The abundance of water and rock in Jupiter as derived from interior models

ICAI~US30, 155--162 (1977)

The Abundance of Water and Rock in Jupiter as Derived from Interior Models ~'~ORRIS P O D O L A K

Princeto~ University Observatory, Princeton, New Jersey 085~0 Received April 9, 1976; revised M a y 10, 1976

Models of Jupiter's interior were computed using a new molecular hydrogen equation of state, and a more precise algorithm for computing gravitational moments. Models with H / H e ratios of 13.5, 11.25, and 10 were computed, and it was found t h a t the models had rocky cores of 16 to 18 E a r t h masses and envelopes with 20 to 50 E a r t h masses of water. Models with temperatures of less t h a n 180°K at 1 bar had water-to-rock ratios which were less than the solar value, thus removing a difficulty of previous computations. Models were also constructed in which rock was p u t into the envelope in place of water. I t was found that, to a good approximation, a given mass of rock will replace an equal mass of water. At present the two cases are indistinguishable.

INTRODUCTION Recently models of the giant planets were computed on the basis of an accretion picture of planetary formation (Podolak and Cameron, 1974; hereafter called Paper I). The basic assumption was that the giant planets formed cores out of condensed material in their neighborhood, and then captured an atmosphere of hydrogen, helium, and other gases from the surrounding primitive solar nebula (Cameron, 1973a). To a first approximation, then, one would expect Jupiter to consist of a core of rock surrounded by an envelope with an elemental composition similar to that of the Sun. If, however, the temperature in the vicinity of the proto-Jupiter dropped below about 170°K, H~O ice would condense (Lewis, 1972) and might be accreted more efficiently than substances which remained in the gaseous state. Thus the H20/H2 ratio in Jupiter might be considerably enhanced with respect to what one would expect for solar composition material. Models of this type were constructed which matched Jupiter's mass, radius,

oblateness, and temperature at 1 bar. It was found that large amounts of H~O were needed in Jupiter to match the observed shape of the planet. It was found, in fact, that the ratio of H~O to rock by mass was about 9.5, while the solar ratio is about 2 (Paper I). This implied that ice was somehow accreted much more efficiently than rock, a situation difficult to envision. In a second paper (Podolak and Cameron, 1975; hereafter called Paper II), some of the assumptions made in Paper I were tested. It was shown that the large H20 abundance could not be lowered by a change in the equation of state of water, a more precise treatment of the adiabat in the metallic hydrogen region, or reasonable changes in the assumed temperature at 1 bar. The water-to-rock ratio could be brought down to the solar value if the hydrogen-to-helium ratio was lowered to 7.7 hydrogen atoms per helium atom, but this is well below the solar value of 13.5 (Cameron, 1973b). It is the purpose of this paper to bring the discussions of Papers 155

Copyright ~ 1977 by Academic Press, Inc. All rights of reproduction in any form reserved.

156

MOll RIS POI)OLAK

I and II up to date by presenting the results of recent computations. MULTIPOLE MOMENTS The external gravitational potential of a rotating fluid body can be written as (Zharkov el al., 1972) 0)

1

= r

/

and 50.1 Earth masses of H2(). The second had a temperature at 1 bar of 155°K, a core of 12.0 Earth masses, and 46.9 Earth masses of H20. Tile water-to-rock ratios were thus reduced to 4.4 and 3.9, respectively. We see then, that using the thirdorder theory removes much of the difficulty, though not all. tIYI)ROGEN EQUATION OF STATE

n~l

/a\

×t;)

2"

-1

<:.)

where G is the gravitational constant, M and a are the mass and equatorial radius of the planet, r and 0 are the coordinates of a point outside tile phmet, P~,~ is the 2nth Legendre polynomial, and J2,, is the 2nth multipole moment. For Jupiter, the Pioneer flybys have given good values for J2 and J4, and an estimate of the value for J6 (Null el aI., 1975). One of the criteria for acceptable models of Jupiter is that they match these observations, hi Papers I and II, J2 and J4 were computed from a theory accurate to second order in ~o'-'a'~GM, where co is the angular velocity (Peebles, 1964). It has been pointed out, however (Zharkov et al., 1972), that this procedure overestimates J2. In the case of Jupiter models this overestimate is typically about 5(~ (Stevenson and Salpeter, 1975). This means that the models of l'apers l and I[ were more oblate than necessary, and one can therefore do with less heavy material in the envelope, and more in the core, thus lowering the water-to-rock ratio. New models were therefore computed, and J2, J4, and J , were calculated with a third-order code using the method of Zharkov and Trubitsyn (1971) kindly supplied by Dr. A. Summers. The best models of Paper II had cores between 5 and 6 Earth masses, and between 50 and 60 Earth masses of water. Of the two new models, one with a temperature of 170°K at 1 bar had a core of 11.5 Earth masses,

