text:
# 10.1 The ellipse

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In this section, you will:
• Write equations of ellipses in standard form.
• Graph ellipses centered at the origin.
• Graph ellipses not centered at the origin.
• Solve applied problems involving ellipses.

Can you imagine standing at one end of a large room and still being able to hear a whisper from a person standing at the other end? The National Statuary Hall in Washington, D.C., shown in [link] , is such a room. Architect of the Capitol. http://www.aoc.gov. Accessed April 15, 2014. It is an oval-shaped room called a whispering chamber because the shape makes it possible for sound to travel along the walls. In this section, we will investigate the shape of this room and its real-world applications, including how far apart two people in Statuary Hall can stand and still hear each other whisper.

## Writing equations of ellipses in standard form

A conic section, or conic , is a shape resulting from intersecting a right circular cone with a plane. The angle at which the plane intersects the cone determines the shape, as shown in [link] .

Conic sections can also be described by a set of points in the coordinate plane. Later in this chapter, we will see that the graph of any quadratic equation in two variables is a conic section. The signs of the equations and the coefficients of the variable terms determine the shape. This section focuses on the four variations of the standard form of the equation for the ellipse. An ellipse    is the set of all points $\text{\hspace{0.17em}}\left(x,y\right)\text{\hspace{0.17em}}$ in a plane such that the sum of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci    ).

We can draw an ellipse using a piece of cardboard, two thumbtacks, a pencil, and string. Place the thumbtacks in the cardboard to form the foci of the ellipse. Cut a piece of string longer than the distance between the two thumbtacks (the length of the string represents the constant in the definition). Tack each end of the string to the cardboard, and trace a curve with a pencil held taut against the string. The result is an ellipse. See [link] .

Every ellipse has two axes of symmetry. The longer axis is called the major axis    , and the shorter axis is called the minor axis    . Each endpoint of the major axis is the vertex    of the ellipse (plural: vertices ), and each endpoint of the minor axis is a co-vertex of the ellipse. The center of an ellipse    is the midpoint of both the major and minor axes. The axes are perpendicular at the center. The foci always lie on the major axis, and the sum of the distances from the foci to any point on the ellipse (the constant sum) is greater than the distance between the foci. See [link] .

In this section, we restrict ellipses to those that are positioned vertically or horizontally in the coordinate plane. That is, the axes will either lie on or be parallel to the x - and y -axes. Later in the chapter, we will see ellipses that are rotated in the coordinate plane.

To work with horizontal and vertical ellipses in the coordinate plane, we consider two cases: those that are centered at the origin and those that are centered at a point other than the origin. First we will learn to derive the equations of ellipses, and then we will learn how to write the equations of ellipses in standard form. Later we will use what we learn to draw the graphs.

how to prroved cos⁴x-sin⁴x= cos²x-sin²x are equal
Don't think that you can.
Elliott
how do you provided cos⁴x-sin⁴x = cos²x-sin²x are equal
What are the question marks for?
Elliott
Someone should please solve it for me Add 2over ×+3 +y-4 over 5 simplify (×+a)with square root of two -×root 2 all over a multiply 1over ×-y{(×-y)(×+y)} over ×y
For the first question, I got (3y-2)/15 Second one, I got Root 2 Third one, I got 1/(y to the fourth power) I dont if it's right cause I can barely understand the question.
Is under distribute property, inverse function, algebra and addition and multiplication function; so is a combined question
Abena
find the equation of the line if m=3, and b=-2
graph the following linear equation using intercepts method. 2x+y=4
Ashley
how
Wargod
what?
John
ok, one moment
UriEl
how do I post your graph for you?
UriEl
it won't let me send an image?
UriEl
also for the first one... y=mx+b so.... y=3x-2
UriEl
y=mx+b you were already given the 'm' and 'b'. so.. y=3x-2
Tommy
Please were did you get y=mx+b from
Abena
y=mx+b is the formula of a straight line. where m = the slope & b = where the line crosses the y-axis. In this case, being that the "m" and "b", are given, all you have to do is plug them into the formula to complete the equation.
Tommy
thanks Tommy
Nimo
"7"has an open circle and "10"has a filled in circle who can I have a set builder notation
x=-b+_Гb2-(4ac) ______________ 2a
I've run into this: x = r*cos(angle1 + angle2) Which expands to: x = r(cos(angle1)*cos(angle2) - sin(angle1)*sin(angle2)) The r value confuses me here, because distributing it makes: (r*cos(angle2))(cos(angle1) - (r*sin(angle2))(sin(angle1)) How does this make sense? Why does the r distribute once
so good
abdikarin
this is an identity when 2 adding two angles within a cosine. it's called the cosine sum formula. there is also a different formula when cosine has an angle minus another angle it's called the sum and difference formulas and they are under any list of trig identities
strategies to form the general term
carlmark
consider r(a+b) = ra + rb. The a and b are the trig identity.
Mike
How can you tell what type of parent function a graph is ?
generally by how the graph looks and understanding what the base parent functions look like and perform on a graph
William
if you have a graphed line, you can have an idea by how the directions of the line turns, i.e. negative, positive, zero
William
y=x will obviously be a straight line with a zero slope
William
y=x^2 will have a parabolic line opening to positive infinity on both sides of the y axis vice versa with y=-x^2 you'll have both ends of the parabolic line pointing downward heading to negative infinity on both sides of the y axis
William
y=x will be a straight line, but it will have a slope of one. Remember, if y=1 then x=1, so for every unit you rise you move over positively one unit. To get a straight line with a slope of 0, set y=1 or any integer.
Aaron
yes, correction on my end, I meant slope of 1 instead of slope of 0
William
what is f(x)=
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
Darius
Thanks.
Thomas
Â
Thomas
It is the Â that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
how do you find the period of a sine graph
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani


text:
## Zucker, Steven Mark

Compute Distance To:
 Author ID: zucker.steven-m Published as: Zucker, Steven; Zucker, Steven M.; Zucker, S. more...less Further Spellings: Цукер Стевен External Links: MGP · Wikidata · Math-Net.Ru · IdRef
 Documents Indexed: 55 Publications since 1976, including 1 Book Biographic References: 3 Publications Co-Authors: 21 Co-Authors with 21 Joint Publications 515 Co-Co-Authors
all top 5

### Co-Authors

 34 single-authored 4 Hain, Richard M. 4 Harris, Michael Howard 2 Brylinski, Jean-Luc 2 Saito, Masa-Hiko 2 Steenbrink, Joseph H. M. 1 Ayoub, Joseph 1 Bell, Steven R. 1 Chirikjian, Gregory S. 1 Cox, David A. 1 Dupont, Johan L. 1 Elzein, Fouad 1 Huckleberry, Alan Trinler 1 Ji, Lizhen 1 Narasimhan, Raghavan 1 Okonek, Christian 1 Saito, Morihiko 1 Sajjadi, Sajdeh 1 Saper, Leslie 1 Schumacher, Georg 1 Shiffman, Bernard 1 Van de Ven, Antonius
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### Serials

 9 Inventiones Mathematicae 2 American Journal of Mathematics 2 Compositio Mathematica 2 Duke Mathematical Journal 1 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 1 Commentarii Mathematici Helvetici 1 Journal of the Mathematical Society of Japan 1 Manuscripta Mathematica 1 Mathematische Annalen 1 Mathematische Zeitschrift 1 L’Enseignement Mathématique. 2e Série 1 Mémoires de la Société Mathématique de France. Nouvelle Série 1 Vietnam Journal of Mathematics 1 The Asian Journal of Mathematics 1 Annals of Mathematics. Second Series 1 Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, Série A 1 Pure and Applied Mathematics Quarterly 1 Acta Crystallographica. Section A
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### Fields

 46 Algebraic geometry (14-XX) 26 Several complex variables and analytic spaces (32-XX) 9 Algebraic topology (55-XX) 6 Differential geometry (53-XX) 5 Number theory (11-XX) 5 Topological groups, Lie groups (22-XX) 4 Group theory and generalizations (20-XX) 4 Manifolds and cell complexes (57-XX) 4 Global analysis, analysis on manifolds (58-XX) 2 History and biography (01-XX) 1 General and overarching topics; collections (00-XX) 1 Category theory; homological algebra (18-XX) 1 Functions of a complex variable (30-XX) 1 Partial differential equations (35-XX) 1 Functional analysis (46-XX) 1 Statistical mechanics, structure of matter (82-XX)

