# Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Elliptic Restricted Three-Body Problem

#### 2.2. Numerical Continuation

#### 2.2.1. Natural Parameter Continuation

#### 2.2.2. Arc-Length Continuation

Algorithm 1: Generic Arc-Length Continuation Algorithm |

#### 2.3. Data Filtering Using Hausdorff Distance

## 3. Problem Formulation

#### 3.1. The HOPE Algorithm

Algorithm 2: Homeomorphic Periodic Revolutions Evaluation (HOPE) Algorithm |

#### 3.2. Station Keeping

Algorithm 3: Station-Keeping Algorithm |

## 4. Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**Left**) Rotating and inertial reference frames in the ER3BP. (

**Right**) Pulsating and rotating reference frames in the ER3BP.

**Figure 5.**Convergence test of estimated HD between the parent EML1 orbit and HPO for ${N}_{\mathrm{rev}}=1,2$.

**Figure 6.**Convergence test of estimated HD between the parent EML3 orbit and HPO for ${N}_{\mathrm{rev}}=1,2$.

**Figure 7.**Bounded $\nu $-amplitudes for ${N}_{\mathrm{rev}}=1,2$ HPO families with different initial geometries.

**Figure 8.**Comprehensive set of HPO configurations for ${N}_{\mathrm{rev}}=1$ at different initial geometries. Unfiltered orbits shown on the left, with corresponding Hausdorff distances shown on the right.

**Figure 9.**Plot depicting the filtered dataset for ${N}_{\mathrm{rev}}=1$ at all initial geometry configurations. Reference halo orbit has ${A}_{z}$ = 93,400 km.

**Figure 10.**Plot depicting the filtered dataset for ${N}_{\mathrm{rev}}=2$ at all initial geometry configurations. Reference halo orbit has ${A}_{z}$ = 93,400 km.

**Figure 12.**Trend capturing the effect of initial geometry on station-keeping costs for sustaining a 1-year lifetime on the selected EML1 northern halo orbit.

**Figure 13.**Cumulative station-keeping costs for the EML1 northern halo orbit over a 1-year lifetime for $\theta ={204}^{\circ}$ (

**left**) and $\theta ={239}^{\circ}$ (

**right**).

**Figure 14.**Pulsating frame trajectories for the EML1 northern halo orbit. Magenta arrows denote the optimal impulsive maneuvers.

**Figure 15.**x-z and y-z projections of the most optimal case plotted in the Moon-Centered Rotating Frame. Magenta arrows denote the optimal impulsive maneuvers.

**Figure 16.**Trend capturing the effect of initial geometry on station-keeping costs for sustaining a 20-year lifetime of the JWST.

**Figure 17.**Cumulative station-keeping costs for the JWST orbit over a 20-year lifetime for $\theta ={50}^{\circ}$ (

**left**) and $\theta ={158}^{\circ}$ (

**right**).

**Figure 19.**x-z and y-z projections of an optimal JWST SK orbit bundle observed in the Earth-Centered Rotating Frame without a depiction of the impulsive maneuvers.

**Table 1.**EML1 halo orbit parameters in normalized units (DU,VU) [4].

Parameter | ${\mathit{x}}_{0}$ | ${\mathit{y}}_{0}$ | ${\mathit{z}}_{0}$ | ${\mathit{x}}_{0}^{\prime}$ | ${\mathit{y}}_{0}^{\prime}$ | ${\mathit{z}}_{0}^{\prime}$ |
---|---|---|---|---|---|---|

Value | $8.392\times {10}^{-1}$ | 0 | $1.553\times {10}^{-1}$ | $-4.273\times {10}^{-16}$ | $2.602\times {10}^{-1}$ | $6.178\times {10}^{-16}$ |

Parameter | ${\mathit{x}}_{0}$ | ${\mathit{y}}_{0}$ | ${\mathit{z}}_{0}$ | ${\mathit{x}}_{0}^{\prime}$ | ${\mathit{y}}_{0}^{\prime}$ | ${\mathit{z}}_{0}^{\prime}$ |
---|---|---|---|---|---|---|

Value | $1.011$ | 0 | $-2.924\times {10}^{-3}$ | $-7.745\times {10}^{-16}$ | $-1.010\times {10}^{-2}$ | $6.547\times {10}^{-16}$ |

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**MDPI and ACS Style**

Alvarado, K.I.; Singh, S.K.
Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem. *Aerospace* **2024**, *11*, 407.
https://doi.org/10.3390/aerospace11050407

**AMA Style**

Alvarado KI, Singh SK.
Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem. *Aerospace*. 2024; 11(5):407.
https://doi.org/10.3390/aerospace11050407

**Chicago/Turabian Style**

Alvarado, Kevin I., and Sandeep K. Singh.
2024. "Exploration and Maintenance of Homeomorphic Orbit Revs in the Elliptic Restricted Three-Body Problem" *Aerospace* 11, no. 5: 407.
https://doi.org/10.3390/aerospace11050407