Radiation Pressure

Shaikhislamov et al.​ estimate the effects of radiation pressure on optically thick gas near 0.045 AU to be negligible for all of their models of the near-planet environment.

Bourrier and des Etangs​ use calculated Lyman-alpha profiles for HD209458 and HD189733 in particle simulations to model radiation pressure on neutral hydrogen escaping the planet. They used a parameter which decreased the effect of stellar gravity, but unlike many studies, their parameter is a function of velocity (Doppler shift, from the stellar Lyman-alpha profiles) and penetration depth (effectively optical depth):

F_{\star,eff} = (1-\beta)F_{\star}$ $\beta(v,\Lambda)=\beta(v,0)e^\frac{{-\sigma_{v0} \lambda_0}}{\Delta v}\int_0^\Lambda n_H(\mu,v)d\mu$

They argue that the absorption signature in the blue wing could be due to radiation pressure accelerating neutral hydrogen away from the star in the radial direction (up to ~120 km/s), but the absorption in the red wing cannot be explained in this manner.

Khodachenko et al.​ use a simplified version of the Lyman-alpha flux to calculate radiation pressure in hydrodynamic simulations. They appear to have radially averaged the radiation pressure — the pertinent equations we would apply are:

x(v_i,T) = \frac{v_i - v_R}{\sqrt{\frac{2kT}{m_P}}}$ $z(x^2) = \frac{x^2-0.855}{x^2+3.42}$ $q(x^2) = \begin{cases} \frac{21+x^2}{1+x^2}z(0.1117+z(4.421+z(5.674z-9.207))), & z(x^2)>0 \\ 0, & z(x^2) < 0 \end{cases}$ $\sigma(v_i, T) \approx 5.9\times 10^{-14}\sqrt{\frac{10^4 \text{K}}{T}}e^{-x^2} + 2.9\times10^{-19}\left (\frac{100 \text{km s}^{-1}}{v_i - v_R}\right )^2 q(x^2)$

where v_i is the Doppler-shifted velocity for each frequency bin, v_R is the radial velocity of the cell, and T is the temperature. They find that radiation pressure is negligible in comparison to charge exchange (in 2D).

Wind Launching

Schneiter et al.​ calculate effects of radiative transfer and photoionization, but do not launch the wind this way. Murray-Clay et al.​ calculate 1D wind from UV flux. Tripathi et al.​ calculate multidimensional wind due to heating from UV flux. Krumholz, Stone, and Gardiner​ present the simulation method used by Tripathi et al in the Athena code.

Summary of Krumholz, Stone, Gardiner: No assumption of equilibrium (thermal or ionization). Usual MHD equations:

\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$ $\frac{\partial}{\partial t}(\rho v) + \nabla \cdot (\rho v v - B B) + \nabla P^* = 0$ $\frac{\partial B}{\partial t} + \nabla \cdot (vB - Bv) = 0$ $\frac{\partial E}{\partial t} + \nabla \cdot [(E+P^*)v-B(B \cdot v)] = \mathcal G - \mathcal L$ $\frac{\partial \rho_n}{\partial t} + \nabla \cdot (\rho_n v) = \mathcal R - \mathcal I$

Still need to specify rates of radiative processes.

Algorithm:

q vector of information now includes neutral densities:

\vec q^n = \begin{pmatrix} \rho^n \\ \vec p^n \\ \vec B^n \\ E^n \\ \rho^n_n \end{pmatrix}

Flow control diagram:

In order to update radiation, start by calculating cells through which rays from each source pass - start with 12, but may split into 4 child rays at any time (ref Abel and Wandelt 2002). Need length of rays that intersect each cell for calculating absorbed flux (usual exponential dependence on optical depth, given by number densities). Similar calculations for photoionization and photoionization heating - calculation ends at edge of domain or when only a small fraction of the photons are left. Sum heating and photoionization for every ray which passes through a given cell. Since collisional ionization, recombination, and optically thin heating and cooling are local effects, just calculate these for current state of fluid. Calculate time step from relative size of changes, then change in q is:

\Delta q = \Delta t \begin{pmatrix} 0 \\ \vec 0 \\ \vec 0 \\ \mathcal G - \mathcal L \\ \mathcal R - \mathcal I \end{pmatrix}

Iterate until particular conditions (relating to time step or change in energy) are met, at which point feed to conservative MHD update.

​Tremblin & Chiang, Computational Charge Exchange

Followup to 2008 and 2010 studies of charge exchange between planetary and stellar winds, which used Monte Carlo simulations of 'meta-particles' that were computationally obstructed by bow shocks. Here they use the hydrodynamic equations (no magnetism, Coriolis and centrifugal forces, or tidal gravity). A slow stellar wind (130 km/s) was chosen to approximate the solar wind, and the isothermal planetary wind was initialized as 80% ionized, following Murray-Clay et al. The planetary wind incorporated photoionization/recombination and advection. To incorporate charge exchange, the hydrodynamic code was augmented with chemical reaction solvers - 4 equations relate x_i, i=1-4 representing each possible combination of hot or cold and neutral or ionized hydrogen, and n_H with beta, the reaction rate. These equations take reverse exchange into account, so as not to overestimate neutral hydrogen too greatly (still slightly overestimated). x_i is also included in the hydrodynamic equations. The simulations appear to reproduce the observed absorption curves well, with asymmetry between the two sides of the Doppler shift.

