Rhea's Internal Structure Inferred from Cassini Gravity and Topography

46th Lunar and Planetary Science Conference (2015) 1069.pdf

RHEA'S INTERNAL STRUCTURE INFERRED FROM CASSINI GRAVITY AND TOPOGRAPHY Doug Hemingway1 ([email protected]), Francis Nimmo1, Paolo Tortora2, Marco Zannoni2, Luciano Iess3, Marzia Parisi3, and Peter Thomas4. 1Department of Earth & Planetary Sciences, University of California Santa Cruz, 1156 High Street, Santa Cruz, California, 95064, USA, 2Dipartimento di Ingegneria Industriale, Università di Bologna, Forlì, Italy, 3Dipartimento di Ingegneria Meccanica e Aerospaziale, Sapienza Università di Roma, Rome, Italy, 4Department of Astronomy, Cornell University, Ithaca, New York, USA.

Summary: Combined analyses of gravity and topography can yield useful information about the interi- or structures of planetary bodies [1–4]. Based on radio tracking of the Cassini spacecraft, we present an estimate of the gravity field of Saturn's large moon Rhea. Combining this with a shape model based on limb profile analyses, we attempt to constrain Rhea's internal structure, concluding that the core likely exhibits some excess flattening. Gravity: During its time in the Saturn system, Cas- sini has performed two close fly-bys of Rhea devoted to gravity investigations: one in November 2005, and another in March 2013. As in previous work [2,4,5], the gravity field of Rhea was estimated by reconstruct-

ing the spacecraft trajectory during these encounters, making use of precise measurements of the Doppler Figure 1: J2 and C22 dimensionless gravitational potential shift of a highly stable microwave radio link between coefficients for Rhea, with crosshairs illustrating the one- Cassini and the ground station antennas of NASA's sigma uncertainties. Expected hydrostatic values of J2 and Deep Space Network. C22 are indicated for various normalized moments of inertia The available data were analyzed in a multi-arc fit along the theoretical hydrostatic line (dashed line). estimating a fully unconstrained quadrupole gravity field for Rhea, the Rhea state vector at a reference (uncertainties are all one-sigma). Figure 2 illustrates epoch, and the spacecraft initial state vector for each how the shape compares with the hydrostatic expecta- fly-by. The stability of the solutions was tested by per- tion for various assumed moments of inertia. The cen- turbing the dynamical model in various ways, includ- tral value is close to the expectation for a nearly ho- ing estimating a full 4x4 gravity field and changing the mogeneous, hydrostatic body, however the uncertain- a priori uncertainties. ties are large. The resulting estimate of Rhea's gravity field ex- Internal Structure: There are a number of ways to hibits a statistically significant departure from the hy- explain the excess oblateness (higher than expected drostatic expectation (Figure 1), with a J2/C22 ratio of J2/C22 ratio) in Rhea's gravity field that are consistent 3.9±0.1 (the hydrostatic ratio is ~3.33). As a result, the with the observed shape. moment of inertia cannot be inferred directly from the Homogeneous Model: If Rhea is of uniform densi- Radau-Darwin approximation. Estimating the moment ty, it can be shown [8,3,4] that the shape required to of inertia directly from the J2 or C22 coefficients yields explain the observed gravity is given by incompatible and unrealistic estimates of ~0.41 and !!!! ! � = � (1) ~0.37, respectively. !" ! !" Shape: Rhea's shape has been determined previ- where R is Rhea's mean radius, and �!" and �!" ously via analyses of limb profiles [6,7]. After incorpo- are spherical harmonic coefficients of degree l and rating the latest available data, we repeat this analysis, order m, representing the dimensionless gravitational obtaining a best-fitting tri-axial ellipsoid with semi- potential and topography, respectively. In addition to axes: the J2 and C22 values estimated from limb profile ob- a = 765.7 ± 0.3 km, servations, Figure 2 shows the values obtained from b = 763.6 ± 0.3 km, equation (1) based on the gravity coefficients illustrat- c = 762.9 ± 0.2 km, ed in Figure 1. The gravity field is thus consistent with from which we obtain the unnormalized degree-2 a homogeneous Rhea of the observed shape, to within coefficients: J2 = 1190 ± 260 m, C22 = 350 ± 120 m the shape model's uncertainties. 46th Lunar and Planetary Science Conference (2015) 1069.pdf

