Earth and Planetary Science Letters, 50 (1980) 197-201 Elsevier Scientific Publishing Company, Amsterdam
197
Printed in The Netherlands
[61
DISSIPATION OF THE RARE GASES CONTAINED IN THE PRIMORDIAL EARTH'S ATMOSPHERE M I N O R U S E K I Y A , KIYOSHI N A K A Z A W A and C H U S H I R O H A Y A S H I
Department of Physics, Kyoto University, Kyoto (Japan)
Received October 16, 1979 Revised version received May 30, 1980
If the Earth was formed by accumulation of rocky bodies in the presence of the gases of the primordial solar nebula, the Earth at this formation stage was surrounded by a massive primordial atmosphere (of about 1 X 1026 g) composed mainly of H 2 and He. We suppose that the ti 2 and He escaped from the Earth, owing to the effects of strong solar wind and EUV radiation, in stages after the solar nebula itself dissipated into the outer space. The primordial atmosphere also contained the rare gases Ne, At, Kr and Xe whose amounts were much greater than those contained in the present Earth's atmosphere. Thus, we have studied in this paper the dissipation of these rare gases due to the drag effect of outflowing hydrogen molecules. By means of the two-component gas kinetic theory and under the assumption of spherically symmetric flow, we have found that the outflow velocity of each rare gas relative to that of hydrogen is expressed in terms of only two parameters - the rate of hydrogen mass flow across the spherical surface under consideration and the temperature at this surface. According to this result, the rare gases were dissipated below the levels of their contents in the present atmosphere, when the mass loss rate of hydrogen was much greater than 1 × 1017 g/yr throughout the stages where the atmospheric mass decreased from 1 × 1026 g to 4 × 1019 g.
1. Introduction The results o f recent investigations o f planetary f o r m a t i o n processes in the solar nebula indicate that possibly the gases o f the nebula played i m p o r t a n t roles in a sequence of the following processes. First, in early stages, dust grains accumulated and sedim e n t e d in the presence o f the gases to form a dust layer which fragmented and condensed into solid bodies with masses o f the order o f 1018 g [ 1 - 3 ] . Second, in stages where these solid bodies were accumulating into larger bodies and finally into the planets, the rate o f growth to the planets or, m o r e exactly, the rate o f radial migration o f accumulated bodies was greater by a factor o f about 10 2 than in a case where the nebula gases were absent [4,5]. The time of growth in the gases is o f the order o f 10 6 and 10 7 yr for the Earth and the solid core of Jupiter, respectively. Finally, all the planets or their solid cores, formed in the above way, were surrounded by gaseous atmospheres c o m p o s e d mainly o f H2 and He,
as long as the gases o f the solar nebula remained to exist. The mass o f the surrounding atmosphere depends u p o n the total mass o f a c c u m u l a t e d solid materials. In the case o f the giant planets, the solid cores as well as the atmospheres became massive e n o u g h so that, finally, the atmospheres became gravitationally unstable to form more tightly-bound envelopes [6]. This will be considered as the origin o f the giant planets. On the o t h e r hand, in the case o f the Earth the atmosphere was less massive and stable [6], since the total mass o f available solid materials was limited to the present Earth's mass. The structure o f this primordial atmosphere o f the Earth, e m b e d d e d in the solar nebula, has been studied in detail by Hayashi et al. [7] and M i z u n o et al. [8]. Their results indicate that, w h e n the Earth grew to the present mass, the atmosphere had a mass o f about 1 ~ 2 × 1026 g and a base temperature o f about 2000 ~ 4 0 0 0 K which was high enough to m e l t solid materials and to form the present core-mantle structure.
