# Solar Sail Trajectories to Earth’s Trojan Asteroids

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Description and Mathematical Model

#### 2.1. Solar Sail Thrust Model

#### 2.2. Spacecraft Dynamics

#### 2.3. Trajectory Optimization

## 3. Numerical Results

#### Case Study

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Notation | |

a | semimajor axis [au] |

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

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

$\mathbb{A}$ | matrix, see Equation (15) |

${A}_{ij}$ | generic entry of matrix $\mathbb{A}$, see Equations (16)–(25) |

$\mathit{b}$ | vector, see Equation (14) |

${B}_{f}$ | non-Lambertian coefficient of the front sail film |

${B}_{b}$ | non-Lambertian coefficient of the back sail film |

$\{{b}_{1},{b}_{2},{b}_{3}\}$ | force coefficients, see Table 6 |

$\mathit{c}$ | auxiliary vector, see Equation (41) |

$\{{d}_{s},{d}_{c}\}$ | auxiliary functions, see Equations (43) and (44) |

e | eccentricity |

$\mathcal{H}$ | Hamiltonian function |

${\mathcal{H}}^{\prime}$ | reduced Hamiltonian function, see Equation (41) |

i | orbital inclination [deg] |

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

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

${\widehat{\mathit{i}}}_{\mathrm{N}}$ | normal unit vector |

J | performance index [days] |

$\widehat{\mathit{n}}$ | unit vector normal to the sail plane |

$\{p,f,g,h,k,L\}$ | MEOEs |

$\{{p}_{1},{p}_{2},{q}_{1},{q}_{2}\}$ | best fit coefficients, see Equation (47) |

