Maneuver Your Satellites - GEO
We'll assume that you can launch directly into the correct plane of the mission orbit, but that your launch vehicle will "park" your satellite in an orbit with an altitude of 500 km. For our purposes here, lets use 35,780 km for the mission orbit altitude, just so we're all working the same problem. (Radius of the earth is 6378 km.)
You'll need to "rendezvous" your satellite with one of the slots in your constellation. This is a very common form of the rendezvous problem. You aren't really going to rendezvous with another spacecraft, but you do need your satellite in a particular set of COEs relative to the other satellites in your constellation. The Hohmann Transfer is the most efficient means of going from one circular orbit to another circular orbit in the same plane.
[This can also work for elliptical orbits, but they must share the same major axis (coapsidal).]
1. The first thing to do is identify the transfer ellipse and its energy. Find the major axis of the transfer ellipse by adding the radius of the orbit you are leaving (the parking orbit) to the radius of the orbit to which you are transferring (your mission orbit). Divide this by two to get the semi-major axis. Calculate the orbital energy of this orbit.
(µ = 398600.5 km3/sec2 for the earth) ε = -µ/2a
2. Now calculate the velocity of this transfer orbit at two places. You want to know the velocity of the transfer ellipse at perigee and at apogee. We know the energy, and we know the radius at perigee, so we can solve for velocity at perigee. We do the same at apogee.
3. Find the velocities of your circular parking orbit and your mission orbit.
4. The size of the Hohmann burns (?V1 and ?V2) are the difference between Parking Orbit velocity and transfer orbit velocity at perigee ?V1; and the difference between the transfer orbit velocity at apogee and your mission orbit velocity ?V2. Calculate these.