Nodal Precession in Closely Spaced Planet Pairs

Planet-planet perturbations can cause planets' orbital elements to change on secular timescales. Previous work has evaluated the nodal precession rate for planets in the limit of low $\alpha$ (semi-major axis ratio, 0$<$$\alpha$$\leq$1). Our simulations show that systems at high $\alpha$ (or low period ratio), similar to multiplanet systems found in the Kepler survey, have a nodal precession rate that is more strongly dependent on eccentricity and inclination. We present a complete expansion of the nodal precession rate to fourth order in the disturbing function and show that this analytical solution much better describes the simulated N-body behavior of high-$\alpha$ planet pairs; at $\alpha\approx$ 0.5, the fourth-order solution on average reduces the median analytical error by a factor of 7.5 from linear theory and 6.2 from a second-order expansion. We set limits on eccentricity and inclination where the theory is precisely validated by N-body integrations, which can be useful in future secular treatments of planetary systems.