The atmospheres of planets are not fixed and static things, but are in constant flux. To take the Earth as an example, the atmosphere we enjoy today bears virtually no resemblance to that of the young Earth. If we were to visit the Earth shortly after its formation, we would see it shrouded in hydrogen and helium from the protostellar nebular. This was quickly lost to space, and replaced with an atmosphere outgassed from the planet’s interior and water vapor delivered by asteroid impacts. The Earth, of course, has ended up with a “tertiary” atmosphere rich in oxygen as the result of biological processes.
From the Earth, we can directly observe the process of atmosphere evolution through the evaporation of planets. It was quickly realised that the high levels of irradiation of the earliest detected hot Jupiters could lead to them having greatly extended exospheres. Guillot et al. (1996) extended their theory of giant planet atmosphere to the case of the then recently discovered 51 Peg b and found, assuming that the stellar characteristics were similar to that of the Sun, that the planet was stable against catastrophic photo-disintigration.
For the hot Jupiters, the rates of evaporation over the lifetime of the planet are small, and typically add up to only a few percent of their total mass (e.g. Sanz-Forcada et al., 2011), but for the intermediate mass planets on short orbits,
Figure 1.13: Dominated by haze top: The instruments required to build the full picture of HD 189733b’s atmosphere (Pont et al., 2013) Bottom: Spectrum of HD 189733b compared to a clear model atmosphere.
that seem to be particularly common, evaporation may be a key sculptor of the observed population. Looking ahead to the characterisation of smaller planets, particularly habitable ones, evaporation may literally be a case of life or death. Too little evaporation may lead to surfaces crushed by the weight of a significant surviving hydrogen envelope, making liquid water impossible (Owen and Wu, 2016), and we have seen the effect of too much evaporation with the fate of Mars, and the atmosphere of Venus is currently evaporating (Brace et al., 1987).
The most straightforward way for a particle to leave the atmosphere is simply for it to have a velocity higher than the local escape velocity, at this point the particle can vanish off into space with no impediment. The velocity distribution of species in an atmosphere are described by the Maxwell-Boltzmann distribution
f(v) = s m 2πkBT 3 4πv2e− mv2 2kB T. (1.21)
The Maxwell-Boltzmann distribution for oxygen and hydrogen is plotted in Figure 1.14. For oxygen at surface temperatures, there is effectively no chance of a molecule reaching escape velocity. Hydrogen on the other hand, being a much lighter molecule has a significantly higher mean velocity, and the high end of the distribution reaches much further. In fact, the exosphere is quite a bit hotter than the surface (∼1500 K), so the distribution is wider still. When this is accounted for a non-trivial fraction of hydrogen atoms at the exosphere will have velocities higher than the escape velocity. The exobase is an important transition point in the atmosphere, moving into the “exosphere”, which is the region where molecules are in principle still gravi- tationally bound to the planet, but the density is low enough that the medium is essentially collisionless. As a result, normal gas laws that dominate the rest of the atmosphere no longer apply. The exobase can be defined as the point where the scale height is equal to the mean free path of a molecule, so that a particle trav- eling through this region would be expected to only have one collision on average. Recalling the definition of a scale height in Equation 1.16,
H= kBTeq
µg (1.22)
then the exobase is the region where the following condition is met,
Hnσc= 1 (1.23)
0 2000 4000 6000 8000 10000 12000 velocity (m/s) 10−20 10−18 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 f(v)
Figure 1.14: The maxwell Boltzmann distribution for an oxygen molecule in the Earths atmosphere at 287 K (blue) and for hydrogen at 287 K (green) and for hydrogen at exosphere temperatures, 1500 K (orange). The escape velocity of the Earth (11.2 km/s )is marked as a dotted line.
both H and n are dependent on the atmospheric temperature. At these altitudes the temperature is controlled largely by the Solar activity level, so the exobase can vary between about 500 and 1000 km. Due to heating from the Solar atmosphere, exosphere temperatures are of order 1000 K.
Starting from the Maxwell-Boltzmann distribution, one can integrate to find the fraction of molecules in the high tail with velocities greater than the escape ve- locity, and from there the total escaping flux of particles can be calculated (JEANS, 1916),
FJ=nc
Vth
√
4π(1 +λc) exp(−λc) (1.24)
whereVth is the most probable velocity from the Maxwell-Boltzmann distribution,
which is Vth= 2kBT m 1 2 (1.25) and λc is the “escape parameter”, which is the square of the ratio of the escape
velocity to the thermal velocity, where the escape velocity is
Vesc = 2GM R 1 2 (1.26)
so Equation 1.28 can be written as FJ=nc Vth √ 4π 1 +kBT R GM m exp −kBT R GM m . (1.27) Note in particular, that the escape rate depends on mass and temperature with an exponential term, so is both extremely sensitive to temperature and also com- paratively inefficient for larger molecules. Recall also that for hot Jupiters, R is approximately constant with respect to mass, so the Jeans flux will scale exponen- tially with the flux. All other things being equal, a 0.5MJ hot Jupiter can have a
Jeans escape flux 100 million times higher than a 10MJ planet.
In Jeans’ time, the high temperature of the exosphere was not known, so the calculated loss rate implied that the Earth could not lose a significant fraction of gas through these means. Jeans hypothesized that since hydrogen was rare in the atmosphere, the Earth must have been hotter in the past. Analogously, when the mass loss rates of hot Jupiters are compared to the calculations from Jeans escape, they are much too low, unless the exosphere of these planets is correspondingly heated to very high temperatures.
The mass loss rate of the hot Jupiter HD 209458b was first measured by Vidal-Madjar et al. (2003), and later by (Linsky et al., 2010) to be a minimum of 1010g/s, by observing UV absorption of metal lines with depths that implied material from the planet extended over the Roche lobe. (See discussion in Chapter 4)
In fact, the mass loss rates on hot Jupiters can become so high that they can reach a state ofhydrodynamic outflow (e.g. Owen and Jackson, 2012). In this situ- ation, heavier elements have been observed to be carried off of the planet, including C, Si and Mg (Linsky et al., 2010; Vidal-Madjar et al., 2013). This is strong evi- dence that a transition to hydrodynamic escape has occurred, due to the exponential dependence on mass in the Jeans escape rate.
Optical light from the photosphere cannot heat the atmosphere to the re- quired temperatures of ∼10,000 K (it would violate the second law of thermody- namics) so the heating is presumably caused by high levels of X-ray and EUV flux from the star. In Chapter 4, I calculate the amount of high energy radiation availible to heat the atmosphere of the evaporating hot Jupiter HD 209458b.
Non-thermal effects are also important for planets in the Solar System, for example, interactions with the Solar wind are believed to have stripped the atmo- sphere of Mars. For hot Jupiters, interactions with the stellar wind could shape the outflowing gas (Bourrier and Lecavelier des Etangs, 2013) and act to remove
material that has been lifted above the magnetosphere of the planet.
The study of mass loss has significant implications for life outside the Solar System, considering that many of the most habitable planets detected to date orbit M dwarfs, which are particularly active (Armstrong et al., 2015).