In this section, the equations and formulations of Appendix A will be applied to specic Super-Earth scenarios. The calculations involve three important quantities relevant for atmospheric escape:
1. Critical temperature Tcrit
Rapid thermal escape, so-called hydrodynamic escape or blow-o, occurs when the exosphere temperature exceeds a critical temperature. This critical temperature depends on planetary characteristics such as gravity.
2. Critical XUV ux ΦXUV,crit
The exosphere temperature is mainly determined by the XUV uxΦXUV of the central star (XUV collectively denotes the X ray and ultraviolet spectral domain). AtΦXUV,crit, the exosphere temperature equals Tcrit. 3. Critical orbit acrit
The critical orbit for hydrodynamic escape is then determined by the orbital distance where the XUV ux received by the planet equals the critical XUV ux. As the XUV output of the star evolves from zero age main sequence (ZAMS) of the central star when it enters the main sequence to the nal age main sequence (FAMS) when it leaves the main sequence towards the giant branches,acrit will change correspondingly.
Thus, a rough estimate of orbital positions will be obtained where one could reasonably expect a Super-Earth atmosphere to be stable against thermal loss.
4.3.1 Critical temperature
The important parameter for thermal escape of a particle with mass mpart is the critical temperature (see also eq. A.4 in the Appendix):
Tcrit = 2 Lfree
GMplanmpart
Rexokb (4.1)
whereLfree is the number of degrees of freedom (atoms: L=3), Gthe gravita-tional and kb the Boltzmann constant, Mplan the planetary mass and Rexo = Rplan +aE the radius of the exosphere.
The critical temperature depends roughly on the ratio Mplan/Rplan because Rplan≈ Rexo. Mass-radius relations (MRR) for Super Earths have been calcu-lated by, e.g., Sotin et al. (2007) and Valencia et al. (2006). In order to obtain a lower limit forTcrit (and thus calculate a conservative lower limit on thermal escape), planetary radii are taken from Sotin et al. (2007), since their radii are slightly larger than the values from Valencia et al. (2006). From the MRR, the critical temperatures for the dierent planet masses could be estimated. The values obtained in this way are listed in Table 4.1.
Table 4.1: Hydrogen critical temperatures Tcrit in K for terrestrial planets and 4 dierent masses (mE mass of Earth).
mass [mE] Tcrit [K]
0.5 2800
1 4800
5 16,100
10 26,800
From Table 4.1, it is evident that thermal escape from Super-Earth atmo-spheres is very slow, since the critical temperatures are very high.
4.3.2 Critical XUV ux
Kulikov et al. (2007) subjected model thermospheres and exospheres of Earth, Venus and Mars to very high XUV uxes and calculated exospheric temper-atures ranging between 10,000-20,000 K, depending on CO2 concentration in units of the present atmospheric level (PAL) on Earth. Their Figure 4 presents exospheric temperatures as a function of XUV ux for dierent assumed CO2
concentrations.
From this Figure 4 in Kulikov et al. (2007), the necessary ΦXUV,crit in terms of ΦXUV,Earth (present-day XUV ux at Earth) required to reach the critical temperature are taken. The values of ΦXUV,crit are summarized in Table 4.2.
Table 4.2: Critical XUV uxes ΦXUV,crit in terms of ΦXUV,Earth as a function of CO2 concentration for terrestrial planets with 4 dierent masses.
Mplan [mE] CO2 [PAL] ΦXUV,crit [ΦXUV,Earth]
0.5 1 2
1 1 3.4
5 1 35
10 1 115
0.5 10 3.2
1 10 5
5 10 35
10 10 115
0.5 100 5
1 100 7.5
5 100 35
10 100 115
0.5 103 8
1 103 12
5 103 46
10 103 115
0.5 3·103 11
1 3·103 15.6
5 3·103 53
10 3·103 115
4.3.3 Critical orbit
In order to calculate the evolution of the critical orbit, the evolution of the stellar XUV activity must be accounted for. As shown by Ribas et al. (2005) (see also Kulikov et al. 2007), ΦXUV of solar-like G-type stars decreases with time, starting at around 100ΦXUV,Earth (ZAMS) to 1ΦXUV,Earth (at 4.6 billion years, 4.6 Gyr, present age of the Solar System). The same decrease of activity has been observed for other star types, e.g. late F (F6 or F8), K and M-type stars (Scalo et al. 2007).