r | radial distance [au] |

$\widehat{\mathit{r}}$ | Sun–spacecraft unit vector |

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

t | time [days] |

$\widehat{\mathit{v}}$ | spacecraft inertial velocity unit vector |

$\mathit{x}$ | spacecraft state vector |

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

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

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

${\u03f5}_{f}$ | emissivity coefficient of the front sail film |

${\u03f5}_{b}$ | emissivity coefficient of the back sail film |

$\lambda $ | adjoint vector, see Equation (39) |

${\lambda}_{y}$ | generic adjoint variable |

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

$\omega $ | argument of periapse [deg] |

$\mathrm{\Omega}$ | right ascension of the ascending node [deg] |

$\rho $ | sail film reflection coefficient |

Subscripts | |

0 | initial, parking orbit |

f | final, target orbit |

Superscripts | |

· | derivative with respect to time |

## References

- Levison, H.F.; Shoemaker, E.M.; Shoemaker, C.S. Dynamical evolution of Jupiter’s Trojan asteroids. Nature
**1997**, 385, 42–44. [Google Scholar] [CrossRef] - Almeida, A.J.C.; Peixinho, N.; Correia, A.C.M. Neptune Trojans and Plutinos: Colors, sizes, dynamics, and their possible collisions. Astron. Astrophys.
**2009**, 508, 1021–1030. [Google Scholar] [CrossRef] - Trilling, D.E.; Rivkin, A.S.; Stansberry, J.A.; Spahr, T.B.; Crudo, R.A.; Davies, J.K. Albedos and diameters of three Mars Trojan asteroids. Icarus
**2007**, 192, 442–447. [Google Scholar] [CrossRef] [Green Version] - Napier, K.J.; Markwardt, L.; Adams, F.C.; Gerdes, D.W.; Lin, H.W. A Collision Mechanism for the Removal of Earth’s Trojan Asteroids. Planet. Sci. J.
**2022**, 3, 121. [Google Scholar] [CrossRef] - Mikkola, S.; Innanen, K.A. Studies on solar system dynamics. II - The stability of Earth’s Trojans. Astron. J.
**1990**, 100, 290–293. [Google Scholar] [CrossRef] - de la Fuente Marcos, C.; de la Fuente Marcos, R. Transient Terrestrial Trojans: Comparative Short-term Dynamical Evolution of 2010 TK7 and 2020 XL5. Res. Notes AAS
**2021**, 5, 29. [Google Scholar] [CrossRef] - Wiegert, P.; Innanen, K.; Mikkola, S. Earth Trojan Asteroids: A Study in Support of Observational Searches. Icarus
**2000**, 145, 33–43. [Google Scholar] [CrossRef] [Green Version] - Connors, M.; Wiegert, P.; Veillet, C. Earth’s Trojan asteroid. Nature
**2011**, 475, 481–483. [Google Scholar] [CrossRef] - Santana-Ros, T.; Micheli, M.; Faggioli, L.; Cennamo, R.; Devogèle, M.; Alvarez-Candal, A.; Oszkiewicz, D.; Ramírez, O.; Liu, P.Y.; Benavidez, P.G.; et al. Orbital stability analysis and photometric characterization of the second Earth Trojan asteroid 2020 XL
_{5}. Nat. Commun.**2022**, 13, 447. [Google Scholar] [CrossRef] - Lei, H.; Xu, B.; Zhang, L. Trajectory design for a rendezvous mission to Earth’s Trojan asteroid 2010 TK
_{7}. Adv. Space Res.**2017**, 60, 2505–2517. [Google Scholar] [CrossRef] - Fu, B.; Sperber, E.; Eke, F. Solar sail technology—A state of the art review. Prog. Aerosp. Sci.
**2016**, 86, 1–19. [Google Scholar] [CrossRef] - Gong, S.; Macdonald, M. Review on solar sail technology. Astrodynamics
**2019**, 3, 93–125. [Google Scholar] [CrossRef] - Zhao, P.; Wu, C.; Li, Y. Design and application of solar sailing: A review on key technologies. Chin. J. Aeronaut. 2022; in press. [Google Scholar] [CrossRef]
- Morrow, E.; Scheeres, D.J.; Lubin, D. Solar Sail Orbit Operations at Asteroids. J. Spacecr. Rocket.
**2001**, 38, 279–286. [Google Scholar] [CrossRef] [Green Version] - Mengali, G.; Quarta, A.A. Rapid Solar Sail Rendezvous Missions to Asteroid 99942 Apophis. J. Spacecr. Rocket.
**2009**, 46, 134–140. [Google Scholar] [CrossRef] - Dachwald, B.; Seboldt, W.; Richter, L. Multiple rendezvous and sample return missions to near-Earth objects using solar sailcraft. Acta Astronaut.
**2006**, 59, 768–776. [Google Scholar] [CrossRef] - Peloni, A.; Ceriotti, M.; Dachwald, B. Solar-Sail Trajectory Design for a Multiple Near-Earth-Asteroid Rendezvous Mission. J. Guid. Control. Dyn.
**2016**, 39, 2712–2724. [Google Scholar] [CrossRef] [Green Version] - Song, Y.; Gong, S. Solar-sail trajectory design for multiple near-Earth asteroid exploration based on deep neural networks. Aerosp. Sci. Technol.
**2019**, 91, 28–40. [Google Scholar] [CrossRef] [Green Version] - Wright, J.L. Space Sailing; Gordon and Breach Science Publishers: Amsterdam, The Netherlands, 1992. [Google Scholar]
- McInnes, C.R. Solar Sailing: Technology, Dynamics and Mission Applications; Springer: Berlin, Germany, 1999; Chapter 2; pp. 46–53. [Google Scholar] [CrossRef]
