# Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits

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## Abstract

**:**

## 1. Introduction

## 2. Mission Description and Models Used

## 3. Transfer Strategy

## 4. Performance Assessment of Two-Phase Strategy

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

a | semimajor axis [au] |

${a}_{c}$ | characteristic acceleration [mm/s^{2}] |

${\mathit{a}}_{\mathrm{p}}$ | propulsive acceleration vector [mm/s^{2}] |

e | orbital eccentricity |

i | orbital inclination [deg] |

${i}_{f}$ | target orbit inclination [deg] |

${i}^{*}$ | maximum inclination with unconstrained transfer [deg] |

$\{{\widehat{\mathit{i}}}_{R},{\widehat{\mathit{i}}}_{T},{\widehat{\mathit{i}}}_{N}\}$ | unit vectors of radial–tangential–normal reference frame |

$\{{\widehat{\mathit{i}}}_{x},{\widehat{\mathit{i}}}_{y},{\widehat{\mathit{i}}}_{z}\}$ | unit vectors of body-fixed reference frame |

n | number of complete revolutions around the Sun |

r | distance from the Sun [au] |

${r}_{c}$ | minimum solar distance constraint [au] |

${r}_{f}$ | target orbit radius [au] |

${r}_{\oplus}$ | reference distance [$1\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$] |

t | time [days] |

$\delta $ | sail clock angle [deg] |

## References

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**Figure 1.**Definition of clock angle $\delta $, radial–tangential–normal reference frame, and direction of propulsive acceleration ${\mathit{a}}_{\mathrm{p}}$.

**Figure 2.**Simulations of optimal trajectories with ${a}_{c}=0.26\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$ to a circular heliocentric orbit of radius $r=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ for different values of final orbital inclination.

**Figure 3.**Illustration of the two-phase strategy with the approach phase and the cranking phase. The green and red solid line represent the orbit at the end of the first and final phase, respectively.

**Figure 4.**Time variation of orbital parameters and control variable in a near-optimal transfer to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ using the two-phase strategy (${a}_{c}=0.26\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; blue triangle → first phase end; dashed red line → minimum solar distance.

**Figure 5.**Near-optimal trajectory to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ using the two-phase strategy (${a}_{c}=0.26\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; blue triangle → first phase end; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.

**Figure 6.**Simulations of optimal trajectories with ${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$ to a circular heliocentric orbit of radius $r=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ for different values of final orbital inclination. Dashed red line → minimum solar distance.

**Figure 7.**Time variation of orbital parameters and control variable in a near-optimal transfer to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ using the two-phase strategy (${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; blue triangle → first phase end; dashed red line → minimum solar distance.

**Figure 8.**Near-optimal trajectory to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ using the two-phase strategy (${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; blue triangle → first phase end; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.

**Figure 9.**Time variation of orbital parameters and control variable in an optimal transfer to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ (${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; dashed red line → minimum solar distance.

**Figure 10.**Optimal trajectory to a circular heliocentric orbit of radius ${r}_{f}=0.32\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and inclination ${i}_{f}=60\phantom{\rule{0.166667em}{0ex}}\mathrm{deg}$ (${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$). Black circle → start; black square → arrival; blue star → perihelion; orange circle → the Sun; blue line → parking orbit; red line → target orbit; black line → transfer trajectory.

**Figure 11.**Comparison of the two-phase strategy (blue line) and the optimal control law (black line) with ${a}_{c}=0.183\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$. Dashed red line → minimum solar distance; filled circle → start; filled square → arrival.

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**MDPI and ACS Style**

Mengali, G.; Quarta, A.A.
Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits. *Appl. Sci.* **2024**, *14*, 2922.
https://doi.org/10.3390/app14072922

**AMA Style**

Mengali G, Quarta AA.
Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits. *Applied Sciences*. 2024; 14(7):2922.
https://doi.org/10.3390/app14072922

**Chicago/Turabian Style**

Mengali, Giovanni, and Alessandro A. Quarta.
2024. "Optimal Trajectories of Diffractive Sail to Highly Inclined Heliocentric Orbits" *Applied Sciences* 14, no. 7: 2922.
https://doi.org/10.3390/app14072922