Frozen orbits in the J2 + J3 problem

An analytical derivation of frozen orbit eccentricities and their location over the range of possible orbital inclinations in the J2 + J3 problem is presented. A gravitational field with only J2 and J3 terms is considered, because the equation defining frozen orbits in this field is an algebraic equation of the third order and an analytical formula for roots of this equation exists. An equation for the frozen orbit eccentricity is derived in a convenient form using only two independent parameters: the inclination and a parameter which is the product of the ratio of the radius of the central body to the orbital semimajor axis and the ratio of the J2 and J3 coefficients. The equation is solved, and, on the basis of its roots, frozen orbits in the J2 + J3 problem are classified.