### Citations contained in zbMATH Open

39 Publications have been cited 560 times in 428 Documents Cited by Year
Hodge theory with degenerating coefficients: $$L_2$$ cohomology in the Poincaré metric. Zbl 0446.14002
Zucker, Steven
1979
Variation of mixed Hodge structure. I. Zbl 0626.14007
Steenbrink, Joseph; Zucker, Steven
1985
L//2 cohomology of warped products and arithmetic groups. Zbl 0508.20020
Zucker, Steven
1982
Intersection numbers of sections of elliptic surfaces. Zbl 0444.14004
Cox, David A.; Zucker, Steven
1979
Unipotent variations of mixed Hodge structure. Zbl 0622.14007
Hain, Richard M.; Zucker, Steven
1987
Generalized intermediate Jacobians and the theorem on normal functions. Zbl 0329.14008
Zucker, Steven
1976
The Hodge conjecture for cubic fourfolds. Zbl 0347.14005
Zucker, Steven
1977
Locally homogeneous variations of Hodge structure. Zbl 0584.14003
Zucker, Steven
1981
Satake compactifications. Zbl 0565.22009
Zucker, Steven
1983
Variation of mixed Hodge structure. II. Zbl 0615.14003
Zucker, Steven
1985
Extendability of normal functions associated to algebraic cycles. Zbl 0545.14017
1984
An overview of recent advances in Hodge theory. Zbl 0793.14005
Brylinski, Jean-Luc; Zucker, Steven
1991
Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety. Zbl 1242.14016
Ayoub, Joseph; Zucker, Steven
2012
Regulators and characteristic classes of flat bundles. Zbl 0976.14005
Dupont, Johan; Hain, Richard; Zucker, Steven
2000
Boundary cohomology of Shimura varieties. III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory. Zbl 1020.11042
Harris, Michael; Zucker, Steven
2001
$$L_ 2$$-cohomology and intersection homology of locally symmetric varieties. II. Zbl 0624.14014
Zucker, Steven
1986
Boundary cohomology of Shimura varieties. II: Hodge theory at the boundary. Zbl 0860.11031
Harris, Michael; Zucker, Steven
1994
Remarks on a theorem of Fujita. Zbl 0503.14002
Zucker, Steven
1982
On the reductive Borel-Serre compactification: $$L^p$$-cohomology of arithmetic groups (for large $$p$$). Zbl 0998.14013
Zucker, Steven
2001
The kernel spectral sequence of vanishing cycles. Zbl 0756.32020
Saito, Morihiko; Zucker, Steven
1990
$$L^ p$$-cohomology and Satake compactifications. Zbl 0753.14017
Zucker, Steven
1991
Boundary cohomology of Shimura varieties. I: Coherent cohomology on toroidal compactifications. Zbl 0860.11030
Harris, Michael; Zucker, Steven
1994
Classification of non-rigid families of K3 surfaces and a finiteness theorem of Arakelov type. Zbl 0697.14024
Saito, Masa-Hiko; Zucker, Steven
1991
The Hodge structures on the intersection homology of varieties with isolated singularities. Zbl 0645.14003
Zucker, Steven
1987
On the reductive Borel-Serre compactification. III: Mixed Hodge structures. Zbl 1082.22008
Zucker, Steven
2004
An introduction to $$L^ 2$$-cohomology. Zbl 0742.32021
Saper, Leslie; Zucker, Steven
1991
On the boundary cohomology of locally symmetric varieties. – Appendix A: No ghosts for $$Sp_4$$. – Appendix B: The Cayley transform (semi-simple or non-arithmetic version). – Appendix C: Equal-rank groups. Zbl 0952.14035
Zucker, Steven
1997
The Cheeger-Simons invariant as a Chern class. Zbl 0790.14013
Zucker, Steven
1989
$$L^ p$$-cohomology: Banach spaces and homological methods on Riemannian manifolds. Zbl 0805.55006
Zucker, Steven
1993
Polar curves, resolution of singularities and the filtered mixed Hodge structure on the vanishing cohomology. Zbl 0644.14004
Steenbrink, Joseph; Zucker, Steven
1987
On the reductive Borel-Serre compactification. II. Excentric quotients and least common modifications. Zbl 1158.32009
Zucker, Steven
2008
A guide to unipotent variations of mixed Hodge structure. Zbl 0622.14008
Hain, Richard M.; Zucker, Steven
1987
$$L_ 2$$-cohomology and intersection homology of locally symmetric varieties. Zbl 0545.55005
Zucker, Steven
1983
Intermediate Jacobians and normal functions. Zbl 0564.14003
Zucker, Steven
1984
Degeneration of Hodge bundles. (After Steenbrink). Zbl 0574.14008
Zucker, Steven
1984
Truncations of mixed Hodge complexes. Zbl 0633.14005
Hain, Richard M.; Zucker, Steven
1987
Erratum: “Boundary cohomology of Shimura varieties. II: Hodge theory at the boundary”. Zbl 1008.11529
Harris, M.; Zucker, S.
1995
L$${}^ 2$$-cohomology of Shimura varieties. Zbl 0707.14018
Zucker, Steven
1990
Complex manifolds. 2nd corrected printing of the 1990 translation. Zbl 0901.32001
Bell, S. R.; Brylinski, J.-L.; Huckleberry, A. T.; Narasimhan, R.; Okonek, C.; Schumacher, G.; Van de Ven, A.; Zucker, S.
1998
Relative Artin motives and the reductive Borel-Serre compactification of a locally symmetric variety. Zbl 1242.14016
Ayoub, Joseph; Zucker, Steven
2012
On the reductive Borel-Serre compactification. II. Excentric quotients and least common modifications. Zbl 1158.32009
Zucker, Steven
2008
On the reductive Borel-Serre compactification. III: Mixed Hodge structures. Zbl 1082.22008
Zucker, Steven
2004
Boundary cohomology of Shimura varieties. III: Coherent cohomology on higher-rank boundary strata and applications to Hodge theory. Zbl 1020.11042
Harris, Michael; Zucker, Steven
2001
On the reductive Borel-Serre compactification: $$L^p$$-cohomology of arithmetic groups (for large $$p$$). Zbl 0998.14013
Zucker, Steven
2001
Regulators and characteristic classes of flat bundles. Zbl 0976.14005
Dupont, Johan; Hain, Richard; Zucker, Steven
2000
Complex manifolds. 2nd corrected printing of the 1990 translation. Zbl 0901.32001
Bell, S. R.; Brylinski, J.-L.; Huckleberry, A. T.; Narasimhan, R.; Okonek, C.; Schumacher, G.; Van de Ven, A.; Zucker, S.
1998
On the boundary cohomology of locally symmetric varieties. – Appendix A: No ghosts for $$Sp_4$$. – Appendix B: The Cayley transform (semi-simple or non-arithmetic version). – Appendix C: Equal-rank groups. Zbl 0952.14035
Zucker, Steven
1997
Erratum: “Boundary cohomology of Shimura varieties. II: Hodge theory at the boundary”. Zbl 1008.11529
Harris, M.; Zucker, S.
1995
Boundary cohomology of Shimura varieties. II: Hodge theory at the boundary. Zbl 0860.11031
Harris, Michael; Zucker, Steven
1994
Boundary cohomology of Shimura varieties. I: Coherent cohomology on toroidal compactifications. Zbl 0860.11030
Harris, Michael; Zucker, Steven
1994
$$L^ p$$-cohomology: Banach spaces and homological methods on Riemannian manifolds. Zbl 0805.55006
Zucker, Steven
1993
An overview of recent advances in Hodge theory. Zbl 0793.14005
Brylinski, Jean-Luc; Zucker, Steven
1991
$$L^ p$$-cohomology and Satake compactifications. Zbl 0753.14017
Zucker, Steven
1991
Classification of non-rigid families of K3 surfaces and a finiteness theorem of Arakelov type. Zbl 0697.14024
Saito, Masa-Hiko; Zucker, Steven
1991
An introduction to $$L^ 2$$-cohomology. Zbl 0742.32021
Saper, Leslie; Zucker, Steven
1991
The kernel spectral sequence of vanishing cycles. Zbl 0756.32020
Saito, Morihiko; Zucker, Steven
1990
L$${}^ 2$$-cohomology of Shimura varieties. Zbl 0707.14018
Zucker, Steven
1990
The Cheeger-Simons invariant as a Chern class. Zbl 0790.14013
Zucker, Steven
1989
Unipotent variations of mixed Hodge structure. Zbl 0622.14007
Hain, Richard M.; Zucker, Steven
1987
The Hodge structures on the intersection homology of varieties with isolated singularities. Zbl 0645.14003
Zucker, Steven
1987
Polar curves, resolution of singularities and the filtered mixed Hodge structure on the vanishing cohomology. Zbl 0644.14004
Steenbrink, Joseph; Zucker, Steven
1987
A guide to unipotent variations of mixed Hodge structure. Zbl 0622.14008
Hain, Richard M.; Zucker, Steven
1987
Truncations of mixed Hodge complexes. Zbl 0633.14005
Hain, Richard M.; Zucker, Steven
1987
$$L_ 2$$-cohomology and intersection homology of locally symmetric varieties. II. Zbl 0624.14014
Zucker, Steven
1986
Variation of mixed Hodge structure. I. Zbl 0626.14007
Steenbrink, Joseph; Zucker, Steven
1985
Variation of mixed Hodge structure. II. Zbl 0615.14003
Zucker, Steven
1985
Extendability of normal functions associated to algebraic cycles. Zbl 0545.14017
1984
Intermediate Jacobians and normal functions. Zbl 0564.14003
Zucker, Steven
1984
Degeneration of Hodge bundles. (After Steenbrink). Zbl 0574.14008
Zucker, Steven
1984
Satake compactifications. Zbl 0565.22009
Zucker, Steven
1983
$$L_ 2$$-cohomology and intersection homology of locally symmetric varieties. Zbl 0545.55005
Zucker, Steven
1983
L//2 cohomology of warped products and arithmetic groups. Zbl 0508.20020
Zucker, Steven
1982
Remarks on a theorem of Fujita. Zbl 0503.14002
Zucker, Steven
1982
Locally homogeneous variations of Hodge structure. Zbl 0584.14003
Zucker, Steven
1981
Hodge theory with degenerating coefficients: $$L_2$$ cohomology in the Poincaré metric. Zbl 0446.14002
Zucker, Steven
1979
Intersection numbers of sections of elliptic surfaces. Zbl 0444.14004
Cox, David A.; Zucker, Steven
1979
The Hodge conjecture for cubic fourfolds. Zbl 0347.14005
Zucker, Steven
1977
Generalized intermediate Jacobians and the theorem on normal functions. Zbl 0329.14008
Zucker, Steven
1976
all top 5

### Cited by 413 Authors

 15 Hain, Richard M. 14 Saito, Morihiko 11 Zucker, Steven Mark 10 Kato, Kazuya 9 Usui, Sampei 9 Voisin, Claire 8 Nakayama, Chikara 7 Sabbah, Claude 6 Arapura, Donu 6 Pearlstein, Gregory J. 6 Schnell, Christian 6 Tankeev, Sergeĭ Gennad’evich 5 Millson, John J. 5 Zuo, Kang 4 Brosnan, Patrick 4 Brylinski, Jean-Luc 4 Griffiths, Phillip Augustus 4 Harris, Michael Howard 4 Libgober, Anatoly S. 4 Shvedov, Igor A. 4 Thompson, Alan Matthew 4 Vaish, Vaibhav 3 Bergeron, Nicolas 3 Bremner, Andrew 3 Carron, Gilles 3 de Cataldo, Mark Andrea A. 3 Dimca, Alexandru 3 Doran, Charles F. 3 Esnault, Hélène 3 Fastenberg, Lisa A. 3 Fujino, Osamu 3 Fujisawa, Taro 3 Harder, Andrew 3 Huang, Chunle 3 Hunsicker, Eugenie 3 Ivorra, Florian 3 Jost, Jürgen 3 Kashiwara, Masaki 3 Kerr, Matthew 3 Kuz’minov, Vladimir I. 3 Lan, Kai-Wen 3 Lewis, James D. 3 Li, Zhiyuan 3 MacPherson, Robert Duncan 3 Maxim, Laurentiu G. 3 Mazzeo, Rafe R. 3 Migliorini, Luca 3 Peters, Christiaan Anthonius Maria 3 Rochon, Frédéric 3 Saito, Masa-Hiko 3 Schoen, Chad 3 Simpson, Carlos Tschudi 3 Ulmer, Douglas L. 3 Wildeshaus, Jörg 3 Yang, Yi-Hu 2 Addington, Nicolas M. 2 Albin, Pierre 2 André, Yves 2 Auel, Asher Natan 2 Brüning, Jochen 2 Burgos, José Ignacio 2 Calegari, Frank 2 Catanese, Fabrizio M. E. 2 Chakrabarti, Debraj 2 Chiarellotto, Bruno 2 Claudon, Benoît 2 Clozel, Laurent 2 Cox, David A. 2 Dan, Ananyo 2 Davison, Ben 2 Deligne, Pierre René 2 Di Cerbo, Luca Fabrizio 2 Dupont, Johan L. 2 Eskandari, Payman 2 Gillet, Henri A. 2 Goldman, William M. 2 Gol’dshtein, Vladimir Mikhailovich 2 Goncharov, Alexander Borisovich 2 Goresky, Robert Mark 2 Green, Mark Lee 2 Gurjar, Rajendra Vasant 2 Hanamura, Masaki 2 Haskell, Peter 2 Hayama, Tatsuki 2 Ji, Lizhen 2 Kamber, Franz W. 2 Kapovich, Michael 2 Kasuya, Hisashi 2 Kaur, Inder 2 Kawai, Takahiro 2 Keum, Jonghae 2 Kim, Inkang 2 Kim, Minhyong 2 Lin, Hsueh-Yung 2 Looijenga, Eduard J. N. 2 Luo, Tie 2 Matsumoto, Makoto 2 Maubon, Julien 2 McConnell, Mark 2 McLaughlin, Dennis A. ...and 313 more Authors
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### Cited in 98 Serials

