What is the minimum energy required to launch a satellite of mass from the surface of a planet of mass and radius in a circular orbit at an altitude of ?
Give your answer as the coefficient of .
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At the desired orbit, we have
Total energy = Potential energy + Kinetic energy = − 2 r G M m = − 6 R G M m ( where r = R + 2 R = 3 R )
At the point of launch near the surface
Total energy = Potential energy = − R G M m
The amount of energy required will be equal to the amount of work done by an external agent to bring the satellite from the surface of the planet to the desired orbit. Thus
Minimum energy required = Final Total energy − Initial Total energy = − 6 R G M m + R G M m = 6 R 5 G M m
Thus the required answer is 6 5 ≈ 0 . 8 3 3 3 .