Jeans formula

In the upper atmospheres of planets, density becomes very low and temperature becomes very high due to absorption of solar ultraviolet light and, for the giant planets, other sources such as the viscous damping of vertically propagating gravity waves (Matcheva and Strobel, 1999; Young et al., 1997). Hot, fast-moving molecules may escape should their kinetic energy (in the vertical direction) exceed their gravitational potential energy: that is,

1 2 GMm

2 r where M is the mass of the planet; G is the gravitational constant; m is the mass of the molecule; v is its velocity in the upward direction; and r is the planetary radius at the altitude of the thermosphere. Since the gravitational acceleration at radius r is given by g = GM/r2, this condition may be rewritten as i v2 > gr (3.2)

Molecules moving upwards with speeds in excess of this calculated escape velocity will, however, only escape if they do not collide with any other molecules on the way. Hence, substantial escape of molecules only occurs when the vertically integrated density of air molecules above a certain critical level, zc, accounts for one mean free path: that is, ana(z) dz = 1 (3.4)

where na(z) is the number density profile of all molecules at an altitude z above a reference level in the planet's atmosphere; and a is the mean collisional cross-section. Assuming the atmosphere to be in hydrostatic equilibrium (Section 4.1.1) such that na(z) = na(zc) exp(—z/H*ca), where H*a is the mean atmospheric number density scale height discussed in Section 3.2.2, Equation (3.4) may be integrated to give ana(z) dz = <rna(zc)

For the giant planet atmospheres, the most abundant molecule in the upper atmosphere is atomic hydrogen, and thus it is this molecule that largely determines the altitude of the critical level, otherwise known as the exobase. Assuming a Maxwell-Boltzmann distribution of speeds, the probability that a molecule of molecular weight m has a speed in the range c to c + dc is given by

where a = m/2RT; R is the gas constant; and T is the temperature. For such a gas, the number of molecules passing upwards through unit area per second at the exobase with speeds in the range c to c + dc is then given by the well-known kinetic theory (e.g., Flowers and Mendoza, 1970) expression dF = ^n(zc)cP(c) dc. (3.7)

Assuming that all such molecules with speed greater than the escape velocity ve = \J2gr will escape the atmosphere, we may integrate Equation (3.7) to calculate the flux of escaping molecules, known as the Jeans' flux a

where the most probable speed U = \J2RT/m; and the escape parameter A is defined as A = v2e/U2.

The rate at which the concentration of molecules is reduced may then be calculated by considering that the total number of molecules per unit area above the exobase (and thus which may escape) is given by N = n{zc)H*, where H* is the number density scale height of the escaping molecule or atom at the critical level. Thus, expressing the Jeans' flux as FJea = ftn(zc), we obtain

From this expression we can see that there is a characteristic escape time for thermal escape given by Te = H*/ft. Alternatively, we can define a mean velocity at which molecules or atoms escape upwards from the exobase, known as the expansion velocity given by vex = H*/Te = ft. The calculated characteristic escape times for various gases in the atmospheres of the giant planets, the Earth, and also Titan and Triton are listed in Table 3.1. As can be seen, compared with the smaller planets, the masses of the giant planets are so large, and their exospheric temperatures so cool, that negligible exospheric escape is calculated and thus these planets have effectively lost none of their atmospheres. This is not the case for the Earth, Titan, and Triton, where significant loss of the lighter atoms is calculated and whose atmospheres have thus significantly evolved over time. The relatively unevolved atmospheres of the giant planets thus offer a unique picture of the composition of the solar nebula at the time of the planets' formation, provided that no other processes have acted to modify the composition. We shall return to this in Sections 3.3 and 3.4.