# Heuristic Technique for the Search of Interception Trajectories to Asteroids with the Use of Solar Sails

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## Abstract

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## 1. Introduction

## 2. Solar Sailing

## 3. Heuristic Technique

- (Step 1) A projection of the position of the spacecraft at instant ${t}_{F}$ is made (${\mathbf{r}}_{sc}^{HIF}\left({t}_{F}\right)$):
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- ${t}_{F}$ is the given moment when the interception of the target position has to occur. Consequently, it is the final integration step.
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- From instant ${t}_{i}$ to ${t}_{F}$, the spacecraft is considered to have an elliptical heliocentric orbit. In other words, only the gravitational influence of the Sun and a null solar radiation pressure acceleration are considered.
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- This projection serves as a preliminary analysis, at each step ${t}_{i}$, in order to quickly understand the ongoing trajectory of the spacecraft without a solar sail and how the sail must be oriented in order to correct this trajectory into a successful interception.

- (Step 2) A “Final Position-Oriented Frame” (FPOF) is defined from ${\mathbf{r}}_{sc}^{HIF}\left({t}_{F}\right)$:
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- Its origin is the position of the Sun;
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- The X-axis points in the direction of ${\mathbf{r}}_{sc}^{HIF}\left({t}_{F}\right)$.
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- The Z-axis points in the direction of the spacecraft heliocentric angular momentum at ${t}_{F}$ (${\mathbf{h}}_{sc}^{HIF}\left({t}_{F}\right)$). In other words, the fundamental plane of FPOF is the spacecraft osculating orbital plane at ${t}_{F}$.
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- The Y-axis is defined by dextrorotation.

- (Step 3) A set of three “guidance properties” ($P1$, $P2$ and $P3$) are defined as seen in Figure 2:
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- They are defined in respect to ${\mathbf{r}}_{sc}^{FPOF}\left({t}_{F}\right)$ and ${\mathbf{r}}_{target}^{FPOF}$, both represented in FPOF.
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- In this study, the target position (${\mathbf{r}}_{target}$) is the position of the asteroid at ${t}_{F}$ (${\mathbf{r}}_{ast}\left({t}_{F}\right)$).
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- ($P1$): Arc length, along the circumference of a circle with radius equal to $1\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$, between ${\mathbf{r}}_{sc}^{FPOF}\left({t}_{F}\right)$ and $pro{j}_{XY}\left({\mathbf{r}}_{ast}^{FPOF}\left({t}_{F}\right)\right)$, which is the projection of the asteroid position at ${t}_{F}$ onto the X–Y plane.
- ∗
- ($P2$): Difference between the magnitudes of ${\mathbf{r}}_{sc}^{FPOF}\left({t}_{F}\right)$ and ${\mathbf{r}}_{ast}^{FPOF}\left({t}_{F}\right)$.
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- ($P3$): Perpendicular distance of ${\mathbf{r}}_{ast}^{FPOF}\left({t}_{F}\right)$ to the X–Y plane.

- (Step 4) Determine the values of $\alpha \left({t}_{i}\right)$ and $\delta \left({t}_{i}\right)$ as a function of $P1\left({t}_{i}\right)$, $P2\left({t}_{i}\right)$ and $P3\left({t}_{i}\right)$.

- (H1): If $P1$ is positive $(P1>0)$, it is necessary to reduce the spacecraft mean angular orbital velocity, and $\alpha $ must be positive $(\alpha >0)$. The opposite is true.
- (H2): If $P2$ is positive $(P2>0)$, it is necessary to reduce the spacecraft orbital energy, and $\alpha $ must be negative $(\alpha <0)$. The opposite is true.
- (H3): If $P3$ is non-zero $(P3\ne 0)$, it is necessary to change the direction of the spacecraft orbital angular momentum with the appropriate $\delta $ value $(\delta \ne 0)$ in accordance to ${\mathbf{r}}_{sc}^{FPOF}\left(t\right)$.

