We present models of Jupiter, Saturn, and Uranus which exactly match ... (molecular hydrogen) layers of Jupiter or Saturn, or if it has, that its effect on the density.
ICARUS 78, 1 0 2 - 1 1 8 (1989)
Optimized Jupiter, Saturn, and Uranus Interior Models W. B. H U B B A R D AND M A R K S. M A R L E Y Department of Planetary Sciences, Lunar and Planetary Laboratory, University of Arizona, Tacson, Arizona 85721 R e c e i v e d A p r i l 25, 1988; r e v i s e d J u l y 15. 1988
We present models of Jupiter, Saturn, and Uranus which exactly match recent accurate determinations of these planets' gravitational harmonics. The models are computed to third order in the rotational disturbance to the total potential and are based upon a method for inverting the gravitational data. For Jupiter and Saturn, a range of gravity models is calculated to test the possibility of a reduction of density in the outer layers due to helium depletion. The results, which are based upon an improved equation of state for molecular hydrogen, indicate that major helium depletion has not occurred in the outer (molecular hydrogen) layers of Jupiter or Saturn, or if it has, that its effect on the density profile is masked by the presence of other, denser, components. Jupiter is found to be slightly enhanced in heavy elements with respect to solar composition, but the density profile of its hydrogen-rich layers generally agrees rather well with a theoretical profile for solar composition. The deviations from such a profile are more pronounced in the case of Saturn. Uranus models have considerable uncertainty; one successful model resembles the ice-rich model of M. Podolak and R. T. Reynolds (1987, Icarus 70, 3136), but is fitted to a newer value of J4. Our Uranus model has a substantial enrichment of heavy elements at depth, but little separation of the ice from the rock component. All of the Jovian planets appear to have central cores of non-hydrogen-helium material which are of similar mass (about 10-15 Earth masses). For Jupiter and Saturn, our calculations yield a gravitational harmonic J6 which is in agreement with observation, but suggest that this quantity, along with harmonics of higher degree and order, is likely to be more useful for constraining the nature of fluid currents in outer layers rather than deep static structure. ©1989 Academic Press, Inc.
I. I N T R O D U C T I O N
The most direct constraints on the interior structure o f the Jovian planets are currently imposed by the mass, radius, rotation period, and gravitational harmonics. Recent Earth-based and spacecraft-based observations of Jupiter, Saturn, and Uranus have provided more precise values for all of these parameters. Campbell and Synnott (1985) published a solution for the Jovian gravity field which gives the mass of the Jovian system to within an uncertainty o f 10-6% and the first three even zonal harmonics J2, J4, and J6 to within uncertainties of 7 × 10 3, 0.9, and 65%, respectively, a substantial improve-
ment on the earlier work of Null (1976). Nicholson and P o r c o (1988) have carried out an analysis of the precession of an eccentric Saturn ringlet to obtain an improved determination of the first three even zonal harmonics of Saturn, to within uncertainties of 0.1, 6.7, and 47%, respectively. Finally, observations of Uranian ring precession rates (French et al. 1988) have yielded the Uranian Jz and J4 to within 0.1 and 1.6%, respectively. The mass, radius, and interior rotation rate of these three planets are now accurately known from spacecraft flyby data. We present here a set of interior models which exactly satisfy the new, more precisely k n o w n gravitational constraints. This 102
0019-1035/89 $3.00 Copyright © 1989 by Academic Press, Inc. All rights of reproduction in any form reserved.
INTERIOR MODELS
103
TABLE I PRIMARY CONSTRAINTS ON STRUCTURE OF JOVIAN PLANETS Planet Jupiter Saturn Uranus Neptune
a (km) 71,492 60,268 25,559 24,992
-+ -+ -+ -+
J2 × 106 4 4 4 30
14,697 16,331 3,516 4,000
-1 18 -+ 3 _+ 300
../4 × 106
,]6 x 106
-584 -- 5 -914 -+ 61 - 3 1 . 9 - 0.5 0a
31 -+ 20 108 -+ 50 0° 0"
q × 106 89,180 154,766 29,513 28,960
-+ 15 -+ 31 -+ 48 -+ 450
a Assumed.
paper also addresses the following questions: (a) How unique are the models which satisfy present constraints, and to what extent can they be used to test chemical evolution models of the Jovian planets? (b) What more can we learn by measuring the zonal harmonics with even more precision? The present study is based upon two earlier investigations. Hubbard and Horedt (1983) presented a computational method for obtaining a planetary interior model in hydrostatic equilibrium which exactly satisfies a given set of gravitational constraints. We have implemented this technique here to test various assumptions about the equation of state and chemical composition of Jupiter, Saturn, and Uranus, using equations of state for H - H e mixtures obtained by Hubbard and DeWitt (1985) and by Marley and Hubbard (1988). Neptune models are not investigated in detail because the Neptune rotation period is still too uncertain to provide a tight constraint on interior structure. II. P R I M A R Y A N D I N D I R E C T C O N S T R A I N T S ON MODELS
A. Primary Constraints
The primary constraints on Jovian-planet interior models are listed in Table I. For Jupiter, the equatorial radius at 1 bar pressure, a, is derived from the equatorial radius at 0.1 bar given by Lindal et al. (1981), a0 = 71,541 --- 4 km. The Jovian zonal harmonics given in Table I correspond to the values given by Campbell and Synnott (1985), but are normalized to a instead of to a radius of 71,398 km. The solid-
body rotation rate to is computed from the magnetospheric rotation period given by Seidelmann and Divine (1977), from which we compute the dimensionless rotation parameter to2a3
q = GM'
(1)
where G is the gravitational constant and M is the planet's mass. For Saturn, the equatorial radius at one bar pressure a is from Lindal et al. (1985), the gravitational moments are from Nicholson and Porco (1988) but renormalized to the adopted value of a, and the solid-body rotation rate is taken to be the magnetospheric rotation rate inferred from periodicities in kilometer-wavelength radio storms (Kaiser et al. 1984). For Uranus, the equatorial radius at one bar pressure a is from Lindal et al. (1987) and the gravitational moments (renormalized to a) are from French et al. (1988). The rotation rate is taken to be the magnetospheric rotation rate deduced by Warwick et al. (1986). The corresponding data for Neptune are much cruder for two principal reasons: (I) precession rates of Neptune rings cannot be readily measured, and thus the only available information about Neptune's J2 comes from observations of the precession of Triton's inclined orbit (Harris 1984); (2) spacecraft measurements of Neptune's magnetospheric rotation rate have not yet been carried out, and thus the interior rotation rate must be inferred much more indirectly (and imprecisely) from the planet's atmo-
104
HUBBARD AND MARLEY