0012-821X/80/0000-0000/$02.25 © 1980 Elsevier Scientific Publishing Company
198 TABLE 1 Tile masses of the rare gases in the present Earth's atmosphere and in the primordial atmosphere with a mass of 1 × 1026 g and with the solar abundance [ 11 ]. In this table 4°Ar is omitted because of its radiogenic origin. Rare gas
Present atmosphere (g)
Primordial atmosphere (g)
Ne 36Ar + 38Ar Kr Xe
6.5 × 1016 2.4 × 1017 1.7 × 1016 2.0 × 1015
1.7 × 1023 1.0× 1022 9.4 × 1018 1.7 × 1018
Now, the existence of this atmosphere gives rise to a question as to how and when it was dissipated into the outer space, since it is generally believed that the present Earth's atmosphere was formed by outgassing from rocky materials in the interior at stages after the formation of the Earth [9]. Especially from the difference of the isotopic ratio of Ar contained at present in the mantle, the crust and the atmosphere, Hamano and Ozima [10] concluded that the outgassing had occurred within 5 X 108 yr after the formation of the Earth. Accordingly, in order that the existence of the primordial atmosphere be consistent with the formation of the present atmosphere as mentioned above, the following conditions must be satisfied. First, since the solar nebula is the basic cause of the primordial atmosphere, the nebula must have been dissipated in a period of time lying between 1 X 107 yr (which is required for the formation of Jupiter with a massive hydrogen envelope) and 5 X 108 yr. Second, the primordial atmosphere itself must have been dissipated in a period less than 5 X 108 yr. Here, it is to be noticed that the major components of this atmosphere are H2 and He and that the amounts of the rare gases contained in the atmosphere of 1.0 X 1026 g with the solar abundances as listed by Cameron [ 11 ] are much greater than those contained in the present atmosphere (Table 1). Especially, Ne had to be depleted by a factor of about 106 before the formation of the present atmosphere. Now, before returning to this depletion problem, we outline the results of recent studies on the mechanism and the rate of dissipation of the solar nebula and of the primordial atmosphere. As to the dissipa-
tion of the solar nebula, Horedt [12] and Elmegreen [ 13] have studied the interaction between the strong solar wind and the solar nebula during the T Tauri phase of the sun, while the present authors [14] have estimated the dissipation time by means of simple energy consideration for a case including the effect of the solar extreme ultraviolet (EUV) radiation. The results of these studies indicate that the dissipation time depends on the intensities of the wind and the EUV radiation which are not well-known at present, but the intensities required to give the time scale mentioned above is not so unreasonable. As to the dissipation of the primordial atmosphere, the present authors [14] have calculated the mass loss of hydrogen due to the heating of the outer layers of the atmosphere by the absorption of the solar EUV radiation and obtained a result indicating that, if the EUV intensity in the T Tauri phase is, for example, a thousand times greater than the present solar value, the time of dissipation of hydrogen of 1 X 1026 g is about 1 X 108 yr. In view of the above results, let us now assume that the major component, H2 and He, of the primordial atmosphere escaped completely in a time required. There still remains the problem for the escape of the rare gases as mentioned above. Obviously, in stages after the dissipation of H2 and He, the escape of the rare gases through Jeans' mechanism is impossible; for instance, in order that the amount of Xe listed in Table 1 be dissipated in 5 X 108 yr, the temperatures at escape layers of the atmosphere must be as high as 5 X 104 K. Then, as a solution to the above rare gas problem, we propose a mechanism that heavy atoms are dragged by the outflow of H2 and He. In general, heavy atoms in the Earth's gravity field tend to differentiate and sediment towards the base of the atmosphere. However, if the outflow velocity of Ha is quite large, heavy atoms gain outward momenta through collisions with H2 molecules and escape from the Earth against the gravitational differentiation. In the following, we study this drag effect to find a condition for the escape of the rare gases. 2. Calculations and results For simplicity, we assume that the atmosphere is spherically symmetric and all the flow velocities of"
