Total Time of Orbit

Classical Mechanics Level 2

A satellite of mass $m$ circularly orbiting a planet of mass $M$ has an angular momentum per unit mass of $a$ . How long does the satellite take to complete a revolution around the planet?

2\pi \frac{a}{GM}

2\pi \frac{a^3}{G^2 M^2}

2\pi \frac{a^{3/2}}{G^2 M^2}

2\pi \frac{a^3}{G^2 M^2 m^2}

1 solution

Matt DeCross
May 13, 2016

Relevant wiki: Characteristics of Circular Orbits

Recall that an object of mass $m$ in circular orbit about an object of mass $M$ at radius $r$ has angular momentum:

$L = m \sqrt{GMr}.$

The given information thus yields:

$a = \sqrt{GMr},$

since $L/m$ is the angular momentum per unit mass. Rearranging, one finds the radius:

$r = \frac{a^2}{GM}.$

Now using the fact that the period of orbit is:

$T = 2\pi \sqrt{\frac{r^3}{GM}},$

and plugging in the $r$ found above, one obtains the answer.

Total Time of Orbit

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