- Murphy, D.; Trautt, T. Solar Sail Propulsion Modeling. In Proceedings of the 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Honolulu, HI, USA, 23–25 April 2007. [Google Scholar] [CrossRef]
- Mengali, G.; Quarta, A.A.; Circi, C.; Dachwald, B. Refined solar sail force model with mission application. J. Guid. Control. Dyn.
**2007**, 30, 512–520. [Google Scholar] [CrossRef] [Green Version] - Dachwald, B.; Mengali, G.; Quarta, A.A.; Macdonald, M. Parametric model and optimal control of solar sails with optical degradation. J. Guid. Control. Dyn.
**2006**, 29, 1170–1178. [Google Scholar] [CrossRef] - Boni, L.; Mengali, G.; Quarta, A.A. Finite Element Analysis of Solar Sail Force Model with Mission Application. Proc. Inst. Mech. Eng. Part G J. Aerosp. Eng.
**2019**, 233, 1838–1846. [Google Scholar] [CrossRef] - Rios-Reyes, L.; Scheeres, D.J. Generalized Model for Solar Sails. J. Spacecr. Rocket.
**2005**, 42, 182–185. [Google Scholar] [CrossRef] [Green Version] - Slade, K.; Belvin, K.; Behun, V. Solar Sail Loads, Dynamics, and Membrane Studies. In Proceedings of the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Denver, CO, USA, 22–25 April 2002. [Google Scholar] [CrossRef] [Green Version]
- Li, Q.; Ma, X.; Wang, T. Reduced Model for Flexible Solar Sail Dynamics. J. Spacecr. Rocket.
**2011**, 48, 446–453. [Google Scholar] [CrossRef] - Dachwald, B.; Macdonald, M.; McInnes, C.R.; Mengali, G.; Quarta, A.A. Impact of optical degradation on solar sail mission performance. J. Spacecr. Rocket.
**2007**, 44, 740–749. [Google Scholar] [CrossRef] - Dachwald, B.; Seboldt, W.; Macdonald, M.; Mengali, G.; Quarta, A.A.; McInnes, C.R.; Rios-Reyes, L.; Scheeres, D.J.; Wie, B.; Görlich, M.; et al. Potential Solar Sail Degradation Effects on Trajectory and Attitude Control. In Proceedings of the AIAA Guidance, Navigation, and Control Conference and Exhibit, San Francisco, CA, USA, 15–18 August 2005. [Google Scholar] [CrossRef] [Green Version]
- Mengali, G.; Quarta, A.A. Optimal three-dimensional interplanetary rendezvous using nonideal solar sail. J. Guid. Control. Dyn.
**2005**, 28, 173–177. [Google Scholar] [CrossRef] - Pezent, J.B.; Sood, R.; Heaton, A.; Miller, K.; Johnson, L. Preliminary trajectory design for NASA’s Solar Cruiser: A technology demonstration mission. Acta Astronaut.
**2021**, 183, 134–140. [Google Scholar] [CrossRef] - Bassetto, M.; Niccolai, L.; Boni, L.; Mengali, G.; Quarta, A.A.; Circi, C.; Pizzurro, S.; Pizzarelli, M.; Pellegrini, R.C.; Cavallini, E. Sliding mode control for attitude maneuvers of Helianthus solar sail. Acta Astronaut.
**2022**, 198, 100–110. [Google Scholar] [CrossRef] - Boni, L.; Bassetto, M.; Niccolai, L.; Mengali, G.; Quarta, A.A.; Circi, C.; Pellegrini, R.C.; Cavallini, E. Structural response of Helianthus solar sail during attitude maneuvers. Aerosp. Sci. Technol.
**2023**, 133, 108152. [Google Scholar] [CrossRef] - Walker, M.J.H.; Ireland, B.; Owens, J. A set of modified equinoctial orbit elements. Celest. Mech.
**1985**, 36, 409–419. [Google Scholar] [CrossRef] - Walker, M.J.H. Erratum: A set of modified equinoctial orbit elements. Celest. Mech.
**1986**, 38, 391–392. [Google Scholar] [CrossRef] [Green Version] - Betts, J.T. Very low-thrust trajectory optimization using a direct SQP method. J. Comput. Appl. Math.
**2000**, 120, 27–40. [Google Scholar] [CrossRef] [Green Version] - Betts, J.T. Survey of Numerical Methods for Trajectory Optimization. J. Guid. Control. Dyn.
**1998**, 21, 193–207. [Google Scholar] [CrossRef] [Green Version] - Forsythe, G.E.; Malcolm, M.A.; Moler, C.B. Computer Methods for Mathematical Computations; Prentice Hall: Englewood Cliffs, NJ, USA, 1976. [Google Scholar]
- Bate, R.R.; Mueller, D.D.; White, J.E. Fundamentals of Astrodynamics; Dover Publications: New York, NY, USA, 1971; Chapter 2; pp. 53–55. [Google Scholar]
- Bryson, A.E., Jr.; Ho, Y.C. Applied Optimal Control; Hemisphere Publishing Corporation: New York, NY, USA, 1975. [Google Scholar]
- Stengel, R.F. Optimal Control and Estimation; Dover Publications: Mineola, NY, USA, 1994; pp. 222–254. [Google Scholar]

