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. usually assumed to be zero. The equations To apply these equations to a dense ther- may be considerably reduced by the use mosphere of hydrogen we make the follow- of dimensionless variables (Chamberlain, ing assumptions: 1961; Noble and Scarf, 1963). Following (1) At a lower boundary located some these authors' treatments we introduce an distance above the homopause the temper- arbitrary reference temperature, T0, and ature is fixed at To, thus ~'(~0) = 1. The define temperature, velocity, and position lower boundary is sufficiently far from the variables as follows homopause that the mixing ratios of heavier T = T/To, gases are much smaller than that of hydro- qt = mu2/kTo, gen, which therefore dominates at all higher altitudes. We treat the case where the at- = GMm/kTor. (6) mosphere is "tightly bound" by gravity at (Note that whereas X is here defined in the lower boundary; i.e., X0 ~> 10 (Parker, terms of an arbitrary temperature, the es- 1964a,b). For H atoms on Earth this condi- cape parameter hj is defined by Eq. (1) tion is satisfied provided To < 750° K; for H2 specifically in terms of the exobase temper- molecules the equivalent condition is To < ature.) The thermal conductivity is parame- 1500°K. For most applications a reasonable terized by value for T Owould be the skin temperature of the planet, i.e., 200-300°K for Venus or K = Ko'r s, Earth. where s is about 0.7 for a neutral gas (Banks (2) The atmosphere is sufficiently dense and Kockarts, 1973). Assuming radial sym- that the optical depth of the lower boundary DYNAMICS OF RAPID ESCAPE 153 to euv is much greater than one. For the Well below the sonic level the velocity term present we further simplify the heating by is negligible. Near the lower boundary, k >> assuming that all the euv is absorbed in a ~" from our first assumption, so the equation narrow region at rl, where the optical depth may be further simplified; is unity. Above this level, Eq. (9b) then reduces to e = Eoo. Below r I the equation (T'gl~)dT/d)~ ~ )k - ~/(~Xl 2) -at- E0o. (15) becomes Using this approximate equation we now = e= -- Srl2/FkTo, (10) show that only values of the escape flux corresponding to complete absorption of all below a certain limit will yield a physical the available energy. Since there is little solution. If the flux variable ~ is small energy deposited above rl and the gas is enough, the r.h.s, of (15) will be negative continually expanding, the temperature de- for all values of ~ less than h0. d,r/dh is then clines as r ~ ~. also negative, meaning that the temperature (3) The pressure at large distances de- increases monotonically from the lower clines toward zero. Because of this the gas boundary out to rl. A temperature profile must expand according to a "critical" solu- resembling curve A of Fig. 1 will then tion of the hydrodynamic equation (Parker, result. Under these conditions a portion of 1963). At the "sonic level," rs, the velocity the energy deposited at rl is conducted passes smoothly from subsonic flow down the temperature gradient and out of (defined by the condition ~ < ~-) to super- the lower boundary. But as ~ is allowed to sonic flow, • > r. We assume that r 1 < rs. increase, a minimum appears in the temper- In the Appendix it is shown that this as- ature profile, at a position given by sumption implies ~/3/(~;,W) - E=. hl/T 1 > 2, (11) which allows us to place limits on tempera- Conducted heat now flows into the mini- tures in the thermosphere. A justification of mum from both above and below, while the this assumption for the cases of interest is expansion of the gas in the gravitational given in the Appendix. We note here that it field tends to cool this region. With further is not necessarily valid if the atmosphere is increases in the escape flux the minimum very strongly heated by uv absorption. Un- becomes progressively deeper. The maxi- der such conditions the assumption of a mum possible flux is obtained as the tem- localized heating function may also break perature tends to zero, as in curve B of Fig. down. 1. A still higher flux would yield negative We now focus our attention on the en- temperatures and is therefore unrealistic. ergy Eq. (9). It is convenient to define the Physically, the escape flux is limited by the following dimensionless variables for the requirement that sufficient energy be con- flux of escaping particles and the euv heat- ducted downward from rl, where it is de- ing: posited, to the inner thermosphere, where = F'(k2To/KoGMrn), (12) it is required to offset the adiabatic cooling of the rapidly expanding gas. fl = S " (GMm/kTo2Ko). (13) Consider now the effect of allowing the Substituting (10) in (9b) and (9b) in (9a), the value of rl to increase. The energy ab- energy equation for the region below rl may sorbed per steradian increases as rl 2, but now be written the greater distance between rl and the tem- perature minimum tends to impede the (~1~) d~'ldh = X -- fl/(~k, 2) downward conduction of energy. If r 1 be- + e® - 5r/2--~/2. (14) comes too great the second effect domi- 154 WATSON, DONAHUE, AND WALKER
