Obliquity Variations of Habitable Zone Planets Kepler-62f and Kepler-186f

Obliquity variability could play an important role in the climate and habitability of a planet. Orbital modulations caused by planetary companions and the planet's spin axis precession due to the torque from the host star may lead to resonant interactions and cause large-amplitude obliquity variability. Here we consider the spin axis dynamics of Kepler-62f and Kepler-186f, both of which reside in the habitable zone around their host stars. Using {\emph{N}}-body simulations and secular numerical integrations, we describe their obliquity evolution for particular realizations of the planetary systems. We then use a generalized analytic framework to characterize regions in parameter space where the obliquity is variable with large amplitude. We find that the locations of variability are fine-tuned over the planetary properties and system architecture in the lower-obliquity regimes ($\lesssim 40^\circ$). As an example, assuming a rotation period of 24 hr, the obliquities of both Kepler-62f and Kepler-186f are stable below $\sim 40^\circ$, whereas the high-obliquity regions ($60^\circ - 90^\circ$) allow moderate variabilities. However, for some other rotation periods of Kepler-62f or Kepler-186f, the lower-obliquity regions could become more variable owing to resonant interactions. Even small deviations from coplanarity (e.g. mutual inclinations $\sim 3^\circ$) could stir peak-to-peak obliquity variations up to $\sim 20^\circ$. Undetected planetary companions and/or the existence of a satellite could also destabilize the low-obliquity regions. In all cases, the high-obliquity region allows for moderate variations, and all obliquities corresponding to retrograde motion (i.e. $> 90^\circ$) are stable.