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On the Bessel solution of Kepler's equation

Riccardo Borghi

Abstract

Since its introduction in 1650, Kepler's equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continues to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

On the Bessel solution of Kepler's equation

Abstract

Since its introduction in 1650, Kepler's equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continues to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.
Paper Structure (6 sections, 38 equations, 4 figures, 1 table)

This paper contains 6 sections, 38 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Bessel' Solution of Elliptic Kepler's Equation
  3. A Constructive Proof That $\mathbb{S}(\epsilon;M)$ Is a Stieltjes Series
  4. A New Integral Representation of KE's Solution
  5. Discussions
  6. Conclusions

Figures (4)

  • Figure 1: The geometry of Kepler's Equation.
  • Figure 2: Behaviour of the function $\theta(t)$, according to Equation (\ref{['Eq:WatsonJ.3.1']}), for $\chi=1/10$ (solid curve), $\chi=1/2$ (dashed curve), and $\chi=1$ (dotted curve).
  • Figure 3: Behaviour of the relative error against $M\in(0,\pi)$. In the above experiments, the "exact value" of the KE solution was evaluated by solving Equation (\ref{['Eq:KE.1']}) via Mathematica's native command FindRoot with the parameter WorkingPrecision set to 50 and the initial guess of $\psi=\pi$. Function $S(\epsilon;M)$ was evaluated by implementing Equation (\ref{['Eq:SolvingKepler.3.1.1.1']}) through the native Mathematica command NIntegrate with different degrees of accuracy, measured by the parameter WorkingPrecision, which was set to 10 (a), 15 (b), 20 (c), and 25 (d).
  • Figure 4: The same as in Figure \ref{['Fig:NumericalSimulation']}, but for $M\in \left[\dfrac{1}{1000},\dfrac{1}{10}\right]$. Note that now the function $S(\epsilon;M)$ is thought of as the imaginary part of $\mathbb{S}(\epsilon;M)$, which is computed, similarly to that in Table \ref{['Tab.Finale.KS']}, via the Weniger $\delta$-transformation with an order of 20 (black circles), 30 (open circles), 40 (open squares), and 50 (black squares).