Q 5 r 1 5
CONDUCTED HEAT
f
FIG. 1. Temperature profiles in a rapidly expanding thermosphere. Curve A: energy is conducted downward from AI to & and upward from A, to infinity. Curve B: at higher escape rates a minimum appears in the profile; the temperature at the minimum approaches zero as the flux approaches the energy-limited rate. Arrows show direction of conducted energy flow. nates and the escape flux must “throttle A, = p A0 down” for lack of energy. ( 1 In the Appendix we utilize Eq. (15) to : [(1 +2J* + tJ1)1’2> (17) obtain simultaneous equations for the en- ergy-limited flux and A,, the value of the radial parameter at the level of energy de- where E’ in (16) is a positive constant position. The equations are derived by ap- defined in the Appendix. plying conditions at the temperature mini- The simultaneous solution of these equa- mum. For the maximum flux case, both r tions for &, and A, must be accomplished by and dr/dh must be zero at that point. Rela- numerical means with specified values of p, tion (11) is used to set limits on the temper- A,,, em, and E’. However, we show in the ature at rl. It should be noted that the en- Appendix that setting em = E’ = 0 can result ergy “throttle” may not operate if this only in an overestimate of the flux. Since temperature rises continuously as rl in- our main interest is to obtain an upper limit creases [i.e., under conditions where (11) is to the rate of escape we may justifiably not valid]. Such conditions require further neglect these terms. With this sim- study. plification, the solution for (,,, as a func- The equations for the maximum flux, &,-,, tion of p is displayed in Fig. 2. Curves are and the associated value of A1 are shown for 5 values of A,,. Figure 3 shows corresponding values for rl/rO (which (A,/2)‘8+“‘* + 1 * equals A,,/A,). As an aid to the physical 5, z 2 (16) s+l A - A, + E’ I ’ interpretation of these figures, Table I 102 i , ,
ko
IO 20-
50
50-
X i " iO-i
,o ,d io~ ,d ~5 ENERGY PARAMETER ie FiG. 2. Dependence of the flux variable ~ on the energy parameter/3 for various values of A~ obtained by numerical solution of Eqs. (16) and (l?) with ~® = ~' = 0. Marked for comparison are results obtained by solution of the full hydrodynamic equations, for ko = 30.
(rl/ro) / , , , ,,,,,, , ..... ,,i , ' , , .... i ' ' ' / I0 I--
6 _ 50
4 20
2
0 I I i I lli,l l i l I i I III I I i I I I I II i i I ,o' ,o2 ,o5 ,o" ENERGY PARAMETER, /9 FIG. 3. Distance of unity optical depth in units of the planetary radius, as a function of/3. Calculated from Eqs. 16 and 17. 155 156 WATSON, DONAHUE, AND WALKER
TABLE 1 solar wind. Wang and Chang used trial-and-
TYPICAL VALUES OF DIMENSIONLESS PARAMETERS error methods to obtain the set of critical FOR THE EARI"HQ solutions which matched a boundary condi- tion of zero temperature at infinity. In our Dimensionless Value for Value for study the subset of these solutions that parameter H atoms H2 molecules match a given temperature at the lower
h0 30.1 60.2 boundary is found, again by trial-and-error 0.043 0.023 integrations beginning at the sonic level. /3 4.5 x 10 a 9.6 x 10 a Each of these doubly matched solutions has a unique number density at the lower aWe take To = 250°K, M = Earth's mass, r 0 = boundary and a unique escape flux. Earth's radius, S = l erg cm -2 sec -1, and F - l0 w particles sr -a sec -~ (or 2.5 x 1011 cm -z see ~ at the Rather than perform the full heating cal- surface). culation, we employ a mean absorption cross section, calculated as a weighted av- erage over the solar euv spectrum. Above shows some typical values of the dimen- the sonic level, the heating term is ne- sionless parameters h0, ~, and fl as applied glected. This simplification is not quite as to the Earth. The figures show that increas- arbitrary as it may appear; at some distance ing the incident euv flux allows the maxi- from the planet the gas must become mum escape rate to rise but tends to de- sufficiently tenuous that recombination pro- crease the "extent" of the atmosphere as cesses can no longer keep