John D. Cook • 6/7/2026

The Laplace limit

This article discusses solving Kepler's equation using power series (Lagrange's approach) versus sine series (Bessel's approach), focusing on the radius of convergence known as the Laplace limit. It explains why the power series only converges for eccentricity e < 0.6627, while the sine series works for all e < 1. The author provides astronomical context (e.g., Sedna's orbit) and Python code using SciPy to compute the Laplace limit. The content is technical, involving mathematics, numerical methods, and orbital mechanics, but is firmly rooted in computational science and programming.

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#Eccentricity #Power Series #Convergence