Origin of isotopic fractionation of terrestrial Xe: hydrodynamic fractionation during escape of the primordial H2He atmosphere

Earth and Planetar3." Science l_z,tter.~, 89 (1988) 323-334

323

Elsevier Science Publishers B.V.. Amsterdam - Printed in The Netherlands

[5]

Origin of isotopic fractionation of terrestrial Xe: hydrodynamic fractionation during escape of the primordial H2-He atmosphere Sho Sasaki ~ and Kiyoshi N a k a z a w a 2 i Geophysical Institute. k2wulty of Science, Unicervitr of Tokyo Bunk.vo-ku, Tokyo 113 (Japan) ~ Department of Applied Physics, FaculO' of Scwnce. I'ol~vo Institute of Technology'. Meguro-ku. Tokyo 152 (Japan) Received September 9, 1987" revised version received May 16. 1988 A systematic fractionation pattern of Xe isotopes of the terrestrial atmosphere can be explained by mixing of two components: hydrodynamically-fractionated Xe of the residual primordial atmosphere and isotopically-unfractionated Xe contained originally inside the Earth. The mixing model can also account for the absence of isotopic fractionation of Kr and Ar between meteorites and the Earth. as well as the difference of Xe patterns between meteorites and the Earth. The isotopic feature of terrestrial Ne also could be explained by hydrtxlynamic fractionation.

1. Introduction :!

X e~ t:;~2;)

The isotopic pattern of terrestrial Xe shows a distinct fractionated feature compared with meteoritic and solar-type (SUCOR) Xe: lighter isotopes are depleted and heavier isotopes are enriched systematically [1,2]. Since there are few processes which can fractionate noble gas isotopes systematically, the problem of the isotopic fractionation of terrestrial Xe is one of the most important keys to clarify the origin of the terrestrial atmosphere. Fig. 1 shows observed isotopic compositions of Xe. In the present study, data of Xe isotopes are normalized to U-Xe, which is a probable candidate of primordial Xe [3,4]. With respect to the problem of the isotopic fractionation of the terrestrial Xe, the following observations should be explained [5]: (1) The degree of Xe isotopic fractionation is very large, about 3.5% per 1 amu (atomic mass unit) (see Fig. 1). (2) Meteorite samples show unfractionated feature of Xe isotopes in general. Though heavy isotopes are enriched in meteoritic Xe compared with certain primordial components such as U-Xe, this enrichment is interpreted by the addition of r-process Xe. 0012-821X/88/$03.50

': 1988 Elsevier Science Publishers B.V.

] Iixl lOC

J

~

EARTH RRAY KENNA -'"

"/

"~

"

""~

0

U'Xe

_ ::

MASS Nt~;l~:_.i~ ?;3 X~ Fig. 1. Observed isotopic compositions of Xe normalized by those of U-Xe (reproduced from table 5.5 of Ozima and Podosek [5]). The solid line denotes the present "Earth's atmosphere; the dashed-and-dotted line is of S U C O R Xe (surfacecorrelated component in a lunar mare soil: the representative of solar composition): patterns of Xe in carbonaceous chondrites Murray and Kenna are also shown.

324 (3) Both Kr and Ar in terrestrial samples are not fractionated isotopically. Strictly speaking, terrestrial Kr shows small fractionation whose trend is opposite to that of Xe: lighter isotopes are more enriched. However, the degree of fractionation of Kr isotopes is much smaller than that of Xe. It is not easy to interpret even one of the above constraints; previous models have not succeeded in satisfying all the above three constraints. The goal of this study is to present a model which can explain all the above constraints on the Xe fractionation. The observed pattern of terrestrial Xe, in particular, the depletion of lighter isotopes, is hardly explained by combinations of anomalous nuclear products. Adsorption by interstellar grains may explain the elemental fractionation pattern (" planetary pattern") of noble gases when temperature is sufficiently low [6], but the adsorption is unlikely to produce large isotopic fractionation because the diffusion coefficient in adsorbers is controlled by the atomic radius of gas molecules which is rather independent of isotopic mass [5]. Electric discharge, which simulated the synthesis of carbonaceous matter, produced isotopic fractionations of Kr and Xe at about 1% per ainu [7], and isotopic fractionations at about 1% per ainu of Xe and Kr would occur by implantation of low-energy (50 to a few 100 eV) ions irrespective of target materials [8,9]. Anyway, such microscopic mechanisms cannot explain all of the above constraints on the Xe fractionation problem, especially the absence of isotopic fractionation in Kr and Ar. Ozima and Nakazawa [10] tried to explain the provenance of the Earth's noble gas by mixing of gravitationally-fractionated gases contained within pores of planetesimals and originally-trapped gases in solid phases. Their mechanism gives a sufficient degree of Xe isotopic fractionation. As for Kr and Ar, the contribution of the fractionated gases is very small to the present atmosphere, and the present atmospheric Kr and Ar come from the isotopically-unfractionated gases in solid phases. And (though the authors did not mention it), Ozima and Nakazawa's model may explain constraint (2): the quantity of gravitationallyfractionated gas would be very small in meteorites, since the ambient nebula density in the asteroid

