Comparison between Laplace-Lagrange Secular Theory and Numerical Simulation

Abstract

The large increase in exoplanet discoveries in the last two decades showed a variety of systems whose stability is not clear. In this work we chose the $\upsilon$ Andromedae system as the basis of our studies in dynamical stability. This system has a range of possible masses, as a result of detection by radial velocity method, so we adopted a range of masses for the planets $c$ and $d$ and applied the secular theory. We also performed a numerical integration of the 3-body problem for the system over a time span of 30 thousand years. The results exposed similarities between the secular perturbation theory and the numerical integration, as well as the limits where the secular theory did not present good results. The analysis of the results provided hints for the maximum values of masses and eccentricities for stable planetary systems similar to $\upsilon$ Andromedae.