text:
\section{Introduction}
Observational campaigns to detect exoplanets have been incredibly successful, having found thousands of possible exoplanets (e.g. Batalha et al. 2013). However, what's surprising is that many of the exoplanets are found to be incredibly close to their parent stars with separations $<0.1$~AU being common. Furthermore, while the original detected planets at small separations were massive (of order a Jupiter mass and higher) - so called ``hot jupiters'' - these are relatively rare (Fressin et al. 2013). With the advent of the {\it Kepler} mission the majority of planets found close to their parent star are small radius, low mass objects, with the majority of stars containing at least one small ($\lesssim 4$~R$_\oplus$) planet with a period shorter than Mercury's (Fressin et al. 2013). This indicates that the formation of close-in small planets is a dominant mode of planet formation.

At such close separations, the environment is very different to planets in our own solar system and irradiation from the central star can play a very important role, particularly when the star is young (e.g. Owen \& Wu, 2013) or during the late stages of stellar evolution (e.g. Bear \& Soker, 2011; 2015). The star's bolometric luminosity can heat-up the planet's atmosphere leading to an increased scale height, as well as preventing the planet from cooling (Guillot et al. 1996, Burrows et al. 2000, Baraffe et al. 2003). Furthermore, the high energy ionizing luminosity from the star can heat-up the upper layers to temperatures of order the escape temperature, leading to mass-loss (e.g. Lammer et al. 2003, Tian et al. 2005, Owen \& Jackson, 2012).

When the star is relatively young ($<$ several 100~Myr old) the ionizing luminosity can represent a significant fraction  of the total output of the star $\sim 10^{-3}\,L_*$ (Jackson et al. 2012). Integrating the total received  ionizing luminosity over Gyr time-scales and comparing it to the binding energy of close in planets it was realised that planetary evaporation could play a significant evolutionary role and actually remove a large fraction of the planet's atmosphere (Lammer et al. 2003, Jackson et al. 2012), especially at low planet masses (Owen \& Jackson 2012, Lopez et al. 2012, Owen \& Wu 2013, Lopez \& Fortney 2013).

Due to the importance of the evolutionary effects of planetary evaporation one requires accurate mass-loss rates. Commonly the approach is to turn a certain fraction of the received ionizing luminosity into work/heating and associated mass-loss, either globally (Lammer et al. 2003, Erkaev et al. 2007) and obtain the well known ``energy-limited'' mass-loss rate; or locally in simplified simulations (e.g. Yelle et al. 2004, Tian et al. 2005, Lammer et al. 2013). Such an approach implicitly assumes that $P{\rm d}V$ work dominates the energy loss and radiative cooling or ionization physics are unimportant.

More detailed studies have since been performed that take into consideration some of the details of radiative transfer and ionization chemistry (Koskinen et al. 2007, 2010, 2014; Chadney et al. 2014); however, these models typically concentrate on a handful of observed hot jupiters.  Owen \& Jackson (2012) considered the role X-rays play and showed heating from secondary ionization and line cooling were important when the sonic point occurred in an area of the flow dominated by X-ray heating and argued that in this case the flow was never ``energy-limited''. Owen \& Jackson (2012) also presented a schematic picture of when UV or X-ray heating dominates and argued that X-rays would dominant for the highest fluxes around low-mass planets.

Furthermore, Murray-Clay et al. (2009) considered the evaporation for a hot jupiter with mass 0.7~M$_J$ and radius 1.4~R$_J$ with a pure UV radiation field at various fluxes. At high fluxes $\gtrsim 10^4$~erg~s$^{-1}$~cm$^{-2}$~, they found the flow to be in approximate radiation-recombination equilibrium, in this case the flow behaved like a HII region, was fully ionized and approximately isothermal with a temperature of $10^4$~K and the cooling was dominated by Lyman-$\alpha$ radiation. As the density in any HII region scales with as $J_0^{1/2}$ (where  $J_0$ is the ionizing flux in photons per unit time), then the mass-loss rate in this regime scale approximately as $J_0^{1/2}$. At fluxes $\lesssim 10^4$~erg~s$^{-1}$~cm$^{-2}$~ the density in the flow was so low that recombination were too slow compared to the flow time and the mass loss rate then scaled linearly with flux. In this case ionizations were balanced approximately by advection, which Murray-Clay et al. 2009 identified as an ``energy-limited'' flow.

The physical understanding of the transition for ``recombination-limited'' to an ``energy-limited'' flow being that once the density drops in the flow such that the recombination time is long compared to the flow time then it can no-longer be in radiative-recombination equilibrium (e.g. Soker, 1999). This obviously depends on planet properties, through the flow time (e.g. planet radius and mass); and as such the transition flux below which the flow cannot be in radiative-recombination equilibrium is not going to be a  unique value. However, several studies of exoplanet evolution (e.g. Jin et al. 2014) take the results of Murray-Clay et al. (2009) for their single planet study and apply it to all planets which is likely to lead to widely inaccurate adopted mass-loss rates.

In this paper we aim to study the properties of UV driven evaporation across the range of observed exoplanets, particularity at low masses which has as yet been unexplored. We aim to put forward a simple physical understanding of the various evaporation regimes and identify three: ``recombination-limited'', ``energy-limited'' and a new regime ``photon-limited''. We present an analytic picture in Section~2, perform a numerical study with a new radiation-hydrodynamic scheme in Section~3. In Section~4 we discuss our results and summarise in Section~5.

\section{Analytic Overview}
The evaporative flow problem is complicated by the fact that in general the flow may not be in local thermodynamic or ionization equilibrium, particularly at low fluxes (e.g. Murray-Clay et al. 2009). However, in the case of irradiation by a spectrum dominated by Hydrogen ionizing photons (EUV) we can make progress by considering ionization balance (e.g. Bertoldi \& Mckee, 1990). We consider a planet with mass $M_p$ and radius $R_p$ close to its central star at a separation $a$. Noting that $R_p/a\ll1$, then the EUV flux is constant on all scales of interest. We consider a flow that is dived into three distinct regions: (i) bolometrically heated planet's atmosphere; where the temperature is set by the flux of optical photons and the optical depth to EUV photons is large such that the ionization fraction ($X$) is very low. (ii) an ``ionization front'' where the photo-ionization rate greatly exceeds the recombination rate and the ionization fraction increases as one moves away from the planet. (iii) a fully ionized region where photo-ionization is balanced by recombinations and the ionization fraction is unity\footnote{Here we consider the case of a pure Hydrogen atmosphere, with the inclusion of other elements the ionization fraction is generally $>1$}.

Since the scale height of the planet's bolometrically heated atmosphere must be $\ll R$ for a bound planet then the transition from the bolometrically heated atmosphere (region i) to the ionization front (region ii) is by construction sharp. In the picture we develop below we take it to occur at an optical depth $\tau_{\rm EUV}=1$ to the EUV photons and call this radius the planet's radius. However, unlike many problems in astrophysics involving EUV radiation, as we shall see we cannot in general assume that the ionization front is thin compared to the flow scale (or even the planet's scale) and in many cases we find a transition from region ii to region iii occurs at a very large radius. In particular, if it occurs at a radius larger than the sonic radius then it is irrelevant in determining the mass-loss and we assume the flow to only contain regions i and ii.