up with pho- measured by rl/ro. The escape flux thus toionization; the gas then becomes progres- increases less rapidly than the incident euv. sively more ionized and the heating due to The atmosphere is generally never so ex- photoionization declines toward zero. For tensive that r 1 exceeds l0 r 0. the numerical solutions studied this nor- Our analysis is not dependent on the den- mally occurred in the neighborhood of the sity of the thermosphere, provided it is sonic level. sufficient for all the euv flux to be absorbed. Figures 4, 5, and 6 show respectively It follows that the escape flux and the ex- temperature, number density and velocity tent of the atmosphere are constrained re- profiles appropriate to a dense thermo- gardless of how much hydrogen is present sphere of hydrogen atoms escaping from near the homopause; the deep temperature the Earth. The efficiency of euv heating is minimum serves to decouple densities in taken to be 15%. The lower boundary is the outer region from those near the lower fixed at 130 km altitude with a temperature boundary. (This property is more clearly of 250 ° K, giving a value of approximately illustrated by the numerical examples of the 30 for ho. The "maximum flux" solution is next section.) approached as the number density at the lower boundary is progressively increased in curves A through E. Figure 4 shows the III. NUMERICAL CALCULATIONS development of a minimum in the tempera- The analytical treatment of the hydrody- ture profile as the flux approaches its maxi- namic equations is only possible under mum allowed value. Note that tempera- highly idealized conditions. We have em- tures are everywhere quite low. This is in ployed numerical methods to obtain tem- marked contrast to the temperatures of tens perature, density, and velocity profiles un- of thousands of degrees which result if a der less restrictive assumptions for a few hydrogen thermosphere is treated as static particular cases. Our method is a modi- (Gross, 1972). As the energy-limited solu- fication of the technique of Wang and tion is approached, the density of the inner Chang (1965), which they applied to the region becomes very large and the velocity DYNAMICS OF RAPID ESCAPE 157
I I I I I I I I I I I I I I I1 I I I I I Ill l I I 1 500 A F=2-.30xlO 28 B F=8.73 x 1028 C F= 1.16 x 1029 400 D F= 1.27x 1029
A E F=1.42 x 1029
i,i
30O - Y B I-- rr" I,I G. 200 - i,i
I00
0 i i ' I I I I I I I I I I llI I I I I I I III J ' ,oz ios ,o4 ,os ALTITUDE (KM) FIG. 4. Numerical solutions to the hydrodynamic equations. Temperature profiles for the escape of a thermosphere of H atoms from the Earth. Fluxes of particles (per steradian per second) are indicated against each curve. Boundary conditions are zero temperature and pressure as k ~ 0 and T = 250° K at the lower boundary. The arrow marks the position of the critical point for higher flux curves. correspondingly small. All the profiles con- tions are plotted on Fig. 2, where they may verge above the altitude of the temperature be compared with the curve for h0 = 30 maximum, the density, temperature, and obtained from the treatment of the previous velocity in the outer regions asymptotically section. Considering the approximations in- approaching a limiting solution as the flux is volved in that analysis, the agreement increased. For that solution, the sonic level seems reasonable. is reached at a distance of about 2 × 105 km Recently, Sekiya et al. (1980a) have pub- or some 30 planetary radii. lished some numerical solutions appropri- In Fig. 7, curve E from Fig. 4 is plotted ate to a massive, very strongly heated pri- together with a comparable temperature mordial atmosphere. Although their cal- profile obtained with the solar euv in- culations take account of the presence of creased by an order of magnitude. At dust and are not therefore strictly compara- higher energies the region of maximum ab- ble to ours, euv is the most important en- sorption moves to lower altitudes and the ergy source. We find that solution of Eqs. inner part of the solution contracts some- (16) and (17) gives good agreement with the what, as might be expected from our pre- fluxes obtained by these authors, generally vious discussion. The values for the flux overestimating the escape by a factor be- parameter ~ obtained from these two solu- tween 1 and 3. 158 WATSON, DONAHUE, AND WALKER
i015 I I I I I I I I I
1012
I (..3 109
I.- O3 Z Ill a 106 d Z
103
I01 I I I l fill I I I I I I l I n J t aLul £ 10 2 10:5 104 105 ALTITUDE (KM)
FIG. 5. Number densities corresponding to the temperature profiles of Fig. 4.
