ICARUS 48, 150--166 (1981) The Dynamics of a Rapidly Escaping Atmosphere: Applications to the Evolution of Earth and Venus ANDREW J. WATSON, THOMAS M. DONAHUE, AND JAMES C. G. WAL KER Space Physics Research Laboratory, University of Michigan, Ann Arbor, Michigan 48109 Received April 17, 1981; revised September 22, 1981 A simple, idealized model for the rapid escape of a hydrogen thermosphere provides some quantitative estimates for the energy-limited flux of escaping particles. The model assumes that the atmosphere is "tightly bound" by the gravitational field at lower altitudes, that diffusion through the lower atmosphere does not limit the flux, and that the main source of heating is solar euv. Rather low thermospheric temperatures are typical of such escape and a characteristic minimum develops in the temperature profile as the escape flux approaches its maximum possible value. The flux is limited by the amount of euv energy absorbed, which is in turn controlled by the radial extent of the thermosphere. Regardless of the amount of hydrogen in the thermosphere, the low tempera- tures accompanying rapid escape limit its extent, and thus constrain the flux. Applied to the Earth and Venus, the results suggest that the escape of hydrogen from these planets would have been energy-limited if their primordial atmospheres contained total hydrogen mixing ratios exceeding a few percent. Hydrogen and deuterium may have been lost in bulk, but heavier elements would have remained in the atmosphere. These results place constraints on hypotheses for the origin of the planets and their subsequent evolution. I. INTRODUCTION a negligible amount of dust. Energy input to the thermosphere is dominated by the ab- Early in their evolution, the atmospheres sorption of solar euv radiation. On the ter- of the terrestrial planets probably contained restrial planets today, the escape of light considerably more hydrogen than they do gases may occur via thermal evaporation today (Holland, 1962; Walker, 1977; (Jeans' escape) or by any of several mecha- Ringwood, 1979; Pollack and Yung, 1980). nisms involving charged particles (Hunten Exactly how much more is still a subject of and Donahue, 1976). These latter, nonther- debate, but some theories advanced for the mal mechanisms are of comparable impor- origin of the Earth envisage the loss of vast tance to Jeans' escape on the Earth, and are quantities of hydrogen and heavier volatiles the major loss process on Venus. However, at an early stage (Ringwood, 1975, 1979; nonthermal processes are not included in Sekiya et al., 1980a,b). On Venus, hydro- our treatment; the hydrodynamic escape gen escape following on a "runaway green- which we discuss is essentially a thermal house" event may explain the low abun- process. Hydrodynamic escape differs from dance of water observed on that planet gas-kinetic evaporation in that in some cir- (Dayhoff et al., 1967; Ingersoll, 1969; cumstances a substantial fraction of the en- Walker, 1975; Pollack and Black, 1979). In tire thermospheric energy budget may be this paper we examine the dynamics of used to power the escape of gas. It is these rapid hydrogen escape in an attempt to circumstances in which we are most inter- place constraints on these theories. ested, and we expect hydrodynamic expan- We consider the case of a planetary atmo- sion to be the dominant loss process under sphere after the solar nebula has dissipated. these conditions. We assume an extensive thermosphere of The process of Jeans' escape has been molecular or atomic hydrogen that contains extensively studied (see, for example, 150 0019-1035/81/110150-17502.00/0 Copyright © 1981 by Academic Press, Inc. All rights of reproduction in any form reserved. DYNAMICS OF RAPID ESCAPE 151 Jeans, 1925; Chamberlain, 1963, and refer- case of the solar wind, where essentially all ences given there). The rate of loss is found of the energy input is near the base of the to be sensitively dependent on the escape atmosphere and the temperature declines parameter, k j, defined by monotonically outward from the Sun. For a planetary atmosphere where there is heat- ~.S = GMm/kTar s. (1) hag from in situ absorption as well as from Here G is the gravitational constant, M is the lower boundary, the use of the poly- the mass of the planet, m is the mass of the trope representation is questionable. Effec- escaping particles, and k is Boltzmann's tively, its use imposes on the energy input a constant. Ta and ra are respectively the tem- particular spatial dependence (Hunten, perature and radius of the exobase. Pro- 1979), which does not necessarily corre- vided the escape rate is slow enough that spond at all with the heating in a planetary the