1.165±90 km, c' reaches about 0.08, implying that the true flattening c~e/sin2(Ç) is even larger, as mentioned at the beginning of this section.

We can relate e to the body spin frequency u>, assuming hydrostatic equilibrium. In the low angular momentum régime, the corresponding oblate Maclaurin spheroid with équatorial and polar radii a where the term A is missing in the denominator. Once (3=b/a is given, Eq. 6 yields 7, which in turn allows to calculate w, using Eq. 7.

In the example given in Fig. 2, we hâve (3=0.771, from which we dérivé 7=0.506, and then a rotation period of 4.4 hours, i.e. a very fast rotator.

In this case, the long axis of the object should be perpendicular to the chords to within ±2 degrees in order to keep the radial residuals of the fit under 4.2 km, as explained at the beginning of this subsection. This has a low probability of ~2% to occur for P randomly distributed between 0 and 180 degrees.

Moreover, Eris must be observed close to pole-on in this case, to avoid a rotational light curve.

The light curve amplitude of such an object is, in magnitude35:

Am = -1.25 • log10 1 + 72 tan2 C 1 + (t//5)2 tan2 ( (8) 
where we recall that C is the angle between the rotation c-axis and the line of sight. With the condition Am ^0.01 imposed by observations13,14, and taking the case (3=0.711, 7=0.506 discussed above (see also Fig. 2), we find that Ç should be smaller than 18 degrees. This has also a low probability of 5% to occur for a randomly oriented rotation axis.

In conclusion, elongated limbs cannot be excluded by our observation, but they require a fast rotating body (period less than ~8 hours) outside the interval i?E=1.165±90 km (Fig. 4S). Moreover, very elongated bodies, for instance with 77e~1,500 km, require a simultaneous fine tuning on the limb and the rotation axis orientations, and thus does not appear as plausible. We finally note that if a rotational light curve is eventually detected for Eris, then c would then be determined from Eq. 4

or Eqs. 6-7, and Fig. 4S can be used to pin down the value of R^.

Constraints from Dysnomia

The satellite is about 60 times fainter than Eris in K' band3G, and 480 times fainter in V band12.

The satellite spectrum is thus very different from Eris' (and quite redder), suggesting a significantly darker surface.

We may assess an upper limit for Dysnomia's mass by considering a very dark satellite with a visible albedo of ~0.05. Then its radius would be Rq ~240 km (using an estimation of 7?e~1.150 km for Eris' radius), its projected surface area would be ~4% of Eris', and its mass ratio to the primary would reach ~0.01 (assuming the same densities for the two bodies). Let us first assume that Dysnomia was formed during a giant impact on Eris and tidally migrated in the primary équatorial plane, from the Roche limit (~2J?e) to its présent orbit (semi-major axis ~37,400 km, eccentricity lower than 0.01 and period 15.77 days12) by extracting orbital angular momentum from the primary rotation.

With the maximum mass estimated above, Dysnomia could hâve decreased Eris' spin rate from a primordial rotation period of ~7.5 hours (typical of rotation periods observed among TNOs3') to a présent period of ~11.5 hours.

At this rate, Eris' flattening would be e/~0.041 (Eq. 5). However, as the angle between Dysnomia orbital pôle (which coïncides with Eris' pôle in the présent scénario) and the line of sight is close to 50 degrees12, the apparent flattening of Eris' limb in the plane of the sky would be reduced to about 0.024. Also, Eris' rotation axis position angle would then be P~10 degrees12. Using these values, we can fit the occultation chords with an effective radius of 7?e=1,143 km, smaller than the value derived from the circular fit (.Re=1,163 km).

If we now assume that Dysnomia is much smaller than considered in the extreme case above, but still revolving in Eris' équatorial plane, we still hâve to fix P~10 degrees, but we can relax the condition on e'. Using solutions 1 and 2, we then obtain an effective radii Re in the range 1,105-1,155 km, smaller again than the circular fit value.

Finally, if Dysnomia has been captured and migrated inward to its présent orbit38, then no constraints on Eris' rotational period and pôle orientation can be derived, and we hâve to turn back to the discussion developed in the previous subsection.

4.5

Comparison with Pluto's size Pluto's radius has been estimated using various techniques, ranging from direct imaging to speckle interferometry, mutual Pluto-Charon events (occultations and éclipsés), and stellar occultations8.

Each of those methods is model-dependent and relies on assumptions on physical parameters that must be independently inferred. Pluto's radius cannot be accurately measured using direct imaging, owing to its small angular diameter, about 0.1 arcsec. Pluto's radius was first pinned down using mutual events observed in the late 1980's, with values ranging from 1,151±4 km (ref. 6) to 1,178±23 km (ref. 7), depending on assumed Charon's orbital radius and Pluto photometrîc properties8.

Pluto stellar occultations cannot reach the usual kilometer accuracy level achieved with airless bodies, as Pluto's atmosphère refracts stellar rays. Actually, during occultations, the stellar image is dimmed but always visible, and in particular, is never blocked by Pluto's opaque limb. However, combining stellar occultation results with Pluto atmospheric models, it is possible to estimate a radius value of 1,152±32 km (ref. 3), recently improved to l,180^o km (re^-5).

Independent results are obtained with high resolution spectroscopy, which provides constraints on the atmospheric CH4 integrated column density and température, yielding an allowed range of 1,159-1,203 km for Pluto's radius, when combined with stellar occultation results4.

As those values are obtained with different techniques and assumptions, it is not possible to average them in order to reduce error bars. At the présent time, we can only say that Pluto's radius safely lies between ~1,150 and 1,200 km. Therefore, it cannot be concluded whether Eris is smaller or bigger than Pluto, until the latter is more precisely measured.

Albedo

Eris' géométrie albedo p is related to its reflectivity I/F and its radius Re (now assuming a circular limb with i?E=l,163±6 km) through the équations:

(I/F)(o) = <I>(a). p = $(a) (AUkm/RE)2 io°-4<He-HE). (9) Here, (I/F)(a) is the ratio of Eris' surface brightness / observed at phase angle a, to F, the surface brightness of a Lambert surface, where nF is the solar flux density at Eris for the relevant wavelength, 4(a) is the phase function (with the convention (I>(0)=1), yl[/km=1.49598xl08 is 1 AU expressed in and a spatially uniform atmospheric pressure. This is not the case for Eris, where the atmosphère is presently at least 3 orders of magnitudes more tenuous than on Pluto, so that the local N2 ice température should be driven by local insolation conditions, not by sublimation/condensation processes.

The dominant rôle of the latent heat flux over thermal flux in establishing the température of N2 can be expressed through the dimensionless parameter I-Lm/asBrP4=LfrP/(cs<tsbT4) where L is the sublimation latent heat, m the sublimation mass flux, P the atmospheric pressure, cs the speed of sound, and fr is the fractional pressure différence between the saturated and actual atmospheric pressure47. For large values of 7, sublimation processes dominate the local energy flux from the Sun (insolation) and the ice is forced to be isothermal. With the conditions prevailing at Pluto, one obtains 7=5,200, using a typical /r~0. 1 (ref. 47). At Eris, using a maximum pressure of ~5 nbar, and T=30 K, we obtain 7 2, showing that the latent heat flux is at most comparable to the thermal radiation flux from the surface.

Hence, the surface température of a N2 ice is likely to be primarily driven by local insolation conditions. In particular, our predicted sub-solar température is at least 32 K (Table 5S), which corresponds to a N2 equilibrium vapor pressure48 larger than 260 nbar, much higher than the upper limit derived here. Consequently, a local N2 atmosphère might exist at Eris' sub-solar point, and that atmosphère could freeze out to undetectable levels (less than a few nbars) at the limb and on the night side, as it is the case for lo49. Detailed models will be required to confirm this point.

Note that if there are régions covered by pure methane ice on Eris, they should not be significantly warmer than N2. The bulk methane abundance on Eris' surface1, 10%, is similar to the solubility limit of CH4 in N2 (ref. 50), suggesting the formation of pure ice grains. Assuming that such pure CH4 ice grains exist but are at the same température as N2, the atmospheric CH4/N2 is given by the ratio of their vapor pressures at ~30 K, i.e. a few times 10~5 at most. If pure methane grains are not présent and the CH4/N2 surface ratio is 10 %, the atmospheric CH4/N2 is further decreased by a factor 10.

We conclude that the probable range of methane abundance in Eris' atmosphère is 10"5-10"6.

Although detailed models would be needed, this is likely unsufficient to lead to a significant strato sphère, and we favor the case where any Eris' atmosphère is isothermal near ~30 K. More precisely, when considering such atmosphères, we hâve assumed that the pressure at the surface corresponds to the saturation value at a given température48. We hâve generated synthetic light curves (see Fig. 3) caused by a tenuous atmosphère using a ray tracing code, as detailed in ref. 29. Assuming a N2 isothermal atmosphère near 30 K, we dérivé an upper limit of about 1 nbar (la limit), see Fig. 3.

Assuming a Pluto-like atmosphère (30 K near the surface and 94 K in the upper atmosphère), we would obtain upper limits larger by a factor of 3-4 compared to the isothermal model.

We can also place upper limits of about 1 nbar for isothermal pure methane or pure argon atmo sphères. In order to sublimate at that pressure48, the methane ice should be at about 37 K. As our surface température estimâtes are comparable to or smaller than those values, it is normal that a CH4 atmosphère escaped détection.

Finally, we note that Eris, if indeed covered by N2 ice, might develop an atmosphère as it approaches perihelion, at 37.8 AU. In that case, the N2 ice température will be buffered by the nitrogen atmo sphère, implying a constant température on night and day sides. Assuming a maximum Bond albedo of 0.7 (typical of bright objects like Enceladus23), we expect a surface equilibrium température of at least ~35 K, large enough to maintain a 2 /rbar or more Pluto-like atmosphère. This atmosphère might collapse when Eris recedes from perihelion, thus explaining its current bright surface caused by a thin nitrogen-ice coating.

Pluto and Eris are icy dwarf planets with nearly identical sizes, comparable densities and similar surface compositions as revealed by spectroscopic studies1,2. Pluto possesses an atmosphère whereas Eris does not; the différence probably arises from their differing distances from the Sun, and explains their different albedos3.

Makemake is another icy dwarf planet with a spectrum similar to Eris and Pluto4, and is currently at a distance to the Sun intermediate between the two. Although Makemake's size (1,420 ± 60 km) and albedo are roughly known5,6, there lias been no constraint on its density and there were expectations that it could hâve a Plutolike atmosphère4,7'8. Here we report the results from a stellar occult ation by Makemake on 2011 April 23. Our preferred solution that fits the occultation chords corresponds to a body with projected axes of 1,430 ± 9 km (1er) and 1,502 ± 45 km, implying a V-band géométrie albedo py = 0.77 ± 0.03. This albedo is larger than that of Pluto, but smaller than that of Eris. The disappearances and reappearances of the star were abrupt, showing that Makemake has no global Pluto-like atmosphère at an upper limit of 4-12 nanobar (1er) for the surface pressure, although a localized atmosphère is possible. A density of 1.7 ± 0.3 g cm-3 is inferred from the data.

Stellar occultations allow détection of very tenuous atmosphères and can provide accurate sizes and albedos9 '10'11,3'12, so The light curves of the occultation are shown in Fig. 1. Fitting synthetic square-well models to the light curves yielded the disappearance and reappearance times of the star (Table 1), from which we calculate one chord in the plane of the sky for each site (see Sup plementary Information section 3). On the basis of analyses of the light curves, taking into account the cycle time between the images and the dispersion of the data, we deduce that there were no secondary occultations, so we can reject the existence of a satellite larger than about 200 km in diameter in the areas sampled by the chords. The resuit is consistent with a deep-image survey that did not find any satellites16. The chords can be fitted with two shape models (Fig. 2).

Our preferred shape, which is compatible with our own and other observations (see Supplementary Information section 8), corresponds to an elliptical object with projected axes of 1,430 ±9 km and 1,502 ± 45 km. By combining this resuit with visible photometry at various phase angles17, we calculated that Makemake has a V-band géométrie albedo ofpy = 0.77 ± 0.03 (see Supplementary Information section 4). This is considerably high compared to albedos of other trans-Neptunian objects (TNOs)5, and is larger than that of Pluto (pv = 0.52)18 but smaller than that of Eris (py = 0.96)3.

The object is large enough to be in hydrostatic equilibrium, so it is possible to use the figures of equilibrium formalism, as done for Haumea19, to analyse the shape of a body that rotâtes with Make make's period of 7.77 h (refs 20, 21). The object could only be a tri axial Jacobi ellipsoid for densities in the range 0.66-0.86 g cm-3 (for example, ref. 22). Such low densities are unrealistic for a body as large as Makemake (see Supplementary Fig. 7). Thus, Makemake must be an oblate Maclaurin spheroid for plausible densities between 1.4 and 2.0 g cm-3 (see discussion in Supplementary Information section 8).

LETTER RESEARCH 1,111,11., 111., [ 11111111, i, 1111 Êj Makemake is, a priori, a good candidate to hâve a fully developed atmosphère4,7,8. Its albedo and distance from the Sun lie between those of Pluto (which has a global atmosphère) and Eris (which does not, at least currently). Makemake may also hâve a similar surface composi tion to Pluto and Eris, on the basis of spectroscopic observations4. At the warm températures of about 50 K expected from two-terrain ther mal models, methane vapour pressure is on the order of a few microbars, whereas nitrogen vapour pressure is around two orders of magnitude higher (as illustrated in Fig. 32 of a work on vapour pressures25).

However, a global Pluto-like atmosphère is ruled out by our occulta tion light curves, which hâve abrupt ingress and egress profiles (Fig. 1).

To get an upper limit on a global atmosphère, it is possible to model its effects on occultation profiles and compare them with observations. The profiles from the New Technology Telescope (NTT) imply an upper limit to the surface pressure of a putative methane atmosphère of only 4-12 nbar at a 1 g confidence level, and 20-100 nbar at the 3er level (see Fig. 3 and Supplementary Information section 6 for a description of the models, which also consider nitrogen).

One possibility that might explain the lack of a global atmosphère is that Makemake has little or no N2 ice, because N2 vapour pressure is well above the microbar level even on the cooler terrain. From an update of the results of the models on rétention of volatiles8, considering new empirical déterminations of the vapour pressures of N2 and CH4 (ref. 25), Makemake would not hâve retained N2 if it were smaller than 1,370 km, which we rule out. With a diameter of 1,430 km, Makemake would hâve to hâve a density ofless than 1.7gcm-3, smal ler than the adopted nominal value of 1.8 g cm-3 (ref. 8), to resuit in complété N2 loss. Considering now that CH4 is abundant on the sur face of Makemake, again from the volatile-retention arguments, its density would hâve to be greater than 1.4gcm"3. Other constraints on the density based on the observed shape and the figures of equilibrium are discussed in Supplementary Information section 8. Another Shadow radius (km) Figure 3 | Observed and synthetic light curves. A comparison of two CH,( pure atmosphère models with data (ingress and egress profiles: green, NTT g' points; red, NTT r' points; blue, VLT J-band) plotted against the distance to Makemake's shadow centre, assuming a circular limb for simplicity. Bars are the radius intervals corresponding to each intégration bin. For better reading, and in contrast to Fig. 1, the fluxes hâve been normalized between zéro (average value of the flux during the occultation) and unity (full stellar flux). The models correspond to a CH4 atmosphère with a surface température of 30 K, a nearsurface température gradient of 17 K km-1 followed by an isothermal profile with T = 100 K for higher altitudes. Solid line: surface pressure of Paurf = 8 nanobar, compatible with the data at 1 a. Dash-dotted line: model with Fsurf= 100 nanobar (compatible with the data at 3cr). See Supplementary Information for a full description of the models. LETTER RESEARCH possibility to explain the lack of a global atmosphère is a nearly pole-on orientation. From a theoretical study7, TNOs with high obliquity are less likely to hâve globally distributed atmosphères.

The remarkably high albedo of Eris (pv = 0.96) is thought to be the resuit of a collapsed atmosphère, which coated Eris with bright, fresh ices3-26. A fully condensed atmosphère on Makemake might hâve resulted in an albedo similar to that of Eris, which is not the case.

However, if Makemake had a local rather than a global atmosphère, some parts of the surface could be fully covered with fresh ice from the collapsed part of the atmosphère and be very bright, and others could remain dark. The overall albedo of Makemake could thus be smaller than that of Eris, but larger than that of Pluto. A local atmosphère would also provide a reason for the two-terrain models needed to explain Makemake's thermal data.

Local atmosphères on TNOs are theoretically plausible7; they can be confined to a subsolar région or a band at the subsolar latitude. It should be noted that a small drop of only 10 K in surface température implies a decrease of three orders of magnitude in the vapour pressure of CH4 and N2 at low températures.

We can investigate whether the presence of a local atmosphère is consistent with our data. The bottom of the occultation light curves should be fiat in an airless body. Flashes in occultations are known to be caused by the focusing effect of an atmosphère when the observer passes near the centre of the shadow27. Thus, the noise level of the light curves at their bottoms can put limits on the local atmosphère that can extend to the limbs. Modelling of central flashes for plausible local atmosphères shows that an atmosphère with surface pressure on the order of several microbars can exist and still be consistent with the data, provided that the atmosphère is confined to spécifie parts of the limb (see Supplementary Information section 7).

From the information gathered on Pluto, Eris and now Makemake using stellar occultations, we hypothesize that the albedos and other surface properties of the largest TNOs are determined by sublimation and condensation processes. In our picture, the largest albedos would resuit from atmosphères that hâve fully condensed (collapsed onto the surface), whereas medium-albedo objects would hâve local atmo sphères and the lower-albedo objects would hâve global atmosphères from sublimation of the volatiles. Future studies will shed light on this possibility and whether sublimation is fully solar driven or is also driven by other mechanisms. The airbome Stratospheric Observatory For Infrared Astronomy, in combination with large aperture télescopes on the ground, would be an excellent tool for this kind of study.

LIST OF ALL THE TELESCOPES THAT OBSERVED OR ATTEMPTED

TO OBSERVE THE OCCULTATION. Table S1. Sites and télescopes that observed the event. In the comments column we include the average signal to noise ratio (SNR) per data point, outside the occultation, for those sites that had a positive resuit. Note that the intégration times were different for each telescope, which explains the various SNR values reached. The synthetic aperture radii used were 1.05, 2.6, 0.6, 2.9, 3.2, 2.4, and 4.0 arcsecs for NTT, TRAPPIST, VLT, Armazones, S. Pedro de Atacama Harlingten, S. Pedro de Atacama ASH2 and Pico dos Dias, respectively. These radii typically corresponded to 2 pixels. (TTa'b'). The géométrie albedo in the R band was obtained from the V-R color of Makemake, implying that HR = -0.395±0.02. Using that resuit for HR and a value of Hsun.R = -27.08 we obtained the following géométrie albedo in the red pR = 0.90±0.02.

THERMAL MODELS OF MAKEMAKE AND EQUILIBRIUM TEMPERA

TURES.

The equilibrium température of an airless spherical body can be computed by using a standard thermal model

F^jl-Alb) G£T]Ç r
where Alb is the bond albedo, Fsun = 1,360 W m'2 is the solar constant (Solar Flux at 1 AU), ct = 5.67x10'8 W m'2 K'4 is the Stefan-Boltzmann constant, and the emissivity (e) accounts for the déviation of pure blackbody émission (usually assumed to be 0.9 based on asteroid studies). The beaming parameter (rj) takes into account some anisotropic émission effects as a resuit of surface roughness and thermal inertia and is usually thought to be around 1.2 for TNOs (ref. 5). The variable r is the heliocentric distance of the body in astronomical units and Ç is a parameter that takes the value of tt if the object is a fast rotator or 2 if it is a slow rotator. If we want to know the température at the subsolar point, this parameter must be set to 1. On the other hand Alb = pvq, where pv is the géométrie albedo of the object, and q is the phase intégral. This parameter is probably close to 0.75 for Makemake: There is an empirical relation between géométrie albedo and phase intégral based on TNO observations29 and also based on observations of the satellites of the outer planets30. From those papers almost identical prédictions for the phase intégral of Makemake are obtained, around 0.75.

Using a value of 0.77 for the géométrie albedo, a phase intégral of 0.75, an emissivity of 0.9, a beaming parameter of 1.2, and the distance at which the occultation by Makemake took place (52.21 AU) we get an equilibrium température for Makemake of only 36 K and 32 K (for the slow rotator and fast rotator cases respectively). However, we know that 0.77 is the average géométrie albedo, and in tact we know that Makemake must hâve a dark terrain to account for the Spitzer and Flerschel measurements5,6,23. The dark terrain must hâve a géométrie albedo of 0.12 or iower. Using the thermal model with this input for the géométrie albedo, we get a température of 44 K for the slow rotator case and 40 K for the fast rotator case. The resuit is 52 K at the subsolar point or even higher if we use a Iower than 0.12 géométrie albedo (see Fig. S4b). This température is high enough for substantial nitrogen and even methane ice sublimation. Makemake has an icy surface with CH4 ice (e.g. ref. 4) and perhaps N2 ice. An atmosphère may then be maintained through vapour pressure equilibrium between the ice and its gaseous counterpart. The ice température Tsurf Controls the atmospheric surface pressure Psurf through sublimation processes, and for pure ices, we may use laboratory measurements to relate Tsurf and Psurf (ref. 25). More complicated laws apply when the ices are mixed but we use here the approximation that a pure ice (either CH4 or N2) is in vapour equilibrium with its gaseous counterpart. The occultation probes Makemake's limb, where the température possibly varies from 35 K down to 10 K (see Fig. S4 where we show an example). We examine the possibilities of a pure CH4 atmosphère, an N2

atmosphère with a small amount of CH4, and a pure N2 atmosphère.

Pure CH4 atmosphère

A pure CH4 atmosphère should start at the surface with a température between 10 and 35 K, but near-IR heating should increase this température within a few kilometers, to reach an isothermal branch, as observed in Pluto's atmosphère31,32.

Considering that Pluto's upper atmosphère is at about 106 K, and that the solar insolation at Makemake is reduced by a factor of about 2.8 with respect to Pluto, a rough estimation of a CH4 atmosphère température at altitude is about 100 K. For the thermal gradient near the surface, we adopt a value of 17 K km'1. This high value is typical of what is expected for methane-rich atmosphères31'32.

the surface pressure of the putative CH4 atmosphère considered here. If one relaxes the 1a limit to a 3a level (increase of x1 to ^iin + 9), we obtain a limit of about 25 nbar. Taking a surface température of 40 K, instead of 30 K, and everything equal besides, we obtain a 1a limit of 11 nbar for Psurf, and no stringent 3a limit up to values of 100 nbar or more (Fig. S2). This is because the deep, denser, part of the atmosphère causes a shallow bottom in the occultation light curve that becomes undistinguishable from the noise in the data (see Fig. 3).

In conclusion, a 1a level upper limit of about 10 nbar can be placed for the surface pressure of a CH4 atmosphère, but the data could accommodate denser atmosphères (up to 100 nbar or more) if a 3a level is considered, or if surface températures larger than 30 K are considered. This said, 10 nbar corresponds the equilibrium vapour pressure of CH4 for a température of about 40 K (ref. 25). This is larger than the température expected along Makemake's limb, and is thus consistent with our non-detection of a global CH4 atmosphère at the 10 nbar limit.

N2 atmosphère

A putative nitrogen atmosphère is likely to be mixed with CH4, considering the détection of the latter species on Makemake's surface (e.g. ref. 4). Thus, the same general thermal structure is expected as for CH4, namely an atmosphère starting from a cold surface, and then reaching a warmer isothermal branch after a few kilometers. In that case, an upper limit of Psurf = 12 nbar can be derived at the 1a level, comparable to the pure CH4 atmosphère case (black curve in Fig. S2), and Psurf of about 35 nbar at the 3a level. The PSUrf = 12 nbar value would correspond to a vapour pressure equilibrium with a N2 surface ice at 29 K, consistent with our starting assumption of Tsurf = 30 K. The possibility still exists, however, that Makemake's atmosphère would be essentially pure N2, since this gas is much more volatile than CH4 by about 5 orders of magnitude. In that case, a 1a upper limit for Psurf is about 4 nbar, and the corresponding 3a limit would be about 20 nbar (blue curve in Fig. S2).

In conclusion, the various atmospheric models considered here place 1a level upper limits of 4-12 nbar for the surface pressure, while less stringents upper limits in the range 20-100 nbar are obtained at the 3a level. We note also that we hâve used a large value of the thermal gradient near the surface (17 K km '1). Lower values of that gradient would provide more stringent limits on Psurf. We hâve actually examined here the less favorable cases, where a thin inversion layer close to the surface remains unnoticed due to the shape of the light curve and the noise in the data (see the dash-dotted curve in Fig. 3) \ WWW.NATURE.COM/NATURE ! 9 separated from it. The image size is 50x72 pixels. Right panel: Same as left image but with the potential artifact removed by interpolation.

In addition to the above results there is also a very marginal indication that light refracted somewhere on the limb might hâve reached the detector at Paranal, because there is a slight departure from a fiat bottom of the curve. However, this feature in the light curve is not significant enough. In the first example (Fig. S5a) Makemake's apparent limb has semi-major and semi-minor axes of 800 km and 714 km respectively, with the semi-minor axis at position angle (PA) of 65°(measured from North and in clockwise sense). This particular shape is taken to create a curvature of the isopycnic (iso-density) atmospheric layers that causes the appropriate caustics near the shadow centre. In fact, what is important for the caustics is the curvature of the isopycnic layers. This curvature is likely to be quite different from that of the limb in the presence of a partial atmosphère. This model is not intended to fit the occultation light curves and it is used only for illustration to see whether the features it produces would be at the noise levels of the light curves.

It is also assumed that Makemake's subsolar latitude is -30°. In that configuration, a local CH4 atmosphère could be maintained at the subsolar point, with surface pressure Psurf = 3-25 pbar, corresponding to the equilibrium vapour pressure of CH4 ice at 50-55 K and then extend to the limb.

We consider this partial atmosphère located between PA's 285°and 350°along the limb (surrounding the subsolar latitude of 30°), and we used a ray-tracing code The x2 values of the fits were 43.3, 22.6, 88.1 and 794.1. The dotted line in the S. Pedro panels is the expected light curve from an airless body.

In the second example, a less elongated Makemake's atmosphère has been used (Fig. S5b). The différences with the other model are the dimensions of the body We point out that the models presented here dépend on several parameters that we do not know. Exploring the possible phase space is clearly out of the scope of this work. This said, the main lesson learnt from our modelling is that a patchy CH4 or N2 atmosphère with Psurf of a few pbar may be présent in the upper and lower left quadrants of the limb and cause structures compatible with the observed light curves. Larger surface pressures would cause too intense features. The élongation direction in the best ellipse fit shown in the main paper is almost perpendicular to the chords direction (Fig. 2). This seems too coincidental. In reality, the élongation in that direction is probably an artificial resuit because the length of the axis in the direction perpendicular to the chords is not well constrained by the observations. This is usually the case for a nearly spherical body when there are no négative chords sufficiently separated from the positive chords. The same effect was also seen in the occultation by Eris3, for which a large set of ellipse solutions was possible with considérable élongation in the direction perpendicular to the chords. Thus, we think that the élongation of the nominal best ellipse fit for Makemake in Fig. 2 is too large to be real and is mainly a resuit of limitations in the observations, similarly to the case of Eris.

RESEARCH SUPPLEMENTARV INFORMATION

We know that the purely circular fit is also unlikely because the general projection of a Maclaurin spheroid is an ellipse and only the pure pole-on orientation can give rise to a circle (but a pure pole-on configuration is very unlikely).

On the other hand, the low rotational variability of Makemake28,20,21 can be explained if the object has a very homogeneous surface with perhaps a latitudinal band, or if its spin axis orientation is not too far from pole-on. Probably the reality is a mix of the two causes (which we show in Fig. S4). Because the object must hâve a Maclaurin shape as explained in the main paper, and the projection of a Maclaurin seen far from the equator-on view is doser to a circle than to a very elongated shape, the low rotational variability would favor a fit doser to the circular one than to the very elongated nominal ellipse fit (which has an axial ratio of 1.15).

Besides, the comparison of the Spitzer+Herschel data with the occultation favors the smaller object, which is the neariy circular fit, because the nominal elliptical solution with an axial ratio of 1.15 implies a larger body than is compatible with the Spitzer+Herschel results at almost 2o. The équivalent diameter of the ellipse with an axial ratio of 1.15 is 1,533 km, away from the maximum diameter allowed for by Spitzer+Herschel (ref. 6).

As mentioned in the main paper, for a spherical Makemake of 1,430 km diameter, a density of ~1.7 g/cm3 is derived to explain the lack of a global atmosphère, using volatile loss models8. In this case the body would not hâve had enough mass to retain the primordial N2. The derived 1.7 g/cm3 density is consistent with the trend observed empirically, as can be seen in Fig. S7. But if Makemake had a projected axial ratio as large as 1.15 (the nominal ellipse fit), the équivalent diameter would be 1,533 km and Makemake would require a smaller density than 1.5 g/cm3 to be depleted in N2, according to the volatile loss models8. Such a low density for a larger object would be further away from the trend of density versus size than the 1.7 g/cm3 value (see Fig. S7). Thus, these considérations also indicate that the circular model (the smaller object) is slightly preferred with respect to the nominal ellipse fit. Nevertheless if the lack of a global atmosphère on Makemake is not a resuit of a surface déplétion in N2 but to a pole-on orientation, this is again in favour that Makemake's projected shape is doser to circular rather than having a Since the discovery of (15760) 1992 QBi (Jewitt & Luu (1993)), about 1,260 transneptunian objects (TNOs) hâve been discovered. Determining their physical and dynamical properties is key to understand the origin and évolution of our Solar System.

They were possibly formed closer to the Sun (near the présent Uranus's and Neptune's orbits), and displaced to their current location through gravitational perturbations during the migration of the giant planets (Gomes et al. 2005).

Their physical characteristics contain information about the primordial protoplanetary nebula. Moreover, the inferred Chemical, thermal and collisional processes that they underwent tell us something about the évolution of the outer Solar System. Due to their large heliocentric distances, our knowledge about their sizes, shapes, albedo, densities and atmosphères remains fragmentary (Stansberry et al. 2008).

Stellar occultations allow accurate déterminations of those parameters. They provide sizes and shapes with accuracies which may reach the kilométré level, and detect atmosphères with a sensitivity of a few nanobars in surface pressure. Unfortunately, they are difficult to be predicted. To overcome this obstacle, we hâve undertaken systematic searches of stars that may be occulted by ten of the largest TNOs, using the 2.2 m telescope of the European Southern Observatory (ESO) at La Silla (Assafin et al. 2010(Assafin et al. , 2012)). Quaoar. Section 2 describes our prédiction method and briefly présents the observations. Data analysis is described in Section 3. Quaoar's size and shape are discussed in Section 4

and their physical implications are presented in Section 5. In Section 6, we give upper limits for the pressure of possible atmosphères. Further constraints (search for satellites and brief comments on the 17 February 2012 and 15 October 2012 occultations) are given in Section 7, before concluding remarks in Section 8. (i)

An alert was triggered at several potential sites, resulting in data collected with a large diversity of instruments (Table 1). Out of sixteen stations distributed in Chile, Uruguay, Argentina and Brazil, thirteen could acquire data, and six of them obtained a positive détection of the occultation. Moreover two stations, Salto/Uruguay and Pico dos Dias/Brazil, were close enough to the shadow to constrain Quaoar's size, even though they did not detecte the event (Figs. 1 and2).

EDITOR: PLACE TABLE 1 HERE.

Analysis

Occultation light curves

The various instruments and detectors setup are listed in Table 2, that also provides the ingress and egress times for the sites where the occultation was detected.

Ail light curves (Fig. 2) were normalized to the unocculted star flux, by applying linear or second degree polynomial fits to the flux. The mid-exposure time of each frame was extracted from the image headers. Only four stations (out of the six that detected the event) hâve an internai accuracy of a few hundredths of a second: Harlingten and ASH2 (at San Pedro de Atacama), Armazones and UEPG. On the other two sites, only the integer part of the second is available, due to the acquisition software used. In order to retrieve the fraction of a second, a linear fit was performed to the set of points (i, U), where i is the image number and U is the corresponding header time. This fit eventually provides the cycle times and mid-exposure times of each frame, once truncation effects are accounted for. This allowed us to retrieve the time of each image with relative accuracies of about 0.05 seconds. Ail sites excepting one set up their docks using robust internet servers for time synchronization, and absolute errors larger than one second are not expected. The exception is the Rivera station, where the computer clock was manually set up one hour before the event by looking at a GPS display, due to the lack of internet access. This point is discussed further in Section 4.3.

EDITOR: PLACE TABLE 2 HERE.

Occultation timing

The start and end times of the occultation were obtained for each light curve by fitting a sharp edge occultation model. This model is convolved by Fresnel diffraction, the CCD bandwidth, the stcllar diameter in kilomètres, and the finite intégration time, see Widemann et al. (2009).

The Fresnel scale (F = yjXD/2) for the présent Quaoar's geocentric distance D = 42.4 AU = 6.34 x 109 km is 1.4 km for a typical wavelength of À = 0.65 /j,m. The star diameter is estimated using the formulae of van Belle (1999). Its B, V and K apparent magnitudes are respectively 16.5, 15.8 and 12.8 in the NOMAD catalogue (Zacharias et al. 2005). This yields a diameter of about 0. (2008). Nevertheless, the formation scénario and collisional history of individual objects are not known, so that we cannot discard solutions that diverge somewhat from the equilibrium figure. We examine here possible ranges of sizes and oblateness for Quaoar, making various extreme assumptions.

