## 1. Introduction

[2] Bulk mass density *ρ* = *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

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 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 determination 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* + *B*)/2]/*C*, and of the second-degree harmonic coefficient of the gravity field*C*_{20} = − [*C* − (*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*)/(4*MR*^{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 *θ* (tilt of the spin axis with respect to the orbit normal) and amplitude of forced libration in longitude *γ*(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 observational evidence that the core of Mercury is molten, and that Mercury occupies Cassini state 1 [*Margot et al.*, 2007]. Here we extend the baseline of observations to a total of 35 epochs spanning 2002–2012, and we provide improved estimates of *θ*, *γ*, *C*, and *C*_{m}.