spheric oblateness (Hubbard et al. 1987). The Neptune equatorial radius at one bar a is computed from the equatorial radius at l p.bar pressure given by Hubbard et al. (1987), subtracting the altitude difference between the two pressure levels according to the atmospheric model of Orton et al. (1987), and assigning an increased error bar because of uncertainties in the model. Comparing the Neptune entries in Table I with the other entries, we conclude that detailed modeling of the interior of Neptune can be deferred until better constraints are available, although we make some general remarks in a subsequent section of this paper. B. I n d i r e c t C o n s t r a i n t s
Considerations of possible bulk composition and distribution of chemical species impose indirect constraints on Jovianplanet interior models. The composition of the atmosphere, like the interior, is dominated by hydrogen, but may not correspond in detail with the bulk interior composition (Stevenson and Salpeter 1977a,b). According to Gautier and Owen (1988), the renormalized helium mass fraction Y of the atmospheric gas is 0.18 -+ 0.04 (Jupiter), 0.06 + 0.05 (Saturn), and 0.26 -+ 0.05 (Uranus). These numbers are renormalized to a gas in which the mass fraction of the non-hydrogen-helium component, Z, is 0.02, as would be the case for solar composition. Proposed values for the primordial solarcomposition value of Y are still somewhat varied, ranging from Y = 0.20 (Cameron 1973) to Y = 0.24 (Anders and Ebihara 1982) to Y = 0.28 (Gautier and Owen 1988). Nevertheless, the observed numbers for the Jovian planets suggest that the helium/hydrogen ratio is likely to be primordial and uniform in Jupiter and Uranus, but strongly nonuniform in Saturn. Stevenson (1975) predicted that heliumhydrogen phase separation might occur in Saturn's metallic-hydrogen zone, and be incipient in Jupiter's metallic hydrogen zone, leading to some atmospheric helium
depletion in both planets. Hubbard and MacFarlane (1985) predicted unlimited solubility of helium in hydrogen in the Jovian and Saturnian interior. The observed atmospheric values of Y favor the Stevenson model. However, one can in principle test the helium-separation prediction further by constructing interior models which match the constraints given in Table I. If helium is severely depleted throughout Saturn's molecular hydrogen envelope, then the predicted pressure-density relation for the envelope should also be consistent with the observed gravitational moments. One may impose a similar test for Jupiter. Strictly speaking, the density differences between hydrogen fluids containing various helium fractions could also be reproduced by hydrogen fluids with a constant helium mixing ratio but variable fractions of nonhydrogen-helium molecules, such as H20 or CH4. In fact, Gautier and Owen (1988) conclude that the atmospheric mixing ratio of CH4 is twice solar in Jupiter, two to six times solar in Saturn, and possibly 25 times solar in Uranus and Neptune. The deep tropospheric NH3 abundance in both Jupiter and Saturn may be enhanced by a factor of order 2 in both planets (Gautier and Owen 1988, de Pater and Massie 1985). If the molecule H20 has a similar enhancement in each planet, then the pressure-density relation could be softened with respect to the solar-composition relation, that is, substantially shifted toward higher densities at a given pressure. Unfortunately, the abundance of H20 is difficult to measure in Jovian planet atmospheres because of pronounced condensation effects, and the results for Jupiter are ambiguous (Bjoraker 1985, Lunine and Hunten 1987). Similarly, NH3 may be depleted in the atmospheres of Uranus and Neptune by condensation effects (de Pater and Massie 1985). lIl. METHOD OF COMPUTING MODELS Conventional models for the formation of Jovian planets (Mizuno et al. 1978, Mizuno 1980, Hubbard and MacFarlane 1980b) vi-
INTERIOR MODELS sualize a process of nucleation of a rockice core with mass Me - 10M, ( M . = 1 Earth mass), which then triggers an instability in the surrounding nebular gas, leading to the accumulation of a massive hydrogen-rich, isentropic envelope. Efficient convection is assumed to homogenize the composition of the envelope. Stevenson (1985) has criticized this conventional nucleation picture, pointing out that the envelope may have a more complicated structure. It may, for example, be enriched in rock-ice materials from late accretion of planetesimals which were not incorporated in the trigger nucleus. The resulting compositional gradients may inhibit convection in the envelope, leading to substantial superadiabaticity. There may not even be a distinct core, due to the inefficiency of gravitational settling. Nevertheless, the conventional nucleation picture offers a simple way to characterize an interior model with a small number of parameters. Thus we shall consider models which are governed by three distinct adiabatic pressure-density relations. The central region is assumed to be composed of either " r o c k " or " i c e , " in which the relation between pressure P and mass density p is given by ("rock") p = p4.406 exp(-6.579 - 0.176p + 0.00202p 2) (2) or
("ice") P
=
,0 3"719
exp(-2.756 - 0.271p + 0.0070102),
(3)
where " i c e " = 56.5% H20, 32.5% CH4, and 11% NH3 by mass; " r o c k " = 38% SiO2,25% MgO, 25% FeS, and 12% FeO; p is in g/cm3; and P is in Mbar. These analytic expressions adequately approximate the theoretical equations of state for warm (temperatures -104°K) adiabatic mixtures of these molecules over the pressure range
105
appropriate to Jovian-planet cores. Because the core region makes little contribution to the gravitational harmonics, and because the composition of the core is poorly constrained, more detailed equations of state are not needed in this region. Thus, a planetary model has a central core of " r o c k , " and, in some variants of the model, this core also has an outer layer of " i c e " with a specified mass ratio of " i c e " to " r o c k . " The total ( " i c e " plus " r o c k " ) core mass is Me. In Jupiter and Saturn models, the central core is overlain with a continuous hydrogen-rich envelope whose pressure-density relation P(p) strongly affects the zonal harmonics. According to the conventional nucleation picture, this envelope should consist solely of hydrogen and helium, together with the solar complement of other volatile molecules which were not incorporated in the core. The latter would not significantly affect the envelope P(p) relation because of their small relative abundance. Thus, if we had available an accurate P(p) relation for a solar-composition mixture of H and He throughout the pressure range spanned by the Jovian envelope, a model calculated from this relation which fitted the constraints would validate the nucleation model. On the other hand, if the constraints were to require a higher density at a given pressure than would be given by the solarcomposition mixture, this would provide support for Stevenson's more complicated scenario. Furthermore, in the case of Saturn, there is evidence for very substantial depletion of helium in the atmosphere. If this depletion extends throughout the molecular hydrogen envelope, the density of the envelope should be substantially lower than the density of a solar-composition envelope at a given pressure level. Such a difference should be apparent in the gravitational moments. Thus, our principal objective is to test for departures from a strictly solar P(p) relation in the envelopes of Jupiter and Saturn.
