Chapter 4 The Interior of Saturn

Transcription

1 Chapter 4 The Interior of Saturn William B. Hubbard, Michele K. Dougherty, Daniel Gautier, and Robert Jacobson Abstract A source of uncertainty in Saturn interior models is the lack of a unique rotation rate to be ascribed to the deep (metallic-hydrogen) interior. As a result, models are not uniquely constrained by measured gravitational multiple coefficients. Further uncertainty is associated with the effect of a multiplicity of rotation periods due to zonal flows of unknown magnitude and depth (and therefore unknown mass). Nevertheless, the inference that Saturn has a large core of mass M E (Earth masses) is robust. The equation of state of dense hydrogen helium mixtures is one area where uncertainty has been much reduced, thanks to new first-principles simulations. However, because there is still uncertainty in Saturn s interior temperature profile, a variety of mantle metallicities and core masses could still fit the constraints, and the question of interior helium separation is still unsettled. Keywords Saturn interior Saturn atmosphere Saturn rotation Jupiter interior 4.1 Diagnostics of Interior Structure and Dynamics Gravity Field and Shape The most direct constraints on Saturn s interior mass distribution come from measurements of the highly-oblate planet s size and mass and its response to rotation, as de- W.B. Hubbard Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ, USA M.K. Dougherty Physics Department, Imperial College, London, UK D. Gautier Observatoire Paris Site de Meudon, France R. Jacobson Jet Propulsion Laboratory, California Institute of Technology, CA, USA termined by its overall shape, and gravitational multipole moments. As a result of further measurements related to the Cassini mission and other work during the two decades between the Voyager encounters and the Cassini mission, some fundamental parameters are known with improved precision, while the rotation period(s) has (have) increased uncertainty. Table 4.1 summarizes the current situation. The observed parameters listed are Saturn s mass M, equatorial radius at 1-bar pressure a, polar radius at 1-bar pressure b, deep (solidbody) rotation period P S, and first three even zonal harmonics J 2, J 4,andJ 6. Values for the zonal harmonics were derived using the methods of Jacobson et al. (2006), but are updated in Table 4.1 to be current as of late The linear response of the second-degree gravity potential of a liquid body to a uniform rotation rate D 2=P S can be written J 2 D ƒ 2 q; (4.1) where q D 2 a 3 =GM (G D gravitational constant) and ƒ 2 is a dimensionless response coefficient that contains information about Saturn s degree of central concentration. For example, a planet of infinite central concentration would have ƒ 2 D 0 and a planet of uniform density (Maclaurin spheroid) would have ƒ 2 D 5=4. As can be seen from Table 4.1, Saturn s ƒ 2 0:11 is distinctly smaller than Jupiter s ƒ 2 D 0:165, direct proof that Saturn s mass distribution is more centrally condensed. By symmetry, all odd terms J 3 ;J 5 ; ::: should be absent in the external potential of a uniformly rotating liquid body, and there is no evidence so far that any such terms are detectable in Saturn s gravity field. According to the theory of nonlinear response to uniform rotation, the leading term in J 4 should go as J 4 ƒ 4 q 2 ; (4.2) with positive ƒ 4. Saturn s axial moment of inertia C is not uniquely constrained by ƒ 2, and the Radau Darwin relation " s C Ma Š C # 5 q 2 e 2 ; (4.3) M.K. Dougherty et al. (eds.), Saturn from Cassini-Huygens, DOI / _4, c Springer Science+Business Media B.V