 37 Duke Mathematical Journal 28 Inventiones Mathematicae 26 Compositio Mathematica 23 Mathematische Annalen 15 Advances in Mathematics 13 Mathematische Zeitschrift 12 Publications of the Research Institute for Mathematical Sciences, Kyoto University 11 Annales de l’Institut Fourier 11 Annales Scientifiques de l’École Normale Supérieure. Quatrième Série 11 Manuscripta Mathematica 10 Journal für die Reine und Angewandte Mathematik 8 Transactions of the American Mathematical Society 8 Bulletin of the American Mathematical Society. New Series 7 Proceedings of the American Mathematical Society 7 Proceedings of the Japan Academy. Series A 7 Journal of Algebraic Geometry 6 Nagoya Mathematical Journal 6 Tôhoku Mathematical Journal. Second Series 6 Journal of the American Mathematical Society 6 The Journal of Geometric Analysis 5 Journal of Algebra 5 Rendiconti del Seminario Matematico della Università di Padova 5 Izvestiya: Mathematics 5 Geometry & Topology 5 Comptes Rendus. Mathématique. Académie des Sciences, Paris 5 Research in the Mathematical Sciences 4 Publications Mathématiques 4 Journal of Number Theory 4 Siberian Mathematical Journal 4 Selecta Mathematica. New Series 4 Annals of Mathematics. Second Series 4 Journal of High Energy Physics 4 Forum of Mathematics, Sigma 3 Communications in Mathematical Physics 3 Mathematics of Computation 3 Journal of Geometry and Physics 3 Geometriae Dedicata 3 Journal of Functional Analysis 3 $$K$$-Theory 3 Annales de la Faculté des Sciences de Toulouse. Mathématiques. Série VI 3 Documenta Mathematica 3 Journal of the Institute of Mathematics of Jussieu 3 Japanese Journal of Mathematics. 3rd Series 2 Communications in Algebra 2 Mathematical Notes 2 Mathematical Proceedings of the Cambridge Philosophical Society 2 Rocky Mountain Journal of Mathematics 2 Annali di Matematica Pura ed Applicata. Serie Quarta 2 Canadian Journal of Mathematics 2 Memoirs of the American Mathematical Society 2 Pacific Journal of Mathematics 2 Topology and its Applications 2 International Journal of Mathematics 2 Differential Geometry and its Applications 2 Geometric and Functional Analysis. GAFA 2 Indagationes Mathematicae. New Series 2 Sibirskie Èlektronnye Matematicheskie Izvestiya 2 International Journal of Number Theory 2 Algebra & Number Theory 1 International Journal of Modern Physics A 1 Israel Journal of Mathematics 1 Letters in Mathematical Physics 1 Arkiv för Matematik 1 Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 1 Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 1 Bulletin de la Société Mathématique de France 1 Illinois Journal of Mathematics 1 Journal of Pure and Applied Algebra 1 Journal of Soviet Mathematics 1 Mathematische Nachrichten 1 Michigan Mathematical Journal 1 Monatshefte für Mathematik 1 Annals of Global Analysis and Geometry 1 Sugaku Expositions 1 Proceedings of the Indian Academy of Sciences. Mathematical Sciences 1 Calculus of Variations and Partial Differential Equations 1 Journal de Théorie des Nombres de Bordeaux 1 Journal of Lie Theory 1 Taiwanese Journal of Mathematics 1 Revista Matemática Complutense 1 Communications in Contemporary Mathematics 1 Algebraic & Geometric Topology 1 Milan Journal of Mathematics 1 SIGMA. Symmetry, Integrability and Geometry: Methods and Applications 1 Complex Variables and Elliptic Equations 1 Frontiers of Mathematics in China 1 Journal of Topology 1 Journal of Topology and Analysis 1 Science China. Mathematics 1 Journal of Singularities 1 Kyoto Journal of Mathematics 1 Annales Mathématiques du Québec 1 EMS Surveys in Mathematical Sciences 1 European Journal of Mathematics 1 Journal de l’École Polytechnique – Mathématiques 1 Research in Number Theory 1 Korean Journal of Mathematics 1 Tunisian Journal of Mathematics
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### Cited in 32 Fields

 332 Algebraic geometry (14-XX) 130 Several complex variables and analytic spaces (32-XX) 76 Number theory (11-XX) 53 Differential geometry (53-XX) 50 Global analysis, analysis on manifolds (58-XX) 35 Algebraic topology (55-XX) 33 Manifolds and cell complexes (57-XX) 25 Topological groups, Lie groups (22-XX) 18 Group theory and generalizations (20-XX) 14 $$K$$-theory (19-XX) 12 Category theory; homological algebra (18-XX) 8 Nonassociative rings and algebras (17-XX) 8 Quantum theory (81-XX) 7 Partial differential equations (35-XX) 6 Commutative algebra (13-XX) 5 Associative rings and algebras (16-XX) 5 Functions of a complex variable (30-XX) 4 Functional analysis (46-XX) 3 Potential theory (31-XX) 3 Special functions (33-XX) 3 Relativity and gravitational theory (83-XX) 2 History and biography (01-XX) 2 Combinatorics (05-XX) 2 Field theory and polynomials (12-XX) 2 Dynamical systems and ergodic theory (37-XX) 2 Operator theory (47-XX) 2 General topology (54-XX) 1 Measure and integration (28-XX) 1 Ordinary differential equations (34-XX) 1 Abstract harmonic analysis (43-XX) 1 Geometry (51-XX) 1 Mechanics of particles and systems (70-XX)

### Wikidata Timeline

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• ### A HST/WFC3 Thermal Emission Spectrum of the Hot Jupiter HAT-P-7b(1805.00424)

May 1, 2018 astro-ph.EP
Secondary eclipse observations of several of the hottest hot Jupiters show featureless, blackbody-like spectra or molecular emission features, which are consistent with thermal inversions being present in those atmospheres. Theory predicts a transition between warmer atmospheres with thermal inversions and cooler atmospheres without inversions, but the exact transition point is unknown. In order to further investigate this issue, we observed two secondary eclipses of the hot Jupiter HAT-P-7b with the Hubble Space Telescope (HST) WFC3 instrument and combined these data with previous Spitzer and Kepler secondary eclipse observations. The HST and Spitzer data can be well fit by a blackbody with $T=2692 \pm 14$ K, and the Kepler data point constrains the geometric albedo to $A_{g}=0.077 \pm 0.006$. We modeled these data with a 3D GCM and 1D self-consistent forward models. The 1D models indicate that the atmosphere has a thermal inversion, weak heat redistribution, and water dissociation that limits the range of pressures probed. This result suggests that WFC3 observations of HAT-P-7b and possibly some other ultra-hot Jupiters appear blackbody-like because they probe a region near the tropopause where the atmospheric temperature changes slowly with pressure. Additionally, the 1D models constrain the atmospheric metallicity ($[\text{M/H}]=-0.87^{+0.38}_{-0.34}$) and the carbon-to-oxygen ratio (C/O $<1$ at 99 % confidence). The solar composition 3D GCM matches the Spitzer data but generally underpredicts the flux in the WFC3 bandpass and cannot reproduce its featureless shape. This discrepancy could be explained by high atmospheric drag or nightside clouds, and may be better understood through further observation with the James Webb Space Telescope (JWST).
• ### Global Climate and Atmospheric Composition of the Ultra-Hot Jupiter WASP-103b from HST and Spitzer Phase Curve Observations(1805.00029)

April 30, 2018 astro-ph.EP
We present thermal phase curve measurements for the hot Jupiter WASP-103b observed with Hubble/WFC3 and Spitzer/IRAC. The phase curves have large amplitudes and negligible hotspot offsets, indicative of poor heat redistribution to the nightside. We fit the phase variation with a range of climate maps and find that a spherical harmonics model generally provides the best fit. The phase-resolved spectra are consistent with blackbodies in the WFC3 bandpass, with brightness temperatures ranging from $1880\pm40$ K on the nightside to $2930 \pm 40$ K on the dayside. The dayside spectrum has a significantly higher brightness temperature in the Spitzer bands, likely due to CO emission and a thermal inversion. The inversion is not present on the nightside. We retrieved the atmospheric composition and found the composition is moderately metal-enriched ($\mathrm{[M/H]} = 23^{+29}_{-13}\times$ solar) and the carbon-to-oxygen ratio is below 0.9 at $3\,\sigma$ confidence. In contrast to cooler hot Jupiters, we do not detect spectral features from water, which we attribute to partial H$_2$O dissociation. We compare the phase curves to 3D general circulation models and find magnetic drag effects are needed to match the data. We also compare the WASP-103b spectra to brown dwarfs and young directly imaged companions and find these objects have significantly larger water features, indicating that surface gravity and irradiation environment play an important role in shaping the spectra of hot Jupiters. These results highlight the 3D structure of exoplanet atmospheres and illustrate the importance of phase curve observations for understanding their complex chemistry and physics.
• ### From thermal dissociation to condensation in the atmospheres of ultra hot Jupiters: WASP-121b in context(1805.00096)

April 30, 2018 astro-ph.SR, astro-ph.EP
A new class of exoplanets has emerged: the ultra hot Jupiters, the hottest close-in gas giants. Most of them have weaker than expected spectral features in the $1.1-1.7\mu m$ bandpass probed by HST/WFC3 but stronger spectral features at longer wavelengths probed by Spitzer. This led previous authors to puzzling conclusions about the thermal structures and chemical abundances of these planets. Using the SPARC/MITgcm, we investigate how thermal dissociation, ionization, H$^-$ opacity and clouds shape the thermal structures and spectral properties of ultra hot Jupiters with a special focus on WASP-121b. We expand our findings to the whole population of ultra hot Jupiters through analytical quantification of the thermal dissociation and its influence on the strength of spectral features. We predict that most molecules are thermally dissociated and alkalies are ionized in the dayside photospheres of ultra hot Jupiters. This includes H$_{\rm 2}$O, TiO, VO, and H$_{\rm 2}$ but not CO, which has a stronger molecular bond. The vertical molecular gradient created by the dissociation significantly weakens the spectral features from water while the $4.5\mu m$ CO feature remain unchanged. The water band in the HST/WFC3 bandpass is further weakened by H$^-$ continuum opacity. Molecules are expected to recombine before reaching the limb, leading to order of magnitude variations of the chemical composition and cloud coverage between the limb and the dayside. Overall, molecular dissociation provides a qualitative understanding of the lack of strong spectral feature of water in the $1-2\mu m$ bandpass observed in most ultra hot Jupiters. Quantitatively, however, our model does not provide a satisfactory match to the WASP-121b emission spectrum. Together with WASP-33b and Kepler-33Ab, they seem the outliers among the population of ultra hot Jupiters in need of a more thorough understanding.
• ### H- Opacity and Water Dissociation in the Dayside Atmosphere of the Very Hot Gas Giant WASP-18 b(1801.02489)

