Abstract
Methods of computer algebra are used to investigate equilibrium orientations of a satellite moving along a circular orbit under the action of gravitational and constant torques. The main focus is placed on the investigation of equilibrium orientations in the cases where the constant torque vector is parallel to the planes formed by the principal central axes of inertia of the satellite. Using methods for Gröbner basis construction, the system of six algebraic equations that determine the equilibrium orientations of the satellite is reduced to one sixth-order algebraic equation in one unknown. Domains with equal numbers of equilibrium solutions are classified using algebraic methods for constructing discriminant hypersurfaces. Bifurcation curves in the space of problem parameters, which define the boundaries of the domains with equal numbers of equilibrium solutions, are constructed. A comparative analysis of the influence of the order of variables in the process of Gröbner basis construction is carried out. Using the proposed approach, it is shown that, under the action of the constant torque, the satellite with unequal principal central moments of inertia has no more than 24 equilibrium orientations in a circular orbit.
Similar content being viewed by others
REFERENCES
Garber, T.B., Influence of constant disturbing torques on the motion of gravity gradient stabilized satellites, AIAA J., 1963, vol. 1, no. 4, pp. 968–969.
Sarychev, V.A. and Gutnik, S.A., Satellite equilibria under the action of gravitational and constant torques, Kosm. Issled., 1994, vol. 32, nos. 4–5, pp. 43–50.
Sarychev, V.A., Paglione, P., and Guerman, A., Influence of constant torque on equilibria of a satellite in a circular orbit, Celestial Mech. Dyn. Astron., 2003, vol. 87, pp. 219–239.
German, A.D., Gutnik, S.A., and Sarychev, V.A., Satellite dynamics under the action of gravitational and constant torques and their stability, Izv. Ross. Akad. Nauk., Teor. Sist. Upr., 2016, no. 3, pp. 142–155.
Gutnik, S.A., Guerman, A., and Sarychev, V.A., Application of computer algebra methods to investigation of influence of constant torque on stationary motions of satellite, Lect. Notes Comput. Sci., Gerdt, V.P., Koepf, W., Seiler, W.M., and Vorozhtsov, E.V., Eds., 2015, vol. 9301, pp. 198–209.
Buchberger, B., Theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull., 1976, vol. 10, no. 3, pp. 19–29.
Buchberger, B., Bazisy Grebnera. Algoritmicheskii metod v teorii polinomial’nykh idealov. Komp’yuternaya algebra. Simvol’nye i algebraicheskie vychisleniya (Gröbner Bases: An Algorithmic Method in Polynomial Ideal Theory. Computer Algebra. Symbolic and Algebraic Computation), Moscow: Mir, 1986, pp. 331–372.
Gutnik, S.A. and Sarychev, V.A., Symbolic-numerical methods of studying equilibrium positions of a gyrostat satellite, Program. Comput. Software, 2014, vol. 40, pp. 143–150.
Gutnik, S.A. and Sarychev, V.A., Application of computer algebra methods for investigation of stationary motions of a gyrostat satellite, Program. Comput. Software, 2017, vol. 43, pp. 90–97.
Gutnik, S.A. and Sarychev, V.A., Symbolic-numeric simulation of satellite dynamics with aerodynamic attitude control system, Lect. Notes Comput. Sci., 2018, vol. 11077, pp. 214–229.
Gutnik, S.A. and Sarychev, V.A., Application of computer algebra methods to investigate the dynamics of the system of two connected bodies moving along a circular orbit, Program. Comput. Software, 2019, vol. 45, pp. 51–57.
Gutnik, S.A. and Sarychev, V.A., Symbolic methods for studying the equilibrium orientations of a system of two connected bodies in a circular orbit, Program. Comput. Software, 2022, vol. 48, pp. 73–79.
Wolfram Mathematica. http://www.wolfram.com/ mathematica.
Hastings, C., Mischo, K., and Morrison, M., Hands-On Start to Wolfram Mathematica and Programming with the Wolfram Language, Wolfram Media Inc., 2020, 3rd ed.
Prokopenya, A.N., Minglibayev, M.Zh., and Mayemerova, G.M., Symbolic calculations in studying the problem of three bodies with variable masses, Program. Comput. Software, 2014, vol. 40, pp. 79–85.
Prokopenya, A.N., Minglibayev, M.Zh., Mayemerova, G.M., and Imanova, Zh.U., Investigation of the restricted problem of three bodies of variable masses using computer algebra, Program. Comput. Software, 2017, vol. 43, pp. 289–293.
Prokopenya, A.N., Minglibayev, M.Zh., and Shomshekova, S.A., Applications of computer algebra in the study of the two-planet problem of three bodies with variable masses, Program. Comput. Software, 2019, vol. 45, pp. 73–80.
Budzko, D.A. and Prokopenya, A.N., Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem, Program. Comput. Software, 2013, vol. 39, pp. 74–80.
Sarychev, V.A., Problems of orientation of artificial satellites, Itogi Nauki Tekh., Ser.: Issled. Kosm. Prostranstva, 1978, vol. 11.
Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, pp. 65–76.
Batkhin, A.B., Parameterization of a set determined by the generalized discriminant of a polynomial, Program. Comput. Software, 2018, vol. 44, pp. 75–85.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors declare that they have no conflicts of interest.
Additional information
Translated by Yu. Kornienko
Rights and permissions
About this article
Cite this article
Gutnik, S.A., Sarychev, V.A. Investigation of the Influence of Constant Torque on Equilibrium Orientations of a Satellite Moving in a Circular Orbit with the Use of Computer Algebra Methods. Program Comput Soft 49, 360–365 (2023). https://doi.org/10.1134/S0361768823020093
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0361768823020093