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Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials
Abstract
Motivated by Kepler's long battle with the orbit of Mars, we present a family of orbits that satisfy his second law, conserving angular momentum around a central mass, and also fulfill Ptolemy's belief in an equant --- a point around which the angular speed is constant. For specific values of the angular momentum, these orbits can be written in terms of Legendre polynomials and functions of the second kind. They are analytic except at perihelion where they are twice differentiable. They resemble the "ovoid" (egg-shaped) or "metopoid" (face-shaped) orbits that Kepler considered on his way to the ellipse.
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BibTeX
@misc{moore2026astronomia,
title = {{Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials}},
author = {Cristopher Moore and Malcolm Smith and Stephan Mertens},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.05358},
eprint = {2610.05358},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.05358},
}
RIS
TY - UNPB
TI - Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials
AU - Cristopher Moore
AU - Malcolm Smith
AU - Stephan Mertens
PY - 2026
DA - 2026/10/04
PB - arXiv
DO - 10.48550/arxiv.2610.05358
UR - https://arxiv.org/abs/2610.05358
AB - Motivated by Kepler's long battle with the orbit of Mars, we present a family of orbits that satisfy his second law, conserving angular momentum around a central mass, and also fulfill Ptolemy's belief in an equant --- a point around which the angular speed is constant. For specific values of the angular momentum, these orbits can be written in terms of Legendre polynomials and functions of the second kind. They are analytic except at perihelion where they are twice differentiable. They resemble the "ovoid" (egg-shaped) or "metopoid" (face-shaped) orbits that Kepler considered on his way to the ellipse.
ER -