As seen in Figs. 2 and3, the chords provide a shape that lias a significant departure from the ellipse. We consider here two extreme possibilities: (1) ail the timings are correct, implying that Quaoar possesses significant topographie features (craters or mountains), a resuit which is not expected for such a big object (Section 4.2); or (2) Quaoar is a spheroid in hydrostatic equilibrium (a Maclaurin object), which implies that there were timing problems at some stations (Section 4.3). We now discuss these two possibilities and their implications. The nominal solution uses the ingress and egress occultation times at face value, and fits an elliptical shape to the chord extremities, accounting for the la error bar of each of them, see Fig. 3. The best fit returns a best value Xpdf= 9-1 clearly indicating that an overall elliptical limb shape is not satisfactory. This is obvious from Fig. 3, and at this point it is meaningless to interpret the physical parameters derived from this fit and to discuss their error bars. We thus examine possible extreme solutions in which Quaoar lias large topographie features, i.e. a deep crater or a high mountain. Another solution consists in a less oblate object, but with an important prominence on its eastern limb (Fig. 6). To obtain this solution, that will be referred to as the mountain solution, we hâve discarded the egress extremity of Rivera, the southernmost positive chord.

So, again we hâve N = 9 data points to fit. The elliptical fit lias a position angle of 7 deg, an oblateness of 0.138, an équivalent radius of 525 km and a Xmin ~15.7 (Xpdf = 3.92),

indicating a fit quality comparable to the crater solution.

EDITOR: PLACE FIGURE 6 HERE.

Fig. 7 shows the radial residuals of tliis fit. The Rivera egress chord extremity displays a residual of more than 140 km, or 24% of the équatorial radius (587 km) obtained here.

This can be compared to the much smaller topographie features, ~20 km at most, expected on an icy body (Johnson & McGetchin 1973). A rocky body could support higher features, about ~150 km, comparable to the value we obtain here. However, the density for this solution would be around 2 g cm-3 (see table 4), implying in a significant fraction of ice.

Thus, the mountain solution seems physically implausible, as the mountain is too big to be sustained at Quaoar surface.

EDITOR: PLACE FIGURE 7 HERE.

Maclaurin solution

Although we do not expect large absolute timing errors at the various stations (Section 3.1), it is difficult to assess timing errors after the fact, as some of the observations are not repeatable. In view of the physically implausible Quaoar's shape obtained in the previous Section, we now allow for time shifts at each station, except for San Pedro de Atacama. For this site, we hâve two observations with independent and consistent ingress and egress times (Table 2), so that a systematic timing error is not expected for the two corresponding occultation chords.

Its known that, in an ellipse, the mid point of a set of parallel chords should be aligned,

i.e. a straight line must go through ail the mid point of the chords. So, as a first exercise, we slide the various chords shown in Fig. 3 along themselves, so as to align the middle of ail chords along a line perpendicular to the chords. Thus, a circle is adjusted to the chord extremities with a radius 555 ± 2.5 km and a value Xmin = 1-33.

To estimate the quality of the fit, however, we hâve to consider that we implicitly added 4 adjustable parameters to our model (the time shifts of ail the chords except San Pedro de Atacama). Besides, we also adjust the radius of the circle RCirc and the coordinates of its centre (fc,9c)• So, we hâve M -7 parameters to be adjusted to the N = 10 chord extremities, implying in a y2df = 0.44. The fact that the x2df is satisfactory is thus an encouraging argument to proceed in our analysis.

Actually, we may now align the middle of the chords, but along a line which makes an angle a relative to the line perpendicular to the chords. Doing so, we transform the original circle (corresponding to a = 0) into an infinity of possible ellipses parametrized by <a, which can vary in principle between -90 and +90 deg, with the convention that a is increasing from celestial north to celestial east2. We will dénoté (3 the angle between the long axis of the ellipse and the direction of the chords (with the same convention of sign as for a).

Note that since we slide the chords along tlieir own direction, ail the ellipses share the same area. Consequently, the radius of the circle lidrc described before actually gives 2Note that a -0 does not mean that the semi-minor axis of the ellipse is pointing to the < celestial north, as the chords are inclined with respect to the east-west celestial direction.

unlikely due to the methane heating process, as discussed above. We obtain a marginal détection of a pure isothermal methane atmosphère at 42 K with a pressure of 54 db 25 nbar.

Unfortunately the light curves from the otlier sites do not hâve the necessary quality to bring further constrains to this resuit.

We note that methane lias a vapour pressure of 32 nbar at 42 K (Fray & Schmitt 2009), consistent with our absence of détection. Nitrogen is more volatile, with a vapour pressure of 176 ^bar at the same température. Thus, the atmospheric upper limit tends to rule out the presence of exposed N2 ice on the surface.

Finally, using the values of the géométrie albedo pv and phase intégral q given earlier, we conclude that the surface température at the sub-solar point can reach 54 K, which could produce a 17 //bar CH4 local atmosphère, that would partially condensate to the nanobar level near the limb, thus being undetectable in our data set. 

Search for secondary events

Considcring its flux ratio to Quaoar and assuming equal albedo, Weywot lias a size ratio of about 1/12 with respect to the primary (Fraser & Brown 2010). From the Quaoar size determined here, we can calculate Weywot équivalent diameter. This yelds to a diameter of about 90 km to the secondary. According to the ephemeris3 calculated by Vachier et al. (2012), the satellite shadow passed at about 800 km north of Quaoar's shadow, and about eleven minutes later. The occultation would hâve lasted for five seconds at most.

No secondary occultation was seen at Belo Horizonte (CEAMIG), neither at Pico do 3available at http://www.inicce.fr/fr/ephemerides/formulaire/form_ephepos.php Dias (OPD), the two nearest sites to the predicted path. However, the observation strategy was not optimized for such a détection, silice the intégration time at CEAMIG was 6 seconds, with a readout time of 3 seconds (Table 1). At OPD, the exposure time was 0.95s, witli about 1.05s of readout time between frames. So, if an occultation by Weywot had occurred during one exposure in CEAMIG, a conspicuous 2.8 magnitude drop would hâve been observed. If it had happened during a readout time interval, a magnitude drop of 1.1 would be expected. No such dimming was detected. If a 5 sec occultation had happened at OPD, it would hâve lasted for at least two complété cycles with a drop of about 3 magnitude, any event like that is seen in the light curve. However, the absence of détection does not bring further constraints on the satellite orbit. 5.

Fifteen sites reported attempts, but the majority was unable to observe due to poor weather or technical issues. The event was detected from four different sites (Table 6), but three sites were separated by 3 km only cross-track. This means that we hâve only two effective chords to fit Quaoar's limb.

EDITOR: PLACE TABLE 5 HERE.

This unfavourable prédiction explains why only four télescopes in Cliile were scheduled for that event. The PROMPT4 instrument was the only one that could eventually monitor the star and detect the event. Two 0.4 m télescopes with identical setups were used with 8 sec intégration times and 1 sec readout times. The acquisitions were offset by 4 seconds, so as to overlap the effective observing intervals and thus avoid information loss caused by readout times.

The event lasted from 00:45:30±2.5 to 00:45:50±2.5 UT, which translates into a chord of 400 km in lentgh. Consequently, this observation does not provide new constraints on Quaoar's size, but we may use this resuit for astrometric purposes. Even not knowing if the chord probed the north or the south part of Quaoar (i.e. we hâve two possible solutions),

this uncertainty is about only 26 mas in the plane of the sky. Table 5 provides the time of Quaoar's closest approach to the star, and its angular distance to the star at that moment.

Conclusion

We observed the first multi-chord stellar occultations by (50000) Quaoar, one from South America on 4 May, 2011 with five chords, and another one from Europe on 17

February 2012, with two chords. A third single-chord event was detected from Cliile on 15

October 2012. The last two occultations confirm, but do not bring further constraints to the results obtained from the first event. Thcy provide, however, astrometric constraints that may be used for the prédiction of future events.

Taking the timings of the 4 May 2011 event at face value, we find topographie features (crater or mountain) that are too large to be maintained on a body as massive as Quaoar.

4Panchromatic Robotic Optical Monitoring and Polarimetry Télescopes, Cerro Tololo (latitude: 30°10' 10".78 S, longitude: 70°48 ' 23".49 W, altitude.: 2,207 m).

i An alternative approach is to allow for time shifts at ail stations, i.e. accept that there are timing errors. Such errors are admittedly bothersome, since we do not hâve clear explanations for them. However, as discussed above, the largest time shift has to be applied to the only station, while the shifts at the other stations remain compatible with the occultation time uncertainties at 2.5cr level.

With this approach, we obtain an variety of satisfactory fits by elliptical limbs. Ail those ellipses hâve the same apparent area, yielding well constrained values for Quaoar's équivalent radius Rcquiv = 555 ± 2.5 km and visible géométrie albedo pv = 0.109 ± 0.007. Adding the further constraint that Quaoar is a Maclaurin spheroid with indeterminate polar aspect angle Ç between 0 and 90 deg, we find that Quaoar has an équatorial radius a = 5691^km, true oblateness e = 0.0897 ± 0.006 (t^oirs) and density p = 1.99 ± 0.14 (± 0.46) g cm"3.

The errors shown within parenthèses on e and p accounts for the current uncertainty on Quaoar's mass M, and can be improved when M is better known.

The position angle of Quaoar's north pôle projected in the plane of the sky is P = -31.8

or 148.4 ± 0.9 deg for our preferred solution (Fig. 9), while Weywot's orbit pôle has P in the range 5-15 deg (Vachier et al. 2012;Fraser et al. 2013). Thus, our preferred solution implies that Weywot is not orbiting into Quaoar's équatorial plane. This is not surprising, considering that Weyw'ot has probably preserved its primordial orbital inclination against tidal damping (Fraser et al. 2013).

The density that we obtain is typical of TNOs, and significantly smaller than the range 2.7 -5.0 g cm"3 given by Fraser et al. (2013). This disagreement fundamentally stems from the fact that our Quaoar's occultât ion-derived diameter is significantly larger than previous indirect size estimâtes. In fact, recent measurements using Herschel far IR data combined with Spitzer provide a preferred équivalent radius of 541 ± 34 km (Fornasier et al. 2013), which agréés with our value at the 0.4a level. The Herschel data also provides a preferred The best elliptical fit to the occultation chords, using the timings of Table 2.

The elliptical limb model does not satisfactorily fit the observed data. The sites that detected the event are indicated as: UEPG: Ponta Grossa; UFSC: Florianôpolis; SPA: San Pedro de Atacama; ARM: Armazones;

RIV: Rivera. 

(

  anexo G, (Ortiz et al. 2012)) e 2003 AZ84 (anexo E, (Braga-Ribas et al. (2011) e Braga-Ribas et al. (2012)). No capftulo 5, um sumârio de todos os resultados jâ obtidos a partir de ocultaçôes estelares por TNOs é apresentado.

Figura 2 . 1 )Figura 2 . 1

 2121 Figura 2.1), pode-se determinar a distancia geocêntrica aparente no piano do céu (a -b) entre a estrela e o objeto. A distancia rmnima entre os dois objetos (C/A), definida por d, pode ser calculada com a equaçâo 2.1. O instante de maior aproximaçâo Lq é definido pela equaçâo 2.2, onde assume-se que t2>ti. A velocidade v relativa entre o objeto e a estrela pode ser calculada com a equaçâo 2.3, onde velocidades positivas significam movimento aparente prôgrado (ascensâo reta crescente). Ainda podeinos définir o ângulo de posiçâo (PA) como sendo o ângulo entre o norte (zéro grau) e a posiçâo de maior aproximaçâo C/A, contado no sentido horârio. Para saber as condiçôes de observabilidade (dia/noite)

Figura 2 . 2

 22 Figura 2.2 Mapa de prediçâo de ocultaçâo: As barras paralelas que cruzam a Terra delimitam o tamanho da sombra projetada pelo objeto ao se deslocar sobre o planeta.

Figura 2 . 4

 24 Figura 2.4 Dériva linear da posiçâo observada de Plutâo em relaçâo à efeméride DE413.

  normal de pedidos e usâ-lo quando uni evento é confirmado. Assiin o fizemos para os telescôpios VLT 8 m (Very Large Telescope) e NTT 3.5 m (New Technology Telescope) do ESO, recebendo 3 horas/semestre em cada instrumento para os semestres 2009a, 2009b, 2010a, 2011a, 2012b. O mesmo vem sendo feito desde 2011 com telescôpios de Calar Alto 1,2m, 0,77 em La Hita, 1,5 m Sierra Nevada, 3,6 m TNG (Telescopio Nazionale Galileo) em La Palma Ilhas Canârias. No caso de nâo termos tempo assegurado em telescôpios profissionais em localizaçâo favorâvel, fazemos pedido de tempo de diretor (DDT, Director's Discretionary Time).

Figura 3 . 7

 37 Figura 3.7 Determinaçâo do instante de ocultaçâo. Dados da ocultaçâo por Makemake em 24 de abril de 2011(Ortiz et al. 2012). O modelo geométrico (linha amarela) é convolvido à difraçâo de Fresnel, ao tamanho da estrela, à banda espectral da observaçâo (linha azul) e ao tempo de integraçâo (segmentos pretos). A curva sintética résultante (linha e circulos vermelhos) é comparada ao dados (pontos pretos) calculando-se o Xmin-

Figura 4 . 2

 42 Figura 4.2 Dois instantes das imagens obtidas pelo Palomar Observatory 1.2 m Oschin

( 2010 )

 2010 mediram um diâmetro de 890 ± 70 km. Do estudo espectroscôpico sabe-se que Quaoar possui gelos de âgua na sua fase amorfa e cristalizada, gelo de metano, e possivelmente gelo de etano e nitrogênio molecular ( Jewitt & Luu (2004), Schaller & Brown (2007b), Guilbert et al. (2009) e Dalle Ore et al. (2009)).

Figura 4 . 5

 45 Figura 4.5 Curvas de luz que detectaram a ocultaçâo por Quaoar em 04 de maio de 2011.Os pontos quadrados correspondem ao fluxo medido em cada imagem, os circulos e linhas vermelhas à curva de luz sintética apôs a convoluçâo, e as linhas prêtas continuas ao modelo geométrico. Todas as curvas foram normalizadas à 1 na parte fora de evento. No grâfîco acima elas foram dcslocadas de quantidades intciras para melhor visualizaçâo, com a exceçâo de S.P. Atacama -50cm.

Figura 4 .

 4 Figura 4.12 À esquerda um esferoide Maclaurin, definido por a = b > c. À direita um elipsoide Jacobi, definido quando a > b > c.

Figura 4 .

 4 Figura 4.13 (a) O parâmetro adimensional fl, associado à velocidade angular, em relaçâo à razâo entre os eixos (c/a) e (b/a), (b) Limites de transiçâo entre Maclaurin e Jacobi e disrupçâo, para densidades p = 1, 2 e 5 g/cm3. Os quadrados pretos correspondem ao periodo de rotaçâo de unico pico em relaçâo à amplitude da variaçào de magnitude de alguns TNOs (Tancredi h Favre 2008).

  F)(a) = $(a) • p = $(a) (uakm//îequiv)2 • 10°-4(H°-Ho) Usando HQy = -26, 74 e os valores obtidos nas soluçôes acima calcula-se os possfveis valores para o albedo de Quaoar apresentados na tabela 4.5. Se considerarmos toda a

  Até hoje, sô se conhecem dois pequenos corpos do Sistema Solar externo que exibem atmosferas sustentadas pela gravidade do corpo, que apresentam estrutura térmica e evoluçâo: Tritâo e Plutâo. Tritâo é provavelmente um TNO que foi capturado por Netuno durante a fase de migraçâo planetâria (McKinnon & Leith (1995) e Nogueira et al. (2011)).

Figura 4 .

 4 Figura 4.21 Retençâo de volâteis pelos TNOs ao longo da idade do Sistema Solar. Objetos à esquerda das curvas de N2, CO e CH4 sâo muito pequenos e/ou quentes para terem retido volâteis, os à direita os mantiveram. Todos os objetos marcados por pontos roxos tiveram CH4 detectado em suas superficies(Brown et al. (2011), atualizadode Schaller & Brown (2007)). Quaoar encontra-se numa zona de transiçào.

Figura 4 .

 4 Figura 4.22 Transiçâo entre escape Jeans ou Hidrodinâmico para uma suposta atmosfera de metano. As curvas correspondem à temperatura mâxima que o corpo poderia se submeter ao mecanismo hidrodinâmico. Para os objetos à esquerda, o metano é perdido pelo mecanismo hidrodinâmico; para os à direita, apenas o escape Jeans atua. Quaoar encontra-se numa zona de transiçâo. (Levi & Podolak 2009).

  ua)-1367 W m~2 c o fluxo solar intcgrado cm todos os comprimentos de onda a 1 ua, e rrSB= 5,670xl0~8 W m"2 K-4 é a constante de Stefan-Boltzmann. No momento daocultaçâo de 2011, Quaoar se encontrava ar = 42,35 ua, entào T0 = 60,5 K. A temperatura de Quaoar no ponto sub-solar Tsubo dépende do albedo (pv = 0,120 ±0,018)

  distancia de Quaoar, o fluxo solar é quase duas vezes menor que em Plutâo, a 31 ua. Para Plutâo, o modelo prevê que a atmosfera superior atinja 106 K à partir de 10 km acima Tabela 4.6 Temperaturas de equilibrio de um corpo sem atmosfera: primeira linha: T0 temperatura no ponto sub-solar de uma superficie lambertiana com emissividade 1; segunda linha: TsubG) temperatura no ponto sub-solar de um dado objeto; terceira linha: Tss objeto com rotaçâo lenta, Tequ objeto com rotaçâo râpida; quarta linha:

  7). Podese entâo verihcar quai séria a pressâo superhcial da atmosfera (teôrica) para diferentes elementos na superficie de Quaoar à T5np = 42 K: N2 = 176 fibar\ CO = 31 fibar; CH4 = 32 nanobar.

Figura 4 .

 4 Figura 4.25 Temperatura versus pressâo de volatilizaçâo e saturaçâo por diferentes modelos para CO, CH4 e N2. Para 0 CO a fase alfa esta compreendida de 14 a 61,55 K e a fase beta de 61,55 a 61,8 K. Para 0N2 a fase alfa esta entre 10 a 35,61 K e a beta de 35,61 a 63,14 K (Fray & Schmitt 2009).

Figura 4 .

 4 Figura 4.27 Curvas de x2 para diferentes distancias do limbo ao centra do corpo (de 550 a 560 km). Atmosfera de metano como descrita acima (Fig.4.24). Um limite superior de pressâo superficial de 21 nanobar (1er), ou 56 nanobar (3a) é obtido. centra de Quaoar. Os cfrculos azuis correspondem à curva sintética convolvida ao tempo de exposiçâo de cada ponto observado.Ajustando apenas os pontos correspondentes à imersâo, tem-se uma detecçâo marginal de uma atmosfera de metano com pressâo de 138^2 nbar (1<t), mas a 3a tem-se apenas um limite superior a 221 nbar (Figura 4.29(a)). Para o ajuste da emersâo isoladamente, obtém-se apenas um limite superior a 145 nbar (Figura 4.29(b)). No entanto pressôes como as obtidas deveriam estar globalmente distribufdas.

Figura 4 .Figura 4 .

 44 Figura 4.28 Curva de luz sintética de uma atmosfera de metano a 21 nbar de pressâo superficial (linhas continuas e prêtas). Pontos verdes c vermelhos correspondem aos dados da imersâo e emersâo respectivamente, os segmentos horizontais representam o tempo de exposiçâo. Circulos azuis sâo obtidos apôs a convoluçâo da curva sintética com o tempo de integraçâo.

  o seguinte, devido à dériva do telescôpio. Foram tomadas 3 imagens com 3 minutos de exposiçâo cada e a ocultaçâo foi detectada em uma delas, como pode-se ver na figura 4.32.Durante a imagem onde detecta-se o infeio do evento obtida em Tourrette-Levens, ocorreu um problema de acompanhamento que fez com que as estrelas ficassem deformadas. Através de processamento da imagem, pode-se determinar que a ocultaçâo ocorreu durante a exposiçâo, pois ainda foi possfvel medir um fluxo parcial da estrela ocultada.Como Tourrette-Levens, TAROT e Valensole situam-se sobre a mesma corda (dentro de 3.1 km), corrigindo da paralaxe de suas posiçôes geogrâficas, espera-se que os instantes de infeio e fim de Tourrette-Levens seguissem os de TAROT e Valensole de 1,25 e 3,85 s, respectivamente. No entanto, o infeio do evento em Tourrette-Levens aconteceu 9,5 ± 3 e 8,5 ± 3 s depois de TAROT e Valensole, ou ainda 8,5±3 ou 4,5±3 s, respectiva mente, depois do instante esperado. O final do evento foi medido, 2,25 ± 2,5 e 1,1 ± 2.2 s antes de TAROT e Valensole ou 3,5 ± 2,5 e 2,75 ± 2,75 s respectivamente, antes do esperado. Portanto, hâ, claramente, um problema de dataçâo nas imagens obtidas em Tourrette-Levens.Para o ajuste da ocultaçâo utilizou-se a corda de Gnosca, e, para a corda mais ao sul, os instantes medidos em Valensole, por apresentar maior precisâo e nenhum problema de dataçâo(Tabela 4.8).

Figura 4 .

 4 Figura 4.32 Imagem traço obtida pelo telescôpio TAROT durante a ocultaçâo. As estrelas se deslocam pelo campo do CCD e a estrela ocultada (no centro da imagem) apresenta uma queda de brilho durante o evento.

Figura 4 .

 4 Figura 4.35 Mapa mostrando uma das soluçôes possiveis para posiçào da faixa de ocultaçâo por Quaoar em 15 de outubro de 2012. O ponto verde corresponde à localizaçâo geogrâfica de Cerro Tololo, onde se situam os telescôpios PROMPT.

Figura 5 . 6

 56 Figura 5.6 Mapa mostrando a disposiçâo dos observatôrios ao longo da faixa de ocultaçâo.

1 Introduction

 1 ) the world-wide campaigns with tens of professional and amateurs observatories, (3) the observation, analysis, and interprétation of the observed occultation by Quaoar and other large transneptunian objects.Since 2009, fourteen occultations by TNOs were already observed, twelve by the Paris/Rio consortium.In addition to the detailed discussion about the Quaoar events, a compilation of ail the other results comprising as size, shape, albedo, density and atmos phère, obtained from TNO stellar occultations is presented. In particular occultations by Eris, Makemake, 2003 AZ84, Varuna and Sedna, ail of which are new.ChapitreLa compréhension du système solaire passe, entre autres, par l'étude des petits corps, comme par exemple les objets trans-neptuniens (OTN). Ce sont des reliques du système solaire primorial qui orbitent au-delà de Neptune. Après Pluton, découvert en 1930, il a fallu attendre 1992 pour découvrir d'autres corps dans cette région(Jewitt & Luu 1993).Leur existence avait été proposée théoriquement (par exemple par Gérard Kuiper), pour expliquer les comètes à courte période. Depuis lors, plus de 1260 OTNs ont été découverts. Ils ont été divisés en plusieurs groupes, tels que la ceinture classique, les objets résonants et les objets dispersés (Gladman et al. 2008). condes (mas), voir Assafin et al. (2010) & Assafin et al. (2012). Ce travail a permis des prédictions fiables des occultations stellaires par les OTNs. Des centaines d'événements sont prévus chaque année, et comme il est impossible de les observer tous, une sélection est effectuée selon des critères prenant en compte l'intérêt scientifique du corps, le flux de l'étoile occultée, la localisation de l'ombre sur Terre et la présence de réseaux d'observateurs. Quelques jours avant l'événement, lorsque l'OTN et l'étoile peuvent être observés sur la même image, les mesures astrométriques deviennent plus précises car on évite alors les erreurs systématiques de catalogues. Des précisions de l'ordre de 20 massont alors atteintes et l'endroit où l'ombre de l'occultation passera est bien définie. Des campagnes internationales sont alors organisées, impliquant des dizaines d'astro nomes professionnels et amateurs. Une page internet pour chaque événement est publiée avec les informations nécessaires1.

  1. http://www.lesia.obspm.fr/perso/braga-ribas/campaigns La thèse, outre la méthodologie de prédiction indiquée ci-dessus, décrit en détail l'en semble de la procédure de réduction des données, avec une description plus détaillée des occultations par Quaoar de 2011 et 2012. L'occultation du 4 mai 2011 pu être détectée de 5 sites différents. Chaque site fournit une courbe de lumière qui est modélisée par le masquage de l'étoile par un corps opaque et sans atmosphère, tenant compte de la dif fraction, et la convolution par la taille de l'étoile et le temps d'intégration du système.Les instants de début et de fin de l'événement fournissent alors ce que nous appelons une corde d'occultation. Les cordes sont alors utilisées pour ajuster une ellipse qui représente limbe du corps, et fournit donc des contraintes sur sa taille, sa forme, et l'orientation du corps. Si les temps d'occultation fournis par l'occultation du 4 mai 2011 sont tous correctes, alors Quaoar devrait avoir avoir une irrégulière. Nous examinons deux solutions : la pre mière se caractérise par un crateère qui serait trop profond. L'autre solution suppose la présence d'une montagne, physiquement peu probable car trop haute, compte tenu de la taille, la masse et la composition de Quaoar. Nous explorons ensuite une troisième solution, dans laquelle on envisage la possibilité de problèmes dans l'enregistrement du temps absolu pour certains des 5 sites. En raison de sa taille, sa rotation et sa composition de surface, on s'attend à ce que Quaoar soit en équilibre hydrostatique, et prenne la forme d'un objet de type Maclaurin. Pour obtenir la forme limbe attendue (une ellipse), les cordes d'occultation sont décalées en temps, c'est-à-dire parallèlement à leur propre direction. Cette procédure conservant l'aire de l'ellipse, nous obtenons une valeur bien contrainte du rayon équivalent de Quaoar (rayon du cercle ayant la même surface que le limbe apparent) : 'RequiV= 555 ± 2.5 km et de son albédo géométrique visuel : py = 0.109±0.007. Par ailleurs chaque solution de limbe fournit le volume de Quaoar et son aplatissement réel, en supposant une valeur donnée de C, l'angle d'aspect polaire, c'est-à-dire l'angle entre l'axe polaire de Quaoar et la ligne de visée. Comme la masse de Quaoar est connue via le mouvement de son satellite Weywot, le volume fourni in fine la densité. Par ailleurs, la relation densité/aplatissement étant connue pour objet de Maclaurin, nous obtenons une solution unique rayon équatorial-densitéaplatissement-orientation dans le plan du ciel, une fois Ç fixé. Comme £ est indéterminé entre 0 et 90°, nous obtenons finalement une densité 1.99=L0.46 g cm-3 et un aplatissement réel de 0.0897=L0.006. De plus, l'orientation de notre solution dans le plan du ciel montre que l'orbite de Weywot ne peut pas être équatorial.

Figure 2 . 2

 22 Figure 2.2 Suite de la page précédente.

  bodies also hâve smaller relative size errors. The accumulation of positive and négative détections also resuit in signifîcant improvements in the TNO ephemeris, thus refining the orbit and substantially increasing the accuracy of the subséquent oc cultation prédictions. Improving the chances of success minimizes the money and time spent in the complex international campaigns, which usually involves large télescopes in wellequipped observatories, smaller mobile instruments, and specialized personnel.Even as one of the most important and larger members of the TNO population, Pluto has only been probed in terms of oc cultations, prior to 2000, almost exclusively by the 1988 mutual events with its larger satellite Charon. The first observed stel lar occultation by Pluto took place only in 1985, when its at mosphère was first discovered(Brosch 1995). The atmosphère could only be more extensively observed in the next occulta tion in 1988(Millis et al. 1993). In 2002, a new stellar occul tation by Pluto was recorded, but with higher time resolution(Sicardy et al. 2003; Elliot et al. 2003), revealing that there had been a dramatic expansion in the atmosphère pressure. Another high signal-to-noise ratio (S/N) light curve was obtained in the June 2006 stellar occultation (Young et al. 2008), which uncovered evidence of the stabilization of the atmosphère pressure in Pluto for the period 2002-2008. This resuit is supported by seven other unpublished occultations observed by our group members in Australia, Brazil, Namibia, and Chile with 0.5 m to 4 m size télescopes in 2006-2010(Sicardy et al., in prep.). Regarding Charon, after its first recorded stellar occultation(Walker 1980), four others hâve so far been observed, one in 11 July 2005(Sicardy et al. 2006; Gulbis et al. 2006), one in 22 June 2008(Sicardy et al. 201 la), and two other unpublished multi-chord events observed in Brazil and Chile(4 June 2011) and in Hawaii(23 June 2011). These last two events, as in 22 June 2008, were double events in which Charon and Pluto occulted the same star.In the 4 June 2011 event in particular, both occultations could be recorded at each site.The population of TNOs in general, hâve been far more poorly studied. Up to the time of writing, only seven stellar oc cultations hâve been successfully recorded, ail with CCD detectors. A double-chord occultation was observed in Hawaii in October 2009 and involved 2002 TX300 and a R = 13.4

  that a stellar occultation by a TNO other than Pluto or Charon was recorded. In February 2010, Varuna was observed to occult a V = 11.1 star from the north-east of Brazil. One chord was obtained. A solution for the size, albedo, and shape was derived after combining the positive with a near négative chord and astrometric and photometric ground-based data(Ortiz et al., in prep.). Our group subsequently observed a multi-chord oc cultation of a R = 16.9 star by Eris in November 2010(Maury et al. 2010; Sicardy et al. 201 lb), a one-chord occultation of a R = 18.2 star by 2003 AZ84 in January 2011, a multi-chord occultation of a R = 18.4 star by Makemake in April 2011(Ortiz et al., in prep.) and a multichord occultation of a R = 15.7 star by Quaoar in May 2011(Braga-Ribas et al., in prep.). The MIT group also reported at the 218th meeting of the AAS in 2011 the observation of a onechord occultation by Quaoar in February of 2011(Person et al. 2011 ;Sallum et al. 2011). Ail seven occultations were predicted in this work. Our group observed them ail (except 2002 TX300 and the February occultation by Quaoar) with 40 cm to 60 cm size télescopes and also with the NTT and VLT in Chile, Brazil, and Uruguay.The main difficulty in deriving reliable prédictions for the stellar occultation of TNOs is their small apparent diameters, even in the case of larger TNOs. Another cause is the lack of accurate orbital éléments, resulting in ephemeris errors as large as a few hundreds of milli-arcseconds (mas). Prédictions based solely on published catalog positions such as the USNO B 1.0(Monet et al. 2003) or the 2MASS(Cutri et al. 2003) usually fail because of the poor précision (or lack) of proper motions and the relatively large zonal and/or random errors in their positions (100-200 mas). Even individual positions for fainter stars in the UCAC2 catalog(Zacharias et al. 2004) may need corrections as large as 70 mas.There are ways to overcome these ephemeris and star posi tion problems. The ephemeris of TNOs must be offset so as to get realistic TNO positions at predicted occultation dates. The straightforward way to do this is to regularly observe TNOs and dérivé astrometrically the ephemeris offsets as a function of time. As successive positive occultations are collected, their ephemerides can be radically improved down to a few mas. Even a one-chord positive occultation helps to improve the ephemeris, since the apparent diameters of TNOs are smaller than 30 mas.In the case of star positions, one strategy is to select possible occultations based on the positions given in any arbitrary as trometric catalog, then perform follow-up observations to im prove the star position and (after applying offsets to the TNO ephemeris) delineate the shadow path.However, this approach to updating star positions has an intrinsic disadvantage in the case of TNOs. One usually wishes to start searching for occultations from an astrometrically sound catalog in order to select only those truly probable events.The shorter the list of candidates, the greater the chance of conducting the follow-up observational programme on dedicated télescopes in due course. Since the highest quiality astrometric catalogs, such as the UCAC2 for example, are more complété for the brighter magnitudes, then stars fainter than say R = 15-16 will always be left out of the candidate search. This is far from idéal in the case of TNOs. In contrast to Pluto occultations, for TNOs we obtain good contrast in the light curves using very modest instruments for stars as faint as R = 18.0, as verified in the recent successfully recorded TNO occultations reported here. This is due to the TNO magnitudes being much fainter than the star brightness and because of the recent developments in CCD detectors. Choosing the USNO B 1.0 or other similar catalogs does not help. The simple addition of fainter stars that do not hâve the minimum required catalog position précision introduces the problem of enlarging the follow-up list with low quality targets. This wastes valuable telescope time in follow-up programs and may lead to the running out of nights to complété the scheduled objects. A more suitable strategy for TNOs is to dérivé local astro metric star catalogs with sufïicient position précision, of say 50 mas at least, for the time span of the occultations, for stars in the magnitude range R = 13-19. In this way, we match the required position précision in the search, preserving faint stars without discarding bright objects. The addition of astrometrically trusted faint stars to the follow-up list, avoiding poor qual ity targets in tum improves the chances of finding more suitable candidates for TNO occultations, owing to the increase in the star density in the sky path. The first noteworthy consistent studies of Pluto stellar oc cultation prédictions were those of Mink & Klemola (1985) for the period 1985-1990 and of McDonald & Elliot (2000a,b) for the years 1999-2009. Two common drawbacks of these works were the astrometric précision of only about 072 and the lack of stellar proper motions leading to uncertainties of the order of the Earth radius in the predicted shadow paths. In addition, these earlier prédictions were degraded by the poorer précision of the older Pluto ephemerides. To overcome the problems faced by these first prédic tion efforts, we used more modem facilities, namely the most up-to-date CCD detectors, reference catalogs, and astrometric procedures, and carried out an observational program at the ES02p2/WFI instrument. We derived local star catalogs with précisé positions for determining accurate prédictions of stellar occultations by Pluto and its satellites Charon, Nix, and Hydra for the period 2008-2015. This work was published in Assafin et al. (2010) and is hereafter referred to as PI. Here, we extend that work to the predict the stellar occultations of the large TNOs Eris, Haumea, Ixion, Makemake, Orcus, Quaoar, Sedna, Varuna, 2002 TX300, and 2003 AZg4 for 2008-2015, using the same observational and astrometric procedures. The local astro metric catalogs obtained are complété down to about 7? = 19, hâve positional errors of 40 mas or smaller, hâve stars with computed proper motions, and are in the UCAC2 frame. The catalogs cover the TNO sky paths with a 30' width in déclination. They can be very useful for the astrometry of small CCD fields for the follow-up of selected candidate stars, and for refining the ephemeris offsets of TNOs for tracking the shadow path across the Earth. The catalogs are also suitable for deriving the photometric properties of calibration and occultation stars. In Sect. 2, we describe the ES02p2/WFI program and ob servations. The astrometric treatment is given in Sect. 3. In Sect. 4, we présent the derived catalogs of star positions along the TNO sky paths. In Sect. 5, we describe the détermination of ephemeris offsets and the candidate star search procedure. The prédictions of the stellar occultations by the large TNOs are finally presented in Sect. 6. An overview is presented in Sect. 7, including discussions about the extemal comparison with actual occultation results and error budget estimâtes, and comments about astrometric follow-up programs for TNO ephemeris imthe ES02p2/WFI program were selected according to their body size, which is usually related to their apparent sizes on the sky. Given the astrometric uncertainties in the star posi tions and the error in the TNO ephemeris, larger apparent diameters correspond to a higher probability of a successful prédic tion. Another sélection criteria was the star field. We preferred to select TNOs currently orbiting dense star régions close to the Galactic plane at Galactic longitudes of interest. The TNOs of intrinsically high science value were also selected, even if the star density along their sky path was low. In these cases, the advantage of using the large FOV of the WFI was even clearer. Observations were made at the ESO 2p2 telescope (IAU code 809) using the Wide Field Imager (WFI) CCD mosaic detector. We used the broad-band R filter ESO#844 with /îc = 651.725 nm and Ad = 162.184 nm (full width at half maximum). The exposure time was 30s. In general, S/N of about 200 were reached for objects of R = 17 without saturating bright (R = 13-15) stars. The limiting magnitude was about R = 21, and the completeness level about R = 19.0. The seeing varied between 076 and 175, being typically 1". The telescope was shifted between exposures in such a way that each star in the path of the TNO was observed at least twice in different CCDs. The program was completed in seven runs, in April, June, September, and October 2007, May/June 2008, and February and December 2009. Observations spanned each TNO sky path from 2008 to 2015; in six cases, this path started in 2009, since their WFI observations only took place in 2009 and tele scope time was limited.

