A United Earth research spacecraft in deep space discovers a huge metallic spher

Question

A United Earth research spacecraft in deep space discovers a huge metallic sphere of diameter d = 1, 316 mi = 2, 118 km. The spacecraft is placed in a circular orbit around the sphere at a height above the surface, h = 2, 660 km. The sphere is perfectly smooth and has no distinguishing marks, hence it is not possible to know when the spacecraft has completed one revolution around the sphere. However, Dr. Ostro, the science officer, devises a way to determine the orbital period of the spacecraft by observing the disappearance and reappearance of a distant star. Use the diagram below to find the mass of the mysterious sphere. The time it takes the spacecraft to move from point A to point B is 27 min 28 sec.

Explanation / Answer

radius of the given body, r = 2118/2 km = 1059 km
height of orbit from surface, h = 2660 km
time period = T

assuming circular orbit, time taken from point A to B is t'
orbital speed = v
v = 2*pi*(r + h)/T = (r + h)*theta-radians/t' = (r + h)*2*arcsin(r/r+h)/t'
=> 2*pi/T = 2*arcsin(r/r+h)/t'
=> T = pi*t'/arcsin(r/r+h) = pi*(27 min 28 sec)/arcsin(1059/1059+2660) = 10.8799*(27 min 28 sec) = 10.8799*(27*60 + 28)sec = 298.836 min

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