2 76 W.B. Hubbard et al. Table 4.1 Parameters constraining Saturn interior structure. Error bars on M=M E are about equally determined by uncertainties in Saturn s mass and Earth s mass (as given at and are not currently an important limitation for interior modeling purposes. Note the inconsistency of the Voyager-era versus Cassini-era value of P S Parameter As of Voyager As of Cassini Reference M.M E / 95:16 0:02 Jacobson et al. (2006) a.km/ Lindal (1992) b.km/ Lindal (1992) P s.s/ Cecconi and Zarka (2005) q D.2 =P s / 2 a 3 =GM 0: : : :00031 J (observed) 16324:19 0:11 Jacobson et al. (2006) ƒ 2 D J 2 =q C=Ma Core mass.m E / J (observed) 939:32 0:98 Jacobson et al. (2006) ƒ 4 D J 4 =q J (theory) J (observed) 91 5 Jacobson et al. (2006) J (assumed) 10 Jacobson et al. (2006) where e D.a b/=a and e=q D.3ƒ 2 C 1/=2 (4.4) is a very poor approximation for Saturn, overestimating its moment of inertia by almost 50%. Nevertheless, the Radau Darwin relation is useful for estimating the impact of a change in Saturn s rotation rate on the inferred C=M a 2. Post-Voyager measurements of Saturn s magnetic-field rotation period, presumably coupled to the conducting metallic-hydrogen envelope, give values longer by about 381 s, or about 1% (see Table 4.1). This discrepancy is related to difficulties in measuring the rotation rate of Saturn s virtually axisymmetric magnetic field, and may not be reducible by further measurements. Thus, it is fair to ask how robust are the inferences of Saturn s C=M a 2. Values of C=M a 2 given in Table 4.1 for the two different proposed rotation periods are computed for representative models using a theory valid to order q 3 and realistic equations of state for a hydrogen helium ice mixture (for Saturn s envelope) and olivine (for Saturn s core). (These models are based on older equations of state and do not represent the current state of the art; they are for demonstration purposes only.) The model values of C=M a 2 differ by , in good agreement with the shift predicted by Radau Darwin. Figure 4.1 shows a profile of mass density as a function of the average radius s of a level surface (s 0 is the average radius of the 1-bar level surface). This profile is computed for a model fitted to observed values of M, a, J 2, and the post-voyager value of P S ; the profile for a model fitted to the Voyager-era value of P S is very similar. Also given in Table 4.1 is the inferred mass of the rock (olivine) core for the two different rotation rates. They differ by only 0.6 M E. We thus conclude that a massive Saturn core, mass one Neptune mass, is a robust result from Saturn modeling and is unlikely to change in the face of continued uncertainty in P S. Fig. 4.1 Profile of a typical Saturn model with an envelope of solarcomposition hydrogen, helium, and hydrides of C, N, and O. The massive olivine core extends to more than 20% of the radius. The envelope equation of state is obtained from the theory of Saumon, Chabrier, and Van Horn (1995) Differential Rotation and Equations of State The rotation state of Saturn s deep interior plays a role in the external gravity and the surface shape of Saturn. In principle, if sufficient mass is involved in differential motions, external gravity coefficients can be affected. Thus, a mismatch between model predictions and observed gravity coefficients could be attributed either to errors in the pressure density relation, or to errors in the assumed rotation rate(s), or both. This problem has recently appeared in a new investigation of the interior of Jupiter using first-principles thermodynamic relations for hydrogen helium mixtures (Militzer et al. 2008). The Jupiter model has constant entropy fixed to the measured entropy at 1 bar, with only the core mass and (constant) mantle metallicity as a adjustable parameters. This model fits Jupiter s M, a, andj 2, but the improved, more precise measurement of J 4 is not fitted within the error bar. As we see in Table 4.1, a similar situation is possibly now

