Two satellites are orbiting the earth in circular orbitsnot

Two satellites are orbiting the earth in circular orbits-not at the same altitude or inclination. What sequence of orbit changes and plane changes would most efficiently place the lower satellite in the same orbit as that of the higher one? Assume only one maneuver can be performed at a time, i.e., a plane change or an orbit transfer.

Solution

TO CHANGE THE ORBIT OF A SATELLITE, SEVERAL APPROACHES CAN BE MADE. BUT THE OPTIMUM APPROACH IN THE REAL WORLD USUALLY CONSISTS OF SPLITTING THE PLANE CHANGE OPTIMALLY BETWEEN THE TWO BURNS IN A CLASSICAL HOHMANN TRANSFER SEQUENCE. IN THIS CASE , THE PERSON IN THE SATELLITE WOULD MAKE A SMALL PORTION OF THE REQUIRED PLANE CHANGE IN CONJUNCTION WITH THE SECOND BURN AT THE APOGEE OF THE TRANSFER ELLIPSE.

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. Thus, as we shall discuss, the Hohmann transfer is a good model for the ”outer” trajectory of an earth-Mars transfer, but we must pay some attention to ”escaping” the earth’s gravitational field before we’re on our way.

It turns out that this transfer is usually optimal, as it requires the minimum vT = |v|+|v| to perform a transfer between two circular orbits.