Abstract
In this study, new analytical solutions to the equations of motion of a propelled spacecraft are investigated using a shape-based approach. There is an assumption that the spacecraft travels a two-dimensional spiral trajectory in which the orbital radius is proportional to an assigned power of the spacecraft angular coordinate. The exact solution to the equations of motion is obtained as a function of time in the case of a purely radial thrust, and the propulsive acceleration magnitude necessary for the spacecraft to track the prescribed spiral trajectory is found in a closed form. The analytical results are then specialized to the case of a generalized sail, that is, a propulsion system capable of providing an outward radial propulsive acceleration, the magnitude of which depends on a given power of the Sun-spacecraft distance. In particular, the conditions for an outward radial thrust and the required sail performance are quantified and thoroughly discussed. It is worth noting that these propulsion systems provide a purely radial thrust when their orientation is Sun-facing. This is an important advantage from an engineering point of view because, depending on the particular propulsion system, a Sun-facing attitude can be stable or obtainable in a passive way. A case study is finally presented, where the generalized sail is assumed to start the spiral trajectory from the Earth’s heliocentric orbit. The main outcome is that the required sail performance is in principle achievable on the basis of many results available in the literature.
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11 February 2022
A Correction to this paper has been published: https://doi.org/10.1007/s42064-022-0136-2
Abbreviations
- a :
-
semimajor axis (km)
- a c :
-
characteristic acceleration (mm/s2)
- a r :
-
propulsive acceleration, with \(a_{r}\triangleq a_{r}\cdot\hat{r}\) (mm/s2)
- {c 1, c 2}:
-
constants of integration, see Eq. (12)
- e :
-
eccentricity vector, with e ≜ ‖e‖
- h :
-
specific angular momentum magnitude (km2/s)
- O :
-
primary body center of mass
- p :
-
semilatus rectum (km)
- r :
-
orbital radius (km)
- r :
-
radial unit vector
- r ⊕ :
-
reference distance 1 au
- S :
-
spacecraft center of mass
- t :
-
time (year)
- \(\cal{T}\) :
-
polar reference frame
- υ :
-
spacecraft velocity vector (km/s)
- υ r :
-
radial component of υ (km/s)
- υ θ :
-
transversal component of υ (km/s)
- α :
-
dimensionless spiral parameter, see Eq. (6)
- β :
-
constant, see Eq. (61)
- γ :
-
dimensionless design parameter, see Eq. (38)
- θ :
-
polar angle (rad)
- \(\hat{\theta}\) :
-
transversal unit vector
- μ :
-
gravitational parameter (km3/s2)
- ν :
-
spacecraft true anomaly (rad)
- χ :
-
dimensionless auxiliary function, see Eq. (14)
- ω :
-
argument of periapsis (rad)
- 0:
-
initial, parking orbit
- max:
-
maximum
- ⊕:
-
the Earth
- ⨀:
-
the Sun
- ·:
-
time derivative
- ∼:
-
threshold value
- *:
-
optimal
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Open Access funding provided by University of Pisa within the CRUICARE Agreement.
Marco Bassetto graduated in aerospace engineering at the University of Pisa in 2016. In 2019, he received his Ph.D. degree in civil and industrial engineering at the Department of Civil and Industrial Engineering of the University of Pisa. He currently holds a post-doc scholarship in the same department. His research activity focuses on trajectory design and attitude control of spacecraft propelled with low-thrust propulsion systems such as solar sails and electric solar wind sails.
Alessandro A. Quarta received his Ph.D. degree in aerospace engineering from the University of Pisa in 2005, and is currently a professor of flight mechanics at the Department of Civil and Industrial Engineering of the University of Pisa. His main research areas include spaceflight simulation, spacecraft mission analysis and design, low-thrust trajectory optimization, solar sail, and E-sail dynamics and control.
Giovanni Mengali received his doctor of engineering degree in aeronautical engineering in 1989 from the University of Pisa. Since 1990, he has been with the Department of Aerospace Engineering (now Department of Civil and Industrial Engineering) of the University of Pisa, first as a Ph.D. student, then as an assistant and an associate professor. Currently, he is a professor of spaceflight mechanics. His main research areas include spacecraft mission analysis, trajectory optimization, solar sails, electric sails, and aircraft flight dynamics and control.
Vittorio Cipolla received his Ph.D. degree discussing a thesis on high altitude-long endurance UAVs powered by solar energy. Between 2011 and 2019 he has participated in several research projects, including “PARSIFAL” (PrandtlPlane Architecture for the Sustainable Improvement of Future Airplanes) and “PROSIB” (hybrid propulsion systems for fixed and rotary wing aircraft). Since 2018 he is a research fellow at the University of Pisa, where he also teaches applied aeroelasticity in M.Sc. course of aerospace engineering.
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Bassetto, M., Quarta, A.A., Mengali, G. et al. Spiral trajectories induced by radial thrust with applications to generalized sails. Astrodyn 5, 121–137 (2021). https://doi.org/10.1007/s42064-020-0093-6
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DOI: https://doi.org/10.1007/s42064-020-0093-6