106
HUBBARD AND MARLEY
However, such a relation can be calculated accurately only in restricted pressure ranges. In the range P -< 760 kbar, we can compute the adiabatic density-pressure relation for a mixture of H2 and He with a helium mass fraction Y using the thermodynamic representation of Marley and Hubbard (1988). This relation is calibrated by shock data (Nellis et al. 1983a,b) to a pressure of 760 kbar, and when extrapolated to higher pressures gives a transition to metallic hydrogen at pressures - 5 Mbar. The starting temperature (at 1 bar pressure) Ti for each adiabat is taken to be 165°K for Jupiter, 135°K for Saturn, and 75°K for Uranus and Neptune. The pressure-density relations (adiabats) are calculated in the metallic hydrogen phase (P > 5 Mbar) using the thermodynamic representation of Hubbard and DeWitt (1985). An alternative theory by Hubbard and MacFarlane (1985) leads to more extreme helium abundances in the models because of a substantial shift in the pressure-density relation for pure hydrogen, and is not used here. A model which is fully self-consistent in the thermodynamic sense could then be calculated using the molecular hydrogen and metallic hydrogen thermodynamic representations, respectively, to calculate the transition pressure and entropy change and thus the complete P(p) relation in the envelope. However, this approach is extremely sensitive to details of the model for intramolecular degrees of freedom in the vicinity of the transition pressure (Marley and Hubbard 1988); hence extrapolation of the shock-calibrated equation of state to pressures near the phase transition, in the range 0.76 Mbar < P < 5 Mbar, is uncertain. The approach taken in this paper is to allow the model-fitting routine to construct a smooth interpolation of the P(p) relation between the molecular hydrogen envelope and the metallic hydrogen zone. The P(p) relation is required to follow a prescribed adiabat in the molecular hydrogen envelope and in the metallic hydrogen zone (with the same spe-
cific entropy in both regions), but is interpolated within a specifed density interval near the phase transition between these zones. The interpolation interval in density is chosen arbitrarily, and the model-fitting program then adjusts the interpolated pressures and core mass to obtain a fit to the constraints given in Table I. In general, the smaller the density interval for interpolation, the larger are the pressure excursions in the interval. Because the hydrogen-rich region is of very limited extent in Uranus and Neptune, the handling of interpolation in models of these planets must be different, and is discussed in a subsequent section. Once a pressure-density relation is specified in each layer, a model planet is calculated by solving the equation of hydrostatic equilibrium, dP = p d ( V + Q),
(4)
V = Gfd3r ' p(r')
(5)
where
I; --7'1
is the gravitational potential and Q(r, O) = f~ w2((')~:'d~:'
(6)
is the centrifugal potential for a given state of differential rotation on cylinders co(~), where ( is the distance from the rotation axis. In the present study, we assume that ~o is constant and given by the magnetospheric rotation rate. We return to the question of nonuniform rotation in a subsequent section. The method used to solve the equation of hydrostatic equilibrium is essentially the same as that presented by Hubbard and Horedt (1983). That is, the third-order level-surface theory for rotating planets in hydrostatic equilibrium presented by Zharkov and Trubitsyn (1978) is used to calculate a model with a given envelope pressure-density relation and core of mass Mc with a pressure-density relation given by Eq. (2) or (3).
INTERIOR MODELS As discussed by Hubbard and Horedt (1983), a model is calculated iteratively until agreement is obtained with the observed values of a, J2, and J4. The value of J6 is still not known with enough precision for any Jovian planet for this quantity to serve as a useful additional constraint; this topic is discussed further below. The adjustable parameters are taken to be Mc and two parameters which describe the envelope pressure-density relation in the chosen interpolation range. These parameters are the values of the logarithm of the pressure at two densities which are equally spaced logarithmically within the interpolation range in p. The pressure at the endpoints of the interpolation range is given by the adiabatic pressure-density relation, from the theory of Marley and Hubbard (1988) at the low end and from the theory of Hubbard and DeWitt (1985) at the high end. The pressure-density relation in the interpolation region is then given by four-point Lagrange interpolation in log P and log p, as specified by Eq. (4) of Hubbard and Horedt (1983). Once a converged model is obtained, we then have available the density and pressure on a set of level surfaces described by the mean radius s, where s is the radius of a sphere enclosing a volume equal to the volume enclosed by the oblate level surface. Following Zharkov and Trubitsyn (1978), the level surfaces are represented by the dimensionless parameter B = S/Sl (where s i is the mean radius of the level surface with equatorial radius a) and the shape of each level surface is characterized by its oblateness e(s) and by the small second- and third-order corrections k(s) and h(s) which describe the departure of each level surface from an ellipsoid of revolution. See Eqs. (30.3)-(30.5) of Zharkov and Trubitsyn (1978) for the explicit relations. IV. R E S U L T S
A. Jupiter Figure 1 shows an inferred pressure-density relation (solid line) for Jupiter's envelope, together with the cumulative mass
107
,~ "~1oo
~
Z1.0
0.5 0.0 log p (g/cm 3)
os m/M
05
FIG. 1. Pressure vs density in the Jovian envelope (in this paper, all logarithms are base lO). Right-hand scale shows relative mass layer of each density point (dots). Light curve shows excursion due to a change in ,/4 constraint. See text for explanation of data points and dashed curves.
