After the NEAR spacecraft passed Mathilde, on several occasions rocket propellan

Question

After the NEAR spacecraft passed Mathilde, on several occasions rocket propellant was expelled to adjust the spacecraft's momentum in order to follow a path that would approach the asteroid Eros, the final destination for the mission. After getting close to Eros, further small adjustments made the momentum just right to give a circular orbit of radius 45 km (45 × 103 m) around the asteroid. So much propellant had been used that the final mass of the spacecraft while in circular orbit around Eros was only 510 kg. The spacecraft took 1.04 days to make one complete circular orbit around Eros. Calculate what the mass of Eros must be.

Mass of Eros = ____________ kg

Explanation / Answer

Kepler’s third law (note that the mass of NEAR is not needed and that the period should be in seconds):

T² = 4²r³/GM

Solved for mass:

M = 4²r³/GT²
= 4²(45,000m)³ / (6.67 x 10¹¹N•m²/kg²)(8.9 x 10^4s)²
= 6.80 x 10^15kg

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