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The geometric origin of the thermal eccentricity law

Year: 2026
Type: article
Abstract: The distribution f(e)=2e is commonly referred to as the `thermal eccentricity distribution' of Keplerian binaries. However, the result does not require thermal equilibrium, and its usual derivations obscure why the distribution is linear in eccentricity. We seek a simple geometrical interpretation of this result and its relation to more general eccentricity distributions. We considered bound Kepler orbits in a Euclidean space of dimension D with a phase-space distribution that depends only on energy, and derived the eccentricity distribution from the Liouville measure. We obtain f_D(e)=(D-1),e,(1-e^2)^(D-3)/2, so that within this family, D=3 is the only case for which the distribution function is linear in e. This follows from the D-1 transverse momentum dimensions and the Kepler period degeneracy. We further show that two known generalisations -- namely a power-law weighting of the normalised angular momentum and a constant velocity anisotropy -- are in fact two representations of the same one-parameter family of eccentricity distributions.
DOI: 10.1051/0004-6361/202662705
Language: English
Open Access status: gold
Estimated APC paid: $1,865
OpenAlex ID: W7214388711
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