region is about 50 times lower than that in the Earth's region [11]. However. the planetesimal size necessary for the Xe fractionation is quite large (/> 600 km), and moreover, it is rather doubtful that gases in pores should be in gravitational diffusion equilibrium. Hunten el al. [12] considered that mass fractionation during hydrodynamic escape of a planetary atmosphere may have formed the present isotopic pattern of terrestrial Xe as well as the elemental abundance pattern of Martian noble gases. Their calculated results show that the hydrodynamic escape process can reproduce the fractionated Xe isotopic pattern of the terrestrial atmosphere. In the present study, we also consider the mass-dependent fractionation in the course of hydrodynamic gas escape from the proto-Earth. And, we show that the mixing of fractionated and unfractionated noble gases can produce Xe fractionation itself (constraint (1)), but also satisfy absence of Xe fractionation in meteorites (constraint (2)) and absence of fractionation in terrestrial Kr and Ar (constraint (3)).

2. Hydrodynamic fractionation during atmospheric escape According to the Kyoto model of planetary formation, terrestrial planets grew in the presence of gas of the solar nebula and the growing Earth attracted gravitationally a dense H , - H e atmosphere with high abundances of noble gas species [13-15]. However, at the T Tauri stage of the proto-Sun, the solar nebula was dissipated owing to the heating by the ultraviolet radiation and the solar wind [11]. Subsequently, the prirnary atmosphere around the proto-Earth was blown off by irradiation of far UV. Sekiya et al. [16,17] studied the process of the atmospheric escape, assuming that it started after the completion of Earth's accretion. Their main result is that the primary atmosphere could be dissipated almost completely during the T Tauri stage (within a period of about 1 × 10 7 y) if far-UV irradiation was 102-103 times as strong as that of the present Sun. During the escape of the atmosphere, hydrogen molecules drag other gas species through molecular collisions; Xe as well as other noble gases in the primordial atmosphere is dragged away from the Earth against gravity in such a manner that lighter (or heavier) isotopes can escape more

325 rapidly (or slowly). Thus, the resulting isotopic patterns of Xe and other noble gases contained in the residual atmosphere are fractionated. The drag effect and resulting fractionation during the gas escape from protoplanets was calculated recently by some authors [12,18-20]. Sekiya et al. [18] estimated the loss of noble gas elements by H 2 drag effect and concluded that all noble gases in the primordial H2-He atmosphere could be dragged away from the Earth completely. Donahue [19] tried to explain isotopes and abundances of light noble gases of terrestrial planets by Jeans' escape from planetesimals. Zahnle and Kasting [20] studied in detail drag effect during hydrodynamic escape of hydrogen considering transonic escape and discussed about loss of water and noble gas pattern of terrestrial planets. And as we introduced in the previous section, Hunten et al. [12] studied the fractionation pattern of terrestrial Xe isotopes as well as Martian elemental abundance pattern of noble gases. In the course of the hydrodynamic fractionation, both the elemental and isotopic patterns of remaining noble gases are changed systematically. We assume that a significant fraction of heaviest gas, Xe, should remain after the escape of the bulk atmosphere, with a fractionated isotopic pattern. On the other hand, the abundances of lighter Kr and Ar of the remaining gas are, though fractionated isotopically, much smaller than the present amounts of these gases. Like Ozima and Nakazawa [10], we assume that the present noble gas inventory is formed by mixing of fractionated remnant of the primordial atmosphere and the isotopically-unfractionated gas which is originally trapped inside the Earth with the so-called "planetary" pattern of fractionated elemental abundance. As for Xe, both sources contribute to the present air. As for the lighter Kr and Ar, however, only the originally-trapped gas without an isotopic fractionation should contribute to the isotopic feature of the present air. With respect to Ne, there remains a possibility that the remnant of the primary atmosphere should contribute to the present air. We will also examine the isotopic pattern of Ne in this paper. 3. Basic equations We here present basic equations which govern the amount of each noble gas isotope during dis-

sipation of the primary atmosphere. The mutual diffusion between helium and any species as well as that between hydrogen and any species is taken into account. On the basis of gas kinetic theory [21], the diffusion of species ./ in a multi-component gas is expressed as:

.~,x,(,,,- u,) /

I

~

X !

(~1 ,

--

°',)

- -

V In P + v x ~ + k T i V In T

(1)

~Z

when the only external force is the gravity. In the above, P is pressure, T is temperature, m / is molecular mass of species j, m is mean molecular mass. and x/ is mole fraction of species j (using number density or partial pressure we have xj = n J n = PI/P). Moreover, u~ is the mean velocity of species j, D,_ j is the mutual diffusion coefficient between molecules i and j, and k ~ is the thermal diffusion ratio. The first, the second, and the third terms in the right-hand side express mass-dependent mutual diffusion, self-diffusion, and thermally-driven diffusion, respectively. In the present formulation, we made the following assumptions: (1) the atmospheric structure is in quasi-steady state: (2) the counter-drag effect by minor components on hydrogen escape is negligible, since the amount of the other species is sufficiently small compared with that of H~: (3) the outflow of the atmospheric gas is spherically symmetric and subsonic; (4) the gas is isothermal; and (5) the diffusion which comes from the gradient of concentration (i.e., self-diffusion) is neglected. Spherical symmetry permits X7 = d / d r (r being radius from the planetary center). In real. the outgoing flow is transonic, but subsonic approximation is applicable to mass fractionation problem and it is rather accurate for heavier dragged species [20]. The subsonic approximation permits us to write: d In P i _ GM~p

dr

r2p

(2)

where G is the gravitational constant, p is the atmospheric density, and M~ is the planetary mass. The assumption of an isothermal atmosphere lets us neglect the last term in the right-hand side of equation (1). In fact, when j is a minor con-

