All three of the observations now known as Kepler's laws may be derived from the fundamental theory put forth by Newton. As an example, we present the derivation for the third of Kepler's observations, which relates the orbital period of a satellite to the radius of its orbit, from the more fundamental Newtonian theory.
This supposes that the mass of the satellite is much smaller than the mass of the central body (this is more or less the definition of the term "satellite"; otherwise, it's simply a two-body system). We'll discuss satellites rotating about the Earth, but the result applies equally well to any tiny body orbiting a large one, such as the moons of Jupiter. For simplicity, we'll take the orbit to be spherical, though Kepler's observation concerned elliptical orbits as well.
Suppose that the satellite is in a circular orbit well above the Earth's atmosphere, so that we can neglect friction effects due to the viscosity of air. The mass m of the satellite is assumably constant, and the total force and the acceleration are in the same direction, towards the center of the Earth. It is also necessary to assume that m is much less than M, so that the satellite's gravitational field does not cause a measurable acceleration of the Earth.
The second law, F = m a, when applied to the centripetal acceleration of the satellite, yields
Solving these algebraic equations for T2 yields
Since by assumption T > 0 and r > 0, this equation has one real solution,
The latter equation is a more precise statement of what we called "Fact 3" above.
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