Marie Yseboodt

Mercury's moment of inertia from spin and gravity data

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, E00L09, doi:10.1029/2012JE004161, 2012 Mercury’s moment of inertia from spin and gravity data Jean-Luc Margot, 1,2 Stanton J. Peale, 3 Sean C. Solomon, 4,5 Steven A. Hauck II, 6 Frank D. Ghigo, 7 Raymond F. Jurgens, 8 Marie Yseboodt, 9 Jon D. Giorgini, 8 Sebastiano Padovan, 1 and Donald B. Campbell 10 Received 15 June 2012; revised 31 August 2012; accepted 5 September 2012; published 27 October 2012. [ 1 ] Earth-based radar observations of the spin state of Mercury at 35 epochs between 2002 and 2012 reveal that its spin axis is tilted by (2.04 AE 0.08) arc min with respect to the orbit normal. The direction of the tilt suggests that Mercury is in or near a Cassini state. Observed rotation rate variations clearly exhibit an 88-day libration pattern which is due to solar gravitational torques acting on the asymmetrically shaped planet. The amplitude of the forced libration, (38.5 AE 1.6) arc sec, corresponds to a longitudinal displacement of

Titan’s obliquity as evidence of a subsurface ocean?

450 m at the equator. Combining these measurements of the spin properties with second-degree gravitational harmonics (Smith et al., 2012) provides an estimate of the polar moment of inertia of Mercury C/MR 2 = 0.346 AE 0.014, where M and R are Mercury’s mass and radius. The fraction of the moment that corresponds to the outer librating shell, which can be used to estimate the size of the core, is C m /C = 0.431 AE 0.025. Citation: Margot, J.-L., S. J. Peale, S. C. Solomon, S. A. Hauck II, F. D. Ghigo, R. F. Jurgens, M. Yseboodt, J. D. Giorgini, S. Padovan, and D. B. Campbell (2012), Mercury’s moment of inertia from spin and gravity data, J. Geophys. Res., 117, E00L09, doi:10.1029/2012JE004161. 1. Introduction [ 2 ] Bulk mass density r = M/V is the primary indicator of the interior composition of a planetary body of mass M and volume V. To quantify the structure of the interior, the most useful quantity is the polar moment of inertia Z r ð x; y; z Þ x 2 þ y 2 dV : C ¼ V In this volume integral expressed in a cartesian coordinate system with principal axes {x, y, z}, the local density is multiplied by the square of the distance to the axis of rotation, which is assumed to be aligned with the z axis. Moments of Department of Earth and Space Sciences, University of California, Los Angeles, California, USA. Department of Physics and Astronomy, University of California, Los Angeles, California, USA. Department of Physics, University of California, Santa Barbara, California, USA. Department of Terrestrial Magnetism, Carnegie Institution of Washington, Washington, D. C., USA. Lamont-Doherty Earth Observatory, Columbia University, Palisades, New York, USA. Department of Earth, Environmental, and Planetary Sciences, Case Western Reserve University, Cleveland, Ohio, USA. National Radio Astronomy Laboratory, Green Bank, West Virginia, USA. Jet Propulsion Laboratory, Pasadena, California, USA. Royal Observatory of Belgium, Uccle, Belgium. Department of Astronomy, Cornell University, Ithaca, New York, USA. Corresponding author: J.-L. Margot, Department of Earth and Space Sciences, University of California, 595 Charles Young Dr. E., Los Angeles, CA 90095, USA. ( jlm@ess.ucla.edu) ©2012. American Geophysical Union. All Rights Reserved. 0148-0227/12/2012JE004161 inertia computed about the equatorial axes x and y are denoted by A and B, with A < B < C. The moment of inertia (MoI) of a sphere of uniform density and radius R is 0.4 MR 2 . Earth’s polar MoI value is 0.3307 MR 2 [Yoder, 1995], indicating a concentration of denser material toward the center, which is recognized on the basis of seismological and geochemical evidence to be a primarily iron-nickel core extending

Influence of an inner core on the long-period forced librations of Mercury

55% of the planetary radius. The value for Mars is 0.3644 MR 2 , suggesting a core radius of

The obliquity of Enceladus

50% of the planetary radius [Konopliv et al., 2011]. The value for Venus has never been measured. Here we describe our determina- tion of the MoI of Mercury and that of its outer rigid shell (C m ), both of which can be used to constrain models of the interior [Hauck et al., 2007; Riner et al., 2008; Rivoldini et al., 2009]. [ 3 ] Both the Earth and Mars polar MoI values were secured by combining measurements of the precession of the spin axis due to external torques (Sun and/or Moon), which depends on [C A (A + B)/2]/C, and of the second-degree harmonic coefficient of the gravity field C 20 = A [C A (A + B)/2]/(MR 2 ). Although this technique is not applicable at Mercury, Peale [1976] proposed an ingenious procedure to estimate the MoI of Mercury and that of its core based on only four quantities. The two quantities related to the gravity field, C 20 and C 22 = (B A A)/(4MR 2 ), have been determined to better than 1% precision by tracking of the MESSENGER spacecraft [Smith et al., 2012]. The two quantities related to the spin state are the obliquity q (tilt of the spin axis with respect to the orbit normal) and amplitude of forced libration in longitude g (small oscillation in the orientation of the long axis of Mercury relative to uniform spin). They have been measured by Earth-based radar observations at 18 epochs between 2002 and 2006. These data provided strong obser- vational evidence that the core of Mercury is molten, and that E00L09 1 of 11

High‐frequency geophysical fluid modeling necessary to understand Earth rotation variability

