The Solution of Optimal Two-Impulse Transfer between Elliptical Orbits with Plane Change

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？a_T,e_T 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.