Although the equation of state of metallic hydrogen is well known, and the various formulations give results which agree well with each other (Hubbard and Smoluchowski, 1973), the equation of state of molecular hydrogen is much more poorly known. Theoretical computations are difficult, and still unreliable at higher pressures. Experimental results at high pressure have only recently become availal)le, and are still uncertain (Gregoriev et al., 1972; Ross, 1974). In Papers I and II a zero-temperature equation of st:~te proposed t)y 1)eMarcus (195S) was used for molecuhu' hydrogen. Recently another equation of state for H2 was proposed by Zharkov et al. (1974, hereafter called ZTM). This e(tuation of state predicts a very small change in density in the transition from molecular to metallic hydrogen, in agreement with the equation of state developed at Livermore (Graboske et al., 1975a) and currently being used in model phmet eah'ulatiolls at NASA Ames Research Center (Summers, 1975, private communication). In fact, sample computations of solar material equations of state from the theory used in the present study agree well with the Livermore calculations. At pressures greater than about 0.08 Mbar, the density predicted by the ZTM equation of state is about 10% higher than that predicted by DeMarcus (1958) (see Fig. 1). With this denser form of H~ in the outer layers of the planet, less H:O should be required to produce the observed oblateness. Model phmet calculations show that this is indeed the case. In these models the

WATER AND ROCK IN JUPITER 7.0

....

I ....

I

'

'

'

be m a d e easily, models were c o m p u t e d with three values of the H / H e ratio: 13.5, 11.25, and 10. Figure 2 and Table I show the results of this computation. As can be seen from the figure, for a given choice of the t e m p e r a t u r e at 1 bar, increasing the helium content lowers the H~O-to-rock ratio. This is consistent with the result found in P a p e r I I . F o r H / H e = 10, the t e m p e r a t u r e at 1 bar m u s t be less t h a n 193°K to h a v e the water-to-rock ratio fall below the cosmogonic constraint. F o r H / H e = 13.5, the t e m p e r a t u r e m u s t be less t h a n 177°K. Current estimates place the t e m p e r a t u r e at the 1 bar level at a b o u t 165 :t: 10°K (Orton, 1975); thus even for high H / H e ratios there is no problem a b o u t a cosmogonically high water-to-rock ratio. A model with a t e m p e r a t u r e at 1 b a r of 165°K and H / H e = 13.5 would h a v e a r o c k y core of 17.8 E a r t h masses, 32 E a r t h masses of water, and a water-to-rock ratio of 1.8, just under the solar value. I t is interesting to note t h a t in the models discussed in P a p e r I I , the addition of helium caused the core mass to increase, while in these models the addition of helium causes the core mass to decrease (see Fig. 3). The difference can be understood b y the following qualitative argument. Let us ignore for the m o m e n t H20, NH3, and CH4. W h a t is necessary for tile

'

H2-Equetion of stote ./

// 6.0

~ u) 5.0 w

_o ..A 4.0 - -

Marcus

De

(1958)

~-- Zh~kov etol (19741 5.0

,I 0

,

,

~

I 0.5

,

,

,

,

I 1.0

,

L

,

, 1.5

(gin cm-5)

DENSITY

FIG. i. Comparison of the DeNIareus equation of state for molecular hydrogen (solid line) with that of Zharkov et al. (dashed line). same procedures were used as in P a p e r s I and II, b u t here again the multipole m o m e n t s were calculated to third order, and the Z T M equation of state for H~ was used. Although in these c o m p u t a t i o n s we h a v e always assumed t h a t the solar ratio of hydrogen to helium is 13.5, other investigators h a v e chosen different values, often as low as 10. To enable a comparison to 5,0

JUPITER .

2.6 2.2 oL i~ 1.8 ~

H/He

=

1

.

157

.

.

.