distribution m / M (dots), where m is the mass enclosed within the level surface at density p. The dots correspond to points which are separated by 0.01Sl in mean radius. The light solid line shows the interpolated pressure-density relation which is obtained when J4 is reduced in absolute value by its error bar (Table I). No converged model was obtained in this case when J4 was increased in absolute value by its error bar. The dashed lines, respectively, show the P(p) relations for hydrogen-helium adiabats with Y = 0.18 (upper) and Y = 0.27 (lower). The horizontal bars show the shock-compression data points for pure deuterium of Nellis et al. (1983a,b), but scaled in density to hydrogen. The temperature values for these shock points are comparable to those on the adiabat at the corresponding pressure. Also shown in Fig. 1 (triangles) are accurate pressure-density points for compression of pure H2 in a diamond cell at room temperature (Mao et al. 1988). The model shown in Fig. 1 was constrained to follow a theoretical adiabat for a pure H - H e mixture with Y = 0.25, and was provided with a small rock core (whose pressure-density relation, given by Eq. (2), is not plotted in Fig. 1). The model-fitting routine adjusted the values of Mc and the
108
HUBBARD AND MARLEY
1
-10
,,"
05 O0 log p ( g / c m 5)
i
/)5
FIG. 2. Illustrationof excursionsin Jovian pressuredensity relation due to nonuniqueness. interpolated pressures in the density range -0 .2 8 -< log p -< 0.20. At the highest pressure point shown, at the base of the metallic-hydrogen envelope, the pressure is 42.5 Mbar, the density is 4.34 g/cm 3, and the temperature (as given by the adiabatic relation) is about 20,000°K. For this model, Mc = 8.3Me. How unique is the model shown in Fig. I? Figure 2 shows a number of alternative models, chosen with various assumptions about the value of Y in the hydrogen-rich envelope, various values for the interpolation interval in the density near the phase transition, and various assumptions about the core composition. For example, if we require Y = 0.18 throughout the envelope, thereby decreasing the density with respect to a Y = 0.24 adiabat, the density correspondingly increases within the interpolation range, and a larger interpolation range is required for a converged model. The choice of Y = 0.25 permits the smallest interpolation range, and gives a model whose overall P(p) relation is closest to theoretical curves with constant chemical composition. The Jovian model shown in Fig. I is essentially consistent with the simplest possible nucleation model. If the rock core of 8.3M, has a solar-composition complement of - 2 5 M e of " i c e " in the envelope, then the entire planet must be enriched with respect to solar composition by roughly one
order of magnitude in non-hydrogen-helium molecules. Assuming that the value Y = 0.25 which we infer for the envelope assuming only hydrogen and helium composition really corresponds to a smaller value of Y with a finite mass fraction of " i c e , " a comparison of pressure-density relations shows that the smaller value of Y should be about 0.21. This is well within the range of uncertainty for the primordial solar helium abundance and for the measured Jovian helium abundance. The envelope mass fraction of " i c e " then corresponds approximately to 0.08, qualitatively consistent with the enhancement of methane with respect to solar composition observed in the atmosphere (Gautier and Owen 1988). According to this m o d e l , there should be a solar complement of H20 (relative to CH4) present as well, and thus the observed depletion of H20 relative to CH4 in the atmosphere must be a superficial effect related to meteorology (Lunine and Hunten 1987). The best-fit static model for Jupiter shows no convincing evidence for substantial chemical gradients. It is consistent with conventional theories for high-pressure hydrogen and helium, with a solar ratio of helium to hydrogen throughout the envelope, with an adiabatic temperature distribution in the envelope, with a modest uniform enhancement of " i c e " in the envelope, and with a rocky core of about 8Me. There is a nonuniqueness in the " i c e " distributions, however. A model which works equally well incorporates all the " i c e " in the core, in an outer layer surrounding the " r o c k . " In this case, the total core has lower mean density, and in order to satisfy the constraints from the gravitational moments, its mass must rise by about 50%, to about 12Me. If there is no significant " i c e " admixture in the envelope (contrary to evidence from observed atmospheric composition), then the helium mass fraction in the envelope must rise to about Y = 0.25 in order to maintain the same density distribution. Despite the improved measurement of
INTERIOR MODELS ,32.5
LO C3 X CO --'3
32.0
FIG. 3. Values of J6 (in first approximation)for Jovian models shown in Fig. 2. Jupiter's J6 by Campbell and Synnott (1985), this quantity still cannot be used to constrain interior models. Figure 3 shows a distribution of results for the calculated first-approximation J6 of our Jovian models which were fitted to the observed a, -/2, and J4. Note that the calculated values of J6 differ only slightly, despite the fact that these models all have different envelope structures. We have investigated other models with even greater differences in envelope structure (not necessarily with realistic pressure-density relations), and find that for most of these models, the value of J6 varies only by +-1.5% for fixed a, .I2, and J4. The effect of various factors on the calculated value of the Jovian J6 is summarized in Fig. 4. This figure shows the percent difference between the observed value of J6 given in Table 1 and the calculated value using our third-order theory. Because our third-order theory of figures only computes ./6 to first approximation, truncation at this order can lead to an underestimate of -/6 by about 5.4%, as determined by comparison of the third-order theory used here with a fourth-order theory (Hubbard and Horedt 1983, Hubbard et al. 1975). Since J6 is still only measured to +-65%, a first-approximation theory for J6 suffices. These results do not include the effect of differential rotation. According to a study
109
carried out by Hubbard (1982), if Jupiter rotates differentially on cylinders, the calculated value of J6 would increase by about 1.9% with respect to the value given here. About the same variation in J6, but in both directions, occurs if J4 is allowed to range across its error bar. As would be expected, the structure of the core has virtually no effect on the value of J6. Figure 4 shows that J6 remains virtually constant as the core varies from a single-component structure of pure " r o c k " to a two-layer structure of " i c e " over " r o c k . " In summary, J6 is not a very clean constraint on the interior structure of Jupiter because it is highly correlated with the values of the lower even zonal harmonics, even more than is J4. If the precision of measurement of the even zonal harmonics could be made arbitrarily high, and one utilized a very high-order theory for calculating them, then the remaining uncertainties indicated in Fig. 4 would be the effects of the envelope equation of state and of differential rotation, which are very similar in magnitude at degree 6. One may anticipate that dynamical effects of outer-layer fluid currents dominate the structure of the very high-order Jovian gravity field. B. S a t u r n
Figure 5 shows an inferred pressure-density relation (solid line) for Saturn's enve-
0.01% core slructure 1.9% X
I
I
I
-,2
40
I J4 error
- 1.9% differential I rotolion -5.4% x truncation
-~
-~
1.5% x I equation of slate
-;,
O6: ( O - C l i O
-~
;
(%)
FIG. 4. Range of variation in calculated value of Jupiter's J6, due to various causes.
