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Mass Fractionation in Hydrodynamic Escape

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Mass Fractionation in Hydrodynamic Escape ICArtUS 69, 532--549 (1987) Mass Fractionation in Hydrodynamic Escape DONALD M. HUNTEN Lunar and Planetary Laboratory, University of Arizona, Tucson, Arizona 85721 ROBERT O. PEPIN School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455 AND JAMES C. G. WALKER Space Physics Research Laboratory, University of Michigan, Ann Arbor, Michigan 48109 Received July 25, 1986; revised October 27, 1986 We show that mass fractionation occurs during the course of hydrodynamic escape of gases from the atmosphere of an inner planet. Light gases escape more readily than heavy gases. The resultant fractionation as a function of mass yields a linear or concave downward plot in a graph of logarithm of remaining inventory against atomic mass. An episode of hydrodynamic escape early in the history of Mars could have resulted in the mass-dependent depletion of the noble gases observed in the Martian atmosphere, if Mars was initially hydrogen rich. Similarly, a hydrodynamic escape episode early in Earth's history could have yielded a mass-dependent fractionation of the xenon isotopes. The required hydrodynamic escape fluxes and total amounts of hydrogen lost from the planets in these episodes are large, but not impossibly so. The theory of the mass fractionation process is simple, but more work will be needed to put together an internally consistent scenario that reconciles a range of data from different planets. © 1987 AcademicPress, Inc. 1. INTRODUCTION molecular flow occurs at an altitude at which the outflow velocity is still subsonic. Many data on the composition of plane- Large escape rates correspond to super- tary atmospheres reveal the tendency for sonic flow at levels where the mean free these atmospheres to be depleted in light path between collisions is still small com- gases relative to a presumed primordial pared to the scale height of the atmosphere. standard. In this paper we suggest that this Under these conditions the continuum the- kind of mass-dependent fractionation may ory provides the most accurate description be a consequence of the hydrodynamic es- of the escape process (Walker, 1977, 1982). cape of large amounts of hydrogen during Conditions on the inner planets today do the early evolution of the planets. not result in hydrodynamic escape, but hy- The escape of gas from a planetary atmo- drodynamic escape from these planets sphere can be approximately described ei- would have occurred if their atmospheres ther by the kinetic theory of gases, in which were ever rich in hydrogen. The low molec- case the escape process is called Jeans es- ular weight and weak gravitational binding cape, or by continuum theory, in which of hydrogen results in large escape rates case the process is called hydrodynamic es- from hydrogen-rich atmospheres around cape. The kinetic theory approach is most the relatively small inner planets. convenient when escape rates are low and The flux of ultraviolet radiation from the the transition from continuum flow to free young Sun was probably considerably 532 0019-1035/87 $3.00 Copyright © 1987 by Academic Press, Inc. All rights of reproduction in any form reserved. MASS FRACTIONATION 533 larger than this flux today (Zahnle and conditions are quite different in an electri- Walker, 1982). Enhanced ultraviolet flux cally neutral gas, for which collision cross could have promoted the photochemical sections are approximately independent of breakdown of water vapor to hydrogen in atomic mass. Under these circumstances the atmospheres of the inner planets early the hydrodynamic escape process is in the history of the Solar System. At the strongly mass selective, with light constitu- same time, enhanced ultraviolet fluxes ents being carried away from the planet and would have provided the source of thermo- heavy constituents being left behind. spheric energy that could have driven large hydrodynamic escape fluxes, particularly 2. EFFECT OF HYDRODYNAMIC ESCAPE ON from Mars with its small gravitational field. HEAVY GASES Other possible sources of hydrogen could A number of theoretical studies have pre- be reduction of H20 in the interior followed sented profiles of density, velocity, and by degassing, or a remnant of solar nebular temperature in an atmosphere undergoing gas, gravitationally condensed to the planet hydrodynamic escape from an inner planet (e.g., Hunten, 1979). (Hunten, 1979; Watson et al., 1981; Kasting We are not able to assert with confidence and Pollack, 1983). Expansion of the out- that hydrodynamic escape occurred from flowing gas keeps temperatures low, even the primitive planets. Such a prediction when the rates of supply of energy to the would require further theoretical study and thermosphere are large. The temperature would depend strongly on assumptions typically rises to a maximum of a few hun- concerning the water vapor or hydrogen dred degrees Kelvin at the level of maxi- contents of their