Astronomia Ova: Kepler's Laws, Ptolemy's Equant, and Legendre's Polynomials

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.