The streamlined and complete set of the nonsingular J2-perturbed dynamic and adjoint equations for trajectory optimization in terms of eccentric longitude

Abstract

The complete set of the dynamic and multipliers differential equations for a nonsingular set of equinoctial elements where the eccentric longitude stands for the sixth or fast orbital element are derived in a more straightforward and streamlined manner as compared to a previous derivation also presented by this author. The mathematics are validated through comparison with the original formulation as well as with two other formulations using the mean longitude and the true longitude elements as the respective sixth element of the equinoctial set. Furthermore, this new formulation is used to derive the costate or adjoint differential equations by fully accounting for the secular first-order perturbative effect of the second zonal harmonic J2, and the complete set of the perturbed dynamic and adjoint system of equations are also validated by direct comparison with the two previously derived formulations using the mean and true longitudes respectively. The present formulation as well as the one using the true longitude as the sixth orbital element remove the need to solve Kepler’s transcendental equation at each integration step, a need that is inevitable when the mean longitude formulation is used, because in the latter case the right-hand sides of the various differential equations cannot be written directly in terms of the mean longitude. The inclusion of the J3 and J4 terms can be similarly accounted for with both the eccentric and true longitude sets and mutually validated also. This particular formulation has been adopted by several aerospace contractors in the United States to build specialized flight guidance software to steer payloads released in high energy orbits to their final destinations using highly efficient low-thrust propulsion.