326

stituent we have kt~ - O(x,) [21], and because the scale height of temperature is larger than that of pressure (Id In T/dr I <1 d In P/dr [, the last term is appreciably, smaller than the first term in the right-hand side of equation (1). The neglect of the term denoting self-diffusion. V',x, in equation (1), implies vertical chemical homogeneity. We assume, for simplicity, that convective motion would stir the atmosphere and keep it chemically homogeneous below the base of the outflow. namely the UV-absorbing zone. And as discussed in Appendix 1, even in the outer region, ~Tx, is much smaller than the term denoting mutual diffusion around the base of the outflow, except for species whose mass is very close to the crossover mass (which will be defined later). Adopting these assumptions equation (1) is simplified to be:

~_, x,x,( u,D,- ul) -x,I ~"ram'-1, '1GM,p , i ,t r~p

H 2 and 4 H e because the abundances of the other components are negligibly small. Then equation (3) is simplified to be: only,

ul~.-uH~. = - l ('mH,: - 1 ) D I t X 1-'1 ,

'

_He

GM, Pr2p

(5)

D?

If the gas is composed only of H~ and minor species .i. we find that equation (4) is identical with equation (12) in Hunten et al. [12]. adopting xu~.=0 and replacing xll 1, (m,/m- 1) by ( m ; m

H.

)/m.

t h e mutual diffusion coefficient between the ith and the j t h species, D,_,. is expressed by the form: f,_,(T)

T)- ~ -

D,_,(P,

J;_,(T) nl,.T

(6)

(3)

-

where f, ,(7") is given by experimental data as a function of temperature [23]. n is the total number density of gas molecules, and /"i~ is Bohzmann constant. Change of f,_, due to rnixing ratio or to difference of isotopes is negligibly small. The estimated values of Jl , between I-t. or He and noble gas species are summarized in Table 1. The difference of f,_, is not large among various species. Therefore, in equations (4) and (5), the drag effect should depend mainly on the difference of molecular mass m~; heavier elements (or isotopes) with large m, have larger velocity differences owing to less efficient drag. The mass escape rate of species j. Q~. is expressed as:

When we estimate the diffusion of minor species other than "~Hc, we take into account the drag by' helium as well as by hydrogen since the helium abundance is not negligible: xuJxtt" = 0.16 (from solar abundance [22]). Assuming three-component diffusion, velocity difference between H , and species j is expressed as: 1 Ill[

-" 14, =

XH2

XIIc

OH:X

DIIc

I

- - - 1

m

I

r 2p

XHc ( /'/11- -- UH~')!' l)tt,"

i

(4)

Qj = 4~r r2p,u/

"

whcrc subscript Hc cxprcsscs 4He. When we calculate diffusion of 4He, we need to consider

(7)

where p, is the density of j t h species. Using the

"I'ABI.[" 1 M u t u a l d i f f u s i o n coefficient b e t w e e n t w o species m u l t i p l i e d by p r e s s u r e : f, , = PD,_, ( b a s e d o n M a r r c r o a n d M a s o n [23]). Values r e l a t i n g to A r a n d Xe at 100 K are o b t a i n e d f r o m e q u a t i o n s w h i c h a r e valid a b o v e 242 K Spcc)es

T = 100 K

(I)

:= H,

i-

He

i,-,H,

T = 1000 K

tie Ne Ar Kr Xe

2.4 1.7 1.0 9.2 X 10 -1 X.3 × 10 -*

1.8 1.2 1.0 79 × 10- I

1.3 x 9.4X 7.1 X 6.1 X 50 x

T .. 10,000 K i-

10: 101 101 10 ~ 101

He

g.7 x 6.2 x 5.3 x 4.4×

101 101 101 101

,=!t, 7.6 x 5.1 x 3.9x 3.5 x 2.6 x

i= He 103 l0 ~ 10 ~ 10 ~ 10 ~

4.g x 3.6 x 3.2X 2.7 x

l0 103 10' 1() ~

327 above, equations (4) and (5) can be rewritten as follows; for minor components: mix I

Qj-

mlt2XH:

x {Qu: -

Putting Q~ -- 0 in equations (8) and (9), we may define a measure, crossover mass rn c which is given from equation (8): xHefH., lie

m,: = m xli:

Xlt,

- -

Xtl,~

-4-- - -

xIt., file /

4TrGM~rnu: (kBT) 2

-X" I I .:

(8)

4~rGM,mH~(mu, - m) I x

QII: - fu:-,t¢

(kBT)2

[

(9) where the dependence on r is eliminated in accordance with the quasi-steady state. The total inventory of the j t h species, M~ is defined as:

