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A Multiple Scales Solution for the Constant Radial Acceleration Problem
Research Areas
  • Aeronautical engineering
The study of the motion of a body around a primary, subjected exclusively to the attraction of said primary and a radial acceleration of constant magnitude, has been given a fair amount of attention. This dynamical system is known to be integrable, although obtaining a general solution in terms of simple functions has proven to be an arduous task. In the classic astrodynamics book by Battin, a solution attributed to Tsien is given in terms of elliptic functions for initially circular orbits. A study based on the Hamiltonian formulation of the problem is developed by San-Juan et al. in [8], reaching a solution in terms of the Jacobi elliptic functions and establishing conditions for bounded and periodic motions. Recently, an exact explicit solution for the general case has been obtained by Izzo et al., in terms of a fictitious time introduced with a Sundmann transformation and the Weierstrass elliptic functions. These solutions can be applied to several practical problems, such as testing the precision of numerical integration schemes, or studying the effects of solar radiation pressure or comet outgassing. In this article, an approximate analytical solution for the two body problem perturbed by a small radial acceleration is obtained, using the regularized formulation of the orbital motion known as Dromo, and the method of multiple scales. Albeit less accurate than the solutions presented in the previous paragraph, it has the advantage of being expressed in terms of much simpler functions, providing additional insight about the physics of the problem in low thrust scenarios. Moreover, unlike the solution by Tsien, it is not restricted to initially circular orbits, requiring only small initial eccentricity. The Dromo orbital formulation was initially introduced by Pel¿aez et al. from the Space Dynamics Group-UPM (formerly Grupo de Din¿amica de Tethers), and has been under active development. It has proven to be an excellent propagation tool, and its suitability for several applications such as low thrust optimal control, formation flying or collision avoidance is being studied. The results obtained with the method of multiple scales reveal that the physics of the problem evolve in two fundamental scales of the Dromo independent variable, related to the true anomaly. A fast one, driving the oscillations of the orbital parameters along each orbit, and a slow one, responsible for the long-term variations in the amplitude and mean values of these oscillations. In the first order solution, the slow independent variable is scaled directly by the perturbing acceleration parameter; an improved higher order approximation can be obtained through a coordinate strain, expressing this scaling through a function of the acceleration parameter. On the other hand, the solution is periodic in both scales, with different periods; this shows that, provided the escape is not reached, the orbital parameters are bounded, and that the motion is periodic in the independent variable only for certain cases. Finally, the multiple scales solution is compared with high precision numerical propagations for several cases, finding a good agreement between them. The asymptotic solution obtained through a regular expansion in the small perturbing acceleration is also included in these comparisons, highlighting the improvement in accuracy and validity range of the asymptotic solution achieved through the method of multiple scales.
International Workshop on Key Topics in Orbit Propagation Applied to Space Situational Awareness
Logroño (Spain)
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  • Autor: Juan Luis Gonzalo Gomez (UPM)
  • Autor: Claudio Bombardelli (UPM)
Research Group, Departaments and Institutes related
  • Creador: Grupo de Investigación: Dinámica Espacial (SDG-UPM)
  • Departamento: Física Aplicada a Las Ingenierías Aeronáutica y Naval
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