atmospheres. Our goal, mum energy input and then falls off gradu- here, is to show how hydrodynamic escape, ally to zero (or a very small value) at very if it took place, could have yielded the great distances from the planet. This tem- mass-dependent fractionation that has been perature variation has an important effect revealed in modern measurements, particu- on the hydrodynamic escape flux of major larly of inert gas concentrations. In what atmospheric constituents but has little im- follows, therefore, we shall assume a value pact on the density or velocity profiles of for the hydrodynamic escape flux of hydro- heavy, minor atmospheric constituents. gen, which we assume to be the major con- For simplicity we shall deal with isothermal stituent of the primitive atmosphere, and atmospheres in our treatment of the heavy shall calculate the impact of this escape flux gases. Temperature gradients can in fact be on the abundances of heavier atmospheric accommodated without difficulty (Hunten, constituents. 1973). Inclusion of eddy diffusion also Mass fractionation in hydrodynamic es- makes very little difference, because the cape has not been previously studied for rapid escape itself tends to nullify diffusive the electrically neutral atmospheres of the separation and to yield a composition inde- planets. The process has received some at- pendent of height. tention in connection with the solar wind The outflow velocity in a hydrodynami- (Geiss et al., 1970; Joselyn and Holzer, cally escaping atmosphere is small at low 1978; Geiss, 1982), but the results turn out altitudes where ambient densities are high. to be quite different for a highly ionized The velocity increases with altitude so as to gas. For such a gas, the collision cross sec- preserve continuity of the escape flux, but tion is a strongly increasing function of the rate of increase of velocity with altitude atomic mass so that heavier atmospheric decreases with increasing altitude. The ve- constituents diffuse less freely than light locity gradient is small at high altitudes. ones. The consequence is that there is little The ambient density closely follows the discrimination with respect to mass in the barometric equation at low altitudes where outflow process. As we shall show here, acceleration terms in the equation of mo- 534 HUNTEN, PEPIN, AND WALKER mentum balance are small. The accelera- dn2 _ mzg 1 tion terms become important at high alti- dr kT n2 + -~ (n2F1 - nlFz) (4) tudes where flow velocities are supersonic. These nonlinear terms have an important in- where F~ = n~wl and F2 = nzwz are the ver- fluence on the hydrodynamic escape flux of tical fluxes of the two gases. Addition of Eqs. (3) and (4) yields the barometric law the major constituents of the atmosphere, but they are not likely to greatly affect the d g escape fluxes of the heavy, minor atmo- d-r (nl + n2) = - (nlml + n2m2)-k--~. (5) spheric constituents that we shall be calcu- lating here. The fluxes of constituents with As discussed below, the density ratio n~/ molecular masses much greater than the nz is independent of altitude when both molecular mass of the ambient atmosphere gases are escaping, except perhaps at very are determined by diffusion processes oc- large distances, so the mean molecular curring at relatively low altitudes where weight is independent of altitude. In an iso- outflow velocities are subsonic. Once the thermal atmosphere the barometric law fluxes are established, they must obey the therefore becomes continuity equation, which is highly restric- tive if the flow is spherically symmetric. na dr Ha (6) These rather intuitive arguments are essen- where na = nl + n2 is the ambient number tially confirmed by a numerical study by density, r the distance from the center of Zahnle and Kasting (1986). the planet, Ha = kT/mago is the ambient at- Consider an atmosphere composed of mospheric scale height at distance r0, ma = two constituents of masses mj and m2, (nlrnl + n2mz)/na is the mean molecular number densities n~ and n2, and vertical weight of the atmosphere, and go is the velocities relative to the planet of w~ and gravitational acceleration at distance r0. w2. Acceleration terms in the equations of The term in the inverse square of the dis- motion of these gases can be neglected pro- tance in Eq. (6) allows for the variation of vided there is little change in mean velocity gravitational acceleration with distance. during the time between collisions. In this The solution of the barometric law is case the relative velocities are determined by diffusion processes n0n--~=exp-[r~a(1-~)]" (7) b[1 dnl m,g] Define the mole fraction X2 by wl - w2 = - --nz L~-dT-r + kT J (1) n2 n2 X2 - -- - • (8) nl + n2 na W 2 -- W 1 = --__ nl ~ + ~ (2) Differentiation of (8) yields 1 dX2 1 dn2 1 dna -- = (9) (Chapman and Cowling, 1970; Hunten, )(2 dr n2 dr na dr" 1973, 1985; Walker, 1977, 1982) where b is the diffusion parameter (the product of dif- For the first term on the right-hand side of fusion coefficient and total number den- Eq.
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