The geometric origin of the thermal eccentricity law
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 consider bound Kepler orbits in a Euclidean space of dimension D≥2, with a phase-space distribution depending only on energy, and derive the eccentricity distribution from the Liouville measure. We obtain fD(e)=(D−1) e (1−e2)(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.
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BibTeX
@misc{pavlik2026geometric,
title = {{The geometric origin of the thermal eccentricity law}},
author = {Václav Pavlík},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30259},
eprint = {2609.30259},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30259},
}
RIS
TY - UNPB
TI - The geometric origin of the thermal eccentricity law
AU - Václav Pavlík
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30259
UR - https://arxiv.org/abs/2609.30259
AB - 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 consider bound Kepler orbits in a Euclidean space of dimension D≥2, with a phase-space distribution depending only on energy, and derive 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.
ER -