The Apollo commandservice module (CSM) traveled in an orbit whose farthest dista
ID: 1493725 • Letter: T
Question
The Apollo commandservice module (CSM) traveled in an orbit whose farthest distance from the surface of the Moon was 310 km. Its closest approach was 110km above the surface of the Moon. After the CSM was placed in the elliptical (310km x 110km) orbit around the Moon, the orbit was circularized to 110 km above the surface.
Find: a) The period of the elliptical orbit
b) The period of the circular orbit
c) Velocity of the CSM in the circular orbit
d) How does the velocity in the circular orbit compare to the velocity in the elliptical orbit at the minimum height of 110 km above the surface?
Explanation / Answer
a) Time period of elliptical orbit = 2 pi sqrt ( a^3 / G M )
a = semi-major axis = 1736.1 + 310 = 2046.11 km
T = (2 x pi ) sqrt [ ( 2046.11 x 10^3 )^3 / (6.67 x 10^-11 x 7.348 x 10^22) ]
T = 8306.65 sec
b) For circular orbit,
T = 2 pi sqrt[ r^3 / G M ]
r = 1736.1 + 110 = 1846.11 km
T = (2 x pi ) sqrt [ ( 1846.11 x 10^3 )^3 / (6.67 x 10^-11 x 7.348 x 10^22) ]
T = 7119 sec
c) v = 2 pi r / T = (2 x pi x 1846.11 x 10^3 ) / (7119)
v = 1629.27 m/s
d) v = sqrt[ G M ( 2/r - 1/a) ]
a = 2046.11 km
r = 1846.11 km
v = sqrt[ (6.67 x 10^-11 x 7.348 x 10^22) { (2 / (1846110)) - (1 / 2046110)} ]
v = 1707.14 m/s
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