April 25, 2018 astro-ph.EP
We present one of the most precise emission spectra of an exoplanet observed so far. We combine five secondary eclipses of the hot Jupiter WASP-18 b (Tday=2900K) that we secured between 1.1 and 1.7 micron with the WFC3 instrument aboard the Hubble Space Telescope. Our extracted spectrum (S/N=50, R=40) does not exhibit clearly identifiable molecular features but is poorly matched by a blackbody spectrum. We complement this data with previously published Spitzer/IRAC observations of this target and interpret the combined spectrum by computing a grid of self-consistent, 1D forward models, varying the composition and energy budget. At these high temperatures, we find there are important contributions to the overall opacity from H- ions, as well as the removal of major molecules by thermal dissociation (including water), and thermal ionization of metals. These effects were omitted in previous spectral retrievals for very hot gas giants, and we argue that they must be included to properly interpret the spectra of these objects. We infer a new metallicity and C/O ratio for WASP-18 b, and find them well constrained to be solar ([M/H]=-0.01 (0.35), C/O<0.85 at 3 sigma confidence level), unlike previous work but in line with expectations for giant planets. The best fitting self-consistent temperature-pressure profiles are inverted, resulting in an emission feature at 4.5 micron seen in the Spitzer photometry. These results further strengthen the evidence that the family of very hot gas giant exoplanets commonly exhibit thermal inversions.
• The James Webb Space Telescope (JWST) presents the opportunity to transform our understanding of planets and the origins of life by revealing the atmospheric compositions, structures, and dynamics of transiting exoplanets in unprecedented detail. However, the high-precision, time-series observations required for such investigations have unique technical challenges, and prior experience with other facilities indicates that there will be a steep learning curve when JWST becomes operational. In this paper we describe the science objectives and detailed plans of the Transiting Exoplanet Community Early Release Science (ERS) Program, which is a recently approved program for JWST observations early in Cycle 1. The goal of this project, for which the obtained data will have no exclusive access period, is to accelerate the acquisition and diffusion of technical expertise for transiting exoplanet observations with JWST, while also providing a compelling set of representative datasets that will enable immediate scientific breakthroughs. The Transiting Exoplanet Community ERS Program will exercise the time-series modes of all four JWST instruments that have been identified as the consensus highest priorities, observe the full suite of transiting planet characterization geometries (transits, eclipses, and phase curves), and target planets with host stars that span an illustrative range of brightnesses. The observations in this program were defined through an inclusive and transparent process that had participation from JWST instrument experts and international leaders in transiting exoplanet studies. Community engagement in the project will be centered on a two-phase Data Challenge that culminates with the delivery of planetary spectra, time-series instrument performance reports, and open-source data analysis toolkits in time to inform the agenda for Cycle 2 of the JWST mission.
• ### The Colorado Ultraviolet Transit Experiment (CUTE): A dedicated cubesat mission to study exoplanetary mass loss and magnetic fields(1801.02673)

Jan. 8, 2018 astro-ph.IM
The Colorado Ultraviolet Transit Experiment (CUTE) is a near-UV (2550 - 3300 Angstrom) 6U cubesat mission designed to monitor transiting hot Jupiters to quantify their atmospheric mass loss and magnetic fields. CUTE will probe both atomic (Mg and Fe) and molecular (OH) lines for evidence of enhanced transit absorption, and to search for evidence of early ingress due to bow shocks ahead of the planet's orbital motion. As a dedicated mission, CUTE will observe more than 100 spectroscopic transits of hot Jupiters over a nominal seven month mission. This represents the equivalent of more than 700 orbits of the only other instrument capable of these measurements, the Hubble Space Telescope. CUTE efficiently utilizes the available cubesat volume by means of an innovative optical design to achieve a projected effective area of 28 sq. cm, low instrumental background, and a spectral resolving power of 3000 over the primary science bandpass. These performance characteristics enable CUTE to discern transit depths between 0.1 - 1% in individual spectral absorption lines. We present the CUTE optical and mechanical design, a summary of the science motivation and expected results, and an overview of the projected fabrication, calibration and launch timeline.
• ### Phase curves of WASP-33b and HD 149026b and a New Correlation Between Phase Curve Offset and Irradiation Temperature(1710.07642)

Dec. 21, 2017 astro-ph.EP
We present new 3.6 and 4.5 $\mu m$ Spitzer phase curves for the highly irradiated hot Jupiter WASP-33b and the unusually dense Saturn-mass planet HD 149026b. As part of this analysis, we develop a new variant of pixel level decorrelation that is effective at removing intrapixel sensitivity variations for long observations (>10 hours) where the position of the star can vary by a significant fraction of a pixel. Using this algorithm, we measure eclipse depths, phase amplitudes, and phase offsets for both planets at 3.6 $\mu m$ and 4.5 $\mu m$. We use a simple toy model to show that WASP-33b's phase offset, albedo, and heat recirculation efficiency are largely similar to those of other hot Jupiters despite its very high irradiation. On the other hand, our fits for HD 149026b prefer a very high albedo and an unusually high recirculation efficiency. We also compare our results to predictions from general circulation models, and find that while neither are a good match to the data, the discrepancies for HD 149026b are especially large. We speculate that this may be related to its high bulk metallicity, which could lead to enhanced atmospheric opacities and the formation of reflective cloud layers in localized regions of the atmosphere. We then place these two planets in a broader context by exploring relationships between the temperatures, albedos, heat transport efficiencies, and phase offsets of all planets with published thermal phase curves. We find a striking relationship between phase offset and irradiation temperature--the former drops with increasing temperature until around 3400 K, and rises thereafter. Although some aspects of this trend are mirrored in the circulation models, there are notable differences that provide important clues for future modeling efforts.
• ### Spitzer Phase Curve Constraints for WASP-43b at 3.6 and 4.5 microns(1608.00056)

Dec. 30, 2016 astro-ph.EP
Previous measurements of heat redistribution efficiency (the ability to transport energy from a planet's highly irradiated dayside to its eternally dark nightside) show considerable variation between exoplanets. Theoretical models predict a positive correlation between heat redistribution efficiency and temperature for tidally locked planets; however, recent HST WASP-43b spectroscopic phase curve results are inconsistent with current predictions. Using the Spitzer Space Telescope, we obtained a total of three phase curve observations of WASP-43b (P=0.813 days) at 3.6 and 4.5 microns. The first 3.6 micron visit exhibits spurious nightside emission that requires invoking unphysical conditions in our cloud-free atmospheric retrievals. The two other visits exhibit strong day-night contrasts that are consistent with the HST data. To reconcile the departure from theoretical predictions, WASP-43b would need to have a high-altitude, nightside cloud/haze layer blocking its thermal emission. Clouds/hazes could be produced within the planet's cool, nearly retrograde mid-latitude flows before dispersing across its nightside at high altitudes. Since mid-latitude flows only materialize in fast-rotating ($\lesssim1$ day) planets, this may explain an observed trend connecting measured day-night contrast with planet rotation rate that matches all current Spitzer phase curve results. Combining independent planetary emission measurements from multiple phases, we obtain a precise dayside hemisphere H2O abundance ($2.5\times 10^{-5} - 1.1\times 10^{-4}$ at 1$\sigma$ confidence) and, assuming chemical equilibrium and a scaled solar abundance pattern, we derive a corresponding metallicity estimate that is consistent with being solar (0.4 -- 1.7). Using the retrieved global CO+CO2 abundance under the same assumptions, we estimate a comparable metallicity of 0.3 - 1.7$\times$ solar. (Abridged)
• ### HST hot-Jupiter transmission spectral survey: Clear skies for cool Saturn WASP-39b(1601.04761)

Aug. 4, 2016 astro-ph.EP
We present HST STIS optical transmission spectroscopy of the cool Saturn-mass exoplanet WASP-39b from 0.29-1.025 micron, along with complementary transit observations from Spitzer IRAC at 3.6 and 4.5 micron. The low density and large atmospheric pressure scale height ofWASP-39b make it particularly amenable to atmospheric characterization using this technique. We detect a Rayleigh scattering slope as well as sodium and potassium absorption features; this is the first exoplanet in which both alkali features are clearly detected with the extended wings predicted by cloud-free atmosphere models. The full transmission spectrum is well matched by a clear, H2-dominated atmosphere or one containing a weak contribution from haze, in good agreement with the preliminary reduction of these data presented in Sing et al. (2016). WASP-39b is predicted to have a pressure-temperature profile comparable to that of HD 189733b and WASP-6b, making it one of the coolest transiting gas giants observed in our HST STIS survey. Despite this similarity, WASP-39b appears to be largely cloud-free while the transmission spectra of HD 189733b and WASP-6b both indicate the presence of high altitude clouds or hazes. These observations further emphasize the surprising diversity of cloudy and cloud-free gas giant planets in short-period orbits and the corresponding challenges associated with developing predictive cloud models for these atmospheres.
• ### No Thermal Inversion and a Solar Water Abundance for the Hot Jupiter HD209458b from HST WFC3 Emission Spectroscopy(1605.08810)

May 27, 2016 astro-ph.EP
The nature of the vertical thermal structure of hot Jupiter atmospheres is one of the key questions raised by the characterization of transiting exoplanets over the last decade. There have been claims that many hot Jupiter's exhibit vertical profiles with increasing temperature with decreasing pressure in the infrared photosphere that leads to the reversal of molecular absorption bands into emission features (an inversion). However, these claims have been based on broadband photometry rather than the unambiguous identification of emission features with spectroscopy, and the chemical species that could cause the thermal inversions by absorbing stellar irradiation at high altitudes have not been identified despite extensive theoretical and observational effort. Here we present high precision HST WFC3 observations of the dayside emission spectrum of the hot Jupiter HD209458b; the first exoplanet suggested to have a thermal inversion. Our observations resolve a water band in absorption at 6.2 sigma confidence. When combined with Spitzer photometry the data are indicative of a monotonically decreasing temperature with pressure over the range 1-0.001 bar at 7.7 sigma confidence. We test the robustness of our results by exploring a variety of model assumptions including the temperature profile parameterization, presence of a cloud, and choice of Spitzer data reduction. We also introduce a new analysis method, "chemical retrieval-on-retrieval", to determine the elemental abundances from the spectrally retrieved mixing ratios with thermochemical self-consistency and find plausible abundances consistent with solar metallicity (0.06 - 10 x solar) and carbon-to-oxygen ratios less than unity. This work suggests that high-precision spectrophotometric results are required to robustly infer thermal structures and compositions of extra-solar planet atmospheres.
• ### 3.6 and 4.5 $\mu$m ${\it Spitzer}$ Phase Curves of the Highly-Irradiated Hot Jupiters WASP-19b and HAT-P-7b(1512.09342)

