Saturn's satellite Rhea is a homogeneous mix of rock and ice

Abstract

[1] A close flyby of Saturn's satellite Rhea by the Cassini spacecraft on 26 November 2005 can be used to infer its mass and quadrupole gravity moments. From these data we derive a mean density of 1233 ± 5 kg m⁻³ for Rhea and an axial moment of inertia consistent with an undifferentiated interior. The small density implies an interior made up of about 25% rock-metal and 75% water ice by mass. The moment of inertia implies a homogeneous mix of these constituents with some compression of the ice and the transition from ice I to ice II at depth.

1. Introduction

[2] With a mean radius of 764.4 ± 1.1 km [Thomas et al., 2006], Rhea is the second largest satellite in the Saturnian system. Titan dominates with a mean radius of about 2575 km, midway in size and density between Jupiter's two large satellites Ganymede and Callisto. Rhea orbits Saturn in a nearly circular orbit in its equatorial plane at a mean distance of 527.07 km and a period of 4.518 days.

[3] The Cassini spacecraft was tracked by the Deep Space Network (DSN) during its flyby of Rhea at a closest approach altitude of 502 km. Both coherent radio Doppler data and Radio ranging data are available. Optical navigation data are also available from the spacecraft's imaging system, mainly for purposes of improving the Rhea ephemeris, and hence the relative location of the spacecraft with respect to Rhea. The successful determination of Rhea's quadrupole gravity moments depends on the Doppler data during the close flyby, but the other data are important for an accurate orbit determination and hence for a minimization of systematic error.

[4] The time of closest approach of the spacecraft to Rhea is 26 November 2005 22:37:36 UTC at the spacecraft, or with the addition of the time required for light to travel from Saturn to Earth, 23:49:37 UTC at the DSN stations. Doppler and ranging data are available for the interval from 26 November 2005 22:14:00 to 27 November 2005 01:36:00. With tracking data available from both DSN Station DSS 34 near Canberra Australia and DSS 55 near Madrid Spain, the Doppler data are essentially continuous over this flyby interval, and the ranging data are nearly continuous. However, for purposes of obtaining the best orbit determination for the Cassini spacecraft in the Saturnian system, including the Rhea flyby, it is important to include tracking data outside the fundamental flyby interval, even though the data outside the flyby interval are not continuous.

[5] The fitting model for the Rhea flyby depends on the gravitational potential V, which is defined by,

here r, θ, ϕ are the usual spherical coordinates, r the radial distance from the center of mass, θ the polar angle (colatitude), and ϕ the azimuthal angle (longitude), with respect to a uniformly rotating right-handed Cartesian system x, y, z defined as follows. The x axis is directed on average from the center of Rhea to the center of Saturn. The z axis is directed along Rhea's spin axis toward its north pole, and y completes the system [Seidelmann et al., 2005]. The Cartesian axes are aligned with Rhea's principal moments of inertia A < B < C, hence the two quadrupole coefficients J₂ and C₂₂ are sufficient to describe the truncated potential function, complete to second degree and order. This orientation of axes, along with the free gravity parameters J₂ and C₂₂, defines the fitting model for the quadrupole field. The two second degree Legendre polynomials of interest are P₂ and P₂₂. The potential is scaled by the third free parameter GM, the gravitational constant G times the mass of Rhea.

[6] We adopt the following values of the gravity parameters as the best fit to the Cassini flyby data. The error bars are expressed in terms of the standard deviation (SD), and they include all sources of both random and systematic error. GM = (153.9396 ± 0.0017) km³ s⁻², J₂ = 0.0009434 ± 0.0000357, C₂₂ = 0.0002335 ± 0.0000172.

