# Bivariate Infinite Series Solution of Kepler’s Equations

## Abstract

**:**

## 1. Introduction

## 2. Methods

## 3. Examples, Discussion and Results

#### 3.1. Bivariate Infinite Series Solution of the Elliptic Kepler Equation around ${e}_{c}=0$, ${M}_{c}=0$

#### 3.2. Bivariate Infinite Series Solution of the Elliptic Kepler Equation around ${e}_{c}=\frac{1}{2}$, ${M}_{c}=\frac{\pi -1}{2}$

#### 3.3. Bivariate Infinite Series Solution of the Hyperbolic Kepler Equation around ${e}_{c}=2$, ${M}_{c}=0$

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Errors ${\mathcal{E}}_{n}(e,M)$ (in logarithmic scale) affecting the approximate polynomial solutions ${S}_{n}(e,M)$ of KE for $({e}_{c},{M}_{c})=(0,0)$ along the diagonal line $M=\pi e$ of the $(e,M)$ plane (thus $\rho =\sqrt{1+{\pi}^{2}}\phantom{\rule{0.166667em}{0ex}}e$ and $tan\varphi =\pi $). The ${S}_{n}$ are obtained by truncating the infinite series of Equation (27) up to degree n, for $n=1,\cdots ,5$. The vertical magenta line at $\rho =1.21$ corresponds to the limit below which convergence is obtained in this direction.

**Figure 2.**Contour levels of the error ${\mathcal{E}}_{5}(e,M)$ affecting the fifth degree polynomial approximation, Equation (27), as a function of the eccentricity e and the mean anomaly M (both in logarithmic scales). The continuous magenta curve marks the boundary of the region of convergence, as estimated with Equations (20) and (26). The vertical dotted line represents the limit of the region of convergence for Lagrange’s univariate series.

**Figure 3.**Contour levels of the error ${\mathcal{E}}_{5}$ affecting the fifth degree polynomial approximation of Equation (30), as a function of the eccentricity e and the mean anomaly M. The continuous magenta curve marks the boundary of the region of convergence, as estimated with Equations (20) and Equation (26). (Notice that here the axes for e and M are linear).

**Figure 4.**Contour levels of the error ${\mathcal{E}}_{5}$ affecting the fifth degree polynomial approximation of Equation (31), as a function of the eccentricity e and the mean anomaly M (both in logarithmic scale). The continuous magenta curve marks the boundary of the region of convergence, as estimated with Equations (20) and (26).

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Tommasini, D.
Bivariate Infinite Series Solution of Kepler’s Equations. *Mathematics* **2021**, *9*, 785.
https://doi.org/10.3390/math9070785

**AMA Style**

Tommasini D.
Bivariate Infinite Series Solution of Kepler’s Equations. *Mathematics*. 2021; 9(7):785.
https://doi.org/10.3390/math9070785

**Chicago/Turabian Style**

Tommasini, Daniele.
2021. "Bivariate Infinite Series Solution of Kepler’s Equations" *Mathematics* 9, no. 7: 785.
https://doi.org/10.3390/math9070785