# A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families

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## Abstract

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## 1. Introduction

- Hansen [8] pioneered in this field, introducing the relative anomalies.
- Sundman, in 1912 [7], introduced a new temporal variable s related to t by means of $r\phantom{\rule{0.166667em}{0ex}}ds=dt$.
- Brumberg [12] introduced the regularized length of arc ${s}^{*}$ by $vd{s}^{*}=dt$, where v is the velocity of the secondary.
- Brumberg and Fukushima [13] introduced the elliptic anomaly w as $w=\frac{\pi z}{2K\left(e\right)}-\frac{\pi}{2}$, where $\phantom{\rule{0.277778em}{0ex}}\mathrm{am}\phantom{\rule{0.166667em}{0ex}}z=g+\frac{\pi}{2}$.

## 2. The Biparametric Family of Anomalies as a Function of Vector Radius and Curvature

## 3. Other Symmetric Variables Not Belonging to the Biparametric Family

## 4. Numerical Examples

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**True f and antifocal ${f}^{\prime}$ anomaly and vector radii $\overrightarrow{r}$ and ${\overrightarrow{r}}^{\prime}$.

**Figure 2.**Section of revolution ellipsoid containing the orbit. Parametric u, geodetic $\Psi $, and geocentric $\Phi $ latitudes.

**Figure 4.**Point distribution for $e=0.7$, $\gamma =0.0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}1,5,\phantom{\rule{0.166667em}{0ex}}2.0$, $\delta =0$.

**Table 1.**Integration errors in position (Km) and velocity (Km/s) for different anomalies in the biparametric family ${\Psi}_{\alpha ,\beta}$ and in their equivalents in the representation ${\Phi}_{\gamma ,\delta}$.

M | g | f | ${\mathit{\tau}}^{*}$ | ${\mathit{s}}^{*}$ | w | ${\mathit{f}}^{\prime}$ | $\mathbf{\Psi}$ | |
---|---|---|---|---|---|---|---|---|

$\alpha $ | 0 | 1 | 2 | $1.5$ | $0.5$ | $1.5$ | 1 | 2 |

$\beta $ | 0 | 0 | 0 | 0 | $-0.5$ | $0.5$ | 1 | 1 |

$\left|\right|\Delta \overrightarrow{r}\left|\right|$ | $9.54\times {10}^{+00}$ | $1.12\times {10}^{-05}$ | $9.49\times {10}^{-10}$ | $2.86\times {10}^{-08}$ | $4.51\times {10}^{-04}$ | $1.07\times {10}^{-07}$ | $2.60\times {10}^{+00}$ | $8.03\times {10}^{-06}$ |

$\left|\right|\Delta \overrightarrow{v}\left|\right|$ | $7.71\times {10}^{-03}$ | $9.01\times {10}^{-09}$ | $3.56\times {10}^{-11}$ | $2.41\times {10}^{-11}$ | $3.64\times {10}^{-07}$ | $4.41\times {10}^{-11}$ | $2.10\times {10}^{-03}$ | $6.51\times {10}^{-09}$ |

$\gamma $ | 0 | 1 | 2 | $1.5$ | 1 | 1 | 0 | 1 |

$\delta $ | 0 | 0 | 0 | 0 | $-0.5$ | $0.5$ | 1 | 1 |

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**MDPI and ACS Style**

López Ortí, J.A.; Marco Castillo, F.J.; Martínez Usó, M.J.
A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families. *Algorithms* **2024**, *17*, 66.
https://doi.org/10.3390/a17020066

**AMA Style**

López Ortí JA, Marco Castillo FJ, Martínez Usó MJ.
A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families. *Algorithms*. 2024; 17(2):66.
https://doi.org/10.3390/a17020066

**Chicago/Turabian Style**

López Ortí, José Antonio, Francisco José Marco Castillo, and María José Martínez Usó.
2024. "A Geometrical Study about the Biparametric Family of Anomalies in the Elliptic Two-Body Problem with Extensions to Other Families" *Algorithms* 17, no. 2: 66.
https://doi.org/10.3390/a17020066