Impulsive orbit correction using second-order Gauss’s variational equations

Based on Gauss’s variational equations (GVEs), the impulsive orbital-element corrections are investigated in orbit transfer problems. Both single impulse and multiple impulses are considered for the first-order and second-order GVEs. For the single impulse, a nonlinear least-squares iteration method for the minimum orbit error is provided to simultaneously solve for the impulse vector and the impulse position. For multiple impulses, a minimum-norm method for the energy cost is proposed to solve the three-impulse and two-impulse corrections for the in-plane and out-of-plane orbital elements, respectively. The impulse positions are analytically derived, and the impulse vectors are obtained by the minimum-norm method. Numerical examples are provided to verify the proposed single-impulse least-squares and multiple-impulse minimum-norm methods. The results show that the nonlinear solutions using the second-order GVEs are more accurate than those using the first-order GVEs.