_{1}

^{*}

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 a
_{1},e
_{1}, a
_{2},e
_{2}, and a
_{T},e
_{T }of the initial orbit, final orbit and transferred orbit respectively. The first impulse Δv
_{1 }at the perigee induces a rotation of the orbital plane by
which will be minimized. The second impulse Δv
_{2 }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.

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. In 1925, Hohmann produced a numerical study showing that the optimum two-impulse transfer path between coplanar circular orbits is a semi-ellipse, tangential at its apsides to both circular orbits, with an impulse occurring at each apse. Hohmann transfer is generalized to the elliptic case (transfer between two coaxial elliptic orbits). A large number of works have been made to optimize non-coplanar transfer between circular or elliptic orbits having collinear major axes [

Any analysis of orbital maneuvers, i.e., the transfer of a satellite from one orbit to another by means of a change in velocity, begins with the energy as

where

where it is evident that

Note that (energy/satellite mass (is dependent only on a, an increases, energy increases. Orbital maneuvers are based on the principle that an orbit is uniquely determined by the position and velocity vector at any point [

The total velocity increment will obtain from the following relations, as seen in

where

Thus, the total increment of the velocity is

For simplicity let

Then (9) will be in the form

By partial differentiation of (11) with respect to

From which we can deduce that

where

After some reduction, we find that

Let

Then

After squaring and some reduction, we may write

Set

Then Equation (16) will be an algebraic equation of degree six in

There is no analytical solution for the Equation (17), but we solved it numerically, using code of MATHEMATICA V10, with no condition on the eccentricity or the semi-major axis of the initial, transformed, and the final orbits. The optimization problem was solved for two angles of rotation of the apsidal lines, using the computed values of the velocities at perigee and apogee, and the transfer angle

We give a complete analytical analysis and numerical solution of optimal two-

50˚ | 8.90678 | 0.0246816 | 0.016775 | 0.0414565 |

60˚ | 17.0631 | 0.0246913 | 0.0165648 | 0.0412561 |

70˚ | 22.4788 | 0.024701 | 0.0160106 | 0.0407116 |

80˚ | 26.8003 | 0.0247105 | 0.0152661 | 0.0399766 |

90˚ | 30.4023 | 0.0247196 | 0.0143589 | 0.0390785 |

100˚ | 33.3842 | 0.024728 | 0.0132935 | 0.0380215 |

110˚ | 35.7109 | 0.024735 | 0.0120627 | 0.0367977 |

120˚ | 37.2327 | 0.0247398 | 0.0106488 | 0.0353886 |

130˚ | 37.6535 | 0.0247411 | 0.0090206 | 0.0337617 |

140˚ | 36.4428 | 0.0247373 | 0.0071325 | 0.0318697 |

150˚ | 32.629 | 0.0247258 | 0.00493346 | 0.0296593 |

160˚ | 24.1032 | 0.0247044 | 0.00241235 | 0.0271167 |

impulse transfer with plane change. Our treatment is based on a changing of transfer velocities concept due to the changing in the energy. We obtained the total velocity increment ^{o},

Youssef, M.H.A. (2017) The Solution of Optimal Two-Impulse Transfer between Elliptical Orbits with Plane Change. International Journal of Astronomy and Astrophysics, 7, 125-132. https://doi.org/10.4236/ijaa.2017.73010