Mercury’s resonant rotation from secular orbital elements

We used recently produced Solar System ephemerides, which incorporate 2 years of ranging observations to the MESSENGER spacecraft, to extract the secular orbital elements for Mercury and associated uncertainties. As Mercury is in a stable 3:2 spin-orbit resonance, these values constitute an important reference for the planet’s measured rotational parameters, which in turn strongly bear on physical interpretation of Mercury’s interior structure. In particular, we derive a mean orbital period of (87.96934962±0.00000037)days\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(87.96934962 \pm 0.00000037)\,\hbox {days}$$\end{document} and (assuming a perfect resonance) a spin rate of (6.138506839±0.000000028)∘/day\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(6.138506839\pm 0.000000028){}^{\circ }/\hbox {day}$$\end{document}. The difference between this rotation rate and the currently adopted rotation rate (Archinal et al. in Celest Mech Dyn Astron 109(2):101–135, 2011. doi:10.1007/s10569-010-9320-4), corresponds to a longitudinal displacement of approx. 67 m per year at the equator. Moreover, we present a basic approach for the calculation of the orientation of the instantaneous Laplace and Cassini planes of Mercury. The analysis allows us to assess the uncertainties in physical parameters of the planet, when derived from observations of Mercury’s rotation.