IV. PLANETARY APPLICATIONS this flux had been maintained throughout the Hadean, some 3.3 x 102a g of hydrogen (a) The Early Earth would have been lost--about twice the total As a starting point for our speculations, amount of hydrogen in the crust and ocean we assume that the energy in the Sun's euv of the present-day Earth. We conclude that spectrum was the same during the Hadean the potential exists for a considerable oxi- (the first billion years of Earth history) as it dation of the planet's crust and atmosphere is today. At altitudes greater than a few during this early period. hundred kilometers the Earth's thermo- The above estimate of the energy-limited sphere consisted almost entirely of molecu- flux is quite insensitive to a number of our lar hydrogen. The temperature near the assumptions. For example, it increases by base of the thermosphere was 250°K and less than a factor of 2 if the temperature at the efficiency of euv heating was 30%. The the lower boundary is raised from 250°K to parameters ~0 and fl are approximately 60 1500°K, the highest value which is consist- and 1.1 x 103 respectively for these condi- ent with the assumption of '+ tight binding." tions. Referring to Fig. 2 we find the en- Neither can the flux be greatly increased if ergy-limited flux parameter ~m to be about we assume that escape occurred simulta- 0.054. This corresponds to an escape rate of neously with accretion; Fig. 8 shows that 2.4 x 102a particles sr -1 sec -1 or about 6 x the escape flux per steradian actually de- l011 cm -2 sec -1 at the Earth's surface. If creases slightly with decreasing mass of the DYNAMICS OF RAPID ESCAPE 159
106
105
j04
A i% 103
~10 2 v >.. i--- 10 °_ql A Ld > 10-1
3 15 2 133
Io-4L IO 2 103 104 105 ALTITUDE (KM) FIG. 6. Bulk velocities for the numerical solutions. planet, if the density of the accreting body gen and the background gas, so that nJnj is is assumed to be constant. The dependence just the H~ mixing ratio. If the background of the flux on planetary mass is shown for gas is N~, the diffusion-limited flux equals several values of the euv-supplied energy. our value of the energy-limited flux when If the mixing ratio of total hydrogen in this mixing ratio is about 2.5%. For larger the lower atmosphere is quite small, the amounts of hydrogen, escape would be en- escape flux may be limited by the rate at ergy-limited, while at lower mixing ratios which the hydrogen is able to diffuse diffusion would be the rate-controlling through the background, static component process. of the atmosphere. Hunten (1973a,b) has Equation (18) may also be used to inves- derived a simple and quite general expres- tigate the conditions under which heavier sion for this diffusion-limited flux. For a gases are "washed out" of the atmosphere minor constituent, by the escaping hydrogen. We apply the equation to the lower thermosphere, where F 1 = ntul r2 hydrogen is the major component and the = b(GM/kT)(mj - m~)(n,/n~), (18) heavier gas is a minor constituent. Sub- scriptj now refers to Hz, while i denotes the where F1 is the maximum diffusive flux per heavy gas. Dividing (18) by the energy-lim- steradian per second and b is a binary colli- ited flux, F = n~u~r 2 = 2.4 × 10~ sr -1 sec -1, sion parameter of order 10TM cm -1 sec-1. Sub- we obtain the ratio of the diffusion-limited scripts i andj refer respectively to hydro- and escape velocities: 160 WATSON, DONAHUE, AND WALKER
700 I I I I I I I I I 1 I I I I I I I ~ I I I I I I II I I I
600
500 ',,t" W n~ 400
w 3OO n
w ~- 2OO
IO0
0 I I I l I I Ill I L I I I I I II I l I I l Ill I02 10:5 I04 I05 ALTITUDE (KM) FIG. 7. Numerically obtained temperature profiles close to maximum flux solutions. The × 1 curve was obtained using present-day solar euv flux with a heating efficiency of 15%. For the × 10 curve the energy input was increased by an order of magnitude. Arrows mark the critical levels.
