Relative Angle at Launch

Here's an example of how to determine the relative angle at launch. For the case of the shuttle launching a GPS satellite, the inner orbit (375 km above the Earth) has a period of about 1.5 hours and the outer orbit (20,200 km above the Earth) has a period of about 12 hours.

For this example, we'll consider a slightly different case.

Assume that the period of the outer orbit (the orbit the satellite is to eventually intersect) is 510 minutes (8.5 hours) and the period of the inner orbit (the shuttle orbit from which the satellite is launched) is 90 minutes. The transfer (elliptical) orbit would have a period of roughly the average of the two, or about 300 minutes (6 hours). This means that it will take our satellite 6/2 = 3 hours to go from the inner orbit (perigee) to the outer orbit (apogee). If our satellite is to reach the target location in the outer orbit, it must take 3 hours to reach that location. For it to reach the target location (let's say on the far left of our illustration), we must launch our satellite when it is at the far right of its orbit in the illustration.

Now we only have to determine where the target spot would be at launch. To see where it would be, we just need to back it up 3 hours from its far left location. A satellite in the outer orbit would travel one complete orbit (circle) in 8.5 hours. During the 3 hours that it takes our satellite to get up to the outer orbit, it would have traveled 3/8.5 of an orbit if it were already in the outer orbit. This fraction, 3/8.5, is .353, which means that the satellite would have only traveled 35.3% of its whole orbit during the time that it took to go from its boost point (in the lower orbit) to its target point (in the outer orbit). If we take 35.3% of a complete orbit (360 degrees), we see that our target has traveled only .353 x 360 = 127 degrees. Our satellite actually traveled 180 degrees (or half of an orbit) during this time, so we see that the difference in angle between the location of the target and the satellite at launch time must be 53 degrees, for this set of inner and outer orbits.

Note that you don't need to have the satellite at the far right before you can launch it. You only need it to be located 53 degrees behind the target when you launch!