The relevant quantity for thermal escape mechanisms is the exospheric kinetic temperature Texo, i.e. the temperature related to the particle velocities.
For terrestrial planets, Texo is determined by radiative heating from stellar γ -ray, X-ray and ultraviolet (UV) radiation (collectively denoted as XUV radi-ation), heating due to exothermic photochemistry, heat conduction, adiabatic cooling due to expansion and IR radiative cooling in the vibro-rotational states of neutral molecules. A detailed summary of these processes is provided by Kulikov et al. (2007), Lammer et al. (2007) and Tian (2009).
An important parameter for thermal escape is the so-called "escape parameter"
λescape, dened as the ratio of potential energy of the particles within the gravity eld of the planet to their thermal (kinetic) energy:
λescape = GMplanmpart RexokbTexo
= Lfree 2
Epot
Ekin (A.2)
where Rexo is the exobase distance to the planet center, G is Newton's gravi-tational constant and Mplan the planetary mass.
The factor of Lfree2 appearing in eq. A.2 must be adapted according to the degrees of freedom of the considered particles (Lfree = 3 for atoms, Lfree = 5 for a diatomic molecules, etc.).
If the escape parameter is smaller than a critical value λcrit, i.e. the kinetic energy of a particle is equal or greater than the potential energy, the particle will be no longer bound to the planet and can move away from the planet.
λescape ≤λcrit = Lfree
2 (A.3)
From eqs. A.2 and A.3, a critical temperature Tcrit can be dened:
Tcrit = 2 Lfree
GMplanmpart
Rexokb (A.4)
Table A.1 shows some values of escape parameters and critical temperatures for typical Earth conditions. Note that the temperatures calculated for the molecules N2, O2 and CO2 are above the respective thermal dissociation tem-peratures.
Table A.1: Escape parameters and critical temperatures for dierent species at Earth's exobase (Texo= 1000 K,Rexo=6700 km)
Species λescape Tcrit [K]
H 7.2 4790
C 86.3 57,500
N 100.7 67,100
O 115.1 76,700
N2 201.4 80,500 O2 230.2 92,000 CO2 316.5 90,400
A.1.1 Jeans escape
Jeans escape is a thermal escape mechanism closely related to a hydrostatic approach in local thermodynamic equilibrium (LTE).
Neither hydrostaticity nor LTE are good approximations in the exosphere (since we are in the collision limit), nevertheless Jeans escape still provides a useful estimate of a lower limit on the escape rate.
The thermal velocities v of gas particles with a temperature T in LTE follow a Maxwell-Boltzmann distribution:
f(v, T) = 4π
( mpart 2πkbT
)3
2
v2e−
mpartv2
2kbT (A.5)
As illustrated in Figure A.1, some of the particles in the so-called Maxwell tail have velocities higher than the escape velocity of the planet and can thus escape into space, even though the mean (or most probable) velocity is lower. This is the general concept of Jeans escape. An atmosphere is slowly "evaporating"
while still being bound to the planet.
Figure A.1: Maxwell-Boltzmann distribution of velocities for hydrogen (red lines) and carbon (black) atoms at T=5000 K (plain line) and T=1000 K (dashed). The vertical green line indicates the Earth's escape velocity of 11,200 m s−1
Integrating the velocity distribution yields the Jeans escape ux:
Φjeans =nexo vt 2√
π ·(1 +λescape)·e−λescape (A.6) wherevt is the thermal velocity of the gas (mpartv2t = 2·kbTexo).
For normal Earth conditions, because of the relatively low temperatures, only atomic hydrogen exhibits signicant Jeans uxes (see Table A.1). With exobase densities of atomic hydrogen of approximately 106 cm−3(Vidal-Madjar 1978), eq. A.6 yields an escape ux of approximately 8· 108 particles cm−2 s−1. All other relevant species (C, N, O) show negligible escape rates.
A.1.2 Hydrodynamic escape
The Jeans escape ux (see eq. A.6) is only a valid approximation below the critical temperature, which is equivalent to λescape > λcrit = Lfree2 .
At higher temperatures, hydrodynamic blow-o occurs. Here, the particles are no longer bound to the planet, as their internal energy exceeds the potential energy. As shown in Figure A.1, for high temperatures a signicant amount of particles have velocities higher than the escape velocity. This means that the atmosphere can freely move away from the planet instead of simply evap-orating, as is the case for Jeans escape. Consequently, the escape rates are much higher. The critical temperature is sometimes referred to as blow-o temperature.