Figure 3: Required J2 and C22 core-mantle-boundary to- Figure 2: Limb-profile-based estimates of the J2 and C22 pography for each of four different mantle thicknesses (5, topography coefficients for Rhea, with crosshairs illustrating 100, 200, & 400 km) relative to the expected hydrostatic the one-sigma uncertainties. Also shown is the surface to- line (dashed black line), assuming the surface conforms to pography required for a homogeneous Rhea to produce the our shape model. For each assumed mantle thickness, the observed gravity field, computed from equation (1). corresponding moment of inertia factor is given in the legend.

Two-Layer Model: Considering that the surface is not significantly overestimated the C22 topography, most likely water ice [9,10], it may be more realistic to then some excess core oblateness (roughly 1 km) is assume some internal layering. Here, we consider a required to account for the observed gravity field. This simple two-layer model: silicate core and H2O mantle. excess core topography implies stresses of ~0.3 MPa, We assume the density of the mantle is 920 kg/m3 and which are readily supported by cold silicate materials. we examine a series of different core densities. For With larger mantle thicknesses, the core-mantle- each assumed core density, we compute the core radius boundary is deeper and so larger amplitudes are re- required to satisfy the observed bulk density constraint quired to overcome the resulting attenuation of the (� = 1236 kg/m3). gravity signal. If the surface conforms closely to the central values Conclusions: The observed shape is consistent of our shape model, then the excess oblateness of the with the shape required for an undifferentiated (uni- gravity field requires an irregular core shape (i.e., ob- form density) Rhea to account for the observed gravity. lateness at the core-mantle-boundary that is beyond the However, because the surface is more likely to be wa- hydrostatic expectation)—similar concepts have been ter ice, a two-layer model may be a better approxima- considered previously for Enceladus [6,11] and Mimas tion (cf. [5]). In this case, and assuming a mantle densi- [12]. With this two-layer model, we can solve for the ty of 920 kg/m3, some ~1 km of excess core oblateness core-mantle-boundary topography required to give rise may be necessary to account for the observed gravity to the observed gravity field (e.g., [11,13]). Given a (Figure 3). A wide range of moments of inertia is al- known gravity field and surface topography, expressed lowed, but models with low moments of inertia (i.e., in spherical harmonics, it can be shown that the re- more differentiation) require greater magnitudes of quired topography at the core-mantle-boundary is giv- excess core topography to satisfy the observations. en by ! ! !!! !!!! !! �!"# = � − � �! (2) References: 1.Wieczorek, M. A. in Treatise Geophys. 10, 165– !" ∆! !!! ! !" ! !" 206 (2007). 2. Iess, L. et al. Science 327, 1367–9 (2010). 3. !"# where �!" is the core-mantle-boundary topogra- Hemingway, D., et al. Nature 500, 550–552 (2013). 4. Iess, L. et al. phy at degree l and order m, ∆� is the density contrast Science 344, 78–80 (2014). 5. Anderson, J. D. & Schubert, G. Geophys. Res. Lett. 34, L02202 (2007). 6. Thomas, P. et al. Icarus at the core-mantle-boundary, d is the depth of the core- 190, 573–584 (2007). 7. Nimmo, F., Bills, B. G. & Thomas, P. C. J. mantle-boundary (� = � − �!"#$), � is Rhea's bulk Geophys. Res. 116, E11001 (2011). 8. Jeffreys, H. (Cambridge density, R is Rhea's full radius, � is the density of the University Press, 1976). 9. Clark, R. N. & Owensby, P. D. The ! infrared spectrum of Rhea. Icarus 46, 354–360 (1981). 10. Stephan, mantle, �!" is the dimensionless gravitational potential K. et al. Planet. Space Sci. 61, 142–160 (2012). 11. McKinnon, W. ! resolved at radius R, and �!" is the surface topogra- B. J. Geophys. Res. Planets 118, n/a–n/a (2013). 12. Tajeddine, R. et phy. al. Science (80-. ). 346, 322–324 (2014). 13. Lefevre, A., et al. Icarus (2014) As illustrated in Figure 3, if the surface conforms to our shape model (Figure 2), and assuming we have