Sect. 4 .

 4 1 ofPl).

Fig. 1 .Fig. 2 .

 12 Fig. 1. Sky région covered by the ES02p2/WFI CCD mosaic obser vations for Varuna. Years 2009-2015 follow from bottom to top. The continuous line is the geocentric sky path of Varuna. Each dashed form represents the 30' x 30' area covered by one single WFI mosaic.

  only to field stars and mean that only one entry was selected according to one of three criteria, in order of priority: a) highest number of used common individual CCD positions (flag f3); b) least (.v, y) measurcment error (flag f4); c) brightest R magnitude (flag f5).

4 .

 4 next in the text) and many faint stars are présent in this région of the sky. A complété description of the astrometric and photometric procedures used here can be found in Sect. 4 of PI, where data for the same WFI CCD mosaic underwent rigorously the same treatment. The catalogs of star positions of each TNO for 2008-2015 sky paths For each TNO, we produced star catalogs following their respective sky paths up to 2015. The paths begin in 2008 for Eris, Orcus, Quaoar, and 2002 TX300, and in 2009 for the other TNOs. The star catalogs consist of the mean (a, J) positions in the ICRS (J2000), proper motions, R magnitudes (also J, H, and K in the case of 2MASS stars), mean epoch of observations, posi tion error at the mean epoch of observation, and the magnitude

  Notes. Number of catalog stars per TNO, position error for faint stars with R at completeness magnitude (errors are estimated from the stan dard déviation of contributing individual CCD positions about the final catalog star positions), R bandpass magnitude completeness, and mag nitude limit. error estimâtes. In ail, 5 356 255 stars were catalogued in the UCAC2 frame. The mean epoch is approximately 2007.75 for the catalogs of TNOs beginning in 2008, and about 2009.5 for the others. Magnitude completeness is about/? = 19. The magni tude limit is about R = 21. Position error is smaller than 40 mas for stars up to magnitude R = 19, and 30 mas up to R = 17. The catalogs are available in electronic form at the CDS. The catalogs are divided in terms of year. Stars that had mul tiple entries within 175 in the global mosaic solutions are flagged (see Table 4). The R magnitudes from PSF photometry were calibrated in the UCAC2 System, thus magnitude zéro-point errors up to 0.3 are expected for R > 17. The position error is esti mated from repeated observations, by evaluating the standard déviation (mean error) in the contributing individual CCD po sitions about the final catalog star positions (last step of global mosaic solution -see Sect. 3). By default, multiple-entry flagged stars hâve no position error estimâtes. Infrared magnitudes (and errors) were extracted from the 2MASS catalog. Error estimâtes for R magnitudes corne from the standard déviation about the mean for individual CCD frames. The sky coverages of the cat alogs hâve widths of at least 30' in déclination and lenghts of a few degrees along the geocentric sky path of each TNO. The proportion of UCAC2, 2MASS, and USNO B 1.0 stars in the WF1 catalogs may be verified from Table

Fig. 4 .

 4 Fig. 4. Position mean errors as a function of R magnitude for ail catalog stars. Position errors arc estimated from the standard déviation in the contributing individual CCD positions about the final catalog star po sitions (last step of global mosaic solution -see Sect. 3). Values were computed over 0.5 mag bins.

  7 of PL 6. Prédictions of stellar occultations by ten large TNOs Following the procedure described in Sect. 5, we identified po tential candidate stars for occultations by Eris, Haumea, Ixion, Makemake, Orcus, Quaoar, Sedna, Varuna, 2002 TX300, and 2003 AZg4. The adopted search radius was 07335 -i.e. 285 mas (approximately the apparent Earth radius at 31 AU, i.e., at Pluto for 2008-2015) plus a safe margin of 50 mas. No prédictions were discarded due to day light at the sub-planet point, as occul tations could, even so, be visible right above the horizon from places still in the dark near the Earth terminator. For each TNO, ail relevant astrometric, photometric, and géométrie informa tion for each potential event found is available in electronic form1. Table 8 lists a sample of prédictions for 2002 TX300, Varuna, Eris, 2003 AZ84, Makemake, and Quaoar. These very prédictions were used to préparé campaigns actually resulting in successful recordings of stellar occultations by our group. The exception was 2002 TX300, for which no observations could be done at Hawaii (the predicted path), because ail télescopes were occupied, must of them having been scheduled to observe 1 Anonymous ftp to cdsarc. u-strasbg. fr the impact of NASA's Lunar Crater Observation and Sensing Satellite (LCROSS; Garner & Dunbar 2009), which occurred about one hour after the predicted central instant of occultation. Prédiction tables contain the date and instant of stellar occul tation (UTC), both the ICRS (J2000) star and TNO estimated coordinates at the event date, the closest apparent geocentric distance between the star and TNO, the position angle of the shadow across the Earth (clockwise, zéro at North), the veloc ity in km s-1, the distance to the Earth (AU), longitude of the sub-solar point, local solar time, applied JPL ephemerides offsets in (RA, Dec) for the central instant, catalog proper motion and multiplicity flags, estimated star catalog position errors, proper motions, and magnitudes R*, J*, H*, and K*. Magnitudes were normalized to a reference shadow velocity of 20 km s

  prédictions for stellar occultations by the large TNOs Eris, Haumea, Ixion, Makemake, Orcus, Quaoar,Sedna, Varuna, 2002 TX300, and 2003 AZg4 for 2008/2009[START_REF] Assafin | Précisé prédictions of stellar occultations by Pluto, Charon, Nix, and Hydra for[END_REF]. We applied the same observational and astrometric pro cedures as a previous study dedicated to the Pluto System and published inAssafin et al. (2010). Many issues were addressed in that work, such as body ephemeris offset détermination, cat alog zéro-point position errors and field-of-view size, long-of predicted events per year for each TNO. prédictions and stellar proper motions, and faint-visual versus bright infrared stars, which also hold true here. The local astrometric catalogs that we obtain are complété to about R = 19 and in the UCAC2 frame. Up to R magnitude completeness brightness, stars hâve positional errors of 40 mas or smaller and computed proper motions. The catalogs cover the TNO sky paths with 30' width in déclination and can be very useful in astrometric follow-up programs on telescopes/CCDs with FOVs of any size. The catalogs are also sources of photometric information for calibration and occultation stars. The astrometry was made by the use of the PRAIA package (Assafin 2006). About 316 GB of science-alone WFI images were treated, not to mention calibration observations to dérivé field distorion patterns, which improved the astrometric results of WFI data to the 40 mas level (see Sect. 9 in PI for a detailed discussion about the field distortion pattern improvements for WFI astrometry). As an important by-product, the high-resolution CCD frames obtained (1 pixel = 07238) form an image bank useful for the Vi sual inspection of the fields of view. For instance, the analysis of small details in the calibration stars, close-by faint objects, and other éléments in the images may prevent problems in the recording and photometry of occultation observations. When it cornes to an event, charts extracted from this image bank are usually posted in our web page, but any particular field around some desired position can be obtained in FITS format on request to the authors. Public access to the original images from the observed WFI CCD mosaics will soon be available through the ES O Image Archive Portai.No threshold in R magnitude was established in the search for occultations, as faint R objects may turn out to be bright IR stars, suitable for observation at adequately equipped IR ground-based facilities or by the SOFIA observatory(Gehrz et al. 2009); H, J, and K magnitudes are readily available in the catalog if the star belongs to the 2MASS. In addition, events that occur at slow shadow speeds of less than 20 km s-1 may become observable. No constraints on a géographie place were applied, as in principle SOFIA observations can be done from any sub-solar point on Earth. Events in daylight at sub-planet point were not excluded either, as they remained observable in the dark, right above the horizon, from places near the Earth terminator. Throughout this paper, we do not distinguish between past and future prédictions, publishing ail found occultations for the sky path covered (or to be covered) by the TNOs over the years 2008/2009-2015. We assume that the WFI prédictions of past occultations may be useful as either reference for ongoing fittings of recently obtained light curves, an aid to deriving ephemeris offsets by comparing expected and observed central

  This limits the search for candidates to about R = 16 stars. The goal of our WFI program was to extend the UCAC2 position précision to stars as faint as about R = 19. The availability of data for such faint stars with about 40 mas position précision improves the chances of finding more suitable candidates for TNO occultations, owing to the increase in the number of suit able stars in the sky path and, consequently, the frequency of appulses. Owing to the recent developments of solid State dé tectons, and because TNOs are usually much fainter than the occulted stars, we can obtain good contrast in the light curves, even using very modest instruments. This has been verified for R = 17-18 stars in the occultations by Eris and 2003 AZg4 (Maury et al. 2010; Sicardy et al. 2011b; Braga-Ribas et al. 2011a and in prep.), observed with 40 cm and 60 cm télescopes in Chile. The sky paths covered by the star catalogs were determined by the chosen time span of our prédictions (2008-2015) in a compromise between the availability of WFI telescope time and the number of selected TNOs. The upper limit coincides with NASA's New Horizons spacecraft arrivai at Pluto in 2015. Given this long time coverage of about eight years, the compu tation of proper motions for the stars was also of concem. That is why we used as first epoch data the best available astrometric resources at the time of completion of this work. In ail, the com parison made with actual events (Table 11 ) and the successful WFI prédictions specifically for the faint stars of Eris, 2003 AZs4 and Makemake occultations indicate that the WFI astrometric solution is sactisfatory in the context of long-term prédictions. Owing to the lack of observations of TNOs for ephemerides, to the few positions sometimes derived without the appropriate astrometric care, and to the relatively short orbital arc length coverage, the actual position of TNOs can differ significantly from their ephemerides. To refine the orbits of large TNOs in the short and long runs, by regularly obtaining précisé astrometric positions, a parallel program involving many télescopes of 60 cm to 2 m class in Chile, France, and Brazil was started in 2007 (see further details in Sect. 5). Using the first results of this program for the prédictions published in this work, we applied average fixed offsets to the current JPL ephemerides of the TNOs in the list (Table

  objects (TNOs) are located beyond Neptune in a région where only small body différentiation is expected with regard to température changes. This makes them 4.5 billion year old interplanetary fossils of the early stages of the formation of the outer solar System. Recent models of planetary migration such as the Nice model (Levisoti & Morbidelli 2003; Gomes et al. 2004, 2005) also indicate that TNOs are a sensitive laboratory for the study of orbital dynamics. Refining the ephemeris and establishing the fundamental properties of TNOs are thus essential for our understanding of the origin and évolution of the solar System. This in turn contributes to the understanding of exoplanetary Systems and helps astrochemical and astrobiological studies (Barucci 2008). ! Article published by EDP Sciences A142, page 1 of 10 observed from many sites allows the shapes and sizes of TNOs to be measured with a précision of a few kilometers. We can probe their atmosphères or place limits on their existence down to the nanobar level. In addition, occultations can eventually lead to the observation or discovery of close binary companions, satellites, or débris around the central body. For size détermi nations, occultations are far more suitable than indirect estima

  high signal-to-noise ratio (S/N) light curve was obtained in the June 2006 stellar occultation(Young et al. 2008), which uncovered evidence of the stabilization of the atmosphère pressure in Pluto for the period 2002-2008. This resuit is supported by seven other unpublished occultations observed by our group members in Australia, Brazil, Namibia, and Chile with 0.5 m to 4 m size télescopes in 2006-2010(Sicardy et al., in prep.). Regarding Charon, after its first recorded stellar occultation(Walker 1980), four others hâve so far been observed, one in 11 July 2005(Sicardy et al. 2006; Gulbis et al. 2006), one in 22 June 2008(Sicardy et al. 201 la), and two other unpublished multi-chord events observed in Brazil and Chile (4 June 2011) and in Hawaii(23 June 2011). These last two events, as in 22 June 2008, were double events in which Charon and Pluto occulted the same star.In the 4 June 2011 event in particular, both occultations could be recorded at each site.The population of TNOs in general, hâve been far more poorly studied. Up to the time of writing, only seven stellar oc cultations hâve been successfully recorded, ail with CCD detectors. A double-chord occultation was observed in Hawaii inOctober 2009 and involved 2002 TX300 and a R = 13.4 

  that a stellar occultation by a TNO other than Pluto or Charon was recorded. In February 2010, Varuna was observed to occult a V = 11.1 star from the north-east of Brazil. One chord was obtained. A solution for the size, albedo, and shape was derived after combining the positive with a near négative chord and astrometric and photometric ground-based data(Ortiz et al., in prep.). Our group subsequently observed a multi-chord oc cultation of a R = 16.9 star by Eris in November 2010(Maury et al. 2010; Sicardy et al. 201 lb), a one-chord occultation of a R = 18.2 star by 2003 AZg4 in January 2011(Braga-Ribas et al. 2011a and in prep.), a multi-chord occultation of a R = 18.4 star by Makemake in April 2011(Ortiz et al., in prep.) and a multichord occultation of a R = 15.7 star by Quaoar in May 2011(Braga-Ribas et al., in prep.). The MIT group also reported at the 218th meeting of the AAS in 2011 the observation of a onechord occultation by Quaoar in February of 2011(Person et al. 2011 ;Sallum et al. 2011). Ail seven occultations were predicted in this work. Our group observed them ail (except 2002 TX300 and the February occultation by Quaoar) with 40 cm to 60 cm size télescopes and also with the NTT and VLT in Chile, Brazil, and Uruguay.The main difficulty in deriving reliable prédictions for the stellar occultation of TNOs is their small apparent diameters, even in the case of larger TNOs. Another cause is the lack of accurate orbital éléments, resulting in ephemeris errors as large as a few hundreds of milli-arcseconds (mas). Prédictions based solely on published catalog positions such as the USNO B 1.0(Monet et al. 2003) or the 2MASS(Cutri et al. 2003) usually fail because of the poor précision (or lack) of proper motions and the relatively large zonal and/or randorn errors in their positions (100-200 mas). Even individual positions for fainter stars in the UCAC2 catalog(Zacharias et al. 2004) may need corrections as large as 70 mas.There are ways to overcome these ephemeris and star posi tion problems. The ephemeris of TNOs must be offset so as to get realistic TNO positions at predicted occultation dates. The straightforward way to do this is to regularly observe TNOs and dérivé astrometrically the ephemeris offsets as a function of time. As successive positive occultations are collected, their ephemerides can be radically improved down to a few mas. Even a one-chord positive occultation helps to improve the ephemeris, since the apparent diameters of TNOs are smaller than 30 mas.In the case of star positions, one strategy is to select possible occultations based on the positions given in any arbitrary as trometric catalog, then perform follow-up observations to im prove the star position and (after applying offsets to the TNO ephemeris) delineate the shadow path.However, this approach to updating star positions has an intrinsic disadvantage in the case of TNOs. One usually wishes to start searching for occultations from an astrometrically sound catalog in order to select only those truly probable events.The shorter the list of candidates, the greater the chance of conducting the follow-up observational programme on dedicated télescopes in due course. Since the highest quiality astrometric catalogs, such as the UCAC2 for example, are more complété for the brighter magnitudes, then stars fainter than say R = 15-16 will always be left out of the candidate search. This is far from idéal in the case of TNOs. In contrast to Pluto occultations, for TNOs we obtain good contrast in the light curves using very modest instruments for stars as faint as R = 18.0, as verified in the recent successfully recorded TNO occultations reported here. This is due to the TNO magnitudes being much fainter than the star brightness and because of the recent developments in CCD detectors. Choosing the USNO B 1.0 or other similar catalogs does not help. The simple addition of fainter stars that do not hâve the minimum required catalog position précision introduces the problem of enlarging the follow-up list with low quality targets. This wastes valuable telescope time in follow-up programs and may lead to the running out of nights to complété the scheduled objects. A more suitable strategy for TNOs is to dérivé local astro metric star catalogs with sufficient position précision, of say 50 mas at least, for the time span of the occultations, for stars in the magnitude range R = 13-19. In this way, we match the required position précision in the search, preserving faint stars without discarding bright objects. The addition of astrometrically trusted faint stars to the follow-up list, avoiding poor qual ity targets in turn improves the chances of finding more suitable candidates for TNO occultations, owing to the increase in the star density in the sky path. The first noteworthy consistent studies of Pluto stellar oc cultation prédictions were those of Mink & Klemola (1985) for the period 1985-1990 and of McDonald & Elliot (2000a,b) for the years 1999-2009. Two common drawbacks of these works were the astrometric précision of only about 072 and the lack of stellar proper motions leading to uncertainties of the order of the Earth radius in the predicted shadow paths. In addition, these earlier prédictions were degraded by the poorer précision of the older Pluto ephemerides. To overcome the problems faced by these first prédic tion efforts, we used more modem facilities, namely the most up-to-date CCD detectors, reference catalogs, and astrometric procedures, and carried out an observational program at the ES02p2/WFT instrument. We derived local star catalogs with précisé positions for determining accurate prédictions of stellar occultations by Pluto and its satellites Charon, Nix, and Hydra for the period 2008-2015. This work was published in Assafin et al. (2010) and is hereafter referred to as PI. Here, we extend that work to the predict the stellar occultations of the large TNOs Eris, Haumea, Ixion, Makemake, Orcus, Quaoar, Sedna, Varuna, 2002 TX300, and 2003 AZr4 for 2008-2015, using the same observational and astrometric procedures. The local astro metric catalogs obtained are complété down to about R = 19, hâve positional errors of 40 mas or smaller, hâve stars with computed proper motions, and are in the UCAC2 frame. The catalogs cover the TNO sky paths with a 30' width in déclination. They can be very useful for the astrometry of small CCD fields for the follow-up of selected candidate stars, and for refining the ephemeris offsets of TNOs for tracking the shadow path across the Earth. The catalogs are also suitable for deriving the photometric properties of calibration and occultation stars. In Sect. 2, we describe the ES02p2/WFT program and ob servations. The astrometric treatment is given in Sect. 3. In Sect. 4, we présent the derived catalogs of star positions along the TNO sky paths. In Sect. 5, we describe the détermination of ephemeris offsets and the candidate star search procedure. The prédictions of the stellar occultations by the large TNOs are finally presented in Sect. 6. An overview is presented in Sect. 7, including discussions about the extemal comparison with actual occultation results and error budget estimâtes, and comments about astrometric follow-up programs for TNO ephemeris improvements.

(

  IAU code 809) using the Wide Field Imager (WFI) CCD mosaic detector. We used the broad-band R filter ESO#844 with Ac = 651.725 nm and A/l = 162.184 nm (full width at half maximum). The exposure time was 30s. In general, S/N of about 200 were reached for objects of R = 17 without saturating bright (R = 13-15) stars. The limiting magnitude was about R = 21, and the completeness level about R = 19.0. The seeing varied between 076 and 175, being typically 1". The telescope was shifted between exposures in such a way that each star in the path of the TNO was observed at least twice in different CCDs. The program was completed in seven runs, in April, June, September, and October 2007, May/June 2008, and February and December 2009. Observations spanned each TNO sky path from 2008 to 2015; in six cases, this path started in 2009, since their WFI observations only took place in 2009 and tele scope time was limited.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Sky région covered by the ES02p2/WFI CCD mosaic obser vations for Varuna. Years 2009-2015 follow from bottom to top. The continuous line is the geocentric sky path of Varuna. Each dashed form represents the 30' x 30' area covered by one single WFI mosaic.

  cases fl and f2 apply only to UCAC2 and 2MASS stars, when only en tries assigned to a catalog werc used. Flag fl means that more than one entry was used (more than one position entry was averaged), whereas flag f2 indicates that only one entry was used. Flags O, f4, and f5 apply only to field stars and mean that only one entry was selectcd according to one of three criteria, in order of priority: a) highest number of used common individual CCD positions (flag f3); b) least (.v, ij) measurement error (flag f4); c) brightest R magnitude (flag f5).

4 .

 4 next in the text) and many faint stars are présent in this région of the sky. A complété description of the astrometric and photometric procedures used here can be found in Sect. 4 of PI, where data for the same WFI CCD mosaic underwent rigorously the same treatment. The catalogs of star positions of each TNO for 2008-2015 sky paths For each TNO, we produced star catalogs following their respective sky paths up to 2015. The paths begin in 2008 for Eris, Orcus, Quaoar, and 2002 TX300, and in 2009 for the other TNOs. The star catalogs consist of the mean (a, ô) positions in the ICRS (J2000), proper motions, R magnitudes (also J, H, and K in the case of 2MASS stars), mean epoch of observations, posi tion error at the mean epoch of observation, and the magnitude A142, page 5 of10 A&A 541, A142 (2012)

  ibrated in the UCAC2 System, thus magnitude zéro-point errors up to 0.3 are expected for R > 17. The position error is esti mated from repeated observations, by evaluating the standard déviation (mean error) in the contributing individual CCD po sitions about the final catalog star positions (last step of global mosaic solution -see Sect. 3). By default, multiple-entry flagged stars hâve no position error estimâtes. Infrared magnitudes (and errors) were extracted from the 2MASS catalog. Error estimâtes for R magnitudes corne from the standard déviation about the mean for individual CCD frames. The sky coverages of the cat alogs hâve widths of at least 30' in déclination and lenghts of a few degrees along the geocentric sky path of each TNO. The proportion of UCAC2, 2MASS, and USNO B1.0 stars in the WFI catalogs may be verified from Table

Fig. 4 .

 4 Fig. 4. Position mean errors as a function of R magnitude for ail catalog stars. Position errors are estimated from the standard déviation in the contributing individual CCD positions about the final catalog star po sitions (last step of global mosaic solution -see Sect. 3). Values were computed over 0.5 mag bins.

6 .

 6 Prédictions of stellar occultations by ten largeTNOs Following the procedure described in Sect. 5, we identified po tential candidate stars for occultations by Eris, Haumea. Ixion,Makemake, Orcus, Quaoar, Sedna, Varuna, 2002 TX300, and 2003 AZg4. The adopted search radius was 07335 -i.e. 285 mas (approximately the apparent Earth radius at 31 AU, i.e., at Pluto for 2008-2015) plus a safe margin of 50 mas. No prédictions were discarded due to day light at the sub-planet point, as occul tations could, even so, be visible right above the horizon from places still in the dark near the Earth terminator. For each TNO, ail relevant astrometric, photometric, and géométrie informa tion for each potential event found is available in electronic form1. Table8 listsa sample of prédictions for 2002 TX300, Varuna, Eris, 2003 AZ84, Makemake, and Quaoar. These very prédictions were used to préparé campaigns actually resulting in successful recordings of stellar occultations by our group. The exception was 2002 TX300» for which no observations could be done at Hawaii (the predicted path), because ail télescopes were occupied, must of them having been scheduled to observe 1 Anonymous ftp to cdsarc.u-strasbg.fr the impact of NASA's Lunar Crater Observation and Sensing Satellite (LCROSS; Garner & Dunbar 2009), which occurred about one hour after the predicted central instant of occultation. Prédiction tables contain the date and instant of stellar occul tation (UTC), both the ICRS (J2000) star and TNO estimated coordinates at the event date, the elosest apparent geocentric distance between the star and TNO, the position angle of the shadow across the Earth (clockwise, zéro at North), the veloc ity in km s-1, the distance to the Earth (AU), longitude of the sub-solar point, local solar time, applied JPL ephemerides offsets in (RA, Dec) for the central instant, catalog proper motion and multiplicity flags, estimated star catalog position errors, proper motions, and magnitudes R*, J*, H*, and K*. Magnitudes were normalized to a reference shadow velocity of 20 km sof 20 km s-1 is typical of events around Pluto opposition. In slow events, M* may become one to two magnitudes brighter than the actual star magnitude, indicating that the observation is feasible after ail, because we can use longer intégration times, and consequently reach reasonably high S/N, without any losses of spatial resolution in diameter measurements. For the same reasoning, we may also probe the atmosphère altitudes in the light curves, despite the faintnesses of the stars. As mentioned in the introduction, in the case of TNOs, occultations involving stars as faint as R = 18.0 hâve been successfully recorded with 50 cm télescopes.

  prédictions for stellar occultations by the large TNOs Eris, Haumea, Ixion, Makemake, Orcus, Quaoar,Sedna, Varuna, 2002 TX300, and 2003 AZ84 for 2008/2009[START_REF] Assafin | Précisé prédictions of stellar occultations by Pluto, Charon, Nix, and Hydra for[END_REF]. We applied the same observational and astrometric pro cedures as a previous study dedicated to the Pluto System and published inAssafin et al. (2010). Many issues were addressed in that work, such as body ephemeris offset détermination, cat alog zéro-point position errors and field-of-view size, long-term of predictcd events per year for each TNO.prédictions and stellar proper motions, and faint-visual versus bright infrared stars, which also hold true here.The local astrometric catalogs that we obtain are complété to about R = 19 and in the UCAC2 frame. Up to R magnitude completeness brightness, stars hâve positional errors of 40 mas or smaller and computed proper motions. The catalogs cover the TNO sky paths with 30' width in déclination and can be very useful in astrometric follow-up programs on telescopes/CCDs with FOVs of any size. The catalogs are also sources of photometric information for calibration and occultation stars. The astrometry was made by the use of the PRAIA package(Assafin 2006). About 316 GB of science-alone WFI images were treated, not to mention calibration observations to dérivé field distorion patterns, which improved the astrometric results of WFI data to the 40 mas level (see Sect. 9 in PI for a detailed discussion about the field distortion pattern improvements for WFI astrometry).As an important by-product, the high-resolution CCD frames obtained ( 1 pixel = 07238) form an image bank useful for the Vi sual inspection of the fields of view. For instance, the analysis of small details in the calibration stars, close-by faint objects, and other éléments in the images may prevent problems in the recording and photometry of occultation observations. When it cornes to an event, charts extracted from this image bank are usually posted in our web page, but any particular field around some desired position can be obtained in FITS format on request to the authors. Public access to the original images from the observed WFI CCD mosaics will soon be available through the ESO Image Archive Portai.No threshold in R magnitude was established in the search for occultations, as faint R objects may turn out to be bright IR stars, suitable for observation at adequately equipped IR ground-based facilities or by the SOFIA observatory(Gehrz et al. 2009); H, J, and K magnitudes are readily available in the catalog if the star belongs to the 2MASS. In addition, events that occur at slow shadow speeds of less than 20 km s-1 may become observable. No constraints on a géographie place were applied, as in principle SOFIA observations can be done from any sub-solar point on Earth. Events in daylight at sub-planet point were not excluded either, as they remained observable in the dark, right above the horizon, from places near the Earth terminator. Throughout this paper, we do not distinguish between past and future prédictions, publishing ail found occultations for the sky path covered (or to be covered) by the TNOs over the years 2008/2009-2015. We assume that the WFI prédictions of past occultations may be useful as either référencé for ongoing fittings of recently obtained light curves, an aid to deriving ephemeris offsets by comparing expected and observed central Table 10. TNO diameters. diameters in km (uncertainties in parenthèses) and as geoccntric apparent sizes in mas for 2012.5. Values were obtained from previous stellar occultations and by modeling visible and IR observa tions (see référencés).

  This limits the search for candidates to about R = 16 stars. The goal of our WFI program was to extend the UCAC2 position précision to stars as faint as about R = 19. The availability of data for such faint stars with about 40 mas position précision improves the chances of finding more suitable candidates for TNO occultations, owing to the increase in the number of suit able stars in the sky path and, consequently, the frequency of appulses. Owing to the recent developments of solid State detectors, and because TNOs are usually much fainter than the occulted stars, we can obtain good contrast in the light curves, even using very modest instruments. This has been verified for R = 17-18 stars in the occultations by Eris and 2003 AZ84 (Maury et al. 2010; Sicardy et al. 2011b; Braga-Ribas et al. 2011a and in prep.), observed with 40 cm and 60 cm télescopes in Chile.

  like radius and a high albedo for the dwarf planet Eris from an occultation B. Sicardy1,2,3, J. L. Ortiz* 4, M. Assafin3, E. Jehin6, A. Maury7, E. Lellouch1, R. Gil Hutton8, F. Braga-Ribas1,9, F. Colas10, D. Hestroffer10, J. Lecacheux1, F. Roques1, P. Santos-Sanz1, T. Widemann1, N. Morales4, R. Duffard4, A. Thirouin4, A. J. Castro-Tirado4, M. Jelinek4, P. Kubânek4, A. Sota4, R. Sanchez-Ramirez4, A. H. Andrei5,9, J. I. B. Camargo5,9, D. N. da Silva Neto9,11, A. Ramos Gomes Jr\ R. Vieira Martins5,9,10, M. Gillon6, J. Manfroid6, G. P. Tozzi12, C. Harlingten13, S. Saravia7, R. Behrend14, S. Mottola13 *, E. Garcia Melendo16,17, V. Péris18, J. Fabregat18, J. M. Madiedo19, L. Cuesta20, M. T. Eibe20, A. Ullân20, F. Organero21, S. Pastor22, J. A. de los Reyes22, S. Pedraz23, A. Castro24,1. de la Cueva23, G. Muler26,1. A. Steele27, M. Cebriân28, P. Montanés-Rodriguez28, A. Oscoz28, D. Weaver29, C. Jacques30, W. J. B. Corradi31, F. P. Santos31, W. Reis31, A. Milone32, M. Emilio33, L. Gutiérrez34, R. Vazquez34 & H. Hernândez-Toledo33

Figure 1 |

 1 Figure 1 | Eris occultation light curves. The plots (black filled circles) show the flux of the star plus Eris, normalized to unity outside the occultation, versus time. No filter was used at any of the télescopes, a, The light curve from the ASH2 40-cm telescope at San Pedro, using a SBIG STL-11000M CCD caméra, with 2X2 pixel binning and a sub-frame of 11.24 X 9.71 arcmin (272 X 235 pixels). The horizontal error bars indicate the total time intervals associated with each point (15 s, while the cycle time was 18.32 s). Those bars are too small to be visible on the other data sets, b, The light curve from theHarlingten 50-cm telescope at San Pedro using an Apogee U42 CCD caméra (2X2 pixel binning; sub-frame, 2.67 X 2.67 arcmin, or 100 X 100 pixels; integration/cycle times, 3 and 3.88 s), c, The light curve from the 60-cm TRAPPIST telescope at La Silla, using a FLI ProLine PL3041-BB CCD caméra (2X2 pixel binning; sub-frame, 3.25 X 3.25 arcmin, or 150 X 150 pixels; integration/cycle times, 3 and 4.55 s), d, The light curve from the 215-cm Jorge Sahade telescope at CASLEO, using a Roper Scientific Versarray 1300B CCD caméra (3X3 pixel binning; sub-frame, 2.62 X 3.50 arcmin, or 77 X 103 pixels; integration/cycle times, 4 and 7 s). The horizontal dashed lines at the bottom of a and c represent Eris' contribution to the flux, showing that the star completely

NFigure 2 |

 2 Figure 2 | Measuring Eris' size. The three oblique solid lines show the star trajectories relative to Eris, as seen from San Pedro, La Silla and CASLEO, with the arrow pointing towards the direction of motion. The San Pedro and La Silla timings provide the lengths of the two occultation segments, or 'chords' (in red); see solution 2 in Supplementary Table3. The médian lines of the two red segments are separated by only 5 km and coincide at that scale with the blue line. Celestial north is up and east is left. Scale bars: 1,000 km and 14.40 mas ( 1 mas corresponds to 69.436 km at Eris). The solid cirde has a radius Re -1,163 km, and is our preferred solution for Eris' size and shape, with the cross marking the position of the centre. The dot near 'P' indicates Dysnomia's orbit pôle direction12 projected onto Eris' surface. The dotted curve is an elliptic limb fitted to our occultation chords, with semi-major and -minor axes of a' = 1,708 and b' = 1,317 km, respectively, that is, an apparent effective radius of Re = 1,500 km, the value of Eris' radius previously derived from thermal measurements11. The long axis of the ellipse should be perpendicular to the occultation chords to within ±2 °in order to match our data points. This has a low probability (2%) of occurring for a random limb orientation between 0 and 180°. Furthermore, the ellipse has an aspect ratio b/a = 0.771 that would require a fast rotator (with a period of 4.4 h) observed pole-on to within 18°to suppress the rotational light curve13,14. This has also a low probability (5%) of occurring for a randomly distributed pôle orientation, making the dotted limb solution unlikely.