[7] The inclination of the Cassini orbit to the Rhea equator is about 10.4°. With just a single near-equatorial flyby, it is not possible to measure J₂ and C₂₂ independently. However, the complete covariance matrix for the three parameters can be used to extract the gravity information contained in the data. According to the covariance matrix from the fit, the correlation between GM and J₂ is 0.318, the correlation is −0.701 between GM and C₂₂, and it is −0.341 between J₂ and C₂₂. We add to this complete information array, including the correlations, the fact that for an equilibrium body 3 J₂ = 10 C₂₂ [Schubert et al., 2004]. A body the size of Rhea, in synchronous rotation with its orbital period and in hydrostatic equilibrium throughout its interior, responds equally to rotational and tidal forces. Even so, it is not necessary to impose the equilibrium constraint exactly. Some deviations from equilibrium can be accommodated by making the constraint inexact with its own SD. The technique we use is to introduce a new observation z = J₂ − 10 C₂₂/3 = 0, with a SD σ_z on the new observation. As it turns out, the revised gravity coefficients, after the introduction of the inexact equilibrium constraint, are not sensitive to σ_z between the limits of zero and 10⁻⁵. For a σ_z of 10⁻⁴, the failure of J₂ and C₂₂ to satisfy the constraint is also at about the 10⁻⁴ level, and the inexactness of the constraint is beginning to affect the result adversely, in the sense that the best-fit values for J₂ and C₂₂ are not consistent with the assumption that Rhea is in hydrostatic equilibrium. However, so long as the equilibrium constraint is satisfied to 10⁻⁵ or less, the equilibrium values for the revised gravity coefficients are stable, and they represent the best possible reduction of the flyby Doppler data to a Rhea gravity field complete through second degree and order. The results are GM = (153.9372 ± 0.0013) km³ s⁻², J₂ = 0.000889 ± 0.000025 and C₂₂ = 0.0002666 ± 0.0000075. We adopt these revised gravity parameters for geophysical interpretation.

[8] The most fundamental datum is the mean density. From our adopted value of GM, the mass can be determined by dividing GM by a modern value of G given by (6.674215 ± 0.000092) × 10⁻¹¹ m³ kg⁻¹ s⁻². Six measurements of G published after 1995 can be found in Gundlach and Merkowitz [2000]. We use the most accurate measurement by Gundlach and Merkowitz [2000] for a percentage error of 13.7 ppm on G, including both systematic error and statistical error. The percentage error in GM is 8.4 ppm, hence the percentage error in the mass determination is 16.1 ppm and M is equal to (2.306447 ± 0.000037) × 10²¹ kg. The volume of Rhea is defined by its mean radius [Thomas et al., 2006] and is equal to (1.8709 ± 0.0081) × 10¹⁸ m³. The corresponding mean density ρ is 1232.8 ± 5.3 kg m⁻³, where the SD corresponds to the relatively large error in the mean radius.

[9] Knowledge of the mean density places a strong constraint on the composition and internal structure of any planetary body. For smaller satellites and asteroids, with low gravity, the role of porosity or void space is the major complicating factor in interpreting density information [Anderson et al., 2005]. Whether significant porosity exists is also closely connected to a body's ability to maintain a non-spherical shape, because both depend on internal stresses and strengths of material [Johnson and McGetchin, 1973]. Satellites with volumes less than ∼ 10¹⁶ m³ generally have arbitrary and highly irregular shapes, with the ratio a/c of the long axis a to the short axis c ranging from 1.1 to 2.0 [Thomas et al., 1986]. Saturn's small satellite Mimas is at the low end of this general rule with a nearly spherical shape [Dermott and Thomas, 1988]. With a mean radius of 198.8 km, Mimas' volume is 3.29 × 10¹⁶ m³ and it apparently has some small mean porosity [Leliwa-Kopystynski and Kossacki, 2000]. Rhea, however, has a volume about 57 times that of Mimas, hence any effects of porosity are most certainly negligible.

[10] The relatively low density of Rhea suggests a relatively low-density material composition, almost certainly icy. Calculations of rock/ice ratios for Ganymede and Callisto imply an uncompressed density of ∼ 1600 kg m⁻³ for the rock/ice material that formed these satellites. (The determination of the interior composition, structure and dynamics of the four Galilean satellites has been summarized, along with a bibliography, by Schubert et al. [2004]). We assume that Rhea, like Ganymede and Callisto, is composed of rock-metal and ice, but because of the lower density, more icy than the outer two Galilean moons. The rock-metal mass fraction m_c, or the ratio of the rock-metal component mass to the total mass of Rhea, follows from the equation

where ρ is the measured mean density, ρ_c is the density of the rock-metal component, and ρ_s is the density of the ice component. We assume Io's mean density for ρ_c (3527.5 kg m⁻³), and with some ice contaminants such as CO₂ a density of 1010 kg m⁻³ for ρ_s, the value of m_c is 0.253_−0.011^+0.050. The nonsymmetrical error reflects one-sigma limits on rock-metal density between 2500 and 4000 kg m⁻³, and a negligible contribution from the uncertainty in ice density. For a lower bound on the rock-metal density of 2500 kg m⁻³, but with the measured mean density of 1232.8 kg m⁻³ and a contaminated ice component of density 1010 kg m⁻³, the mass fraction m_c of the rock-metal component is 0.303. The assumption of an Io rock-metal mass ratio for Rhea could be wrong, but at least it accounts for a possible metallic component in Rhea's composition, and the assumed one-sigma limits on its density are reasonable. The low density of Rhea suggests that a density of 1010 kg m⁻³ is perhaps too large for the ice component. In that case it could lead to an underestimate of the fraction of ice II in Rhea.