**Figure 4.**Simulation results (red circle) and best fit interpolation (dashed black line) of the flight time as a function of the characteristic acceleration.

**Figure 5.**Optimal Earth–ETa transfer trajectory (black line), assuming an optical force model, as a function of the characteristic acceleration. Blue line → Earth’s orbit, red line → asteroid orbit, blue circle → start, red circle → arrival, blue star → Earth’s orbit perihelion, red star → asteroid orbit perihelion, yellow circle → Sun.

**Figure 6.**Optimal trajectory in an Earth–asteroid 2020 XL${}_{5}$ transfer, with ${a}_{c}=0.12\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$ and an optical force model. The figure legend is consistent with that reported in the label of Figure 5.

**Figure 7.**Time variation of the osculating orbit elements and propulsive acceleration components (in an RTN reference frame) in an Earth–asteroid 2020 XL${}_{5}$ transfer, with ${a}_{c}=0.12\phantom{\rule{0.166667em}{0ex}}\mathrm{mm}/{\mathrm{s}}^{2}$ and an optical force model. Blue circle → start; red circle → arrival.

**Table 1.**Reflective film optical coefficients in an ideal case, and in a sail membrane with a highly reflective aluminum coated front side and a highly emissive chromium-coated backside.

Force Model | $\mathit{\rho}$ | s | ${\mathit{B}}_{\mathit{f}}$ | ${\mathit{B}}_{\mathit{b}}$ | ${\mathit{\u03f5}}_{\mathit{f}}$ | ${\mathit{\u03f5}}_{\mathit{b}}$ |
---|---|---|---|---|---|---|

ideal | 1 | 1 | $2/3$ | $2/3$ | 0 | 0 |

optical | $0.88$ | $0.94$ | $0.79$ | $0.55$ | $0.05$ | $0.55$ |

**Table 2.**Force coefficients for an ideal and an optical force model obtained through Equations (3)–(5).

Force Model | ${\mathit{b}}_{1}$ | ${\mathit{b}}_{2}$ | ${\mathit{b}}_{3}$ |
---|---|---|---|

ideal | 0 | 2 | 0 |

optical | $0.1728$ | $1.6544$ | $-0.0109$ |

Celestial Body | a [au] | e | i [deg] | $\mathit{\omega}$ [deg] | $\mathbf{\Omega}$ [deg] |
---|---|---|---|---|---|

Earth | $1.0008$ | $1.5940\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | $3.0225\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | $302.9781$ | $159.8640$ |

2010 TK${}_{7}$ | $1.0001$ | $1.9076\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $20.8847$ | $45.8665$ | $96.5194$ |

2020 XL${}_{5}$ | $1.0007$ | $3.8721\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | $13.8467$ | $87.9847$ | $153.6008$ |

**Table 4.**Earth–asteroid 2010 TK${}_{7}$ minimum-time performance as a function of the characteristic acceleration and the sail force model.

${\mathit{a}}_{\mathit{c}}$ [mm/s${}^{2}$] | Force Model | ${\mathit{t}}_{\mathit{f}}$ [Days] | $\mathit{\nu}\left({\mathit{t}}_{0}\right)$ [deg] | $\mathit{\nu}\left({\mathit{t}}_{\mathit{f}}\right)$ [deg] | N |
---|---|---|---|---|---|