We choose to work in one-dimensional spherically symmetric picture along a streamline connecting the star and planet (see Stone \& Proga, 2009 and Owen \& Adams, 2014 for the role of multi-dimensional flows). For the sake of clarity we neglect the contribution from the tidal gravity due to the star. For a steady-state, spherically symmetric flow the governing equations of hydrodynamics are:
\begin{eqnarray}
\frac{\partial}{\partial r}\left(\mu r^2nu\right)&=&0\label{eqn:mass}\\
u\frac{\partial u}{\partial r}{ +}\frac{1}{\rho}\frac{\partial P}{\partial r}{ +}\frac{GM_p}{r{ ^2}}&=&0\label{eqn:mom}\\
{\frac{1}{r^2}}\frac{\partial}{\partial r}\left[{ u}r^2\left(\rho E+P-\frac{GM_p\rho}{r}\right)\right]&=&q\label{eqn:energy}
\end{eqnarray}
where $\mu$ is the mean-molecular weight given by $\mu=m_H/(1+X)$, with $X$ the ionization fraction; $G$ is the gravitational constant; $n$ is the number density { of Hydrogen nuclei}; $\rho=n\mu$ the mass density; $u$ the gas velocity; $P=k_bnT$ the gas pressure, where $T$ is the temperature; $E$ is the {\it total} specific energy density and $q$ is the net volumetric heating rate, { which is detailed in Section 3.1}. Furthermore steady state ionization balance requires { (e.g. Bertoldi \& McKee 1990, their Equation 2.1)}:
\begin{equation}
J_0={ {\mathcal{F}_{\rm IF}}}-\int_\infty^{R_p}\!\!\!\!\!{\rm d}R \,\alpha_RX^2n^2\label{eqn:IB}
\end{equation}
where $\alpha_R$ is the recombination rate { and $\mathcal{F}_{\rm IF}$ is the flux of ionizing photons into the ionization front. Since recombination are negligible inside the ionization front then $\mathcal{F}_{\rm IF}$ must balance the flux of neutral particles into the ionization front (Churchwell et al. 1987, Bertoldi \& McKee 1990). Namely:
\begin{equation}
\mathcal{F}_{\rm IF}=n(R_p)u(R_p) \label{eqn:PL1}
\end{equation}
Note since the radiation field is plane parallel, Equation~\ref{eqn:IB} is written in plane parallel form. Since the flow is approximately spherical we must be careful combining Equation~\ref{eqn:IB} with the hydrodynamic terms, something we discuss further in Section 2.1.1. Specifically, we cannot trivially integrate Equation~\ref{eqn:PL1} over the surface of the planet by multiplying by $4\pi R_p^2$ to find the mass-loss rate, since the total number of ionizing photons entering the ionizing front does not necessarily equal the total number reaching the base of the atmosphere, as some may pass through the flow without interacting with the it.} Equation~\ref{eqn:energy} can be recast into a limiting mass-loss, the ``energy-limited'' mass-loss rate where uncertainties in heating and cooling are encapsulated into an efficiency parameter $\eta$ which can only be determined through numerical calculations with values typically in the range 0.01-0.3 (e.g. Murray-Clay et al. 2009; Owen \& Jackson, 2012).
\begin{equation}
\dot{M}\le\eta\frac{\pi R_p^3J_0h\bar{\nu}_h}{4GM_p}\label{eqn:EL}
\end{equation}
where $h\bar{\nu}_h$ is the mean photon energy that goes into heating (rather than ionization). { The origin of the factor 4 in the denominator comes from averaging the received flux by a $\pi R_p^2$ disc over the $4\pi R_p^2$ surface area of the planet. We choose to this form as it is the commonly used form in the exoplanet literature and allows comparisons of the efficiency ($\eta$) values, although we note some authors do not perform such an averaging (e.g. Murray-Clay et al. 2009).} For a flow that can be considered close to being energy-limited requires that $q$ is predominately controlled by heating and radiative cooling is a small contribution. Therefore, energy losses are dominated by $P{\rm d}V$ work or advection. The main contribution to the radiative cooling rate $\Lambda$ is Ly$\alpha$ radiation with cooling rate (Black 1981):
\begin{equation}
\Lambda=7.5\times10^{-19}{\rm ~erg~cm^{-3}~s^{-1}}n_en_0\exp\left(\frac{{ -}118348~{\rm K}}{T}\right)
\end{equation}
The ionization state of the gas is controlled by the recombination rate; adopting the on-the-spot approximation such that $\alpha_R=\alpha_B$ where $\alpha_B$ is the case-B recombination coefficient given by $\alpha_B=2.6\times10^{-13} {\rm ~cm^3~s^{-1}}(T/10^4~{\rm K})^{-0.7}$ so that the recombination rate $\Lambda_{\rm ion}$ is:
\begin{equation}
\Lambda_{\rm ion}=\alpha_Bn_e^2
\end{equation}
where $n_e$ is the electron density. Noting the ratio of the cooling time-scale ($t_c$) to the recombination time-scale ($t_{ r}$) is given by:
\begin{equation}
\frac{t_c}{t_r}=\frac{k_bT\Lambda_{\rm ion}}{\Lambda}
\end{equation}
and is purely a function of temperature and ionization fraction. When the gas is close to $\sim10^4$~K and roughly in ionization-recombination equilibrium then $t_c/t_r\approx0.05$. However, if the flow were no longer in  {\it local} ionization equilibrium and the temperature falls below $10^4$~K. { The temperature falls because the heating rate and ionization rate scale identically with the ionizing flux; therefore, since the ionizing photons are unable to obtain ionization equilibrium they are unable to heat the gas to $10^{4}~K$ temperature (i.e. the ratio of the heating to ionization time-scale is a constant). Thus, $P{\rm d}V$ cooling becomes the dominant cooling mechanism (e.g. Murray-Clay et al. 2009)}, so we find  $t_c/t_r\sim\exp(-10^5~{\rm K}/T)$. Thus, $t_c$ quickly becomes longer than the recombination time when the flow is no longer in {\it local} ionization equilibrium. Since {\it local} ionization { balance} breaks down when when the recombination time becomes longer than the flow time-scale $t_{\rm flow}$, then one would expect that in the majority of the cases $t_c>t_{\rm flow}$ when $T<10^{4}$~K and the flow will be close to ``energy-limited'' (c.f. Soker 1999, Bear \& Soker 2011). This does not mean that Equation~\ref{eqn:EL} will provide an accurate description of the mass-loss rate over the range of parameters of interest as $\eta$ may still vary considerably due to hydrodynamic effects.

\subsection{Limiting cases}

Inspection of Equation~\ref{eqn:IB} \& \ref{eqn:EL} indicate that three physical { processes} may limit the mass-loss. Either the number of incoming photons, recombinations or energy considerations. Equating $J_0$ with the advection term in Equation~\ref{eqn:IB} indicates that the flow will be limited by the number of incoming ionization photons --- a processes we term ``photon-limited". Equating $J_0$ with the recombination term in Equation~\ref{eqn:IB} results in a density in the ionization front that is determined only by radiation-recombination balance --- a processes we will term ``recombination-limited". Finally, insisting that the flow satisfies Equation~\ref{eqn:EL} will result in a flow that is limited by the available energy, this has become known commonly as ``energy-limited'' evaporation. In this case the mass-loss rate is described by taking Equation~\ref{eqn:EL} as an equality where $\eta$ is yet to be determined.

\subsubsection{Photon limited flows}\label{sec:photon_lim}

This corresponds to the case where the ionization front encompasses the entire flow (including the sonic point) and as such recombinations are completely negligible in the flow {  so that $\mathcal{F}_{\rm IF}=J_0=n(R_p)u(R_p)$}. { Additionally the pull of gravity is weak and the inequality in Equation~6 is easily satisfied. Note we are implicitly assuming that the gas temperature is high enough that the gas is able to escape freely.} . Therefore, we are limited by the number of incoming ionizing photons: namely we can only loose one atom per incoming photon. { Note we are implicitly assuming that the gas temperature reached in the ionization front is high enough that escape from the planet's potential is possible; if not the flow becomes ``energy-limited'' as we will discuss in Section~2}. However, here we are complicated by geometric effects. For uniform spherical irradiation then the total number of absorbed photons is the area  out to the ionization front, not the planet's radius itself. However, planet evaporation does not occur with spherical irradiation, but rather plane parallel. Furthermore, since the ionization front is marginally optically thin ionizing photons, then any photons with an impact parameter much larger than the planets radius, but smaller than the ionization front radius are likely to pass straight through the flow. Therefore, we can approximate the effective absorbing area of the planet as $\sim \pi R_p^2$. Therefore, in order to obtain the photon limited mass-loss rate we drop the recombination integral in Equation~\ref{eqn:IB} and equating $J_0$ and the advection allows us to calculate the mass-loss rate approximately as:
\begin{equation}
\dot{M}_{\rm PL}\approx \pi R_p^2m_HJ_0\label{eqn:mdot_PL}
\end{equation}
We note that we do not need to make assumptions about the energy considerations here as we do not need to know the gas temperature to find the mass-loss rates. Such a mass-loss limit has not previously been discussed in the context of planet evaporation. Thus,  it will corresponds to cases where planetary gravity is not strong and the flux is low, so namely low-mass planets at large separations and early times. Unfortunately, due to the fact any 1D model essentially assumes spherically symmetric irradiation any photon limited mass-loss rate is likely to be much higher than this value (since the effective absorbing area in such a calculation is $\pi R_{\rm IF}^2$ and $R_{\rm IF}> R_p$ often with $R_{\rm IF}\gg R_p$). Therefore, in our numerical calculations presented in Section~3 we will not be able to study this limit of planet evaporation. Any numerical constraints on ``photon-limited'' evaporation must be obtained with 2/3D numerical calculations. Although, Equation~\ref{eqn:mdot_PL} should roughly represent an appropriate estimate for the mass-loss rate, {  and is similar to the 1+1D numerical calculations of Bertoldi \& McKee (1990)}.