[11] The gravity coefficient C₂₂ can be used along with the theory of equilibrium figures to derive the normalized axial moment of inertia for Rhea. For synchronous rotation the mean orbital period is equal to the mean rotation period. We adopt a mean angular velocity ω of 16.0978436 × 10⁻⁶ rad s⁻¹, and the smallness parameter q_r for the equilibrium figure follows by [Schubert et al., 2004]

where the error is dominated by the error on the mean radius, not GM. The error contribution from ω, derived from Rhea's known orbital period, can be neglected [Seidelmann et al., 2005].

[12] Rhea's secular Love number for static equilibrium k_f is determined by the value of C₂₂ according to k_f = 4 C₂₂/q_r [Schubert et al., 2004] and is equal to 1.418 ± 0.040. The normalized axial moment of inertia can be approximated by the first order Radau relationship [Kaula, 1968],

The normalized moment of inertia C/MR² is 0.4 for an interior of constant density. Our experimentally determined value is significantly smaller by about two SD. This can be understood if compression of the ice and rock toward Rhea's center and the possible existence of water ice in the denser phase II near the center [Ellsworth and Schubert, 1983] are taken into account. In addition, the three principal axes can be evaluated from equilibrium theory [Hubbard and Anderson, 1978] and compared to the measured values. The radial values along Rhea's principal axes are c < b < a, with the equatorial radius a along x equal to a = c [1 + 2 (1 + k_f) q_r] = (766.0 ± 1.1) km, the equatorial axis along y equal to b = c [1 + (1/2) (1 + k_f) q_r] = (763.9 ± 1.1) km and the polar axis c equal to (763.2 ± 1.1) km. These computed axes from the gravity field are constrained by the measured mean radius R, and are in agreement with the measured values of a = 767.6 km, b = 762.5 km and c = 763.2 km [Thomas et al., 2006]. The polar axes agree exactly. The difference in axes a – c is (2.733 ± 0.048) km for the equilibrium axes and (4.4 ± 2.7) km for the measured axes, an excellent agreement and consistent with a body in hydrostatic equilibrium. Furthermore, our calculations for a reasonable range of the rock-metal mass fraction m_c indicate that Rhea has enough rock that long-term radiogenic heating would melt an ice-rock mixture not cooled by subsolidus convection. Hence, Rhea could not have a rock fraction increasing with depth. Such a distribution of rock would preclude convective cooling and lead to melting and total differentiation. Rhea is essentially an undifferentiated homogeneous mixture of ice and rock, with perhaps a thin rock-free shell near the surface.

[13] The size of Rhea suggests that the ice near its center is probably ice II, not the ice I expected nearer the surface or in a thin outermost shell of pure ice. This phase change in the ice is consistent with a homogeneous interior, but with a normalized moment of inertia slightly less than a value of 0.4 for constant density. The excess density in the core, caused by a denser ice phase, can be expressed as an overall normalized core excess density 〈δρ〉 given by (ρ_c − ρ_s)/ρ_s, where ρ_c is the density in the ice II-rock core and ρ_s is the density in the ice I-rock shell. The approximate size of an ice II-rock core in Rhea can be determined from the measured moment of inertia factor C/MR² divided by the same factor for constant density, or 0.3911/.4 = 0.9777. This measurement imposes a constraint between the normalized core radius r_c/R and the normalized core excess density 〈δρ〉 given by

The normalized ice II core radius is plotted in Figure 1 over a reasonable range of 〈δρ〉. The ice II core is large, between about 0.5 and 0.7 of Rhea's radius.

Figure 1

Radius of Rhea's ice II core normalized by the satellite radius vs. the core excess density normalized by the density of Rhea's outer ice shell.