Figure 3 |

 3 Figure3| Upper limit on Eris' atmosphère. Each data point (filled coloured circles) obtained at three of the stations shown in Fig.2has been projected onto a radial scale (distance from Eris' centre), using the circular solution 2 displayed in Fig.2. The horizontal bars indicate the fmite radial resolution associated with finite intégration intervals; the vertical dotted line shows the adopted Eris radius, RE = 1.163 km. The black solid line is a model light curve obtained at 1-km resolution, using an isothermal pure N2 atmosphère. Black crosses mark the expected flux associated with each data point, once the convolution with the finite intégration intervals has been performed. The fit minimizes the différences between the black crosses and the corresponding data points (filled circles). The model shown here is the 3<7-level upper limit of an isothermal N2 atmospheric profile, with T = 27.7 K and a surface pressure of 2.9 nbar. Most of the constraint on the model cornes from the two data points obtained at La Silla (the two green filled circles just right of the vertical dotted line), corresponding to the data points obtained just before and just after the event (Fig.1). The two closest San Pedro data points (red) hâve only a small contribution to the y2 value, while the CASLEO data points (blue) are too far away fforn Eris' edge (~200km) to effectively constrain the atmospheric pressure. Using solution 1 instead of solution 2 for Eris' shape would hâve a minimal impact on the atmospheric upper limit, as this would slightly displace the San Pedro data points in the plot, leaving the La Silla points where they are shown here.dormant Pluto twin, with a bright icy surface created by a collapsed atmosphère. Detailed models are required, however, to confirm this model.

  withall the télescopes at different Image rates and with different deadtimes, asshown. Ail the observations were madein the visible, except for the Paranal lightcurve, which was obtained with ISAAC, a near-infrared instrument111, The sequences werestarted typically 20 min before the nominal occultation time, and finished around 20 min later. The images were biassubtracted and flatfield corrected using calibration trames taken before or after the occultation. From the image sequences, fluxes of the combined light source formed by Makemakeand theblended star were obtained. The fluxes were obtained by means of synthetic circular-aperture photometry techniques, and the fluxes of other stars in the field of view were extracted using the DAOPHOT package15. The fluxes as a function of time constitute what we call light eu rves. These were divided by the fluxes of other stars to compensate for transparency fluctuations in the terrestrial atmosphère. The resulting light curves were divided by the average value of the unocculted part of the light eu rveto obtain a normalized flux. The uncertainties in the fluxes wereobtained from the standard déviation of the data outside the occultation drop. The computers thatcontrolled the caméras were ail periodically synchronized with ut time servers, except for ULTRACAM at the 3.5-m NTT, the timing of which was directly synchronized by means of a global positioning System that provided a timeaccuracy betterthan 1 ms. We tested theaccuracy of the timing of the Internet-synchronized computers by checkingtheerror logs. The maximum déviations of the computer docks were ail less than 10 ms. Thus we adopt this value as a conservative estimate of the error in the times of the images.

  -î-^-i-I-^-f-^^-f-I î î -î--î-' < 1 I * >-(> i i | i i . h i i ) >-. i I .«H;i . i*. .A,.« .««* *»'-* i ** «* *1 . * r »*-.

Figure 1 |

 1 Figure 1 | Light curves of the Makemake event observed from seven télescopes on 2011 April 23. Note that the brightness drop in the Pico dos Dias light curve happens earlier than the rest because the observatory is at a very different longitude (see map in Supplementary Fig. 1). Also note that the ULTRACAM caméra provided two channels of useful data (one in the red and the other in the green part of the spectrum). The light curves show the sum of the star and Makemake fluxes, arbitrarily normalized to unity outside the occultation. The R-band star magnitude is about 18.22 according to the

Figure 2 |

 2 Figure 2 | Occultation chords obtained at five different sites plotted in the projected plane of the sky. The axis marked g indicates the north-south direction in the projected plane of the sky,/ indicates the east-west direction.

i 6 .

 6 LIMITS ON A GLOBAL ATMOSPHERE FROM THE NTT AND VLT IN-GRESS AND EGRESS PROFILES.

Fig

  Fig. S5: a) Ray-tracing with a partial CH4 atmosphère in the limb between 285°and 350 °of Position Angle (PA), and between the symmetrical angles with respect to the minor axis (a proxy for the rotation axis). The semi-major and semi-minor axes lengths are 800 and 714 km, respectively. b) Idem, but with PAs between 275°and 350°(plus symmetrical angles), and 717 and 714 km of semimajor and semi-minor axes lengths, respectively. Both models use a température profile of 3 K/km until 100K is reached beyond which the atmosphère is isothermal. The surface pressure and température are 5 pbar and 50K. The gray scale represents the light intensity. The blue lines mark the chords at the sites indicated with labels, and the green lines indicate the local curvature of the atmosphère. The red line in the left panel marks the centers of curvature of the limb, where caustics are expected due to ray focusing. These models are used only for illustration. The modeled occultation light curves for the left and right cases are shown in Fig. S6 and compared with the observations.

Fig

  Fig. S6: a) Expected fluxes (black line) at S. Pedro, Pico dos Dias, Paranal and La Silla for a surface pressure Psurf ~5pbar using the model in the left panel of Fig. S5. (The Armazones datahâve not been considered here as the corresponding chord is very close to Paranal, but with a much coarser time resolution, see Fig.1.) The fluxes from the model hâve been binned to the time resolution of the observed light curves (blue, brown, red, and green curves). Note that in the Pico dos Dias light curve (brown) the alleged cosmic ray has been removed. The x2 values of the fits were36.2, 13.2, 91.7 and 738.3 for 25, 17, 99, and 540 data points respectively. Thus, the square root of x2/N is close to 1 in ail cases, indicating that the fits are reasonably good. However, these models are just for illustration, b) Same as panels a, but with the shape model in the right panel of Fig. S5 (note this model can explain a central flash at Pico dos Dias). Note that the Pico dos Dias light curve (brown) is shown as it was initially obtained, with no removal of a potential cosmic ray.

( 1 ,

 1 428 km and 1,434 km for the projected minor and major axes) and the position angles (275°to 350°) for the local atmosphère. The synthetic light curves from the local CH4 atmosphère are plotted in the right panels of Fig. S6. This model reproduces a flash at Pico dos Dias while preserving the fluxes at S. Pedro and Paranal low. Similar models but with élongations up to 1.06 can give rise to a flash in Pico dos Dias while creating small signais at S. Pedro and Paranal.

  The first successful observation of a stellar occultation by a TNO (besides Pluto and Charon) was the 9 October 2009 occultation by 2002 TX30o (Elliot et al. 2010). Since tlien, the following objects hâve been measured by stellar occultations: Varuna (19 February 2010, Sicardy et al. (2010)), the dwarf planet Eris (6 November 2010, Sicardy et al. (2011)), 2003 AZ84 (8 January 2011, Braga-Ribas et al. (2011)), Quaoar(11 February 2011, Person et al. (2011)), the dwarf planet Makemake[START_REF] Ortiz | The steilar occultation by Makemakc on[END_REF], Ortiz et al. (2012)), Quaoar(4 May 2011, 17 February 2012 and 15 October 2012, this paper), 2003 AZ84 (3 February 2012, In this paper, we présent results derived from the 4 May 2011 stellar occultation by

4

 4 May 2011 occultation was identified in a systemat.ic search made at the 2.2 m telescope of ESO, using the Wide Field Imager (WFI). The paths of ten large TNOs for the period 2008-2015 were covered by WFI, providing local astrometric catalogs with an accuracy of 40 mas, for stars with magnitudes as faint as R = 18-19. The positions were measured against the UCAC2 (Second US Naval Observatory CCD Astrograpli Catalog, Zacharias et al. (2004)) catalog, see Assafin et al. (2010) and Assafin et al. (2012) for details.Astrometric updates were performed with several télescopes some weeks before the event, at Pic du Midi (lm telescope), La Silla (TRAPPIST 0.6 m telescope), San Pedro de Atacama (ASH2 0.4 m telescope), and Pico dos Dias (0.6 m telescope). On the last few nights prior to the occultation, both Quaoar and the star were observed in the same field of view. This reduced the effects of systematic errors from the reference catalog, thus improving the prédiction accuracy. Those updates confirmed that Quaoar's shadow would cross the Southern part of South America. A compilation of our measurements provides the » following ICRF/J2000 star position: a = +17h 28m 50.801s ± 0.007" à = -15°27' 42.770" ±0.014" 

  5 km projected at Quaoar's distance. The smallest intégration time used in the positive observations was three seconds, which translates to almost 55 km in the plane of the sky. Therefore, the occultation light curves are largely dominated by the intégration times, not by Fresnel diffraction nor by the star diameter.

  Fig.4has a depth of roughly 200 km, about 45% of the crater diameter. Consequently, such a crater appears to be too deep, unless Quaoar has unexpected cohésion properties.

  2012, another multi-chord stellar occultation by Quaoar was observed from Europe. Also identified in the ESO/WFI program(Assafin et al. 2012), the occulted star has magnitude R= 15.2 and K-11.6. Astrometric updates could only be made several months in advance, before its conjunction with the Sun. The small solar élongation just prior to the event, prevented last minute prédictions. The final ICRF/J2000 star position is given in Table

Fig. 3 .

 3 Fig.3.-Nominal solution: The best elliptical fit to the occultation chords, using the timings of Table2.

Fig. 4 .Fig. 6 .Fig. 7 .

 467 Fig. 4.-Crater solution. Left: a 2D-map of x2 obtained from various apparent oblateness e' and équivalent radius Reqmv in the crater case (see Section 3.3 for details). The darker régions correspond to better solutions (smaller x2)-The value of x2 is coded not only in colors (see the vertical bar at the right of the panel), but also in size: the larger the value of x2> the smaller the dot. Right: the best fit for the crater solution, where the San Pedro de Atacama egress point (left) has been discarded from the fit. It lias P -21 deg, e' = 0.29 and Rcqujv = 557 km.

Fig. 8 .

 8 Fig.8.-Red line: the Maclaurin equilibrium curve relating the density and true oblateness (Eq. 6) for a body rotating with a period of 8.8394 hours. Gray segments: the crater and mountain solutions with their lu error bars. In this case, the oblateness is actually the apparent oblateness of the limb. Solid blue lines: the relation between p and e derived from Eqs. 5. They correspond to spheroids that ail hâve an apparent équivalent radius Rcqmv -555 km and a mass M = 1.40 ± 0.21 x 1021 kg(Fraser et al. 2013). In this figure, we plot the true oblateness, which is derived from the apparent oblateness using Eq. 4. Since the relationship between density and oblateness also dépends on the polar aspect angle Ç, we hâve plotted the extreme equator-on (£ = 90 deg) and pole-on ((,' = 0 deg) geometries. The dashed lines are obtained by varying Quaoar's mass over its error interval. The spheroid solutions must résidé inside the V-shaped blue limits, and the Maclaurin solutions must furthennore be on the intersection with the red line, plotted as the dotted blue segment.

Fig. 9 .

 9 Fig. 9.-The best fit for the Maclaurin solution. It corresponds to the middle point of the blue dotted segment in Fig. 8. It lias a position angle P = 148.4 ± 0.9 deg, an équivalent radius Rcquiv = 555 ± 2.5 km,an équatorial radius ReqUa = 569117 km and apparent oblateness e' = 0.0486lo'o486-Once projection effects are accounted for, this corresponds to true oblateness e = 0.0897±0.006 and density p = 1.99±0.14 g cm-3, see text. To obtain this fit, the cliords were shifted as follows (leaving the San Pedro de Atacama timing unchanged), Rivera: -9.1 s, Armazones: -4.6 s, Florianôpolis: -2.4 s and Ponta Grossa: -3.8 s.

Fig. 10 .

 10 Fig.10.-The adopted thermal structure of a Quaoar's CH4 atmosphère. It starts at 42 K near the surface with a gradient of 5.7 K km-1 and ramps up to an isothermal profile at 102 K in the upper atmosphère.

Fig. 11 .

 11 Fig. 11.-Curves of x2 for different centre-to-limb radii (from 550 to 560 km), using a pure methane atmosphère with the thermal structure shown in Fig.10. An upper limit of 21 nbar (1er), or 56 nbar (3<r) atmosphère is derived. The radius is indicated for some of the curves.

Fig. 13 .

 13 Fig. 13.-The 2012 chords are compatible with the 2011 Maclaurin solution but do not bring further constraints. It corresponds to the middle point of the blue dotted segment in Fig. 8, projected to the 17 February 2012. It has a position angle P= 146.9 deg, an équivalent radius Rcquiv -555 km, an équatorial radius ReqUa = 569 km, apparent oblateness e' = 0.0428 and a Xpdf = 0.23.

  Depuis 2009, onze occultations par des OTNs ont été observées : neuf par le consortium dont fait partie le groupe du LESIA, et deux par une équipe du MIT. Dans cette thèse on possible atmosphères up to the nanobar level. Keywords: Stellar occutlation, TNO, KBO, 50000 Quaoar, Maclaurin object.

		Definiçôes, acrônimos e constantes
	Sumârio Lista de Tabelas ua G <=> Capitulo 1
	^SB	
			2.1	Catâlogos estelares
	IAU
	1	Introduçâo 4.1 Circunstâncias dos observatôrios utilizados -Quaoar 04/05/2011
	2 Prediçâo de Ocultaçôes 4.2 Instantes de inlcio/fim da ocultaçâo
		2.1	Catâlogos estelares 4.3 Soluçôes obtidas para o tamanho e forma de Quaoar
		2.2	Ôrbitas 4.4 Posiçâo de Quaoar 04/05/2011
			4.5	Albedo e Densidades derivados
			4.6	Temperaturas de Equilibrio
			4.7	Pressâo de vapor para entre 30 a 60 K para CO, CH4 e N2
			4.8 Telescôpios e instantes da ocultaçâo por Quaoar 17/02/2012
			4.9 Soluçâo Quaoar 17/02/2012
			4.10 Posiçâo de Quaoar 17/02/2012
			4.11 Posiçâo de Quaoar 15/10/2012
			5.1	Resultados obtidos de todas as ocultaçôes estelares por TNOs
			é

  . Estes catâlogos apresentam precisôes astrométricas e quantidades de estrelas muito diferentes (ver Tabela 2.1), fatores que sâo muito impor tantes quando se considéra a busca por estrelas candidatas à serem ocultadas.

	Catâlogo	N°estrelas	mov. prôprio	Mag. limite	precisâo astrométrica
	HIPPARCOS	118218	sim	9	1 mas
	TYCHO-2	2.5 x 108	sim	12	32 mas
	2MASS	4,71 x 108	nâo	16*	100 mas
	USNO B1.0	1,05 x 10y	sim	21	200 mas
	UCAC4	1,13 x 108	sim	16	40 mas

  Com este equipamento, observamos o caminho que o sistema de Plutâo, Éris, Haumea, Makemake, Quaoar, Varuna, Sedna, tempo no telescôpio Max-Planc 2.2 m para observarmos o caminho de 2012 a 2014, de mais 39 TNOs e centauros (programas: 088.C-0434, 089.C-0356, 090.C-0118 e 091.C-0454). Estas observaçôes jâ foram realizadas e reduzidas, e as prediçôes de ocultaçôes

	Isto nos permitiu gerar catâlogos estelares para cada objeto com posiçôes de alta
	precisâo astrométrica (<30 mas) e bastante densos. Estes catâlogos serâo referidos ao
	longo do texto como catâlogos WFI. Por consequência as prediçôes de ocultaçôes geradas
	à partir dcles sâo mais confiâveis, o que permite que conccntrcmos nossos esforços sobre
	eventos com maior potencial de serem efetivamente observados.

Ixion, Orcus, 2003 AZ84 e 2002 TX30o farâo no céu de 2008 a 2015 (Figura 2.3). Estes 1Europcan Southern Observatory/Max-Planck-Gelscllschaft objetos forain selecionados entre os mai ores e cientificamente mais relevantes na época do pedido. RA (hours) Figura 2.3 Caminho de Plutâo entre 2008 e 2015 (AR, DEC) como exemplo de caminho observado com o telescôpio do ESO 2,2 m para 11 objetos. Os retângulos representam o campo de 30; x 30' da caméra WFI, efetivamente observados. A linha continua représenta o movimento de Plutâo no céu ao longo do perîodo. Estas observaçôes se estenderam de 2007 a 2009 (programas: 077.C-0283, 075.C-0154, 079.A-9202, 079.C-0354) com mais de 7,5 x 106 estrelas identificadas, com magnitudes R entre 11 e 21. As observaçôes foram feitas em filtro R de banda larga (À = 651,7 nm, AA = 162,2 nm). A posiçâo de cada estrela foi determinada tendo o catâlogo UCAC2 como referência, e apresentam erros de 30 mas para estrelas entre magnitudes R 11 e 18. Os movimentos prôprios foram utilizados diretamente do catâlogo UCAC2, quando dispomvel, ou calculados tendo as posiçôes dadas pelos catâlogo 2MASS ou USNO Bl, nesta ordem de preferência, como posiçôes de primeira época. Mais detalhes sobre estas observaçôes podem ser encontrados nas publicaçôes de Assafin et al. (2010) & Assafin et al. (2012). Desde que o Brasil se tornou membro2 do ESO, pedimos e fomos contemplados com mais

  Outra alternativa é a utilizaçào de caméras de video de alta sensibilidade. Alguns modelos, como a Watec -120N+, permitem o contrôle do tempo de integraçâo pelo usuârio. Elas precisam de um sistema paralelo que escreve o horârio proveniente de uma antena GPS diretamente sobre a imagem (ITV, Incrustador de Tempo em Video). As imagens precisam ser digitalizadas para pôs-processamento e fotometria. O pesquisador Frédéric Vachier do Observatoire de Paris / IMCCE desenvolveu, o programa "Aquisiçâo e Tratamentos de Ocultaçôes Estelares"(ATOS, do francês Acquisition et Traitement des Occultations Stellaires), no quai colaborei, para implementaçâo do ajuste das curvas de luz para obtençâo dos instantes de infcio e fim de evento.

			1	1	r	1	1	|	1	1	.	1		i	!	1
	X													
	JD												_	*	*
	_çn													
	UJ	-													-
	+	.					San Pedro de Atacama				.
	L_													
	o													
							Harllngten 50cm				
		"												
	g O													
	V)													
	"5													
	E														.
	o													
	Z													
							*"								.
		.	1 I	1	I	, 1	i .	1 I	I	±U	I	1	1	. I	1
			8300				8350					8400	
			Time (sec after Nov. 6, 2010, 00:00:00.0 UTC)
	Mas, como ele é inevitâvel em algumas câmeras, procura-se fazer o tempo de exposiçâo
	ao menos duas vezes mais longo que o tempo-morto, assim ele representarâ nâo mais que
	33% do tempo total do ciclo (integraçào + tempo-morto), diminuindo a probabilidade do
	evento iniciar ou terminar durante tal intervalo.				

Figura 3.3 Exemplo de curva de luz obtida da ocultaçâo por Éris observada em 06 de novembro de 2010

(Sicardy et al. 2011)

. Os pontos pretos sâo o fluxo medido em cada imagem, os segmentos de traços sobre eles representam o tempo de integraçào de cada imagem, o traço vermelho é a modelagem geométrica da ocultaçâo.

3.3.1 Câmeras

Como as ocultaçôes sâo eventos râpidos que necessitam de imagens sequenciais, câ meras especiais sâo necessârias. Infelizmente as câmeras mais comumente encontradas nos observatôrios nâo sâo adequadas por diversos fatores. O principal deles é a existência de um intervalo entre cada integraçào. Este intervalo, conhecido como tempo-morto, é o tempo que a câmera CCD précisa para descarregar a imagem para a memôria e salvâ-la em disco. Isto représenta uma real perda de informaçâo, jâ que durante este intervalo nâo temos nenhuma informaçâo sobre a possfvel variaçâo do fluxo da estrela. Se, por exem plo, a ocultaçâo se iniciar durante o tempo-morto, haverâ uma incerteza sobre o infcio do evento igual ao tempo-morto. Se, por exemplo, o tempo-morto for de 2 segundos num evento a 20 km/s, a incerteza sera de até 40 km sobre o limbo do objeto. Algumas açôes podem diminuir o tempo-morto, como, por exemplo, binar o CCD ou selecionar apenas uma parte da matriz, ou ainda selecionar um modo de leitura râpido. Uma câmera que tem sido bastante utilizada, por ter sido concebida para as necessidades observacionais de ocultaçôes estelares, é a Raptor/247 Merlin. Foi desenvolvida pela empresa Raptor photonics em conjunto com pesquisadores do Observatoire de Paris-Meudon Com um sensor EM-CCD (Electron Multiplying -Charging Coupled Device) de > Magnitude R Figura 3.4 Histograma mostrando a quantidade de eventos previstos pelo programa caminhos por magnitude R entre os anos 2009 e 2015. Ha apenas 2 eventos envolvendo estrelas de magnitude 11 a 12 e 4 envolvendo estrelas de mag. 12 a 13. Como o catâlogo é completo até magnitude 19, hâ uma queda em quantidade de eventos a partir de magnitude 20.

alta sensibilidade e ruido de leitura menor que 1 elétron, opéra em frame transfer, o que permite ter tempo-morto desprezivel. Com dimensôes menores que 9 cm e menos de 400 g pode ser facilmente transportada e instalada em qualquer tipo de telescôpio. O programa de aquisiçâo de imagem também foi otimizado para a observaçâo astronômica. Por exemplo, registra no cabeçalho de cada imagem o exato instante em que a imagem foi tomada e permite visualizar a imagem em tempo real. Figura 3.5 O conjunto da caméra Raptor/Merlin à direita e alguns acessôrios, como a antena GPS à esquerda, fundamental para o registro dos instantes corretos

  Uma vez as cordas recombinadas, considera-se uma elipse como o caso mais gérai para descrever o perfil do objeto. Esta elipse sera ajustada às cordas para entâo obter-se os parâmetros: tamanho, elipticidade, centro e ângulo de posiçâo do corpo.

	Outro parâmetro que pode-se calcular à partir da soluçâo do ajuste de uma elipse é
	o raio équivalente (Requiv) dado pela equaçâo 4.1. Ele corresponde ao raio de um cfrculo
	com mesma area superficial de uma elipse de semi-eixo maior a e semi-eixo menor c. Ou-
	tros métodos de determinaçâo de tamanhos de objetos celestes costumam apresentar o
	raio équivalente como resultado. Alguns destes métodos, como o utilizado pelos telescô-
	pios espaciais Spitzer e Herschel (Stansberry et al. (2008) e Santos-Sanz et al. (2012)),
	baseiam-se sobre o fluxo observado do objeto, portanto nâo podem medir sua elipticidade.
	Apresentam entâo o raio que um objeto circular teria conforme o fluxo medido, o raio
	équivalente.	
		Pequiv	y/CL C	(4.1)
	4.1	Quaoar	
			Belo Hori-
			zonte/BR
			ltajuba/3R
			Soo José dos
			Campos/BR
			UEPG/BR
			UFSC/BR
			Son Pedro de
			Atocomo/CH
			Armozones/CH
			Rivera/JY
			Solto/UY
	Figura 4.1 Cordas observadas na ocultaçâo de Quaoar de 04 de maio de 2011. Cada
	observatôrio obtém uma medida do perfil do objeto numa dada latitude (linhas azuis),
	e ao recombina-las obtém-se a forma do corpo. A direita os nomes dos observatôrios.

Os segmentos vermelhos correspondem à barra de erro da determinaçâo do instante de imcio/fim do evento. As linhas verdes correspondem à observatôrio que obtiveram dados mas nâo detectaram o evento. Em Sâo José dos Campos (linha tracejada), alta umidade impediu a observaçâo do evento. Para discussâo da metodologia de obtençâo dos parâmetros ffsicos do objeto e as conclusôes deles derivadas, apresta-se os resultados obtidos a partir da anâlise da ocultaçâo estelar por Quaoar em 04 de maio de 2011, 17 de fevereiro e 15 de outubro de 2012. Estes resultados sâo apresentados e discutidos no artigo (Braga-Ribas et al. 2013). Quaoar foi descoberto em junho de 2002 em imagens do Palomar Observatory 1,2 m Oschin Telescope (Williams et al. 2002) (Figura 4.2), recebendo a denominaçâo 2002 LM60. Apôs a confirmaçâo de sua ôrbita, a Uniâo Astronômica Internacional (UAI) o designou o nûmero 50000 em comemoraçâo pela descoberta de um dos maiores objetos da regiâo. Apôs, objetos maiores como Éris, Makemake e Haumea foram encontrados, mas suas numeraçôes foram dadas seguindo a ordem normal de descoberta. O semieixo-maior de Quaoar é de 43.51 ua, com excentricidade de 0,035, inclinaçâo de 7,9°e perfodo orbital é de 287 anos. Quaoar é classificado dinamicamente como um objeto quente clâssico do cinturâo de Kuiper, também chamados de cubewanos. Em 2007, descobriu-se um satélite em torno de Quaoar (Brown h Suer 2007). Denominado

Weywot, ele orbita à uma distancia média de 14,500 km em aproximadamente 12 dias.

  Qi,caic sâo as coordenadas das extremidades correspondentes da elipse ajustada. O valor do x2 por grau de liberdade {x%/) é calculado como Xpdf = Xmin/(N -M) e também pode ser usado como um critério de qualidade do ajuste.

		N				
		X -^) j^(/i,obs 7c,cale) "b (S^obs 5c,cale) j /^i,ri	(4.3)
		1				
	O ajuste é feito pelo método Downhill Simplex com a rotina Amoeba (Press et al. 1990),
	no quai parâmetros iniciais sâo fornecidos, e os instantes derivados das curvas de luz
	(Tabela 4.2), considerados como os pontos a serem ajustados. Por uma série de passos,
	o programa procura minimizar uma funçâo, que neste caso é o x2-Para explorar todo o
	Tabela 4.2 Instantes de imcio/fim da ocultaçâo dos sitios que detectaram o evento e
	circunstâncias de observaçâo.				
	Obscrvatôrio	Instrumente	Tclcscôpio	Exposiçâo		
		Escala de Plaça	Abertura	Tcmpo-Morto		
		(")	Filtro	S/R		
	Pico dos Dias	Andor/Ixon	Cari Zciss	0,95 s	Inicio obs.:	Fim obs.:
	-Brasôpolis	EMCCD	0,60 m	1,05 s	2:26:10	2:58:20
	/ BRA	1,11	nâo	S/R=85		Negativo
	Ponta Grossa -	SBIG	RCX 400	14 s		
	UEPG / BRA	STL-6303E	0,40 m	4,25 s	2:41:47,2±2,7	2:42:42,1±3,1
		1,74	V	S/R=20		
	Florianôpolis	SBIG	CM-1100	10 s		
	-UFSC/ BRA	ST-7	0,254 m	2,75 s	2:41:28,7±2,8	2:42:31,9±2,9
		1,48	nâo	S/R=12		
	San Pedro de	Apogée	Harlingten	3,0 s		
	Atacama	U42	0,5 m	1,1 s	2:42:33,1±0,13	2:43:31,4±0,15
	/ CHL	1,61	nâo	S/R=52 Ortiz		
	San Pedro de	SBIG	ASH2	9,0 s		
	Atacama	STL11000	0,4 m	7,1 s	2:42:32,5±1,0	2:43:30,30±4,3
	/ CHL	1,21	nâo	S/R=22 Ortiz		
	Armazones	SBIG	OCA	7,0 s		
	/ CHL	STL6303E	0,84 m	2,4 s	2:42:41,5±2,2	2:43:26,3±2,0
		0,57	nâo	S/R=42		
	De cada corda tem-se dois pontos que sâo usados para determinar o perfil de Quaoar,
	Rivera	SBIG	LX200			
	um ponto no ingresso e outro no egresso. Cada ponto tem seu erro correspondente (cq,r)
	que é dado pela incerteza em tempo sobre a determinaçâo de tais instantes, multiplicada
	pela velocidade perpendicular ao limbo do TNO. Para se ajustar a elipse utiliza-se os

N=10 pontos para ajustar M=5 parâmetros, buscando o x2 minimo da equaçâo 4.3. Os valores fi^s e gij0bs sâo as coordenadas das extremidades das cordas, e os valores /itCa/c e Imcrsâo ttempo (UT) h:mm:ss±ss Emcrsâo tempo (UT) h:mm:ss±ss

  , o ajuste foi feito coin valores pré-definidos de PA e e'. Variou-se PA entre 0 e 180°com passos de 3°e, para cada PA explorou-se e' de 0 a 0,5 com passos de 0,004.Portanto, os parâmetros a serem ajustados sâo o raio équatorial a' e o centro da elipse fc e gc. Para cada conjunto de parâmetros, a soluçâo com seu correspondente Xmin ® salvo.O conjunto de parâmetros com o menor x%f entre todas as soluçôes é tido como o melhor ajuste, e o erro da soluçâo a la é obtido tomando-se os valores de Xmin + E Com todos estes ajustes cria-se um mapa 2D em cores, no quai o x2 de cada conjunto de parâmetros é apresentado em funçâo do achatamento e raio équivalente. As cores sâo codificadas, sendo os pontos pretos a regiâo de la, e os amarelos todas as soluçôes acima de 3a. Os tamanhos dos pontos sâo inversamente proporcionais ao valor do seu x2-Estes mapas apresentam uma forma em V. Isto acontece porque as maiores cordas impôem que

	espaço de fase					
				4,0 s	Inicio obs.:	Fim obs.:
	/ URY	516wsg	0,305 m	2,7 s	2:20:29	2:54:04
		1,22	nâo	S/R=18		Negativo

a' seja maior que um valor L, prôximo ao valor da corda mais longa. O alinhamento destas cordas é prôximo ao alinhamento do eixo-maior da elipse, o que impôe que b' seja menor que L. Entâo o raio équivalente sera limitado por L\J 1 -e' < Requiv < L/y/1 -e'. Estes limites correspondem às bordas à esquerda à e direita, respectivamente, da forma em V, e

  que os ajustes apresentam menor valor de Xmim cores escuras, que correspondem aos

	melhores ajustes. diferença entre as extremidades das cordas e o limbo da elipse ajustada. Para San Pedro -25 seja fisicamente improvâvel.
	de Atacama esta diferença é de ~-200 km, ou 46% do diâmetro estimado para a cratera. -24
										Il 23
		0.5 r-•	•	1	1	.	•	1	•-	-	25
										22
	O b l a t e n e s s									21 20
				E						
				^o o						19 18 17 16
				i						
				o						
											f0 (km)
	Figura 4.9 Soluçâo Montanha: 0 grâfico da esquerda corresponde ao mapa de Xmin
				0		2		4		6	8	10
	500 para cada soluçâo testada em funçâo do achatamento e do raio équivalente. O ponto do 600 700 800 Equivalent Radius (km) Point index
	Figura 4.7 Soluçâo Cratera: O grâfico da esquerda corresponde ao mapa de Xmin Para cada soluçâo testada em funçâo do achatamento e do raio équivalente. O ponto do egresso egresso de Rivera, corda mais ao sul, foi desconsiderado para esta soluçâo. A regiâo prêta Figura 4.8 Soluçâo Cratera: Diferença entre as extremidades das cordas e o limbo da équivale à valores a la (veja cabeçalho da figura 4.7 para mais detalhes). No grâfico à elipse ajustada. Do sul para o norte, Rivera é representada pelos indices 1 e 2, Armazones 3 e 4, San Pedro de Atacama 5 e 6, Florianôpolis 7 e 8, e Ponta Grossa 9 e 10, onde indices direita tem-se a elipse correspondente ao melhor ajuste. A sudeste, a elipse nâo se ajusta
	de San Pedro de Atacama foi desconsiderado para esta soluçâo. A regiâo prêta équivale à impares correspondem aos pontos a oeste e pares a leste. Tal diferença para o ponto do às cordas de Rivera e Armazones, dando lugar à uma grande proeminência.
	valores de Xmin +1 > isto é, egresso em San Pedro de Atacama (indice 4) é de ~200 km. A palheta de cores foi ajustada de forma que todas as cores,
		que nâo amarelo, correspondem à valores dentro da regiâo de 3a. O tamanho dos pontos
	também é inversamente proporcional ao valor do seu Xmin• Pode-se ver que as melhores soluçôes se restringem à uma pequena gama de valores. No grâfico à direita tem-se a elipse Figura 4.11 Mâxima carga estâtica topogrâfica que pocle ser suportada pela força dos Considerando os pontos discutidos acima, pode-se concluir que, ou a soluçâo é fisi-materiais em funçâo do raio planetârio. Proeminências observadas na Terra, Mercurio,
	correspondente ao melhor ajuste. As linhas azuis sâo as cordas observadas, os segmentos camente impossivel, ou o modelo de Leliwa-Kopystynski et al. (2008) nâo é complexo o Venus, Marte e Lua sâo mostradas pelos pontos sôlidos. Os satélites galileanos e os
	vermelhos os erros associados à cada ponto e as linhas verdes cordas negativas. suficiente, subestimando os valores mâximos das caracteristicas das crateras. Uma pos-maiores satélites também sâo indicados. Valores da rugosidade h/R sâo apresentados
	sivel explicaçâo para esta ultima hipôtese séria uma inesperada forte coesâo interna em como parâmetro, onde hé a escala da proeminência e R o raio do planeta (Johnson &
	Observando a distribuiçâo de tamanhos das crateras no Sistema Solar, e com ensaios de u McGetchin 1973). Quaoar.
					i					
					O					
		colisôes em laboratôrio Leliwa-Kopystyhski et al. (2008) determinaram uma relaçâo para
	calcular o diâmetro mâximo de uma cratera sobre um corpo de gelo, ou de rocha, sem que 4.7.2 Soluçâo: Montanha
	4.8	ele seja quebrado pelo impacto de origem. Seja R o raio do corpo, o diâmetro mâximo Objeto Maclaurin com instantes deslocados
	da cratera serâ dado por DC)ge|0 ~1,2R se o objeto for predominantemente formado por 0 2 A 6 8 10 A présente soluçâo foi obtida eliminando-se o ponto do egresso da corda de Rivera, a Point index
	na tabela 4.3. gelo, ou i}C)rocha ~1,6A se ele for formado por silicatos. mais ao sul a detectar o evento. A soluçâo é menos alongada, e apresenta uma importante Figura 4.10 Soluçâo Montanha: Diferença entre as extremidades das cordas e o limbo da Como dito e sera explicado em detalhe a seguir, espera-se que Quaoar tenha a forma
	Considerando o RequiV = 557 km obtido nesta soluçâo, tem-se DC)ge\0 ~668 km e proeminência no seu bordo a sudeste, por isto é chamada de Montanha. No painel direito elipse ajustada. Para o egresso em Rivera (indice 2), esta diferença é de ~144 km. Para o de um objeto Maclaurin. As soluçôes acima nâo condizem com esta expectativa. Entâo,
	4.7.1 da figura 4.9 pode-se ver a elipse que corresponde ao melhor ajuste. Ela apresenta um Soluçâo: Cratera rocha ~891 km. Do painel direito da figura 4.7, pode-se estimar um diâmetro de egresso em Armazones, Florianôpolis e Ponta Grossa (indices 4, 8 e 10 respectivamente), assume-se que o horârio absoluto registrado por alguns si'tios apresentem deslocamentos
	~430 km para a suposta cratera. Este valor é compatlvel com os diâmetros mâximos a diferença é de aproximadamente 80 km. em relaçâo ao tempo universal, para entâo obter a soluçâo correspondente à um objeto achatamento de 0.138, ângulo de posiçâo de 7,5°, raio équivalente de 525 km e xldfmin = A primeira soluçâo apresentada foi obtida eliminando-se o ponto do egresso em San é a soluçâo mais alongada dentre as apresentadas e tem uma grande "falha" no seu bordo leste, por isto é chamada de Cratera. Ela tem um achatamento de 0,29, ângulo de posiçâo a 21°, raio équivalente de 557 km e xldf,min -3,8 (Tabela 4.3). Na figura 4.7 pode-se respectivamente, à 108, 133 e 111 km para a cratera de 430 km. Na figura 4.8, tem-se a Em (Johnson & McGetcliin 1973) pode-se verificar na Figura 1 (reproduzida aqui na suportadas em seu relevo. Isto pode significar que a montanha mostrada nesta soluçâo respectivamente, em relaçâo ao diâmetro da cratera. Estas proporçôes correspondem, de ~165 km o que équivale à 31% do raio équivalente. implica em uma fraçâo importante de gelo, portanto grandes proeminências nâo seriam mente por H20, CO2 ou silicatos, as profundidades mâximas variam de 25%, 31% e 26%, soluçâo sâo mostradas na figura 4.10. Para o ponto do egresso em Rivera, a diferença é como sera mostrado, esta soluçâo corresponde à uma densidade prôxima a 2 g/cm3, o que investigado em Leliwa-Kopystyhski et al. (2008). Sendo o corpo formado predominante As diferenças entre as extremidades das cordas e o limbo da elipse ajustada para esta por silicatos com o mesmo raio, a maior amplitude séria de ~150 km. No entanto, A profundidade mâxima de uma cratera sobre um corpo de gelo ou silicato também é (ver seçâo 4.7.1 para maiores explicaçôes). a maior proeminência suportada sobre a superficie é de ~20 km. Em um corpo formado Pedro de Atacama, considera-se entâo que tem-se N=9 pontos para serem ajustados. Esta calculados. 3,9 (Tabela 4.3). No painel esquerdo da figura 4.9 também é apresentado o mapa de Xmin Maclaurin.
	Figura 4.11) que, em um corpo formado predominantemente por gelo, com 525 km de raio,

. Para corpos formados principalmente por gelo, o diâmetro crftico varia entre D ~200 -900 km; para corpos rochosos, a variaçâo se dâ entre D ~500 -1200 km. Na seçâo 4.8 é feita uma discussâo detalhada deste tema. Como o histôrico da formaçâo e colisâo, bem como a estrutura interna de Quaoar nâo sâo conhecidos, nâo pode-se descartar soluçôes que nâo correspondam às formas de equilibrio. Sendo o objetivo maior do trabalho, melhorar o conhecimento sobre o tamanho e forma de Quaoar e utiliza-los para derivar albedo e densidade, assume-se casos extremos, explorando assim todas as possfveis formas para o corpo.