199 the gases have components only in the radial direction. We also assume that the outflow of H2 is steady; since most of the atmospheric mass concentrates near the Earth's surface, the flow in the outer layers of the atmosphere can be regarded as steady. Furthermore, we simplify the problem by considering a drag process in a binary gas mixture composed of a large amount of H2 molecules and a very small amount of a rare gas of species i, He atoms being neglected because the number density is less than a fifth of that of H2 molecules. Let the mean radial velocities of H2 molecules and rare gas atoms i be denoted by Uo and ui, respectively, and the velocity difference by Aui. Then, if the condition: (1)
At'/i ~"//0 -- Ui < < UO
is satisfied, the rare gas atoms are dragged almost completely by I-Iz molecules and the chemical homogeneity is retained. According to the two-component gas kinetic theory [15], the velocity difference at a surface with radius r from the Earth's center is, in general, given by: AU i =
noni
Di
-D°Oi (F
np
Fo) + kT ~rln T 1
mi O Aui = -~Zo Di ~r In P
(3)
Generally, the diffusion coefficient, Di, may be expressed approximately in the form:
Di = fi(T)/P
(4)
where jS(T) is a function of temperature determined by Mason and Marrero using experimental data [16, 17]. If the outflow velocity of H2 molecules is small compared with the sound velocity (this condition will be satisfied almost everywhere in the atmosphere except for the outermost escape layers), the pressure gradient is approximately determined by the equation of hydrostatic equilibrium, that is:
O Jr
--
In P -
GMmo r2 k T
(5)
where G is the gravitational constant and M is the mass of the Earth. Further, the rate of hydrogen mass loss is expressed in the form:
? (n;i +-no,,;(mo- mi)- -0- In P
[_Jr\ n /
kT0 in T/Jr ( k r i s found to be of the order ofni/n in the present case) in equation (2) are both negligibly small as compared with the pressure-gradient term, and we have finally:
Jr
(2)
Here, n, p and m are the number density, the mass density and the molecular mass, respectively, the subscripts 0 and i refer to H2 and the rare gas i, respectively, the symbols without a subscript denote the total quantities (for example, p = Po + Pi), and P and T are the total pressure and the temperature, respectively. Further, Fo and k•. are the external forces which act on the two gases of unit mass and in the present case of the Earth's gravity we have simply Fo = Fi. Finally, D i and kT represent the diffusion coefficient and the thermal diffusion ratio (i.e., the ratio of the diffusion coefficient to the thermal diffusion coefficient), respectively, for the gas mixture under consideration. Now, in order to simplify equation (2), we consider a case where the chemical homogeneity is nearly retained in the flow. In this case, we have no > > ni and Po > > [i besides m i ) > mo. Then, we find that the first term O(ni/n)/Or and the last term
~Ig =
-4rrr2 puo
(6)
and this rate is independent of the radius r for the steady flow under consideration. From equations (3) through (6) and the equation of state, p = nkT, we have finally:
Aui _ 4rrGMrnorn~.(T) "o
I/~/gl(k T) 2
(7)
It is to be noticed that the above expression for 2xui/uo does not include any quantities, except for the temperature, which depend on the radius r, i.e., on the structure of the atmosphere. Further, equation (7) can also be derived from Hunten's diffusion equation of a static atmosphere (see equation (A1) in Hunten [18] or equation (20)in Hunten [19]). In our case, Hal > > Hei ~ rZkT/GMmi, therefore his diffusion equation reduces to the following form; i.e.:
Au i -
nD iGMm i nor2kT
Using equations (4) and (6), and the equation of
(8)
200 I O n'
I
IMgl (g/yr)
I
I
i .....
Xe
-
Kr
I
I
I
I
I
[
TABLE 2 The mass of the primordial atmosphere with the solar abundance; which just contains the same mass of a rare gas as contained in the present atmosphere Ne 36Ar+38Ar Kr Xc
1016
I O is
i
i
i
i
I
I O0
I
i
I
I
1000
10000
m (K) Fig, 1. T h e c r i t e r i o n
f o r t h e escape o f each rare gas due to the
drag effect of outflowing hydrogen molecules. If the mass loss rate I~/g I of hydrogen is much greater than the value indicated by the solid curve, the rare gas can escape with hydrogen molecules without fractionation. A dashed curve indicates an extrapolated value using the formula of diffusion coefficient by Mason and Marrero {16,171.
state, we have again e q u a t i o n (7). N o w , using e q u a t i o n (7) we consider the c o n d i t i o n for the drag o f the rare gases. The solid curves in Fig. 1 represent the marginal c o n d i t i o n , Aui/uo = 1, for I 0 IE
q
F
T
1
q
"
(g/yr) .
.
.
.
.
.
i
-
.I
.