ul/uj = (bGM/FkTo)(mj - mO. therefore result in a large enrichment of -~ - 1.6(mJmj - 1). (19) deuterium on the planet. The question of the "washout" of heavy (The ratio is negative because the heavier volatiles has previously been investigated gas tends to diffuse downwards relative to by Hunten (1979) and Sekiya et al. (1980b). the escaping hydrogen.) If the absolute These authors considered cases in which value ofu~/u~ is much greater than one, the the energy available to power escape is heavier gas diffuses downward fast enough many orders of magnitude greater than can to remain in the atmosphere. Conversely, if be supplied by modern-day solar euv radia- the absolute value is much less than one, tion. Hydrogen may then escape rapidly the minor constituent is entrained with the enough to carry with it all the minor constit- flow of escaping gas. It is apparent that uents regardless of their molecular weights. gases as heavy or heavier than atomic nitro- The above discussion is relevant to theo- gen, for which mJm~ - 7, are not lost from ries for the origin of the Earth which call for the planet under our assumptions. How- the loss of huge amounts of volatiles at an ever, HD molecules, for which mi/m j = earlier period. For example, the "modified 1.5, are sufficiently light that they might be homogeneous accretion theory" (Ring- washed out of the atmosphere. Escape of wood, 1975, 1979) requires that the material copious quantities of hydrogen need not which condensed to form the Earth con- DYNAMICS OF RAPID ESCAPE 161
I I I I i I I I I
=¢/1 It.,. .0.- 1/1 HEATING, x30 v O3 LLJ _J xlO fie J x" x3 _J IJ.. J LU xl ft. J f o3 IJJ d I I I I I I I I I 0.0 0.1 0.2 0.5 04 0.5 0.6 0.7 0.8 0.9 1.0 MASS OF ACCRETING PLANET, M E FIG. 8. Energy-limited escape flux of H2 per steradian per second from an accreting Earth, for several values of the euv parameter. × 1 curve refers to present-day solar euv flux, 30% heating efficiency. The escape rate remains nearly constant regardless of the mass of the accreting planet. tained roughly 5 × 1024g hydrogen, 1.5 × be of the same order as its gravitational 1024 g nitrogen and 2 x 1035 g carbon. By energy even near the surface; i.e., h0 - 2.5. contrast the present complements of these The implied surface temperature is of order elements in the crust, ocean and atmo- 6000°K. The temperature may be reduced sphere are about 1.9 × 10=agofH, 5.0 × to 2000°K if we are prepared to accept 1021gofN and 1.4 x I02agofC (Anders Ringwood's hypotheses that the Earth was and Owen, 1977). Under our assumptions, rotating rapidly at that time and the atmo- solar-powered escape accounts for only sphere corotated like a rigid body out to 10% of the implied hydrogen loss, while C several planetary radii. The high surface and N do not escape at all. temperatures would be maintained only if If such copious escape did indeed occur, the planet accreted very rapidly; an accre- it seems unlikely that the energy source tion time of about l0 a years is needed if was solar euv; that would require the sun to 6000°K is the relevant temperature, or 105 have sustained a level of euv emission sev- years if it is 2000° K (Jakosky and Ahrens, eral orders of magnitude greater than at 1979; Ringwood, 1970). These periods are present for periods of 10s-109 years. A considerably shorter than the commonly more promising mechanism is strong heat- quoted estimate of 10a years (Safronov, ing at the base of the atmosphere, due to 1972), but at present the accretion time is the release of gravitational energy during poorly known, and a period as short as 105 the accretion of the planet, as discussed by years is not out of the question. The impor- Ringwood (1975). The very rapid escape tant conclusion is that theories requiring requires that the internal energy of the gas the loss of an early massive atmosphere are 162 WATSON, DONAHUE, AND WALKER rather firmly committed to such rapid ac- and Hunten, 1969), but subsequent obser- cretion. They are not consistent with longer vations disproved this hypothesis (Wallace time scales, as has been claimed (Ring- et al., 1971). wood, 1979). However, if the low abundance of hydro- We may also note that Ringwood's the- gen now observed is attributed exclusively ory requires that the Earth ultimately re- to escape of the gas, some enrichment of tained 1% or less of its original