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A satellite of mass m orbits the Earth (mass M, radius R) in a circular orbit of radius r. Due to weak air resistance it loses mechanical energy at a constant rate C joules per second. Find the time taken for the satellite to spiral down to the Earth's surface.

1. (GMm/2C)(1/R - 1/r)
2. (GMm/C)(1/R - 1/r)
3. (GMm/2C)(1/r - 1/R)
4. (GMm/2C)(r - R)

Correct answer: (GMm/2C)(1/R - 1/r)

Solution

The total energy at radius r is -GMm/(2r). As the satellite descends from r to R, the magnitude of energy it must lose equals the difference of these total energies. Dividing by the constant loss rate C gives the time.

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