Velocity corrections in generalized coplanar coaxial Hohmann and bi-elliptic impulsive transfer orbits

We shall investigate the problem of the differential corrections of the orbital elements a,ea,e of the final elliptic orbit, in order that we might obtain the required very precise one. This will be achieved by the application of differential motor thrust impulses ΔνAΔνA, ΔνBΔνB at peri-apse A and apo-apse B, to induce correctional ΔaΔa, ΔeΔe increments. We apply differential thrusts at peri-apse and apo-apse of the generalized Hohmann and bi-elliptic transfer impulsive systems of orbits. Our purpose is to find the differential corrections in major axes and eccentricities of the two considered transfer systems. Thus we can obtain a very precise final elliptic orbit, throughout differential velocity increments perpendicular to the coaxial major axes and tangential to peri-apse and apo-apse of the elliptical terminal orbits. We find out the four relationships between Δa1Δa1, ΔaTΔaT, Δe1Δe1, ΔeTΔeT and ΔνAΔνA, ΔνBΔνB the velocity increments at the points A and B , for the four cases of the Hohmann type trajectories . We notice that Δa1Δa1 and Δe1Δe1 are related to ΔνAΔνA whilst ΔaTΔaT and ΔeTΔeT are combined with , ΔνBΔνB. This occurs for the four Hohmann configurations. Referring to the bi-elliptic type of impulsive transfer we have: ΔνA=f(Δa1)ΔνA=f(Δa1); ΔνC=f(ΔνA)=ψ(Δa1)ΔνC=f(ΔνA)=ψ(Δa1); ΔνB=f(ΔνA)=ψ(Δa1)ΔνB=f(ΔνA)=ψ(Δa1), ΔaT=f(a1,a2,e1,e2)ΔaT=f(a1,a2,e1,e2) for all four feasible cases.ΔaT′=f(ΔνA)=ψ(Δa1)ΔaT′=f(ΔνA)=ψ(Δa1) for all four bi-elliptic cases. In addition for all feasible bi-elliptic cases we have Δe1=f(ΔνA)Δe1=f(ΔνA).