M,(t) = f~ 4~rr-x,(t)m;nj(t) dr

(10)

where t is time, and r~ and r,, are the radii of the bottom of the atmosphere (i.e., planetary radius) and of the UV-absorbing shell, respectively. We have Q j = - d M J d t ; we need not care about change of ro because Q~ is independent of r. Sincc chemical homogeneity within r,, is assumed, we can obtain M/(t) and x~(t) (= [Mfft)/ m~]/Y.[(Mfft)/m,)] by numerically integrating equations (8) and (9) with respect to t. The initial values of M/ and xj are estimated from the assumed atmospheric mass before escape M,,(0) and its solar chemical composition. The abundance ratio of isotope j to isotope k, then, is expressed as:

x,(t) -

-

xk(t)

m~,M,(t) -

m/Mk(t )

(11)

and the isotopic ratio normalized by its initial (solar) value is:

x,(t)/xk(t ) x i(O)/xk(O)

M~(t)/M,(O) Mk(t)/Mk(O)

(12)

(1,3)

and from equation (9): QH:

m~ = m + and for 4He: mHeXIle QItc -- - mlI2XH2

XFI c

. QH,(kBT 2 fu,-, fu~- I 4qrGM,rnu, xu,

+

× [.,~ - m + xHe/lt'-II"I ( mlle -- m

(rn tk. - rn)

(kBT) 2

47rGM,mll, fu:-j

(14)

When m / > m~, species j cannot escape by the drag effect and u / b e c o m e s zero. In the case where mj < rn c. gas molecules can escape; then from equations (4) or (5) we find that the outflow velocity is approximately proportional to ( m ~ m/) [12]. Another measure can be introduced: the minimum value of the hydrogen escape flux above which an assigned species could gain positive outflow velocity [18]. This limiting hydrogen flux QI can be obtained by setting Q~ = 0 in equations (8) and (9); we have for minor species:

QI =

xu, XH

_ _

'

+

fu.,-., I'

4~rGM~mu, xu~_~

(kl~ T

)2

fu~,

X, rn t - m +

XlnJ't:-'n*(mm'-m)) Xll-fllc-t

(15)

and for 4He: 4,-'rGM, m H:( m / - m )

Q, =fu.-i

(kBT)2

(16)

Fig. 2 shows QI for various noble gas isotopes as functions of temperature. When H , flux is far larger than Qi, isotopic fractionation of the assigned species is not efficient because the second term denoting diffusion is smaller than the first term expressing drag in the right-hand side of equations (8) and (9). and the species escapes in accordance with H,. As Qu~ approaches QI, isotopic fractionation of the species becomes dominant. The escape rate of the species becomes smaller and reaches zero when QH: < Qn- This

328

..-C.101~ >,,, _ . .

!!6Xe

_.-~..~...

86Kr

_Ia>

x

=]1013 LI--

oq

..1_ L.D ~10 ~2

~___

~He

._J

102 10 3 10z' TEMPERATURE [K] Fig. 2. Lirniting hydrogen escape rate QI for escape of the heaviest isotope of cach noble gas when M, - I Mt:. Horizontal axis is temperature. The solid curves express the case where drag b3 4He is considered, whereas in the dashed cur.'es two-component approximation is used and drag effect bv helium is neglected in correspondence with fig. 1 of Sekiya et al. [171.

figure indicates that the effect of t e m p e r a t u r e is not seriously i m p o r t a n t : the limiting flux shows only a slight decrease with increasing t e m p e r a t u r e since the flux is p r o p o r t i o n a l to f , _ , / T ~- which does not d e p e n d largely on T.

4. Fractionafion of noble gases during atmospheric escape Using the a b o v e e q u a t i o n s , we c a l c u l a t e n u m e r i c a l l y the f r a c t i o n a t i o n of noble gas isotopes. A d o p t e d a s s u m p t i o n s a n d p a r a m e t e r s arc as follows. The initial c o m p o s i t i o n of the p r i m o r dial a t m o s p h e r e is taken to be solar [22]; noble gas species have s o l a r - t y p e elemental a n d isotopic a b u n d a n c e s . Especially, we a s s u m e that the initial Xe isotopic c o m p o s i t i o n is the s a m e as that of U-Xe. A n d d u r i n g the a t m o s p h e r i c escape there is no gas s u p p l y such as degassing from the interior. W e also assume that the a t m o s p h e r i c escape started after the accretion: mass of the p r o t o - E a r t h M~ is 1MF. T h e a t m o s p h e r i c mass is 1.0 × 102~ kg (0.017Mr_) before start of the escape [17]. G a s t e m p e r a t u r e is a s s u m e d to be 100 K which represents a region near the b o t t o m of the U V - a b s o r b i n g layer [18]. The intensity of f a r - U V radiation, which heats up a n d dissociates h y d r o g e n molecules through a series of H 2 0 - c a t a l y t i c reac-