\subsubsection{Recombination limited flows}
``Recombination-limited'' flows are perhaps the easiest to understand as in this case the radiative transfer and hydrodynamics problems are essentially de-coupled and the basics in the context of planetary evaporation were discussed in Murray-Clay et al. (2009). When the recombination time is short compared to the flow time-scale the ionization front becomes thin compared to the flow scale and essentially the transition from region i to region iii can be considered infinitesimally thin. Furthermore, the gas is thermostated to approximately $10^4$~K  and approximately isothermal. For cases when the planet's gravity is strong enough that the scale height in $10^4$~K is small compared to the planet's radius then one can approximate the density structure close to the planet as hydrostatic and the integral in Equation~\ref{eqn:IB} becomes (see also Murray-Clay et al. 2009):
\begin{equation}
\int_\infty^{R_p}\!\!\!\!\!{\rm d}R \,\alpha_RX^2n^2\approx\alpha_Bn_b^2\left(\frac{c_s^2R_p^2}{2GM_p}\right)\label{eqn:sg}
\end{equation}
where $n_b$ is the density at the ionization front. However, in the case that the planet's gravity is weak then the density profile will be roughly that of a constant velocity outflow and will fall of as $r^{-2}$. Therefore, in this case the integral in Equation~\ref{eqn:IB} becomes (see Johnstone et al. 1998):
\begin{equation}
\int_\infty^{R_p}\!\!\!\!\!{\rm d}R \,\alpha_RX^2n^2\approx\alpha_Bn_b^2\left(\frac{R_p}{3}\right)\label{eqn:wg}
\end{equation}
Writing the term in parenthesis in Equations~\ref{eqn:sg} and \ref{eqn:wg} as a scale height $H$ then we crudely set:
\begin{equation}
H=\min\left[\frac{R_p}{3},
\left(\frac{c_s^2R_p^2}{2GM_p}\right)\right]
\end{equation}
Therefore,  we can write the base density as:
\begin{equation}
n_b=\sqrt{\frac{J_0}{\alpha_BH}}
\end{equation}
Now we can write the recombination-limited mass-loss rate, noting in fully ionized flow $\mu=m_H/2$ as:
\begin{equation}
\dot{M}_{\rm RL}=2\pi R_p^2m_Hn_bc_s\mathcal{M}_b\label{eqn:mdot_RL}
\end{equation}
where $\mathcal{M}_b$ is the Mach velocity at the base of the flow. For an isothermal flow $\mathcal{M}_b$ is given by the well known Parker wind solution (Parker, 1958, Cranmer 2004) where:
\begin{equation}
\mathcal{M}_b=\sqrt{-W_0\left[-\frac{R_p}{R_s}^{-4}\exp\left(3-\frac{4R_s}{R_p}\right)\right]}\label{eqn:mach}
\end{equation}
with $W(x)$ the Lambert $W$ function and $R_s=GM_p/2c_s^2$, if $R_p/R_s<1$ and $\mathcal{M}_b=1$ otherwise. For tightly bound planets Equation~\ref{eqn:mach} has an exponential fall off of the form:
\begin{equation}
\mathcal{M}_b\approx\left(\frac{R_p}{R_s}\right)^{-2}\exp\left(-\frac{2R_s}{R_p}\right)
\end{equation}
Therefore for $R_p/R_s$ close to unity $\dot{M}_{RL}\propto R^{3/2}$; however for $R_p/R_s\ll 1$ it has a steep (exponential) dependence on mass and radius.

\subsection{Picture of Evaporation across parameter space}

For fixed mass, since $\dot{M}_{\rm EL}\propto\eta R{ _p}^3$ and $\dot{M}_{\rm PL}\propto R_{ p}^2$, ({ i.e. they intercept once}) then we note that there are two possible cases. Either the flow is ``recombination-limited'' at all possible radii (high masses and high fluxes) or the flow is ``recombination-limited'' at both large radii (where $\dot{M}_{RL}\propto R_{ p}^{3/2}$) and at small radii where it has an steep dependence on radius. Furthermore, since ``photon-limited'' mass-loss has a shallower dependence on radius than ``energy-limited'' then photon limited flows will dominate at larger planetary radii and lower masses than ``energy-limited''. The transition between ``photon-limited''  and ``energy-limited'' flows can be found by taking Equation~\ref{eqn:EL} as an equality and equating it with Equation~\ref{eqn:mdot_PL} to find:
\begin{equation}
M_p=\eta\left(\frac{h\bar{\nu}_h}{4Gm_h}\right)R_p
\end{equation}
{ Note this expression indicates that when the photon energy is much larger that the depth of the gravitational potential, then the photon limit is the relevant one at low fluxes (i.e. we can assume heating is sufficient to allow the atmosphere to escape as we adopt in Section 2.1.1). Whereas, when the photon energy is much less than the depth of gravitational potential, the energy limit is the relevant one at low fluxes as one cannot assume the gas can escape freely.}

One can find the transition between ``recombination-limited'' and both ``photon-limited'' and ``energy-limited'' by equating Equation~\ref{eqn:mdot_RL} with Equations~\ref{eqn:mdot_PL} \& \ref{eqn:EL} respectively to find:
\begin{equation}
W_0\left[-\frac{R_p}{R_s}^{-4}\exp\left(3-\frac{4R_s}{R_p}\right)\right] = - \frac{J_0\alpha_B H}{4c_s^2}\label{eqn:tran1}
\end{equation}
for the transition between ``recombination-limited'' and ``photon-limited'' and
\begin{eqnarray}
W_0\left[-\frac{R_p}{R_s}^{-4}\exp\left(3-\frac{4R_s}{R_p}\right)\right] =&& -\eta \frac{J_0\alpha_B H}{4c_s^2}\nonumber \\&\times&\left(\frac{R_ph\bar{\nu}_h}{GM_pm_h}\right) \label{eqn:tran2}
\end{eqnarray}
for the the transition between ``recombination-limited'' and ``energy-limited'' where we note the last term in brackets is the ratio of the photon energy to binding energy. Equations \ref{eqn:tran1} \& \ref{eqn:tran2} posses either two solution or no solutions and cannot be presented in closed form. Therefore, we proceed graphically.

In computing the various mass-loss regimes we assume that the mass-loss efficiency is constant with planet radius and mass. As lowering the efficiency reduces the energy limited mass-loss rate this makes this limit more restrictive and increases the range of parameters over which the flow can be characterised by energy-limited evaporation. For our fiducial case we take $h\bar{\nu}_h=6.4$~eV, $\eta=0.1$ and use an EUV flux of $3\times10^{4}$~erg~s$^{-1}$~cm$^{-2}$~ and the various mass-loss regimes are shown in top left panel of  Figure~\ref{fig:regions}. In our panels of Figure~\ref{fig:regions} we also show a recent census of the exoplanet population as points.
\begin{figure*}
\centering
\begin{tabular}{cc}
\includegraphics[width=0.5\textwidth]{F3e4_eff10_2} &
\includegraphics[width=0.5\textwidth]{F3e3_eff10_2} \\
\includegraphics[width=0.5\textwidth]{F3e5_eff10_2} &
\includegraphics[width=0.5\textwidth]{F3e4_eff30_2}
\end{tabular}
\caption{The various mass-loss regions are shown as a function of planet radius and mass for various incident EUV fluxes and mass-loss efficiencies. The crossed green region shows ``photon-limited'' mass-loss, the blue striped region show ``energy-limited'' mass-loss and the remaing parameter space is recombination limited. The thin lines show lines of constant density at 0.1, 1.0 \& 10 g cm$^{-3}$ and the points show currently observed exoplanets obtained from the open exoplanet catalogue - data obtained on 5th February 2015 - (Rein, 2012). The top left panel shows our fiducial case for a flux of $3\times10^{4}$~erg~s$^{-1}$~cm$^{-2}$~ and efficiency $\eta=0.1$. The bottom left and top right panels show fluxes of $3\times10^{5}$ \&  $3\times10^{3}$~erg~s$^{-1}$~cm$^{-2}$~ respectively. The bottom right panel shows a flux of $3\times10^{4}$~erg~s$^{-1}$~cm$^{-2}$~ and efficiency $\eta=0.3$. }\label{fig:regions}
\end{figure*}

This moderate flux corresponds roughly to a separation of 0.2~AU around a young star and 0.04~AU for a 1~Gyr old star (Jackson et al. 2012). In this example we see that jupiter mass and above planets are evaporating in the recombination limit, whereas most sub-jupiter mass planets are losing mass in an energy limited fashion. For very low mass planets ($\sim$M$_\oplus$) we find that evaporation is photon limited. We note that while most planets evolve to the left in the diagram as they shrink this is only a minor adjustment for massive planets. However, planets with envelope mass-fractions of order 10s of percent can be much larger at early times, and can often have radii 5-10 times large than at Gyr ages when they are observed today (Lopez \& Fortney 2013; Owen \& Wu 2013).

In the bottom left and top right panels of Figure~\ref{fig:regions} we explore the role incident flux plays, with fluxes of $3\times10^{3}$ and $3\times10^5$~erg~s$^{-1}$~cm$^{-2}$~ respectively. As one expects at lower fluxes the parameter space dominated by energy-limited and photon-limited flows, with the majority of planets loosing mass in an energy limited fashion and a small minority in a photon limited picture. While massive ($>{\rm M}_J$) planets still remain in the recombination limit.

Alternatively at high fluxes no planets are in the photon-limited picture and the energy limited flow occupies a small fraction of parameter space and most planets are evaporating in the recombination limited scenario. Since at this flux the majority of planets that sit in the energy limited-region are low mass. In order to experience fluxes of this magnitude requires the planets to been irradiated at early time ($<100$~Myr - Jackson et al. 2012, Owen \& Wu 2013) and as such the planets plotted here would have been considerably larger and most would sit to the right of the energy-limited region.

Finally, we investigate how the mass-loss efficiency parameter in the energy-limited flows affects the parameter space. In the bottom right panel of Figure~\ref{fig:regions} we show our fiducial model, but with an efficiency parameter of $\eta=0.3$ corresponding to the upper range found by Owen \& Jackson (2012). As expected increasing the efficiency parameter reduces the importance of the energy-limit and the majority of low mass planets are now in the photon-limit. Whereas energy-limited evaporation is constrained to intermediate mass-planets.

