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