The moon is a sphere with radius of 92 km. Determine an equation for the ellipse if the distance of the satellite from the surface varies from 206 km to 943 km.

Question

A satellite is to be put into an elliptical orbit around a moon. The moon is a sphere with radius of 92 km. Determine an equation for the ellipse if the distance of the satellite from the surface of the moon varies from 206 km to 943 km.

Answer 1

I will work this out in the perifocal coordinate system.

 

r(φ) = a(1 - e²) / (1 + e cosφ)

 

r(φ) = distance from center of moon to satellite 

a = semi-major axis

e = eccentricity

φ = true anomaly

c = distance from center of moon (at a focus) to center of ellipse = ea

h_p = height at perigee = 206 km

h_a = height at apogee = 943 km

R = radius of moon = 92 km

 

We need to figure out a and e.

 

a = c + R + h_p = ea + R + h_p

 

e = (a - R - h_p) / a

 

2a = h_a + 2R + h_p = [943 + 2(92) + 206] km = 1333 km

 

a = (1333 km)/2 = 666.5 km

 

e = (666.5 - 92 - 206) / 666.5 = 0.5529 < 1 (e < 1 for an ellipse)

 

a(1 - e²) = 666.5(1 - 0.5529²) km = 462.75 km

 

r(φ) = [462.75 / (1 + 0.5529cosφ)] km