Abstract On the basis of gravity and radar observations with the Cassini spacecraft, the moment of inertiaof Titan and the orientation of Titan’s rotation axis have been estimated in recent studies. Accordingto the observed orientation, Titan is close to the Cassini state. However, the observed obliquity isinconsistent with the estimate of the moment of inertia for an entirely solid Titan occupying theCassini state. We propose a new Cassini state model for Titan in which we assume the presenceof a liquid water ocean beneath an ice shell and consider the gravitational and pressure torquesarising between the di erent layers of the satellite. With the new model, we nd a closer agreementbetween the moment of inertia and the rotation state than for the solid case, strengthening thepossibility that Titan has a subsurface ocean. 1 Introduction On the basis of Cassini radar images, [6] and [7] precisely measured the orientation of therotation axis of Titan. Using the orientation of the normal to the orbit of Titan given in the IAUrecommendations (Seidelmann et al. 2007), they determined the obliquity to be about 0:3

The long-period forced librations of Titan

Abstract The planetary perturbations on Mercury’s orbit lead to long-period forced librations of Mercury’s mantle. These librations have previously been studied for a planet with two layers: a mantle and a liquid core. Here, we calculate how the presence of a solid inner core in the liquid outer core influences the long-period forced librations. Mantle–inner core coupling affects the long-period libration dynamics mainly by changing the free libration: first, it lengthens the period of the free libration of the mantle, and second, it adds a second free libration, closely related to the free gravitational oscillation between the mantle and inner core. The two free librations have periods between 2.5 and 18y depending on the internal structure. We show that large amplitude long-period librations of a few tens of arcsec are generated when the period of a planetary forcing approaches one of the two free libration periods. These amplitudes are sufficiently large to be detectable by spacecraft measurements of the libration of Mercury. The amplitudes of the angular velocity of Mercury’s mantle at planetary forcing periods are also amplified by the resonances, but remain much smaller than the current precision of Earth-based radar observations unless the period is very close to a free libration period. The inclusion of mantle–inner core coupling in the rotation model does not significantly improve the fit to the radar observations. This implies that it is not yet possible to determine the size of the inner core of Mercury on the basis of available observations of Mercury’s rotation rate. Future observations of the long-period librations may be used to constrain the interior structure of Mercury, including the size of its inner core.

Evolution of Mercury's obliquity

Abstract The extraordinary activity at Enceladus’ warm south pole indicates the presence of an internal global or local reservoir of liquid water beneath the surface. While Tyler (Tyler, R.H. [2009]. Geophys. Res. Lett. 36(15), L15205; Tyler, R.H. [2011]. Icarus 211(1), 770–779) has suggested that the geological activity and the large heat flow of Enceladus could result from tidal heating triggered by a large obliquity of at least 0.05–0.1°, theoretical models of the Cassini state predict the obliquity to be two to three orders of magnitude smaller for an entirely solid and rigid Enceladus. We investigate the influence of an internal subsurface ocean and of tidal deformations of the solid layers on the obliquity of Enceladus. Our Cassini state model takes into account the external torque exerted by Saturn on each layer of the satellite and the internal gravitational and pressure torques induced by the presence of the liquid layer. As a new feature, our model also includes additional torques that arise because of the periodic tides experienced by the satellite. We find that the upper limit for the obliquity of a solid Enceladus is 4.5 × 10 - 4 degrees and is negligibly affected by elastic deformations. The presence of an internal ocean decreases this upper limit by 13.1%, elasticity attenuating this decrease by only 0.5%. For larger satellites, such as Titan, elastic effects could be more significant because of their larger tidal deformations. As a consequence, it appears that it is easier to reconcile the theoretical estimates of Titan’s obliquity with the measured obliquity than reported in previous studies wherein the solid layers or the entire satellite were assumed to be rigid. Since the obliquity of Enceladus cannot reach Tyler’s requirement, obliquity tides are unlikely to be the source of the large heat flow of Enceladus. More likely, the geological activity at Enceladus’ south pole results from eccentricity tides. Even in the most favorable case, the upper limit for the obliquity of Enceladus corresponds to about two meters at most at the surface of Enceladus. This is well below the resolution of Cassini images. Control point calculations cannot be used to detect the obliquity of Enceladus, let alone to constrain its interior from an obliquity measurement.

Remaining error sources in the nutation at the submilliarc second level

Improvements in diurnal atmospheric and oceanic modeling are needed to calculate excitations of high-frequency Earth orientation parameters.The study of Earth rotation variability is useful not only for practical application involving areas of astronomy, satellite geodesy, and spatial navigation, but also for obtaining constraints on parameters of Earths internal structure. Modeling Earth rotation requires an interdisciplinary approach, involving subjects as diverse as magneto-hydrodynamics, oceanography, atmospheric science, and celestial mechanics. When studying Earth rotation, we are dealing with its non-constant rate and the position of the rotation axis, observed both from an inertial reference frame and from a frame rotating with the Earth.

Atmospheric excitation of the Earth's nutation: Comparison of different atmospheric models

A moon in synchronous rotation has longitudinal librations because of its nonspherical mass distribution and its elliptical orbit around the planet. We study the librations of Titan with periods of 14.7y and 29.5y and include deformation effects and the existence of a subsurface ocean. We take into account the fact that the orbit is not Keplerian and has other periodicities than the main period of orbital motion around Saturn due to perturbations by the Sun, other planets and moons. An orbital theory is used to compute the orbital perturbations due to these other bodies. We numerically evaluate the amplitude of the long-period librations for many interior structure models of Titan constrained by the mass, radius and gravity field. Measurements of the librations may give constraints on the interior structure of the icy satellites.