May 27, 2016 astro-ph.EP
We analyze full-orbit phase curve observations of the transiting hot Jupiters WASP-19b and HAT-P-7b at 3.6 and 4.5 $\mu$m obtained using the Spitzer Space Telescope. For WASP-19b, we measure secondary eclipse depths of $0.485\%\pm 0.024\%$ and $0.584\%\pm 0.029\%$ at 3.6 and 4.5 $\mu$m, which are consistent with a single blackbody with effective temperature $2372 \pm 60$ K. The measured 3.6 and 4.5 $\mu$m secondary eclipse depths for HAT-P-7b are $0.156\%\pm 0.009\%$ and $0.190\%\pm 0.006\%$, which are well-described by a single blackbody with effective temperature $2667\pm 57$ K. Comparing the phase curves to the predictions of one-dimensional and three-dimensional atmospheric models, we find that WASP-19b's dayside emission is consistent with a model atmosphere with no dayside thermal inversion and moderately efficient day-night circulation. We also detect an eastward-shifted hotspot, suggesting the presence of a superrotating equatorial jet. In contrast, HAT-P-7b's dayside emission suggests a dayside thermal inversion and relatively inefficient day-night circulation; no hotspot shift is detected. For both planets, these same models do not agree with the measured nightside emission. The discrepancies in the model-data comparisons for WASP-19b might be explained by high-altitude silicate clouds on the nightside and/or high atmospheric metallicity, while the very low 3.6 $\mu$m nightside planetary brightness for HAT-P-7b may be indicative of an enhanced global C/O ratio. We compute Bond albedos of 0 ($<0.08$ at $1\sigma$) and $0.38\pm 0.06$ for WASP-19b and HAT-P-7b, respectively. In the context of other planets with thermal phase curve measurements, we show that WASP-19b and HAT-P-7b fit the general trend of decreasing day-night heat recirculation with increasing irradiation.
• ### Transiting Exoplanet Studies and Community Targets for JWST's Early Release Science Program(1602.08389)

April 21, 2016 astro-ph.EP, astro-ph.IM
The James Webb Space Telescope will revolutionize transiting exoplanet atmospheric science due to its capability for continuous, long-duration observations and its larger collecting area, spectral coverage, and spectral resolution compared to existing space-based facilities. However, it is unclear precisely how well JWST will perform and which of its myriad instruments and observing modes will be best suited for transiting exoplanet studies. In this article, we describe a prefatory JWST Early Release Science (ERS) program that focuses on testing specific observing modes to quickly give the community the data and experience it needs to plan more efficient and successful future transiting exoplanet characterization programs. We propose a multi-pronged approach wherein one aspect of the program focuses on observing transits of a single target with all of the recommended observing modes to identify and understand potential systematics, compare transmission spectra at overlapping and neighboring wavelength regions, confirm throughputs, and determine overall performances. In our search for transiting exoplanets that are well suited to achieving these goals, we identify 12 objects (dubbed "community targets") that meet our defined criteria. Currently, the most favorable target is WASP-62b because of its large predicted signal size, relatively bright host star, and location in JWST's continuous viewing zone. Since most of the community targets do not have well-characterized atmospheres, we recommend initiating preparatory observing programs to determine the presence of obscuring clouds/hazes within their atmospheres. Measurable spectroscopic features are needed to establish the optimal resolution and wavelength regions for exoplanet characterization. Other initiatives from our proposed ERS program include testing the instrument brightness limits and performing phase-curve observations.(Abridged)
• ### Benchmark Transiting Brown Dwarf LHS 6343 C: Spitzer Secondary Eclipse Observations Yield Brightness Temperature and mid-T Spectral Class(1603.09343)

March 30, 2016 astro-ph.SR, astro-ph.EP
There are no field brown dwarf analogs with measured masses, radii, and luminosities, precluding our ability to connect the population of transiting brown dwarfs with measurable masses and radii and field brown dwarfs with measurable luminosities and atmospheric properties. LHS 6343 C, a weakly-irradiated brown dwarf transiting one member of an M+M binary in the Kepler field, provides the first opportunity to probe the atmosphere of a non-inflated brown dwarf with a measured mass and radius. Here, we analyze four Spitzer observations of secondary eclipses of LHS 6343 C behind LHS 6343 A. Jointly fitting the eclipses with a Gaussian process noise model of the instrumental systematics, we measure eclipse depths of 1.06 \pm 0.21 ppt at 3.6 microns and 2.09 \pm 0.08 ppt at 4.5 microns, corresponding to brightness temperatures of 1026 \pm 57 K and 1249 \pm 36 K, respectively. We then apply brown dwarf evolutionary models to infer a bolometric luminosity log(L_star / L_sun) = -5.16 \pm 0.04. Given the known physical properties of the brown dwarf and the two M dwarfs in the LHS 6343 C system, these depths are consistent with models of a 1100 K T dwarf at an age of 5 Gyr and empirical observations of field T5-6 dwarfs with temperatures of 1070 \pm 130 K. We investigate the possibility that the orbit of LHS 6343 C has been altered by the Kozai-Lidov mechanism and propose additional astrometric or Rossiter-McLaughlin measurements of the system to probe the dynamical history of the system.
• ### A Search for Water in the Atmosphere of HAT-P-26b Using LDSS-3C(1511.08226)

Dec. 24, 2015 astro-ph.EP, astro-ph.IM
The characterization of a physically-diverse set of transiting exoplanets is an important and necessary step towards establishing the physical properties linked to the production of obscuring clouds or hazes. It is those planets with identifiable spectroscopic features that can most effectively enhance our understanding of atmospheric chemistry and metallicity. The newly-commissioned LDSS-3C instrument on Magellan provides enhanced sensitivity and suppressed fringing in the red optical, thus advancing the search for the spectroscopic signature of water in exoplanetary atmospheres from the ground. Using data acquired by LDSS-3C and the Spitzer Space Telescope, we search for evidence of water vapor in the transmission spectrum of the Neptune-mass planet HAT-P-26b. Our measured spectrum is best explained by the presence of water vapor, a lack of potassium, and either a high-metallicity, cloud-free atmosphere or a solar-metallicity atmosphere with a cloud deck at ~10 mbar. The emergence of multi-scale-height spectral features in our data suggests that future observations at higher precision could break this degeneracy and reveal the planet's atmospheric chemical abundances. We also update HAT-P-26b's transit ephemeris, t_0 = 2455304.65218(25) BJD_TDB, and orbital period, p = 4.2345023(7) days.
• ### A continuum from clear to cloudy hot-Jupiter exoplanets without primordial water depletion(1512.04341)

Dec. 14, 2015 astro-ph.EP
Thousands of transiting exoplanets have been discovered, but spectral analysis of their atmospheres has so far been dominated by a small number of exoplanets and data spanning relatively narrow wavelength ranges (such as 1.1 to 1.7 {\mu}m). Recent studies show that some hot-Jupiter exoplanets have much weaker water absorption features in their near-infrared spectra than predicted. The low amplitude of water signatures could be explained by very low water abundances, which may be a sign that water was depleted in the protoplanetary disk at the planet's formation location, but it is unclear whether this level of depletion can actually occur. Alternatively, these weak signals could be the result of obscuration by clouds or hazes, as found in some optical spectra. Here we report results from a comparative study of ten hot Jupiters covering the wavelength range 0.3-5 micrometres, which allows us to resolve both the optical scattering and infrared molecular absorption spectroscopically. Our results reveal a diverse group of hot Jupiters that exhibit a continuum from clear to cloudy atmospheres. We find that the difference between the planetary radius measured at optical and infrared wavelengths is an effective metric for distinguishing different atmosphere types. The difference correlates with the spectral strength of water, so that strong water absorption lines are seen in clear-atmosphere planets and the weakest features are associated with clouds and hazes. This result strongly suggests that primordial water depletion during formation is unlikely and that clouds and hazes are the cause of weaker spectral signatures.
• ### A Detection of Water in the Transmission Spectrum of the Hot Jupiter WASP-12b and Implications for its Atmospheric Composition(1504.05586)

Oct. 17, 2015 astro-ph.EP
Detailed characterization of exoplanets has begun to yield measurements of their atmospheric properties that constrain the planets' origins and evolution. For example, past observations of the dayside emission spectrum of the hot Jupiter WASP-12b indicated that its atmosphere has a high carbon-to-oxygen ratio (C/O $>$ 1), suggesting it had a different formation pathway than is commonly assumed for giant planets. Here we report a precise near-infrared transmission spectrum for WASP-12b based on six transit observations with the Hubble Space Telescope/Wide Field Camera 3. We bin the data in 13 spectrophotometric light curves from 0.84 - 1.67 $\mu$m and measure the transit depths to a median precision of 51 ppm. We retrieve the atmospheric properties using the transmission spectrum and find strong evidence for water absorption (7$\sigma$ confidence). This detection marks the first high-confidence, spectroscopic identification of a molecule in the atmosphere of WASP-12b. The retrieved 1$\sigma$ water volume mixing ratio is between $10^{-5}-10^{-2}$, which is consistent with C/O $>$ 1 to within 2$\sigma$. However, we also introduce a new retrieval parameterization that fits for C/O and metallicity under the assumption of chemical equilibrium. With this approach, we constrain C/O to $0.5^{+0.2}_{-0.3}$ at $1\,\sigma$ and rule out a carbon-rich atmosphere composition (C/O$>1$) at $>3\sigma$ confidence. Further observations and modeling of the planet's global thermal structure and dynamics would aid in resolving the tension between our inferred C/O and previous constraints. Our findings highlight the importance of obtaining high-precision data with multiple observing techniques in order to obtain robust constraints on the chemistry and physics of exoplanet atmospheres.
• ### 3.6 and 4.5 $\mu$m Phase Curves of the Highly-Irradiated Eccentric Hot Jupiter WASP-14b(1505.03158)

Aug. 28, 2015 astro-ph.EP
We present full-orbit phase curve observations of the eccentric ($e\sim 0.08$) transiting hot Jupiter WASP-14b obtained in the 3.6 and 4.5 $\mu$m bands using the \textit{Spitzer Space Telescope}. We use two different methods for removing the intrapixel sensitivity effect and compare their efficacy in decoupling the instrumental noise. Our measured secondary eclipse depths of $0.1882\%\pm 0.0048\%$ and $0.2247\%\pm 0.0086\%$ at 3.6 and 4.5 $\mu$m, respectively, are both consistent with a blackbody temperature of $2402\pm 35$ K. We place a $2\sigma$ upper limit on the nightside flux at 3.6 $\mu$m and find it to be $9\%\pm 1\%$ of the dayside flux, corresponding to a brightness temperature of 1079 K. At 4.5 $\mu$m, the minimum planet flux is $30\%\pm 5\%$ of the maximum flux, corresponding to a brightness temperature of $1380\pm 65$ K. We compare our measured phase curves to the predictions of one-dimensional radiative transfer and three-dimensional general circulation models. We find that WASP-14b's measured dayside emission is consistent with a model atmosphere with equilibrium chemistry and a moderate temperature inversion. These same models tend to over-predict the nightside emission at 3.6 $\mu$m, while under-predicting the nightside emission at 4.5 $\mu$m. We propose that this discrepancy might be explained by an enhanced global C/O ratio. In addition, we find that the phase curves of WASP-14b ($7.8 M_{\mathrm{Jup}}$) are consistent with a much lower albedo than those of other Jovian mass planets with thermal phase curve measurements, suggesting that it may be emitting detectable heat from the deep atmosphere or interior processes.
• ### Spitzer Secondary Eclipse Observations of Five Cool Gas Giant Planets and Empirical Trends in Cool Planet Emission Spectra(1508.00902)