0.1 | optical | 5032.8 | 280.3 | 196.2 | 15 |

0.1 | ideal | 4830.9 | 264.4 | 214.3 | 13 |

0.2 | optical | 2530.9 | 281.1 | 193.9 | 7 |

0.2 | ideal | 2215.7 | 306.3 | 158.3 | 7 |

0.3 | optical | 1644.5 | 311.3 | 154.2 | 5 |

0.3 | ideal | 1494.0 | 347.3 | 129.7 | 5 |

0.4 | optical | 1271.4 | 284.5 | 188.2 | 3 |

0.4 | ideal | 1119.3 | 304.6 | 160.1 | 3 |

0.5 | optical | 1110.7 | 266.8 | 245.0 | 3 |

0.5 | ideal | 910.2 | 291.8 | 176.5 | 2 |

0.6 | optical | 838.4 | 306.4 | 155.6 | 2 |

0.6 | ideal | 758.4 | 318.6 | 133.5 | 2 |

0.7 | optical | 733.8 | 336.4 | 126.3 | 2 |

0.7 | ideal | 723.9 | 291.0 | 276.3 | 2 |

0.8 | optical | 710.1 | 317.5 | 289.9 | 2 |

0.8 | ideal | 643.3 | 5.2 | 286.5 | 1 |

0.9 | optical | 640.4 | 32.2 | 278.5 | 1 |

0.9 | ideal | 564.4 | 110.5 | 279.0 | 1 |

1 | optical | 535.1 | 86.7 | 189.6 | 1 |

1 | ideal | 471.4 | 103.8 | 168.0 | 1 |

**Table 5.**Earth–asteroid 2020 XL${}_{5}$ minimum-time performance as a function of the characteristic acceleration and the sail force model.

${\mathit{a}}_{\mathit{c}}$ [mm/s${}^{2}$] | Force Model | ${\mathit{t}}_{\mathit{f}}$ [Days] | $\mathit{\nu}\left({\mathit{t}}_{0}\right)$ [deg] | $\mathit{\nu}\left({\mathit{t}}_{\mathit{f}}\right)$ [deg] | N |
---|---|---|---|---|---|

0.1 | optical | 4008.9 | 150.5 | 195 | 12 |

0.1 | ideal | 3478.8 | 213.8 | 119.5 | 11 |

0.2 | optical | 1868.1 | 195.2 | 127.2 | 6 |

0.2 | ideal | 1724.2 | 224.9 | 116.2 | 6 |

0.3 | optical | 1233.4 | 205.5 | 122.2 | 4 |

0.3 | ideal | 1140.7 | 237.4 | 113.2 | 4 |

0.4 | optical | 919 | 214.0 | 119.1 | 3 |

0.4 | ideal | 849.6 | 245.4 | 112.0 | 3 |

0.5 | optical | 710.2 | 185.1 | 132.4 | 2 |

0.5 | ideal | 643.7 | 205.6 | 119.0 | 2 |

0.6 | optical | 608.6 | 228.0 | 113.4 | 2 |

0.6 | ideal | 561.6 | 256.8 | 108.4 | 1 |

0.7 | optical | 547.9 | 282.2 | 110.2 | 1 |

0.7 | ideal | 504.3 | 337.5 | 111.1 | 1 |

0.8 | optical | 598 | 225.0 | 292.8 | 1 |

0.8 | ideal | 559.6 | 246.7 | 286.6 | 1 |

0.9 | optical | 568.2 | 248.2 | 292.1 | 1 |

0.9 | ideal | 533.8 | 272.0 | 285.3 | 1 |

1 | optical | 546.6 | 269.3 | 290.8 | 1 |

1 | ideal | 514.7 | 294.9 | 283.6 | 1 |

**Table 6.**Best fit coefficients in Equation (47) as a function of the mission scenario and the sail force model.

Scenario | Force Model | ${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | ${\mathit{q}}_{1}$ | ${\mathit{q}}_{2}$ |
---|---|---|---|---|---|

Earth-2010 TK${}_{7}$ | ideal | 12,285 | 41,589 | $107.6$ | $-1.896$ |

Earth-2010 TK${}_{7}$ | optical | $\mathrm{275,545.8}$ | $\mathrm{3,314,602.08}$ | 6648 | $0.1183$ |

Earth-2020 XL${}_{5}$ | ideal | $362.83$ | $-109.47$ | $-0.297$ | $-1.425\phantom{\rule{3.33333pt}{0ex}}\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

Earth-2020 XL${}_{5}$ | optical | $\mathrm{663,873.84}$ | $\mathrm{1,060,354.05}$ | 3738 | $-93.04$ |

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**MDPI and ACS Style**

Quarta, A.A.; Mengali, G.
Solar Sail Trajectories to Earth’s Trojan Asteroids. *Universe* **2023**, *9*, 186.
https://doi.org/10.3390/universe9040186

**AMA Style**

Quarta AA, Mengali G.
Solar Sail Trajectories to Earth’s Trojan Asteroids. *Universe*. 2023; 9(4):186.
https://doi.org/10.3390/universe9040186

**Chicago/Turabian Style**

Quarta, Alessandro A., and Giovanni Mengali.
2023. "Solar Sail Trajectories to Earth’s Trojan Asteroids" *Universe* 9, no. 4: 186.
https://doi.org/10.3390/universe9040186