The picture described here is consistent with the study of Murray-Clay et al. (2009) who studied a 0.7~M$_J$ planet with a radius of $10^{10}$~cm as a range of fluxes and found that the flow transitions from recombination limited at high fluxes to energy limited at low fluxes around $\sim10^{4}$~erg~s$^{-1}$~cm$^{-2}$~, consistent with our calculations that predict a transition flux of $\sim 2\times10^{4}$~erg~s$^{-1}$~cm$^{-2}$~.

Now we can build up a picture for the mass-loss histories of both low mass planets ($\lesssim 20$M$_\oplus$) and high mass planets ($\sim$M$_J$). Close-in high mass planets begin evaporating in the recombination limited regime, as they evolve and the EUV flux falls with time they may transition to energy limited at some low flux, for typical parameters this is about $10^4$~erg~s$^{-1}$~cm$^{-2}$~; however, we note that this is a strong function of planet mass and radius (as the transition occurs when $R_p/R_s\ll1$). Similarly, close-in low mass planets will begin evaporating in the recombination limit as they contract significantly due to cooling and the flux drops they may make a transition to photon-limited evaporation; however as the flux drops further and they contract even more they will eventually transition to energy limited evaporation. Whether or not low-mass planets experience photon-limited evaporation and when they make the transition from recombination to photon to energy limited is non-trivial. What is certain is there is not a unique value of the flux that one transitions from one regime to the other as assumed in previous evolutionary studies (e.g. Jin et al. 2014). Furthermore, our analysis presented here makes no effort to quantify what the efficiency of energy limited flow is and that can only be answered by numerical calculations such as those we preform in the next Section.

\section{Numerical Simulations}

In the previous section we have put forward the basic physical picture of planetary evaporation driven by EUV radiation, and their scalings. However, the actual mass-loss rates are still unknown as the details of the picture remain unquantified. In particular, the efficiency of any energy limited flow, or the exact temperature profile of any radiation recombination limited flow is sensitive to the exact details of the heating and cooling.

Furthermore, since we only expect the ionization front to be thin in the case of a flow which is recombination limited, and as we shall see it often the scale of the planet, one cannot assume a monochromatic irradiation field (as done by Murray-Clay et al. 2009) and radiation hardening will need to be taken into account. Thus, the only way to obtain an accurate picture of planetary evaporation we must proceed and perform numerical calculations.

We follow many other previous approaches and consider 1D spherically symmetric outflow, where we solve the problem along a streamline connecting the star and planet, neglecting the Coriolis force. Along this streamline we solve the radiation-hydrodynamic problem including EUV irradiation, and non-equilibrium heating/cooling and Hydrogen ionization chemistry. We use the {\sc zeus-MP} hydrodynamics code (Stone \& Norman, 1992, Hayes et al. 2006), which has been previously used for planetary evaporation studies (Owen \& Adams, 2014) and add to it a new module to perform the radiative-transfer, ionization structure and heating and cooling update.

As discussed in Section~\ref{sec:photon_lim}, our numerical setup will result in a ``photon-limited'' mass-loss rate that is too high compared to reality. In fact in all simulations we perform we find that the photon limited rate of our spherically symmetric setup is much higher than any measured mass-loss rates. As such in our numerical study here we only investigate the transition between energy and recombination limited and we must leave an investigation of the transition to photon limited flows to future multi-dimensional calculations.

\subsection{Ionizing radiative transfer}
Ionizing radiative transfer can be solved most efficiently in a causal manner where one performs the ray tracing calculation proceeding away from the source (towards the planet in our case) in a step-wise manner. We operator split our radiative transfer calculation from the hydrodynamics and perform it between the ``source'' and ``transport'' steps of the standard {\sc zeus} scheme. We consider only pure Hydrogen and the equations for ionization and energy due to this update are given by:
\begin{eqnarray}
\frac{{\rm d}X}{{\rm d}t}=(1-X)(\Gamma_i+n_eC_H)-Xn_e\alpha_B \label{eqn:ion_frac} \\
\frac{\partial e}{\partial t}=-P\nabla\cdot{\bf u} +\Gamma_h-\Lambda \label{eqn:eng}
\end{eqnarray}
where $\Gamma_i$ is the photoionization rate, $C_H$ is the collisional ionization rate, $e$ is the internal energy of the gas, $\Gamma_h$ is the heating rate and $\Lambda$ is the cooling rate. We include heating from photo-ionizations and cooling from recombination, collisional excitation, collisional ionization and free-free emission (Katz et al. 1996). { However, we neglect heat transport due to conduction as it known to be sub-dominant by many orders of magnitude (e.g. Murray-Clay et al. 2009)}. We follow (Mellema et al. 2006) and pre-tabulate the heating and ionization rates as a function of neutral hydrogen column density for our input spectrum, thereby capturing the non-monochromatic nature of our problem. Equations~\ref{eqn:ion_frac} \& \ref{eqn:eng} are then solved explicitly by sub-cycling then on time-scales which can be shorter than the hydrodynamic time-step. We note that our scheme requires that we resolve the ionization front, something we have enough resolution to do.

\subsection{Numerical setup}
We perform a series of simulations in order to better understand the regimes which planetary evaporation exists in. Since a full exploration of the entire parameter space numerically is beyond the scope of this work, we instead choose to focus on low-mass planets (where evaporation is more evolutionary important - see Owen \& Wu 2013; Lopez \& Fortney 2013). In all simulations we use an underlying bolometrically heated atmosphere of 1000~K, which is forced to be isothermal by means of a temperature floor. We adopt a grid with 192 non-uniformly spaced cells with a resolution of $\sim$20~km at the base of the grid, and a radial extent from $R_p$ to $30~R_p$. The simulation is initiated with a neutral hydrostatic density structure for a 1000~K atmosphere. { At the lower boundary the ghost cells are reset to the neutral, hydrostatic density and temperature structure every time-step. At the upper boundary we use standard outflow boundary conditions, which are perfect in 1D super-sonic outflow (see Stone \& Norman 1992; Hayes et al. 2006, for their implementation in the {\sc zeus} code). In order to choose the density at the inner boundary we set the density, such that the initial neutral hydrostatic atmosphere has an optical depth of $\tau=100$ to 13.6eV photons, which typically results in densities of order $\sim 10^{-11}-10^{-12}$g~cm$^{-3}$. We find that such a choice is sufficient to keep the inner regions of the active domain part of the bolometrically heated atmosphere rather than UV heated atmosphere.}  The domain is then irradiated by the UV and a  transonic thermal wind is launched. In all cases the flow approaches steady-state and no variability is found. All results are analysed after $\sim 100$ sound crossing times after steady-state is firmly established, { with the mass-flux constant at the $\sim 10^{-3}$ level (see Figure~2)}.

We perform two series of simulations: one at constant planet mass with a mass of 10~M$_\oplus$ where we vary the planet radius and one set at constant radius of 3.14~R$_\oplus$ where we vary the planet's mass. In both cases the radius/mass are varied such that the smallest/most massive planets have a density of 10~g~cm$^{-3}$ and the largest/least massive have at ratio of planet radius to the ``Parker'' radius\footnote{The ``Parker'' radius is the sonic radius of an isothermal flow, where we take the temperature of the underlying bolometrically heated atmosphere and is given by: $GM_p/2c_s^2$; planets larger than this radius are not hydrostatically bound over evolutionary time-scales.} of 0.2. For each set we perform 24 for individual simulations at { 12 different fluxes varying between $\sim 10$ and $\sim10^6$~erg~s$^{-1}$~cm$^{-2}$~ (where we take the EUV luminosity contained between 13.6 and 30eV)}. Our input spectrum was calculated by Ercolano et al. (2009) and has been used in previous exoplanet evaporation studies by Owen \& Jackson (2012) and extends out to several 100eV (although photons with high energies have an incredibly small photo-ionization cross-section with Hydrogen, $\sigma\propto E^{-3}$).

\subsection{Results}

As expected we identify two limiting flow types: recombination limited at high fluxes and energy limited at low fluxes. Note as discussed above ``photon-limited'' flow cannot be found for realistic parameters in 1D spherically symmetric simulations. We show the different flow topologies for recombination limited (top panels) and energy limited flow (bottom panels) in Figure~\ref{fig:topology}. Both simulations are for a planet with a radius of 3.14~R$_\oplus$ and mass of 6.25~M$_\oplus$. The recombination limited flow has a flux of $10^{6}$~erg~s$^{-1}$~cm$^{-2}$~ and the energy limited flow has a flux of $10^{3}$ ~erg~s$^{-1}$~cm$^{-2}$~.

\begin{figure*}
\centering
\begin{tabular}{c}
\includegraphics[width=1.\textwidth]{RL3}\\
(a)\\
\includegraphics[width=1.\textwidth]{EL3}\\
(b)
\end{tabular}
\caption{ Flow toplogy for a planet with radius of 3.14~R$_\oplus$ and mass of 6.25~M$_\oplus$ receiving an ionizing flux of $10^{6}$~erg~s$^{-1}$~cm$^{-2}$~ (a) and $10^{3}$~erg~s$^{-1}$~cm$^{-2}$~ (b). The top left panels show the gas temperature; bottom left panels show the ionization fraction; top middle panels show the velocity (solid) and sound speed (dashed); bottom middle panels show the mass-flux (i.e. $4\pi r^2 \rho u$); top right panels show the density and bottom right panels show the volumetric heating rate ($q$) multiplied by $r^3$ to indicate where most of the heating is taking place.}\label{fig:topology}
\end{figure*}

In the ``recombination-limited'' case (top panels), we see that the size of the ionization front is small compared to the planet's radius and the flow is fully ionized at large radius indicating recombination-ionization equilibrium. The temperature quickly reaches $10^{4}$~K at the ionization front and the sonic point is very close to the planet's surface. { Furthermore, we note that most of the heating is concentrated close to the planet's surface, as indicated by the heating rate profile ($qr^3$ - multiplied by $r^3$ to indicate the radius where the majority of heating takes place) shown in the bottom right panel of Figure~2a.} We find that at these low masses the flow does adiabatically cool slightly at large radius although this is in the super-sonic regime of the flow it doesn't affect the mass-loss rate when compared to the simple estimates presented above.