Figura 4.6 Soluçâo nominal: O ajuste sobre todos os pontos nâo é satisfatôrio. Nem todas as extremidades das cordas sâo contempladas pela elipse ajustada. 4.7 Soluçôes extremas Para investigar soluçôes extremas, elimina-se pontos especificos artificialmente. Para isto, considera-se que Quaoar pode ter grandes caracteristicas sobre sua superficie, o que o difere de uma elipse. Isto é feito ao se associar à eles erros muito grandes, e, portanto tais pontos passam a ter peso muito pequeno sobre o ajuste (Eq. 4.3). Os pontos escolhidos para serem "eliminados" sâo aqueles que têm maior peso sobre a forma da elipse previamente ajustada. Seguindo este procedimento, dois casos sâo apresentados a seguir. Um terceiro caso também é estudado, onde considera-se que os instantes registrados estâo em desacordo com o Tempo Universal Coordenado e que Quaoar encontra-se em equilîbrio hidrostâtico. Os parâmetros derivados, bem como suas barras de erro sâo apresentados ver a elipse correspondente ao melhor ajuste, bem como um mapa de Xmin com os valores correspondes para cada soluçâo testada. No mapa de Xmin pode-se visualizar a regiâo O b l a t e n e s s em

  Caracteiisticas de observaçâo nos sftios que detectaram o evento. Os instantes de ingresso e egresso sào dados em tempo universal, em 17 de fevereiro 2012. t Mctodo "traço", acompanhamento a 96,4%, resoluçâo temporal de 6 s por pixel. t Durante os primeiros minutos da observaçâo o tempo de exposiçâo foi de 10,24 s, c foi reduzido para 5,12 s cerca de 60 s antes do inîcio do evento.

	Obscrvatôrio	Latitude	Abcrtura	Exposiçâo	Tempo (TU)
	Cidade	Longitude	Instrumente	S/R	Imersâo
	Obscrvador	Altitude	Escala de Plaça	Fonte de Tempo	Emersâo
	Gnozca	46°13' 53",2 N	0,4 m	10,24 s	h:mm:ss,s±ss
	Bellinzona/CHE	09°01' 26",5 L	Watcc 120N+	4,4	4:28:37,6±4,5
	S. Sposeti	260 m	1",9	ITV IOTA GPS	4:29:09,4±5,5
	TAROT	43°45' 07",3 N	0,25 m	180+ s	h:mm:ss,s±ss
	Calcrn/FRA	06°55' 25",1 L	CCD Andor	4,6	4:28:08,1±3,0 s
	A. Klotz &	1270 m	3",3	GPS	4:29:06,5±3,5 s
	E. Frappa				
		43°51' 52",4 N	0,21 m	5,12* s	h:mm:ss,s±ss
	Valensolc/FRA	06°00' 23",0 L	Watcc 120N+	4,0	4:28:09,2±2,1 s
	J. Lecachcux	622 m	0",84	TIM-10 GPS	4:29:03,2±2,2 s
	Tourrctte-	43°47' 22",2 N	0,356 m	10 s	h:mm:ss,s±s,s
	Levcns/FRA	07°15' 47",2 L	Apogée Alta U1		

de luz foi obtida por fotometria diferencial e o instante de cada imagem lido diretamente do cabeçalho da imagem. O programa de captura utilizado foi o "MaxImDL", entâo os instante registrados foram truncados no segundo, e o procedimento descrito na seçâo 3.4.2 aplicado. Finalmente, no sitio TAROT3, a observaçâo foi realizada pelo método "traço".

Neste método, o motor de acompanhamento sidéral do telescôpio é desligado, entâo ao se adquirir imagens com alguns segundos de duraçâo, as estrelas caminham pelo campo da caméra, deixando traços nas imagens. Se a ocultaçâo ocorrer, haverâ queda de brilho ao longo do traço da estrela ocultada. Para esta ocultaçâo, por ser uma estrela muito fraca para o instrumento, o acompanhamento foi regulado para 96,4% da velocidade de acompanhamento sidéral. Com isto, a estrela ficaria 6 segundos sobre cada pixel, antes 3TAROT: Télescopes à Action Rapide pour les Objets Transitoires

Tabela 4.8 

  . O instante de maior aproximaçâo em relaçâo ao geocentro, dado em h:mm:ss,s ± ss, de 17 de fevereiro de 2012. O offset corresponde ao centro da soluçâo circular em relaçâo à efeméride JPL#21/DE405, dado em mas, onde lmas équivale à 31,561 km à distancia de 43,5208 ua. A posiçâo de Quaoar em ascensâo reta e declinaçâo no instante de C/A e aposiçâo da estrela ocultada também sâo dadas.

			Data		17 de fevereiro 2012
			Instante do C/A	04:30:43,73 ± 0,11 s
			Distancia do C/A	161 ± 12 mas Norte
			offset AR		-101 ± 25 mas
			offset DEC		-92 ± 24 mas
			Quaoar AR		17/l 34m 2T,8438 ± 0S,0023
			Quaoar DEC		-15°42' 10",4270 ± 0",012
		400	500 Estrela AR 600	700	800 I7h 34m 21*',8453 ± 05,022 900 1000 1100 1200
			Estrela DEC		Radius (km)
			Raio		fc	9c	v2 . A. 771Z 71
	4.13.2	(km) 15 de Outubro de 2012	(km)	(km)
			688îîf4	-3182ij«	-2895^32	0,02
	Uma nova ocultaçâo por Quaoar foi detectada no dia 15 de outubro de 2012. Inicial-

10, o instante de maior aproximaçâo relativa ao geocentro e a dis tancia sâo dados para o dia 17 de fevereiro de 2012. Todas as curva de luz deste evento apresentam tempos de exposiçâo muito elevados (>5,12 s) e/ou S/R muito baixo para trazer restriçôes aos resultados obtidos do evento de 2011. Tabela 4.9 Soluçâo circular obtida com as cordas de Gnosca e Valensole. Obtém-se um raio minimo para o caso de um corpo esférico. Barras de erro a la

(Figura 4.33)

. Figura 4.33 Curva de x2 do ajuste de um objeto circular às cordas observadas. Pode-se verificar que as soluçoes sâo limitadas (la) à raios maiores que 534 km, o que représenta um limite inferior para o raio de Quaoar, no caso esférico. f0 (km) Figura 4.34 O ajuste circular sobre as duas cordas efetivas observadas na ocultaçâo de 17 de fevereiro de 2012. O ajuste tem como soluçâo Rq = 688^54 km, estabelecendo um raio mrnimo para o caso de Quaoar ser circular. Comparado aos resultados de 2011, conclui-se que o raio de Quaoar delimita-se entre 534 e 557.5 km, caso esférico. mente o evento era previsto para 0 norte do Chile, Bolivia e Argentina. A estrela, com magnitudes R*= 17,3 e K*= 14,4 séria um bom alvo para telescôpios localizados nesta regiâo. Atualizaçôes apontaram um grande deslocamento da estrela, colocando o evento Quaoar-15°42' 10",5860 ± 0",0008 no sul do do Chile e Argentina. Tratando-se de uma regiâo desprovida de telescôpios adequados e de clima imprôprio, nâo se realizou uma campanha, mas os colaboradores do centro-norte do Chile foram alertados para realizarem a observaçâo, sabendo da baixa probabilidade de detecçâo. Quatro sitios, San Pedro de Atacama, La Silla (TRAPPIST4) e Cerro Tololo (PROMPT5) e Cerro Burek (ASH6) participaram da observaçâo. Apenas o PROMPT teve condiçôes técnicas e climâticas para monitorar o evento.

  Exceto o evento envolvendo o objeto 2002 TX30o, todos os outros foram observados por colaboradores do nosso grupo. Um sumârio dos resultados é dado na tabela 5. evento ou nào puderam observar. Apesar de ser esperado que o objeto tenha um leve achatamento devido à sua rotaçâo e curva de luz de rotaçâo (P = 8,14 horas, Am = 0,14, até hoje nâo se detectou nenhuma variaçâo de brilhos em curvas de luz de rotaçâo de Éris, seu periodo de rotaçâo nâo é conhecido. Acredita-se que ele esteja em equilibrio hidrostâtico devido ao seu tamanho. Entâo sô poderia apresentar a forma de um objeto Maclaurin ou Jacobi. Esta ûltima é descartada, jâ que exige que seu eixo de rotaçâo esteja orientado exatamente na direçâo de visada, para nâo promover curva de Eris como um dos mais brilhantes objetos do Sistema Solar. Investigou-se a presença de uma atmosfera em torno de Eris e determinou-se o limite superior de 1 nanobar para uma atmosfera de N2 isotérmica a 30 K. Como o espectro de Éris é bastante similar ao de Plutâo, com presença de N2 e CH4 na superficie(Tegler et al. 2010), imagina-se que ao se aproximar do Sol, elementos de sua superficie sublimem formando uma atmosfera sazonal. Ao se distanciar, essa atmosfera condensa sobre a superficie, o que justifica o alto albedo e a nâo existência de variaçâo de brilho sobre Éris(Sicardy et al. 2011). se limitaram a poucos telescôpios. O evento foi detectado em San Pedro de Atacama por dois telescôpios, mas nâo foi detectado de La Silla (TRAPPIST), cerca de 610 km, no piano do céu, ao sul.

	Thirouin et al. (2010)), uma soluçâo circular com Regu^= 143 ± 5 km foi obtida. Possi-Apenas uma estaçâo, localizada em Sâo Luiz do Maranhâo, observou com sucesso a ocultaçâo, obtendo uma corda de 1003 km. Outra estaçâo, em Quixadâ, cerca de 200 km do Sul. Uma campanha observacional envolvendo 26 observatôrios foi realizada, mas apenas très telescôpios no Chile detectaram o evento. Dois dos telescôpios encontravam-se no mesmo sitio, San Pedro de Atacama, e o terceiro em La Silla (TRAPPIST). Vârios te massa é conhecida M = 1,66 ± 0 , 02 x 1022 kg (Brown h Schaller 2007). A partir do raio obtido da ocultaçâo, pôde-se entâo calcular sua densidade p = 2,52 ± 0,05g/cm3. Este valor sugere que Éris seja um corpo rochoso no seu interior, coberto por uma camada de quanto a observaçâo Com uma unica corda, apenas um limite inferior para seu diâmetro pode ser obtido
	velmente por algum problema na dataçâo de um dos telescôpios, os instantes centrais das gelo.
	cordas tinham cerca de 32 segundos de separaçâo no piano do céu. Por isto, os autores ao sul, obteve dados mas nâo detectou o evento. Isto trouxe fortes restriçôes ao tamanho de Varuna. climâticas (Sicardy et al. 2011). Do novo diâmetro obtido para Éris, calculou-se o albedo no vislvel pv = 0,961q/4, lescôpios no Brasil nâo puderam observar, apesar de preparados para tal, por questôes Dmin = 573 ±21 km (Braga-Ribas et al. 2011).
	alinharam as cordas para obter uma soluçâo circular.
	Com este raio eles obtiveram um albedo de 0,88io'oe-Este alto valor é explicado pelos No momento da ocultaçâo, Varuna apresentava seu brilho mâximo, isto é, maior super 5.4.2 2003 AZ84 -03/02/2012
	ficie aparente. Entâo, considerando fatores como o mâximo achatamento esperado para
	autores como consequência da presença de gelo de âgua na superficie do corpo. Como
	Varuna, limitaçôes sobre seu tamanho obtido por outras técnicas, seu albedo e as cordas
	nâo hâ elementos leves como N2 e CH4 detectados sobre a superficie de 2002 TX300,
	positiva e negativa observadas na ocultaçâo, pode-se obter uma soluçâo com limite supe-
	investigou-se a presença de uma atmosfera de elementos pesados, como o Xe, nas curvas
	rior para a = 860 km e b' -375 km (Requiv -567 km), para um albedo de 0,04 (Sicardy et
	O de luz de ocultaçâo obtidas. Obteve-se um limite superior de 2 x 1015 cm-3 para uma al. 2010) (Figura 5.2). Ainda nâo foi realizado nenhum estudo sobre a assinatura de uma CM -4000 -4500 -5000 -5500
	atmosfera isotérmica com temperatura de 49 K.	
	5.2 5.4	Varuna -19/02/2010 e 08/01/2013 2003 AZ84 -08/01/2011 e 03/02/2012
			dark (pv=0.04)				dark (pv=0.04)
			least elongated				most elongated
			probability -0				probability -0.3
			610x525 km				860x375 km v, ..,---,	F
	satélite.	&orb (u3.)	q (ua)	c	i(°)	P (anos)
			39.64	32.73	0.174	13.54	249.60
			brightest possible			
			(Pv=°.12)			
	5.1	probability -0 502x218 km &orb (u.3.) 2002 TX30o -09/10/2009 q (ua) c i(°) &orb (va) q (ua) e i(°) 45.55 38.27 0.1599 29.01	P (anos) P (anos) 307.48
			43.2	40.9	0.053	17.1	283.82
	Figura 5.2 Soluçôes extremas para o tamanho e forma de Varuna, considerando limitaçôes
	Varuna é um candidato a planeta-anâo (Tancredi & Favre 2008), uma vez que seu sobre o seu tamanho, o achatamento e o albedo obtidos por outras técnicas, e as cordas
	tamanho, curva de luz (Am = 0,42) e velocidade de rotaçâo (P= 6,34 h) condizem com observadas na ocultaçâo.
	Descoberto em outubro de 2002 pelo programa de busca de objetos prôximos à Terra um objeto Jacobi em equilibrio hidrostâtico (Lacerda & Jewitt 2007). Ajustando uma
	do Observatôrio Monte Palomar, 2002 TX30o teve seu tamanho estimado em 640 ± 250 km algumas suposiçôes sobre o albedo, Lacerda & Jewitt (2007) encontram a densidade de (Stansberry et al. 2008) a partir de observaçôes com o telescôpio Spitzer. É classificado curva de luz de um objeto Jacobi sobre a curva de luz de rotaçâo de Varuna, e fazendo 5.2.2 Varuna -08/01/2013
	como objeto quente clâssico do cinturâo de Kuiper. E um dos maiores membros da farmlia Uma nova ocultaçâo por Varuna, envolvendo uma estrela de mag. R= 16,5, foi ob-&orb (v3.) q (ua) e i(°) P (anos) 0,992^015 g/cm3. Com observaçôes do telescôpio Spitzer, seu diâmetro équivalente foi 67.96 38.2 0.437 43.88 560.23
	colisional de Haumea (Brown et al. 2007) e possui gelo de âgua em sua superficie (Peixinho estimado em 621 ± 180 km. servada no dia 08 de janeiro de 2013. Apôs observaçôes astrométricas realizadas com os
	observadas e, por vezes, em eventos duplo com Plutâo. Alguns dos resultados mais recen
	et al. 2012). Possui os seguintes elementos orbitais (Chamberlin 2008): Duas ocultaçôes estelares por Varuna jâ foram observadas, uma em 2010 e outra telescôpios Calar Alto 1,2 m e Pic du Midi 1 m, as atualizaçôes da prediçâo apontaram Encontra-se atualmente numa regiâo do céu bastante desprovida de estrelas (conste-tes sobre a atmosfera de Plutâo podem ser vistos em (Lellouch et al. (2009) e Zalucha & Gulbis (2012)). A descoberta de Plutâo a mais tempo (1930), e o fato de estar transitando &orb (^a) q (ua) c i(°) P (anos) 43.4 38.1 0.122 25.8 286.03 recentemente em 2013, apresentadas a seguir. uma ocultaçâo sobre China e Japâo. Très telescôpios no Japâo detectaram o evento, um Figura 5.4 Imagem artistica de Éris mostra as cordas de ocultaçâo observadas, sobrepostas laçâo da Baleia). Isto, aliado ao seu lento deslocamento devido à grande distancia ao à um objeto bastante brilhante sem atmosfera. A imagem fui publicada no site European em Hiroshima e dois em Shiga (figura 5.3). Sol, faz com que ocultaçôes estelares por ele promovidas sejam muito raras. Em 06 de Southern Observatoryt na ocasiâo da publicaçâo do artigo (Sicardy et al. 2011).
	5.2.1 novembro de 2010, uma ocultaçâo de uma estrela de mag. R = 17 foi confirmada depois Varuna -19/02/2010 Reduçôes preliminares apontam para um objeto achatado com a -817 km e b' = 578 km, ^http://www.eso.org/public/news/esol142/ (acessado cm 04/02/2013)
		Como Éris possui uma lua, Dysnomia, com diâmetro que pode chegar a 650 km, sua

primeira ocultaçào por um TNO foi observada em outubro de 2009, envolvendo o objeto 2002 TX300, e as detecçôes positivas aumentaram em numéro rapidamente. Em 2010 mais duas ocultaçôes foram detectadas; mais quatro em 2011; mais cinco em 2012; e mais duas sô em janeiro de 2013, totalizando 14 eventos que envolvem apenas 8 objetos. Do total de 14 eventos, 12 foram observados pelo grupo no quai me enquadro ou por colaboradores, dois nâo estavam no catâlogo de eventos do programa caminhos e dois foram detectados pela équipé do MIT (seçâo 5.1).

Vale ressaltar que nâo estâo sendo consideradas aqui as ocultaçôes envolvendo Plutâo e Caronte (Figura 5.1), jâ que suas ocultaçôes vêm sendo observadas desde 1985

(Brosch 1995)

. Depois de 1985, novas ocultaçôes foram observadas em 1988, 2002 e anualmente, desde entâo, mostrando a evoluçâo sazonal da atmosfera de Plutâo. Caronte foi observado pela primeira vez numa ocultaçào em 1980

(Walker 1980)

. Em 2005 realizou-se a primeira observaçâo multi-corda para este corpo, quando foi possivel determinar seu tamanho

(Sicardy et al. 2006b)

. Mais recentemente, novas ocultaçôes por Caronte foram uma regiâo do céu que tem como fundo o piano da galâxia, sâo fatores que contribmram para que eventos envolvendo este sistema jâ tenham sido observados em maior numéro.

Nas seçôes a seguir cada um dos eventos observados envolvendo os TNOs é brevemente relatado com seu contexto e principais resultados. As ocultaçôes envolvendo Quaoar foram discutidas em detalhes no capitulo 4.

UT (sec), orange: Santa Martina, red: TRAPPIST, green: LNA. blue: San Pedro Figura 5.1 Ocultaçâo dupla envolvendo Plutâo e Caronte ocorrida em 04 de junho de 2011 e observada de très si'tios no Chile e um no Brasil

(Sicardy et al. 2012)

. É clara a diferença entre uma ocultaçâo por um objeto desprovido de atmosfera (Caronte à esquerda) e de um com uma atmosfera de cerca de 12fibar (Plutâo à direita). Ocultaçôes duplas sâo oportunidades ûnicas para medir com alta precisâo a distancia entre um objeto e seu Em 9 de outubro de 2009, detectou-se a primeira ocultaçâo por um objeto transnetuniano (Elliot et al. 2010). Ela envolveu o objeto 2002 TX300 e foi observada pela équipé do MIT (Massachusetts Institute of Technology). A ocultaçâo envolveu uma estrela de magnitude R= 13,1 e foi detectada por dois telescôpios no Havai. Outros 19 telescôpios participaram da campanha, mas nâo detectaram o Descoberto em novembro de 2000 pelo projeto Spacewatch, Varuna recebeu designaçâo provisôria 2000 WRi06-Recebeu a numeraçâo 20000 numa forma de comemoraçâo à sua descoberta, jà que era o maior objeto de sua classe até aquele momento, e também por ter ocorrido 200 anos apôs a descoberta de Ceres. Recebe a classificaçâo dinâmica de um objeto do cinturâo de Kuiper clâssico quente, e possui gelo de âgua em sua superficie (Peixinho et al. 2012). Alguns de seus principais elementos orbitais sâo (Chamberlin 2008):

A ocultaçâo de uma estrela de magnitude R= 12, prevista pelo programa caminhos, foi atualizada com observaçôes astrométricas realizadas poucas semanas antes do evento.

no nordeste do Brasil e Narmbia (Africa). atmosfera na curva de luz, mas a ocultaçâo parece ser bastante abrupta, o que implica num limite na ordem de poucos nanobar à pressâo de uma possivel atmosfera. o que équivale à um Reguiî; = 687 km e e = 0,29. Neste caso, seu albedo deve ser ainda Figura 5.3 Mapa da ocultaçâo observada no Japâo em 08 de janeiro de 2013. O pontos verdes indicam locais em que o evento foi detectado, em azul onde os dados nâo apontam uma ocultaçâo e em vermelho sitios que nâo puderam adquirir dados. menor que 0,04. Os dados foram obtidos com tempos de exposiçâo na ordem de 10 s, e, portanto nâo devem trazer limitaçôes sobre a presença de atmosfera em Varuna. 5.3 Éris -06/11/2010 O planeta-anâo Éris foi descoberto 2005 a partir de imagens tomadas em outubro de 2003 do Monte Palomar, recebendo a designaçâo 2003 UB3i3 (Brown et al. 2005). Seu raio foi estimado em 1.200 ±100 km e 1.500 ± 200 km, sendo potencialmente maior que Plutâo. Sua descoberta foi um dos motivos que levou a IAU a criar a classificaçâo de planetas-anôes (Tancredi & Favre 2008). É atualmente o objeto observâvel mais distante no Sistema Solar, a 97 ua, e apresenta uma ôrbita bastante excêntrica, o que faz com que no periélio chegue a 38 uado Sol, tâo perto quanto a distancia de Plutâo. Seus elementos orbitais (Chamberlin 2008): de uma intensa campanha astrométrica para a atualizaçâo da ocultaçâo. Foi apenas nos ultimos dias que as prediçôes mostraram que o evento poderia ser observado da América Como luz de rotaçâo. Como as cordas observadas estâo praticamente sobre a mesma mediana, assume-se que a forma de Éris seja prôxima à uma esfera. Com o ajuste circular obtém-se Requiv =1,163±6 km (Figura 5.4). Tamanhos maiores sâo possfveis se ajustes elipticos forem permitidos, mas os ajustes tornam-se rapidamente muito achatados, exigindo râpida rotaçâo o que nâo é observados em suas curvas de luz. Este tamanho é comparâvel ao tamanho de Plutâo R = 1,152 ± 32 km (Lellouch et al. 2009). colocando Objeto pertencente à classe dos plutinos, em ressonância 2:3 com Netuno, 2003 AZ84 foi descoberto em janeiro de 2003 a partir de imagens do Monte Palomar. Possui os seguintes elementos orbitais (Chamberlin 2008): ksi Figura 5.5 Soluçâo eliptica para 2003 AZ84 a partir das cordas da ocultaçâo de fevereiro de 2012. Uma elipse de e = 0, 37 ± 0,1 e RequiV = 343 ± 7 km é encontrada. Os traços amarelos sâo as cordas observadas, os segmentos vermelhos representam a barra de erro de cada ponto. Na corda mais ao sul, o erro do ingresso é grande porque ele ocorreu durante um tempo-morto de 14,5 segundos.(Braga-Ribas et al. 2012) Makemake pertence à classe dinâmica dos objetos clâssicos quentes do cinturâo de Kuiper, jâ que sua ôrbita é bastante inclinada. Alguns de seus elementos orbitais sâo (Chamberlin 2008):

  para Éris, divido a seu lento deslocamento no céu, ocultaçôes estelares por Sedna sâo bastante raras (<3/ano). Uma ocultaçâo no dia 13 de janeiro de 2013, envolvendo um a estrela de magnitude R = 18, foi prevista para a regiâo da Austrâlia. Sumârio dos principais resultados obtidos a partir de cada ocultaçâo estelar por TNO jâ observada. A data corresponde à data do evento. O numéro de telescôpios que detectaram cada evento pode ser maior que o numéro de cordas apresentadas à esquerda na coluna 'N°de cordas', jâ que por vezes hâ mais de um telescôpio por sitio. O numéro de telescôpios que nâo detectaram(ou nâo puderam observar) o evento é dado à direita. O simbolo à Trente dos valores na coluna 'RequiV representam limites inferiores para o raio do objeto, assim como '<' representam limites superiores para as colunas 'e, 'Albedo' e Atmosfera'. A distancia do objeto no momento da ocultaçâo é dada na coluna 'r\ Na coluna 'Classe' as letras representam as seguintes classe de objetos transnetunianos: S -espalhados; D -destacados; HC -clâssicos quentes; CC -clâssicos frios; R -ressonantes. As referências encontram-se na ultima coluna. Os resultados que ainda estâo sob anâlise sâo denotados por "preliminar". Assim, esta diferença é aplicada sobre a efeméride. Com boas posiçôes das estrelas e dos TNOs, gera-se prediçôes de ocultaçôes mais précisas, com anos de antecedência. Este trabalho esta sintetizado em duas publicaçôes: Assafin et al. (2010) & Assafin et al. Outros eventos observados que merecem destaque foram as ocultaçôes por Éris e Makamake. Calculou-se que Eris tem um tamanho similar ao de Plutâo (indistinguivel quai é

	o maior), com um albedo e densidades bastante elevados. Ainda determinou-se um limite					
	(2012).											
	Uma vez que uma ocultaçâo favorâvel é selecionada, inicia-se outra etapa, a de atua-Atualizaçôes foram realizadas apontando uma ocultaçâo para o norte australiano. Dois superior para a pressâo superficial de uma atmosfera global de 1 nanobar. Possivelmente,					
	Eris possui uma atmosfera sazonal, que atualmente esta condensada sobre a superficie, observadores na cidade de Cairns observaram o evento, com exposiçôes de 60 e 180 segun-estrelas, e efemérides que se desviam gradualmente da posiçâo real do objeto, é preciso dos. Ambos detectam a ocultaçâo em pelo menos uma imagem. Numa anâlise preliminar, lizaçôes prôximas à data do evento. Devido a possfveis erros em movimento prôprio das o que explicaria seu alto brilho e nào detecçâo de curva de luz de rotaçâo (Sicardy et Apêndice A				
	observar tanto a estrela candidata, quanto o ocultador em épocas prôximas ao evento, se for considerado que a ocultaçâo nâo pode ter durado mais do que 60 segundos, coloca-se al. 2011). Para Makemake, o terceiro maior objeto da classe, mediu-se seu tamanho com					
	um limite inferior para o diâmetro de Sedna de 700 km. Objeto Data Local N°de precisâo de alguns quilômetros e nào detectou-se uma atmosfera global, apesar de ser um H-equii; e Albcdo cordas (km) para corrigir tais erros e melhorar ainda mais as prediçôes. As prediçôes mais précisas Densidade (g cm-3) sâo alcançadas dias antes da ocultaçâo, quando a estrela e o corpo podem ser observados dos melhores candidatos a possuir uma. No entanto, uma atmosfera local nâo pode ser Classificaçâo dinâmica dos TNOs	Atmosfera (nanobar)	r (ua)	Classe	Rcfcrência
	5.9 descartada. Seu albedo também é elevado, o que pode ser causado por uma atmosfera Sumârio dos resultados (136199) Éris 06/11/2012 CL 2+| 24-1,163+6 0 0 Qfi+0 09 U'JO-0.04 sobre a mesma imagem. Assim, erros sistemâticos de catâlogos sâo eliminados, e as predi 2,52 + 0,05	Ï 1 (N2)	94.74	S	(Sicardy et al. 2011)
	(136472) Makemake 23/04/2011 çôes podem ser tâo précisas quanto a um diâmetro do objeto. Sendo a prediçâo favorâvel, CL,BR 5+|9-732+9 0,05 + 0,02 0.77+0.03 parcialmente condensada sobre a superficie (Ortiz et al. 2012).	1.7+0.3	<12(CH4)	51.50	HC	(Ortiz et al. 2012)
	(20000) Varuna Ocultaçôes por objetos como Varuna, Sedna, 2003 AZ84 jâ foram detectadas, e anâlises 19/02/2010 BR 1+| >7->565 <0.56 -	-	-	42.75	HC	(Sicardy et al. 2010)
	grandes campanhas observacionais sâo lançadas.								
	(20000) Varuna	08/01/2013	JP	2+| >10-	-685	0.29	<0.04	-	-	42.67	IIC	preliminar
	(50000) Quaoar	11/02/2011	US	1+	>380	-	-	-	-	43.62	HC	(Person et al. 2011)
	(50000) Quaoar	04/05/2011 CL,UY,BR 5+| 11-	555+2.5^0 .0895lo!oi6 0.109+0.007 1.99+0.42	< 21 (CH4) 42.35	HC	(Braga-Ribas et al. 2013)
	(50000) Quaoar	17/02/2012	FR,CH	2+|<10-	G85+ÎJg	-	-	-	-	43.52	HC	(Braga-Ribas et al. 2013)
	(50000) Quaoar	15/10/2012	CH	1+|3-	>200	-	-	-	-	43.56	HC	(Braga-Ribas et al. 2013)
	(90377) Sedna	13/01/2013	AU	2+	>700	-	-	-	-	86.20	D	preliminar
	(208996) 2003 AZ84 08/01/2011	CL	1+|1-	>286+11	-	-	-	-	44.32	R	(Braga-Ribas et al. 2011)
	(208996) 2003 AZ84 03/02/2012	IN,IL	3+|>20-	343+7	0.37+0.1	0.147+0.007	0.76-0;i7	-	44.26	R	(Braga-Ribas et al. 2012)
	(119951) 2002 KXi4 26/04/2012	ES	1+|2-	>207+1	-	-	-	-	38.55	CC	preliminar
	(55636) 2002 TX300 09/10/2009	HW	2+ | 19-	143+5	-	0.88±g;è|	-	<33/iibar (Xe) 40.61	HC	(Elliot et al. 2010)
	2005 TVi89	13/11/2012	CZ	l+(visd) >28+3 & >22+6	-	<0.22	-	-	31.26	HC	
	Ao menos duas grandes descobertas ainda podem ser feitas a partir de ocultaçôes					
	estelares por objetos transnetunianos:									
	1. Detecçâo de uma atmosfera global sobre algum objeto transnetuniano (além de					
	em tempo absoluto, 4 das 5 cordas observadas. Usando propriedades dos elipsoides, como Plutâo). Alguns dos melhores candidatos jâ foram detectados em ocultaçôes, e				
	o posicionamento de seus instantes centrais em uma linha reta, e associando tais proprie apenas limites superiores na ordem de alguns nanobars foram determinados. Estes					
	dades à curva de equilibrio (e, p) de um corpo Maclaurin, pôde-se determinar achatamento, resultados podem trazer implicaçôes sobre as teorias que explicam os processos de					
	Para superar este problema, realizou-se um programa de observaçôes astrométricas R-equzu , ângulo de posiçâo, densidade e albedo de Quaoar com precisôes sem precedentes retençâo/perda de volâteis por estes objetos.				
	do caminho que 10 entre os maiores TNOs realizarâo no céu até 2015. Com isto, além para este objeto. Investigou-se também a presença de uma atmosfera global, e o limite				
	de melhorar a posiçâo de estrelas relativamente brilhantes, aumentou-se sensivelmente superior de 21 nbar foi dado. No entanto, a existência de atmosfera local nâo pode ser 2. Detecçâo de um satélite de um TNO. Determinar os parâmetros da lua pode ajudar				
	a quantidade de estrelas com posiçôes précisas o suficiente para se realizar prcdiçôes de descartada. Analisou-se também outras duas ocultaçôes promovidas por Quaoar em fe-a determinar sua origem, seja ela capturada ou formada ao mesmo tempo que o					
	10,273 ± 0,002 h (Gaudi et al. 2005). Estimativas de tamanho ocultaçôes confiâveis. Estrelas de magnitude R <18 tiveram suas posiçôes determinadas corpo principal. Isso pode ajudar a compreender os processos dinâmicos aos quais					
	os TNOs foram submetidos.										

inicialmente 0 colocavam como um objeto com até 1600 km de diâmetro. Com dados do Spüzer e Herschel seu diâmetro foi calculado em D= 995 ± 80 km

[START_REF] Pal | TNOs are Cool": A survey of the trans-Neptunian région. VII. Size and surface characteristics of (90377) Sedna and[END_REF]

.

