Explicit Solution to the Constant Radial Acceleration Problem

Izzo, Dario; Biscani, Francesco

doi:10.2514/1.g000116

Cited by 19 publications

(16 citation statements)

References 14 publications

(43 reference statements)

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“…The Hamiltonian-based solution by San-Juan et al (2012) shows that the three kinds of Jacobi elliptic integrals are intrinsic to the problem. This is consistent with the solution for the most general case in terms of Weierstrass's elliptic and related functions by Izzo and Biscani (2015), as they can be expressed as a combination of the three Jacobi elliptic functions. One key aspect of Izzo and Biscani (2015) is that they provide a solution for the state as a function of time (explicit solution), a feature also shared by the slightly more recent, Dromo-based work by Urrutxua et al (2015), whether previous exact solutions only provided the time as a function of the state (implicit).…”

Section: Comparison With Other Solutionssupporting

confidence: 89%

“…One common characteristic of all the exact solutions, either for the general problem [see Izzo and Biscani (2015)] or for parts of it like time of flight determination, is that they rely on the use of elliptic functions. The Hamiltonian-based solution by San-Juan et al (2012) shows that the three kinds of Jacobi elliptic integrals are intrinsic to the problem.…”

Section: Comparison With Other Solutionsmentioning

confidence: 99%

“…However, Quarta and Mengali (2012) highlight a number of practical limitations to these solutions when introducing their Fourier-based approximate model. On the one hand, they mention the time-implicit nature of exact solutions (note that the timeexplicit exact solutions by Izzo and Biscani (2015) and Urrutxua et al (2015) are posterior to the work by Quarta and Mengali). On the other hand, they stress how the use of elliptic functions hinders their practical application for mission design due to the lack of physical insight and more complex analytical manipulation.…”

Section: Comparison With Other Solutionsmentioning

confidence: 99%

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Multiple scales asymptotic solution for the constant radial thrust problem

Gonzalo

Bombardelli

2019

Celest Mech Dyn Astr

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An approximate analytical solution for the two body problem perturbed by a radial, low acceleration is obtained, using a regularized formulation of the orbital motion and the method of multiple scales. Formulating the dynamics with the Dromo special perturbation method allows us to separate the two characteristic periods of the problem in a clear and physically significative way, namely the orbital period and a period depending on the magnitude of the perturbing acceleration. This second period becomes very large compared to the orbital one for low thrust cases, allowing us to develop an accurate approximate analytical solution through the method of multiple scales. Compared to a regular expansion, the multiple scales solution retains the qualitative contributions of both characteristic periods and has a longer validity range in time. Looking at previous solutions for this problem, our approach has the advantage of not requiring the evaluation of special functions or an initially circular orbit. Furthermore, the simple expression reached for the long period provides additional insight on the problem. Finally, the behavior of the asymptotic solution is assessed through several test cases, finding a good agreement with high-precision numerical solutions. The results presented not only advance in the study of the two body problem with constant radial thrust, but confirm the utility of the method of multiple scales for tackling problems in orbital mechanics.

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Section: Comparison With Other Solutionssupporting

confidence: 89%

Section: Comparison With Other Solutionsmentioning

confidence: 99%

Section: Comparison With Other Solutionsmentioning

confidence: 99%

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Multiple scales asymptotic solution for the constant radial thrust problem

Gonzalo

Bombardelli

2019

Celest Mech Dyn Astr

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“…The explicit solution found by Battin in terms of elliptic integrals is also noteworthy [7], as well as the accurate approximations of the spacecraft trajectory and the flight time found by Quarta and Mengali in terms of a Fourier series [8]. Gonzalo involve asymptotic expansions [9], while Izzo and Biscani showed that a solution relating the state variables to a time parameter can be expressed in terms of Weierstrass elliptic and related functions [10].…”

Section: Introductionmentioning

confidence: 99%

Spiral trajectories induced by radial thrust with applications to generalized sails

et al. 2020

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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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“…The explicit solution to the constant radial thrust problem is found in terms of elliptic integrals (Battin, 1999, pp. 408-418), and alternative solutions involving the Weierstrass elliptic functions (Izzo and Biscani, 2015), approximate methods (Quarta and Mengali, 2012) and asymptotic expansions (Gonzalo and Bombardelli, 2014) have been proposed. Urrutxua et al (2015b) solved the Tsien problem in closed form relying on the Dromo formulation (see Chap.…”

Section: Discussionmentioning

confidence: 99%

Regularization in Astrodynamics: applications to relative motion, low-thrust missions, and orbit propagation = Regularización en Astrodinámica: aplicaciones al movimiento relativo, misiones de bajo empuje, y propagación de órbitas

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Explicit Solution to the Constant Radial Acceleration Problem

Cited by 19 publications

References 14 publications

Multiple scales asymptotic solution for the constant radial thrust problem

Multiple scales asymptotic solution for the constant radial thrust problem

Spiral trajectories induced by radial thrust with applications to generalized sails

Regularization in Astrodynamics: applications to relative motion, low-thrust missions, and orbit propagation = Regularización en Astrodinámica: aplicaciones al movimiento relativo, misiones de bajo empuje, y propagación de órbitas

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