The optimizing total velocity increment Δv needed for orbital maneuver between two elliptic orbits with plane change is investigated. Two-impulse orbital transfer is used based on a changing of transfer velocities concept due to the changing in the energy. The transferring has been made between two elliptic orbits having a common centre of attraction with changing in their planes in standard Hohmann transfer with the terminal orbit which is elliptic orbit and not circular. We develop a treatment based on the elements of elliptic orbits a1,e1, a2,e2, and？aT,eT of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv1 at the perigee induces a rotation of the orbital plane by ？which will be minimized. The second impulse Δv2 at apogee is induced an angle ？to product the final elliptic orbit. The total plane change required . We calculate the total impulse Δv and minimize by optimizing angle of plane’s variation . We obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits ？and . The solution obtained numerically, using programming code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. We find that there are constrains on the transfer angles and α. For α it must be between 40° and 160°, and there is no solution if α is less than 40° and bigger than 160° and ？takes the values less than 40°. The minimum total velocity increments obtained at the value of ？less than 25° and& alpha; equal to 160°. This is an interesting result in orbital transfer problem in which the change of orbital plane is necessary for the transferring.
Sun, F.T. (1969) Analytic Solution for Optimal Two-Impulse 180° Transfer between Noncoplanar Orbits and the Optimal Orientation of the Transfer Plane. AIAA Journal, 7, 1898-1904. https://doi.org/10.2514/3.5478