In the energy limited case (bottom panels), we find that the ionization front essentially encompasses the entire flow and the flow is still partly neutral at 10 planetary radii. This results in a much colder flow only reaching $\sim 5000$~K at 2 planetary radii, before adiabatically cooling at large radii. This results in the sonic point being pushed to a larger radius. { This flow topology is emphasised by the fact the heating rate peaks at larger radii than in the recombination limited case and is more spatially extended, as shown in the bottom right panel of Figure~2b.}

We can investigate the transition between energy-limited and recombination limited in Figure~\ref{fig:tran} where we plot the normalised mass-loss rate (where the mass-loss rate for all models is normalised such the mass-loss rate is unity at the lowest flux of $\sim 10$~erg~s$^{-1}$~cm$^{-2}$~) against flux (top panel) and mass-loss rate divided by flux (bottom panel) for a planet with several different masses: 5.12, 6.25, 10.31, 13.92~M$_\oplus$ (with a radius of 3.14~R$_\oplus$). In the insert we show the logarithmic derivative of mass-loss rate with flux. At low fluxes the mass-loss rate scales linearly with flux following the dashed line as expected for energy limited flow; however, at large fluxes the mass-loss rate scales approximately as the square root of flux (following the dotted lines), indicating recombination limited flow. We find that the transition point varies with mass and perhaps more interestingly the transition is slow, taking a large range in flux.The predicted transition fluxes from Section~2 -- as indicated by the vertical lines in the insert of Figure~\ref{fig:tran} -- pick out the end of the transition. As expected planets with higher masses become recombination limited at lower fluxes due to there increased flow-time scales.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mdot_flux_increase_mass_3}
\caption{The top panel shows the normalised mass-loss rate (the mass-loss rate for all cases is scaled to equal unity at the lowest flux of $\sim$10~erg~s$^{-1}$~cm$^{-2}$~) as a function of flux shown as solid lines for a planet with a radius of 3.14~R$_\oplus$ and masses of (from top to bottom) 5.12, 6.25, 10.31, 13.92 M$_\oplus$. The dashed line shows the scaling expected for a pure ``energy'' limited flow. The dotted line show lines propotional to the square root of flux, the scaling expected from recombination limited flows. The bottom panel shows the same mass-loss rates divided by flux (again scaled so that the ratio of mass-loss rate and flux is one at the lowest flux of $\sim$10~erg~s$^{-1}$~cm$^{-2}$~. The insert shows the logarthmic derivative of mass-loss rate by flux at the different masses 5.12 (solid), 6.25 (dashed), 10.31 (dotted), 13.92 (dashed) M$_{\oplus}$, the vertical lines show the analytical parscription would predict the flow transitions from energy limited to recombination limited.}\label{fig:tran}
\end{figure}

We can see this transition in more detail if we plot the mass-loss rate as a function of flux for various planet masses, for the full simulated range of incident flux. The results are shown in Figure~\ref{fig:mdot_mass} with the $1/M_p$ scaling for energy-limited flows shown as dotted lines. We find at low fluxes the simulation is well matched by an energy limited scaling; however, at higher fluxes the trend becomes much flatter indicating recombination limited flows (Noting that in this parameter range the planet's escape temperature is $<10^{4}$~K so mass plays a limited role as discussed above). The dashed line in Figure~\ref{fig:mdot_mass} shows the predicted analytic transition from recombination to energy limited flows. Above the line is predicted to be recombination limited and below is predicted to be energy limited. We see this prediction agrees well with the numerical results.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mdot_mass_2}
\caption{Mass-loss rate as a function of planet mass for the full ranges of fluxes (bottom to top) logarithmically spaced between $10$ and $10^{6}$~erg~s$^{-1}$~cm$^{-2}$~. The dotted lines show the $1/M_p$ scaling expected from ``energy'' limited flows. The dashed line shows the predicted transition between recombination limited flows and energy limited flows from Section~2, above the line is predcited to be recombination limited and below is predicted to be energy limited.}\label{fig:mdot_mass}
\end{figure}

Finally, in order to measure the efficiency of the flow we measure the mass-loss rate as a function of planet radius (due to its steep dependence this should provide a good range of mass-loss rates over which to tune this parameter). The discussion in Section~2 indicates that at low fluxes and low masses the flow should be strongly energy limited, which is indeed what we find. In Figure~\ref{fig:mdot_rad} we show the mass-loss rate as a function of radius for a planet of mass 10~M$_\oplus$ and fluxes of $\sim 10$-100 (points). We find that an mass-loss rate with an energy efficiency of $20$\% agrees well with the simulations\footnote{Note in order to compare with Equation~\ref{eqn:EL}, we must drop the factor 4 in the denominator that arises from geometrical averaging which does not occur in spherically symmetric simulations.} and as such represents a suitable measure of the mass-loss efficiency in the region of parameter space.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{mdot_rad_eff}
\caption{Mass-loss rate as a function of planet radius. Simulation results are shown as points for fluxes (bottom to top) of $\sim$ 10, 30, 100 erg~s$^{-1}$~cm$^{-2}$~. The solid lines shows the ``energy'' limited mass-loss rate with an efficiency of $ 20\%$.}\label{fig:mdot_rad}
\end{figure}

\section{Discussion}\label{sec:discussion}

We have argued that evaporation due to UV radiation can exist in three limiting cases: ``recombination-limited'' - where the flow is in radiative-recombination equilibrium and approximately isothermal at $10^4$~K; ``energy-limited'' where incoming radiation is balanced by mechanical luminosity; and a new regime:``photon-limited'' - where the mass-loss is limited by the fact that each ionizing photon can allow one hydrogen atom to escape.  Each distinctive limit has a different scaling with planet mass, radius as well as the incident flux. We have shown that there is no unique transition flux from one regime to the other and such a limit depends on the planetary parameters.

Schematically if the flux is high enough such the the density in the flow is large (and correspondingly the recombination time is short) or the planet is so big or massive that the flow time-scale is long then the problem is in the radiative-recombination limited, where the recombination time is much faster than the flow-time-scale. As argued by Murray-Clay et al. (2009); Owen \& Jackson (2012) in such a scenario it behaves as a HII region that is approximately isothermal  and the density at the ionization front scales as $J_0^{1/2}$, in this region the mass-loss rate (in a 1D calculation) is analytically tractable and we present such a calculation in Section~ 2.1.2.

If one decreases the flux, the density in the ionization front will drop and the recombination time will increase. Once it reaches the point where the recombination time is longer than the flow time-scale then recombinations can be ignored.  When the recombination time is much longer than the flow time then we also find the cooling time is longer than the recombination time and subsequently the flow time-scale. In this case radiative cooling can be ignored. This is what one thinks off as standard ``energy-limited'' flow. However, we note that when the recombination time is of order the flow-time scale then the cooling time is shorter than the recombination time and thus shorter than the flow time-scale. Thus, one expects the transition from recombination to energy limited to happen smoothly and broadly over a large range of fluxes. This is indeed what our numerical calculations indicate, that the transition occurs over several orders of magnitude in flux. This is different to the case of X-ray heating discussed by Owen \& Jackson (2012), where since the heating and cooling times scaled identically with flux then the cooling time was always comparable or shorter than the flow time-scale. Thus, in the case of X-ray heating Owen \& Jackson (2012) argued that the flow was never ``energy-limited''.

If when one reaches the point where the recombination time is becoming longer than the flow time-scale and the planet is not tightly bound (such that the energy limit is not restrictive) then the mass-loss rate will be limited by the number of ionizing photons. Such that one can only lose one hydrogen atom per ionizing photon. Obviously, in this limit planet mass is completely irrelevant and the mass-loss rate just scales as the flux $\times$ area.  Such a case is similar to the evaporation of low-mass neutral clumps by nearby young massive stars discussed by Bertoldi \& Mckee (1990). However, this regime will only be restricted to a small (but perhaps important) regime of parameter space for low-mass large radii planets, similar to the masses and radii of the {\it Kepler} planets at young 10s Myr ages (e.g. Lopez et al. 2012, Owen \& Wu, 2013). Unfortunately, such a problem could not be investigated with realistic planet parameters in 1D spherically symmetric simulations. This is because such simulations assume spherical irradiation, in this case the absorbing area would be $4\pi R_{\rm IF}^2$. However, in the reality of planet-parallel irradiation and the fact the ionization front is marginally optically thin then we suspect the absorbing area is more likely $\sim\pi R_p^2$. In this regime the ionization front is thick and by construction has a radius $R_{\rm IF}\gg R_p$ and as such the ``photon-limited'' mass-loss rate from the simulations is much higher than reality and any other limit. One needs to perform 2/3D simulations to investigate the role of ``photon-limited'' mass-loss in planetary mass-loss.

