The effect of post-Newtonian spin precessions on the evolution of exomoons' obliquity

Putative natural massive satellites (exomoons) has gained increasing attention, where they orbit Jupiter-like planets within the habitable zone of their host main sequence star. An exomoon is expected to move within the equatorial plane of its host planet, with its spin ${\boldsymbol S}_\mathrm{s}$ aligned with its orbital angular momentum $\boldsymbol L$ which, in turn, is parallel to the planetary spin ${\boldsymbol S}_\mathrm{p}$. If, in particular, the common tilt of such angular momenta to the satellite-planet ecliptic plane, assumed fixed, has certain values, the latitudinal irradiation experienced on the exomoon from the star may allow it to sustain life as we know it, at least for certain orbital configurations. An Earth--analog (similar in mass, \textcolor{black}{radius, oblateness} and obliquity) is considered, which orbits within $5-10$ planetary radii $R_\mathrm{p}$ from its Jupiter-like host planet. The de Sitter and Lense--Thirring spin precessions due to the general relativistic post-Newtonian (pN) field of the host planet have an impact on an exomoon's habitability for a variety of different initial spin-orbit configurations. Here, I show it by identifying long--term variations in the satellite's obliquity $\varepsilon_\mathrm{s}$, where variations can be $\lesssim 10^\circ-100^\circ$, depending on the initial spin-orbit configuration, with a timescale of $\simeq 0.1-1$ million years. Also the satellite's quadrupole mass moment $J_2^\mathrm{s}$ induces obliquity variations which are faster than the pN ones, but do not cancel them.