Aug. 4, 2015 astro-ph.EP
In this work we present Spitzer 3.6 and 4.5 micron secondary eclipse observations of five new cool (<1200 K) transiting gas giant planets: HAT-P-19b, WASP-6b, WASP-10b, WASP-39b, and WASP-67b. We compare our measured eclipse depths to the predictions of a suite of atmosphere models and to eclipse depths for planets with previously published observations in order to constrain the temperature- and mass-dependent properties of gas giant planet atmospheres. We find that the dayside emission spectra of planets less massive than Jupiter require models with efficient circulation of energy to the night side and/or increased albedos, while those with masses greater than that of Jupiter are consistently best-matched by models with inefficient circulation and low albedos. At these relatively low temperatures we expect the atmospheric methane to CO ratio to vary as a function of metallicity, and we therefore use our observations of these planets to constrain their atmospheric metallicities. We find that the most massive planets have dayside emission spectra that are best-matched by solar metallicity atmosphere models, but we are not able to place strong constraints on metallicities of the smaller planets in our sample. Interestingly, we find that the ratio of the 3.6 and 4.5 micron brightness temperatures for these cool transiting planets is independent of planet temperature, and instead exhibits a tentative correlation with planet mass. If this trend can be confirmed, it would suggest that the shape of these planets' emission spectra depends primarily on their masses, consistent with the hypothesis that lower-mass planets are more likely to have metal-rich atmospheres.
• ### A giant comet-like cloud of hydrogen escaping the warm Neptune-mass exoplanet GJ 436b(1506.07541)

June 24, 2015 astro-ph.EP
Exoplanets orbiting close to their parent stars could lose some fraction of their atmospheres because of the extreme irradiation. Atmospheric mass loss primarily affects low-mass exoplanets, leading to suggest that hot rocky planets might have begun as Neptune-like, but subsequently lost all of their atmospheres; however, no confident measurements have hitherto been available. The signature of this loss could be observed in the ultraviolet spectrum, when the planet and its escaping atmosphere transit the star, giving rise to deeper and longer transit signatures than in the optical spectrum. Here we report that in the ultraviolet the Neptune-mass exoplanet GJ 436b (also known as Gliese 436b) has transit depths of 56.3 +/- 3.5% (1 sigma), far beyond the 0.69% optical transit depth. The ultraviolet transits repeatedly start ~2 h before, and end >3 h after the ~1 h optical transit, which is substantially different from one previous claim (based on an inaccurate ephemeris). We infer from this that the planet is surrounded and trailed by a large exospheric cloud composed mainly of hydrogen atoms. We estimate a mass-loss rate in the range of ~10^8-10^9 g/s, which today is far too small to deplete the atmosphere of a Neptune-like planet in the lifetime of the parent star, but would have been much greater in the past.
• ### Spitzer Secondary Eclipses of the Dense, Modestly-irradiated, Giant Exoplanet HAT-P-20b Using Pixel-Level Decorrelation(1411.7404)

June 3, 2015 astro-ph.EP
HAT-P-20b is a giant exoplanet that orbits a metal-rich star. The planet itself has a high total density, suggesting that it may also have a high metallicity in its atmosphere. We analyze two eclipses of the planet in each of the 3.6- and 4.5 micron bands of Warm Spitzer. These data exhibit intra-pixel detector sensitivity fluctuations that were resistant to traditional decorrelation methods. We have developed a simple, powerful, and radically different method to correct the intra-pixel effect for Warm Spitzer data, which we call pixel-level decorrelation (PLD). PLD corrects the intra-pixel effect very effectively, but without explicitly using - or even measuring - the fluctuations in the apparent position of the stellar image. We illustrate and validate PLD using synthetic and real data, and comparing the results to previous analyses. PLD can significantly reduce or eliminate red noise in Spitzer secondary eclipse photometry, even for eclipses that have proven to be intractable using other methods. Our successful PLD analysis of four HAT-P-20b eclipses shows a best-fit blackbody temperature of 1134 +/-29K, indicating inefficient longitudinal transfer of heat, but lacking evidence for strong molecular absorption. We find sufficient evidence for variability in the 4.5 micron band that the eclipses should be monitored at that wavelength by Spitzer, and this planet should be a high priority for JWST spectroscopy. All four eclipses occur about 35 minutes after orbital phase 0.5, indicating a slightly eccentric orbit. A joint fit of the eclipse and transit times with extant RV data yields e(cos{omega}) = 0.01352 (+0.00054, -0.00057), and establishes the small eccentricity of the orbit to high statistical confidence. Given the existence of a bound stellar companion, HAT-P-20b is another excellent candidate for orbital evolution via Kozai migration or other three-body mechanism.
• ### Characterizing the Habitable Zones of Exoplanetary Systems with a Large Ultraviolet/Visible/Near-IR Space Observatory(1505.01840)

May 7, 2015 astro-ph.SR, astro-ph.EP
Understanding the surface and atmospheric conditions of Earth-size, rocky planets in the habitable zones (HZs) of low-mass stars is currently one of the greatest astronomical endeavors. Knowledge of the planetary effective surface temperature alone is insufficient to accurately interpret biosignature gases when they are observed in the coming decades. The UV stellar spectrum drives and regulates the upper atmospheric heating and chemistry on Earth-like planets, is critical to the definition and interpretation of biosignature gases, and may even produce false-positives in our search for biologic activity. This white paper briefly describes the scientific motivation for panchromatic observations of exoplanetary systems as a whole (star and planet), argues that a future NASA UV/Vis/near-IR space observatory is well-suited to carry out this work, and describes technology development goals that can be achieved in the next decade to support the development of a UV/Vis/near-IR flagship mission in the 2020s.
• ### Low False-Positive Rate of Kepler Candidates Estimated From A Combination Of Spitzer And Follow-Up Observations(1503.03173)

March 11, 2015 astro-ph.EP
(Abridged) NASA's Kepler mission has provided several thousand transiting planet candidates, yet only a small subset have been confirmed as true planets. Therefore, the most fundamental question about these candidates is the fraction of bona fide planets. Estimating the rate of false positives of the overall Kepler sample is necessary to derive the planet occurrence rate. We present the results from two large observational campaigns that were conducted with the Spitzer telescope during the the Kepler mission. These observations are dedicated to estimating the false positive rate (FPR) amongst the Kepler candidates. We select a sub-sample of 51 candidates, spanning wide ranges in stellar, orbital and planetary parameter space, and we observe their transits with Spitzer at 4.5 microns. We use these observations to measures the candidate's transit depths and infrared magnitudes. A bandpass-dependent depth alerts us to the potential presence of a blending star that could be the source of the observed eclipse: a false-positive scenario. For most of the candidates (85%), the transit depths measured with Kepler are consistent with the depths measured with Spitzer as expected for planetary objects, while we find that the most discrepant measurements are due to the presence of unresolved stars that dilute the photometry. The Spitzer constraints on their own yield FPRs between 5-40%, depending on the KOIs. By considering the population of the Kepler field stars, and by combining follow-up observations (imaging) when available, we find that the overall FPR of our sample is low. The measured upper limit on the FPR of our sample is 8.8% at a confidence level of 3 sigma. This observational result, which uses the achromatic property of planetary transit signals that is not investigated by the Kepler observations, provides an independent indication that Kepler's false positive rate is low.
• ### The Atmospheric Circulation of the Hot Jupiter WASP-43b: Comparing Three-Dimensional Models to Spectrophotometric Data(1410.2382)

Jan. 12, 2015 astro-ph.EP
The hot Jupiter WASP-43b has now joined the ranks of transiting hot Jupiters HD 189733b and HD 209458b as an exoplanet with a large array of observational constraints on its atmospheric properties. Because WASP-43b receives a similar stellar flux as HD 209458b but has a rotation rate 4 times faster and a much higher gravity, studying WASP-43b serves as a test of the effect of rotation rate and gravity on the circulation when stellar irradiation is held approximately constant. Here we present 3D atmospheric circulation models of WASP-43b using the SPARC/MITgcm, a coupled radiation and circulation model, exploring the effects of composition, metallicity, and frictional drag. We find that the circulation regime of WASP-43b is not unlike other hot Jupiters, with equatorial superrotation that yields an eastward-shifted hotspot and large day-night temperature variations (~600 K at photospheric pressures). We then compare our model results to observations from Stevenson et al. which utilize HST/WFC3 to collect spectrophotometric phase curve measurements of WASP-43b from 1.12-1.65 microns. Our results show the 5x solar model lightcurve provides a good match to the data, with a phase offset of peak flux and planet/star flux ratio that is similar to observations; however, the model nightside appears to be brighter. Nevertheless, our 5x solar model provides an excellent match to the WFC3 dayside emission spectrum. This is a major success, as the result is a natural outcome of the 3D dynamics with no model tuning, and differs significantly from 1D models that can generally only match observations when appropriately tuned. In sum, these results demonstrate that 3D circulation models can provide important insights in interpreting exoplanet atmospheric observations, even at high spectral resolution, and highlight the potential for future observations with HST, JWST and other next-generation telescopes.
• ### Thermal structure of an exoplanet atmosphere from phase-resolved emission spectroscopy(1410.2241)

Oct. 8, 2014 astro-ph.EP, astro-ph.IM
Exoplanets that orbit close to their host stars are much more highly irradiated than their Solar System counterparts. Understanding the thermal structures and appearances of these planets requires investigating how their atmospheres respond to such extreme stellar forcing. We present spectroscopic thermal emission measurements as a function of orbital phase ("phase-curve observations") for the highly-irradiated exoplanet WASP-43b spanning three full planet rotations using the Hubble Space Telescope. With these data, we construct a map of the planet's atmospheric thermal structure, from which we find large day-night temperature variations at all measured altitudes and a monotonically decreasing temperature with pressure at all longitudes. We also derive a Bond albedo of 0.18 +0.07,-0.12 and an altitude dependence in the hot-spot offset relative to the substellar point.
• ### A Precise Water Abundance Measurement for the Hot Jupiter WASP-43b(1410.2255)

Oct. 8, 2014 astro-ph.EP
The water abundance in a planetary atmosphere provides a key constraint on the planet's primordial origins because water ice is expected to play an important role in the core accretion model of planet formation. However, the water content of the Solar System giant planets is not well known because water is sequestered in clouds deep in their atmospheres. By contrast, short-period exoplanets have such high temperatures that their atmospheres have water in the gas phase, making it possible to measure the water abundance for these objects. We present a precise determination of the water abundance in the atmosphere of the 2 $M_\mathrm{Jup}$ short-period exoplanet WASP-43b based on thermal emission and transmission spectroscopy measurements obtained with the Hubble Space Telescope. We find the water content is consistent with the value expected in a solar composition gas at planetary temperatures (0.4-3.5x solar at 1 $\sigma$ confidence). The metallicity of WASP-43b's atmosphere suggested by this result extends the trend observed in the Solar System of lower metal enrichment for higher planet masses.


text:
# Rotate an object defined by intrinsic angles with a quaternion, how to?

