Rotation and libration angles of Mercury: definitions and models

Librations describe the oscillatory motion of a body’s orientation, particularly relevant for tidally locked objects such as Mercury, which is in a 3:2 spin-orbit resonance. However the definition of libration is not unique. We compare several libration definitions for Mercury, highlighting their implications for observed longitudinal libration amplitudes.
In Yseboodt et al. (2010, 2013), the rotation angle ϕ(t) is defined from a fixed reference point (the intersection of two inertial planes). Writing the equation of motion in an inertial frame is crucial to avoid spurious terms that arise in rotating frames. The libration angle is defined as a small deviation from the orbital forcing term 3/2 M(t)+Ω(t)+ω(t).
Alternatively, another libration angle can be defined as a small increment from a uniform rotation. These two approaches yield different dynamical equations and expressions for libration amplitudes. We compute the transformations between the two formulations and analyze their behavior in regimes where the forcing frequency is much smaller or larger than the natural (free) frequencies, showing how the sensitivity of the amplitude is affected by interior structure parameters.
We also provide a frequency decomposition of the orbital perturbations acting on each part of the equations of motion, along with examples of libration models.
A third rotation angle, W(t), is defined from the intersection of Mercury’s equator and the ICRF equator (as in the IAU reports, e.g., Archinal et al. 2018). We link it with the previously defined libration angles and with the orbit orientation in the ICRF and the expressions in Stark et al. (2015)..
The trends of ϕ(t) and W(t), corresponding to mean rotation rates, differ slightly due to the precession of Mercury’s orbit. We quantify these differences numerically.
This study is important for an accurate interpretation of spacecraft and radar observations of Mercury’s rotation (Xiao et al., this meeting, and Rivoldini et al, this meeting).