Compare these with the equations used by Christie et al (see below) for incorporating charge exchange:

​Christie paper

2.5D spherical simulations of planetary and stellar wind interactions, including charge exchange, were performed. Hydrodynamic simulations were performed, with density fixed at the base of the planetary wind and an inflow boundary condition on one half of the simulation serving to emulate the stellar wind. In addition to charge exchange, advection, photoionization and recombination, and collisional ionization were included. The escape parameter lambda was used to categorize the models; it was found that there were two distinct regimes, with a transition region between. With lambda ⇐ 4 (high planetary temp), the planetary wind becomes transonic before colliding with the stellar wind, creating a large tail that takes a significant amount of time to mix. With lambda ≥ 6 (low planetary temp), the planetary wind has no chance to become transonic before it encounters the stellar wind, and the winds mix turbulently rather than collide, resulting in a well-mixed, barely evident tail. The transition region between these is also shown clearly in the calculated mass-loss rates of the simulations.

​Schneiter paper (2016)

Paper makes synthetic observations of Lyman-alpha absorption in tails created by interacting solar and planetary winds. Simulations are performed in 3d spherical coordinates, using the Guacho hydrodynamics code, with photoionization of hydrogen included (no magnetism). They have nineteen models of varying stellar UV flux (photoionization rate), stellar wind conditions, and the mass-loss rate of the planet. The planetary wind is initialized self-consistently in order to obtain the desired mass-loss rate, at 3R_p. Both the stellar and planetary winds are isotropic, ignoring effects of tidal locking and atmospheric mixing. They approximate the radiation pressure from the star by reducing stellar gravity.

They find that by including photoionization, a smaller neutral tail is formed, leading to less absorption; they also find a lower time to a stationary state than in previous models without photoionization. By numerically integrating to determine the optical depth, it is seen that the most absorption in in the blue-shifted side, between -130 and -40 km/s. This absorption is most dependent on the mass loss rate of the planet (with more material, there is more absorption) and on the ionizing flux (more ionization, less absorption, in an approximately linear relationship). By comparing their models to observation, the heat efficiency of HD 209458b (the planet modelled for simulations) can be predicted to be less than 50%. In addition, it can be seen that the observed Lyman-alpha absorption does not necessarily require charge exchange to accelerate the neutral hydrogen sufficiently.

​Murray-Clay paper

Authors seek to numerically determine validity of hypothesis that hot Jupiters could be evaporated down to their rocky cores over the planetary lifetime. They use a one-dimensional model that includes heating/cooling terms, tidal gravity, and the effects of ionization on the mass-loss rate, and focus on the substellar point, at which tidal gravity and UV flux are greatest, thereby putting an upper limit on the possible mass-loss rate of the planet (by extending the one-dimensional result over the surface of the planet). They assume a planet of 1.4 R_J and 0.7 M_J and ignore the Coriolis force, under the assumption that the Lyman-alpha radiation from excited H is the only significant cooling term. Numerically, they use a relaxation solver, and find solutions iteratively by removing simplifying conditions one at a time.

They find that, for main-sequence stars, about 20% of H is still neutral at the sonic point, and place an upper bound of ~3.3*10¹⁰ g/s on the mass loss rate. For hotter (T Tauri) stars, they find an upper bound of ~6.4*10¹² g/s. At low flux, the mass loss is energy-limited, while for higher flux, the mass loss is radiation/recombination-limited. The assumption of a hydrodynamic wind is shown to be self-consistent, and they estimate that, due to the directionality of the tidal gravity and the UV irradiation, the maximum rate is an overestimate by ~4x. By reducing the wind speed to subsonic values and including a stellar wind, the day-side wind may be reduced or completely suppressed - they hypothesize that this may lead to night-side outflows, due to circulation of hot gases from the day-side.

They compare observations to estimates from their model, and note that the disagreement in Lyman-alpha lines could be due to a variety of factors, including some missing physics or a cause unrelated to absorption by the planetary wind. A promising candidate is cited as acceleration of neutral hydrogen due to charge exchange. They note that modelled spectrally-unresolved measurements appear to be in agreement with observation.

​Stone-Proga paper

Paper is a comparison of 2D simulations (of close-in hot Jupiters) to spherically symmetric simulations run by others. They characterize escape from the planet with the hydrodynamic escape parameter (lambda), ratio of gravitational potential to thermal potential - a small lambda suggests a thermal wind. They use no magnetism (hydrodynamic rather than MHD), and ignore heating and small-scale effects at the base of the wind. The wind is made self-consistent by fixing the density and energy at the base.

In models with no stellar wind and an isothermal outflow, they find that the sonic surface of the wind is closer to the planet, with a slower radial velocity on the night side of the planet, and a very evident shock in the |v|/v_s plot at various angles. For larger values of gamma, this shock produces a delta T, and the sonic surface is farther from the planet (but still nearer than in the spherically symmetric models). Introducing a stellar wind creates a back-swept profile, but has little effect on the sonic surfaces or mass-loss rate.