An analytic solution to the Kozai–Lidov evolution equations

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