3 4 The Interior of Saturn 77 emerging with Saturn. The error bar on Saturn s observed J 4 is remarkably small, only 0.1%, and the discrepancy with the simple interior models (fitted to J 2, as was done with Jupiter) is far larger. Moreover, the sign of the discrepancy is the same for both planets. In principle, one might resolve the discrepancy for Saturn (and for Jupiter) by assuming that the planet rotates as a solid body but with a different rotation period P S than either of the values presented in Table 4.1. In this approach, we fix the response coefficients ƒ 2 and ƒ 4 and adjust the value of q to match J 2 and J 4. As demonstrated in Table 4.1, this procedure would lead to a value of P S that differs substantially from any directly measured value, and would therefore be essentially ad hoc. On the other hand, we have ample evidence for large and possibly variable zonal winds in Saturn s visible atmosphere, and if these winds are deep, they would involve enough mass to affect gravitational coefficients. The surface shape of Saturn can be measured with enough precision, via occultation techniques, to shed some light on this matter. Figure 4.2 shows the shape of the 100-mbar surface as measured by several spacecraft occultations and one stellar occultation. The shape surfaces in Fig. 4.2 are referenced to a constantpotential surface defined by Saturn s J 2, J 4, J 6,... andsolidbody rotation with the Voyager-era period P S (Table 4.1) passing through Saturn s equatorial 100-mbar atmosphere. These surfaces are computed by integrating the @` C @z ; (4.5) where P is the pressure, V is the external gravitational potential, and ` and z are respectively coordinates perpendicular and parallel to the rotation axis. Eq. 4.5 gives the alignment of isobars as determined by the inertial rotation rate and by itself gives no information about the depth of zonal flows. On the other hand, occultation measurements are directly sensitive to and not P, so the fact that such measurements yield an overall shape surface matching isobars as determined through Eq. 4.5 means that on a planetary scale, Saturn s 100-mbar surface is also an isopycnic surface. This result implies that the Poincaré Wavre theorem (Tassoul 1978) applies, meaning that D.`/ only (rotation on cylinders), suggesting that the large equatorial uprise (amplitude 100km when referencedto the Voyager-era period P S ) extending over a broad range of latitudes, is indeed deep-seated and would necessarily involve significant mass. As Fig. 4.2 makes evident, we have spacecraft- and stellaroccultation measurements of Saturn s shape over planetocentric latitudes ranging from 65 ı north to 70 ı south, together with a high density of occultation data points near the equator (Fig. 4.3). Data in our Figs. 4.2 and 4.3 are exhibited slightly differently from the corresponding Fig. 4.9 of Lindal et al. (1985). In our Figs. 4.2 and 4.3, we reference Fig. 4.2 Shape of Saturn s 100-mbar surface as predicted by measured winds (thin solid and dashed curves are from Voyager-era measurements of windspeeds, while the heavy solid curve is based on 2005-era measurements of reduced equatorial windspeeds), compared with radiooccultation data points (triangles; Lindal et al. 1985) and a measurement of a 1989 stellar-occultation central flash (dashed line between crosses; Nicholson et al. 1995) Fig. 4.3 Same as Fig. 4.2, but for the 2- bar surface. Data points are from the July 1989 occultation of 28 Sgr by Saturn (Hubbard et al. 1997)

4 78 W.B. Hubbard et al. the atmospheric distortions to an atmosphere in uniform rotation with Voyager-era period P S, and we further reference those distortions to the (maximum) equatorial value so as to clearly exhibit the curvature of the equatorial uplift as a function of the zonal windspeed model. The key point, as shown in Fig. 4.9 of Lindal et al. (1985) and in Figs. 4.5 and 4.6 of Hubbard et al. (1997) is that the equatorial curvature of Saturn s atmosphere, as well as the 1-bar values a and b, are well determined from multiple, consistent spacecraft and ground-based observations spanning the 10 year interval Anderson and Schubert (2007) chose to relax any observational constraint on P S and sought instead to fit uniformly-rotating interior models to Saturn s a, b and gravity field by varying the parameter q of Table 4.1, leading to P S D s, a value considerably smaller than either value in Table 4.1, and leading to virtual disappearance of the equatorial excess bulge. Helled et al. (2009) recently investigated Saturn interior structure via models with no interior differential rotation allowed and with P S treated as a free parameter. Figure 4.2 suggests that the reported 2005 decrease in equatorial windspeeds (Sanchez-Lavega 