Coplanar Maneuvers Transfer for Mission Design with Lowest ∆ʋ using Series Solution

Orbital maneuver transfer time is traditionally accomplished using direct numerical sampling to find the mission design with the lowest delta-ʋ requirements. The availability of explicit time series solutions to the Lambert orbit determination problem allows for the total delta-ʋ of a series of orbital maneuvers to be expressed as an algebraic function of only the individual transfer times. Series solution was applied for Hohmann transfer and Bi-elliptic transfer and comparing between Hohmann transfer and Bi-elliptic transfer for long distance. It has been concluded that Hohmann transfer is more appropriate when the ratio of radius of final orbit to initial orbit ( ) is less than 11.94. The purpose of this work is to minimize total full requirements, as well known that no refueling station in space, then using the computed ∆ʋ for determining the mass propellant consumed , at different specific impulse of the propellants, help us to carefully plane a mission to minimize the propellant mass carried on the rocket.

. The orbit transfer maneuvers considered accomplished by ideal impulsive velocity changes. It was assumed that the velocity required achieving certain mission objectives could be attained instantaneously. The concept of an impulsive velocity change can be exploited to provide an excellent rocket engine steering law which is applicable for a wide variety of orbit transfers [5]. One of the important characteristics of a space maneuver (and a space mission) is the change of characteristic velocity needed to realize the maneuver/mission, the so-called delta-ʋ (∆ʋ). Any rocket or spacecraft possesses its ideal velocity-the maximal change of speed it can provide to its payload using the fuel onboard. So, delta-ʋ of any maneuver (and any mission in total) is limited by the ideal velocity of the vehicle. As it has already been mentioned, characteristic velocity should be treated as exponential cost of the mission in terms of mass. To provide heavier payloads and more complicated mission, it is critical to use the limited reserve of ideal velocity as efficiently as possible, thus seeking for maneuvers with smaller delta-ʋ [6].
Orbital transfers are usually achieved using the propulsion system onboard the spacecraft.
Since the propellant mass on board is limited, it is very crucial for mission planning to estimate the propellant required for every transfer. The overall need for propulsion is usually expressed in terms of spacecraft total velocity change, or (delta-ʋ) budget as shown in Figure   (1). The propulsion was assumed is applied impulsively, i.e. the velocity change will be acquired instantaneously. This assumption is reasonably valid for high-thrust propulsion. magnitude for a series of orbital maneuvers to be written as a single algebraic expression, an explicit function of only the individual transfer times [9].
The purpose of this work is to minimize total full requirements, as well known that no refueling station in space, then using the computed ∆ʋ for determining the mass propellant consumed , at different specific impulse of the propellants, help us to carefully plane a mission to minimize the propellant mass carried on the rocket.

COPLANAR MANEUVERS
Coplanar maneuvers don't change the orbital plane, as the name implies, so the initial and final orbits lie in the same plane. These maneuvers can change the orbit's size and shape and the location of the line of apsides. Coplanar burns are either tangential or non-tangential. The burns allowed doing two types of coplanar changes: Hohmann transfers (two tangential burns) and general transfers (two non-tangential burns). Consider the simple tangential transfer of The terms are Pochhammer symbols, which are defined by To determine the semi-major axis, the reciprocal of the series may be found [10], then The coefficients can be treated as a vector, where And the elements of could be found from the following recursive expressions [2]: The total delta-ʋ that required for transformation is the sum of the vector differences. The sum of the magnitudes of the total delta-ʋ is then given by Velocity and there component is given by The goal is how to minimize the to obtain the transfer orbit with less fuel consumed as well as is the time of transfer is not very long.
Let is a function of the velocity vector ⃑ , therefore ⋯ ⋯ ⋯

APPLICATION AND DISCUSSION
For coplanar transfer the vectors input data are ( = 8839.683 km and = 18689.09 km) and using the Lambert's theorem at a given flight time the velocity computed and given in Table (1). There is a unique value for semi-major axis associated with the arc of conic section, express the major axis in terms of transfer time to solve orbital cases, Table (2) show the computed velocity component and the total minimum ∆ʋ by using the analytical optimization with assistant of Matlab program. Table (3) show the computed ( ⁄ ) at different .  The purpose of this work is to minimize total full requirements, as well known that no refueling station in space, then using the computed ∆ʋ in Table (3) for determining the mass propellant consumed , at different specific impulse of the propellants, help us to carefully plane a mission to minimize the propellant mass carried on the rocket,   In Tables (7,8,9) list position vector sin two different or bits with ratio much than , the change in velocity required by Bi-elliptic transfer is smaller than that computed in Hohmann transfer also, therefore from the above concludes that for ratio less than the Hohmann transfer is Appropriate, while for greater than the Bi-elliptic saves with an extreme in time.  The results in Table (7, 8, and 9) show that the Bi-elliptic transfer requires much longer transfer time computed to the Hohmann transfer. However Bi-elliptic is more efficient for long distance the change in velocity for long distance orbit transfer at different (the ratio of apogee radius of transfer orbit to initial orbit) in Bi-elliptic at the same value of the ration .
The results agree with concept that when is increase the ∆ʋ decrease as given in the reference [1].

CONCLUSION
With a complete set of series solutions available for every case of Lambert's Theorem, it is possible to apply it for coplanar orbit transfer and gives good results in magnitude of change of velocity. Using analytical methods for multiple-impulse missions to minimize total fuel requirements. Computed change in velocity for different types of maneuvers (Hohmann and Bi-elliptic). The results show that the Hohmann transfer is saves fuel by reduce the change in velocity and extreme in time.