tions [17], is taken to be c o n s t a n t a n d 0.22 W / m 2, which is 10 times as large as the present value. The mass loss rate of h y d r o g e n c o r r e s p o n d i n g to the a b o v e flux, Q u , , is c a l c u l a t e d as a function of p l a n e t a r y mass a c c o r d i n g to Sekiya et al. [17] a n d illustrated in Fig. 3. In the early' stage of the escape the radius of U V - a b s o r b i n g sphere is much larger than the p l a n e t a r y radius r,; large effective cross section of UV a b s o r p t i o n (rrr~ >> ~rr,2) resuits in large Qu-. As the a t m o s p h e r i c mass decreases, Q u . decreases a n d a p p r o a c h e s a c o n s t a n t value with d e c r e a s i n g effective cross section which w o u l d a p p r o a c h vrr,2. Fig. 4 shows the c h a n g e in isotopic fractionation p a t t e r n of Xe m the residual a t m o s p h e r e whose mass is expressed by the d e p l e t i o n factor f , (which is a ratio of the total mass of the r e m a i n i n g a t m o s p h e r e to its initial mass). The degree of isotopic f r a c t i o n a t i o n increases with decreasing f , . In the early stage ( f , > 1 × 10-2), h y d r o g e n outflow flux is large (QN. >> QI and m~. >> m, for all n o b l e gas isotopes) a n d the resulting isotopic f r a c t i o n a t i o n is not a p p r e c i a b l e . Then Q u . decreases a n d a p p r o a c h e s QI for Xe isotopes and, hence, the isotopic f r a c t i o n a t i o n becomes d o m i nant (the effect of diffusion b e c o m e s c o m p a r a b l e with the d r a g effect by H2), and at the last stage (f,, < 1 × 10-~), as for the heavier isotopes of Xe, f r a c t i o n a t i o n as well as escape stops because Q u . < QI (or r n / > m~.). As seen in Fig. 4, the resulting degree of Xe isotopic f r a c t i o n a t i o n is large enough to explain the f r a c t i o n a t i o n of the present ter-

~_ 101~ "¢"

10

%

"%10;

d

Fuv = 10-l[W/m 2]

101~ i

10!a

,

i

i

•

i

i

1616 10m 102c 1022 Ma [ kg ]

Fig. 3. Hydrogen escape rate Qu.. as a function of tile total mass of the remaining atmosphere. The calculation is based on Sekiya et al. I17]. The dashed curves express calculated Qua(M,) for far-UV intensity 0.1. 1. and 10 W/m e in the case of M, ~ 1M~!.The solid curve shows Qu. corresponding to our assumed far-UV flux, 0.22 W/re'. The dotted curve expresses Qu. conforming to the case M,(0)-10 aM t and M,= 0.6,~4~..

329

;: x.:i 2

)

~':'!

I i

! X- ' '~

. i

x

'

C 8

'-',::~ X ~.1.. ,,

. ""

--:~,

f'a , .',,

i,

/::.-

-!2

I:; ~

:'.."

~2~ :~. :~: ~2~ :3MASS NUI4BER 0? Xe

~s~

similar to the isotopic feature of the present terrestrial Xe. In contrast to Xe, light noble gases are lost with decreasing atmospheric mass even after the termination of Xe escape, and their amounts in the remaining primordial atmosphere become much smaller than amounts in the present Earth (denoted by dashed lines in Fig. 6). Residual lighter noble gases of the primordial atmosphere, though they are also isotopically fractionated, do not contribute to those in the present atmosphere. The present Kr and Ar should derive from the originally-trapped components whose isotopes are not fractionated. Thus, our model can explain constraint (3): the absence of isotopic fractionation in terrestrial Kr and Ar. As for Ne. there is a possibility that a certain fraction of the present Ne should come from the remnant of the primordial atmosphere. We will discuss about this later. Our model is also able to explain constraint (2). The mass of the primary atmosphere attracted by a protoplanct depends largely on the planetary

Fig. 4. Fractionation pattern o f X e isotopes in the remaining atmosphere. Isotopic ratios are normalized by those of U-Xe. The number attached to each curve denotes depletion factor of the primordial atmosphere, f;,: the dashed line connecting cross marks denotes normalized Xe isotopic ratio in the present atmosphere (which includes radiogenic component).

TOTA~IAIR MASSI02CHa [kg] i

,

,

,

,

I

,

,

ly0 restrial Xe. Then our mechanism may explain constraint (1): the high degree of fractionation of Xe isotopes. The change in the amount of each noble gas species during the atmospheric escape is shown in Fig. 5. At the early stage of escape all the noble gases flow out and their amounts decrease monotonically. In the later stage when Q . : approaches Qt for Xe isotopes, mass loss rate of heavy Xe becomes smaller, and finally, a large amount of Xe comparable to the present amount is left in the remaining atmosphere. Fig. 6 shows Xe isotopic pattern of a gas mixture of the hydrodynamically fractionated and trapped unfractionated components. We assume that originally-trapped Xe has the composition of U-Xe. The total Xe mass after the mixing is assumed to equal the mass of the present atmospheric Xe, 2.02 x 1012 [kg]. When the depletion factor fj is 1 x 10 - 9 t o 1 x 10-10 we obtain a fractionation pattern which is fairly

,,.J

rl i./) IJ_

CD c/)

s Ar

z LD Z

Ne Kr Xe

z

Xe

,,Z,l@0 ,,.r I

i

'

I

I

I

I

I0-I0 i0-5 DEPLETION FACTOR

I

I

fa

Fig. 5. Change of mass of each noble gas species M, (H 2 and noble gases) contained in the remaining atmosphere. The scale on the top shows the absolute amount of the atmosphere ( M, ). while the scale on the bottom is mass depletion factor (f.,) which is ratio of M a to the initial atmospheric mass. The horizontal dashed lines show masses of noble gas species (except radiogenic components) contained in the present Earth's atmosphere.