Our numerical calculations indicate that the hydrodynamic efficiency if fairly low for low-mass planets $\sim 10-20$~\%, comparable to the values used in planetary evolution studies (e.g. Lopez et al. 2012, Lopez \& Fortney 2013) indicating that much of the absorbed energy remains as kinetic/thermal energy in the flow rather than being converted to mass-loss. When the flow transitions to energy or photon limited the ionization front is thick; essentially encompassing the entire the flow. Furthermore, we note that not all the heating is concentrated at the planet's surface as can be seen in the bottom panels of Figure~\ref{fig:topology}, where the maximum temperature is reached at $\sim 2$ planetary radii. The ionization fraction increases monotonically with radius but still contains a significant neutral fraction out to many planetary radii. Therefore simplified calculations assuming infinitesimally thin ionization fronts are not appropriate in this limit. This means photon hardening can play an important role in pre-heating material; { this occurs when harder photons penetrate deeper than the standard ionizing ($\sim 13.6$~eV) photons resulting in an average photon energy that is harder than the total spectrum, these photons 'pre-heat' the underlying gas}. Thus, one requires a non-monchromatic numerical method to take this into account (unlike Murray-Clay et al. 2009, who assume a spectrum with just 20~eV photons).

Finally, we note we have presented a new numerical algorithm for the inclusion of UV heating in planetary evaporation simulations. We have included the frequency dependence of the ionizing spectrum for pure Hydrogen atmospheres and note this can easily be extended to include more species (Friedrich et al. 2012). Such an algorithm can be incorporated into multi-dimensional calculations that include ray-tracing (e.g. Owen \& Adams, 2014) to investigate planetary evaporation in 2/3D. Such a further step is important, not only to investigate the flow properties, in particular ``photon-limited'' evaporation identified here; but, also to produce realistic calculations that can be compared to observational indicators of planetary evaporation. This includes measured transits in Lyman-$\alpha$ (Vidal-Madjar et al. 2003, Ben-Jaffel et al. 2007, 2008, Lecavelier Des Etangs et al. 2010) and other metal lines (Vidal-Madjar et al. 2004, Ballester \& Ben-Jaffel 2015).

\section{Summary}
We have investigate planetary evaporation under the influence of irradiation by ionizing UV photons at close enough separations such that evaporation takes place in the hydrodynamic limit. We argue that there are three main evaporation limits. The first two were demonstrated by Murray-Clay et al. (2009), at high fluxes the flow is in radiative-recombination equilibrium and the mass-loss rate scales as the square-root of the ionizing flux; the second where recombinations are negligible and the mass-loss rate is limited by the amount of incoming energy and as such scales linearly with the ionizing flux. Finally, we argue that there is a third limiting factor that must be taken into account, which we term ``photon-limited'' mass-loss. In photon-limited mass-loss one can only lose a single hydrogen atom per incoming ionizing photon and is important when the planets gravity is weak (low density, low mass planets).

We present a basic physical picture of the three different evaporation regimes and show that transitions from one regime to another does not happen a specific UV flux, as assumed in some recent evolutionary studies (e.g. Jin et al. 2014). The different evaporation regimes dominate different planetary parameters and different flux and we present a framework that can be incorporated in to planet evolution studies, provided a mass-loss efficiency is chosen for the energy-limited flows.

We have presented a new numerical algorithm for performing radiation-hydrodynamic simulations that takes into account the frequency dependence of the irradiation spectrum. It does not require the ionization front to be thin, or approximately isothermal and self consistently finds the correct ionization and thermodynamic structure of the flow. This radiative transfer scheme has been incorporated into the {\sc zeus} hydrodynamic code.

Using 1D spherically symmetric numerical simulations we have investigated the transition between ``energy-limited'' flows and ``recombination-limited'' flows. We take into account the non-monochromatic nature of the ionizing photons unlike previous works (e.g. Murray-Clay et al. 2009). The transition from recombination limited to energy limited occurs when the recombination time becomes longer than the flow time-scale and the ionization front encompasses the entire flow scale. In this case radiation hardening from the full spectrum is an important contribution to the thermodynamics.

We find that the transition from recombination limited to energy limited is broad and takes several orders of magnitude in flux. Planets with deeper potentials transition to energy-limited flows at higher fluxes consistent with our analytically expectations. Furthermore, we find that in the`energy-limited case efficiencies around 10-20\% provide a good fit to mass-loss rates.

Finally, we could not use our 1D spherically symmetric simulations to investigate ``photon-limited'' flows and future mutli-dimensional calculations are required to investigate the importance of this type of flow. Something that is possible with our presented algorithm and we plan to incorporate this into the multi-dimensional ray-tracing scheme for planetary evaporation presented by Owen \& Adams, (2014).

\acknowledgments
We thank the two anonymous referees for constructive reports that improved the article. We are grateful to Noam Soker and Ealeal Bear for comments on the original manuscript. JEO acknowledges support by NASA through Hubble Fellowship grant HST-HF2-51346.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. JEO is grateful to CITA for hospitality during the completion of this work. Several of the numerical calculations were performed on the Sunnyvale cluster at CITA, which is funded by the Canada Foundation
for Innovation.

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Victor Alicea
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"The present is the only pathway that connects our past with our future," Dr. Victor G. Alicea, is often heard to remark. Always present in the "here and now" Victor Gabriel Alicea has worked tirelessly for the education and flourishing of all people of New York City. He is the founder and president of Boricua College for the past 46 years.
Dr. Victor G. Alicea was born in Ponce, Puerto Rico and spent his childhood in New York City's East Harlem "El Barrio" and in Harlem. He attended the City's public school system and upon graduation entered City College in 1956 as a pre-engineering student. After three years he transferred to Columbia University, where he completed a Bachelor of Science degree in Experimental Psychology (1963), Masters in Clinical Social Work (1966), Masters in Philosophy (1977) and Ph.D. in Urban Planning (1978). He also completed a three year post-graduate program in Psychotherapy at the Gestalt Psychotherapy Institute (1980).
Victor Alicea has worked throughout his life combining intellectual strength and his vast knowledge of education and human polity with the caring he has for people. He has participated and led numerous institutes in public administration, group dynamics, community organization, health and mental health care services and is a practicing social worker, educator, planner, consultant to various community development programs and is a psychotherapist in private practice.
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Gun-rights groups aim for another bear hunt
TALLAHASSEE, Fla. (NSF) - As a renewed debate builds about the issue, the National Rifle Association and the Unified Sportsmen of Florida are urging the Florida Fish and Wildlife Conservation Commission to hold a bear hunt later this year.
The gun-rights organizations' influential Tallahassee lobbyist, Marion Hammer, sent a letter Tuesday to the commission that also called for steps such as increasing the number of days to hunt.
"Bears continue to terrorize homeowners and prevent families from allowing children to play outside in some areas," Hammer, a past president of the NRA and the executive director of the Unified Sportsmen of Florida, wrote. "And while FWC (the commission) is working to educate people about securing trash and is trying to move dangerous bears out of residential areas, those programs are helpful but cannot succeed without hunts to reduce the population."
Commissioners, who approved a controversial bear hunt last year but have not made a decision about another hunt, are expected to receive a staff recommendation prior to a meeting June 22 in the Franklin County community of Eastpoint.
The October 2015 hunt --- the first in the state in more than two decades --- was scheduled for seven days, but ended after two days as hunters killed 304 bears. The state agency had put a 320-bear quota on the hunt and later acknowledged it "underestimated the hunter success for the first day."
The state didn't put a cap on the number of permits that could be issued in the 2015 hunt, charging state residents $100 to participate. Hunters from out of state had to pay $300.
Money raised from the permits helps pay for community efforts to manage the bear population through programs such as bear-proof trash containers.
Kate MacFall, Florida state director for The Humane Society, said while hunt opponents push for non-lethal means to reduce human-bear conflicts and see opposition growing against another hunt, they believe commissioners have already decided.
"It's trophy hunting, which we certainly don't support," MacFall said. "Floridians love bears. Their (the bears) subpopulations are already fragmented. They're already having a tough time with habitat destruction, huge developments, so many people moving to Florida … there are so many challenges that these bears already face."
Commission staff members have been holding a series of webinars that recap the 2015 hunt, offer the latest estimates on the numbers of bears in Florida, summarize efforts to reduce incidents involving humans and bears and take public input. The final webinar is planned for Thursday night.
Harry Dutton, leader of the commission's hunting and game management division, said last week that "for a possible future hunt" officials are looking at the length and time of year for the hunt and limiting the number of permits. Also, they are looking at how check stations are monitored, rules for hunting on wildlife management areas, the prohibition on baiting bears, the use of dogs to track bears and the minimum size of bears that could be killed.
In last year's hunt, there was a 100-pound minimum as bears under that weight are considered cubs.
The agency estimates, based on recent surveys, 4,220 bears are in the state, up from 2,640 in 2002. The population growth has been called robust as the estimated bear count was as low as 300 to 500 in the 1970s, when bears were put on the state's list of threatened species. Bears were removed from the list in 2012.
Local government officials from Seminole, Miami-Dade and Volusia counties have voiced opposition to a repeat of the 2015 hunt.
News Service of Florida Assignment Manager Tom Urban contributed to this report.
Information taken from The News Service of Florida.


