New Analytic Solution with Continuous Thrust: Generalized Logarithmic Spirals

A new fully analytic solution to the motion of a continuously accelerated spacecraft is presented. It is demonstrated that the same tangential control law that generates a logarithmic spiral yields an entire family of generalized spirals. The system admits two integrals of motion that follow from the equations of the energy and angular momentum. Three different subfamilies of spiral trajectories are obtained, depending on the sign of the constant of the generalized energy: elliptic, parabolic, and hyperbolic. Elliptic spirals are bounded, never escape to infinity, and the trajectory is symmetric. Parabolic spirals are equivalent to logarithmic spirals. There are two subfamilies of hyperbolic spirals, classified in terms of their generalized angular momentum. The first family has only one asymptote, whereas spirals of the second type are symmetric and exhibit two asymptotes. The new family of solutions is obtained when rigorously solving the equations of motion with no prior assumptions. Closed-form expres...

[1]  Massimiliano Vasile,et al.  Preliminary Design of Low-Thrust Multiple Gravity-Assist Trajectories , 2006 .

[2]  D. J. Benney.  Escape From a Circular Orbit Using Tangential Thrust , 1958 .

[3]  R. Noomen,et al.  Hodographic-Shaping Method for Low-Thrust Interplanetary Trajectory Design , 2015 .

[4]  Derek F Lawden,et al.  Optimal trajectories for space navigation , 1964 .

[5]  Massimiliano Vasile,et al.  Extended analytical formulas for the perturbed Keplerian motion under a constant control acceleration , 2015 .

[6]  Victoria Coverstone-Carroll,et al.  CONSTANT RADIAL THRUST ACCELERATION REDUX , 1998 .

[7]  Frederick W. Boltz,et al.  ORBITAL MOTION UNDER CONTINUOUS TANGENTIAL THRUST , 1992 .

[8]  Massimiliano Vasile,et al.  On the optimality of a shape-based approach based on pseudo-equinoctial elements , 2006 .

[9]  Ralph Hoyt Bacon,et al.  Logarithmic Spiral: An Ideal Trajectory for the Interplanetary Vehicle with Engines of Low Sustained Thrust , 1959 .

[10]  Dario Izzo,et al.  Lambert's Problem for Exponential Sinusoids , 2006 .

[11]  Massimiliano Vasile,et al.  Designing optimal low-thrust gravity-assist trajectories using space pruning and a multi-objective approach , 2009 .

[12]  Anatomy of the Constant Radial Thrust Problem , 2001 .

[13]  Massimiliano Vasile,et al.  Semi-Analytical Solution for the Optimal Low-Thrust Deflection of Near-Earth Objects , 2009 .

[15]  J. Roa,et al.  Orbit propagation in Minkowskian geometry , 2015 .

[16]  A. Quarta,et al.  New Look to the Constant Radial Acceleration Problem , 2012 .

[17]  Anastassios E. Petropoulos,et al.  Design and Optimization of Low-Thrust Trajectories with Gravity Assists , 2003 .

[18]  Ossama Abdelkhalik,et al.  Initial three-dimensional low-thrust trajectory design , 2016 .

[19]  DROMO formulation for planar motions: solution to the Tsien problem , 2015 .

[20]  T. Tsu.  Interplanetary Travel by Solar Sail , 1959 .

[21]  Colin R. McInnes,et al.  Solar Sailing: Technology, Dynamics and Mission Applications , 1999 .

[22]  Anastassios E. Petropoulos,et al.  Shape-Based Algorithm for Automated Design of Low-Thrust, Gravity-Assist Trajectories , 2004 .

[23]  Hsue-shen Tsien,et al.  Take-Off from Satellite Orbit , 1953 .

[24]  Massimiliano Vasile,et al.  Direct transcription of low-thrust trajectories with finite trajectory elements , 2012 .

[25]  J. M. Hedo,et al.  A special perturbation method in orbital dynamics , 2007 .

[26]  Alessandro Antonio Quarta,et al.  Escape from Elliptic Orbit Using Constant Radial Thrust , 2009 .

[27]  Daniel Novak,et al.  Improved Shaping Approach to the Preliminary Design of Low-Thrust Trajectories , 2011 .

[28]  J. Roa,et al.  Singularities in Dromo formulation. Analysis of deep flybys , 2015 .

[29]  Asymptotic solution for the two-body problem with constant tangential thrust acceleration , 2011 .

[30]  Bradley J. Wall,et al.  Shape-Based Approach to Low-Thrust Rendezvous Trajectory Design , 2009 .

[31]  O. Abdelkhalik,et al.  Fast Initial Trajectory Design for Low-Thrust Restricted-Three-Body Problems , 2015 .

[32]  Javier Roa,et al.  Spiral Lambert’s Problem , 2016 .

[33]  Frank M. Perkins.  Flight Mechanics of Low-Thrust Spacecraft , 1959 .