Kepler’s Third Law for \( N \)-Body Periodic Orbits: Exact and Approximate Forms

For a bound orbit of Newtonian point masses, the product \( T|E|^{3/2} \) of the period and the \( 3/2 \) power of the energy is the only scale-invariant combination of the two; for two bodies it is fixed by Kepler's third law. Two closed-form generalisations to \( N\ge3 \) bodies have been proposed on dimensional grounds: a cube-sum form, supported by numerical data for planar three-body orbits, and a pair-sum form, which reproduces Semay's envelope-theory result for self-gravitating identical bosons. We show that the pair-sum form is exact for a definite class of orbits. In a homographic motion the configuration keeps its shape while rotating and pulsating, and the equations of motion reduce to a Kepler problem in which the moment of inertia \( I(\bf{a}) \) of the configuration acts as the mass and its potential \( U(\bf{a}) \) as the coupling constant. It follows that \( T|E|^{3/2}=\tfrac{\pi}{\sqrt2}\,G\,U(\bf{a})\sqrt{I(\bf{a})} \) for every eccentricity; for the equilateral triangle with arbitrary masses this coincides with the pair-sum formula. The same relation yields \( 5\pi/2 \) for the equal-mass Euler orbit, closed-form results for rings of co-orbital satellites, and the two-body limit for Trojan configurations. The semiclassical spectrum of the reduced problem is hydrogen-like, \( E_n=-K^2/(\pi\hbar n)^2 \), where \( K \) is the classical constant, and the correspondence principle returns \( T|E|^{3/2}=K \). This accounts for the agreement between Semay's quantum period and a classical law, and for the factor \( \binom N2 \) that appeared in his comparison. For non-homographic orbits we find that the moment of inertia of the figure-eight varies by only \( 0.2\% \) over a period, while its period exceeds that of a relative equilibrium of equal size and energy by \( 14\% \). We also test the cube-sum form, interpreted as a period per free-group letter, against published data for about 1600 planar orbits. The agreement is within \( 5 \)--\( 15\% \) for comparable masses, but the form fails when one body is light, since the period per letter must vanish as that mass tends to zero. Neither form is a universal \( N \)-body law, and the exact result implies that no mass function alone can be.