# A Parallelized Genetic Algorithm to Evaluate Asteroid Impact Missions Using Electric Propulsion

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## Abstract

**:**

## 1. Introduction

## 2. NEXT-C Ion Thruster

_{in}(t) = P

_{source}/( $\left(t\right)/1\phantom{\rule{4pt}{0ex}}\mathrm{AU}{)}^{2}$, where $\left(t\right)=\sqrt{r{\left(t\right)}^{2}+z{\left(t\right)}^{2}}$ is the magnitude of separation. The data from Fisher et al. [19] was used to set the propellant flow rate as $\dot{m}\left({P}_{\mathrm{in}}\left(t\right)\right)$ and the thrust efficiency as $\eta \left({P}_{\mathrm{in}}\left(t\right)\right)$. When $\dot{m}\left(t\right)>0$, the thrust power was ${P}_{\mathrm{th}}\left(t\right)=\eta \left(t\right){P}_{\mathrm{in}}\left(t\right)$ and the available thrust was ${F}_{\mathrm{th}}\left(t\right)=\sqrt{2{P}_{\mathrm{th}}\left(t\right)/\dot{m}\left(t\right)}$.

## 3. Mission Parameters

## 4. Parallelized Evaluation Process

#### 4.1. Launch

#### 4.2. Impact

#### 4.3. Heliocentric Trip

## 5. Genetic-Algorithm-Based Optimization of Parameters

#### 5.1. Crossover

#### 5.2. Mutations

#### 5.3. Annealing

## 6. Results and Discussion

#### 6.1. Baseline

#### 6.2. Comparable Trip Time

#### 6.2.1. Varying Available Propellant Mass

#### 6.2.2. Varying Available Power

#### 6.2.3. Varying Survivor Selection

#### 6.2.4. Varying Launch Energy

#### 6.3. Short Trip Time

## 7. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Nomenclature

${a}_{\mathrm{th}}$ | Acceleration of the spacecraft due to thrust (AU s${}^{-2}$) |

${f}_{\mathrm{coast}}$ | Function used to optimize coasting (#) |

${F}_{\mathrm{th}}$ | Thrust exerted on the spacecraft (kg AU s${}^{-2}$) |

${m}_{\mathrm{sp}}$ | Mass of the spacecraft (kg) |

$r,\phantom{\rule{4pt}{0ex}}\theta ,\phantom{\rule{4pt}{0ex}}z$ | Radial, angular, and off-plane position of the spacecraft (AU, rad, AU) |

Magnitude of separation of the spacecraft from the Sun (AU) | |

ESOI | Earth’s sphere of influence |

T | Trip time from leaving ESOI to impact (s) |

${V}_{r},\phantom{\rule{4pt}{0ex}}{V}_{\theta},\phantom{\rule{4pt}{0ex}}{V}_{z}$ | Radial, tangential, and off-plane velocity of the spacecraft (AU/s) |

$\alpha $ | Angular location in the ecliptic plane at ESOI (rad) |

$\beta ,\phantom{\rule{4pt}{0ex}}\zeta $ | In-plane and off-plane angles of velocity at ESOI (rad) |

$\gamma ,\phantom{\rule{4pt}{0ex}}\tau $ | In-plane and off-plane directions of thrust (rad) |

$\eta $ | Thrust efficiency (#) |

$\mathsf{\Phi}$ | Fraction of “parents” chosen based on proximity vs. impact speed (#) |

$\psi $ | Input function to ${f}_{\mathrm{coast}}$ (rad) |

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**Figure 1.**Variation of the impact speed with propellant mass (${C}_{3}=4.676\phantom{\rule{4pt}{0ex}}{\mathrm{km}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$, ${P}_{\mathrm{source}}=7330$ W, and $\mathsf{\Phi}=0.5$).

**Figure 2.**Variation of the impact speed with power as a % of ${P}_{\mathrm{max}}$ (${C}_{3}=4.676\phantom{\rule{4pt}{0ex}}{\mathrm{km}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$, ${m}_{\mathrm{prop}}=150$ kg, and $\mathsf{\Phi}=0.5$).

**Figure 3.**Variation of the impact speed with the selection variable, $\mathsf{\Phi}$ (${C}_{3}=4.676\phantom{\rule{4pt}{0ex}}{\mathrm{km}}^{2}\phantom{\rule{4pt}{0ex}}{\mathrm{s}}^{-2}$, ${P}_{\mathrm{source}}=7330$ W, and ${m}_{\mathrm{prop}}=150$ kg).

**Figure 4.**Variation of the impact speed with launch energy as a % of the baseline (${m}_{\mathrm{prop}}=150$ kg, ${P}_{\mathrm{source}}=7330$ W, and $\mathsf{\Phi}=0.2$).

**Figure 5.**Orbit of the spacecraft, with the thrust vectors, from the earth to the asteroid for a very short trip. This option had a high value of $\zeta $ at launch, $T=52.98$ days, ${m}_{\mathrm{prop},\mathrm{used}}=16.6$ kg, and $\mathsf{\Delta}{V}_{\mathrm{imp}}=7.46$ km/s.

Variable | Range |
---|---|

T | 1.0–1.5 yrs |

$\alpha $ | ±$\pi $ rad |

$\beta $ | 0–$\pi $ rad |

$\zeta $ | ±$\pi /2$ rad |

$\gamma $ | ±$\pi $ rad |

$\tau $ | ±$\pi /2$ rad |

$\psi $ | ±$\pi $ rad |

$\mathit{\alpha}$ (Rad) | $\mathit{\beta}$ (Rad) | $\mathit{\zeta}$ (Rad) | T (Days) | $\mathsf{\Delta}{\mathit{V}}_{\mathbf{imp}}\phantom{\rule{4pt}{0ex}}(\mathbf{km}/\mathbf{s})$ |
---|---|---|---|---|

−1.8 | 1.7 | −0.9 | 435.69 | 6.64 |

−1.6 | 1.7 | −0.9 | 435.44 | 6.63 |

−1.5 | 1.8 | −0.9 | 435.33 | 6.62 |

−0.8 | 1.8 | −0.9 | 434.55 | 6.58 |

−0.6 | 1.9 | −0.9 | 434.16 | 6.57 |

−0.3 | 1.9 | −0.9 | 433.72 | 6.57 |

0.1 | 1.9 | −0.9 | 432.77 | 6.59 |

0.3 | 1.9 | −0.9 | 432.41 | 6.60 |

0.5 | 1.9 | −0.9 | 432.21 | 6.62 |

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**MDPI and ACS Style**

Sankaran, K.; Griffith, S.A.; Thompson, N.C.; Lochridge, M.D.; O’Kins, A.S.
A Parallelized Genetic Algorithm to Evaluate Asteroid Impact Missions Using Electric Propulsion. *Aerospace* **2022**, *9*, 116.
https://doi.org/10.3390/aerospace9030116

**AMA Style**

Sankaran K, Griffith SA, Thompson NC, Lochridge MD, O’Kins AS.
A Parallelized Genetic Algorithm to Evaluate Asteroid Impact Missions Using Electric Propulsion. *Aerospace*. 2022; 9(3):116.
https://doi.org/10.3390/aerospace9030116

**Chicago/Turabian Style**

Sankaran, Kamesh, Scott A. Griffith, Noah C. Thompson, Matthew D. Lochridge, and Andrew S. O’Kins.
2022. "A Parallelized Genetic Algorithm to Evaluate Asteroid Impact Missions Using Electric Propulsion" *Aerospace* 9, no. 3: 116.
https://doi.org/10.3390/aerospace9030116