2005) would not be consistent with occultation data on Saturn s equatorial curvature. More evidence bearing on this matter is presented in Fig. 4.3, which shows the corresponding 2- bar surface with stellar-occultation data points (Hubbard et al. 1997). The importance of Fig. 4.3 is that the shape of Saturn s high atmosphere in 1989 was consistent with the high-speed equatorial winds modeled by Lindal et al. (1985) based on 1980 and 1981 Voyager data and the earlier Pioneer 11 data. The heavy curve based on the much slower equatorial winds observed by Sanchez-Lavega et al. (2005) is not consistent with the 1989 occultation data. In this connection, we note that Choi et al. (2009) report recent Cassini observations showing that slower equatorial winds are underlain by faster winds consistent with the data. We conclude that (a) Saturn s excess equatorial bulge was consistently present when observed over a baseline of approximately a decade; (b) the fact that the bulge is present as nearly constant-density surfaces over many scale heights implies that it is deep rooted and may have a gravitational signature; and (c) measurements of Saturn s atmospheric shape at the present epoch could help to elucidate whether the bulge is in fact time-variable. 4.2 Evolution of Saturn There is a long-standing problem related to Saturn s intrinsic luminosity: over the 4.5 Gyr lifetime of Saturn, not enough thermal energy can be stored to account for the planet s observed luminosity at present. Stevenson (1975) and Stevenson and Salpeter (1977a, b) proposed the standard solution for this problem. The solution involves a proposed phase diagram for binary mixtures of hydrogen and helium, causing separation of the fluid H He mixture into a He-depleted phase which rises, a He-enriched phase which sinks, and consequent gradual conversion of gravitational potential energy into heat. An initial solar mixture would have a sufficiently large abundance of He relative to H to provide an adequate energy source to prolong Saturn s cooling by the required amount. The problem is that the Stevenson Salpeter binary phase diagram, which is based on extrapolation of a model of fully pressure-ionized H and He to the relatively low pressures in Saturn.10 Mbar/, turns out to not have the right behavior to jointly explain the evolution of Saturn and Jupiter. Figure 4.4 (Fortney and Hubbard 2003) illustrates possible phase diagrams for dense hydrogen with helium impurities. In Fig. 4.4, the upper boundary of the hatched region marked HD delineates where, according to the model of Hubbard and DeWitt (1985), a solar mixture of H and He first phase-separates; the HD model is equivalent to the Stevenson and Salpeter (1977a) model. The hatched region marked Pfaffenzeller shows where, according to Pfaffenzeller et al. (1995), phase separation occurs. The hatched region lying between the Jupiter and Saturn adiabats shows a phase-separation region modeled by Fortney and Hubbard (2003) which successfully prolonged Saturn s cooling with no prolongation of Jupiter s cooling. Figure 4.5 (from Fortney and Hubbard 2003) shows results of two cooling/phase-separation models. The Fortney and Hubbard models predict a present-day helium abundance for Saturn which could be investigated with Cassini data. The predictions are self-consistent in the sense that they use reasonable input thermodynamics and are fitted to the observed present-day Saturn luminosity, but they are based upon an ad-hoc model for the H He phase diagram. With these caveats, the predicted Saturn atmosphere He abundance is Y 0:185 to 0.200, where Y is the helium mass fraction. This number may be compared with the result derived by Conrath and Gautier (2000) from a reanalysis of Voyager data: Y D 0:18 to Given that the thermodynamics of dense H He mixtures can now be calculated from first-principles simulations, the Fortney and Hubbard scenario needs to be updated. Initial investigations of jovian-planet structure based on the new simulations are just beginning to be published, and numerous discrepancies based on modeling discrepancies are beginning to appear.

