What is the relationship between Keplers third law of planetary motion and Newtons law of gravitation?

We shall derive Kepler's third law, starting with Newton's laws of motion and his universal law of gravitation. The point is to demonstrate that the force of gravity is the cause for Kepler's laws (although we will only derive the third one).

Let us consider a circular orbit of a small mass

The net external force on mass is gravity, and so we substitute the force of gravity for

The mass cancels, yielding

The fact that cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius

Substituting this into the previous equation gives

Solving for

Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields

This is Kepler's third law. Note that Kepler's third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body cancel.

Now consider what we get if we solve

If and are known for a satellite, then the mass of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio should be a constant for all satellites of the same parent body (because

It is clear from Table 6.2 that the ratio of is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes -uncertainties in the and data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be-and have been-used to predict the location of new planets and moons. This is another verification of Newton's universal law of gravitation.

Newton's universal law of gravitation is modified by Einstein's general theory of relativity, as we shall see in Particle Physics. Newton's gravity is not seriously in error - it was and still is an extremely good approximation for most situations.

Einstein's modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions.

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