# On the Bessel Solution of Kepler’s Equation

## Abstract

**:**

## 1. Introduction

## 2. Bessel’ Solution of Elliptic Kepler’s Equation

## 3. A Constructive Proof That $\mathbb{S}(\mathbf{\u03f5};\mathbf{M})$ Is a Stieltjes Series

**Proof**

**of**

**Theorem**

**1.**

`InverseFunction`. In Figure 2, the behaviour of $\theta \left(t\right)$, evaluated according to Equation (23), is plotted for different values of the aspect ratio $\chi $.

## 4. A New Integral Representation of KE’s Solution

`FindRoot`with the parameter

`WorkingPrecision`set to 50 and an initial guess of $\psi =\pi $. Function $S(\u03f5;M)$ was evaluated by implementing Equation (36) using 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).

## 5. Discussions

`FindRoot`with the parameter

`WorkingPrecision`set to 50 and an initial guess of $\psi =\pi $, but now the function $S(\u03f5;M)$ is thought of as the imaginary part of $\mathbb{S}(\u03f5;M)$. The latter is computed, as done in Table 1, via the Weniger $\delta $-transformation with different orders.

## 6. Conclusions

In common with almost any scientific problem which achieves a certain longevity and whose literature exceeds a certain critical mass, the Kepler problem has acquired luster and allure for the modern practitioner. Any new technique for the treatment of trascendental equations should be applied to this illustrious case; any new insight, however slight, lets its conceiver join an eminent list of contributors.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Behaviour of the function $\theta \left(t\right)$, according to Equation (23), 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 (1) via Mathematica’s native command

`FindRoot`with the parameter

`WorkingPrecision`set to 50 and the initial guess of $\psi =\pi $. Function $S(\u03f5;M)$ was evaluated by implementing Equation (36) 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 3, but for $M\in \left[{\displaystyle \frac{1}{1000}},{\displaystyle \frac{1}{10}}\right]$. Note that now the function $S(\u03f5;M)$ is thought of as the imaginary part of $\mathbb{S}(\u03f5;M)$, which is computed, similarly to that in Table 1, via the Weniger $\delta $-transformation with an order of 20 (black circles), 30 (open circles), 40 (open squares), and 50 (black squares).

**Table 1.**Resummation, via Weniger $\delta $-transformation [46], of the (divergent) Kapteyn series $\sum _{m\ge 1}\phantom{\rule{0.166667em}{0ex}}{\displaystyle \frac{{z}^{m}}{m}}\phantom{\rule{0.166667em}{0ex}}{J}_{m}\left(m\u03f5\right)$, evaluated for $z=10\phantom{\rule{0.166667em}{0ex}}\mathrm{exp}(\mathrm{i}\pi /3)$. The value obtained by numerically computing the integral into the right side of Equation (38) is: −1.001838…+ 1.238765…i.

Order | Partial Sum Sequence | Weniger $\mathit{\delta}$-Transformation |
---|---|---|

1 | 2.02 + 3.51 i | 0.112240 + 1.211289 i |

10 | (4.4 − 10 i) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{8}$ | −1.003096 + 1.238166 i |

20 | (−3.1 + 32 i) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{18}$ | −1.001839 + 1.238763 i |

30 | (7.7 + 10 i) $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{27}$ | −1.001838 + 1.238765 i |

… | … | … |

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Borghi, R.
On the Bessel Solution of Kepler’s Equation. *Mathematics* **2024**, *12*, 154.
https://doi.org/10.3390/math12010154

**AMA Style**

Borghi R.
On the Bessel Solution of Kepler’s Equation. *Mathematics*. 2024; 12(1):154.
https://doi.org/10.3390/math12010154

**Chicago/Turabian Style**

Borghi, Riccardo.
2024. "On the Bessel Solution of Kepler’s Equation" *Mathematics* 12, no. 1: 154.
https://doi.org/10.3390/math12010154