nitrogen and deuterium is bound to have occurred. From carbon complement. Since these elements Eq. (19), it can be shown that the escape cannot have escaped once the temperature velocity of hydrogen is insufficient to re- dropped below -2000°K, it follows that move deuterium from the atmosphere once less than 1% of the volatiles were brought the flux (referenced to the planet's surface) in during the final stages, as the accretion drops below about 4 × 1011 cm -2 sec -1. Us- rate declined towards zero. A very abrupt ing Eq. (18), we find that diffusion through cessation of accretion is implied, in con- the dense CO2 component of the atmo- trast to the pronounced tail-offwhich might sphere limits the flux to this value when the be expected as the last of the planetesimals mixing ratio of hydrogen declines to about were swept up by the newly formed planet. 2%. But the present-day content of total We conclude that the loss of large quanti- hydrogen in the Venusian atmosphere is, at ties of heavy volatiles requires a highly most, about 0.1% (Oyama et al., 1979). specific accretion history for the planets. Therefore, the escape of the last remnant of hydrogen would have enriched by at least a (b) Venus factor of 20 the present-day ratio of deute- Most of the above discussion for the rium to hydrogen. Measurements of the Pi- Earth applies equally to Venus. For the oneer Venus neutral mass spectrometer, same conditions as before, we calculate an while not conclusive on this issue, indicate energy-limited H~ flux of 1.1 × 10 TM cm -2 that no very large difference exists between sec -1 at the planet's surface. Loss of the the D/H ratios of the Earth and Venus equivalent of an ocean of hydrogen would (Hoffman et al., 1980). It seems most likely, occur in a mere 280 my at this rate, so that therefore, that the last few percent of the the water-rich atmosphere which would 'qost" hydrogen on Venus did not escape have resulted from a Cytherean "runaway the planet, though the mechanism by which greenhouse" may have lasted for only a it was removed from the atmosphere re- small fraction of the planet's history. As on mains an unsolved problem. Of course it is Earth, the hydrogen would not escape so possible that, as Lewis (1969, 1980) has rapidly that significant quantities of heavy maintained, Venus was all but devoid of volatiles would be lost from the planet, so water from the time of its formation and no the oxygen released by dissociation of the runaway greenhouse event ever occurred. water must still be on Venus; it may have In conclusion, this work illustrates that it reacted with the surface (Walker, 1975) or is important to consider the energy budget with reduced gases in the atmosphere (Day- when dealing with the escape of large quan- hoffet al., 1967). tities of gases. After the accretion of the Just as on Earth, on Venus we need not planets, the energy available to power es- expect deuterium to have been left behind cape processes is rather limited, probably by the rapid escape of hydrogen. The es- being confined to solar euv heating. The cape of oceans of water does not, therefore, dynamics of the escape process allow only imply a huge residue of deuterium on the a limited amount of this energy to be inter- planet. Such a residue was at one time pro- cepted and converted into escape energy. posed to explain the two-component Ly-a During accretion, some gravitational en- emission observed by Mariner 5 (McElroy ergy may be available to power escape, but DYNAMICS OF RAPID ESCAPE 163 only if the time scale for formation of the position of the temperature minimum. Ap- planet is very short. Energy considerations plying this condition we obtain a relation do not rule out the loss of considerable between the unknowns ~m and kl; quantities of hydrogen and deuterium from [2/(1 + s)l;m]1/2 the primordial Earth and Venus, but re- = h o --O/(~mhl 2) 4" I~®. (A4) moval of heavier elements is much less likely to have occurred. A second relation may be derived as fol- lows. We obtain an estimate of the temper- APPENDIX: DERIVATION OF Eqs. (16) AND (17), AND JUSTIFICATION OF ASSUMPTIONS ature near the level of unity optical depth by integrating Eq. (15) from the tempera- We first show that the assumption that ture minimum to hi. Close to hi, the tem- the level of unity optical depth is below the perature may be high enough that it is no sonic level implies relation (11): longer accurate to neglect the term 5T/2 in kl/'r I > 2. (A1) the full energy Eq. (14). The approximate solution therefore gives only a lower limit We then derive Eqs. (16) and (17). Finally for T1. Performing this integration yields we examine the validity of our assump- tions. [2/(1 + S)~m]ll2T1 (l+s)12
