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Question

Explain the third law of Kepler.

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Solution

Kepler discovered that the size of a planet's orbit (the semi-major axis of the ellipse) is simply related to the sidereal period of the orbit. If the size of the orbit (a) is expressed in astronomical units (1 AU equals the average distance between the Earth and the Sun) and the period (P) is measured in years, then Kepler's Third Law says:
P2a3
After applying Newton's Laws of Motion and Newton's Law of Gravity we find that Kepler's Third Law takes a more general form:
P2=4π2G(M1+M2)a3
where M1 and M2 are the masses of the two orbiting objects in solar masses. Note that if the mass of one body, such as M1, is much larger than the other, then M1+M2 is nearly equal to M1. In our solar system M1=1 solar mass and this equation become identical to the first.
Kepler's Third Law

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