Friends and I are building a "simple" physics engine, but we are running into trouble. All of our objects' angles are defined by a vector which represents an Euler intrinsic rotation of type z-y'-x''.

Problem: we have a system of points which represent an object, each point have a mass and a relative position to the center of mass (barycenter) of the object. Because we have all informations about forces on each point, we can easily calculate the moment (or torque, or couple), we obtain a vector which represents the axis of rotation and the torque value (given by the length of the vector).

After a lot of try and retry, looking at forums and applying different transformations (see here), we can't make this work.

Because the torque is defined by a vector which is the axis of "rotation force" of the system and everything else in our work is defined by Euler angles, we have to use some tricks. There is an example :

(0 multiply rotation speed by the delta time) - 1/ convert the rotation speed (quaternion) to a rotation matrix 3x3 - 2/ convert the current object rotation (Euler intrinsic angles) to rotation matrix - 3/ multiply the two matrix //This is the only way to "add" a rotation - 4/ convert this new matrix to an Euler intrinsic rotation vector //Then we have our new position

Question: Do you agree with this procedure ? And do you know if there is a faster way to do what we want to do ?

To be more precise, the precedent procedure may be correct, but it is a big source of error, and we have currently somme errors that we can't identify. The system for instance is rotating with in the good way but after 45° of rotation it bounce / jump back and "fall" again.

(In the gif bellow, the instantaneous change of orientation is due to the gif loop, it's not a bug, the bug is only the bounce effect) Bounce effect GIF White vectors represent forces (inverted), cubes are the "points" and blues links are the links between points.

I think the error is here, we use this documentation4 to get the intrinsic rotation from the rotation matrix, we have theses equations : $$w = tan^{-1}(R_{21} / R_{11}) = atan2(R_{21},R_{11})\\ v = -sin^{-1}(R_{31})\\ u = tan^{-1}(R_{32} / R_{33}) = atan2(R_{32},R_{33})$$

But when we tests the invert conversion of the conversion ("matrix to intrinsic" of the "intrinsic to matrix" operation) given a arbitrary vector (a,b,c) ; we obtain : $$(a \equiv \pi,b \equiv \pi,c \equiv \pi)\\ or \\(a \equiv \pi, - b \equiv \pi,c \equiv \pi)$$ And I think this "or" is our current problem, but I am not sure.

UPDATE : Finally it seems to be a limitation to the conversions, I working on it tomorrow. It's because I use the quaternion length to save the angle of rotation or the angular velocity. So when I compute angular acceleration to the angular velocity to have a new angular velocity, it's via rotations matrix that limit angle of rotation to PI radians, that is ok for orienting objects, but not with speed, because if PI and 0 are the same as angle, their are not exactly the same as angular velocity. For now I found the solution to divide the size of the vectors by a number in order to have every time something between 0 and PI, then after operation I multiply the result vector by this same number, not very beautiful...

Thank you a lot, I'm going to have a headache soon... Sorry for my english, don't hesitate to ask if you don't understand something !

Context: We use openGL for visualization, and Java as language. You can find the code here. The main test is in main.TestsCentripete for those who want to test the code !

Below is my current code, I figured out that the wikipedia article had a mistake, so I found another solution: In the part Rotation matrix ↔ quaternion, qr was given by

qr = arccos( 0.5 * (A11+A22+A33-1));

According to this document: I can have the rotation matrix from quaternion and the invert function (quaternion from matrix). I also have the conversion from Euler angles to rotation matrix and the invert function.

/**
* Convert a quaternion axis to matrix of rotation (VERIFIED - OK)
* @param quaternion
* @return
*/
public static Matrix quaternionToMatrix(Vect3D quaternion){

double theta = quaternion.size();

double uUn = quaternion.x;
double vUn = quaternion.y;
double wUn = quaternion.z;

//Normalisation
double u = uUn/theta;
double v = vUn/theta;
double w = wUn/theta;

double u2 = u*u;
double v2 = v*v;
double w2 = w*w;
double cosT = Math.cos(theta);
double oneMinusCosT = 1-cosT;
double sinT = Math.sin(theta);

Matrix res = new Matrix();

if(Math.abs(theta)<=1E-9){
res.setCoef(1, 1, 1);
res.setCoef(2, 2, 1);
res.setCoef(3, 3, 1);
return res;
}

// Build the matrix entries element by element.
res.setCoef(1, 1, u2 + (v2 + w2) * cosT);
res.setCoef(1, 2, u*v*oneMinusCosT - w*sinT);
res.setCoef(1, 3, u*w*oneMinusCosT + v*sinT);

res.setCoef(2, 1, u*v * oneMinusCosT + w*sinT);
res.setCoef(2, 2, v2 + (u2 + w2) * cosT);
res.setCoef(2, 3, v*w * oneMinusCosT - u*sinT);

res.setCoef(3, 1, u*w * oneMinusCosT - v*sinT);
res.setCoef(3, 2, v*w * oneMinusCosT + u*sinT);
res.setCoef(3, 3, w2 + (u2 + v2) * cosT);

return res;
}

/**
* Convert an intrinsec axis to matrix of rotation (VERIFIED - OK)
* @param quaternion
* @return
*/
public static Matrix intrinsecToMatrix(Vect3D intrinsec){
return rotMatrixModule_Space(intrinsec);
}

/**
* Convert a rotation matrix to quaternion (VERIFIED - OK)
* @param quaternion
* @return
*/
public static Vect3D matrixToQuaternion(Matrix matrix){

double qr = Math.acos(0.5 * (matrix.getCoef(1, 1)+matrix.getCoef(2, 2)+matrix.getCoef(3, 3)-1));

double qi = (matrix.getCoef(3, 2)-matrix.getCoef(2, 3))/(4*qr);
double qj = (matrix.getCoef(1, 3)-matrix.getCoef(3, 1))/(4*qr);
double qk = (matrix.getCoef(2, 1)-matrix.getCoef(1, 2))/(4*qr);

double l = Math.sqrt(qi*qi + qj*qj + qk*qk);

if(l<=1E-9 || qr<=1E-9){
return new Vect3D();
}

double x = qi * qr / l;
double y = qj * qr / l;
double z = qk * qr / l;

return new Vect3D(x,y,z);
}

/**
* Convert a rotation matrix to intrinsec angles
* @param quaternion
* @return
*/
public static Vect3D matrixToIntrinsec(Matrix matrix){

double z = Math.atan2(
matrix.getCoef(2,1),
matrix.getCoef(1,1)
);

double y = -Math.asin(
matrix.getCoef(3,1)
);

double x = Math.atan2(
matrix.getCoef(3,2),
matrix.getCoef(3,3)
);

if(y>=Math.PI/2-(1E-9)){
z = 0;
x = Math.atan2(
matrix.getCoef(1,2),
matrix.getCoef(1,3)
);
}

if(y<=-Math.PI/2+(1E-9)){
z = 0;
x = Math.atan2(
-matrix.getCoef(1,2),
-matrix.getCoef(1,3)
);
}

return new Vect3D(x,y,z);

}

/**
* Compute the rotation matrix between the spaceship and the space
*
* @return The rotation matrix between the spaceship and the space
*/
public static Matrix rotMatrixModule_Space(Vect3D orientation) {
Matrix rotx = new Matrix();
rotx.setCoef(1, 1, 1);
rotx.setCoef(2, 2, Math.cos(orientation.x));
rotx.setCoef(3, 3, Math.cos(orientation.x));
rotx.setCoef(2, 3, -Math.sin(orientation.x));
rotx.setCoef(3, 2, Math.sin(orientation.x));

Matrix roty = new Matrix();
roty.setCoef(2, 2, 1);
roty.setCoef(1, 1, Math.cos(orientation.y));
roty.setCoef(3, 3, Math.cos(orientation.y));
roty.setCoef(1, 3, Math.sin(orientation.y));
roty.setCoef(3, 1, -Math.sin(orientation.y));

Matrix rotz = new Matrix();
rotz.setCoef(3, 3, 1);
rotz.setCoef(1, 1, Math.cos(orientation.z));
rotz.setCoef(2, 2, Math.cos(orientation.z));
rotz.setCoef(1, 2, -Math.sin(orientation.z));
rotz.setCoef(2, 1, Math.sin(orientation.z));

return Matrix.multiply(Matrix.multiply(rotz, roty), rotx);
}

• I edited the post, thanks for your time, sorry for the bad presentation, I tried to improve it. What I call a moment is the torque of our system at a specific time according to the current forces present on each point of the system. – Romaric Mollard Jan 9 '17 at 22:45

This is the normal procedure:

1) Compute body-fixed torques $\bf\tau$.

2) Compute body-fixed rotational accelerations, $\alpha = \bf I^{-1}\bf \tau$

3) Integrate accelerations to body-fixed rotational velocity, $\bf\omega$.

4) Convert $\bf\omega$ to quaternion derivative, $\dot{\bf q} = \frac{1}{2}\bf q \left[ \begin{array}{c} 0 \\ \bf\omega \end{array} \right]$.

5) Integrate $\dot{\bf q}$ to update $\bf q$

6) Normalize $\bf q$ as $\bf q \leftarrow \bf q/\|\bf q\|$

7) Convert $\bf q$ to Euler angles as needed (not necessary).

