Constraining Mercury’s interior structure by using rotational parameters in a Bayesian framework

Introduction: Despite major advances from the MESSENGER mission, important questions remain regarding Mercury’s rotational state and internal structure. Independent analyses of Earth-based radar observations (Margot et al., 2007, 2012), radio tracking data (Mazarico et al., 2014; Verma and Margot, 2016; Genova et al., 2019; Konopliv et al., 2020), and laser altimetry (Stark et al., 2015; Bertone et al., 2021, Xiao et al., 2025) have yielded different estimates of Mercury's spin pole orientation. These orientation parameters can be combined with the gravity field under the assumption of the Cassini state to derive the polar moment of inertia C/MR² (Peale et al., 2002), widely used to constrain interior models (e.g. Genova et al., 2019; Goossens et al., 2022). However, measurements show that Mercury's spin pole deviates from the Cassini plane, suggesting that the exact Cassini state assumption is not fully consistent with observations. The observed offset thus represents an independent and meaningful constraint on Mercury's internal structure. MacPherson and Dumberry (2022) showed that the observed upper bound on the Cassini state offset implies a bulk mantle viscosity no smaller than approximately 10¹⁷ Pa s if a Maxwell rheology is assumed, a constraint that may be difficult to reconcile with the low mantle viscosities of 10¹⁶–10¹⁸ Pa s typically required to reproduce the observed k₂ values in the range 0.53–0.57 derived by Genova et al. (2019) and Konopliv et al. (2020). In this work, we incorporate the rotational model of MacPherson and Dumberry (2022) into a Markov Chain Monte Carlo (MCMC) framework to jointly use Mercury’s mass, 88-day libration amplitude, obliquity, Cassini state offset, and k₂ as observational constraints. This formulation replaces the moment of inertia of the planet, C/MR², derived under the Cassini state assumption, and the silicate-shell-to-planet moment of inertia, C_cr+m/C, with quantities directly tied to rotational and tidal measurements. We focus on the solutions of Genova et al. (2019) and Konopliv et al. (2020), which both provide k₂ estimates but differ significantly in their spin-pole orientation parameters, and assess the consistency between the geophysical observables and the corresponding interior models.

Methods: We develop a Bayesian inference framework for the interior structure inversion of Mercury based on a MCMC approach. The inversion jointly uses Mercury’s mass, obliquity, offset from the Cassini plane, libration amplitude, and tidal Love number k₂ as observational constraints for the planet. Mercury is modeled as a multi-layered body, with layer radii, densities, shear moduli, and viscosities treated as free parameters. These parameters are sampled iteratively using the Metropolis-Hastings algorithm, producing an ensemble of interior models consistent with the adopted prior distributions and observational uncertainties. After verifying the convergence of all chains, we compute the marginal posterior distributions of the model parameters, which constrain the range of internal structures consistent with the full set of geophysical observables.  

Mercury Interior Model Inversion: The internal structure of Mercury is constrained using geophysical parameters derived from MESSENGER mission data. In this work, we compare inferred interior structure models obtained from two sets of orientation parameters reflecting significantly different rotational states. The first set is from Genova et al. (2019), who estimated an obliquity of 1.968 ± 0.027 arcmin and a Cassini state offset of 0.004 ± 1.52 arcsec, placing Mercury nearly on the Cassini plane. The second set is from Konopliv et al. (2020), who report an obliquity of 2.04 ± 0.1 arcmin and an offset of 8.9 ± 4.49 arcsec, indicating a measurable departure from the exact Cassini state. Both studies also provide estimates of the tidal Love number k₂, obtaining values of 0.569 ± 0.025 and 0.53 ± 0.03, respectively. The 88-day libration amplitude is taken from Margot et al. (2012), who reported a forced libration of 38.5 ± 1.6 arcsec. Librations are modeled following the formulation of Van Hoolst et al. (2012), accounting for the effect of gravitational coupling with a solid inner core. Mercury's obliquity and spin-axis deviation from the Cassini plane are computed using the rotational model of MacPherson and Dumberry (2022). This model accounts for viscous and electromagnetic friction at the core-mantle boundary and inner core boundary, as well as deformations resulting from tidal forcing and differential rotation between internal layers.

Mercury is modeled as a four-layer body composed of a solid inner core, a fluid outer core, a mantle, and a crust. The radii of all layers are assumed as free parameters, while the mantle thickness is determined from the known planetary radius. The densities of all layers, the shear moduli, and viscosities of the solid regions are included as free parameters. Starting from a spherical description of the interior, the flattening of each internal interface is computed by imposing the observed surface flattening, gravitational equipotential surfaces at the inner- and outer-core boundaries, and consistency with the degree-2 gravity field coefficients C₂₀ and C₂₂.

Preliminary results suggest that the joint use of librations, obliquity, Cassini state offset, and k₂ as independent observational constraints reveals inconsistencies between the geophysical observables associated with the two rotational solutions. These inconsistencies are partly masked when the interior inversion relies only on the moment of inertia derivded under the Cassini state assumption.