Icarus 181 (2006) 327–337 www.elsevier.com/locate/icarus Evolution of Mercury’s obliquity Marie Yseboodt ∗, Jean-Luc Margot Department of Astronomy, Cornell University, Ithaca, NY 14853, USA Received 24 June 2005; revised 3 November 2005 Available online 17 February 2006 Abstract Mercury has a near-zero obliquity, i.e. its spin axis is nearly perpendicular to its orbital plane. The value of the obliquity must be known precisely in order to constrain the size of the planet’s core within the framework suggested by Peale [Peale, S.J., 1976. Nature 262, 765–766]. Rambaux and Bois [Rambaux, N., Bois, E., 2004. Astron. Astrophys. 413, 381–393] have suggested that Mercury’s obliquity varies on thousand-year timescales due to planetary perturbations, potentially ruining the feasibility of Peale’s experiment. We use a Hamiltonian approach (free of energy dissipation) to study the spin–orbit evolution of Mercury subject to secular planetary perturbations. We can reproduce an obliquity evolution similar to that of Rambaux and Bois [Rambaux, N., Bois, E., 2004. Astron. Astrophys. 413, 381–393] if we integrate the system with a set of initial conditions that differs from the Cassini state. However the thousand-year oscillations in the obliquity disappear if we use initial conditions corresponding to the equilibrium position of the Cassini state. This result indicates that planetary perturbations do not force short-period, large amplitude oscillations in the obliquity of Mercury. In the absence of excitation processes on short timescales, Mercury’s obliquity will remain quasi-constant, suggesting that one of the important conditions for the success of Peale’s experiment is realized. We show that interpretation of data obtained in support of this experiment will require a precise knowledge of the spin–orbit configuration, and we provide estimates for two of the critical parameters, the instantaneous Laplace plane orientation and the orbital precession rate from numerical fits to ephemeris data. Finally we provide geometrical relationships and a scheme for identifying the correct initial conditions required in numerical integrations involving a Cassini state configuration subject to planetary perturbations. 2006 Elsevier Inc. All rights reserved. Keywords: Interiors; Mercury; Planetary dynamics; Rotational dynamics; Resonances 1. Introduction Dyce, 1965) revealed Mercury’s unusual 3:2 spin–orbit reso- nance: the orbital period (∼88 days) is exactly 3/2ofthespin Mercury’s orbit is not fixed in space but precesses because of period (∼59 days). torques exerted by planets exterior to its orbit. To first order, if Provided that the planet is in a Cassini state, Peale (1976) has we consider perturbations of the planets on Mercury but neglect shown that the knowledge of the second degree coefficients of mutual interactions between planets, Mercury’s orbit precesses the gravity field, the 88-day libration amplitude, and the obliq- with a period of about 235 kyr and a constant inclination with uity can be used to determine the state and size of the Hermean respect to a plane called the Laplace plane. Mercury’s spin vec- core. While the knowledge of the 88-day libration amplitude tor is also expected to precess if the planet is a Cassini state, an and C22 gravitational harmonic are sufficient to distinguish a evolved rotational state where the spin axis, orbit normal, and molten core from a solid core (Margot et al., in preparation), the normal to the Laplace plane are coplanar while the obliquity re- combination of all four quantities above is needed to evaluate mains constant (Colombo, 1966; Peale, 1969, 1988). In order to the moment of inertia C and the ratio Cm/C between the mo- maintain coplanarity in the Cassini state, the spin axis precesses ment of inertia of the mantle and the moment of inertia of the at the same rate as the orbital plane (see, e.g., Ward, 1975; entire planet. This ratio with assumptions on mantle and core Gladman et al., 1996). Radar measurements (Pettengill and densities can be used to constrain the size of the core. Therefore a good description of the obliquity behavior is essential to infer * Corresponding author. the radius of Mercury’s core, which is itself critical to further E-mail address: [email protected] (M. Yseboodt). our understanding of terrestrial planet formation and evolution. 0019-1035/$ – see front matter 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2005.11.024 328 M. Yseboodt, J.