~ /

5

1.4 1.0 0.6 1 140

I 160

I

I 180

i

TI bar

I 200

I

I 220

i 240

(K)

Fl(J. 2. Variation of water-to-rock ratio with temperature at t bar for various values (,f tile H/He ratio. The solar value is about 2.

158

MORRIS PODOLAK TABLE I J u p i t e r Models H/He"

7'1 b,,, (°K)

M¢,,r,, ~

M ~(lr,,g ....

2]~fheli .....

13.5 13.5 13.5 13.5

224.5 202.7 182.0 157.7

15, S 16,S 17,4 l S, (i

196.30 199.19 202.46 207.2;{

57. S0 58.65 5!1.62 (il . 02

46. () 41 .t-; 36. N 30.0

5.94 2.4~ 5.12 I . (iS

11.25

208. ti

15.6

I !15.65

2.4~

182.4

16.7

199.66

31 .9

1 .91

1 I. 25

164.4

17.5

202. b;1

(i9.13 70.5,5 71.66

3S. ti

I 1.25

26.7

1.53

10.0

222,4

14.7

I N,().93

] 0.0 l 0.0

t 77.7 160.1

16.3 17.0

196. N() 19!1.92

7,5.50 7S. 23

:iN. 7 26. N

5.64 1 . 65

79.47

21

] , 26

H/ite"

T l I,:,r ( ° K )

3I~., ,,.

zl/l,y,l,. .......

:ll~,,.li .....

3/r,,,.k d

13.5 13.5

191 . 2 174.2

11 . 4: 12.7

205.90 208. lil

13.5

157.7

13.9

210.56

50.33 61.00 61.70

36.9 32.ti 58.3

M~x:lt,,r

•4

Waier/ro('k ~

" By immber.

b All m a s s e s are given in terms of i h e E a r l h ' s mass ((i X 1027 g). c By mass. Solar ratio is ~ 2 . Only the rock in the envelope.

mass and radius of tile model to remain the same is that the density remain roughly the same as a function of radius. If hydrogen is fairly dense, then the addition of, say, 10 Earth masses of helium can be offset by removing a little hydrogen from each v o l u m e element to keep the density distribution invariant. This involves the removal

of less than 10 Earth masses of hydrogen, and therefore some of the rocky core must also be removed. If hydrogen is less dense, more of it needs to be removed at each radial point to keep tile density constant. If more than 10 Earth masses are removed, some rock must t)e added to the core. A p p a r e n t l y the l)eMarcus equation of

22

i

i

JUPITER 20 18 H/He = 13.5 16

~o

=E

14 12 10 140

~Rock l

160

I

I

I

180

I

200

I

I

220

I

240

t i b e r (K)

FIG. 3. Variation of core nlass with t e m p e r a t u r e at 1 bar for various values of the II/]Fle ratio. T h e c u r v e l a b e l e d " R o c k " h a s rock in t h e envelope in place of w a t e r (see text).

WATER AND ROCK IN JUPITER state and the ZTM equation of state are sufficiently different for this effect to be observed. SOLAR H20 MODELS With the cosmogonic difficulty of a high water-to-rock ratio removed, one is tempted to stop here, but there are other points which must be considered. In particular, there is the observation of Larson et al. (1975) of an H20-to-H2 mixing ratio in Jupiter of ~10 -6, which means an H20 abundance of about 10-3 that of the Sun. These observations were made at 5 tLm, presumably in the 5 t~m hot spots that have been observed on Jupiter (Westphal et al., 1974). These hot spots are as small as 1 '~rcsec in extent, with temperatures as high as 300°K. Apparently they are clear spots in Jupiter's cloud deck which enable us to see to greater depths (and higher temperatures) in the planet. On this basis Danielson (1975, private communication) has suggested that strong downdrafts might be causing these holes in the cloud deck. These downdrafts, composed of high-altitude gas, would be expected to be poor in water vapor, thus providing a ready explanation for the observations. An alternative explanation, of course, is that there is actually very little water present on Jupiter. In this case some other material of high molecular weight is needed in Jupiter's envelope to provide the observed oblateness. One suggestion (Paper II) has been that rocky material falling into Jupiter in the last stages of its accumulation was vaporized, and somehow remained suspended in the envelope. This rocky material may make up some of the excess mass that we have tentatively identified as water vapor. Cameron (1975, private communication) has pointed out that there are other good reasons for suspecting that Jupiter may not be rich in water• The decrease in density of the Galilean satellites with increasing distance