330

-!.

i

,

Fig. 6 and the present pattern would become smaller. This is rather favorable for our aim in the present problem.

i'.i

5.2. Missing Xe -,,~.~.

•i i " f~

Fig. 6. F r a c t i o n a t i o n p a t t e r n of Xe isotopes after m i x i n g of f r a c t i o n a t e d Xe from the r e m a i n i n g a t m o s p h e r e a n d o r i g i n a l l y - t r a p p e d u n f r a c t i o n a t e d Xe. The factorial n u m b e r beside each curve is d e p l e t i o n factor of the p r i m o r d i a l a t m o sphere, f.,. a n d the n u m b e r parenthesized is m i x i n g ratio of the fractionated Xe to the total Xe.

mass (M~, cx M~3 4 [15]). If the planetary radius is less than 1000 km, we can say that no attracted atmosphere has yet existed. Parent bodies of meteorites would hardly have attracted nebular gas as an atmosphere because of their small masses. Hence the above fractionation process could not occur and meteoritic Xe would keep unfractionated feature of the originally-trapped gas. Thus, our scenario on the origin of the terrestrial noble gas can clear all of the constraints mentioned in section 1. 5. Discussions and remarks

5.1. The initial isotopic composition of Xe We have assumed that the originally-trapped gas as well as the primordial atmosphere before escape would have U-Xe composition. Even if we adopt another candidate (AVCC- or solar-Xe) instead, our results would not be essentially altered, or correctly, since heavier isotopes are a little more enriched in AVCC- or solar-Xe than in U-Xe, difference between the mixing curves in

In our mixing model, the total Xe mass in tile Earth is put equal to that in the present atmosphere. But the amount of Earth's Xe in the atmosphere is about a tenth of the extrapolated value of "planetary" abundance pattern: it has been advocated that a significant amount of Xe is now stored somewhere inside the Earth though not yet observed [24]. Even if the missing Xc exists inside the Earth, our result does not change so long as isotopic composition of the missing Xc is not fractionatcd. We also executed calculation of a model where the present Earth has Xe 10 times more than the atmosphere and hiding Xc has also the fractionated isotope feature. Escape rate of species depends on the mass escape rate of hydrogen or irradiated UV flux; a slight decrease in QH, would leave more Xe as residue with fractionatcd isotopic feature, and an obtained result could still explain constraints on the Xc problem. We have no observation of unfractionated missing Xe inside the present Earth. But in the case of Nc, solar type No, whose isotopic composition is different from the present terrestrial atmosphere, is observed recently in samples from mantle [25,26]. If originally-trapped Xe and fractionated Xe from the primordial atmosphere was not mixed completely, unfractionated Xe would be left insidc the present Earth. We expect future observation of less fractionated Xe in mantle, which may support our model with two sources (fractionatcd and unfractionated) for terrestrial Xc.

5.3. Escape time of the primordial atmosphere Tile result of our calculation is very sensitive to change of UV-intensity, in other words, hydrogen escape rate Qlt.. When UV-intensity is 10 times larger, Xe as well as lighter noble gas should escape from the Earth rapidly completely. For our result that amount of fractionated Xe is comparable with that in the present atmospheric Xe, Qtt, should decrease as small as Qi (Xe). But if QH. should be fairly smaller than Q~, Xe could not escape or become fractionated? Sekiya et al. [18] concluded that all noble gases could be dragged away from the protoearth,

331 whereas our result indicates that a small fraction of Xe would remain and contribute to the present gas. The difference of the results lies in difference of choice of UV intensity. The assumed far-UV flux in this study (10 times as large as the present one) gives a smaller hydrogen escape rate than the EUV flux used by Sekiya et al. [18]. Our calculation (where we assume M~(0) = 1.0 × 10 `'3 kg, M~ = IMI~ and the solid curve in Fig. 3 for Qu,) gives the escape time of 1.1 × 10 ~ y. Although this could be shorter than the interval between accretion and secondary degassing (_< 5 × 10 s y, [27]), the time scale is much longer than 1 × 10 7 y, which is the possible upper bound of T Tauri duration of a low mass ( - 1Mo) star. How can we reconcile this difference of time scales? One explanation is that the initial air would be massive and duration of escape would be really longer. Recent observations suggest that the T Tauri stage is followed by a fairly active stage which continues long and has far-UV intensity

;,

xo,"'

:

]

:<

[cc,

fa • --3 / /

. [ :-~

!::o' : u : o T U '

fa

:24

l~-: i2., ~.':.'. l ~ ; ~ i~6 MASS NUMBER OF Xe Fig. 7. Fractionation pattern of Xe isotopes after mixing of fractionated Xe of the remaining atmosphere and originallytrapped unfractionated Xe. Different from Fig. 6, planetary mass Ms is assumed to be O.6ME, the initial atmospheric mass to be 1 x lO-4Mu, and hydrogenoutflow flux to be 50% of the solid curve in Fig. 3. A factorial number beside each curve is depletion factor of the primordial atmosphere, .f~.

