# Periodic Orbits of the Restricted Three-Body Problem Based on the Mass Distribution of Saturn’s Regular Moons

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Best-Fitting Distribution of the Regular Moons’ Mass

## 3. Dynamic Equations and Approximate Periodic Orbits

#### 3.1. Dynamic Model with Variable Mass

#### 3.2. Periodic Orbits near the Lagrangian Point ${L}_{3}$

## 4. Numerical Simulation

#### 4.1. Influence of the Scale Parameter $\sigma $

#### 4.2. Influence of the Three-Body Interaction Parameter k

#### 4.3. Influence of the Variable-Mass Parameter $\gamma $

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The mass CDF of Saturn’s regular moons and the corresponding CDF of the best-fitting distribution.

**Figure 2.**Approximate periodic orbits near the Lagrangian point ${L}_{3}$ when (

**a**) $\sigma $ = 1, (

**b**) $\sigma $ = 2, (

**c**) $\sigma $ = 7.82406, and (

**d**) $\sigma $ = 100, respectively.

**Figure 3.**Third-order periodic orbits and their planar projections under different $\sigma $-values.

**Figure 4.**Approximate periodic orbits near the Lagrangian point ${L}_{3}$ when (

**a**) k = $-0.1$, (

**b**) k = 0, (

**c**) k = 0.3, and (

**d**) k = 0.7, respectively.

**Figure 6.**Approximate periodic orbits near the Lagrangian point ${L}_{3}$ when (

**a**) $\gamma $ = 0.05, (

**b**) $\gamma $ = 0.1, (

**c**) $\gamma $ = 0.6, and (

**d**) $\gamma $ = 1, respectively.

**Figure 7.**Third-order periodic orbits and their planar projections under different $\gamma $-values.

Names of Saturn’s Regular Moons | Mass (kg) | Names of Saturn’s Regular Moons | Mass (kg) |
---|---|---|---|

Aegaeon | 59,946,737,324 | Methone | 8,992,010,598,583 |

Anthe | 1,498,668,433,097 | Mimas | 37,505,676,206,690,400,000 |

Atlas | 6,594,141,105,627,650 | Pallene | 32,970,705,528,138 |

Calypso | 2,547,736,336,265,230 | Pan | 4,945,605,829,220,740 |

Daphnis | 77,930,758,521,054 | Pandora | 138,476,963,218,181,000 |

Dione | 1,095,745,430,185,280,000,000 | Polydeuces | 4,496,005,299,292 |

Enceladus | 107,944,591,230,692,000,000 | Prometheus | 160,956,989,714,639,000 |

Epimetheus | 526,032,620,017,115,000 | Rhea | 2,307,089,151,289,080,000,000 |

Helene | 11,389,880,091,538,700 | S/2009 S1 | —— |

Hyperion | 5,585,537,250,153,240,000 | Telesto | 4,046,404,769,362,420 |

Iapetus | 1,805,952,411,282,580,000,000 | Tethys | 617,551,805,221,061,000,000 |

Janus | 1,892,818,231,001,750,000 | Titan | 134,552,523,083,241,000,000,000 |

Beta | Birnbaum– Saunders | Burr | Exponential | Extreme Value | Gamma | Generalized Extreme Value | Generalized Pareto | Half Normal | Inverse Gaussian | Logistic | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mass ($kg$) | h | null | 1 | 0 | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 |

p | 0.0345 | 0.8643 | 0.0000 | 0.0000 | 0.3145 | 0.0000 | 0.1723 | 0.0000 | 0.0000 | 0.0000 | ||

parameter | $\beta $ = 8.9038 × ${10}^{16}$ $\gamma $ = 370.466 | $\alpha $ = 3.76388 × ${10}^{20}$ c = 0.170255 k = 2.83657 | $\mu $ = 6.11012 × ${10}^{21}$ | $\mu $ = 2.40706 × ${10}^{22}$ $\sigma $ = 4.92323 × ${10}^{22}$ | a = 0.0760157 b = 8.03796 × ${10}^{22}$ | k = 0.569397 $\sigma $ = 1.32889 × ${10}^{16}$ $\mu $ = 7.07519 × ${10}^{15}$ | k = 11.3686 $\sigma $ = 1.55618 × ${10}^{13}$ $\theta $ = 0 | $\mu $ = 0 $\sigma $ = 2.8064 × ${10}^{22}$ | $\mu $ = 6.11012 × ${10}^{21}$ $\lambda $ = 1.29748 × ${10}^{12}$ | $\mu $ = 6.11012 × ${10}^{21}$ $\sigma $ = 1.54407 × ${10}^{22}$ | ||

