Abstract
The dynamics of a spacecraft propelled by a continuous radial thrust resembles that of a nonlinear oscillator. This is analyzed in this work with a novel method that combines the definition of a suitable homotopy with a classical perturbation approach, in which the low thrust is assumed to be a perturbation of the nominal Keplerian motion. The homotopy perturbation method provides the analytical (approximate) solution of the dynamical equations in polar form to estimate the corresponding spacecraft propelled trajectory with a short computational time. The accuracy of the analytical results was tested in an orbital-targeting mission scenario.
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Abbreviations
- {a 1,a 2,a 3, a 4}:
-
dimensionless coefficients
- a R :
-
propulsive acceleration magnitude (mm/s2)
- E :
-
dimensionless error
- \({\cal H}\) :
-
homotopy
- k :
-
embedding parameter
- \({\cal L}\) :
-
linear operator
- \({\cal N}\) :
-
nonlinear operator
- r :
-
radial coordinate (km)
- t :
-
time (s)
- ∊ :
-
dimensionless propulsive acceleration
- θ :
-
polar angle (deg)
- μ :
-
gravitational parameter (km3/s2)
- ρ :
-
dimensionless radial coordinate
- ρ i :
-
i-th coefficient of power series
- \({\bar \rho }\) :
-
linear approximation of ρ
- τ :
-
auxiliary parameter
- app:
-
analytical approximation
- 0:
-
initial parking orbit
- f:
-
final
- num:
-
evaluated through orbital propagator
- ′:
-
derivative w.r.t. θ
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Lorenzo Niccolai received his Ph.D. degree in industrial engineering (aerospace curriculum) from the University of Pisa in 2018. He was a research assistant at the Department of Civil and Industrial Engineering of the University of Pisa from 2019 to 2020. He is currently an assistant professor of spaceflight mechanics at the same department. His research interests include mission design, low-thrust trajectory analysis and control, with special attention on innovative propulsive concepts such as solar sails and electric solar wind sails. E-mail: lorenzo.niccolai@ing.unipi.it.
Alessandro A. Quarta received his Ph.D. degree in aerospace engineering from the University of Pisa in 2005 and, currently, he is 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, and solar sail and E-sail dynamics and control. E-mail: a.quarta@ing.unipi.it.
Giovanni Mengali received his Doctor 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 space flight mechanics. His main research areas include spacecraft mission analysis, trajectory optimization, and solar sails, electric sails and aircraft flight dynamics and control. E-mail: g.mengali@ing.unipi.it.
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Niccolai, L., Quarta, A.A. & Mengali, G. Application of homotopy perturbation method to the radial thrust problem. Astrodyn 7, 251–258 (2023). https://doi.org/10.1007/s42064-022-0150-4
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DOI: https://doi.org/10.1007/s42064-022-0150-4