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L17 Orbit Transfers and Interplanetary Trajectories

L17 Orbit Transfers and Interplanetary Trajectories - S...

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S. Widnall, J. Peraire 16.07 Dynamics Fall 2008 Version 2.0 Lecture L17 - Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to construct an interplanetary trajectory. One of the assumptions that we shall make is that the velocity changes of the spacecraft, due to the propulsive e ff ects, occur instantaneously. Although it obviously takes some time for the spacecraft to accelerate to the velocity of the new orbit, this assumption is reasonable when the burn time of the rocket is much smaller than the period of the orbit. In such cases, the v required to do the maneuver is simply the di ff erence between the velocity of the final orbit minus the velocity of the initial orbit. When the initial and final orbits intersect, the transfer can be accomplished with a single impulse. For more general cases, multiple impulses and intermediate transfer orbits may be required. Given initial and final orbits, the objective is generally to perform the transfer with a minimum v . In some situations, however, the time needed to complete the transfer may also be an important consideration. Most orbit transfers will require a change in the orbit’s total specific energy, E . Let us consider the change in total energy obtained by an instantaneous impulse v . If v i is the initial velocity, the final velocity, v f , will simply be, v f = v i + v . If we now look at the magnitude of these vectors, we have, v f 2 = v i 2 + v 2 + 2 v i v cos β , where β is the angle between v i and v . The energy change will be 1 E = v 2 + v i v cos β . 2 From this expression, we conclude that, for a given v , the change in energy will be largest when: - v i and v are co-linear ( β = 0), and, - v i is maximum. 1
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For example, to transfer a satellite on an elliptical orbit to an escape trajectory, the most energy e cient impulse would be co-linear with the velocity and applied at the instant when the satellite is at the elliptical orbit’s perigee, since at that point, the velocity is maximum. Of course, for many required maneuvers, the applied impulses are such that they cannot satisfy one or both of the above conditions. For instance, firing at the perigee in the previous example may cause the satellite to escape in a particular direction which may not be the required one. Hohmann Transfer A Hohmann Transfer is a two-impulse elliptical transfer between two co-planar circular orbits. The transfer itself consists of an elliptical orbit with a perigee at the inner orbit and an apogee at the outer orbit. The fundamental assumption behind the Hohmann transfer, is that there is only one body which exerts a gravitational force on the body of interest, such as a satellite. This is a good model for transferring an earth-based satellite from a low orbit to say a geosynchronous orbit. Inherent in the model is that there is no additional body sharing the orbit which could induce a gravitational attraction on the body of interest.
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