Second-order solution of artificial satellite theory without air drag

Abstract

Second-order periodic perturbations with third-order secular perturbations in motions of artificial satellites moving in the gravitational field of the earth without air drag are derived by von Zeipel's method (1916). The potential of the earth is developed into a series of zonal spherical harmonics by assuming that / 2 is a small quantity of the first order, / 3 and J± are of the second order, and J-n , / 6 , / 7 , and / 8 are of the third order. The final expressions of the short-periodic perturbations are given in the radius, the argument of latitude, the inclination, and the longitude of the ascending node. Arguments appearing in the first-order expressions are transformed from / and g to V and g' to simplify the second-order expressions. The long-periodic perturbations are given in Kepler's elements with an argument g".