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 .