Analytical solution of the Kozai resonance and its application

When Kozai (1962) studied the secular resonance of asteroids, he found the so-called Kozai resonance and expressed the analytical solution with the use of Weierstrass p. Here we discuss the case where the disturber is outside a disturbed body and give the analytical solution of the eccentricity, the inclination and the argument of pericenter with the use of the Jacobi elliptic functions, which are more familiar than the Weierstrass p. Then we derive the Fourier expansion of the longitude of node and the mean anomaly. The analytical expressions obtained here can be used for any value of the eccentricity and the inclination. Finally we applied these analytical expressions to several dynamical systems - Nereid, that is a highly eccentric satellite of Neptune, and newly discovered retrograde satellites of Uranus.