An Analytic Solution to the Kozai-Lidov Evolution Equations

TL;DR

This paper derives an exact analytic solution for the Kozai-Lidov oscillations of a test particle in a binary system within the quadrupole approximation, simplifying the understanding of orbital evolution.

Contribution

It provides the first exact analytic solution for zero initial eccentricity in Kozai-Lidov evolution equations, applicable within the quadrupole approximation.

Findings

Solution involves simple trigonometric and hyperbolic functions.

Accurately describes orbits with small initial eccentricity.

Error remains below 1% for certain initial conditions.

Abstract

A test particle in a noncoplanar orbit about a member of a binary system can undergo Kozai-Lidov oscillations in which tilt and eccentricity are exchanged. An initially circular highly inclined particle orbit can reach high eccentricity. We consider the nonlinear secular evolution equations previously obtained in the quadrupole approximation. For the important case that the initial eccentricity of the particle orbit is zero, we derive an analytic solution for the particle orbital elements as a function of time that is exact within the quadrupole approximation. The solution involves only simple trigonometric and hyperbolic functions. It simplifies in the case that the initial particle orbit is close to being perpendicular to the binary orbital plane. The solution also provides an accurate description of particle orbits with nonzero but sufficiently small initial eccentricity. It is accurate over a range of initial eccentricity that broadens at higher initial inclinations. In the case of an initial inclination of pi/3, an error of 1% at maximum eccentricity occurs for initial eccentricities of about 0.1.