i 10
HUBBARD AND MARLEY
19 05
r,//M
O0
2
I
10
05
'og
O.O
,o
05
(g/cm ~)
Fro. 5. Pressure vs density in Saturn's envelope. Light curves show excursions due to changes in J4 constraint. See text for explanation of data points and dashed curves.
lope, together with the cumulative mass distribution m / M (dots), where m is the mass enclosed within the level surface at density p. The dots correspond to points which are separated by 0.01s~ in mean radius. The light solid lines show interpolated p r e s s u r e - d e n s i t y relations which are obtained when J4 is reduced in absolute value by its error bar (line which initially curves up, then down), and when J4 is increased in absolute value by its error bar (line which initially curves down, then up). Note that the higher the absolute value of J4, the more rapid the increase of density with pressure in the outermost layers of the planet, in agreement with the approximate theory of H u b b a r d (1974). The dashed lines, respectively, show the P(p) relations for h y d r o g e n - h e l i u m adiabats with Y = 0.06 (upper), Y = 0.27 (middle), and Y = 0.50 (lower). Also shown in Fig. 5 (large triangles) are p r e s s u r e - d e n s i t y points for compression of pure H2 in a diamond cell at room temperature (Mao et al. 1988), and (small triangles) p r e s s u r e - d e n sity points for compression of HeO in a diamond cell at room temperature (Hemley et al. 1987). The model shown in Fig. 5 represents an attempt to confirm a Saturn interior model which has a helium-depleted H2 outer envelope, and a helium-enriched metallic-hy-
drogen inner envelope, in accordance with the prediction o f Stevenson (1975) and Stevenson and Salpeter (1977a; see also Hubbard and Stevenson 1984). Thus, the model P(p) relation was required to follow an adiabat with Y = 0.4 in the metallic hydrogen core, for log p > 0.20, and an adiabat with Y = 0.06 (the observed atmospheric composition) for log p < - 1.0. The model-fitting program adjusted Mc and the pressure in the interpolation region to obtain a fit to the constraints in Table I. However, as is evident from Fig. 5, the interpolated P(p) shows considerable softening at pressures greater than a few hundred kilobars, even in the pressure range where the hydrogen equation of state has been calibrated by shock and static experimental data. In other words, a model inferred from fitting the gravitational constraints is not consistent with the picture of a molecular hydrogen envelope which is e v e r y w h e r e heavily depleted in helium, and which has no other significant c o m p o n e n t besides hydrogen and helium. In an attempt to explore whether a model could be found in which Y = 0.06 throughout the molecular-hydrogen envelope, we systematically shifted interpolation limits and made various assumptions about the value of Y in the molecular hydrogen and metallic hydrogen envelopes. Figure 6 shows an ensemble of such models, all of which satisfy the constraints of Table I. We z 1
, / , ' f, x) / -"/" " /
l
"C"
9
l
!
I0
,;~> ,~>
-05
O0
05
FIG. 6. Illustration of excursions in Saturn pressure-density relation due to nonuniqueness.
