On non-coplanar Hohmann transfer using angles as parameters

We study a more complex case of Hohmann orbital transfer of a satellite by considering non-coplanar and elliptical orbits, instead of planar and circular orbits. We use as parameter the angle between the initial and transference planes that minimizes the energy, and therefore the fuel of a satellite, through the application of two non-tangential impulses for all possible cases. We found an analytical expression that minimizes the energy for each configuration. Some reasonable physical constraints are used: we apply impulses at perigee or apogee of the orbit, we consider the duration of the impulse to be short compared to the duration of the trip, we take the nodal line of three orbits to be coincident and the three semimajor axes to lie in the same plane. We study the only four possible cases but assuming non-coplanar elliptic orbits. In addition, we validate our method through a numerical solution obtained by using some of the actual orbital elements of Sputnik I and Vanguard I satellites. For these orbits, we found that the most fuel-efficient transfer is obtained by applying the initial impulse at apocenter and keeping the transfer orbit aligned with the initial orbit.