# E-Sail Optimal Trajectories to Heliostationary Points

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## Abstract

**:**

## 1. Introduction

## 2. Mission Description and Mathematical Problem Statement

#### 2.1. E-Sail Performance Requirements

#### 2.2. Spacecraft Dynamics and Trajectory Optimization

## 3. Numerical Simulations and Parametric Analysis

**Direct transfer (DT).**During the optimal transfer, the Sun–spacecraft distance r continuously increases with time until the vehicle reaches the target HP, so that the perihelion distance of the optimal transfer coincides with the radius ${r}_{\oplus}$ of the circular parking orbit; see Figure 5a.**Solar wind assist with inactive perihelion constraint (iSWA).**In this case, the minimum-time transfer trajectory contains a phase where the spacecraft approaches the Sun to increase its propulsive acceleration magnitude according to the thrust model of Equation (1). Paralleling the nomenclature used for a solar sail mission case [36], this behavior can be seen as a sort of solar wind assist (SWA) because of the thrust increase due to the variation of solar wind plasma density. The approaching phase ends when the spacecraft reaches a perihelion distance $min\left(r\right)$ that, in this case, is greater than the minimum admissible value ${r}_{p}=0.33\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$, so that the inequality constraint (14) is naturally satisfied (case of inactive constraint); see Figure 5b.**Solar wind assist with active perihelion constraint (aSWA).**This case is similar to the preceding one, with the only difference that the perihelion distance of the transfer trajectory is equal to ${r}_{p}=0.33\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ as sketched in Figure 5c. In other terms, $min\left(r\right)={r}_{p}$, and thus the constraint (14) on the solar distance becomes active at the perihelion.

#### 3.1. Two-Dimensional Trajectory

#### 3.2. Three-Dimensional Scenario

#### 3.3. Mission Application

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

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

${a}_{r}$ | radial component of $\mathit{a}$ [mm/s${}^{2}$] |

${a}_{\theta}$ | transverse component of $\mathit{a}$ [mm/s${}^{2}$] |

${a}_{\varphi}$ | azimuthal component of $\mathit{a}$ [mm/s${}^{2}$] |

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

$\mathcal{H}$ | Hamiltonian function |

h | distance from the ecliptic [au] |

${\widehat{\mathit{i}}}_{r}$ | radial unit vector |

${\widehat{\mathit{i}}}_{\theta}$ | transverse unit vector |

${\widehat{\mathit{i}}}_{\varphi}$ | azimuthal unit vector |

J | performance index [days] |

O | Sun’s center of mass |

r | radial distance [au] |

${r}_{p}$ | minimum perihelion radius [au] |

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

${S}_{w}$ | switching function |

t | time [days] |

$\mathcal{T}(O;r,\theta ,\varphi )$ | spherical reference frame |

${\mathcal{T}}_{S}(O;x,y,z)$ | heliocentric-ecliptic reference frame |

${v}_{r}$ | radial component of the spacecraft velocity vector [km/s] |

${v}_{\theta}$ | transverse component of the spacecraft velocity vector [km/s] |

${v}_{\varphi}$ | azimuthal component of the spacecraft velocity vector [km/s] |

$\alpha $ | sail cone angle [rad] |

$\mathsf{\Delta}t$ | flight time [days] |

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

$\theta $ | ecliptic longitude [rad] |

${\lambda}_{r}$ | variable adjoint to r |

${\lambda}_{\theta}$ | variable adjoint to $\theta $ |

${\lambda}_{\varphi}$ | variable adjoint to $\varphi $ |

${\lambda}_{{v}_{r}}$ | variable adjoint to ${v}_{r}$ |

${\lambda}_{{v}_{\theta}}$ | variable adjoint to ${v}_{\theta}$ |

${\lambda}_{{v}_{\varphi}}$ | variable adjoint to ${v}_{\varphi}$ |

${\mu}_{\odot}$ | Sun’s gravitational parameter [km${}^{3}$/s${}^{2}$] |

$\tau $ | dimensionless switching parameter |

$\varphi $ | ecliptic latitude [rad] |

Subscripts | |

0 | initial, parking orbit |

f | final |

HP | heliostationary point |

p | perihelion |

Superscripts | |

· | derivative with respect to time |

${}^{\prime}$ | function of control variables |

## Appendix A

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**Figure 2.**HP potential locations, as a function of $\{{r}_{\mathrm{HP}},{\varphi}_{\mathrm{HP}}\}$, in the heliocentric-ecliptic reference frame.

**Figure 3.**Characteristic acceleration necessary for a spacecraft to maintain an HP as a function of its solar distance ${r}_{\mathrm{HP}}$; see also Equation (3).

**Figure 6.**The minimum flight time as a function of the Sun–HP distance in a two-dimensional mission scenario.

**Figure 7.**Optimal transfer trajectory characteristics as a function of ${r}_{\mathrm{HP}}$ in a two-dimensional mission scenario.

**Figure 8.**Optimal transfer trajectories in a two-dimensional mission scenario when ${r}_{\mathrm{HP}}\in \{2,4,6,8\}\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$. The radial distance is in astronomical units, black circle → start, and black square → arrival/equilibrium point.

**Figure 9.**Minimum flight time and final ecliptic longitude as a function of the HP ecliptic latitude in a three-dimensional mission scenario with ${r}_{\mathrm{HP}}=2.5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}\to \mathrm{black}\phantom{\rule{4.pt}{0ex}}\mathrm{line}$, ${r}_{\mathrm{HP}}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{au}\to \mathrm{red}\phantom{\rule{4.pt}{0ex}}\mathrm{line}$, and ${r}_{\mathrm{HP}}=3.5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}\to \mathrm{blue}\phantom{\rule{4.pt}{0ex}}\mathrm{line}$.

**Figure 10.**Optimal DT trajectories in a three-dimensional mission scenario as a function of ${\varphi}_{\mathrm{HP}}$ when ${r}_{\mathrm{HP}}=3\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$. Black circle → start, and black square → arrival/equilibrium point.

**Figure 11.**The minimum flight time and optimal transfer trajectory to reach a polar HP as a function of the solar distance.

**Figure 12.**Tie variation of the control variables $\{\tau ,\phantom{\rule{0.166667em}{0ex}}\alpha ,\phantom{\rule{0.166667em}{0ex}}\delta \}$ when ${r}_{\mathrm{HP}}=2.5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$. Black circle → start, and black square → arrival/equilibrium point.

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**MDPI and ACS Style**

Quarta, A.A.; Mengali, G.
E-Sail Optimal Trajectories to Heliostationary Points. *Aerospace* **2023**, *10*, 194.
https://doi.org/10.3390/aerospace10020194

**AMA Style**

Quarta AA, Mengali G.
E-Sail Optimal Trajectories to Heliostationary Points. *Aerospace*. 2023; 10(2):194.
https://doi.org/10.3390/aerospace10020194

**Chicago/Turabian Style**

Quarta, Alessandro A., and Giovanni Mengali.
2023. "E-Sail Optimal Trajectories to Heliostationary Points" *Aerospace* 10, no. 2: 194.
https://doi.org/10.3390/aerospace10020194