# Trajectory Analysis and Optimization of Hesperides Mission

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Spacecraft Characteristics and Baseline Mission Description

^{2}. The electric propulsion system uses xenon propellant exhausted at a constant mass flow rate and its specific impulse is ${I}_{sp}=4000\phantom{\rule{0.166667em}{0ex}}\mathrm{s}$. An estimate of the masses of the main spacecraft subsystems is reported in Table 1 and is illustrated in the pie chart of Figure 3.

**Pre-perihelion phase.**The spacecraft is launched from Earth toward an heliocentric orbit with perihelion distance ${r}_{P}=0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ and aphelion distance ${r}_{A}=2.5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$. The required initial hyperbolic excess velocity is provided by an upper stage of the launcher, and the spacecraft reaches the target orbit with a flight time of about 2 years without using the solar sail or the electric thruster. In particular, the spacecraft mass in this phase is constant and equal to 450 kg.**Solar sail acceleration.**When the spacecraft is at the perihelion of the heliocentric orbit (end of phase 1) the solar sail is unfurled with a Sun-facing attitude [20,21,22,23], that is, its nominal plane is oriented normal to the Sun. In this phase, the spacecraft is continuously accelerated by the solar sail induced thrust until it reaches a solar distance of 5 au, when the sail propulsive acceleration becomes ineffective and is therefore jettisoned. During this whole phase, the spacecraft mass remains equal to 450 kg. The flight time of this phase is about 1 year.**Radioisotope-electric propulsion.**This phase starts when the solar sail is discarded, so that the spacecraft mass suddenly reduces to 249 kg. The radioisotope-electric thruster is switched on and is continuously operated until the whole available xenon propellant is exhausted at a constant mass flow rate $\tau =3.17\times {10}^{-7}\phantom{\rule{0.166667em}{0ex}}\mathrm{kg}/\mathrm{s}$. Since the total propellant mass is 100 kg, the phase (time) length is 10 years. During this phase the induced thrust is nearly along the radial direction, so that the spacecraft (heliocentric) hyperbolic excess velocity increases from about 35.5 km/s to 56 km/s, at a distance of about 100 au away from the Sun. At the end of this phase, the spacecraft mass is reduced to 149 kg.**Cruise to 600 au.**The last (baseline) trajectory part is a coasting phase with a constant velocity (with respect the Sun), equal to that reached at the end of phase 3. The spacecraft continues its (Keplerian) flight for about 42.5 years until it reaches a target distance of 600 au from the Sun. The spacecraft mass in this phase is constant and equal to 149 kg.

## 3. Mission Optimization

#### 3.1. Phase 1

#### 3.1.1. Case a

#### 3.1.2. Case b

#### 3.1.3. Case c

#### 3.2. Phase 2

#### 3.3. Single Optimization of Phases 1 and 2

#### 3.4. Phase 3

#### 3.5. Phase 4

## 4. Parametric Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Clusters of small satellites in a string-of-pearls arrangement towards the Sun’s gravitational lens. Courtesy of and reprinted with permission of The Aerospace Corporation.

**Figure 3.**Mass breakdown model for Hesperides spacecraft [18].

**Figure 4.**Graphical sketch and spacecraft mass variation of Hesperides baseline mission [18].

**Figure 10.**Minimum-time transfer trajectory to reach a heliocentric distance of $0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ with zero radial velocity (case c).

**Figure 11.**Osculating parameters of the transfer orbit, spacecraft velocity components and optimal steering law to reach a distance of $0.2\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ with zero radial velocity (case c).

**Figure 12.**Osculating parameters of the transfer orbit, spacecraft velocity components and optimal steering law to reach a distance of $5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ with ${V}_{\infty}=35.5\phantom{\rule{0.166667em}{0ex}}\mathrm{km}/\mathrm{s}$.

**Figure 13.**Minimum-time transfer trajectory to reach a heliocentric distance of $5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ with a hyperbolic excess velocity ${V}_{\infty}=35.5\phantom{\rule{0.166667em}{0ex}}\mathrm{km}/\mathrm{s}$.

**Figure 14.**Heliocentric distance, velocity components and optimal steering law to reach a distance of $5\phantom{\rule{0.166667em}{0ex}}\mathrm{au}$ with ${V}_{\infty}=35.5\phantom{\rule{0.166667em}{0ex}}\mathrm{km}/\mathrm{s}$.

**Figure 16.**Flight time necessary to complete phases 1 and 2 as a function of the hyperbolic excess velocity at the end of phase 2.

**Figure 17.**Hyperbolic excess velocity at the end of phase 3 as a function of that at the end of phase 2.

**Figure 18.**Total mission time as a function of the hyperbolic excess velocity at the end of phase 2.

**Table 1.**Hesperides spacecraft subsystems mass [18].

Subsystem | Mass (kg) |
---|---|

Solar sail | 201 |

Ion thruster | 10 |

Radioisotope electric generator | 40 |

Propellant | 100 |

Payload | 99 |

Total | 450 |

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Mengali, G.; Quarta, A.A.
Trajectory Analysis and Optimization of Hesperides Mission. *Universe* **2022**, *8*, 364.
https://doi.org/10.3390/universe8070364

**AMA Style**

Mengali G, Quarta AA.
Trajectory Analysis and Optimization of Hesperides Mission. *Universe*. 2022; 8(7):364.
https://doi.org/10.3390/universe8070364

**Chicago/Turabian Style**

Mengali, Giovanni, and Alessandro A. Quarta.
2022. "Trajectory Analysis and Optimization of Hesperides Mission" *Universe* 8, no. 7: 364.
https://doi.org/10.3390/universe8070364