5 log T ( K ) 4 The Interior of Saturn Jupiter Saturn laser shock 50% PPT? Pfoffenzeller 3.6 liquid H 2 liquid H + HD log P (Mbar) Fig. 4.4 Interior adiabats for present-day Jupiter and Saturn (heavy curves), together with possible regions of He H phase separation (hatched regions). The trajectory marked laser shock shows an experimentally-accessible regime. The dashed curve marked PPT shows a putative plasma phase transition, a first-order transition from undissociated hydrogen to ionized hydrogen. The curve marked 50% shows where, according to one model, molecular hydrogen is 50% pressure-dissociated Fig. 4.5 Saturn s effective temperature T eff (total luminosity is proportional to T eff 4 )versus age of Saturn, for three different cooling models. Circles denote homogeneous (no phase separation) cooling, while models 8 and 9 have He separation with different He-solubility constants (model 8 is more realistic). The horizontal dashed line shows Saturn s observed effective temperature 4.3 Coupling of Detailed Evolutionary Models for Saturn to the Helium Partitioning Problem, and Comparison with Jupiter Jupiter and Saturn are the two giant planets predominantly of hydrogen and helium, so a proper theoretical synthesis of their interior structures must fit numerous simultaneous observational and theoretical constraints, beginning with a consistent thermodynamic model of hydrogen helium mixtures. This is a daunting agenda, which is only beginning to be addressed. In 2008, the first papers based on realistic simulations of dense H He appeared (Nettelmann et al. 2008; Militzer et al. 2008), and the discrepancies which emerged point out a number of critical issues. Both of these initial papers were devoted to Jupiter, but Saturn models should appear in the near future. The Jupiter model published by Militzer et al. (2008) indicates that the so-called PPT sketched in Fig. 4.4 does not exist. Instead, hydrogen gradually metallizes over a range of pressures in the megabar range. Militzer s hydrogen simulations include He impurities at approximately solar concentration. The effect of the helium is to moderate a noticeable depression of the adiabatic temperature gradient in pressure range corresponding to gradual metallization, but the temperature increase in the Jupiter interior model is still much slower than depicted in Fig An interesting consequence

6 80 W.B. Hubbard et al. of the lower interior temperatures is that Militzer et al. infer a Jupiter core mass similar to the Saturn core masses given in Table 4.1. Since the present-day Jupiter model of Militzer et al. (2008) is substantially colder than the model depicted in Fig. 4.4, it is possible that a new Jupiter cooling model will require an additional heat source such as He separation. A consistent treatment of Saturn evolution based on the same approach will require further high-precision mapping of the phase diagram, at even lower temperatures. We note that so far such a first-principles phase diagram has not yet been incorporated in Saturn and Jupiter models, and generating such a diagram is a difficult problem, for it requires sufficient accuracy and precision in the simulation data points to accurately calculate second derivatives of thermodynamic variables. We do have a clue: The Fortney and Hubbard investigation demonstrates that the phase-separation locus in temperature pressure space must have a maximum temperature limit that increases with pressure, which rules out the behavior indicated by the Stevenson Salpeter (or Hubbard DeWitt) model. 