The tting of XUV ux can be done with a power law:
ΦXUV(t) =bn·t−a·fXUV (4.2) where t is in Gyr and fXUV the present XUV ux at a 1 AU orbit. bn is is a normalization factor, i.e. bn·4.6−a=1 .
Table 4.3 summarizes the parameters chosen in this study for the approxi-mation of XUV activity, as described by eq. 4.2. The values are taken from Lecavelier Des Etangs (2007)
Table 4.3: Values of parametersaand bn in eq. 4.2 Star type a bn
G 1.2 6.24
K 0.94 4.2
other 1 4.6
In order to obtain absolute values for the XUV ux from eq. 4.2, uxes for dierent star types with ages comparable to our Sun are needed, i.e. values must be assigned to fXUV.
Lecavelier Des Etangs (2007) assumed values of fXUVSun =4.6·10−3 Wm−2 for G and F8 type stars, fXUVM =2.9·10−3 Wm−2 for M stars and fXUVK =14.7·10−3 Wm−2 for K and F6 stars, at a planet-star distance of 1 AU. These values are derived from two dierent sources. The "Sun in Time" measurement program (Ribas et al. 2005, Table 4) provides observations for Sun-like G-type stars.
For F, K and M stars, XUV uxes are obtained as a median from ROSAT observations of a large set of stars (Hodgkin and Pye 1994, Table 4).
The calculation of the evolution of the critical orbit proceeds in three steps:
1. Critical ux in relation to a planet orbiting a G star at 1AU
The values for the critical uxΦXUV,crit, as obtained from Table 4.2, are for the Earth around the Sun at 1 AU. They are converted into a critical XUV ux ΦstarXUV,crit for a dierent central star at 1 AU:
ΦstarXUV,crit = fXUVSun
fXUVstar ·ΦXUV,crit (4.3)
For example, choosing a 10 mE terrestrial Super-Earth with 10 PAL CO2 around an M star yields ffXUVSunM
XUV
= 4.62.9··1010−−33 = 1.586. Then, one nds ΦMXUV,crit=1.586·ΦXUV,crit=1.586· 115=182 (from Table 4.2).
2. Critical orbit position att=t0=4.6 Gyr
This ux ΦstarXUV,crit is then converted into a critical orbital distance a0crit by assuming an r−2-dependence.
For the example above, this results in a value ofa0crit=(1821 )0.5=0.074 AU.
3. Critical orbit positions with time
Assuming the time evolution of the XUV ux of the stellar type, the evolution of the critical orbitacrit can be calculated by:
acrit(t) =a0crit√
bn·t−a (4.4)
4.3.4 Examples for change of critical orbits
Figure 4.1: Critical orbit over stellar age for dierent stellar types (color-coded) and planet masses. CO2 concentration 1 PAL. Vertical dashed lines indicate positions of S=1 for example stars Sun, AD Leo,ϵEridani and σ Bootis
Figures 4.1 and 4.2 show examples of how acrit changes for terrestrial planets over the stellar main-sequence lifetime (eq. 4.4). Several parameters were varied, such as CO2 concentration (1 and 3·103 PAL), planet masses (0.5, 1, 5 and 10 mE) and central stars (M, K, G, F6 and F8). The dashed vertical lines indicate orbits where planets around example stars (AD Leo as an M star, ϵ Eridani as a K star and σ Bootis as an F star) would receive 1,360 Wm−2, i.e. the same amount as the present Earth at 1 AU around the Sun (stellar constant S=1). This corresponds roughly to a position in the middle of the Habitable Zone (HZ) of these stars.
In Fig. 4.1, one can clearly see the eect of planetary mass on the critical orbit. As planetary mass increases (here, from 0.5 to 10 mE), the critical orbit moves towards the star (e.g., from 0.7 AU to 0.1 AU for a planet orbiting a K star). For planets below 10 mE, the critical orbit distances decrease with increasing CO2, as expected. Also, as could already be inferred from Table 4.2, for the 10 mE planet, the critical orbit is insensitive to CO2 concentration.
For an Earth-mass planet, critical orbits for K and M stars lie outside the S=1 orbit for a signicant part of the stellar lifetime, regardless of the CO2
concentration. For almost all combinations of parameters, the critical orbit lies outside 0.1 AU.
Figure 4.2: As Fig. 4.1, but with 3·103 PAL CO2