Abstract
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.
Highlights
The problem of the optimal impulsive transfer between two orbits is almost seventy years old, but the question, how many impulses are still open despite of the theories and a lot of numerical works developed in this field
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 obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits θ1 and θ2= α −θ1
Summary
The problem of the optimal impulsive transfer between two orbits is almost seventy years old, but the question, how many impulses are still open despite of the theories and a lot of numerical works developed in this field. A polynomial equation of six degrees on the generalized Hohmann transfer with plane change using energy concepts is obtained without analytical solution [6]. The necessary condition for optimality is reduced to a polynomial equation of the eighth degrees on the semi-latus rectum and with the fixed transfer angle, for which no solution has been found in two-impulse transfer between two elliptic coplanar orbits [8]. We calculate the total impulse ∆v and minimize by optimizing angle of plane’s variation θ1 , we obtain a polynomial equation of six degrees on the two transfer angles between neither two elliptic orbits θ1 and θ2 with any restrictions on their eccentricities and semi-major axis, nor any restrictions on the terminal distances and the initial and final orbital velocities. The total velocity impulse is minimized by optimizing angle of plane’s variation numerically under some constrains of the transfer angles
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