3-20 times as large as the present sun at 5 × l0 s y [28]. This flux is large enough and the duration is long enough to make Xe fractionation during the escape of rather massive ( - 10-2MF) atmosphere. Another explanation is given by assuming the initial atmospheric mass M:,(0) smaller (~< 10 ~M~) than the value used in the above calculation. The constraint from dissolved mass of Ne prefers less massive atmosphere a n d / o r earlier start of atmospheric dissipation [14]. We find that the fractionatcd Xe isotopic pattern can bc formed (see Fig. 7) when we use M , , ( 0 ) = I × 1 0 4My.= 6 . 0 × 1 0 :° kg, M~=O.6M v, and the hydrogen escape rate (denoted by the dotted curve in Fig. 3) is 50% smaller. The escape time of the atmosphere is 9.6 x 10 ~' y, which is compatible with the duration of the T Tauri stage (though predicted UV flux is low: 5 times as large as that of the present Sun).

5.4. Origin of terrestrial Ne We will add a comment on the origin of the terrestrial Ne, of which the isotopic composition ( 2 ° N e / 2 2 N e = 9.8) is different from representative Ne compositions, i.e., the solar value (solar wind: 2 ° N e / 2 2 N e = 13.6) or planetary-type meteoritic value (Ne-A: 2°Ne/22Ne = 8.2) [5]. The Earth's Ne is usually explained by isotopic fractionation or mixing between solar Ne and Ne-A, though detailed mechanisms are not clarified. We here explain also terrestrial Ne by mass fractionation during the escape of the primordial atmosphere. Let us assume that the initial Ne of the primordial atmosphere, namely, Ne of the solar nebula is solar-type ( 2 ° N e / 2 2 N e = 13.6). When f,, is 1 × 10 -s, the ratio of the remaining Ne decreases to be 9.4, which is a little smaller than the present ratio (Fig. 8), and the amount of remaining Ne is still comparable with that of the present Ne in the Earth's atmosphere (Fig. 5). At this stage, isotopic pattern of mixed Xe has become fractionated (Fig. 6), whereas contribution of fractionated Ar and Kr can be neglected since amounts of remaining Ar and Kr have already become about 70 times smaller than their present atmospheric amounts (Fig. 5). Based on 2°Ne/22Ne data of deep-sea basaltic glass and volcanic gas slightly higher than that of the present atmosphere, Craig and Lupton [29] suggested that the atmospheric Ne is formed by

332 i

15

Appendix l--Neglect of the self-diffusion

i

fa lo -2 ~ t -

SOLAR-Ne

10-~, /*Ne-B J 10-8 ~/JEARTH-Ne 10 -6

zlo c~

10-m ~-Ne-A

~3J Z

We here present the condition that the selfdiffusion from Vxj can be neglected in calculating mutual diffusion [19], and we will show that this neglect is applicable to our problem. For simplicity, let us consider two-component case. Under subsonic and isothermal assumptions, equation (1) is rewritten by: UII

Du,-, ( [ m, :t x ' [ ~ XH,X,

- - L/, =

-

--

i GM~p dx, ] 1] + rZp dr I -

-

(A-l)

Ne-E t 0

I

t

I

i

0.002 0.004 21Ne/22Ne

Fig. g. Change of Ne isotopic composition during escape of the primordial atmosphere. The initial ratios are those of solar-Ne. The number attached to the line is depletion factor of the primordial atmosphere f.,. The adopted parameters are the same :is those used in obtaining Figs. 4-6.

Using equations (6) and (7), we have from the above:

4rrr

2dx~

1 (Qlt,

~

- ~, ~ . -

, re+l,

Q~

]

----

m]

')

ii,

,

4rrGM~(m,- rn) k J"

x,

(A-2)

Rearranging the above, we get:

fractionation during gas escape. Inside the protoEarth there may exist dissolved Ne, whose isotopic ratio reflects a value of the initial atmosphere ( - 13.6) [30], and recent observations suggest that solar-type Ne should exist inside the present Earth [25,26]. We may tentatively say that the present atmospheric Ne also could be produced by mixing of the fractionated residue of the primordial atmosphere with low 2°Ne/22Ne and other components with high 2°Ne/Z2Ne {e.g. dissolved Ne a n d / o r trapped Ne).

Acknowledgements We thank Prof. M. Ozima, Dr. G. Igarashi, Dr. M. Sekiya, Prof. O.K. Manuel and Prof. Yu.A. Shukolyukov for their valuable discussions and Prof. N. Sugiura for reviewing manuscript. We also thank Dr. K. Zahnle for useful comments. This work is supported by the Grant-in-Aid for Scientific Research of Japanese Ministry of Education, Science and Culture (Nos. 61790127 and 61460043).

dr

d .,c/

re

AI.~."I - Bj

(A-3)

where:

Qu,kl3T

GM~(m, - m) A / = 4~r£mun: kHT

(A-4)

and:

xu:Qjkt~T 8,=

4~f,m,

(A-5)

When A, = 0 , we find m] =rn~ from equation (12). We concentrate ourselves on escaping species where m] < rn~, we have Aj > 0. Since the air outside r = r,, is in steady state (Q, being constant) and isothermal, both Aj and B, are independent of r. Integrating equation (A-3) by r and adopting r--, m to the outermost boundary, we obtain:

x,(r,,) -

x,(~c)

¢ -

-

A,

333

where xj(~e) should be some finite value between 0 and 1. So long as: exp - r~- << 1

(A-7)

we have:

e, x,(r,,)

-

= 0

(A-8)

This can be reduced to: UH, -- u I = -

--1

XH, \ m

Dtl _ j

r2p

.