-L. Margot / Icarus 181 (2006) 327–337 One of the goals of both the MESSENGER (Solomon et including one simulation that reproduces the oscillations in the al., 2001) and Bepi–Colombo missions (Milani et al., 2001) obliquity seen by Rambaux and Bois (2004). We find that inte- is to determine these quantities (gravity field, librations, obliq- grations started in the Cassini state do not exhibit oscillations. uity) for Mercury. Earth-based radar data can also be used to In Section 6, we present a practical way to numerically find determine the libration amplitude and the instantaneous spin the position of the Cassini state and the associated obliquity. orientation. The equilibrium obliquity—important for the interpretation of The idealized situation in Peale’s experiment can be compli- future Mercury geodesy data—depends on the values of the mo- cated by the presence of free modes of rotation. A free mode ments of inertia and other parameters, which we investigate in has an arbitrary phase and amplitude but its frequency is fixed. Section 7. Possible free modes are a libration in longitude which produces a variation of the rotation angle about the spin axis with a pe- 2. Method riod of order 15 years (Peale, 1974) and a precession which moves the spin axis around the Cassini state with a period of One way to study the spin–orbit motion of Mercury is to order 1000 years (Colombo, 1966; Peale, 1974). A free libra- numerically integrate the motions of the planets of the Solar tion in longitude would not affect the obliquity nor the ability System with a relativistic integrator (e.g., Rambaux and Bois, to distinguish a molten from a solid core. But a free preces- 2004). Another way is to use a simplified analytical approach. sion would change the orientation of the spin axis in space and In a masterful paper, D’Hoedt and Lemaitre (2004) obtain the therefore the obliquity value, potentially ruining the ability to equilibrium solutions and the frequencies of the spin-axis mo- determine the radius of the core in Peale’s experiment. tion by using Hamilton’s equations. In order to maintain a Peale (2005) examined the free rotational motions of Mer- tractable Hamiltonian with two degrees of freedom, those au- cury and their damping by considering tidal friction and dis- thors make the following simplifications: principal axis rota- sipation at the core–mantle interface. He derived damping tion, a description of the gravity field limited to second degree timescales for these modes of about 105 years, much shorter and order terms, no planetary perturbations, no tides nor damp- than the age of the Solar System. If the free modes are detected ing, and all the perturbations with period equal or smaller to the in the spin state measurements, it will indicate an active or re- revolution period (88 days) are neglected. cent excitation mechanism, as the amplitude of the free modes For the convenience of the reader, we briefly describe the ref- should decay on ∼105 year timescales. erence frames and the Andoyer and Delaunay variables that are Rambaux and Bois (2004) have suggested that plane- used in the development of the Hamiltonian. Additional details tary perturbations cause obliquity variations on thousand-year can be found in D’Hoedt and Lemaitre (2004). The reference 5 timescales, much less than the 10 year damping timescales. frame X0, Y0, Z0 is based on the ecliptic plane at epoch J2000. Their numerical integrations show that Mercury’s obliquity is This choice of reference frame allows us to easily add planetary not constant with time. The authors find variations of a few arc- perturbations. The frame tied to the orbital plane is denoted by minutes around a mean value of about 1.6 arcmin at a proper X1, Y1, Z1, the frame tied to the spin orientation is denoted by frequency of 1066 years, corresponding to the free precession X2, Y2, Z2, and the frame tied to the principal axes of inertia is period for the values of moments of inertia they chose. For a denoted by X3, Y3, Z3. rigid body, the free precession frequency νP is given to first The Andoyer variables (l,g,h,L,G,H) (Deprit, 1967) de- order by the equation scribe the rotation of Mercury. Three lowercase symbols rep- C − (A + B)/2 resent angles and three uppercase symbols represent conju- νP = n (1) gated momenta: g + l is the angle between X and the axis C 2 of minimum inertia X , h is the angle between X and X which follows from Euler’s equations for rigid-body rotation. 3 0 2 (the longitude of the ascending node of the spin in the ecliptic In this equation, n is the mean motion, A, B and C the mo- frame), G is the norm of the spin angular momentum of Mer- ments of inertia. An expression for the precession frequency cury, H = G cos K is the projection of the angular momentum that takes the triaxial shape and the resonant rotation of Mer- vector on the inertial axis Z where we define K as the angle cury into account is given by Peale (2005). The free libration 0 between Z and Z (the inclination of the spin axis with respect period similarly depends on the moments of inertia.