## 4. Preliminary Examples

- Spacecraft in a heliocentric orbit with an acting solar sail (with SRP acceleration).
- Initial circular orbit with radius ${r}_{0}=1\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$.
- Target position at a distance of ${r}_{target}=1.5237\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ to the Sun on the opposite direction of the spacecraft initial position (transfer angle of ${180}^{\circ}$).
- The time of transfer is initially set as equal to a Hohmann transfer with the same initial and final positions (${t}_{transf}={t}_{hoh}=258.87\phantom{\rule{0.166667em}{0ex}}\mathrm{days}$).

- ${t}_{transf\#1}=1.0\times {t}_{hoh}$.
- ${t}_{transf\#2}=1.05\times {t}_{hoh}$.
- ${t}_{transf\#3}=0.95\times {t}_{hoh}$.
- ${t}_{transf\#4}=2.0\times {t}_{hoh}$.
- ${t}_{transf\#5}=1.0\times {t}_{hoh}$ and $i={2}^{\circ}$.

- $to{l}_{lw}=[1.0\times {10}^{6},1.0\times {10}^{3},1.0\times {10}^{3}]\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$
- $to{l}_{up}=[1.0\times {10}^{11},1.0\times {10}^{10},1.0\times {10}^{9}]\phantom{\rule{0.166667em}{0ex}}\mathrm{m}$

## 5. Case Study: Didymos System

#### 5.1. Earth Escape

- Escape time: ${t}_{esc}=6.198505\phantom{\rule{0.166667em}{0ex}}\mathrm{days}$.
- Initial speed increase: $\Delta {v}_{0}=3.2634\phantom{\rule{0.166667em}{0ex}}\mathrm{km}/\mathrm{s}$.

- Initial speed increase: $\Delta {v}_{0}=3.6195\phantom{\rule{0.166667em}{0ex}}\mathrm{km}/\mathrm{s}$.
- Economy: $9.84\%$ (when using a solar sail to reach the asteroid system).

#### 5.2. Interception Trajectories

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Transfer #1 spacecraft heliocentric trajectory with an evolution of the final position projection.

**Figure 5.**Transfer #4 guidance properties (in linear and logarithmic scale) and sail-orientation angles over time.

**Figure 6.**Transfer #5 guidance properties (in linear and logarithmic scale) and sail-orientation angles over time.

**Figure 8.**(October 2021) Guidance properties (in linear and logarithmic scale) and sail-orientation angles over time.

**Figure 9.**(November 2021) Guidance properties (in linear and logarithmic scale) and sail-orientation angles over time.

Transfer | ${\mathit{\theta}}_{\mathit{lw}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{\theta}}_{\mathit{up}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $\mathit{os}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{m}\right)$ |
---|---|---|---|

#1 | $[35,90,0]$ | $[0,35,35]$ | $[1.911096915\times {10}^{11},-11.2\times {10}^{3},0\times {10}^{3}]$ |

#2 | $[30,90,0]$ | $[0,30,35]$ | $[4.627248276\times {10}^{10},0.3359\times {10}^{6},0\times {10}^{6}]$ |

#3 | $[90,90,0]$ | $[35,35,35]$ | $[1.926755143\times {10}^{10},0.20329\times {10}^{6},0\times {10}^{6}]$ |

#4 | $[90,90,0]$ | $[25,16.3,35]$ | $[3.524816787\times {10}^{11},0.5711\times {10}^{6},0\times {10}^{6}]$ |

#5 | $[35,90,0]$ | $[10,35,20]$ | $[5.201761063\times {10}^{10},0.911\times {10}^{3},0.229\times {10}^{3}]$ |

Orb.Elem. | Earth | Didymos |
---|---|---|

$a\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{au}\right)$ | $1.00000011$ | $1.644324083929969$ |

e | $0.01671022$ | $0.3839233231470776$ |

$i\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $0.00005$ | $3.407876986118815$ |