103

Assim como

Aqui apresenta-se a tabela 5.1, com um sumârio dos principais resultados jâ obtidos a partir de todas as ocultaçôes estelares por objetos transnetunianos jâ observadas. Tabela 5.1 preliminar (^Assumindo a soluçâo Maclaurin. *Observaçâo visual.) Capltulo 6

Conclusâo

A compreensâo do Sistema Solar, da sua formaçâo e de sua evoluçâo, passa pelo estudo dos objetos transnetunianos. Estes corpos guardam informaçôes sobre a evoluçâo dinâmica e sâo compostos por elementos primordiais da nuvem protoplanetâria. Portanto, conhecer seus parâmetros flsicos, como distribuiçâo de tamanho, densidade e estrutura interna, é fundamental para remontai' a histôria do nosso sistema estelar.

Empregou-se o método de ocultaçôes estelares para determinar, com precisôes sem precedentes, o tamanho, a forma, o albedo, a densidade e a presença de atmosfera sobre alguns TNOS. Como discutido, algumas dificuldades precisaram ser superadas antes de se lograr sucesso em observaçôes de tais ocultaçôes. Os primeiros resultados positivos sô começaram a ser obtidos a menos de 3,5 anos, no final de 2009. Desde entâo, 14 eventos jâ foram observados, dos quais 12 com participaçâo efetiva de nossa équipé.

A primeira etapa do método consiste em predizer tais eventos. Para isto, utiliza-se as efemérides dos objetos, que sâo comparadas à catâlogos estelares, para determinar eventos de conjunçâo. No entanto, em consequência de sua grande distancia do Sol (>30 ua), o movimento dos TNOs no céu é bastante lento.

Para aumentar o nûmero de eventos posslveis, recorre-se a catâlogos estelares menos precisos em termos astrométricos. Suas ôrbitas também nâo apresentam as precisôes necessârias para uma boa prediçâo. Assim, os erros envolvidos, oriundos das posiçôes das estrelas e do objeto, sâo superiores a vârios diâmetros do objeto, dificultado a detecçâo do evento. em relaçâo ao catâlogo astrométrico de referêneia UCAC2. Os TNOs também foram observados para se determinar quai a diferença entre sua efeméride e sua posiçâo real no céu. Nos ûltimos très anos, mais de 40 campanhas foram realizadas. Para isto, paginas na internet dedicadas a cada evento foram criadas com informaçôes e dicas aos observadores. Além disso, milhares de e-mails foram trocados com colaboradores para pedir por observaçôes, e explicar o procedimento observacional. Desde o inicio de 2010, 12 eventos, de um total de 14, j a foram observados positivamente utilizando esta metodologia. Apôs a observaçâo e detecçâo positiva do evento, com as cordas de ocultaçâo obtidas, inicia-se o estudo de tamanho e forma do objeto. Aqui apresentou-se em detalhes a reduçâo da ocultaçâo por Quaoar em 04 de maio de 2011. Como, segundo as cordas obtidas, Quaoar teria uma forma bastante irregular, estudou-se os limites fisicos em que estruturas como cratera e montanhas poderiam ser sustentadas sobre a superficie destes objetos. Como as estruturas observadas sâo fisicamente improvâveis, investigou-se o caso de Quaoar apresentar uma figura de equilibrio hidrostâtico. Formas de equilibrio hidrostâtico Maclaurin e Jacobi foram apresentadas, bem como suas regiôes de transiçâo. Assim determinou-se que Quaoar deve ter a forma de um elipsoide Maclaurin, onde dois de seus semi-eixos sâo iguais (a=b>c). Neste caso, deslocou-se, vereiro e outubro de 2012, mas nenhuma nova restriçâo sobre as propriedades fisicas do objeto pode ser obtida (Braga-Ribas et al. 2013). preliminares jâ apontam para resultados intéressantes como formas achatadas e determinaçâo de densidade para o caso de objetos em equilibrio hidrostâtico (Braga-Ribas et al. 2012). Intéressante notar que, para cada evento, diferentes propriedades fîsicas pré-determinadas a partir de outras técnicas sâo aplicadas para derivar seus parâmetros fîsicos, como al bedo e densidade. Destacam-se entre estas propriedades a massa, a magnitude absoluta, o periodo de revoluçâo e a composiçâo superficial. A mctodologia de prediçâo se mostrou muito eficiente, ha,ja visto a quantidade de detecçôes positivas nos ûltimos anos. A partir das observaçôes das ocultaçôes, importantes parâmetros fîsicos e suas implicaçôes puderam ser determinados para alguns dos maiores e mais importantes objetos transnetunianos. O projeto esta sendo expandido para mais 39 TNOs (Camargo et al. 2013). O primeiro objeto transnetuniano a ser descoberto foi (15760) 1992 QBx por Jewitt & Luu (1993). Desde entâo, mais de 1260 objetos transnetunianos jâ foram descobertos. Eles se dividem em alguns grupos dinâmicos, de acordo com sua distancia e parâmetros orbitais (Gladman et al. 2008). A seguir apresenta-se os grupos dinâmicos e suas subdivisées: Objetos Transnetunianos: todos os objetos que orbitam o Sol à uma distancia maior que 30 ua:

1. O cinturâo de Kuiper é uma regiâo que se espalha de 30 a ~55 ua. Ela se divide em dois subgrupos:

• Ressonantes que se encontram em ressonâncias com Netuno como por exemplo, 1:2 (twotinos) e 2:3 (plutinos);

• Clâssicos (ou cubewanos) com ôrbitas quase circulares nào perturbadas por Netuno. Estes ainda se dividem em duas famüias:

  Felipe Braga-Ribas dnas la spécialité Astronome et Astrophysique, aussi bien an titre de l'Observatoire de Paris que de l'Observatôrio Nacional de Rio de Janeiro, et avec mention très honorable. OTN) sont des corps primitifs du système solaire qui orbitent au-delà de la planète géante Neptune. Plus de 1200 objets ont été découverts depuis 1992, et on compte dans cette population les planètes naines Pluton, Makemake et Eris. Ce sont des corps primitifs, et à ce titre, leur dynamique orbitale, leur composition interne et surfacique et leur distribution en taille nous renseignent sur la formation et , l'analyse et l'interprétation de plusieurs occultations par Quaoar et d'autres gros OTNs. Depuis 2009, quatorze occultations par des OTNs ont été observées : douze par le consortium dont fait partie le groupe de Paris/Rio, et deux par une équipe du MIT. Dans cette thèse, outre la discution détaille sur les occultations par Quaoar, un résumé des résultats sur la taille, forme, albédo, densité et sur l'atmosphère, à partir des occultation par d'autres TNOs est présenté. En particulier les occultations par Eris, Makemake, 2003 AZg4, Varuna et Sedna, qui son toutes des premières. Mots clefs : occultations stellaires, TNO, KBO, 50000 Quaoar, objet de Maclaurin.AbstractTransneptunian objects (TNOs) are fossils of the Solar System beyond Neptune. Over 1200 TNOs hâve been found since 1992, and among them we find dwarf-planets such as Pluto, Makemake and Eris. The dynamics, internai and surface composition, size and mass distribution may hâve information about the early âges and évolution of the Solar

	MlNISTÉRIO DA ClÊNCIA, TECNOLOGIA E INOVAÇÂO Ministère de L'Enseignement Supérieur et de la Recherche Doctorat
	Observatôrio Nacional Observatoire de Paris	Spécialité
	61 avenue de l'observatoire	Astronomie et Astrophysique
	75014 Paris	École Doctorale
	Doutorado em Astronomia Service de la Scolarité d'Astronomie & Astrophysique d'Ile-de-France
	Tél. : 01 40 51 21 70	
	Le jury, dont la liste est donné au recto du présent document, a assisté ce Résumé
	Apêndice B jour (26 février 2013) à la soutenance de thèse de M. Felipe Braga-Ribas, sur Observatoire de Paris
	Résumé Français de la Thèse École Doctorale RAPPORT DE SOUTENANCE DE THÈSE le sujet suivant : "Exploration des objets trans-neptuniens par la méthode Astronomie et Astrophysique D'Ile-de-France Doctorat par M. Felipe BRAGA-RIBAS Astronomie et Astrophysique Thèse soutenue le mardi 26 Février 2013 à 14h00 des occultations stellaires : prédictions, observations, Quaoar et les premiers résultats". Les objets trans-neptuniens (System.
	Sujet de la thèse : Etude d'occultations stellaires par de objets transneptuniens
	Felipe Braga Ribas Président du Jury : Rodney GOMES, Chercheur Titulaire/ON/au MCTI au Brésil
	EXPLORANDO OS OBJETOS
	TRANSNETUNIANOS PELO MÉTODO DE
	OCULTAÇÔES ESTELARES :
	PREDIÇAO, OBSERVAÇAO, QUAOAR E OS PRIMEIROS
	RESULTADOS.
	EXPLORATION DES OBJETS
	TRANS-NEPTUNIENS PAR LA MÉTHODE DES
	OCCULTATION STELLAIRES :
	prédiction, observation, Quaoar et les premiers
	résultats.	
	3. Objetos destacados sâo aqueles que no seu periélio nâo chegam prôximos à in
	fluência gravitacional de Netuno, nem de nenhum outro planeta. Por isto parecem Fait à Rio de Janeiro
	destacados do Sistema Solar.	

4. Objetos da nuvem de Oort, sâo todos aqueles que orbitam o Sol a distancias supe riores a a.>2000 ua.

Le jury a apprécié la présentation orale du candidat, qui a répondu de manière tout à fait satisfaisante aux questions du jury.

Le manuscrit est également apprarue comme bien structuré et très com plet. Le jury a souligné la quantité et la qualité du travail effectué. Le candidat a mi en place tout ma chaîne qui permet de prédire les événements, organiser les campagnes d'observation, et finalement analyser les donnés et en déduire les propriétés des objets étudié : taille, forme, densité, albédo, et contraints sur la présence d'atmosphère éventuelles. Par toutes ces raisons, le jury est très heureux de décerner le titre de docteur à l'évolution du système solaire dans son ensemble. De nombreuses observations photométriques, spectroscopiques et thermiques (IR loin tain) ont fourni des renseignements sur leurs périodes de rotation, leur composition de surface et des estimations de leurs tailles et de leurs albédos.

vation

Table 2 .

 2 Enfin, nous trouvons une limite supérieure de 21 nanobar pour la présence d'une atmosphère globale de meéthane autour de Quaoar. Cependant, nous ne pouvons pas exclure la présence d'une atmosphère locale.La première occultation par OTN a été observée à la fin 2009 par un groupe américain occultations par Varuna(19 Février ) et Éris (6 Novembre 6) (?) ont été détectées par notre groupe. En 2011, nous avons détecté des événements par 2003 AZ84 (8 Janvier), Makemake 1 Résumé des principaux résultats de chaque occultation stellaire par OTN déjà observé. La date correspond à la date de l'événement. Le nombre de télescopes qui ont détecté chaque événement peut être plus grand que le nombre de cordes présentées sur la colonne "Numéro de cordes", parce que parfois il y a plusieurs télescopes par site. Le symbole ">" en face des valeurs dans la colonne "Requiv ", représentent des limites inférieures pour le rayon de l'objet, ainsi que "<" représente des limites supérieures pour les colonnes e, "Albédo" et "Atmosphère". La distance de l'objet au moment de l'occultation est donnée dans la colonne "r". Dans la colonne "Classe" les lettres représentent la classe d'objets transneptuniens suivantes : S -dispersés, D -détaché ; HC -classiques chaud, CC -classiques froid, R -résonnants. Les références sont donnés dans la dernière colonne. Les résultats qui sont en cours d'analyse, sont marqués comme préliminaires.

	(23 Avril) (Ortiz et al. 2012) et Quaoar (4 mai). En 2012, de nouvelles occultations par
	2003 AZ84 (3 Février), Quaoar (17 Février et 15 Octobre) ont été enregistrées, ainsi qu'une
	autre par 2002 KX14 (26 Avril) et possiblement par 2005 TVi89 (13 novembre). Enfin en
	2013, nous avons détecté à nouveau Varuna (8 Janvier) et pour la première fois, Sedna
	(13 Janvier). Ces occultations sont discutées à la fin de la thèse.

A l'exception de Pluton et son satellite Charon, qu'ont eu plusieurs occultations obser vées, les premières occultations par objets transneptuniens ont commencé a être détecté seulement à la fin de l'année 2009. Dans l'occasion, le groupe américain du MIT a détectée l'occultation par l'objet 2002 TX30o. En 2010, les occultations par Varuna et Éris (Sicardy et al. 2012) ont été détectées par notre groupe. En 2011, les nous avons détecté les événe ments par 2003 AZ84 (Braga-Ribas et al. 2011), Makemake (Ortiz et al. 2012) et Quaoar (Braga-Ribas et al. 2013). En 2012, en plus des nouvelles occultations par 2003 AZ84 et Quaoar, nous avons également observé un événement par 2002 KX14 et 2005 TVi8g. Que en janvier 2013, deux occultations on été déjà observé, une par Varuna et outre par Sedna.

Dans la table 2.1 ci-dessus un sommaire des principaux résultats est présenté. La liste des campagnes auxquelles j'ai participé est donnée dans les figures 2.1, 2.2.

Table 1

 1 

	for 2009-2015. A total of 1167 WFI mosaics or 9336 individ-
	ual CCD frames were acquired for science, resulting in about
	316 GB of photometrically calibrated processed images. The
	two WFI field distortion patterns derived in PI were used here.
	The pattern associated with the September 2007 run was used
	in ail earlier runs; the October 2007 pattern was used in ail sub
	séquent runs. The used instruments and observation procedures
	are described in detail in Sect. 3 of PI.

shows the approximate limits of the sky paths observed for each TNO, hence indicates the pe riod covered. Figure

1

illustrâtes the sky path covered by Varuna

Table 1 .

 1 Approximate (a,ô) limits and sky path periods covered by ES02p2/WFI mosaics for each TNO.

	TNO	(a, S) (ICRF)	(a, ô) (ICRF)	Time span
		h m		h m		yr
	Eris	01 38	-04 48	01 42	-02 58	2008-2015
	Haumea	13 40	+ 1857	14 01	+ 1732	2009-2015
	Ixion	16 55	-23 58	17 28	-27 15	2009-2015
	Makemake	12 30	+28 25	12 52	+25 37	2009-2015
	Orcus	09 40	-05 09	10 04	-08 42	2008-2015
	Quaoar	17 10	-15 25	17 48	-15 34	2008-2015
	Sedna	03 27	+06 28	03 41	+07 17	2009-2015
	Varuna	07 30	+25 45	08 03	+26 35	2009-2015
	2002 TX3oo	00 32	+26 59	01 07	+32 43	2008-2015
	2003 AZ84	07 36	+ 12 17	08 02	+09 58	2009-2015

Table 2 .

 2 Astrometry of individual CCD frames of WFI mosaics.

		Mean errors	No. of	No. ref.
	TNOs	«(Aa'costf)	(t(A<S)	frames	stars/frame
		mas	mas		
	Eris	97	95	576	27
	Haumea	51	50	784	8
	Ixion	54	53	976	108
	Makemake	48	51	800	7
	Orcus	58	55	1528	14
	Quaoar	57	57	904	121
	Sedna	54	57	576	11
	Varuna	49	53	848	31
	2002 TX300	47	50	1184	20
	2003 AZ84	50	50	1160	48

The (a, ij) centers were pre-corrected by the ficld distortion patterns (see details of these pre-corrcctions in Sect. 4.1 of PI).

Table 3

 3 

	lists the mean errors of UCAC2 stars
	before and after the WFI mosaic global solutions. The listed val
	ues are averages taken over ail sky path years. In addition to
	eliminating zéro-point errors présent in the positions in the indi
	vidual CCD frames, the overall standard déviations with respect
	to the UCAC2 catalogue were generally smaller, as can be seen

in Table

3

. Detailed information about the global réduction pro cedures are given in Sect.

4.3 of PI. 

In this final step, multiple entries coming from distinct CCD frames for the same objects are resolved (see

Sect. 4.4 

of P1 for details). Table

4

displays the statistics for resolved mul tiple entries. The listed values are taken over the global solutions Notes. Multiplicity flag statistics for each and ail TNOs. Percentages refer to the total number of mosaic positions for each TNO and for ail TNOs together (last line). No flag (good astrometry) means no mul tiple entries within 175 for the same object after mosaic solutions. Flag cases fl and f2 apply only to UCAC2 and 2MASS stars, when only en tries assigned to a catalog were used. Flag fl means that more than one entry was used (more than one position entry was averaged), whereas flag f2 indicates that only one entry was used. Flags f3, f4, and f5 apply

Table 5 .

 5 Proper motion computations from 2MASS, USNO B 1.0, and ES02p2/WFI global mosaic star positions.

		UCAC2	2MASS	USNO B 1.0	No
	TNOs	p.m.	p.m.	p.m.	p.m.
		%	%	%	%
	Eris	8.4	35.5	32.9	23.2
	Haumea	6.3	24.2	41.6	27.9
	Ixion	1.7	22.3	13.5	62.5
	Makemake	6.4	25.2	38.8	29.6
	Orcus	15.6	50.3	22.1	12.0
	Quaoar	5.9	41.3	21.6	31.2
	Sedna	12.2	49.0	22.8	16.0
	Varuna	10.7	34.3	34.7	20.3
	2002 TX300	18.2	48.0	24.6	09.2
	2003 AZ84	10.9	27.8	34.5	26.8

Table 6 .

 6 Star catalogs for each TNO sky path for 2008/2009-2015.

Table 7 .

 7 TNOs offsets with respect to JPL ephemerides.

	TNO	No. stars	o~(ï	(Tù	R magnitude	R magnitude
			mas	mas	completeness	limit
	Eris	25 685	33	37	19.0	21.0
	Haumea	77 860	41	36	19.5	21.5
	Ixion	3452 835	41	29	19.5	21.5
	Makemake	64 487	42	39	19.5	21.5
	Orcus	56 042	40	39	18.5	20.5
	Quaoar	1 150375	41	37	18.5	20.5
	Sedna	27464	33	31	19.0	21.0
	Varuna	140041	38	39	19.0	21.0
	2002 TX300	82746	33	37	18.0	20.0
	2003 AZ84	278 720	33	34	19.0	21.5

Table 7

 7 

		Observed -JPL ephemeris	No.	JPL
		Au cos ô	Aô	Elt	Es	nights	ephemeris
		mas	mas	mas	mas		version
	Eris	-020	-068	084	055	09	28
	Haumea	-090	-098	060	081	21	37
	Ixion	+020	+ 191	092	093	15	09
	Makemake	-039	-165	060	058	20	40
	Orcus	-087	+001	059	046	19	13
	Quaoar	-144	-078	098	043	22	17
	Sedna	-043	-160	040	080	05	17
	Varuna	-045	+005	015	015	-	21
	2002 TX300	-084	-053	040	040	-	14
	2003 AZ84	+232	+006	003	015	-	11
	Notes. Average "observed minus JPL ephemeris" position offsets.
	Errors corne from the standard déviation about the mean of individ
	ual nightly averages. TNO positions in the UCAC2 System corne from
	a parallel astromctric program of CCD observations carried out during
	2007-2010 at a variety of 0.6 m to 2.2 m télescopes in Chile, Brazil, and

displays the ephemeris offsets information for each TNO. The TNO observed positions used to dérivé the offsets were in the UCAC2 System. They came

Table 8 .

 8 Sample from prédiction tables for stellar occultations by large TNOs. The complété table sets of 2008/2009-2015 prédictions for Eris, Haumea, Ixion, Makemake, Orcus, Quaoar,

	Year m d h m s	RA (ICRS) Dec		C/A P/A	u	D
		h m	s	°'	"	mas °km s"1 AU
	Notes. In entry order, the above samples correspond to the prédictions for 2002 TX30o (October 2009), Varuna (February 2010), Eris (November
	2010), 2003 AZ84 (January 2011), Makemake (April 2011), and Quaoar (May 2011), rcspectively. Prédiction tables list event date and instant
	(UTC), the ICRS (J2000) star coordinates at occultation, the closest apparent distance between star and TNO (C/A), the position angle (P/A) of
	the shadow across the Earth (counter-clockwise, zéro at north), the velocity in km s-1, the distance (D) to the Earth (AU), /?*, /*, H\ and K*
	star magnitudes normalized to a reference shadow velocity (v) of 20 km s-1 (50.0 means no magnitude available), longitude (d) of the sub-solar
	point, local solar time (LST), (A^", A^) JPL ephemeris offsets in (a,ô) for the central instant (see Sect. 5), catalog cross-identification (uc =
	UCAC2, 2m = 2MASS, fs = field star), proper motion existence and multiplicity flags (see Sect. 3), estimated star-catalog position errors (E", E6),
	and proper motions	Positive/ncgative v means, respectively, prograde/retrograde velocities, that is TNO's geocentric right ascension is
	increasing/decreasing, respectively.		

Table 9

 9 

	displays the total number of predicted events for
	each TNO for the period 2008/2009-2015.

Table 10 .

 10 TNO diameters.

			TNO diameter
		km	mas	Reference
	Eris	2326 (+/-12)	33	Sicardy et al. (2011b)
	Haumea	1324 (+/-167)	36	Lellouch et al. (2010)
	Ixion	0650 (+250/-220)	22	Stansberry et al. (2008)
	Makemake	1455 (+/-12)	38	Ortizel al. (2011)
	Orcus	0850 (+/-90)	25	Lim, et al. (2010)
	Quaoar	1170 (+/-40)	34	Braga-Ribas et al. (201 Ib)
	Sedna	1200-1600	22	Brown (2008)
	Varuna	1003(+/-09)	32	Sicardy et al. (2010)
	2002 TX300	0143 (+/-05)	05	Elliot et al. (2009)
	2003 AZ84	0910 (+/-60)	28	Mueller et al. (2010)
	Notes. TNO diameters in km (uncertainties in parenthèses) and as geo-
	centric apparent sizes in mas for 2012.5. Values were obtained from
	previous stellar occultations and by modeling visible and IR observa
	tions (see references).		

Table 11 .

 11 Comparison between WFI-based and actual observed occul tations: central instants and C/As.

		Star	Prédiction minus Occultation
	TNO	mag (R)	Central instant		C/A
			s	mas	mas	km
	2002 TX30o	12.9	-15	-13	-09	-265
	Varuna	11.0	-20	-12	-63	-1953
	Eris	16.9	-604	-229	+39	+2708
	2003 AZS4	18.2	+0	+0	+0	+0
	Makemake	18.4	+ 196	+ 116	+40	+ 1494
	Quaoar	15.7	+ 130	+78	-44	-1351
	average (moduli)		193	90	39	1554
	average (Eris out)		90	55	39	1266
	Notes. Comparison between central instants and C/As from WFI-based
	(Table 8) and actual observations (fit to data) for 6 (out of seven) suc-
	cessfully recorded TNO stellar occultations to date for which data is
	available. Using P/A, distance and the relative speed, central instant
	différences are expressed in seconds and in mas, and C/A différences
	in mas and in km. Averages for the modulus of the différences are com-
	puted with and without Eris. 2003 AZ«4 is excluded from the analysis
	as the fit to the actual occultation alrcady uses the WFI star position,
	and the current WFI prédictions alrcady uses the offsets derived from
	the occultation fitting.					

They are given in km and as geocentric apparent sizes in mas for 2012.5. Values were obtained from previous stellar occul tations and by modeling visible and IR observations (see référ encés in Table

10

).

The ES02p2/WFI prédictions can be directly compared to actually observed stellar occultations in order to estimate the external accuracy and précision of prédictions. Table

11

displays a direct comparison between prédictions (ES02p2/WFI-based) and actually observed occultations, for which P/A, central in stant and C/A values were obtained from fits to the light curves of the chords. We used six out of the seven TNO stellar occul tations known to date, for which data is available

(Elliot et al. 2009; Sicardy et al. 2011b; Braga-Ribas 2011, priv. comm.)

.

Using P/As, distances and relative speeds, we computed both the predicted minus fitted central instant and C/A différences.

Différences in central instants are expressed in seconds and in mas (in order to be comparable with C/A results), and C/A différences are given in both mas and kilometers. We give in Table

11

the averages for the modulus of the différences. We rection of the chords are usually east-west). One possible cause is geocentric parallax error (see discussion about the follow-up program and ephemeris offset détermination in the next paragraphs). In ail, for R = 19 stars (catalog magnitude completeness) and 40 mas errors in the WFI positions, we may assume a bulk error of about 80 mas for C/A, that is dominated by the

Table 1

 1 

	shows the approximate limits

Table 1 .

 1 Approximatc (a, ô) limits and sky path pcriods covered by ES02p2/WFI mosaics for each TNO.

	TNO	(a, <5) (ICRF)	(a, 6) (ICRF)	Time span
		h m		h m		yr
	Eris	01 38	-04 48	01 42	-02 58	2008-2015
	Haumea	13 40	+ 1857	1401	+ 1732	2009-2015
	Ixion	16 55	-23 58	17 28	-27 15	2009-2015
	Makemake	12 30	+28 25	12 52	+25 37	2009-2015
	Orcus	09 40	-05 09	10 04	-08 42	2008-2015
	Quaoar	17 10	-15 25	17 48	-15 34	2008-2015
	Sedna	03 27	+06 28	03 41	+07 17	2009-2015
	Varuna	07 30	+25 45	08 03	+26 35	2009-2015
	2002 TX30o	00 32	+26 59	01 07	+32 43	2008-2015
	2003 AZs4	07 36	+ 12 17	08 02	+09 58	2009-2015

Table 2 .

 2 Astrometry of individual CCD frames of WFI mosaics.

		Mean errors	No. of	No. ref.
	TNOs	cr(Aû' cos ô)	cr(AJ)	frames	stars/frame
		mas	mas		
	Eris	97	95	576	27
	Haumea	51	50	784	8
	Ixion	54	53	976	108
	Makemake	48	51	800	7
	Orcus	58	55	1528	14
	Quaoar	57	57	904	121
	Sedna	54	57	576	11
	Varuna	49	53	848	31
	2002 TX300	47	50	1184	20
	2003 AZx4	50	50	1160	48

Table 3

 3 

	lists the mean errors of UCAC2 stars
	before and after the WFI mosaic global solutions. The listed val
	ues are averages taken over ail sky path years. In addition to
	eliminating zéro-point errors présent in the positions in the indi
	vidual CCD frames, the overall standard déviations with respect
	to the UCAC2 catalogue were generally smaller, as can be seen
	in

Table 3

 3 

	Eris	86.1	4.3	1.3	2.8	5.4	0.1
	Haumea	93.1	0.9	0.4	0.7	4.8	0.1
	Ixion	93.8	0.3	1.3	2.6	2.0	0.0
	Makemake	90.3	1.9	0.6	1.2	5.9	0.1
	Orcus	82.3	9.1	3.2	1.9	3.5	0.0
	Quaoar	92.9	1.8	1.0	1.7	2.6	0.0
	Sedna	92.3	3.0	0.9	1.0	2.8	0.0
	Varuna	93.9	1.0	0.5	1.2	3.4	0.0
	2002 TX300	94.2	2.0	1.0	0.4	2.3	0.1
	2003 AZ84	95.6	0.3	0.2	1.1	2.7	0.1
	Ail TNOs	93.5	0.8	1.1	2.2	2.3	0.1
	Notes. Multiplicity fiag statistics for each and ail TNOs. Percentages
	refer to the total number of mosaic positions for each TNO and for

. Detailed information about the global réduction pro cedures are given in Sect. 4.3 of PI. In this final step, multiple entries coming from distinct CCD frames for the same objects are resolved (see Sect. 4.4 of P1 for details). Table 4 displays the statistics for resolved mul tiple entries. The listed values are taken over the global solutions ail TNOs together (last line). No flag (good aslrometry) means no mul tiple entries within 175 for the same object after mosaic solutions. Flag

Table 5 .

 5 Proper motion computations from 2MASS, USNO B 1.0, and ES02p2/WFI global mosaic star positions.

		UCAC2	2MASS	USNO B 1.0	No
	TNOs	p.m.	p.m.	p.m.	p.m.
		%	%	%	%
	Eris	8.4	35.5	32.9	23.2
	Haumea	6.3	24.2	41.6	27.9
	Ixion	1.7	22.3	13.5	62.5
	Makemake	6.4	25.2	38.8	29.6
	Orcus	15.6	50.3	22.1	12.0
	Quaoar	5.9	41.3	21.6	31.2
	Sedna	12.2	49.0	22.8	16.0
	Varuna	10.7	34.3	34.7	20.3
	2002 TX3(X)	18.2	48.0	24.6	09.2
	2003 AZ84	10.9	27.8	34.5	26.8

Table 6 .

 6 Star catalogs for each TNO sky path for 2008/2009-2015.

Table 7 .

 7 TNOs offsets with respect to JPL cphemcrides.

	TNO	No. stars	(Ta	(T&	R magnitude	R magnitude		Observed -JPL ephemeris	No.	JPL
			mas	mas	completeness	limit		Acr cos ô	Aô	Ea	E6	nights	ephemeris
	Eris	25 685	33	37	19.0	21.0		mas	mas	mas	mas		version
	Haumea	77 860	41	36	19.5	21.5	Eris	-020	-068	084	055	09	28
	Ixion	3452 835	41	29	19.5	21.5	Haumea	-090	-098	060	081	21	37
	Makemake	64487	42	39	19.5	21.5	Ixion	+020	+ 191	092	093	15	09
	Orcus	56 042	40	39	18.5	20.5	Makemake	-039	-165	060	058	20	40
	Quaoar	1 150 375	41	37	18.5	20.5	Orcus	-087	+001	059	046	19	13
	Sedna	27 464	33	31	19.0	21.0	Quaoar	-144	-078	098	043	22	17
	Varuna	140 041	38	39	19.0	21.0	Sedna	-043	-160	040	080	05	17
	2002 TX300	82746	33	37	18.0	20.0	Varuna	-045	+005	015	015	-	21
	2003 AZ84	278 720	33	34	19.0	21.5	2002 TX300	-084	-053	040	040	-	14
							2003 AZ84	+232	+006	003	015	-	11
	Notes. Number of catalog stars per TNO, position error for faint stars							
	with R at completeness magnitude (errors are estimated from the stan	Notes. Average "observed minus JPL ephemeris" position offsets.
	dard déviation of contributing individual CCD positions about the final	Errors corne from the standard déviation about the mean of individ
	catalog star positions), R bandpass magnitude completeness, and mag	ual nightly averages. TNO positions in the UCAC2 System corne from
	nitude limit.						a parallel astrometric program of CCD observations carried out during
							2007-2010 at a varicty of 0.6 m to 2.2 m télescopes in Chile, Brazil, and
	error estimâtes. In ail, 5 356 255 stars were catalogued in the							
	UCAC2 frame. The mean epoch is approximately 2007.75 for							
	the catalogs of TNOs beginning in 2008, and about 2009.5 for							
	the others. Magnitude completeness is about/? = 19. The magni							
	tude limit is about R = 21. Position error is smaller than 40 mas							

Table 8 .