159

from Jupiter is good evidence for a hot proto-Jupiter that kept temperatures in its vicinity sufficiently high that H:O remained as a vapor, and did not accumulate on the innermost satellites (Pollack and Reynolds, 1974; Cameron and Pollack, 1976). A proto-Jupiter this hot could well have caused ice to vaporize off infalling debris, so that only the rocky portion was accreted into the planet. Strong convection in the proto-Jovian atmosphere may also have acted to mix any water that was collected back into the solar nebula before the final hydrodynamic collapse (Perri and Cameron, 1974). Finally, evolutionary models of Jupiter (Graboski et al., 1975b) indicate that central temperatures were at one time considerably higher than they are now. Such high temperatures may have vaporized part of the core. All these arguments indicate the need for some exploratory models of Jupiter which contain large amounts of rocky materials in the envelope instead of water. The problem is complicated by uncertainties in the chemistry and physics of rocky material under the appropriate conditions of temperature and pressure. Consider, for example, the chemistry of silicon, one of the most abundant rock-forming elements. Rock, in the form of SiO~, can react with hydrogen in the following way: SiO2 + 4H2 ~---2H20 + Sill4.

(2)

Under ordinary conditions the reaction goes violently to the left. One can, however, compute the partition functions of the molecules, assuming that they act as if they were in the gaseous state, and from that compute the equilibrium constant for the reaction. The result is shown in Fig. 4 for an assumed hydrogen to water ratio of 103 (i.e., solar). The approximate pressure-temperature relations for Jupiter and Saturn have also been plotted. It can be seen that Si02 is t h e . d o m i n a n t species below 2000°K, land SIH4 above. Fhls agrees with the result of Lewis (1969). •

~J

r

•

160

MOIIRIS PODOLAK DENSITY

(gin crn -5)

lO-15 i0 -Io

10-5 /0 -5

I0

~

,

,

i0 -I i

T = 5018

K

2100 K lO08k

0

505 K r

~ ,¢ N

-IO

o

gg I

.P

I

_o

-

20

~'~

233 K

-50

-40

-50

/ IO

L 20

30

FIe-. 4. Straight lines show the vm'ialion of nsinJnsloo with hydrogen number densily (nu2) for H=/H~O = 103 at various temperatures. Curves show approximate n~12-temperature relations for Jupiter and Saturn. The other a b u n d a n t rock-forming element is magnesium. Its chemistry is less well understood, however. In chemical equilibrium one would expect that Mg and M g I t are the dominant species at 2000°K, but MgO has such a low w~por pressure even at this temperature that. MgO condensation provides an efficient sink which allows only about 1% of the solar abundance of magnesium to remain in the v'tpor phase (Lewis, 1969). We can, therefore, a t t e m p t to model the distribution of rocky material in Jupiter in a rough way, by assuming t h a t it acts like Sill4. Although the equation of state of SitI~ is poorly known, the additive volume law ensures t h a t tt substance of high density which comprises a small fraction of the total mass of material will have a negligible effect on the tot'd e
particular about the accuracy of the equation of state of Sill4. For reasons of convenience we have, therefore, used the same equation of state as was used for rock in the core. It was also assumed that (following tile behavior of SiHt) the mixing ratio of rocky material to hydrogen remained constant for temperatures above 2000°K, and went linearly to zero between 2000 and 1000°K. This interval is quite arbitrary, but numerical tests show t h a t the results are insensitive to the precise range over which the cutoff is operative. In computing the adiabat through this material, we have ignored any energy released by a chemical or phase change of the material. This is a fair approximation for two reasons. First the rock constitutes only about ..,n5(,//~ of the total number of particles even in the most rock-rich case. Thus even though the energy released per rock particle m a y be large, that energy per gas p a r t M e must be small compared to the relevant thermal energy of the gas p.trticles. The second reason is t h a t the composition of the rock is not well known, nor is its thermodynamics. Including the effects of possible chemical changes and phase transitions would only add to the uncert'dnties already present in the problem. The results of the computation are shown in Fig. 3 by the curve labeled "rock," and in Table I. Perhaps the most i m p o r t a n t thing to be seen from the table is that to an excellent approximation a given mass of water in the envelope can be replaced t)y an e(lual mass of rock. It cannot, in general, be replaced by an equal mass of helium. This is due to the fact that helium, because of its greater abundance, contributes much more than water or rock to the total equation of state. In all cases the mass of condensable material is about 50 E a r t h masses, which is of the right magnitude t'()r the Perri-(Jaineron instability to (tevclop (Perri and Cameron, 1974).

