Paper
On “generalized Kepler’s third law” and mass dependence of periods of three-body orbits
Published Mar 19, 2021 ·
Meccanica
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Abstract
Five years ago a generalization of Kepler’s third law was formulated (Dmitrašinović and Šuvakov in Phys Lett A 379:1939–1945, 2015) for three-body orbits in Newtonian gravity with equal masses and vanishing angular momentum. In the meantime this linear relation between the scale-invariant period and the algebraic description of the orbit’s topology has been confirmed (Li and Liao in Sci China Phys Mech Astron 60(12):129511, 2017) in about 700 topologically distinct orbits with equal masses and vanishing angular momentum and theoretically explained (Dmitrašinović et al. in J Phys A Math Theor 51:315101, 2018) as a consequence of holomorphy of the action integral. More recently, Li et al. (Publ Astron Soc Jpn 70:64, 2018) presented 1349 unequal-mass periodic 3-body orbits with vanishing angular momentum. On that basis a “generalized Kepler’s third law” for unequal-mass orbits was proposed with a specific mass dependence of the “general” scale-invariant period. First, we show that Euler’s 3-body orbit does not satisfy this “generalized Kepler’s third law”. Then we discuss the general implications of scaling symmetry, and of the virial theorem in particular, on periods of 3-body orbits in Newtonian gravity, to show that in the absence of a universal \(S_3\) symmetry under permutations of three bodies in all periodic orbits there can be no truly universal Kepler’s third law for 3-body orbits. Orbits with definite permutation symmetries, however, may have similar, though not necessarily identical mass dependences of their periods, in the absence of angular momentum. All 1349 orbits in Li et al. (Publ Astron Soc Jpn 70:64, 2018) are symmetric under permutations of two equal-mass bodies, though not of the third one, even when the third body has a mass that is equal to the other two. That reduced \(S_2\) permutation symmetry allows for the possibility that this class of orbits has a common mass dependence. We have re-analyzed the mass dependence of these 1349 orbits subject to the normalization condition that all orbits must have the same sum of masses. The extracted mass-ratio dependencies of the scale-invariant period (normalized by the topology) for various families of orbits are generally different for different topological families of orbits. Thus there is no generalized Kepler’s third law for 3-body orbits with different masses either universally or in this subset of orbits.
There is no universal "generalized Kepler's third law" for three-body orbits with different masses, as there is no universal scaling symmetry in all periodic orbits.
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