Above ra, the energy flow term, ~ in (9), is >/3/(GX, 2) - ~ - X,. positive and equal to ~=. In this region, Eqs. (8) and (9) are identical in form to those Zl can be eliminated from this expression by used in the
d~m _ 0~m/0~ + (0~m/(~Xl)(0Xl/(~IE) ~1 < 0.025 T12. dE 1 - (0~m/0Jkl)(0Xl/0~m) For all values of rx less than 40, ~t~1 is is then negative. Since ~' and ~ are positive therefore less than Zl. We know that q~ < r definite, fluxes obtained with ~' -- ~= = 0 everywhere below r s and q~ > r everywhere must be greater than those obtained with above rs, so it follows that rl < rs for these any other values. conditions. We conclude that the assump- tion is justified over the range of parameters EXAMINATION OF ASSUMPTIONS utilized in the text, but that it may break (a) rl Lies below the Sonic Level down if the atmosphere is so strongly The conditions under which this assump- heated as to allow very high escape rates tion is valid may be investigated by obtain- simultaneously with high temperatures in ing an upper limit on ~, the velocity vari- the thermosphere. able at unity optical depth. If the atmosphere were isothermal and static, the condition (b) Boundary Conditions at Infinity for unity optical depth would be In the text it is assumed that the pressure at infinity is zero and the gas accordingly nlHicr = 1, expands supersonically. In reality the pres- where or is an appropriately averaged cross sure at large distances is set by conditions section to euv and in the interplanetary medium. However, the effect of nonzero pressures at infinity HI = k"ciTora2/GMm (A8) can only be to slow the escape of the gas. is the equilibrium scale height at rl. Be- Since our purpose is to obtain upper limits cause the gas is accelerating and the tem- for the escape, the assumption does not perature is declining beyond ra, the actual affect the validity of the results. scale height is less than the equilibrium The analysis does not assume any value, thus specific value for the temperature at infinity. It does, however, assume that en- nlHlcr > 1. (A9) ergy deposited above ri cannot be con- Using the definition of ~ [Eq. (12) in the ducted downward into the inner thermo- text] with F = nmulr~ 2, we may write ni as sphere. At low escape rates this requires that a temperature maximum exists near n I = KoGMm ~/k2TorlZal. (AI0) the position of unity optical depth, so that Substituting (AI0) and (A8) in (A9) there temperatures decline above that region. At results high loss rates conduction becomes pro- gressively less able to redistribute the ab- KO~'l~O'/ku 1 ~ 1 sorbed energy; under such conditions the from which, since ~a = mua2/kTo, we ob- temperature profile above the minimum is tain largely irrelevant to the problem. The posi- tions of unity optical depth and the temper- W1 < (m/k3To)(KoT"l~o') 2. (A 11) ature minimum must then be essentially co- Now the cross section of hydrogen mole- incident. cules to euv is, at its maximum, 10 -17 cm 2. We restrict our attention to cases where ~ is (c) Convective Instability 10 or less, and take To = 1001Y K, in which An assumption which is implicit in the case K0 = 4.45 × 104 ergs cm -a sec -1 °K -a analysis is that temperatures in the inner- (Hanley et al., 1970). This choice, which is most region never decline so rapidly as to made deliberately to yield a generous upper be superadiabatic. At high escape rates this limit on ~, gives is not necessarily true. A portion of the DYNAMICS OF RAPID ESCAPE 165 temperature profile should then be replaced HUNTEN, D. M. (1973a). The escape of H2 from Titan. by an adiabat to take account of convective J. Atmos. Sci. 30, 726-732. motions. A temperature minimum still HUNTEN, D. M. (1973b). The escape of light gases from planetary atmospheres. J. Atmos. Sci. 30, results and the situation may be analyzed 1481-1494. by a technique very similar to the convec- HUNTEN, D. M. (1979). Capture of Phobos and De- tively stable case. 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