(A-9)

Thus as long as relation (A-7) holds, the self-diffusion can be neglected. N o w let us consider relation (A-9). We have:

of xenon ( m ~ - m ~ >> mH). At the later stage of atmospheric escape when mass of xenon should correspond to m c, r,, should approach r~ ( - 6 × 10 6 m) and A i r o is still large. Therefore (A-7) is effective in general. The exception, if exists, is for an isotope whose mass is almost the same with m c (difference being smaller than 10 Im~t). In this case the species may ooze owing to ~'xj. But we find, from the condition x~ should not become zero throughout the atmosphere, that in our case the escaping rate of the oozing species is very small ( u j < < u H , ) and its effect on the isotopic pattern is s m a l l Thus, during the whole stage of the dissipation of the primordial atmosphere we can neglect the effect of self-diffusion.

References

r,, -

4 ~ f l m H ro .•

1 -

QHz(kr~T)"

(A-10)

=/3,(I - a,)

When m~ = rn~ (crossover mass), we get 1 - a~ = 0 from equation (14). Since f~ is not largely changed a m o n g species, we can deduce: QH.kB T

flJ

GM~(rn~ - m)

4,rrfjn ii r,, =

kBTr °

_ 5 × 10 ~ ( m ) m ~ -

ro

m

(A-11)

m ]t

where m H is mass of hydrogen atom, and M, = 1M~ and T = 100 K are adopted. A n d 1 - a~ is also expressed, using m ~ , as: m c -- m ;

1 - a/

(A-12) D'1c -- ~1

Thus we have: A/ ro

5 × 1 0 8 (m) m ~ - m j ro

mH

(A-13)

This suggests that relation (A-7) is effective unless m~ = m~. Even if species j is lighter only by 1 mass n u m b e r (i.e., m ~ - r n j - - - m H ) , we can approximate as (A-7) when r,, < 1 × 10 ~ m. As to lighter species (rn~ << m~), the validity of (A-9) becomes much better. In the present problem, at the initial stage of the atmospheric escape where 6, might be as large as 3 × 10 8 m [16], QH, is larger than the limiting flux and m~ is larger than mass

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334 14 H. Mizuno. K. Nakazawa and C. Hayashi. Gas capture and rare gas retention by accreting planets in the solar nebula, Planet. Space Sci. 30, 765-771, 1982. 15 S. Sasaki. Structure and evolution of the primordial H 2 He atmosphere surrounding the accreting Earth and the origin of the terrestrial noble gases. 229 pp., Ph.D. Thesis, University of Tokyo, Tokyo, 1987. 16 M. Sekiya, K. Nakazawa and C. Hayashi, Dissipation of the primordial terrestrial atmosphere due to irradiation of the solar EUV. Prog. Theor. Phys. 64, 1968-1985, 1980. 17 M. Sekiya. C. Hayashi and K. Nakazawa, Dissipation of the primordial terrestrial atmosphere due to irradiation of the solar far-UV during T Tauri stage. Prog. Theor. Phys. 66, 1301-1316, 1981. 18 M. Sekiya, K. Nakazawa and C. Hayashi, Dissipation of the rare gas contained in the primordial earth's atmosphere, l-arth Planet. Sci. Left. 50, 197-201, 1980. 19 T.M. Donahue, Thermal escape from accreting planetesireals, Icarus 66, 195-210. 1986. 20 K.J. Zahnle and J.F. Kasting, Mass fractionation during transonic escape and implications for loss of water from Mars and Venus, Icarus 68, 462-480, 1986. 21 D. C h a p m a n and T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 423 pp., University of Cambridge Press, Cambridge, 1970.

22 E. Anders and M. Ebihara, Solar-system abundances of the elements. Geochim. Cosmochim. Acta 46. 2363-2380, 1982. 23 T.R. Marrero and E.A. Mason, Gaseous diffusion coefficients, J. Phys. Chem. Ref. Data 1, 3--118, 1972. 24 M. Ozima, Looking for missing xenon, Nature 322. 813-814. 1986. 25 M. Ozima and S. Zashu, Solar type Ne in Zaire cubic diamonds, Geochim. Cosmochim. Acta 52. 19-26, 1988. 26 T. Staudacher, Upper mantle origin for Harding ('ounty well gases. Nature 325. 605-607, 1987. 27 Y. l l a m a n o and M. Ozima, Earth-atmosphere evolution model based on Ar isotopic data, in: Advances in Earth and Planetary Sciences 3, E.C. Alexander and M. Ozima. eds., pp. 155-171, ('enter for Academic Publications. Tokyo. 1978. 28 J.E. G a u s t a d and S.N. Vogel, High energy solar radiation and the origin of life. Origin of Life 12, 3-8, 1982. 29 1I. Craig and J.E. Lupton, Primordial neon, helium, and hydrogen in oceanic basalts, Earth Planet. Sci. l,ett. 31. 369-385. 1976. 30 1t. Mizuno, K. Nakazawa and ('. Hayashi. Dissolution of the primordial rare gases into the molten Earth's material, Earth Planet. ,~i. l,ett. 50. 202 210, 1980.