INTERIOR MODELS 75
tO O
x | cO
72 FIG. 7. Values o f J6 (in first approximation) for Saturn m o d e l s s h o w n in Fig. 6.
were unable to find a model in which the followed the Y = 0.06 adiabat to pressures higher than 200 kbar. This result is of course dependent upon the value of q given in Table I; if the measured rotation rate o f the deep interior of Saturn were to be revised significantly, this conclusion might need to be revised also. Adopting the model shown in Fig. 5 as a baseline Saturn model for discussion purposes, we find that the planet has an allrock core with mass Mr = 9.8Me, and a metallic hydrogen envelope of about 30% of the planet's total mass, heavily enriched in helium (or some other dense component). The mass density p is 2.6 g/cm 3 at the base o f the metallic hydrogen envelope, the pressure is 12 Mbar, and the temperature, assuming adiabatic stratification, is about 12,000°K. In the molecular hydrogen envelope, there is a systematic softening of the pressure away from the Y = 0.06 adiabat, implying that the mixing ratio of some denser constituent gradually increases with depth. The mixing ratio of this constituent is significant even at pressures o f a few hundred kilobars, and it is apparently too dense to be only helium. A likely candidate is " i c e , " or perhaps H20 only (see Fig. 5). If the solar complement o f " i c e " to the rocky core were distributed throughout Saturn's enve-
P(p) relation
111
lope, it would comprise on the order of 1030% o f the e n v e l o p e ' s total mass. Comparing the empirical P(p) relation with a Y = 0.21 adiabat, we obtain an integrated envelope mass fraction of about 20% " i c e , " consistent with such an assumption. A strong gradient in the concentration of this significant c o m p o n e n t would obviously act to suppress convection in Saturn's envelope. Moreover, the gravitational settling of this c o m p o n e n t would probably be even more significant for Saturn's energy balance than the gravitational settling of helium (if it occurs). If gravitational settling occurs, there may be mass-dependent depletions of molecules in Saturn's atmosphere. Deuterium (in the form of HD) might be a useful tracer. Note that, as in the case of Jupiter's core, Saturn's dense core b e c o m e s larger in mass if its mean density decreases, as would be the case if a significant amount of " i c e " is included in its makeup. Depending on assumptions about the composition of the core and the p r e s s u r e - d e n s i t y relation in the envelope, we obtain Saturn core masses ranging from 9M~ to 20M~. Nicholson and Porco (1988) have obtained the first direct measurement of Saturn's J6 (Table I). Although we have not fitted models to this parameter because of its larger error bars, our results are all consistent with the measured value. Figure 7 shows a distribution of results for the calculated first-approximation J6 of our Saturn models which were fitted to the observed a, J2, and ./4. The calculated values o f J6 differ by at most 3%. It is interesting to note that our calculated values for Saturn's J6, all for the value o f J4 given in Table I, do not satisfy a further constraint on Saturn's gravitational harmonics, although they do easily satisfy the formal error bars on J6. This further constraint is given by Eq. (41) of Nicholson and Porco: •/4 - 1.304J6 = ( - 1 0 5 0 --- 12) × 10 -6
(6)
(with their normalization to a = 60,330 km).
112
HUBBARD AND MARLEY
As can be readily verified from this relation, if we choose a model with J4 equal to the central value in Table I, then this model would require a ,]6 close to 100 x 10 -6, substantially larger than the values shown in Fig. 7. From analogy to the effects shown for Jupiter in Fig. 4, we expect that the effects of differential rotation and higher approximations in the calculated value of J6 will tend to increase J6 and thus reduce the discrepancy from the constraint given by Eq. (6). However, such a detailed study lies beyond the scope of this paper. We have investigated the effect of differential rotation of Saturn on cylinders (Hubbard 1982). When such rotation is included for a typical Saturn model, the calculated absolute value of J4 increases by 2.5% and the value of J6 increases by about 10% with respect to a model rotating uniformly with the magnetospheric period. Thus, the effect of differential rotation on ,/6 is considerably more important than the effect of various pressure-density relations, which according to Fig. 7 is only about 3%. Thus, as in Jupiter, dynamical effects of outer-layer fluid currents probably dominate the structure of Saturn's gravity field above degree 4. C. U r a n u s
Interior models of Uranus have undergone considerable revision recently because the Voyager 2 flyby provided an accurate interior rotation rate via measurement of the magnetospheric rotation rate. Models calculated by Hubbard and MacFarlane (1980a) assumed structures analogous to the Jupiter model presented above, that is, that Uranus is separated into three distinct layers whose pressure-density relations are, respectively, given by " r o c k " (Eq. (2)), " i c e " (Eq. (3)), and, outermost, a solar-composition adiabat. Such models are now clearly ruled out by the constraints of Table I, for they are too centrally condensed. A recent model (Podolak and Reynolds 1987) was fitted to earlier values of the gravitational harmonics, derived
by French et al. (1986), for which the value of J4 was larger in absolute value than the figure given in Table I. The Podolak and Reynolds model has three layers, but the rocky core is very small, the intermediate " i c e " layer dominates the planet, and the hydrogen-rich envelope has more than an order of magnitude enhancement in " i c e " relative to solar composition. Gudkova et al. (1988) have a Uranus model fitted to the same gravitational harmonics, in which the rocky core is either very small or nonexistent, but an important difference is that the overall ratio of " r o c k " to " i c e " is roughly solar rather than much less than solar. Because the hydrogen pressure-density relation does not dominate the interior of either Uranus or Neptune, there will be considerable nonuniqueness in relations which would satisfy the constraints. Our approach is to allow the model-fitting program to find a structure which would satisfy the constraints, allowing a very broad interpolation region. Thus we consider a class of models in which there exists a " r o c k " core of mass M~, an " i c e " layer over the core, and finally an atmosphere composed primarily of hydrogen and helium in agreement with observations. However, the atmosphere and the " i c e " layer are not separated by a distinct interface. Rather, the program is allowed to construct an interpolated pressure-density relation between the " i c e " layer and the atmosphere over the density range - 1 . 0 -< log P -< 0.5. The interpolation constants and Mc are adjusted to fit the constraints. Figure 8 shows the resulting pressuredensity relation (solid line) for Uranus' " i c e " - a t m o s p h e r e layer, together with the cumulative mass distribution m / M . The dashed lines are (left to right) a solar-composition adiabat, the same adiabat mixed with an " i c e " mass fraction of 0.2 and with an " i c e " mass fraction of 0.7, a pure " i c e " adiabat, and the " r o c k " pressure-density relation. Also shown, for comparison, are the room-temperature H20 data of Hemley et al. (triangles), and the "artificial
INTERIOR MODELS
1
. . . . . . . . . . . . . . . .
,,'1 / /5" / ,'1 /
.
-
1.0
-0.5 m / M
g
Zo
,%1
i
o.o
/
o_
....,,~ .'" ~
---1
J -2 -1
/ 05
.~
/~
/z
//
,@
~ ,
/ 0.0
0.5
tog p ( 9 / c m 3)
FIG. 8. P r e s s u r e vs d e n s i t y in U r a n u s ' envelope. Light c u r v e s h o w s e x c u r s i o n due to change in ,/4 constraint. See text for explanation of data points and dashed curves.
