Identify the needed variable values. Let us consider an example problem where a satellite orbit has a semi-major axis of 7,500 kilometers and a semi-minor axis of 7,450 kilometers. We will determine the altitude of the satellite at apogee and perigee and also at the point on the orbit where the satellite radius makes an angle of 60 degrees with the major axis.
Determine the eccentricity of the elliptical path of a satellite given the semi-major axis "a" and the semi-minor axis "b." Eccentricity "e" is calculated with the formula e = sqrt (1 -- (b*b)/(a*a)). :
Our example looks like this:
a = 7,500 kilometers, b = 7,450 kilometers and e = sqrt (1 -- (7,450*7,450)/(7,500*7,500)) = 0.115.
Find the apogee radius of the satellite "ra" using the formula ra = a(1+e), where "a" is the semi-major axis and "e" is the eccentricity of the elliptical path. Hence the apogee altitude is a(1+e) - R, where "R" is the radius of the Earth (about 6,370 kilometers on average). Here is our example equation:
Apogee altitude = 7,500 kilometers (1 + 0.115) - 6,370 kilometers = 1,992 kilometers.
Find the perigee radius of the satellite "rp" using the formula rp = a(1-e), where "a" is the semi-major axis and "e" is the eccentricity of the elliptical path. Hence, the perigee altitude is a(1- e) - R, where "R" is the radius of the Earth. Here is the equation for our example:
Perigee altitude = 7,500 kilometers (1 - 0.115) - 6,370 kilometers = 268 kilometers.
Find the radius at a point on the orbit that forms an angle "A" with the major axis, using the formula
rA = a(1-e*e)/(1+ecosA), where "a" is the semi-major axis and "e" is the eccentricity of the elliptical path. Hence the altitude of the satellite at this point is given by a(1-e*e)/(1+ecosA) - R, where "R" is the radius of the Earth. Notice that, at perigee, A = 0 degrees and cosA = 1. In this case, the given formula reduces to the one given in the previous step. At apogee, A = 180 degrees and cosA = -1. You can check and see that making these substitutions will give you the formula for the altitude of the satellite at the apogee position. Since A = 60 degrees in our example, here is our formula:
Altitude = 7,500 kilometers (1 - 0.115*0.115)/(1 + 0.115*co(60)) - 6,370 kilometers = 628 kilometers.