.I .1~ . I
I0"
Ar i0,~ I I 0 ~9
I
I~le
i O z0
i i 0 z~
Kr
I
i1
i 0 z2
r e 23
i O s*
i i O z~
i O z~
Mg(g)
Fig. 2, The criterion for the escape of all the rare gases below" the levels of their contents in the present atmosphere. The bottleneck temperature is taken to be 100 K. If the mass loss of hydrogen proceeds, until the mass decreases to 4 X 1019 g, along a path lying in regions outside the hatched area, all the rare gases escape below their contents in the present atmosphere. The dashed curve and the dot-dashed curve indicate the change of mass loss rates calculated by the present authors for cases where solar EUV intensities are 1000 and 100 times greater than the present value, respectively.
3.9 2.3 1.8 1.2
X 1019 X 1021 X 1023 X 1023
g g g g
ep..ch of the rare gas elements. If the mass loss rate IMgl is m u c h greater than that indicated by the solid curve for a rare gas considered, this gas o u t f l o w s with nearly the same velocity as that o f H2 molecules. Generally, a b o t t l e n e c k in the gas flow lies at a point where the temperature takes a m i n i m u m value. The calculated result of the present authors [14] on the steady o u t f l o w o f H2 molecules, as m e n t i o n e d in section 1, indicates that the t e m p e r a t u r e is about 280 K at the p h o t o s p h e r e o f the a t m o s p h e r e (where the solar visible light is incident) and it decreases outwards until it takes a m i n i m u m value o f about 100 K at a point near the base o f an EUV-absorbing layer. A c c o r d i n g to this result, we fix the b o t t l e n e c k temperature at 100 K in the following discussions. Let us now consider a case where all the rare gases escape with nearly the same velocities as that o f H2 molecules. Even in this case, in order that the a m o u n t o f each rare gas decrease below that contained in the present atmosphere, the total m a s s M g of the atmosphere has to decrease, at least, to the values listed in Table 2. Then, these values o f Mg and the marginal values o f 13~Ig[ at 100 K are p l o t t e d in a diagram as shown in Fig. 2. Generally, the mass loss rate will decrease with time, i.e., with the decrease o f the mass o f the atmosphere, owing to the decrease in the absorption cross section for the solar wind and radiation. The diagram implies that the mass loss rate o f the primordial atmosphere, as a f u n c t i o n o f its mass, must be greater than that indicated by the hatched region in Fig. 2, in order that a consistency be held with the f o r m a t i o n o f the present atmosphere. The dashed curve and the dot-dashed curve in Fig. 2 indicate calculated results of the present authors [14] for cases where the solar E U V luminosities are 2400 and 240 erg/cm 2 s, i.e., about 1000 and 100 times greater than the present solar value, respectively.
201 Besides the mass loss due to t h e E U V r a d i a t i o n , t h e r e will b e a loss due to t h e solar wind w h i c h has n o t yet b e e n s t u d i e d in details b y us. A t p r e s e n t , the i n t e n s i t i e s o f the E U V r a d i a t i o n and the w i n d e m i t t e d b y T T a u r i stars are n o t w e l l - k n o w n , b u t the abovem e n t i o n e d result o f o u r i n v e s t i g a t i o n i n d i c a t e s t h a t the existence o f t h e p r i m o r d i a l E a r t h ' s a t m o s p h e r e is n o t i n c o n s i s t e n t w i t h the f o r m a t i o n o f the p r e s e n t a t m o s p h e r e w h i c h o c c u r e d w i t h i n 5 X 10 8 yr after the E a r t h grew to the p r e s e n t mass.
Acknowledgement T h e a u t h o r s wish t o t h a n k Prof. D.M. H u n t e n for his useful advice. This w o r k is s u p p o r t e d p a r t l y b y the G r a n t - i n - A i d for Science R e s e a r c h o f the Ministry o f E d u c a t i o n , Science a n d C u l t u r e ( P r o j e c t No. 4 5 4 0 4 7 ) . N u m e r i c a l c a l c u l a t i o n s were p e r f o r m e d b y FACOM-M 190 at t h e D a t a Processing C e n t e r o f Kyoto University.
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