 8 Samplc from prédiction tables for stellar occultations by large TNOs.

	Year m d h m s	RA (ICRS) Dec		C/A P/A	v	D
		h m	s	°'	"	mas °km s"1 AU
	Notes. In entry order, the above samples correspond to the prédictions for 2002 TX30o (October 2009), Varuna (February 2010), Eris (November
	2010), 2003 AZ84 (January 2011), Makemake (April 2011), and Quaoar (May 2011), respectivcly. Prédiction tables list event date and instant

  3), estimated star-catalog position errors (Ea, Es), and proper motions (pitt,Es)-Positive/negative v means, respectivcly, prograde/retrograde velocities, that is TNO's geocentric right ascension is increasing/decreasing, respectivcly. The complété table sets of 2008/2009-2015 prédictions for Eris, Haumea, Ixion, Makemake, Orcus, Quaoar,

Table 9

 9 

	displays the total number of predicted events for
	each TNO for the period 2008/2009-2015.

Table 11 .

 11 Comparison between WFI-bascd and actual observed occul tations: central instants and C/As.

		Star	Prédiction minus Occultation
	TNO	mag (R)	Central instant		C/A
			s	mas	mas	km
	2002 TX30o	12.9	-15	-13	-09	-265
	Varuna	11.0	-20	-12	-63	-1953
	Eris	16.9	-604	-229	+39	+2708
	2003 AZ84	18.2	+0	+0	+0	+0
	Makemake	18.4	+ 196	+ 116	+40	+ 1494
	Quaoar	15.7	+ 130	+78	-44	-1351
	average (moduli)		193	90	39	1554
	average (Eris ont)		90	55	39	1266

  suggests low conductivity, porous surfaces that are unlikely to support large reliefs. We also note that other icy bodies like the Uranian satellite Titania (radius 789 km) hâve local topographie features of ±2 km at most (see the discussion in ref.29). Another large icy satellite, Triton (radius 1,353 km), has even smaller reliefs, with amplitude of less than ~0.5 km (ref.33) We thus consider that an amplitude of ±3 km is a safe maximum for local topographie features on Eris' surface. Generating circular fits to the chords obtained with solution 2, and allowing for residuals up to 3 km, we obtain values of 7?e in the range 1,163±6 km. This error bar may be combined quadratically with the formai error bar of ±6 km found above, yielding a final error bar of ±9 km for Eris' radius, in the circular limb assumption with (probably overestimated) reliefs of ±3

km.

Table 1 .

 1 The occultation was successfully recorded from seven télescopes, listed in Table1, at five sites. From the images obtained, we made photometric measurements as a fonction of time (light curves).

	we embarked

Table 11

 11 Details of the successful observations on 2011 April 23

	Site and telescope	Pixel scale	Intégration	Filter name	Dead time	Instrument name/	Immersion time (ut)	Emersion time (ut)	Longitude/Latitude/Height
		O	time (s)		(s)	detector			
	La Silla, 3.5 m NTT	0.35	0.272	r'	0.0036	ULTRACAM	1:35:44.59 ±0.07	1:36:43.51 ±0.08	289'15'58.5" E/
						channel r'			29' 15'31.8" S/2,345.4 m
	La Silla, 3.5 m NTT	0.35	0.272	g'	0.0036	ULTRACAM	1:35:44.64 ±0.04	1:36:43.66 ±0.07	289" 15' 58.5'E/
						channel g'			29°15'31.8" S/2,345.4m
	La Silla, 0.6m	0.64	5	Clear	1.435	FU-PL3041BB	1:35:46.82 ± 1.6	1:36:45.47 ± 1.6	289'15'38.2" E/
	TRAPPIST								29'15' 16.6" S/2,317.7 m
	Paranal, 8 m VLT	0.148	1.521	J	0	ISAAC	1:35:46.00 ±0.35	1:36:49.60 ±0.35	289°35'50.1" E/
									24°37' 30.3" S/2,635 m
	Armazones, 0.84 m	0.57	10	Clear	3.5	SBIG-STL6303E	1:35:46.30 ± 1.1	1:36:48.52 ±3	289°48' 13.6" EJ
									24°35'51.9" S/2,705.7 m
	San Pedro de Atacama, 1.61	5	Clear	1.048	Apogee U42	1:35:37.86 ±2.7	1:36:43.56 ±3.1	291°49' 13.0" E/
	0.5 m Harlingten								22°57' 12.2" S/2,305 m
	San Pedro de Atacama, 1.21	15	Clear	5.966	SBIG-STL11000	1:35:38.66 ±4	1:36:41.16±2	291°49' 13.0" E/
	0.4 m ASH2								22°57' 12.2" S/2,305 m
	Pico Dos Dias, 0.6 m	1.98	5	Clear	3.851	SITe SI003AB	1:33:57.27 ± 1.6	1:35:01.08 ± 2.2	314°25' 02.5" E/
	Zeiss								

  Units are milliarcseconds (mas). Note that the Paranal and Armazones chords almost overlap. The Paranal, Armazones, Pico dos Dias and San Pedro chords sampled the central part of Makemake. The star disappearance takes place on the left. The chord extremities can be fitted by two different models. The first (pictured in brown) is a circle of diameter 38.28 ± 0.22 mas (lrr level), équivalent to 1,430 ± 9 km, with areduccd yj of 1.032. The second (pictured in black) is an ellipse with a minor axis of 1,428 ±17 km and an axial ratio of 1.15 ± 0.17, with the long axis tilted by 9 ± 24°(1er level) with respect to the local celestial north. The reduced y1 of the fit is 1.027. The dashed line shows the axes of the best-fitting ellipse. As discussed in the Supplementary Information, the best shape is between the two models. Makemake was 51.5 au from Earth and 52.21 au from the Sun at the time of the occultation.

	object is rotating nearly pole-on, if the dark terrain is spread uniformly
	in longitude (a banded configuration) or by a combination of both
	conditions.
	output at 24 pm (ref. 6), which requires a warm terrain on the order of
	50 K (see Supplementary Information section 5). The two terrains and
	Makemake's low rotational variability20,21,24 can be reconciled if the

  It is at just 1.8 a.

	Regardless of whether the features discussed are significant or not, local
	atmosphères are plausible in TNOs7, especially in Makemake, given that it has a
	fraction of its surface covered with a dark terrain5,6,23 where high enough
	températures are reached and can sustain at least sublimation driven local
	atmosphères. Hence it makes sense to explore what kind of a local atmosphère
	would be consistent with the data and to see what limits we can put on such a local
	atmosphère.							
	7.1	MODELLING	OF	THE	REFRACTED	LIGHT	FROM	A	LOCAL

ATMOSPHERE

We considered a pure CH4 atmosphère localized above patches of warmer methane ice, assuming that the subsolar point température reaches 50-55 K as discussed in section 5. A sketch of the location of a possible local atmosphère is show in Fig.

S4

. Note this is an example of a localized atmosphère, and that other configurations with other orientations and shapes could be examined. The aim of this model is to show that our data are compatible with the existence of a local atmosphère, in spite of the non-detection of a global one.

  1.15 axial ratio. According to section 6.2 of a theoretical study on TNO atmosphères7, TNOs with orientation close to pole-on are less likely to develop Observatôrio do Valongo / UFRJ, Rio de Janeiro, Brazil 8Observatoire de Genève, Switzerland 9San Pedro de Atacama Celestial Explorations, San Pedro de Atacama, Cliile 10Universidade Estadual de Ponta Grossa, Ponta Grossa, Brazil nUniversidade Fédéral de Santa Catarina, Florianopolis, Brazil 12Instituto de Astronomia, Universidad Catôlica del Norte, Antofagasta, Chile 13Instituto Nacional de Pesquisas Espaciais, DAS, Sâo José dos Campos, Brazil 14Centro de Estudos Astronômicos de Minas Gérais (CEAMIG), Belo Horizonte, Brazil 15Complejo Astronomico El Leoncito k San Juan National University, Argentina 16Department of Electrical Engineering and Center of Astro-Engineering, Pontificia Uni versidad Catôlica de Chile, Av. Vicuna Mackenna 4860, Santiago, Chile 17Institut d'Astrophysique de l'Université de Liège, Belgium 18Gnosca Observatory, Switzerland 19Laboratoire Lagrange, Université de Sophia Antipolis, Observatoire de la Côte d'Azur,

	1Observatôrio Nacional, Rio de Janeiro, Brazil
	2Observatoire de Paris, LESIA, Meudon, France
	3niversité Pierre et Marie Curie, Paris, France
	4Inst,ituto de Astroffsica de Andalucfa -CSIC, Granada, Spain
	5Observatorio Astronomico Los Molinos, Uruguay
	6Observatoire de Paris, IMCCE, Paris, France
	global atmosphères. ' CNRS UMS7293
	20Université de Toulouse, UPS-OMP, IRAP, France
	21Euraster, IB cours J. Bouchard, St-Etienne, France
	22Rede de Astronomia Observacional, Brasilia, Brazil
	23Unidad de Astronomia, Universidad de Antofagasta, Antofagasta, Chile
	24Observatorio Los Algarrobos, Salto, Uruguay
	(

Table 2 .

 2 Circumstances of observations for the sites that detected the event (or were very close to it).

	Site	Instrument	Télescope	Exposurc	Ingrcss"	Egrcss"
		Pixel scale	Aperture	Dcad Time	time (UT)	time (UT)
		(arcsec)	Filter	SNR		
	Pico dos Dias	Andor/Ixon	Cari Zeiss	0.95 sec	Obs. start:	Obs. end:
	-Brasôpolis	EMCCD	0.60 m	1.05 sec	2:26:10	2:58:20
	/ BRA	1.11	Clcar	SNR=85		No occ.
	Ponta Grossa -	SBIG	RCX 400	14 sec		
	UEPG / BRA	STL-6303E	0.40 m	4.25 sec	2:41:47.2±2.7	2:42:42.1±3.1
		1.74	V	SNR=20		
	Florianôpolis	SBIG	CM-1100	10 sec		
	-UFSC/ BRA	ST-7	0.254 m	2.75 sec	2:41:28.7±3.6	2:42:31.9±3.8
		1.48	Clcar	SNR=12		
	San Pedro de	Apogée	Harlingten	3.0 sec		
	Atacama	U42	0.5 m	1.1 sec	2:42:33.1±0.13	2:43:31.4±0.15
	/ CHL	1.61	Clcar	SNR=52		
	San Pedro de	SBIG	AS H 2	9.0 sec		
	Atacama	STL11000	0.4 m	7.1 sec	2:42:32.5±1.0	2:43:30.30±4.3
	/ CHL	1.21	Clcar	SNR=22		
	Armazones	SBIG	OCA	7.0 sec		
	/ CHL	STL6303E	0.84 m	2.4 sec	2:42:41.5±1.2	2:43:26.3±1.8
		0.57	Clcar	SNR=42		
	Rivera	SBIG	LX200	4.0 sec		
	/ URY	ST-7	0.305 m	2.3 sec	2:41:58.2±0.4	2:42:32.5±0.4
		2.57	Clcar	SNR=26		
	Salto -OLASU	QSI	LX200	4.0 sec	Obs. start:	Obs. end:
	/ URY	516wsg	0.305 m	2.7 sec	2:20:29	2:54:04
		1.2	Clcar	SiNR=18		No occ.

"Error bars are given at la level

Table 3 .

 3 Paramcters for the best fits shown inFigs. 4, (i and 9. Error bars are at 1 a level. For the cratcr and mountain solutions, thèse quantities arc the apparent semi-major axis and oblatencss of the limb. For the Maclaurin solution, they arc the truc équatorial radius and oblatencss, sec text.

	Solution:	Crater	Mountain	Maclaurin
	Semi-major axis" (km)	656 ± 12	568 ± 12	5691Ô
	blatencss0	0.29 ± 0.04	0.14 ±0.05	0.0897i°;°i7!
	fc (km)	-2647 ±19	-2712 ± 16	-2710 ±4
	9c (km)	-1253 ±20	-1202 ±32	-1216 ±5
	Position angle (dcg.)	21 ±3	7.5 ± 11	148.4 ± 0.9
	Xmin	15.1	15.7	1.33
	^pdf	1.68b	1.74b	0.44b
	"			

bThe calculation of Xpjf from X^in explicited in the text.

sftios diferentes. Cada sitio obtém uma curva de luz que é modelada por uma ocultaçâo de um corpo sem atmosfera, convolvida com a difraçâo e tamanho da estrela e o tempo de integraçâo da observaçâo. Assim, obtém-se os instantes de imcio e fim do evento, o que caracteriza uma medida do perfil de Quaoar, cliamada de corda. As cordas sâo utilizadas 1Europcan Southern Obscrvatory/Max-Planck-Gelscllschaft 2http://www.lesia.obspm.fr/perso/braga-ribas/campaigns <

A adcsâo oficial ainda nâo foi sancionada! Mas tempo de telescôpio ja sâo concedidos aos brasilciros. >

Com perlodo de rotaçâo P = 6,49 h e amplitude Am = 0,07[START_REF] Thirouin | Short-term variability of a sample of 29 trans-Neptunian objects and centaurs[END_REF], 2003 AZ84 deve estar em equillbrio hidrostâtico, assumindo uma forma de equillbrio Maclaurin(Tancredi &; Favre 2008). Em sua superficie observa-se gelo de âgua e possivelmente metanol(Guilbert et al. 2009).Estimativas de tamanho para 2003 AZ84 variam de D = 686 ± 95 km(Stansberry et al. 2008) a D = 727 ± 66 km[START_REF] Mommert | TNOs are cool: A survey of the trans-Neptunian région. V. Physical characterization of 18 Plutinos using Herschel-PACS observations[END_REF]. Em 2007 uma lua de cerca de 70 km foi descoberta com imagens do Hubble Space Telescope(Brown h Suer 2007).Infelizmente novas observaçôes nâo detectaram novamente a lua. Portanto, ainda nâo hâ uma determinaçâo de ôrbita, de onde poderia se obter a massa do primârio. Dois eventos envolvendo 2003 AZ84 jâ foram observados, sendo o primeiro de apenas uma corda, e o segundo com très cordas. Eles sâo apresentados a seguir. 5.4.1 2003 AZ84 -08/01/2011 Uma ocultaçâo de uma estrela de mag. R = 18,4 por 2003 AZ84 foi prevista para o dia 08 de janeiro 2011. Por se tratar de uma estrela muito fraca, tanto as atualizaçôes

Em 03 de fevereiro de 2012, uma ocultaçâo envolvendo uma estrela mag. R = 15,2 foi objeto de uma campanha envolvendo maisde 20 telescôpios na Europa, norte da Africa, India e Indonésia. Très telescôpios detectaram o evento, dois na India (Mount Abu Observatory e IUCAA Girawali Observatory) e um em Israël (Weizmann Istitute). Dois telescôpios separados por aproximadamente 1000 km no piano do céu, obtiveram dados mas nâo detectaram o evento (Alicante e Liverpool), ajudando a limitar o tamanho do TNO. Devido ao ruido nas imagens oriundas de Mt. Abu, obtida com filtro H, duas soluçôcs sâo possfveis para o egresso. No entanto, apenas uma das soluçôes satisfaz uma elipse ao se considerar as outras cordas. No observatôrio de Girawali, as observaçôes foram feitas com 2 segundos de exposiçâo e 16,5 segundos de ciclo. Uma das imagens apresenta uma queda maior que uma magnitude provocada pela ocultaçâo. Buscou-se entâo o tamanho mâximo de uma corda durante o tempo-morto antes e depois desta imagem ocultada. Com as outras cordas, foi possivel determinar que a imagem ocultada corresponde ao egresso de 2003 AZ84 naquele sltio.Com as consideraçôes acima, ajustou-se uma elipse sobre as cordas sobre 2003 AZ84, obtendo um objeto com achatamento e = 0,37 ± 0,1 e Kequiv = 343 ± 7 km (Figura 5.5).Seu albedo no visivel também foi calculado em pv -0,147 ± 0,007. Como dito, esperase que 2003 AZ84 seja um objeto Maclaurin, entâo compara-se o achatamento obtido à curva de equilfbrio de objeto Maclaurin com periodo de rotaçâo P = 6 h, obtendo assim a densidade do objeto p -0,76lo,i7 (Braga-Ribas et al. 2012). 5.5 Makemake -23/04/2011 Com cerca de 2/3 o tamanho de Plutâo, Makemake é o terceiro maior objeto transnetuniano. Descoberto em março de 2005 com imagens do Observatôrio Monte Palomar, recebeu o nome provisôrio de 2005 FY9. Ele é um dos poucos entre os grandes objetos transnetunianos a nâo possuir um satélite, o que dificulta a determinaçâo de sua densi dade.

Roques, F., Moncuquet, M.,& Sicardy, B. 1987, Stellar occultations by small bodies -Diffraction effects, The Astronomical Journal, 93, 1549 Santos-Sanz, P.,Lellouch, E., Fornasier, S., et al. 2012, "TNOs are Cool": A survey ofthe trans-Neptunian région. IV. Size/albedo characterization of 15 scattered disk and detached objects observed with Herschel-PACS, Astronomy &: Astrophysics, 541, A92 Schaefer, B. E., Rabinowitz, D. L., & Tourtellotte, S. W. 2009, The Diverse Solar Phase Curves of Distant Icy Bodies II. The Cause of the Opposition Surges and Their Corrélations, The Astronomical Journal, 137, 129 Schaller, E. L., & Brown, M. E. 2007b, Détection of Methane on Kuiper Belt Object (50000) Quaoar, The Astrophysical Journal, Letters, 670, L49 Schaller, E. L., & Brown, M. E. 2007, Volatile Loss and Rétention on Kuiper Belt Objects,The Astrophysical Journal, Letters, 659, L61Sicardy, B., Ferri, F., Roques, F., et al. 1999, The Structure of Titan's Stratosphère from the 28 Sgr Occultation, Icarus, 142, 357Sicardy, B., Widemann, T., Lellouch, E., et al. 2003, Large changes in Pluto's atmosphère as revealed by recent stellar occultations, Nature, 424, 168Sicardy, B., Bellucci, A., Gendron, E., et al. 2006, Charon's size and an upper limit on its atmosphère from a stellar occultation Nature, 439, 52

Cependant, au vu de leurs distances (plus de 30 unités astronomiques) et de leur taille, il est actuellement impossible de résoudre leur diamètre angulaire (en général bien inférieur à 0.1 arcsec) et de détecter directement leur atmosphère.La méthode d'occultation stellaire consiste à observer le passage d'OTNs devant une étoile. Les durées précises du phénomène, observé en différents endroits, permettent en suite de retrouver la taille et la forme des objets avec des précisions kilométriques. Ainsi, des valeurs précises de l'albédo peuvent être déduites, et pour les corps dont la masse est connue via le mouvement de satellites, des valeurs précises de la masse volumique peuvent également être calculées. Par ailleurs, des atmosphères éventuelles provoquent des baisses de signal juste avant et après l'occultation, permettant des détections à des niveaux de pression de quelques nanobars.Cette thèse décrit (1) les méthodes de prédiction des occultations par une cinquantaine de OTNs, avec des précisions de l'ordre de 0.03 arcsec, (2) l'organisation de campagnes mondiales faisant intervenir des dizaines de sites professionnels et amateurs, (3) l'obser-

Ces groupes portent la signature de l'évolution du système solaire. Ainsi, il se peut qu'au cours d'une migration des planètes géantes, plusieurs OTNs aient été capturés dans la résonance 3/2 avec Neptune, tandis que d'autres étaient dispersés(Gomes 2003). La formation de Neptune a dû se produire en environ 10 millions années, ce qui signifie que les OTNs se sont formés plus rapidement que ce dernier. Cependant, la densité actuelle du disque qui compose ces objets n'est pas compatible avec cette formation rapide(McKinnon et al. 2008). Différentes théories de formation ont été suggérées, et les propriétés de ces objets apportent des informations sur les processus thermiques et chimiques qu'ont subis ces corps, ou permettent au moins de discriminer différentes théories. Par ailleurs, les distributions de taille et de masse sont également liées au processus de formation.Ces propriétés physiques sont difficiles à déterminer. Des observations photométriques, spectroscopiques et thermiques fournissent des contraintes sur leur périodes de rotation, la taille, l'albédo et leur composition. Par ailleurs, environ 4,5% des OTN doivent avoir des satellites ou être binaires, et de nombreux cas sont déjà connus. Ceci permet entre autres de déterminer la masse de ces systèmes, et leur densité, si les tailles sont connues.En utilisant la technique des occultations stellaires, qui consiste à observer un OTN lors de son passage devant une étoile, nous cherchons à déterminer les dimensions et formes de ces corps avec des précisions de l'ordre du kilomètre. Nous en déduisons les albédos et les densités avec une grande précision. Ceci a des implications directes sur la composition de surface et sur la structure interne de ces corps. Aujourd'hui, les occultations stellaires sont la technique la plus puissante pour détecter des atmosphères à des niveaux de pression de l'ordre du nanobar autour de ces corps. Ces atmosphères éventuelles (ou leur absence) ont elles-mêmes des implications pour les processus de sublimation et les propriétés thermiques de ces objets. Toutefois, de par leurs longues périodes orbitales (> 200 ans), et de leurs faibles éclats dûs à leur grande distance au Soleil (> 30 UA), leurs orbites présentent de barres d'erreur en général dix fois plus grande par rapport à ce qui est nécessaire pour le calcul des occultations. Pour pallier ce problème, nous faisons des mesures astrométriques de ces objets afin de déterminer les différence avec les positions données par ses éphémérides. Ces différences sont ensuite uilisées pour affiner les prédictions d'occultations. En outre, le lent déplacement des OTNs dans le ciel rend les occultations rares. Cela requiert des catalogues stellaires plus denses, avec des étoiles plus faibles. Cela cause un autre problème, car la précision des positions astrométriques de ces étoiles est également inférieure à celle qui est requise pour une bonne prédiction des occultations. Pour contour ner ce problème, nous avons mis sur pied un programme utilisant le télescope Max-Planck de 2.2 m et la caméra WFI (Wide Field Imager) de l'ESO (European Southern Observatory). Nous avons ainsi observé les bandes stellaires balayées dix OTNs parmi les plus gros entre de 2008 à 2015. En utilisant des techniques astrométriques appropriées, nous avons créé des catalogues jusqu'à la magnitude R=18, avec une précision de 40 millise

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Acknowledgements. M.A., J.I.B.C., and R.V.M. acknowledge CNPq grants 482080/2009-4, 306028/2005-0, 478318/2007-3, 151392/2005-6 and 304124/2007-9. M.A. and J.I.B.C. thank FAPERJ for grants E-26/170.686/2004 and E-26/110.177/2009. F.B.R thanks the (inancial support of CDFB/CAPES.

The authors acknowledge J. Giorgini (JPL) for his help in the use of NAÏF tools.

The authors are grateful to F. Colas and F. Vachier for the kind access to unpublished TNO observations and/or positions from a number of télescopes, which in part contributed to the détermination of the provisional TNO ephemeris offsets used in this work. Ackiiowledgements. M.A., J.I.B.C., and R.V.M. acknowledge CNPq grants 482080/2009-4, 306028/2005-0, 478318/2007-3, 151392/2005-6 and 304124/2007-9. M.A. and J.l.B.C. thank FAPERJ for grants E-26/170.686/2004 and E-26/110.177/2009. F.B.R thanks the financial support of CDFB/CAPES. The authors acknowledge J. Giorgini (JPL) for his help in the use of NAÏF tools. The authors are grateful to F. Colas and F. Vachier for the kind access to unpublished TNO observations and/or positions front a numbcr of télescopés, which in part contributed to the détermination of the provisional TNO ephemeris offsets used in this work. Acknowledgements We thank W. M. Owen and S. Preston for providing astrometric updates on the occulted star just before the observations, and I. Belskaya and M. E. Brown for discussions when writing the paper. We acknowledge support from the French grant 'Beyond Neptune' and from the Institut Universitaire de France. J.L.O., AJ.C.-T., L.C. and M.T.E acknowledge funding from Spanish AYA grants and FEDER funds. TRAPPIST is a project funded by the Belgian Fund for Scientific Research (FRS-FNRS) with the participation of the Swiss National Science Foundation (SNF). J.I.B.C. acknowledges CNPq and FAPERJ grants. F.B.-R. acknowledges the support of CDFB/CAPES, Brazil. WJ.B.C., W.R. and F.P.S. thank the Brazilian Agency FAPEMIG. ©2011 Macmillan Publishers Limited. AU rights reserved F.l Suplementos Acknowledgements These results were based partially on observations made with European Southern Observatory Télescopes at the La Silla and Paranal Observatories under programme 287C-5013. J.L.O. acknowledges funding from Spanish and Andalusian grants and the European Régional Development Fund (FEDER). B.S. acknowledges support from French National Research Agency (ANR) grant 'Beyond Neptune', and from the Institut Universitaire de France. E.U.-S. acknowledges the support from the Chilean National Commission for Scientific and Technical Research

(Gemini-CONICYTfunds),and from the North Catholic University of ChileVicerectorate of Research and Technology Development (UCN-VRIDT).TRAPPISTis a project funded by the Belgian Fund for Scientific Research (FRS-FNRS) with the participation of the Swiss National Science Foundation (SNF). J.I.B.C. acknowledges grants by the Brazilian National Council for the Development of Science and Technology (CNPq), and the Foundation for Research Support of the State of Rio de Janeiro (FAPERJ). P.S.-S. acknowledges financial support by the Centre National de la Recherche Scientifique (CNRS). R.G.-H. acknowledges partial financial support by the Argentinian National Scientific and Technical Research Council (CONICET). F.B.-R. acknowledges the support of the French-Brazilian Doctoral College Coordination of Improvement of Graduated Personnel programme (CDFB/CAPES). AA.-C. acknowledges support from the Marie Curie Actions of the European Commission (FP7-COFUND). S.P.L., V.S.D. and T.R.M. acknowledge funding for ULTRACAM from the UK Science and Technology Facilities Council. R.D. acknowledges support from Spanish Ministry of Economies and Competitiveness through a Ramôn y Cajal contract. ©2012 Macmillan Publishers Limited. Ail rights reserved G.l Suplementos profile to the surface température (Fig. 10). EDITOR: PLACE FIGURE 10 HERE.

We assume an equilibrium surface température of tsurf = 42 K. This is within the température range of a slow or fast rotator body. This crude estimation is sufficient for our purpose. Fig. 10 summarizes the general structure of the adopted température profile for a CH4 atmosphère.

-24-

The equilibrium surface température of an airless spherical body dépends on its

, Chercheur Titulaire/ON/Tese (Doutorado) -

 2were responsible for the observations. Ail authors were given the opportunity to review the results and comment on the manuscript.

Author Information Reprints and permissions information is available at www.nature.com/reprints.The authors déclaré no competingfinancial interests.

Readers north and south America (see www.lesia.obspm.fr/perso/bruno-sicardy/predic_occn_10/Eris_2010).

This was the unique event predicted for 2010. Actually, stellar occultations by Eris remain rare events as the dwarf planet moves far away from the galactic plane, in poor stellar fields in Cetus constellation (no event involving Eris is expected in our list before 2013).

Measurements made at the 2.2m telescope of the European Southern Observatory (see ref. 15 for the method) provide the following ICRF (loosely speaking J2000) star position as given by the UCAC2 reference frame:

f a = 01h 39min 09.94215 S = -04°2T 12.119" (1) with lcr errors of about 50 milli-arcsec (mas) in both right ascension and déclination.

Astrometric updates were conducted from October 18, 2010 onward to pin down the prédiction Twenty six stations were eventually involved in our campaign, most of them being clouded out and/or outside the shadow path (Fig. 1S). The occultation was detected from two sites in Chile, one with two télescopes separated by about 20 meters (0.5m-Harlingten and 0.4m-ASH2) at San Pedro de Atacama (San Pedro for short), and the other one at the ESO La Silla site with the new 0.6m TRAnsiting Planets and Planeteslmals Small Telescope2' (TRAPPIST). A third station in Argentina (Complejo Astronomico El Leoncito, CASLEO, 2.15-m Jorge Sahade telescope) was sufficiently close to the shadow edge (~200 km, see Fig. 2 and1S) and with a sufficient SNR to be used for potentially placing an upper limit of Eris' atmosphère. The general geometry of the event is shown in Fig. 2S. Besides the Eris events detected at San Pedro and La Silla (see below), no secondary events were detected in the light curves that we hâve analysed. Note that Dysnomia predicted position was far away from the stellar tracks that scanned Eris' surrounding (Fig. 2S). The UT time of each image was taken from the frame headers. However, for the three stations where the event was detected (Harlingten, ASH2 and TRAPPIST), only the integer part of the second is available, due to the truncated format provided by the MaximDL software used at those three stations. To retrieve the correct timing of each image, we fitted a linear model to the set of points (i,th,i), where i is the image number, and f-h,i is the corresponding time written in the show a regular sawtooth pattern confined between -0.5 and +0.5 seconds. This (1) confirms that the truncation to the integer part of the second is indeed présent, and (2) shows that the acquisition cycle was regular. To obtain the mid-exposure time of each frame, we finally added to the fitted SUPPLEMENTARY INFORMATION

The internai accuracy of those linear fits ensures that the absolute times of individual images can be retrieved to better than 0.05 s. Furthermore, the computers used at San Pedro and La Silla were set up against UT time every 15 minutes at least, using a time server and the 'Dimension 4' synchronisation software, which ensures an absolute time accuracy better than 0.05 s for ail computers. In summary, the time associated with each image has an absolute accuracy of about 0.07 s, corresponding to an accuracy of ± 2km along Eris' shadow track, which is negligible compared to the error associated with the model fitting described below.

Occultation timing

The star dis-and re-appearance times are obtained by fitting an abrupt edge shadow model to the light curves, after convolving that shadow by Fresnel diffraction, stellar diameter28 projected at Eris, finite bandwith of the CCD and finite intégration time of the instrument29. The final synthetic light curve is largely dominated by finite intégration time, which was 3 s or more for ail data sets, see 

where 4>j-0bs (resp. T>jca|) is the observed (resp. calculated) flux at point i, a-, is the 1er error bar at that point, estimated as explained earlier, and N is the total number of points considered. The la error bar on t.occ is estimated by varying tocc so that to increase x2 from the best value x^m t0 V2 . +1.

Amin 1

The best fits are shown in Fig. 3S, and the corresponding occultation times are listed in Tables IS and3S. When adjusting the star re-appearance at San Pedro/Harlingten, two equally satisfactory fits (called solution 1 and solution 2, in chronological order) are obtained. This is because it occurred near one of the 0.88 s gaps between consecutive exposures (Table IS). Consequently, the x2 residual exhibits two local minima for two values of t.occ separated by 1.2 s, corresponding to a différence of 32 km in the two chord lengths, see Fig. 3S and Table 3S.

Occultation geometry and Eris' size

The occultation times given in Although they confirm the event (Fig. 3S), they are not included in the analysis below. So, we finally obtain four extremities of the two occultation segments (or 'chords'), derived from the timings at San Pedro/Harlingten and La Silla/TRAPPIST télescopes, see the red segments in Fig. 2.

The most general limb shape considered here is an ellipse projected in the plane of the sky, with body, so that R^=\/a'b'.

Our problem has M=5 free parameters: the apparent semi-major axis a', the apparent flattening e'-{a' -b')/a', the ellipse orientation in the sky (position angle P of the semi-minor axis, counted eastward from the local celestial north), and the two coordinates of its centre /c, gc. Note that it is sufficient to consider 0 < P < 180 degrees, as other values of P yield the same solutions by symmetry.

We fit the iV=4 chord extremities by finding the minimum value Xmjn °f (the other parameters being free) so as to change x2 from Xmin t0 Xmin + 1-The value of x2 per degree of freedom for the best fit, Xpdf (see Table 3S) is given by Xpdf=Xmin/(-^-M). With iV=4 chord extremities, and M-5 free parameters, there is an infinity of possible solutions and x^f 's not defined. In order to obtain a preferred solution, we hâve to rely on independent observations and on physical arguments.

Circular fit

We first note in Fig. 2 that the two chords hâve almost the same médian lines, i.e. the lines going through the middle of the segments, and perpendicular to them, see the blue line in Fig. 2.

More precisely, the two médian lines are separated by 21 km for solution 1, and 5 km for solution 2.

Consequently, a circle can be fitted to the two chords to within our error bars, especially for solution 2.

This strongly suggests that Eris is indeed close to spherical, unless a very spécial geometry occurred, as discussed later. A circular fit has M-3 free parameters (i?E=a/=6/, fc and gc), so that N -M=1

and Xpdf=Xmin-Table 3S gives the values of Re, fc and gc corresponding to the best fits using solutions 1 and 2, respectively. Note that due to the high value of x2df 'n the case of solution 1, we do not provide error bars, since the assumption of a circular limb is not valid in this case, see main text. For solution 2, we obtain a radius i?E=l,163±6 km for the best circular fit to the chords extremities. The position of Eris' centre, (fc,gc) corresponds to the offset to apply to Eris ephemeris in order to fit the occultation chords (Fig. 2). Note that since we actually détermine Eris' position relative to the star at a given moment, this offset dépends (1) upon the adopted ephemeris (here DE405/JPL28) and ( 2) upon the adopted star position (Eq. 1). Thus, if one uses an ephemeris different from JPL28 for Eris and/or a star position different from the one mentioned here, the offset should be corrected, taking into account both the différences between the ephemerides and star positions.

Using the star position in Eq. 1 and the ephemeris DE405/JPL28, we obtain offsets of +212±50 and -56±50 mas for Eris, in right ascension and déclination, respectively, where the error bar is dominated by the uncertainty in the star position. Alternatively, one may use the time and distance of geocentric closest approach (C/A) in the plane of the sky to calculate those offsets (see Table 3S), once a star position and a reference ephemeris is given, knowing that Eris was at a position angle of 167.57 degrees relative to the star at C/A, as seen from the geocentre. and c, respectively, must satisfy34:

where G is the gravitational constant, M is the mass of the body, and coa(ip)=c/a-l -e. Note that this équation assumes that the object is incompressible (uniform density). If compressible, complications arise and the body will rotate with slightly different periods. For small flattenings e,

Eq. 4 provides the approximation u2a3/GM~4e/5, yielding a rotation period of:

taking a=l,163 km and M=(1.66±0.02)xl022 kg (ref. 12).