$\mathsf{\Omega}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $-11.26064$ | $73.19326428620921$ |

$\omega \phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $102.94719$ | $319.3188977070352$ |

${t}_{p}$ | 2022-Jan-4.161111 | 2022-Oct-21.76418056 |

Launch | ${\mathit{\theta}}_{\mathit{lw}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{\theta}}_{\mathit{up}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | $\mathit{os}\phantom{\rule{0.166667em}{0ex}}\left(\mathbf{m}\right)$ |
---|---|---|---|

September 2021 | $[90,90,0]$ | $[15,5,35]$ | $[1.884540031\times {10}^{10},0.964420\times {10}^{6},4.498857\times {10}^{6}]$ |

October 2021 | $[90,90,0]$ | $[15,5,35]$ | $[3.033811596\times {10}^{10},1.880184\times {10}^{6},8.59387\times {10}^{6}]$ |

November 2021 | $[90,90,0]$ | $[35,55,35]$ | $[1.446875557\times {10}^{10},1.2367\times {10}^{6},3.7153\times {10}^{6}]$ |

December 2021 | $[90,90,0]$ | $[45,25,35]$ | $[3.914802412\times {10}^{10},2.122068\times {10}^{6},1.496168\times {10}^{6}]$ |

January 2022 | $[90,90,0]$ | $[55,35,25]$ | $[3.197588329\times {10}^{10},1.005404\times {10}^{6},1.335357\times {10}^{6}]$ |

Launch | $|{\mathit{v}}_{\mathit{sc}}\left({\mathit{t}}_{\mathit{F}}\right)|\phantom{\rule{0.166667em}{0ex}}(\mathbf{km}/\mathbf{s})$ | $\mathbf{\Delta}\mathit{v}\phantom{\rule{0.166667em}{0ex}}(\mathbf{km}/\mathbf{s})$ | $\mathit{\theta}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{\theta}}_{\mathit{xy}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ | ${\mathit{\theta}}_{\mathit{z}}\phantom{\rule{0.166667em}{0ex}}{(}^{\circ})$ |
---|---|---|---|---|---|

Lamb. (November 2021) | $28.128$ | $6.434$ | $4.89$ | $4.52$ | $+1.88$ |

September 2021 | $28.259$ | $11.753$ | $19.05$ | $18.36$ | $+5.15$ |

October 2021 | $28.350$ | $11.401$ | $18.36$ | $18.21$ | $+2.36$ |

November 2021 | $30.136$ | $11.091$ | $18.70$ | $18.70$ | $-0.01$ |

December 2021 | $29.645$ | $11.674$ | $19.65$ | $19.56$ | $-1.94$ |

January 2022 | $31.620$ | $13.567$ | $23.51$ | $23.34$ | $-2.88$ |

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## Share and Cite

**MDPI and ACS Style**

Meireles, L.G.; Prado, A.F.B.d.A.; de Melo, C.F.; Pereira, M.C.
Heuristic Technique for the Search of Interception Trajectories to Asteroids with the Use of Solar Sails. *Symmetry* **2023**, *15*, 617.
https://doi.org/10.3390/sym15030617

**AMA Style**

Meireles LG, Prado AFBdA, de Melo CF, Pereira MC.
Heuristic Technique for the Search of Interception Trajectories to Asteroids with the Use of Solar Sails. *Symmetry*. 2023; 15(3):617.
https://doi.org/10.3390/sym15030617

**Chicago/Turabian Style**

Meireles, Lucas Gouvêa, Antônio Fernando Bertachini de Almeida Prado, Cristiano Fiorilo de Melo, and Maria Cecília Pereira.
2023. "Heuristic Technique for the Search of Interception Trajectories to Asteroids with the Use of Solar Sails" *Symmetry* 15, no. 3: 617.
https://doi.org/10.3390/sym15030617