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text:
Charles Saunders, KB (c.1715 – 7 de dezembro de 1775) foi um almirante da Marinha Real Britânica durante a Guerra dos Sete Anos e depois Primeiro Lorde do Almirantado.

Ele comandou a esquadra que trouxe o general James Wolfe para Quebec em 1759 e consolidou a vitória do falecido general na Batalha das Planícies de Abraão

A ilha Saunders, localizada no arquipélago das ilhas Sandwich do Sul, foi assim nomeada em sua homenagem.

Ligações externas
 Biografia em Dictionary of Canadian Biography Online

Políticos do Reino Unido
Navegadores do Reino Unido
Saunders
Naturais de Doncaster

redpajama_set_name: RedPajamaWikipedia


text:
'Chicago PD' 8×04 Review: "Unforgiven"
The team continues to play their cases completely by the book. Even when it comes to the murder of an officer that Voight knew very well. But I struggled to remain invested in the case of the week, which means it's a good thing that there were some amazing Upstead moments to keep me hooked! And a deeper look at how Hailey's past has affected her present was a nice bonus to see. Plus, Jay supporting Hailey unconditionally had me swooning! So keep reading for my Chicago PD 8×04 review!
Staying Inside the Lines
Even though the team has been following what seems like all of the rules lately, I still find myself surprised. And also waiting for the moment when the shoe will drop and Voight will backslide into some of his old tendencies. When a cop that he said he knew well was killed, I definitely thought this situation would be the one to push Voight back into old ways. But he and the rest of the team continued to keep things above board. And the longer they do that, the more I think that maybe the team, and Voight, are really evolving for the better. So I'll remain cautiously optimistic, just in case.
Image via NBC
When it came to Mike Blaine, he certainly had a lot of enemies. Which meant a lot of suspects. And for what felt like the first time in a long time, I genuinely spent most of the episode with no idea who killed Blaine. But I also found myself kind of uninterested in the latest case of the week. Honestly, I would have been more interested in Maria's story. Especially since I may not have had much of an idea on who killed Blaine, but I had a feeling she was innocent and I would have liked to see the team helping her instead. It was a nice twist to have Maria's ex-boyfriend be the one to kill Blaine.
Deeper Connections
I really like that we are learning more about Hailey as time goes on. And I do like the way that her story and background seem to blend with the case of the week that's happening. Plus, Hailey's history and experiences allowed her to form a deeper bond with Maria. Watching her struggle on whether or not to see her father was hard to watch. But regardless of what other people say or tell her to do, Hailey has to do what's right for her. If she goes to see her father because her family thinks she should then she's not doing it for herself. And it certainly isn't going to give her any closure.
At first, I didn't know how I was going to feel about Samantha Miller. But as this season has progressed, I actually find myself liking her. I think Voight needs someone who isn't afraid to disagree with him and tell him he's wrong about something. But Samantha also isn't afraid to listen to Voight about things when the time is right. Also, I know I've said this before, but there is some serious chemistry between Voight and Samantha. Chemistry that I find myself completely fine with. It's way past time that Voight got a love interest.
All the Upstead
Okay, now I'm going to gush about Hailey and Jay! Their opening scene was freaking magical and had me grinning from ear to ear. They may even be the reason why I wasn't as invested as usual in what was going on. But that's because I need more and this episode didn't provide me with nearly enough about the two of them. Are they officially together now? Because I need them to be. And the way Jay supported Hailey about her father had me freaking swooning! Hailey made a not-so-great choice in fibbing to Jay about visiting her father in the end, but I think she just hasn't been in a relationship in a while and it's hard for her to completely trust anyone. But I don't doubt she'll let Jay in.
Gif via Tumblr, Property owned by NBC
What did you think of Chicago PD 8×04? Were you excited that Jay and Hailey finally seem to be together? What did you think about the case of the week? Do you also see the chemistry between Voight and Samantha? Or are they going to remain colleagues? Did you know who killed Blaine? Or were you as unsure as we were? Find and follow us on Twitter @capeandcastle to share all of your Chicago PD 8×04 thoughts with us! Be sure to check out our site for more content on all things Disney and beyond!
Chicago PD, Chicago PD 8x04, Chicago PD 8x04 Recap, Chicago PD 8x04 Review, Chicago PD Recap, Chicago PD Review, Jason Beghe, Jesse Lee Soffer, LaRoyce Hawkins, Marina Squerciati, NBC, Patrick John Flueger, Review, Tracy Spiridakos


text:
AACR Minority and Minority-serving Institution Faculty Scholar Awards
Scholar-in-Training Awards: Annual Meeting
Global Scholar-in-Training Awards: Annual Meeting
Women in Cancer Research Scholar Awards
Minority and Minority-serving Institution Faculty Scholar Awards
Minority Scholar Awards
Undergraduate Scholar Awards
Scholar-in-Training Awards: Other AACR Meetings
CSCO Young Investigator Travel Awards
AACR Minority and Minority-serving Institution Faculty Scholar in Cancer Research Awards
The American Association for Cancer Research (AACR) is extremely pleased to announce the availability of scholar awards in cancer research for full-time minority faculty members and faculty members of minority-serving institutions (historically black colleges and universities, Hispanic-serving institutions, and other minority institutions as defined by the US Department of Education). The purposes of this award program are to increase the scientific knowledge base of minority faculty members and faculty members at minority-serving institutions, and to encourage them and their students to pursue careers in cancer research.
AACR Minority and Minority-serving Institution Faculty Scholar in Cancer Research Awards are presented by the American Association for Cancer Research to scientists who are full-time underrepresented minority faculty members (underrepresented minorities as defined by the NCI include African-Americans, Hispanic Americans, Native Americans, Native Pacific Islanders/Hawaiians, and Alaska Natives) and faculty members of minority-serving institutions (as defined by the U.S. Department of Education) at the level of assistant professor or above, who are engaged in meritorious basic, clinical, or translational cancer research, and are the presenters of meritorious abstracts at the AACR Annual Meeting or special conferences.
Applications for the AACR Annual Meeting 2020 are now OPEN.
Candidates must have completed doctoral studies or clinical fellowships and hold full-time faculty status at an institution designated as a minority-serving institution or be full-time underrepresented minority faculty members at another institution.
Candidates must have acquired doctoral degrees in fields relevant to cancer research.
Candidates must be citizens or permanent residents of the United States or Canada.
Candidates must be the presenters of meritorious abstracts at the AACR Annual Meeting or special conferences.
Candidates need not be members of the AACR to apply for this award.
Applications for the AACR Annual Meeting 2020 must be completed through myAACR.
Log in to the myAACR portal. If you do not have a myAACR account, select, "Create New Account."
Once you are logged in, click on, "Applications/Awards." After you've been transferred to the awards portal, click on, "View All Activities," and choose this award application.
Required Application Materials:
Official application form/online application. Provide all information requested. The applicant must list one mentor in a supervisory capacity (i.e. dean, department head, etc.) at the candidate's institution who could certify the information in the application upon request.
Personal statement written by the candidate (no longer than two pages in length) describing the benefit he/she expects to derive from attending the conference and the reasons for the candidate's interest in the conference.
One letter of reference from a faculty member in a supervisory role at the applicant's institution. The letter should be addressed to the MMSI Faculty Scholar in Cancer Research Awards Committee, and should comment on the applicant's academic and personal qualifications for this award. Letter of recommendation should be on letterhead with visible signature.
A copy of the abstract submitted for presentation at the meeting. Candidates must provide the abstract control number.
Curriculum vitae, including a complete bibliography (list of publications).
AACR Special Conferences
AACR special conference complimentary registration or registration reimbursement will be provided to all awardees. A stipend to support participation in the selected AACR special conference will be distributed to awardees for expenses such as travel, accommodations, and meals. If an awardee is unable to attend the meeting for which the award is given, the award must be forfeited.
Award applications for participation in an AACR special conference:
The deadline and application form for applying for an AACR Minority and Minority-serving Institution Faculty Scholar Award for participation in an AACR special conference is listed in each conference website on the financial support for attendance page.
Please also review carefully the abstract submission deadline for the conference you are interested in attending. Usually the award application and the abstract are due on the same day, and these are separate processes.
AACR special conferences on focused topics in cancer research have gained wide recognition as unique opportunities for in-depth discussion of important scientific issues in attractive, informal resort environments.
Download and complete the application by visiting the specific conference webpage and include the required application materials. Complete applications must be submitted on or before the application deadline and emailed to micr@aacr.org.
A copy of the abstract submitted for presentation at the meeting. Candidates must indicate the temporary abstract number.
Awardees will be selected on the basis of their qualifications, references from mentor, abstracts, and an estimation of the potential professional benefit to the awardees.
An advisory committee consisting of senior members of AACR Minorities in Cancer Research (MICR) carefully reviews applications. Please note that when reviewing applications, the award selection committee will also strongly consider the following factors:
Candidates who are presenters of an abstract submitted for presentation at the meeting (or who have students presenting an abstract on which the candidate is an author);
First-time applicants; and
Financial need for travel funds.
Please keep in mind that because of these factors the total number of awards given to any one applicant will be restricted.
Award Conditions and Reporting
In addition to scientific sessions, awardees must attend all special events organized for them.
Awardees must submit a written report commenting on the scientific sessions that they attended and the MMSI Faculty Scholar Award Program.
According to IRS regulations, the award is subject to federal income tax. Award recipients will receive additional related information.
Generously supported by the AACR and National Cancer Institute's Center to Reduce Cancer Health Disparities
The National Cancer Institute's Center to Reduce Cancer Health Disparities provides opportunities and support for research training and career development of minorities and for involving minority institutions in cancer research, research training, education, and outreach.
Please note that this award, although funded by the National Cancer Institute (NCI), is administered by the AACR. All communications must go through AACR and not the NCI. Therefore, should you have any questions regarding the status of awards, deadlines, and award requirements, please contact AACR at micr@aacr.org or 215-440-9300.
2019 Awardees
Congratulations 2019 Minority and Minority-serving Institution Faculty Scholars in Cancer Research.
AACR Minority and Minority-serving Institution Faculty Scholar in Cancer Research Awards
615 Chestnut Street, 17th Floor
Email: micr@aacr.org