[14] The relatively old, heavily cratered surface of Rhea supports our conclusion that the satellite's interior is a homogeneous, undifferentiated mixture of ice and rock. Nevertheless, there are variations in the appearance of Rhea's surface. The leading hemisphere is highly cratered and uniformly bright, while the trailing hemisphere is darker, has fewer craters, and has bright streaks. Some features on Rhea might be of tectonic origin [Thomas, 1988]. For example, Moore et al. [1985] suggested that the pit chain Pu Chou Chasma is a tectonic feature, but more recently Moore et al. [2004] interpreted this and similar pit chasms as secondary craters from large impacts such as Tirawa. Moore et al. [2004] also suggest that seismic shaking and focusing of seismic energy antipodal to large impacts could have played an important role in modifying Rhea's surface. Examples of surface modification include degradation of craters and modification of crater shapes, features seen in the undulating terrain antipodal to Tirawa [Moore et al., 2004].

[15] There is little if any evidence on Rhea's surface of the endogenic activity that might be expected of a differentiated satellite. In contrast, Enceladus shows ample evidence of such internal activity [Schubert et al., 2007]. Cassini magnetometer observations [Khurana et al., 2006] show that the interaction of Rhea with the plasma in Saturn's magnetosphere is similar to that of the Moon's interaction with the plasma of the solar wind. Rhea is an absorber of plasma, unlike Enceladus, which is a source of plasma. The plume in the southern hemisphere of Enceladus [Porco et al., 2006] is the likely source of the plasma and a sign of internal activity in a differentiated satellite [Schubert et al., 2007]. Thermal history calculations for Rhea indicate that subsolidus ice convection could occur inside Rhea, and contribute to the modification of its surface, but differentiation of ice from rock is precluded [Ellsworth and Schubert, 1983].

[16] Finally, for comparison purposes, we show in Table 1 the calculation of the bulk densities for all the larger inner satellites. The GM values are taken from pre-Cassini estimates for Mimas, Tethys, and Dione [Jacobson, 2004], and from Cassini data for Enceladus Porco et al., 2006], [Schubert et al., 2007]. The mean radii are taken from Thomas et al. [2006]. The masses are computed from GM with the same value of G used above for Rhea [Gundlach and Merkowitz, 2000].

Table 1. Masses M Bulk Densities ρ for Five of Saturn's Satellites

Satellite	GM, km^3 s^−2	R, km	M, 10^20 kg	ρ, kg m^−3
Mimas	2.55 ± 0.05	198.3 ± 0.6	0.3821 ± 0.0075	1170 ± 25
Enceladus	7.210 ± 0.011	252.1 ± 0.2	1.0803 ± 0.0016	1609.6 ± 4.6
Tethys	41.21 ± 0.08	533.0 ± 1.0	6.175 ± 0.012	973.5 ± 5.8
Dione	73.12 ± 0.02	561.6 ± 0.5	10.9556 ± 0.0030	1476.6 ± 4.0
Rhea	153.9372 ± 0.0013	764.4 ± 1.1	23.06447 ± 0.00037	1232.8 ± 5.3

[17] The ratio m_c of rock-metal to ice by mass in each of these satellites can be calculated from equation 2, with the same assumptions on the densities as for Rhea. It has been suggested that Mimas may contain some small porosity [Leliwa-Kopystynski and Kossacki, 2000], and with 4% of the total volume made up of pores, the ratio m_c is 0.240_−0.011^+0.047. For Enceladus with no porosity at all, m_c is 0.522_−0.023^+0.103. For Tethys, the relatively low density requires a porosity of 4% in order to produce a sample density equal to contaminated water ice and with no rock-metal component. For Dione with no porosity m_c is equal to 0.443_−0.020^+0.087. Dione, like Enceladus, appears to be a source of plasma [Khurana et al., 2006]. Accordingly, like Enceladus, it might be a differentiated and internally active body [Schubert et al., 2007]. Enceladus and Dione are in resonance and they have the highest rock fractions, and therefore the highest contents of heat producing radiogenic elements, of any of the mid-size Saturnian moons. These characteristics are likely involved in explaining the interior structure and activity of both satellites.

Acknowledgments

[18] J. D. A. acknowledges support by the Cassini Project at the Jet Propulsion Laboratory, California Institute of Technology, under contract with NASA. G. S. acknowledges support by grants from NASA through the Planetary Geology and Geophysics program. We acknowledge G. Giampieri for helpful discussions while he held an ORAU/NASA Senior Research Associateship Award at JPL. We thank the entire Cassini Navigation Team, but in particular R. Mackenzie, P. G. Antreasian, R. A. Jacobson, D. Parcher, F. Pelletier, and D. C. Roth for access to their results from fitting the Cassini data, and for insights into the appropriate fitting procedures for the Rhea flyby. We also thank J. Palguta for drawing Figure 1.