• Ok, thank you, but in the case of the simulator, we are not in a continuous space/time. So I obtain the new velocity with $$v = a * \Delta t$$ For linear acceleration. So for angular acceleration I do the same thing, but I was not able to add the two vectors (acceleration and current velocity), it was needed to convert the two vectors (which are actually quaternions) into matrix and multiply them. I don't know if this make a difference with your procedure I'll update my post in a couple of minutes. But thank you, my problem is finely resolved ! – Romaric Mollard Jan 10 '17 at 8:10
• In discrete time your update law would be: $\bf\omega_{k+1} = \bf\omega_{k} + \bf\alpha \Delta t$ (i.e. both $\bf\alpha$ and $\bf\omega$ are vectors. Same law applies for $\bf q$, namely $\bf q_{k+1} = \bf q_{k}+ \dot{\bf q}\Delta t$. Also, you do not need to convert quaternions into matrices to multiply them. There are laws for direct multiplication. – Tpofofn Jan 11 '17 at 1:25
• Ok, interesting, but in the case where my quaternion's length is equal to my quaternion angle, I have to normalize the vectors, then add them, re-normalize and what is the new rotation ? The sum of the angles ? (quaternion is defined by $$(qi, qj, qk, qr)$$ where qr is an angle usely, and $$(qi, qj, qk)$$ a normalized vector, but I have only a vector). – Romaric Mollard Jan 11 '17 at 7:02
• I am not sure what you mean by quaternion angle. The quaternion should always have unit length. $\dot{\bf q}$ should always be perpendicular to $\bf q$ so the renormalization is a small adjustment to keep it on the unit sphere. Also, to combine rotations represented by quaternions, you multiply them, not add them. If you are properly using quaternions, should not need to use angles at all. That is the beauty of them. – Tpofofn Jan 11 '17 at 16:28
• Also, note that the whole quaternion is normalized, not just the vector part, so $\| (qi, qj, qk, qr) \| = 1$ – Tpofofn Jan 11 '17 at 16:29


text:
written in

With a bit of tweaking, you can create rich documents with R markdown that combine text, figures, equations and tables that are generated by R on the fly. Now if only blogs could get a DOI…

Tables:

Tables can be used with the knitr::kable function.

mpg cyl disp hp drat wt qsec vs am gear carb
Mazda RX4 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4
Mazda RX4 Wag 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4
Datsun 710 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1
Hornet 4 Drive 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1
Hornet Sportabout 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2

Math

Equations are handled in LATEX format by MathJax: $$\frac{1}{n} \sum{i=i}^{n} x{i}$$

Embedded Images:

Links to websites can be included like this.

Embedding images with control over the size and resolution is a bit tricky. I like to import my PNG images as R objects, that way they are treated like any other plot.

Citations:

It is also straightforward to include inline citations like this using the package knitcitations package (Abrams et al. (2012); Bolnick et al. (2010)).

And include a dynamically generated list of references:

Works Cited:

• Peter A. Abrams, Lasse Ruokolainen, Brian J. Shuter, Kevin S. McCann, (2012) Harvesting Creates Ecological Traps: Consequences of Invisible Mortality Risks in Predator–Prey Metacommunities. Ecology 93 281-293 10.1890/11-0011.1
1. Bolnick, T. Ingram, W. E. Stutz, L. K. Snowberg, O. L. Lau, J. S. Paull, (2010) Ecological Release From Interspecific Competition Leads to Decoupled Changes in Population And Individual Niche Width. Proceedings of The Royal Society B: Biological Sciences 277 1789-1797 10.1098/rspb.2010.0018

Source code for this document is available here


text:
The University of Edinburgh is committed to equality of opportunity for all its staff and students, and promotes a culture of inclusivity.

Welcome

This website has been established to provide a hub for information about equality and diversity across the University.

About the Equality and Diversity (E&D) Unit and its function.

Strategy and Outcomes

The Equality and Diversity Strategy, Outcomes and Action Plan and how the University vision is a continuing commitment to equality and diversity for both students and staff.

What are equality and diversity?

What are equality and diversity, and how can we promote them?

Legislation, Policies & Publications

The equality bill which received royal assent on 8 April 2010, becoming the Equality Act 2010 was enforced as at 1st October 2010. The Act consolidates current anti-discrimination legislation and also introduces new measures that will have direct implications for higher education institutions (HEIs).

Contacts

Who to contact for equality and diversity advice and assistance.

Find us

Maps, directions and postal address for Equality and Diversity.


text:
# Derive a seventh order power series representation of the general solution to Airy's equation $y^n(x)=xy(x)$

Derive a seventh order power series representation of the general solution to Airy's equation $y^n(x)=xy(x)$. Initial conditions to the general solution $y(0)=1,$ $y'(0)=1.$ (We don't need to write a general formula for the $n-th$ term. Just write all the terms up to and including $x^7$).
I have done it but cannot write it as a power series of seventh order.
My attempt:I take as usual $y=\sum_{k=0}^{\infty} a_kx^k$ and then $y^n=\sum_{k=n}^{\infty} \frac{k!}{(k-n)!}a_kx^{k-n}$. And then rearranging the terms and using initial conditions we get $a_0=1$ and $a_1=1$ and $a_{k+n}=\frac{k!}{(k+n)!}a_{k-1}$ and obviously $a_n=0$. Then from here we get $a_{n+1}=\frac{1}{(n+1)!}a_0$$=$$\frac{1}{(n+1)!}$, $a_{n+2}=\frac{2}{(n+2)!}a_1$=$\frac{2}{(n+2)!}$, proceeding similarly. But according to the question we have to write upto and including $x^7$. Please help me to find the required power series representation as I failed it to form without using $a_n$.

• Check for typos in your task, it should be $y''$, not $y^n$. This mistake happens from time to time one way or the other. See also other sources for Airy equation or Airy function. – LutzL Jul 13 '18 at 6:24
• @LutzL. sir I am also surprised. But it $y^n$ there. – abcdmath Jul 13 '18 at 6:28
• Then ask your tutor, it is probably a mistake in reading the hand-written original. You could write down your general solution and then add the numbers for $n=2$. – LutzL Jul 13 '18 at 7:12
• The Airy equation is $y''=xy$ – Dylan Jul 13 '18 at 7:56
• – cgiovanardi Jul 13 '18 at 17:55


text:
Question

# Find the HCF and LCM of $120$ and $144$ by the fundamental theorem of Arithmetic.

Hint:Fundamental theorem of Arithmetic states that every composite number can be factored uniquely as a product of primes. HCF is the highest factor common to two given natural numbers. LCM is the smallest multiple common to two given natural numbers. To find HCF and LCM, first we will find prime factorization of the given natural numbers. Then, observe the common prime factors with smallest power to find HCF and observe all prime factors with greatest power to find LCM.

First we need to find the prime factorization of given numbers $120$ and $144$. These both numbers are even so we can start prime factorization with prime numbers $2$.

Therefore, $120 = 2 \times 2 \times 2 \times 3 \times 5 = {2^3} \times {3^1} \times {5^1}$

Therefore, $144 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 = {2^4} \times {3^2}$

Now we can see that $2$ and $3$ are common prime factors in the prime factorization of $120$ and
$144$. Also we can see the smallest powers of $2$ and $3$ are $3$ and $1$ respectively.
Therefore, HCF of $120$ and $144$ is the product of ${2^3}$ and ${3^1}$.
That is, HCF of $120$ and $144$ is ${2^3} \times {3^1} = 8 \times 3 = 24$.
Now if we observe prime factorization of both numbers then we can see that there are three prime factors $2,3$ and $5$. Also we can see the greatest powers of $2,3$ and $5$ are $4,2$ and $1$ respectively. Therefore, LCM of $120$ and $144$ is the product of ${2^4},{3^2}$ and ${5^1}$. That is, LCM of $120$ and $144$ is ${2^4} \times {3^2} \times {5^1} = 16 \times 9 \times 5 = 720$.
Hence, HCF and LCM of $120$ and $144$ are $24$ and $720$ respectively.

Note:Fundamental theorem of Arithmetic is also known as unique factorization theorem. In this example, the product of two given numbers $120$ and $144$ is $17,280$. If we take a product of HCF and LCM of these two numbers then we will get the same answer. That is, $24 \times 720 = 17280$. Note that the product of given numbers is equivalent to the product of their HCF and LCM. This is the useful property of HCF and LCM.


text:
# 12.6. 👩💻 Decoding a Function¶

In general, when you see a function definition you will try figure out what the function does, but, unless you are writing the function, you won’t care how it does it.

For example, here is a summary of some functions we have seen already.

• input takes one parameter, a string. It is displayed to the user. Whatever the user types is returned, as a string.

• int takes one parameter. It can be of any type that can be converted into an integer, such as a floating point number or a string whose characters are all digits.

Sometimes, you will be presented with a function definition whose operation is not so neatly summarized as above. Sometimes you will need to look at the code, either the function definition or code that invokes the function, in order to figure out what it does.

To build your understanding of any function, you should aim to answer the following questions:

1. How many parameters does it have?

2. What is the type of values that will be passed when the function is invoked?

3. What is the type of the return value that the function produces when it executes?

If you try to make use of functions, ones you write or that others write, without being able to answer these questions, you will find that your debugging sessions are long and painful.

The first question is always easy to answer. Look at the line with the function definition, look inside the parentheses, and count how many variable names there are.

The second and third questions are not always so easy to answer. In Python, unlike some other programming languages, variables are not declared to have fixed types, and the same holds true for the variable names that appear as formal parameters of functions. You have to figure it out from context.

To figure out the types of values that a function expects to receive as parameters, you can look at the function invocations or you can look at the operations that are performed on the parameters inside the function.

Here are some clues that can help you determine the type of object associated with any variable, including a function parameter. If you see…

• len(x), then x must be a string or a list. (Actually, it can also be a dictionary, in which case it is equivalent to the expression len(x.keys()). Later in the course, we will also see some other sequence types that it could be). x can’t be a number or a Boolean.

• x - y, x and y must be numbers (integer or float)

• x + y, x and y must both be numbers, both be strings, or both be lists

• x[3], x must be a string or a list containing at least four items, or x must be a dictionary that includes 3 as a key.

• x['3'], x must be a dictionary, with ‘3’ as a key.

• x[y:z], x must be a sequence (string or list), and y and z must be integers

• x and y, x and y must be Boolean

• for x in y, y must be a sequence (string or list) or a dictionary (in which case it’s really the dictionary’s keys); x must be a character if y is a string; if y is a list, x could be of any type.

Check your understanding: decode this function definition

How many parameters does function cyu3 take?

def cyu3(x, y, z):
if x - y > 0:
return y -2
else:
z.append(y)
return x + 3

• 0
• Count the number of variable names inside the parenetheses on line 1.
• 1
• Count the number of variable names inside the parenetheses on line 1.
• 2
• Count the number of variable names inside the parenetheses on line 1.
• 3
• x, y, and z.
• Can't tell
• You can tell by looking inside the parentheses on line 1. Each variable name is separated by a comma.

What are the possible types of variables x and y?

def cyu3(x, y, z):
if x - y > 0:
return y -2
else:
z.append(y)
return x + 3

• integer
• x - y, y-2, and x+3 can all be performed on integers.
• float
• x - y, y-2, and x+3 can all be performed on floats.
• list
• x - y, y-2, and x+3 can't be performed on lists.
• string
• x - y and y-2 can't be performed on strings.
• Can't tell
• You can tell from some of the operations that are performed on them.

What are the possible types of variable z?

def cyu3(x, y, z):
if x - y > 0:
return y -2
else:
z.append(y)
return x + 3

• integer
• append can't be performed on integers.
• float
• append can't be performed on floats.
• list
• append can be performed on lists.
• string
• append can't be performed on strings.
• Can't tell
• You can tell from some of the operations that are performed on it.

What are the possible types of the return value from cyu3?

def cyu3(x, y, z):
if x - y > 0:
return y -2
else:
z.append(y)
return x + 3

• integer
• y-2 or x+3 could produce an integer.
• float
• y-2 or x+3 could produce a float.
• list
• y-2 or x+3 can't produce a list.
• string
• neither y-2 or x+3 could produce a string.
• Can't tell
• You can tell from the expressions that follow the word return.