Uranus" shock-compression points (squares) of Nellis et al. (1988). The light solid line in Fig. 8 shows an interpolated pressure-density relation which is obtained when ./4 is replaced by its older value, from French et al. (1986). Within the errors of measurement of Uranus' gravitational harmonics, a model with no discrete interfaces is admissible. A qualitatively similar model was also found by Stevenson (1987). All recent modeling studies consistently find that major separation of Uranus into its individual " r o c k " and " i c e " components has apparently not occurred, although there appears to be a rapid increase in the heavy-element component at pressures greater than about 100 kbar. Calculation of the chemical composition of our Uranus model is nonunique, since a variety of mixtures could give the same effective pressure-density relation. Figure 9 shows two possible interpretations of the P ( p ) relation of Fig. 8. The upper curve (dashed) shows the mass fraction of " i c e " at each mass shell in the envelope, as calculated by referencing the empirical P ( p ) to Eq. (3) and to a solar-composition adiabat, assuming additive volumes. The lower curve (solid) shows the result for the mass fraction o f " i c e " plus " r o c k " at each mass shell, obtained by assuming that " i c e " and
113
" r o c k " are in solar proportions at each point, and that the remaining mass fraction is solar-composition hydrogen and helium. The modest jump in the curve near m / M = 0.5 is caused by interpolation from the molecular hydrogen to the metallic hydrogen equation of state and has no physical significance. The total mass of free hydrogen and helium in Uranus is given by the area above the curve in Fig. 9. The dashed curve gives the result for no " r o c k " in the envelope, an unrealistic limit since the " r o c k " core is vanishingly small. For this case we obtain a total mass fraction for free hydrogen and helium in Uranus equal to 0.09, or 1.3Me. In the more realistic case of an envelope with solar proportions of " i c e " and " r o c k , " free hydrogen is present even to rather high pressures, where it would comprise approximately 10% of the total mass and be in the metallic phase. In this case, the total mass fraction for free hydrogen and helium in Uranus equals 0.14, or 2.0Me. These numbers are comparable to the mass fractions found by Podolak and Reynolds, and by Gudkova, Zharkov, and Leontyev. Is such "unseparated" structure for Uranus reasonable? In the absence of more detailed models for the accumulation of planetesimals, formation of a trigger nucleus, and capture of nebular gas, it is diffi-
0.8
+uO.6 L L
~04 ! o
0.0
O0
'
L
on2
'
Q h4
'
0/6
m/U
'
0~8
10
FIG. 9. M a s s fraction o f the n o n - h y d r o g e n - h e l i u m c o m p o n e n t in U r a n u s as a function of m a s s shell. D a s h e d curve s h o w s result w h e n this c o m p o n e n t is " i c e " only, while the solid curve s h o w s result when this c o m p o n e n t is a solar composition mixture of " i c e " and " r o c k " ( " i c e " / " r o c k " = 2.7).
114
HUBBARD AND MARLEY
cult to quantify the amount of nebular gas which might be entrained during the initial formation of the nucleus. However, the unconventional scenario presented by Stevenson (1985) envisions the formation of a planet such as Uranus not as a well-separated two-stage process of (a) formation of a dense nucleus, and (b) capture of nebular gas, but rather as a process of continued capture of large, solid objects into a giant planet which already contains major amounts of the gaseous component. Such a scenario is not inconsistent with our Uranus model. D. N e p t u n e
We shall not present a Neptune model at this time because the constraints (Table I) have considerably greater uncertainty than for any other Jovian planet. As discussed by Harris (1984), the value o f J2 is inferred from the precession of Triton's inclined orbit about the total angular momentum vector of the Neptune-Triton system. Lack of an accurate Triton mass determination is therefore one of the major causes of uncertainty in the value of J2, although there is an indication from the orientation of Neptune's spin axis (Hubbard et al. 1987) that the mass of Triton has been overestimated. Neptune's q is likewise very uncertain, because the rotation period of Neptune's deep interior cannot be directly measured by ground-based observations. One may obtain an atmospheric rotation period by measuring the movements of atmospheric albedo features (Cruikshank 1985), but these movements are unlikely to be closely connected to the rotation of the bulk of the planet. Alternatively, measurement of the planet's atmospheric oblateness (French et al. 1985, Hubbard et al. 1987) can be combined with the equation of hydrostatic equilibrium (4) to solve for q. But again, this method yields some average value for the atmospheric rotation rate, which might differ from that of the deep interior. The method of choice would be to determine the rotation rate of Neptune's magnetosphere,
?,
•
i x I
FIG. 10. Distribution of density in Jupiter, Saturn, and U r a n u s as a function of relative radius/3.
which should be rooted in the deep, conducting layers of the planet. If the planned Voyager 2 flyby of Neptune in 1989 is successful, measurements of Triton's mass, Neptune's J2, and the magnetospheric rotation period may be obtained. It would then be appropriate to subject the planet to modeling studies similar to those presented here. In the meantime, we remain with the conclusion reached by French et al. (1985), and by Hubbard et al. (1987), that presently available data imply that a satisfactory Neptune model may need to be substantially less centrally condensed than the Uranus model presented in this paper. V. S U M M A R Y
Figure 10 shows a combined plot of the distributions of density as a function of relative radius /3 in Jupiter, Saturn, and Uranus. These distributions correspond to the models shown in Figs. 1, 5, and 8. Of the three distributions, Saturn has the smallest density in its envelope, not because of a major difference in composition from Jupiter, but because the hydrogen is less compressed. On the other hand, Uranus is substantially denser than either Jupiter or Saturn in its deepest layers, but has a density lower than Saturn in the outer 20% of its radius because of less compression. This structure seems to be required by Uranus' J4.
INTERIOR MODELS
115
25
J
20
•
~
(•)
~
15 O (D
:
z
S
U
A~
m(~
rnolec l l q
metGI
liq.
i
i
10
5
--
i
i
i
i
O
S
U
N
Lx Azi
Fie. 11. Mass of dense (non-hydrogen-helium) core in Jovian planets, for a range of models which satisfy the constraints of Table I.
