# Planetary Systems and the Hidden Symmetries of the Kepler Problem

## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Transformations of Planetary Orbits

- $-\infty $:Me, 0:V, 1:E, 2:Ma, 3:A, 4:J, 5:S, 6:U, 7:P.

- 1:Me, 2:V, 3:E, 4:Ma, 5:A, 6:J, 7:S, 8:U, 9:N, 10:P.

#### 2.2. Symmetries of the Kepler Problem

## 3. Results

## 4. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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Planets | T (Day) | Smax (AU) | ecc. |
---|---|---|---|

TRAPPIST-1b | 1.51087 | 0.0111 | 0.081 |

TRAPPIST-1c | 2.42182 | 0.0152 | 0.083 |

TRAPPIST-1d | 4.04961 | 0.0214 | 0.070 |

TRAPPIST-1e | 6.09962 | 0.0282 | 0.085 |

TRAPPIST-1f | 9.20669 | 0.0371 | 0.063 |

TRAPPIST-1g | 12.35294 | 0.0451 | 0.061 |

TRAPPIST-1h | 18.76700 | 0.0630 | 0.000 |

HD 10,180 b | 1.17766 | 0.0222 | 0.001 |

HD 10,180 c | 5.75969 | 0.0641 | 0.073 |

HD 10,180 d | 16.35700 | 0.1286 | 0.131 |

HD 10,180 e | 49.74800 | 0.2699 | 0.051 |

HD 10,180 f | 122.74400 | 0.4929 | 0.119 |

HD 10,180 g | 604.67000 | 1.4270 | 0.263 |

HD 10,180 h | 2205.00000 | 3.3810 | 0.095 |

HD 40,307 b | 4.31230 | 0.0468 | 0.200 |

HD 40,307 c | 9.61840 | 0.0799 | 0.060 |

HD 40,307 d | 20.43200 | 0.1321 | 0.070 |

HD 40,307 f | 51.76000 | 0.2470 | 0.020 |

HD 40,307 g | 197.80000 | 0.6000 | 0.290 |

Kepler-20 b | 3.69612 | 0.0463 | 0.030 |

Kepler-20 e | 6.09852 | 0.0639 | 0.280 |

Kepler-20 c | 10.85409 | 0.0949 | 0.160 |

Kepler-20 f | 19.57758 | 0.1396 | 0.320 |

Kepler-20 g | 34.94000 | 0.2055 | 0.150 |

Kepler-20 d | 77.61130 | 0.3506 | 0.600 |

Kepler-296 c | 5.84164 | 0.0521 | 0.330 |

Kepler-296 b | 10.86438 | 0.0790 | 0.330 |

Kepler-296 d | 19.85029 | 0.1180 | 0.330 |

Kepler-296 e | 34.14211 | 0.1690 | 0.330 |

Kepler-296 f | 63.33627 | 0.2550 | 0.330 |

Kepler-444 b | 3.60011 | 0.0418 | 0.160 |

Kepler-444 c | 4.54588 | 0.0488 | 0.310 |

Kepler-444 d | 6.18939 | 0.0600 | 0.180 |

Kepler-444 e | 7.74349 | 0.0696 | 0.100 |

Kepler-444 f | 9.74049 | 0.0811 | 0.290 |

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Cseh, J.
Planetary Systems and the Hidden Symmetries of the Kepler Problem. *Symmetry* **2020**, *12*, 2109.
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Cseh J.
Planetary Systems and the Hidden Symmetries of the Kepler Problem. *Symmetry*. 2020; 12(12):2109.
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2020. "Planetary Systems and the Hidden Symmetries of the Kepler Problem" *Symmetry* 12, no. 12: 2109.
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