energy lost with the escaping particles atmosphere. For this reason we do not at- does not appreciably affect temperatures tempt to use the polytrope approximation. near the exobase, values for Ta and rj may The rapid escape of matter from an atmo- be found by treating the atmosphere as sphere requires a concomitant expenditure static (Chamberlain, 1962; McElroy, 1968). of energy to lift the gas in the gravitational When applied to a dense hydrogen thermo- field (McElroy, 1974). Let us suppose that sphere on one of the inner planets, this this energy is supplied by euv heating at a technique fails completely, as Gross (1972) rate S ergs cm -2 sec -1, where S has been has shown; one obtains exospheric temper- suitably averaged over a sphere and cor- atures in excess of 10,000°K and values of rected for the less-than-unity efficiency of k a less than one. But if the escape parame- euv heating. Let the radius of the planet be ter is less than 1.5 (or 2.5 for a diatomic gas) r0 and the ultraviolet be mostly absorbed the internal energy of the gas is greater than near r 1, the level where the optical depth is the depth of the gravitational potential well unity. Equating the energy input to the en- in which it is situated. The gas should there- ergy of escape then yields fore flow off the planet in bulk rather than F = Sr12ro/GMm (2) merely evaporate from the exobase (0pik, 1963; Shklovskii, 1951). Jeans' formula is for the escape flux of particles per steradian not applicable under these circumstances per second. An upper limit to the escape and a hydrodynamic treatment becomes rate is thus immediately obtained, provided necessary. the value of rl can be specified. However, The hydrodynamic escape of a light gas fixing r~ is not such a simple matter. from a planetary atmosphere is a process Superficially, we might expect that if the similar in many respects to the supersonic density at the base of the thermosphere is flow of plasma from the Sun (Parker, 1963). increased, the position of unity optical The solar wind approach to the planetary depth will move outward, allowing progres- problem was developed by Gross (1974), sively higher escape rates. In the following and applied by Hunten (1979) to the dissipa- section we set up a simple model for the tion of a primordial atmosphere on Mars. thermosphere which allows us to solve si- Gross and Hunten both used a polytrope multaneously for the value of r I and the law to model the escaping gas, according to maximum escape flux associated with it, as which the temperature varies as n ~-~, a function of the incident euv. We find that where n is the number density and a is a under many conditions the escape flux is constant index. Use of a polytrope greatly constrained regardless of the amount of hy- simplifies the solution of the hydrodynamic drogen which is present. In the third sec- equations (Parker, 1963). Furthermore, tion, results of numerical solutions to the good first-order results are obtained for the hydrodynamic equations are displayed and 152 WATSON, DONAHUE, AND WALKER compared with the simple model. In the last metry, (3) may be immediately integrated to section some applications to the evolution give of Earth and Venus are discussed. F = nur 2, (7) II. THE ENERGY-LIMITED FLUX while (4) and (5) may be reduced to the Consider a dynamically expanding, non- following expressions (Noble and Scarf, viscous gas of constant molecular weight, 1963, Eqs. (8) and (9)). in which the pressure is isotropic. The (1 - ~'/xIt) datt/dk steady-state equations of mass, momen- = 2(1 - 2~-/k - d~./dh). (8) tum, and energy conservation are respec- tively, (Ko GMm/k2ToF)~(d"r /dh) = E + h- 5T/2- ~/2 (9a) V • (nu) = 0, (3) mn(u. V)u + V(nk73 = nmg, (4) ~ = ~ - (1/FkTo) f: qr2dr, (9b) V . (KV 73 = V • [~nkTu where we have been careful to retain q 4= 0. + n(mu2/2)u] - nmg.u - q, (5) The quantity ~ is the outward energy flow per escaping particle, expressed in units of where u is the bulk velocity of the gas, g is kTo. ~ is the energy flow at infinity. Though the acceleration of gravity, r is the thermal ~ may in general be either positive or nega- conductivity, and q is the volume heating tive, ~ cannot be less than zero unless rate. The factor ~ in (5) is specific to a there exists an energy source at infinity monatomic gas and would be replaced by ½ (Parker, 1964a). For the special case where for diatomic particles. E~ = 0, the outward flow of energy is just The equations are identical to those used sufficient to lift the gas out of the gravita- to model the solar wind, with the exception tional field, leaving no excess to be con- that in those studies the heating term q is ducted or advected at very large distances.