An estimation of the hydrodynamic escape rate can be obtained from two lim-its, the energy limit and the diusion limit (Watson et al. 1981). Both limits are approximately independent of atmospheric structure, so are particularly useful for estimates of escape rates on exoplanets.
In the energy limit (Watson et al. 1981), where a major constituent escapes, the escape is only limited by the maximum possible energy deposition in the exosphere which is needed to overcome the potential well of the planet. In their approximation, Watson et al. (1981) only considered heating by the cen-tral star. However, other energy sources might be present in planetary atmo-spheres. For example, in the case of the giant gas planets of the Solar System, planetary waves are thought to contribute to the exospheric energy budget, as well as internal energy from contraction (Yung and deMore 1999).
When only stellar heating is taken into account, the energy-limited escape ux (in particles of mass mpart per second) is given by:
Fen= Φen·πR2exo =πR2exoSϵ Rplan
GMplanmpart (A.7) where S is the stellar XUV energy ux reaching the exosphere, ϵ the heating eciency and Rplan is the planetary radius. However, eq. A.7 may over-estimate the escape rate because it neglects atmospheric processes such as adiabatic cooling associated with expansion.
The diusion limit takes the atmospheric structure into account, at least to a certain degree. In the diusion limit (Watson et al. 1981), a minor constituent (mass mmin) escapes. In order to do so, the particles must diuse through a main background gas which poses a barrier to overcome. The diusion-limited escape ux (in particles of massmmin per second) is given by:
Fdiff =π·b˙(mmain−mmin)· nmin nmain
GMplan
kbTexo (A.8)
where the indices i and j refer to the minor and main gas respectively, b is a collision parameter describing the diusion of gas i through gas j and mmin and mmain are the molecular masses. The value ofb is about 1021 (m s)−1 (see table 1 in Hunten 1973).
The energy-limited ux provides a theoretical upper limit for the escape ux.
Whether the escape ux of a species is diusion- or energy-limited is
deter-mined by a critical concentrationccrit above which Fdiff > Fen. In the absence of energy sources other than the stellar radiation, this condition is cannot be reached. Hence, the escape ux is energy-limited. For concentrations lower than the critical concentration, the escape ux is limited by the diusion of the escaping component through the atmosphere.
Assuming an XUV ux of 4.6·10−3 J m−2 s−1 for Earth (Lecavelier Des Etangs 2007), a heating eciency of 0.3 (Kulikov et al. 2007) and a temperature at the exobase of 1,000 K, the critical concentration of hydrogen (diusing through molecular nitrogen) is about 0.5 vmr. Thus, escape rates would be more or less diusion-limited. However, for a 2 Earth radii Super-Earth with 10 Earth masses at 1 AU around the Sun, this value changes to approximately 0.01, hence scenarios of energy-limited escape could be possible.
A.1.3 Dragging
If the hydrodynamic escape of light species proceeds very fast, heavier gases (which are themselves stable against thermal escape in the atmosphere) can be dragged away by the escaping lighter constituents. The importance of this process can be estimated by comparing the upward escape velocityulight of the light gas with the downward diusion (relative to the lighter, escaping gas) velocity uheavy of the heavier gas (Watson et al. 1981):
ulight
uheavy = (mlight−mheavy)·π·b Fen
GMplan
kbTexo (A.9)
When the fractionfv = light
heavy is close to or smaller than−1, the heavier compo-nent would remain in the atmosphere because the absolute velocity would be directed downwards. However, iffv >−1or even close to0, the heavy species is dragged away with the escaping light gas.
For example, consider mlight = 1.67·10−27 kg for atomic hydrogen. On Earth assuming an exosphere temperature of Texo = 1000 K, one obtains Fen ≈ 1.9· 1030 particles s−1 (ϵ=0.3). Then, eq. A.9 yields fv = −0.1
(mheavy
mlight −1 ). Consequently, in this case, helium (mass 4 mH, fv ≈ −0.3) could be dragged away at about 70% of the hydrogen escape velocity. But at these temperatures, hydrogen is well below its critical temperature (see Table A.1), so the overall ux would be very low. However, at temperatures of T=5,000 K, one nds fv =−0.02
(mheavy
mlight −1
), and so even carbon, nitrogen or oxygen atoms (with their respective masses of 12, 14, and 16mH) could be lost.