Mechanics of trajectory optimization using nonsingular variational equations in polar coordinates

The complete set of the state and adjoint differential equations for the nonsingular equinoctial orbit elements expressed in polar coordinates and written in terms of the true longitude is presented. Previous formulations adopted the equinoctial frame as the orbital frame for component resolution of the various perturbation accelerations. The consideration of the position dependency of the J(2) acceleration in the context of precision integrated orbital transfer trajectories led us to adopt the more convenient polar frame coupled with the use of the true longitude as the accessory variable needed in the description of the variational equations, inasmuch as it is more difficult to generate the contribution of the J(2) perturbation to the adjoint equations if the variational equations are left in terms of the eccentric longitude.