Figure 11 summarizes our conclusions about possible masses of the dense (i.e., non-hydrogen-helium) cores of the Jovian planets. Each dot represents a model which fitted the constraints of Table I. For Jupiter and Saturn, the lower core masses are for purely " r o c k " cores, while the larger core masses generally represent cores of " r o c k " with an outer layer of " i c e " in solar proportions. In the case of Uranus, the dots represent the mass remaining after hydrogen and helium are deducted, for the two cases shown in Fig. 10. For Neptune, the dot shows the approximate mass remaining after deduction of the (small) fraction of hydrogen and helium. Figure 11 confirms that the dense cores of the Jovian planets are similar in mass, in each case comprising 10Me-15Me of material, consistent with the nucleation hypothesis. In Table II we summarize the structure of the planetary models in a different way. In this table we present the polar moment of inertia C and the equatorial moment of inertia A of the entire planet, as calculated from
2 -1
0
log
1
P
(Mbar)
FIG. 12. A phase diagram for hydrogen, showing approximate adiabatic temperature distributions in Jupiter, Saturn, and Uranus (light lines). Dashed phase boundaries are particularly uncertain. Solid triangles show shock compression data points of Nellis et al. (1983a,b), while open triangles show data of Mao et al. (1988).
the most plausible interior model• These numbers should be almost model independent. In addition, the table gives ac, the equatorial radius (in km) of a rock core (the models presented have a rock-only core), and the corresponding moments of inertia Cc and Ac of the core. Figure 12 displays a phase diagram for hydrogen, together with assumed temperature distributions in Jupiter, Saturn, and Uranus• Two possible boundaries between the molecular hydrogen liquid and the metallic hydrogen liquid are shown; these correspond to the boundaries calculated by Marley and Hubbard (1988), and show the sensitivity of the phase boundary to assumptions about intramolecular degrees of freedom. The three adiabats are shown as continuous paths across this phase boundary, although in the case of an actual first-
TABLE II MOMENTS
OF
INERTIA
OF JOVIAN
PLANETS
Planet
C/Ma 2
A/Ma 2
a¢ (km)
MJM
C¢/Mca~
Ac/Mca~
Jupiter Saturn Uranus
0.26401 0.22037 0.22680
0.24931 0.20404 0.22328
8387 9771 1397
0.0261 0.1027 0.0012
0.390 0.380 0.40
0.384 0.374 0.40
116
HUBBARD AND MARLEY
order transition from molecular to metallic hydrogen, the temperature would have to either drop along the phase boundary (in the case of the left-hand boundary) or rise along the phase boundary (right-hand boundary) until it joined with the adiabat in the metallic phase. In the real planet, the entropy across the jump would be discontinuous rather than the temperature (Stevenson and Salpeter 1977b). Recent quantum Monte Carlo calculations of the pressure for the transition from solid molecular hydrogen to solid metallic hydrogen give a value of 3.0 -+ 0.4 Mbar (Ceperley and Alder 1987, Ceperley 1988), and the phase boundary between these two phases is therefore shown tending to this value at low temperature (the boundary must become vertical at very low temperatures). Our Jovian-planet models show a tendency for the pressure-density relation to soften as the pressure exceeds about 1 Mbar, approximately where the slope of the adiabats is changed in Fig. 12. This pressure lies just above the range of the maximum accurate shock experiments (triangles in Fig. 12). Some unconventional phase diagrams for hydrogen (e.g., Robnik and Kundt 1983; see also Hubbard 1988) have a phase boundary in this region, and it is possible that the band gap in molecular hydrogen narrows to zero at about this pressure (Ceperley 1988), opening the possibility of a transition to an intermediate phase of metallic molecular hydrogen in this region. Our models, although not unique, provide guidance about the class of pressuredensity relations which are admissible for the deep interiors of Jovian planets. We find that Jupiter lies closest to the classical picture of a solar-composition mixture overlying a dense rocky core. The derived deviations of the Jovian composition from solar composition are moderate, and are in general agreement with the results of earlier studies (e.g., Zharkov and Trubitsyn 1978, Hubbard and Horedt 1983, Zharkov 1986). In the case of Saturn, we find no confirmation from the gravity field for a model in
which the entire molecular envelope is heavily depleted in helium. Rather, our model shows substantial chemical gradients in Saturn involving an abundant, dense component other than, or in addition to, helium. Whether this model is consistent with Saturn's observed heat flow requires further study. Uranus also could have a diffuse distribution of heavier material, with a uniform ice-rock mixture concentrated toward the center. Our interpretation is consistent with the results obtained by Pollack et al. (1986) and Podolak et al. (1988), who found that, due to the dissolution of incoming planetesimals in the envelopes of the growing giant planets, a significant portion of the outer planets' high-Z material may be contained in their envelopes. These studies found the depth of planetesimal penetration to be strongly dependent on composition. Rock and iron planetesimals reach almost to the core, while ice planetesimals may deposit their mass further from the center of the growing planet. However, this envelopeenrichment scenario may face difficulties in explaining our inferred homogeneity of the Jovian envelope and heterogeneity of the envelopes of Saturn and Uranus. ACKNOWLEDGMENTS This research was supported by NASA Grants NAGW-192 (Jupiter Data Analysis Program), NGT50049 (Training Grant), and NSG-7045 (Planetary Astronomy). We thank Morris Podolak and Vladimir Zharkov for comments on the manuscript. REFERENCES ANDERS, E., AND M. EaIHARA 1982. Solar-System abundances of the elements. Geochim. Cosmochim. Acta 46, 2363-2380. BJORAKER, G. L. 1985. The Gas Composition and Vertical Cloud Structure o f Jupiter's Troposphere Derived .from Five Micron Spectroscopic Observations. Ph.D. dissertation, University of Arizona. CAMERON, A. G. W. 1973. Abundances of elements in the Solar System. Space Sci. Rev. 15, 121-146. CAMPBELL, J. K., AND S. P. SYNNOTT 1985. Gravity field of the Jovian system from Pioneer and Voyager tracking data. Astron. J. 90, 364-372. CEPERLEY, D. M. 1988. Quantum Monte Carlo simulations of systems at high pressure. In Simple Molecu-
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