confidence interval | $\beta $∈ [3.5716 × ${10}^{16}$, 1.4236 × ${10}^{17}$] $\gamma $ ∈ [263.409, 477.524] | $\alpha $∈ [664.59, 2.13166 × ${10}^{38}$] c ∈ [0.0739575, 0.39194] k ∈ [0.0204874, 392.734] | $\mu $∈ [4.21915 × ${10}^{21}$, 9.63871 × ${10}^{21}$] | $\mu $∈ [2.54927 × ${10}^{21}$, 4.5592 × ${10}^{22}$] $\sigma $ ∈ [3.85631 × ${10}^{22}$, 6.28534 × ${10}^{22}$] | a∈ [0.049786, 0.116065] b ∈ [1.72057 × ${10}^{22}$, 3.75508 × ${10}^{23}$] | k∈ [0.494556, 0.644238] $\sigma $ ∈ [7.47124 × ${10}^{15}$, 2.36367 × ${10}^{16}$] $\mu $ ∈ [1.14168 × ${10}^{15}$, 1.30087 × ${10}^{16}$] | k∈ [6.12161, 16.6156] $\sigma $ ∈ [1.31349 × ${10}^{12}$, 1.84372 × ${10}^{14}$] $\theta $ = 0 | $\mu $ = 0 $\sigma $ ∈ [2.18117 × ${10}^{22}$, 3.93671 × ${10}^{22}$] | $\mu $∈ [−Inf,Inf] $\lambda $ ∈ [−Inf,Inf] | $\mu $∈ [−Inf,Inf] $\sigma $ ∈ [−Inf,Inf] | ||

Log-logistic | Lognormal | Nakagami | NegativeBinomial | Normal | Poisson | Rayleigh | Rician | t Location-Scale | Weibull | Stable | ||

Mass($kg$) | h | 0 | 0 | 0 | null | 1 | 1 | 1 | 1 | 1 | 0 | 1 |

p | 0.9759 | 0.9889 | 0.2574 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.0000 | 0.7818 | 0.0000 | ||

parameter | $\mu $ = 39.1312 $\sigma $ = 4.58867 | $\mu $ = 39.1272 $\sigma $ = 7.82406 | $\mu $ = 0.0357998 $\omega $ = 7.8759 × ${10}^{44}$ | $\mu $ = 6.11012 × ${10}^{21}$ $\sigma $ = 2.80064 × ${10}^{22}$ | $\lambda $ = 6.11012 × ${10}^{21}$ | B = 1.98443 × ${10}^{22}$ | s = 1 $\sigma $ = 2.71315 × ${10}^{8}$ | $\mu $ = −1.54662 × ${10}^{15}$ $\sigma $ = 1.81001 × ${10}^{16}$ $\nu $ = 0.177628 | A = 4.42076 × ${10}^{18}$ B = 0.142598 | $\alpha $ = 0.4 $\beta $ = 0.987279 c = 3.62044 × ${10}^{18}$ $\mu $ = 2.24553 × ${10}^{18}$ | ||

confidence interval | $\mu $∈ [35.7976, 42.4648] $\sigma $ ∈ [3.29169, 6.39669 | $\mu $∈ [35.7438, 42.5106] $\sigma $ ∈ [6.05109, 11.0738] | $\mu $∈ [0.0236219, 0.0542557] $\omega $ ∈ [9.08296 × ${10}^{43}$, 6.82924 × ${10}^{45}$] | $\mu $∈ [−6.00076 × ${10}^{21}$, 1.8221 × ${10}^{22}$] $\sigma $ ∈ [2.166 × ${10}^{22}$, 3.96389 × ${10}^{22}$] | $\lambda $∈ [6.11012 × ${10}^{21}$, 6.11012 × ${10}^{21}$] | B∈ [1.64901 × ${10}^{22}$, 2.49241 × ${10}^{22}$] | s = 1 $\sigma $ ∈ [2.71315 × ${10}^{8}$, 2.71315 × ${10}^{8}$] | $\mu $∈ [−Inf,Inf] $\sigma $ ∈ [−Inf,Inf] $\nu $ ∈ [−Inf,Inf] | A∈ [2.12356 × ${10}^{17}$, 9.20299 × ${10}^{19}$] B ∈ [0.104361, 0.194845] | $\alpha $∈ [0,2] $\beta $ ∈ [−1,1] c ∈ [0,Inf] $\mu $ ∈ [−Inf,Inf] |

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**MDPI and ACS Style**

Cheng, H.; Gao, F.
Periodic Orbits of the Restricted Three-Body Problem Based on the Mass Distribution of Saturn’s Regular Moons. *Universe* **2022**, *8*, 63.
https://doi.org/10.3390/universe8020063

**AMA Style**

Cheng H, Gao F.
Periodic Orbits of the Restricted Three-Body Problem Based on the Mass Distribution of Saturn’s Regular Moons. *Universe*. 2022; 8(2):63.
https://doi.org/10.3390/universe8020063

**Chicago/Turabian Style**

Cheng, Huan, and Fabao Gao.
2022. "Periodic Orbits of the Restricted Three-Body Problem Based on the Mass Distribution of Saturn’s Regular Moons" *Universe* 8, no. 2: 63.
https://doi.org/10.3390/universe8020063