An analytic solution to the Kozai–Lidov evolution equations

Abstract

ABSTRACT A test particle in a non-coplanar 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 non-linear 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 non-zero 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 π/3, an error of 1 per cent at maximum eccentricity occurs for initial eccentricities of about 0.1.