WATER AND ROCK IN JUPITEI~ CONCLUSIONS These models of Jupiter which have been computed were chosen to fit the mass, radius and J~ of Jupiter to three significant figures. A more accurate algorithm for computing the multipole moments and an improved equation of state for H2 do not change the conclusion, arrived at in Papers I and II, that large amounts of heavy material are required in Jupiter's envelope (about 30 to 40 Earth masses). The problem of having too large a water-to-rock ratio is no longer present, however. In the waterrich models the water-to-rock ratio is still near the solar value, indicating that if water alone is enhanced, the temperature in Jupiter's vicinity dropped below 170°K (the ice condensation temperature) quite early in the planet's history. The solar-H20 models indicate, however, that a hot nebula in Jupiter's vicinity cannot be ruled out on this basis. The two compositions give very similar models, which hardly differ even in J4. Thus a more accurate measurement of J4 alone will not help to distinguish between the two compositions, but rather a probe may be required. Mention of J4 brings out another interesting point. For all of the models computed above, J4 is in the neighborhood of -0.00066. The observed value is --0.00058 ± 0.00004 (Null et al., 1975). The algorithm used for computing J4 probably introduces an error of about 3% (Stevenson and Salpeter, 1975). Thus the disagreement between the observed and computed values of J4 may be absorbed in the combined uncertainties of computation and measurement. Certainly this points out the need for both a more accurate algorithm for computing J4 [such an algorithm has been developed for continuous density distributions (Hubbard et al., 1975)~, and more precise measurements from spacecraft. A more intriguing explanation for the discrepancy is possible, however. J4, "~s Hubbard (1974) has shown, is related to the density at a depth of

161

about 3000 km in Jupiter. In fact, the density we have computed may be somewhat high, and a slightly lower density would bring J4 more into accord with the observed value. Numerically, the mass fraction of hydrogen is about 0.68, that of helium is about 0.20, while that of 40 times the solar complement rock is about 0.12. For a zero-temperature pressure of 30 kbar, the densities of hydrogen, helium, and rock are 0.22, 0.68, and 3.2 g/cm a, respectively. For a mixture of the three, the additive volume law gives a density of 0.29 g/cm 3. For a mixture of hydrogen and helium only, the density is 0.26 g/cm 3. We see that the rock contributes about 10% to the total density, and since J4 is proportional to the density at the 3000 km level, there is the possibility of affecting the computed value of J4 by roughly this amount. If some of the rocky material has different properties than Sill4, and its mixing ratio goes to zero at depths greater than 3000 kin, it might still be possible to obtain the required J2, as well as the observed J4. Such an explanation would not be possible with a water-rich envelope, and may serve as a method of distinguishing between the two possibilities. In view of the present uncertainties in theory :rod observation, we will have to wait a considerable time before it can be decided whether such an explanation is indeed necessary. It must be pointed out here that higher temperatures at the 3000 km level would also achieve the same result. Anderson et al. (1974) have shown this, but they run into the problem that higher temperatures at 100 kbar mean higher temperatures at 1 bar. Their value of about 250°K at 1 bar is higher than the observed value. We see, then, that Jupiter's interior is still not completely understood. In addition to better values for the multipole moments, and the temperature at 1 bar, a knowledge of the composition in the upper layers will prove to be very informative. If it turns

162

MORRIS PODOLAK

o u t t h a t t h e w a t e r - r i c h m o d e l s are u n t e n a b l e , it will be n e c e s s a r y to d e t e r m i n e a c c u r a t e p a r a m e t e r s for r o c k y m a t e r i a l s t h a t are likely to be f o u n d in J u p i t e r ' s interior. ACKNOWLEI)GMENTS It is a great pleasure to thank Drs. A. (~. W. Cameron, R. E. Danielson, J. Gelfand, and R. Smoluchowski for many stimulating discussions and comments. In addition I would like to thank I)rs. R. Reynolds and A. Summers for their hospitality at the NASA Ames Research Center, where some of the computations were done. This work was supported by NSF Grant GP-39055. REFERENCES

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