4.4 Summary The study of Saturn s interior has reached a new phase in both theory and observation. Old paradigms that have been overturned include (a) a precise Saturn deep-rotation period with a claimed uncertainty of 7 s; (b) the original Stevenson He-immiscibility diagram with He miscibility increasing as a function of pressure; (c) hydrogen phase diagrams with a first-order transition associated with hydrogen metallization. The consequence of (a) is not so severe, because we still know Saturn s primary rotational disturbance with enough precision to infer the existence of a massive core. The revision to (b) can in principle be verified by high-precision many-body simulations, a process that is ongoing. The disappearance of (c) does not necessarily rule out discontinuities in the hydrogen-rich outer layers of Jupiter or Saturn, for a He-immiscibility boundary could produce such a discontinuity. The concentration of He and C, N, O hydrides on either side of such a boundary should properly follow from consistent thermodynamics rather than from ad hoc assumptions. It is premature to quote theoretical predictions for such concentrations for Saturn, since Saturn has not yet been modeled with the improved, first-principles hydrogen helium equations of state. When such modeling has been reported, it will be possible to compare the results with determinations of atmospheric abundances from Cassini observations, such as, e.g., Hersant et al. (2008), and to update predictions such as Mousis et al. (2006). The question of whether Saturn s higher-order gravity harmonics can be used to constrain interior structure, or whether they will be primarily sensitive to envelope dynamics, is so far unresolved. Relevant data should come from the 2016 Jupiter orbiter mission, Juno. Hubbard (1999) argued that Jupiter gravity harmonics J n with n greater than about 8 should be dominated by the dynamics of Jupiter s outer layers. Militzer et al. (2008) argued that dynamical effects may even enter starting with n D 4. We have argued in this chapter that a similar situation may apply to Saturn, implying that Juno-like gravity measurements at Saturn will be most illuminating. Acknowledgments We thank M. Podolak and another referee for helpful comments. Figures 4.4 and 4.5 are reprinted from Icarus 164, J. J. Fortney and W. B. Hubbard, Phase separation in giant planets: inhomogeneous evolution of Saturn, pp , Copyright 2003, with permission from Elsevier. References Anderson, J. D., and Schubert, G. 2007, Science 317, 1384 Cecconi, B., and Zarka, P. 2005, J. Geophys. Res. 110, A12203 Choi, D. S., Showman, A. P., and Brown, R. H. 2009, J. Geophys. Res., 114, Issue E4, CiteID E Conrath, B. J., and Gautier, D. 2000, Icarus 144, Fortney, J. J., and Hubbard, W. B. 2003, Icarus 164, Helled, R., Schubert, G., and Anderson, J. D. 2009, Icarus 199, Hersant, F., Gautier, D., Tobie, G., and Lunine, J. I Planetary Space Sci. 56, Hubbard, W. B., Porco, C. C., Hunten, D. M., Rieke, G. H., Rieke, M. J., McCarthy, D. W., Haemmerle, V., Haller, J., McLeod, B., Lebofsky, L.A., Marcialis, R., Holberg, J. B., Landau, R., Carrasco, L., Elias, J., Buie, M. W., Dunham, E. W., Persson, S. E., Boroson, T., West, S., French, R. G., Harrington, J., Elliot, J. L., Forrest, W. J., Pipher, J. L., Stover, R. J., Brahic, A., and Grenier, I. 1997, Icarus 130, Hubbard, W. B., and DeWitt, H. E. 1985, Astrophys. J. 290, Hubbard, W. B. 1999, Icarus 137, Jacobson, R. A., Antreasian, P. G., Bordi, J. J., Criddle, K. E., Ionasescu, R., Jones, J. B., Mackenzie, R. A., Meek, M. C., Parcher, D., Pelletier, F. J., Owen, W. M., Jr., Roth, D. C., Roundhill, I. M., and Stauch, J. R. 2006, Astron. J. 132, Lindal, G. F., Sweetnam, D. N., and Eshleman, V. R. 1985, Astron. J. 90, Lindal, G. F. 1992, Astron. J. 103, Militzer, B., Hubbard, W. B., Vorberger, J., Tamblyn, I., and Bonev, S. A. 2008, Astrophys. J. Lett. 688, L45-L48 Mousis, O., Alibert, Y., and Benz, W. 2006, Astron. Astrophys. 449, Nettelmann, N., Holst, B., Kietzmann, A., French, M., Redmer, R., and Blaschke, D. 2008, Astrophys. J. 683,

7 4 The Interior of Saturn 81 Nicholson, P., McGhee, C. A., and French, R. G. 1995, Icarus 113, Pfaffenzeller, O., Hohl, D., and Ballone, P. 1995, Phys. Rev. Lett. 74, Sanchez-Lavega, A. 2005, Science 307, Saumon, D., Chabrier, G., and Van Horn, H. M. 1995, Astrophys. J. Suppl. 99, Stevenson, D. J.1975, Phys. Lett. A 58, Stevenson, D. J., and Salpeter, E.E. 1977a, Astrophys. J. Suppl. 35, Stevenson, D. J., and Salpeter, E. E. 1977b, Astrophys. J. Suppl. 35, Tassoul, J.-L., Theory of Rotating Stars, Princeton University Press, 1978.