For e>e/~0.08, we obtain T<8.2 hours. In other words, values of i?E outside the interval 1,165±90 km require a fast rotating Eris. Rotation periods (more precisely, their upper bounds, as e>e')

corresponding to a few values of t1 are shown in Fig. 4S. They show that the required rotation period rapidly decreases as R^deviates from 1,165 km, i.e. Eris must be a faster and faster rotator.

We may consider more extreme situations. For instance in Fig. 2, we may reconcile our results with IRAM flux11 (yielding i?E=1.500 km) by considering a very elongated limb with b1 /a'-0.771 (and a'=1708 km, £'=0.229). This can be achieved with a fast rotator, where Eris has a triaxial shape (Jacobi ellipsoid) with semiaxes a>b>c, observed pole-on to avoid rotational light curve. The ratios SÜPPLEMENTARY INFORMATION km, IIq is the Sun apparent magnitude (at 1 AU), and 77e is Eris' absolute magnitude, that would be observed if the object were at 1 AU from the Sun and observed at 1 AU from Earth at exact opposition (rv=0). In the visible band, ref. -0.05 (10) Note that the error bars are dominated by the photometric uncertainties attached to 77e,v and 77e,r, not by the uncertainty on 77e-Note also that Eris' phase angle a varies between ~0.15 and 0.6 degrees as seen from Earth, and that the values of T7e.v and 77e,r given above assume ordinary opposition effect40. If Eris has a strongly peaked opposition surge below 0.15 degrees (which is not known), then we would obtain higher géométrie albedos than given in Eqs. 10. More generally, a review of TNO and Centaur photometric properties41 actually shows that the largest TNOs (Eris, Haumea, Makemake and Pluto) hâve shallower phase slope when compared with smaller TNOs, possibly associated with active (sublimating atmosphère) surfaces and/or collisional history. However, the lack of data at very small (<$.l degrees) phase angles prevents conclusions to be drawn about the existence of opposition surges for TNOs. An exception is Varuna, for which a surge has clearly been observed at a~0.06-0.1 degrees (ref. [START_REF] Belskaya | The Solar System Beyound Neptune[END_REF].

The géométrie albedos derived here for Eris can be compared to the visual géométrie albedos of Enceladus,pv~T4 and Tethys,py^l.2 (ref. 20). However, because those satellites exhibits strong opposition surge of more than 30% for a^0.2 degrees, and because Eris' photometric behavior is not known in this domain, we may instead compare their reflectivity (7/F)(a) in the same range of phase angle. For instance, Eris has 7/F~0.91 near o-=0.15 degrees in visual band10, while Enceladus has 7/F~1.2 in the same conditions42, about 30% larger than for Eris.

Surface température

We now use the radius 7?e and the géométrie albedo pv derived above, in combination with thermal fluxes at 70 and 1,200 //.m (measured by Spitzer22 and IRAM11, respectively), to constrain Eris' surface température and global photometric properties. We consider two limiting cases: (1) slow rotation (or equivalently, pole-on orientation, or equivalently, zéro thermal inertia), referred to as the 'Standard Thermal Model' (STM) and ( 2) fast rotation with equator-on geometry and Isothermal Latitude Model (ILM). The first model provides the 'warm' equilibrium situation, where only the illuminated side of Eris is heated by the Sun, while the second model yields the 'cool' situation,

where the température is constant along a given latitude (on both the lit and unlit sides). Actual models should lie between these extreme cases.

In each model, we use the formulae in Table 4S to calculate the sub-solar température Tss for the STM and the équatorial température Teq for the ILM. In that table, the angle 0 is the local solar zénith angle (STM) and 4> is the local latitude (ILM). The local température T given in the first line is introduced in the intégral of the second line in Table 4S. This yields Tss (STM) and Teq (ILM), introducing the emissivity e (not to be confused with the flattening e defined earlier) and using Planck's law B", once Eris' flux density Su at frequency u and geocentric distance A is measured (note that A=96.28 UA for the IRAM data and A=96.41 UA during the Spitzer observations).

In the calculations, we neglect Dysnomia's contribution to the total thermal flux, as its projected surface area is probably more than 25 times smaller than Eris', see above. 5S and refs. 11,22) in millijansky are compared to various models discussed in the text, assuming a radius Re-1,163 km and a standard emissivity22 c=0.9 for the dwarf planet. Note that only an upper limit is available at 24 pm (arrow). Green: expected flux from a black body with uniform brightness température Xt,=31 K. Red: expected flux using a Standard Thermal Model (STM), or slow rotator, with a sub-solar température Tss=35 K. Blue: expected flux using a Isothermal Latitude Model (ILM), or fast rotator, with an équatorial température Teq=31.7 K. Ail models agréé to within 

The third line of Table 4S provides the thermal equilibrium équations that relate Tss and Teq to Bond's albedo A=pvq, where py is the géométrie albedo in visible band (where most of the solar flux is emitted), q is the phase intégral and q is the beaming factor describing the effects of surface roughness (which enhance the thermal radiation at low phase angles), following the NEATM43 and Hybrid STM/ILM formalisms22. The quantity Tq is the equilibrium sub-solar température of a perfectly diffusing and completely absorbing surface with unit emissivity at Eris distance, To=[Fq(t)/ctsb]1^,

where T©(r) is the solar flux integrated over ail wavelengths at distance r from the Sun, and (7sb=5.670x10"8 W m~2 K'4 is the Stefan-Boltzmann constant. Using F©(1 AU)=1367 W m"2, and r=96.9 AU, we obtain 7o=40.0 K. Solutions in terms of Tss and Teq are found in Table 5S.

Finally, the fourth line of In the slow rotation model, the 70 pm flux is fit with a sub-solar température Tss=33.5±1.4 K, while the 1,200 pm flux gives Tss=46.8±9.3 K. A satisfactory compromise, matching both the Spitzer and IRAM measurements close to their respective lcr error bars, is achieved for Tss~35 K (Fig. 5S).

In the fast rotation model, the 70 pm flux implies Teq=31.7±1.3 K, that satisfactorily fits the 70 and 1,200 pm data (Fig. 5S).

Finally, our occultation measurement provides the géométrie albedo pv=0.96lo o4-Adopting a standard value £=0.9 (ref. 22), the only two free model parameters are p and q in the third line of Table 4S. Using the STM, i.e. Tss~35 K and a plausible range from 77=1 (no roughness) to 0.7 (large surface roughness), leads to ç=0.49-0.66, which is fully consistent with Saturn's brightest icy satellites Mimas, Enceladus, Tethys, Dione and Rhea (q = 0.51-0.63), as inferred from their géométrie and bolometric albedos20,23, see main text.

The ILM in contrast leads to extreme conditions for q. Even at the lower possible value Teq=31.7 K and with 0.9>r;>0.7, we obtain 0<ç<0.24, an implausible range as bright objects also hâve large phase intégrais24. The situation is even worse if we note that the lower bound of 0.7 for 77 is actually unrealistic in the absence of diurnal température variations44. From the fitted times at the different sites, one can generate chords in the projected plane of the sky, which represent the motion (in milli-arcsecs) of the star relative to the TNO as seen from the different sites. One can fit a shape to the extremities of the chords. We fitted elliptical and circular shapes by minimizing a X2 function defined as follows:

\'(fi.obs fç) . (Sj.nbs Sç)~tr\\ r"r «V*

where /) and g/ are the positions of the star in the plane of the sky relative to Makemake as seen from each observing site at ingress and egress times, a and b are the semiaxes of the ellipse (in a circular fit both axes are equal). In this case our are the errors of the extremities, which were derived from the errors in the retrieved ingress and egress times and N is the number of extremities. Last, fc and gc are the centers of the ellipse in the f,g plane. The best fit was obtained by means of a Marquardt-Levenberg minimization routine. The retrieved a,b parameters in milliarcsecs (mas) were translated into length in km by using the known distance of Makemake from Earth, according to its ephemerides (which gave 37.35 km/mas).

The velocity of Makemake with respect to the star was 22.12 km/s. The 1a uncertainties in the parameters were obtained from the Marquardt-Levenberg fitting routine.

GEOMETRIC ALBEDO COMPUTATIONS.

Once the apparent area (A) of Makemake was obtained from the limb fit to the oc cultation chords, we used the expression that relates the géométrie albedo (pv), absolute magnitude (Hobj), magnitude of the sun (Hsun) and the projected area of a solar System body: lO0-4(H",-Hobj)

to dérivé pv. We should take into account that the area A in the above équation is expressed in square astronomical units (AU), with 1 AU = 149598000 km. We used an absolute magnitude Hv = 0.091 ±0.015 from ref. atmosphère with some CH4 to generate a température gradient. The green line is the same as the red line but for a higher surface température. Note that most of the constraints corne from the NTT data since the VLT data hâve a lower time resolution.

A ray-tracing program provides the expected ingress and egress occultation light curves caused by such an atmosphère, keeping the surface pressure PSUrf as a free parameter. We hâve adjusted this mode! to the NTT data points (g1 and r' channel), as well as to the VLT observations (J-band). Only the NTT and VLT data hâve been used because they présent the highest signal to noise and hâve good enough time resolution. A standard x2 analysis provides the goodness of the fits.

We used N = 1984 data points in total, obtaining a minimum x2 value of ^iin= 2218 for Psurf = 0 nbar, which corresponds to a value of x2Pd/= 1-'l Per degree of freedom (red curve in Fig. S2). Thus, the data are consistent with a complété absence of atmosphère, at least at the points of the limb probed by the occultation.

The increase of x2 to + 1 actually provides a 1er upper limit of about 8 nbar for From the initial inspection of the light curves we realized that there were apparently some signais at the bottom of a few light curves, which might be indicative of refracted light by a local atmosphère somewhere on the limb of Makemake.

Because the features were seen in different sites whose chords were nearly central to Makemake, they did not seem a mere coincidence and required doser scrutiny.

We inspected the residuals of a square well fit to the occultation light curve of the S. Pedro de Atacama 0.5m telescope. There is a brightness increase near 5,760s

UT, at about the middle of the occultation, when a central flash would be expected.

The significance level of this potential spike is 3.1o, but because the statistics are usually not Gaussian we do not consider this a firm détection.

In Fig. S3 we show the residuals of a square-well fit to the occultation light curve from Pico dos Dias. In this case there is a clear brightening event that also takes place at the middle of the occultation phase and is well above the noise, at a safe 4.8a significance level. But contrary to the S. Pedro de Atacama 0.5m telescope data, the event takes place in a single image. Visual inspection of the image revealed a bright single pixel slightly away from the core of the point spread function, which raises the suspicion that this might be a cosmic ray hit or a single pixel event of some sort in the detector. If the flux from this pixel is removed from the total flux computation, the light curve spike disappears. Nevertheless we computed the likelihood of such an event by searching for similar single pixel events of the required brightness in the image sequence. It turns out that there were only 24 events with a similar flux level (within 20%) or higher than the alleged single pixel event, in 200 images. We computed the likelihood that such an event took place within a 4x4-pixel box (which would contain Makemake) at the précisé image in the middle of the occultation. Given that the images were 50x72 pixels in size, the probability would be only 0.05%. There are alternative explanations to a cosmic ray hit: Because there was image shift of a few pixels in previous images (due to wind, telescope jitter or seeing), this could hâve also occurred during the acquisition of the image. Hence a very brief central flash while the telescope was shaking might hâve produced a point source displaced from the core of the point spread function and no smearing of the stars. This seems feasible because the pixel scale was very coarse (1.98 arcsec/pixel), but this would require a very brief and intense central flash (much shorter than the intégration time). The seeing in the 5-s images was typically 2 to 2.5 arcsecs (full width at half maximum). to generate fluxes in the shadow (Fig. S5a). We adopt a température profile starting from 50 K at the surface with a gradient of 3 K/km, ramping up to 100 K.

The main resuit is that a Psurf = 5 pbar CH4 atmosphère (corresponding to an ice température of 51 K) is able to cause some bumps in the light curves at Paranal, '33,34,35,36,37'38,39,40 . Two density estimâtes for Makemake from this paper are shown in red and yellow. The red spot value was obtained by assuming that Makemake is depleted in N2 ice and using the model in ref.

8 to détermine the density below which N2 would not be retained (using the size determined from the circular fit to the occultation chords). The yellow dot is the same as the red one but using the size derived from the best ellipse fit in Fig. 2 of the main paper, which required an axial ratio of 1.15 and implied a larger body. In reality, the dots indicate upper limits because lower densities also resuit in no N2 rétention. On the other hand, because methane is retained in Makemake (it is clearly seen in its spectra), the densities must be higher than 1.4 and 1.2 g/cm3 for the red and yellow dots (using again the model in ref.

8, but this time for methane). This is marked by the arrows. The yellow spot is far from the observed trend and appears less likely. The dashed line is a linear fit to the data, and the continuous line is an exponential fit.

A Maclaurin spheroid of 1.7 g/cm3 density, spinning at 7.77 hours, would hâve a true (not projected) axial ratio of 1.17 according to the figures of equilibrium formalism, but when viewed with an aspect angle of 35 degrees (a reasonable angle according to the above discussions) the object would give a projected axial ratio of 1.05. A maximum aspect angle of 45 degrees (corresponding to an orientation exactly in the middle between pole-on and equator-on) would give rise to a projected axial ratio of slightly less than 1.08.

Also, the model of a possible central flash in Pico dos Dias, described in section 7, requires a limb less elongated than 1.06 (in axial ratio) including the atmosphère. Then, if the flash is real, the object would hâve to be even less elongated than 1.06 in projected axial ratio.

SUPPLEMENTARY INFORMATION

In summary, there are six different indications to favor the solution between the circular model and the nominal ellipse fit: The degeneracy in the direction perpendicular to the chords, the low rotational variability, the fact that for a plausible aspect angle and a plausible density the projected axial ratio would be close to 1.05 (for a body rotating at 7.77h), the lack of a global atmosphère, the comparison with Spitzer+Herschel data, and the possible central flash. These indications point to a projected shape between the circular model and the best ellipse fit, although doser to the circle. The best compromise is for a projected axial ratio of 1.05 ± 0.03. For ail the above, the preferred solution from the occultation of Makemake is an ellipse with minor axis 1,430±9 km and major axis 1,502±45 km.

Concerning the preferred density, even though the volatile rétention arguments point to a density of 1.7 g/cm3, the models of volatile loss and rétention are too simplistic to account for ail the processes, and therefore this resuit is only approximate (but consistent with the value derived from the density vs size trend shown in Fig. S7). A possible indication of the accuracy of this value may be obtained from the dispersion in Fig. S7. Because the dispersion is on the order of 0. oblateness of e = 0.087lo;oi?5> an équatorial radius of 569117 km and a density of 1.99 ± 0.46 g cm-3. The orientation of our preferred solution in the plane of the sky implies that Quaoar's satellite Weywot cannot hâve an équatorial orbit.

Finally, we detect no global atmosphère around Quaoar, considering a pressure upper limit of about 20 nbar for a pure methane atmosphère.

Subject headings: KBO, Maclaurin Object, Quaoar, Stellar occultation, TNO The transneptunian object ( 50000) Quaoar (also known as 2002 LM60) was discovered by Brown & Trujillo (2004) in June 2002 . Its diameter was estimated to be 1260 ± 190 km from direct Hubble Space télescopes (HST) imagining (Brown & Trujillo 2004). The Spitzer and new HST yielded a smaller diameter of about 850-890 km (Stansberry et al. 2008;Fraser & Brown 2010). With a semi-major axis of 43.51 AU, an orbital period of 287 years, an orbital eccentricity of 0.035 and inclination of 7.98 deg, it is a classical object. Quaoar lias a small moon, Weywot, discovered in 2007 (Brown & Suer 2007). Its orbital motion provides Quaoar's mass (Fraser & Brown 2010;Vachier et al. 2012), with the most recent value of M = (1.40 ± 0.21) x 1021 kg derived by Fraser et al. (2013).

Quaoar lias retained water ice in crystalline and amorphous phases, and volatiles such as methane on its surface (Jewitt & Luu 2004;Schaller & Brown 2007b). More tentative détections of ethane and molecular nitrogen are also reported (Guilbert et al. 2009;Dalle Ore et al. 2009). As such, and due to its size and mass, Quaoar might retain a tenuous atmosphère.

The occultation fits consist in minimizing the x2 value for each light curve:

where $i)0bs and TijCai are the observed and calculated fluxes at point z, respectively, and (j; is the 1er error bar of point i. The value of cq is assumed to be constant for ail points, and was derived from the standard déviation of the full stellar flux before and after the occultation. The summation is performed over the N data points around the expected ingress or egress time. The free parameter to adjust is the ingress (disappearance) or egress (re-appearance) time Locc, which provides the minimum value of x2, noted Xmin-The 1er error bar on tocc is estimated by varying tocc so as to increase x2 from Xmin to Xmin + T

The best fits to the occultation light curves are shown in Fig. 2 and the derived occultation times are listed in Table 2. The error bars are not necessarily symmetric due to the presence of gaps (caused by readout time intervals) between images. No information is gathered during those gaps, causing local plateaux in the x2 function, and hence asymmetric error bars.

Limb fitting

The occultation times provide the chords shown in Fig. 2. The most general shape for Quaoar's limb that we considered here is an ellipse characterized by five adjustable parameters: the coordinates of the body centre, relative to the star in the plane of the sky (/c 9c)] the apparent semi-major axis a'; the apparent oblateness e' = (a' -b')/a' (where b'

is the apparent semi-minor axis) and the position angle P of the semi-minor axis b'. The quantities J'c and gc, expressed in kilometers, are positive towards the local celestial east and north, respectively. They are calculated using the JPL#21 Quaoar's ephemeris (Giorgini et al. 1997) at each site, and the star position given in Eq. 1. The position angle P is counted positively from the direction of local celestial north to celestial east.

EDITOR: PLACE FIGURE 2 HERE.

The six positive observations provide N = 12 chord extremities. Among the six chords, two were obtained at San Pedro de Atacama, with the Harlingten and ASH2 télescopes.

Although consistent with the Harlingten results, the ASH2 timing has significantly higher error bars (Table 2), so that we kept only N = 10 independent points along the limb, and M = 5 free parameters to be adjusted. The elliptical fit to the N = 10 chord extremities is obtained by minimizing the y2 function:

where (fi^bs> 9i,obs) are the coordinates of the chord extremities, and crijr are the corresponding radial uncertainties. The latter are obtained by multiplying the la uncertainties on the occultation times by the star velocity perpendicular to the local Quaoar's limb. The value of x2 per degree of freedom (or unbiased x2) is given by y2df = Xmm/{N ~M).

As discussed later, in the hypothesis of hydrostatic equilibrium and in the small angular momentum régime, the apparent oblateness e' is the resuit of the projection of an oblate Maclaurin spheroid with semi axes a = b > c, where a and c are the true équatorial and polar radii, respectively1. The true oblateness e = 1 -c/a is then related to the apparent

^ote that in this case, the apparent semi-major axis of the limb, a', is equal to the \ équatorial radius a of the spheroid.

where £ is the angle between the polar c-axis and the line of sight. We will call this quantity the "polar aspect angle", with £ = 0 (resp. £ -90 deg) corresponding to the pole-on (resp.

equator-on) geometry.

To explore the parameter space, we hâve fitted the chord extremities with ellipses, imposing prescribed values of the position angle P and oblateness e'. We hâve then varied P between 0 and 180 degrees by steps of 3 degrees, and e1 from 0 to 0.5 by steps of 0.004.

For each pair of prescribed values (F, e4 * * 7), we hâve adjusted and stored the parameters a7, /c, gc and the resulting y2. This allowed us to construct 2D colored maps (see Figs. 4 and6),

where the y2 value is shown as a function of oblateness o! and apparent équivalent radius (instead of a'). This quantity is defined by Fcquiv = \Ja!b! -a!\J 1 -e', and is the radius of the disk that has the same area as that enclosed by the apparent limb. The la error bars on Rcquiv and e are obtained by varying y2 from its minimum value Xmin Xmin + 1 •

We note that the colored points are confinod inside a V-shaped boundary. This is because the longest occultation cliords impose that a' be larger than some value L, wliich is close to the lengths of those longest chords. Also, those chords are roughly aligned with the longer axis of the fitted ellipse (Fig. 4), which imposes that b' be smaller than ~L. Thus, we must hâve LyJ 1 -e1 < Rcqniv < L/y/1 -e', corresponding to the V-shaped domain seen in Figs. 4 and6. The value of L dépends on the particular iriodel considered to fit the chords, and varies in a range from about 500 to 680 km, see next Section.

Quaoar's shape

We expect thousand-kilometer class objects to be in hydrostatic equilibrium, with relaxed to Maclaurin spheroidal or Jacobi ellipsoidal shapes. The critical diameter to reach equilibrium can be determined under some assumptions, as shown in Tancredi & Favre i Quaoar's équivalent radius, Rcqu\v = 555 ± 2.5 km, common to ail the ellipses. The next step is to deduce Quaoar's actual shape in space from its apparent elliptical limb.

Quaoar lias a small amplitude rotational light curvc with Am = 0.133 ± 0.028. Its rotation period is 8.8394 ± 0.0002 hours, if it lias a single peaked rotational light curve (Ortiz et al. 2003). The period can be 17.6788 ± 0.0004 hours in case of a double peaked light curve and can be caused by albedo features, shape effects, or a combination of the two.

Quaoar's large size and density suggest a body in hydrostatic equilibrium, i.e. a Maclaurin spheroid. We note that a Jacobi-type equilibrium (Lacerda & Jewitt 2007) is not expected here because a 17.7-hour rotation period would require a very low density smaller then 0.6 g cm-3 and an oblateness larger than 0.42.

Consequently, we will assume from now on that Quaoar is a Maclaurin spheroid of équatorial radius a, polar radius c, true oblateness e = 1 -c/a, observed with a polar aspect angle £ and with a rotation period of 8.8394 hours (so that the amplitude of its rotational light curve is entirely due to albedo features). Then, it can be shown from Eq. 4 that

Quaoar's apparent surface area is S = 7ra2y/cos2(Q 4-(c/a)2 sin2(Ç), where S = 7rR2qu[v.

Moreover, Quaoar's density is given by p = 3M/(47ra2c). Elementary calculations show that:

tan2(/?) = 1 -e\ with sgn(/3) = -sgn(a).

where the apparent oblateness e' is given as a function of the true oblateness e by Eq. 4.

The condition 0 < (1 -e)2 < 1 imposes Rcquiv < a < Rcquiv/y/\ cos(C)|. By varying a inside this allowed interval, the first two équations above yield an infinity of solutions (p, e) that fit equally well the occulting chord extremities (after applying the necessary time shift to each chord). These (p, e) solutions are shown as blue fines in Fig. 8. Tfiey define a V-shape domain inside wfiich tfie actual solution must résidé, with tfie limit corresponding to tire equator-on (£ = 90 deg) and pole-on (Ç = 0 deg) geometries.

Tfie blue fines are parametrized by R0quiv = 555 ±2.5 km (obtained from our observations) and M = 1.40 ± 0.21 x 1021 kg, derived from Weywot's orbit (Vachier et al. 2012;Fraser et al. 2013). We note that tfie errors on tfie position of tfie V-sfiaped domain (the dasfied blue fines) are dominated by tfie uncertainties on A/, not on Rcquiv.

The final condition tfiat we impose to tfie solutions is that tfiey satisfy the Maclaurin

wfiere oj is the spin frequency, G is tfie constant of gravitation and cos(xjj) = 1 -e (Plummer 1919;Sicardy et al. 2011). Considering the single peak case (i.e. a rotation period of 8.8394 fiours as discussed above), the équation above provides the red curve plotted in Fig. 8.

Tfie intersection between the Maclaurin equilibrium curve and the V-shaped domain finally provides Quaoar's true oblateness e -0.0897 ± 0.006 and density p = 1.99 ± 0.14 g cm-3, where the error bars corne from the indétermination of (. Also considering tfie uncertainty in Quaoar's mass détermination (see above), we obtain larger

error bars e = 0.0897lo;cn75 an<5 P -1-99 ± 0.46 g cm-3.

EDITOR: PLACE FIGURE 8 HERE.

From the oblateness, density and équivalent radius given above, tfie second of Eqs. 5 provides Quaoar's équatorial radius a -569li7 km. The tfiird équation indicates that (3 must be négative, since a is positive, so (3 --44.1 ± 0.9 deg (see the position angle of the centers of the chords in Fig. 9). Since the chords are themselves inclined bv 12. This satisfactory limb fitting cornes at a price, that of applying time shifts of several seconds to the chords, see the values given in Fig. 9. The problem is particularly acute for the Rivera station, where we hâve to apply a time shift of -9.1 s. However, we note that this is the only site where the computer clock was not systematically corrected by an internet server, due to the lack of internet access at that station. It was set up with respect to a GPS display, half an hour before the event. Such a procedure usually provides an accuracy of a fraction of a second. Its impossible to accesses the errors after the fact, but, although improbable, we cannot exclude a time drift of the clock during the observation or even that the GPS display may not reflect the actual UTC time, due to unknown acquisition problem.

The docks at the other stations were ail connected to an internet time server via NTP.

We note that the time shifts applied to the Florianopolis and Ponta Grossa chords are within the 1er level obtained for the fitted occultation times, see Section 3.2, so the shift could be explained by the errors in the occultation timing. Concerning Armazones, the shift is at about the 2.5a level of the occultation times, a discrepancy which is more difficult to explain. Finally, we assumed that the San Pedro de Atacama timing is the most reliable because it is confirmed by two experiments. But, although unlikely, an identical systematic error for the two experiments cannot be discarded.

Summary of Quaoar's physical properties

We now briefly discuss the various physical parameters derived from our analysis, assuming a Maclaurin shape for Quaoar. Table 3 lists the various parameters of the best lits using each of the models described above, as well as the quality parameter Xpdf-Table 4 provides sonie physical parameters derived from tliose hts.

EDITOR: PLACE TABLE 3 HERE.

If Quaoar is a Maclaurin spheroid, its density is p -1.99 ± 0.14 (±0.46) g cm"3,

where the first error bar measurcs our internai accuracy, while the error bar in parenthèses takes into account the uncertainty on Quaoar's mass (Fraser et al. 2013). The crater and mountain solutions would imply a range of densities of 1.40 -2.46 g cm-3.

Similarly, the true oblateness of the Maclaurin solution is e = 0.0897 ± 0.006 (îo.orrs)» with a possible range of 0.0722 -0.33, taking account the crater and mountain solutions.

The Maclaurin solution has a well defined équatorial radius of a = 569t37 km and a well defined équivalent radius RCquiv = 555 ± 2.5 km at the date of occultation. We can now i dérivé Quaoar's géométrie albedo p through: P = (AUyJR^Ÿ • (7) where AUkm = 1.49598 x 106 * 8 km, //0 is the Sun magnitude at 1 AU, and Ii is Quaoar's absolute magnitude. We find at least two measurements of its absolute magnitude Rabinowitz et al. (2007) 

Atmosphère

Spectroscopic studies show that the largest TNOs retained water ice and volatiles on their surfaces. Water ice (H20), mostly in crystalline phase, was detected on many objects, as well as methane (CH4), ammonia (NH3), molecular nitrogen (N2) and even methanol (CH3OH) and ethane (C2H6) for a few of them (Barucci et al. 2005;Guilbert et al. 2009). In contrast, the smaller objects only reveal water ice bands or mere fiat spectra.

A simplistic approach is thus to split the TNOs in two groups: large volatile-rich bodies and small volatile-poor objects (Schaller & Brown 2007a).

Consequently, large TNOs may undergo out-gassing from their interiors, or suffer ongoing volatile loss at their surface, leading to the presence of a thin global or local atmosphère. Seasonal sublimation and recondensation can even be expected, causing surface changes, as seems to be the case for Eris (Sicardy et al. 2011).

Only water ice in crystalline and amorphous phase and methane ice are firmly detected on Quaoar's surface (Jewitt & Luu 2004;Schaller & Brown 2007b). More tentative détections were made on ethane and molecular nitrogen (Guilbert et al. 2009;Dalle Ore et al. 2009).

To investigate the stability of volatile on TNOs, Schaller &Brown (2007a) andBrown et al. (2011) developed a simple model of atmospheric loss through Jeans escape, including N2, CH4, and CO ices. In the case of Quaoar, given the bulk properties of the object (diameter, mean température), they found that N2 may be too light to hâve been retained over the âge of the Solar System, wliile CH4 would be near its limit of stability. These results are consistent with the observed surface composition where only CH4 is firmly detected, but maybe not N2 or CO.

However, the Jeans mechanism gives only a lower limit to the escape rates. Since crystalline water is observed in many TNOs including Quaoar, this implies that their surface température was larger than ~100 K at some recent epoch (i.e. within the last 107 years (Cooper et al. 2003;Jewitt & Luu 2004). This motivated Levi h Podolak (2009) to investigate the transition between the Jeans and hydrodynamic escapes. For a température of ~100 K, they found that N2 is firmly in the Jeans régime, while hydrodynamic escape could be significant for CH4. However, in a global hydrodynamic régime, the escape rates of the various species dépends on their atmospheric mixing ratios, rather than on their individual vapour pressures. Therefore, a partially hydrodynamic régime could potentially enhance the CH4 escape rates. Nonetheless, the established presence of CH4 ice on Quaoar' surface indicates that these effects are not strong enough to hâve entirely depleted Quaoar's surface of methane ice.

Quaoar's mass (Fraser et al. 2013;Vachier et al. 2012) and our preferred équivalent radius of 555 km imply a surface gravity of about 0.4 m s~2. This is intermediate between the values of 0.64 and 0.30 m s-2 obtained for Pluto and Charon, respectively, based on their current mass déterminations (Tholen et al. 2008). Also, the mean température of Quaoar is similar to that of Pluto, due to their respective albedo and heliocentric distances. Thus, a Pluto-like atmosphère for Quaoar is plausible, considering both its surface composition, température and gravity. In this context, ground-based stellar occultations are currently the most powerful technique to detect (or put upper limits on) very thin atmosphères of a few nanobars, see e.g. Sicardy et al. (2011) for the case of Eris.

Quaoar's atmospheric thermal structure is unknown, but we would expect it to be mainly composed of CH4 (and/or N2, if présent at the surface). The upper atmosphère température can be obtained from the methane "thermostat-model" (Yelle & Lunine 1989),

where the near-IR heating is balanced through the émission in the 7.7 gm methane band.

With an incoming solar flux at Quaoar twice weaker than at Pluto, this model predicts a

Quaoar's upper atmosphère température of roughly 102 K (vs 106 K for Pluto) at about 10 km above the surface, and a rapid drop below that altitude, connecting the thermal géométrie albedo pv, phase intégral q, emissivity of e, beaming factor 77 and heliocentric distance r, see for instance the équations in Table 4S of Sicardy et al. (2011). Using Pv = 0.120 ± 0.018, e = 0.9, q = 0.52, 77 = 1.7 (Fornasier et al. 2013) and, r = 42.35 AU, we obtain Tss = 54 K for the equilibrium sub-solar température in the case of a slow rotator, and Tcq = 40 K for the equilibrium équatorial température of a fast rotator.

EDITOR: PLACE FIGURE 11 HERE.

With that prescribed atmosphère, we hâve modelled synthetic occultation liglit curves, using a ray tracing code as described in (Sicardy et al. 2006b;Widemann et al. 2009). We compare these synthetic curves with our best light curve obtained with the San Pedro de Atacama 0.5 ni telescope.

Because there are various possible solutions for Quaoar's shape, we do not know the exact distance of the sub-occultation points to the body centre. To overcome this problem, we hâve generated several synthetic light curves for a range of centre-to-limb distances.

For eacli distance, we obtain a x2 curve that provides its own détection threshold. The envelope of the various x2 curves yields an upper limit of 21 nbar (1er), or 56 nbar (3<r)

for a méthane atmosphère, see Fig. 11. The profile expcctcd from a 21 nbar atmosphère is shown in Fig. 12. Fitting only the ingress data points (green dots) in Fig. 12 does provide a détection of an atmosphère with a pressure of 138132 nbar (1er), but at the 3<j level this remains an upper limit of 221 nbar. In conclusion, we can not exclude the presence of a local atmosphère.

EDITOR: PLACE FIGURE 12 HERE.

For completeness, we hâve also studied the isothermal model, even though it is

The observations were performed with video and CCD caméras. Drift scan mode was used with the TAROT instrument, with a 6 sec/pixel résolution, resulting in a low accuracy chord length. At Tourrette-Levens a sequence of CCD images was acquired, but during ingress a tracking problem occurred. Moreover, the measured chord is too short compared to the other two along the same track, possibly pointing toward a timing problem. As the stations TAROT, Tourrette-Levens and Valensole observed the same chord, we use the occultation times given by Valensole observation, as it présents the best accuracy among them (Table 6).

EDITOR: PLACE TABLE 6 HERE.

With two chords at hand, wc can pcrform a circular fit, with radius R and centre (fc,gc) as free parameters. This provides R = 685^55 km (1er), and thus a lower limit of Rmin = 530 km. The observed chords are compatible with the Maclaurin solution obtained from the 4 May 2011 event, but does not bring further constraints (Figure 13).

This event can be used, however, to improve Quaoar's ephemeris by giving its time of closest approach to the star, its distance to the star at that moment and its offset relative to the JPL#21/DE405 ephemeris, see Table 5.

EDITOR: PLACE FIGURE 13 HERE. Quaoar's géométrie albedo, 0.13 ± 0.01, which also agréés with our value at the 1.6cr level.

We put an upper limit of about 20 nbar (la) to the presence of a global methane The bigger red dot is the closest geocentric approach, at 01:38:37 UT. The s dots are the sites where the occultation was detected. The blue dots are the sites that had obtained data but did not detect the event,

and the wliite ones are the sites that were clouded out (Table 1). ^Assuming a Quaoar radius of R = 535 km, the middle value of the global range of RcqUiv in Table 4 "The label "N" (resp. "S") means that the shadow center passed north (resp. south) of geocenter.

**Quaoar's offset relative to the JPL#21/DF405 ephemeris dThe observation started with 10,24 sec of intégration time, but about one minute before the beginning of the event it was set to 5.12 sec.