text:
require 'spec_helper'
describe "analytics code snipets" do
  describe "gauge" do
    it "generate gauge snippet if site exists" do
      Monologue::Config.gauge_analytics_site_id = "gauge id here"
      visit root_path
      page.should have_content("t.setAttribute('data-site-id', 'gauge id here');")
    end

    it "does not generate gauge snippet if no site id has been set" do
      Monologue::Config.gauge_analytics_site_id = nil
      visit root_path
      page.should_not have_content("t.setAttribute('data-site-id'")
    end
  end

  describe "google analtycs" do
    it "generate GA snippet if site exists" do
      Monologue::Config.google_analytics_id = "GA id"
      visit root_path
      page.should have_content("_gaq.push(['_setAccount', 'GA id'")
    end

    it "does not generate GA snippet if no site id has been set" do
      Monologue::Config.google_analytics_id = nil
      visit root_path
      page.should_not have_content("_gaq.push(['_setAccount', ")
    end
  end

end

redpajama_set_name: RedPajamaGithub


The vision I have for "Experiences from Life" is to help others learn. My goal is to add more value to your world by giving you tools that you can immediately use. Enabling others more time to be productive instead of going around in circles or spending hours researching. Through me, I want to help others cut through all the nonsense and get straight to it. I am here to help you in every possible way I can. My commitment is to bring you the very best of what I'm living and learning and to keep it real and honest.
Daily mistakes are made so grace is needed here! I am sincere and want to gain your trust. Please know I am giving you the best I possibly can to enable you to achieve much more than you imagined. I find a lot of satisfaction in helping others and watching them grow.
This blog is not set on one theme, such as baking cakes. Instead, it is the whole bakery, if you will. Here you will find posts such as How Formula Stopped Acid Reflux, 6 Steps to Finding a Family Church, How to Create a Daily Routine, Get Collections off Your Credit Report, and Alternative to Cage Dryer for Dogs. Being a perfectionist, I researched for a year on how to start a blog. I am finally taking the plunge and cannot wait to share with you all that I learn.
I was working for a delightful attorney (they do exist!) and we had an exceptional working relationship. His entire family became my family. Due to health conditions, I had to resign. It literally took God two years to convince to do that. The day I drove to work and became dizzy at the wheel was the day I said enough. No matter what, my health comes first. The next year I focused on healing my body. To help with the healing process, I completely redecorated my house. During one of my Pinterest searches, I came across a post regarding starting a blog. The researcher in me went to work and a year later "Experiences from Life" was born.
I am a wife, mother, and child of God who struggles daily. There is no doubt in my mind that I am where I am because of God. My family is my rock and inspire me in so many ways. In the 1990's I started traveling the world with my family due to my father's job. The experiences I gained from overseas were some of the most influential in my life. I have always enjoyed writing and reading. In fact, I would rather write than try to verbally express myself. Writing helps me think and get to the real problem instead of focusing on my emotions. Growing spiritually, learning, music, dancing, the beach, helping those in need, having everything in its place, goals, being financially stable are some tidbits about me. Above all, helping others is my passion. I pray this blog does just that.
Thank you so much for stopping by. I hope you leave this site with a newly learned skill, idea, or a feeling of encouragement. Feel free to contact me at rissa@experiencesfromlife.com with any question. Together we will grow!


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The Bowls England Greens Maintenance Advisory Scheme is a service to Bowling Clubs and the hard working people who maintain, often in a voluntary capacity, that hallowed, treasured green. The green is the focal point of every club and as such has a vital role in the club's life.
The principal aim of the scheme is to offer practical assistance to the club whose green needs attention, in the quickest and simplest manner possible. This is being achieved by a system of Regional Co-ordinators and County Co-ordinators who by reason of their work, experience and constant involvement in green maintenance work, will provide a fund of local knowledge. This will include not only a degree of practical knowledge of greenkeeping, but also recognition of problems and where the expertise and machinery to solve that problem is readily available.
If your club requires help, please contact your County Co-ordinator, and you will immediately be in touch with a sympathetic and knowledgeable ear. He may refer you to a more local source of help or more likely get someone who can advise you on your green – quickly. In all cases the scheme is geared to getting advice promptly – the remedy and the work entailed will be the responsibility of the club.
It is envisaged that the majority of your problems, particularly the 'panic' situations, will be dealt with locally. In all cases requiring more specialist knowledge, your county representative will be able to call on the experience and knowledge of the Regional Co-ordinator and his advisors.
This group of people in conjunction with the commercial concerns assisting the Bowls England in this scheme, will also arrange local seminars on such matters as early season preparation work, machinery, irrigation and laying up the green for the winter. At the national level, the scheme's administrator will advise his regional co-ordinators on contacts with interested commercial concerns and finance. He will generally oversee the scheme and direct its development so that it meets the demands of all bowling clubs.
With this constant review and improvement in mind, your County Co-ordinator, Regional Co-ordinator and National Administrator will welcome your constructive suggestions and comments.
For more information on greens maintenance, please visit the Bowls England website.


Peter Johnston is synonymous with understated, effortless elegance. "We have created a unique cut, simple yet classic. Eschewing bold logos and unnecessary aesthetics, our fine tailoring is defined by four key principles: the finest quality materials, first-rate craftsmanship, traditional attention to detail and timeless functionality."
Peter Johnston offers clients the full wardrobe of bespoke tailoring.
BUSINESS TAILORING: Suiting or smart casual dress down, ready for business either in the boardroom or informal meeting environments.
CASUAL TAILORING : Sports coats, blazers, top coats and separate trousers comfortable for the weekend, holidays or for business travel.
FORMAL TAILORING : Evening dress (black tie), Morning dress (wedding or court) and Full Evening dress (white tie / tailcoat).
We take great pride in sourcing the finest Scottish, English and Irish cloths.
Using only the finest quality worsted suitings from the textile mills of Huddersfield, woollen flannels from the West of England and the world's best cashmere and tweeds from the Scottish Borders, highlands and the islands of Harris or Shetland. Having access to thousands of Scottish clan tartans; ancient, modern, hunting and dress tartans.
Hand-cut and hand tailored using only the finest craftsmanship.
Peter Johnston has an experienced team of cutters, tailors, finishers and pressers. We hand-cut and hand tailor all of our clothing, ensuring that the highest standards of craftsmanship are maintained at every stage.
Traditional dress code for all occasions.
Peter Johnston is very well versed on the traditions and etiquette of Gentlemen's dress codes. Traditional occasions in Scotland and England often requires a multitude of different dress codes. Whether it be traditional Evening Dress (black tie), Morning Dress (wedding or Racing) or even the very formal Full Evening Dress (tailcoat / white tie). Every client is given the confidence that they are dressed correctly for every occasion.
Exquisite, hand worked details throughout each tailored garment.
Hand-worked details are added as each piece of tailoring reaches completion - fine hand sewn buttonholes, real horn buttons, hand sewn edge or top stitching and the discreet hand sewn on labels.
Bespoke tailoring that is a reflection of a Scottish background.
Peter Johnston was born, raised, works and lives in Scotland. The townhouse showroom is located in the very prestigious "New Town" district of Edinburgh, a UNESCO World Heritage site.
A contemporary style that stands the passing of time.
Peter Johnston's style is timeless, not affected by changes in the whims of fashion. A distinctive house cut delivering a classic shape, bringing balance, proportion and enhancing the body's natural outline.


In September 2018, Murray Feldman and crew went to Oklahoma City for the annual IPT Forum. Experts from around the country to review how far U.S. organizations have come with regards to investor education and protection and look at what needs to be done in the future to ensure all Americans have the information they need to make wise and safe investment decisions. View keynote addresses and interviews.