Skip to main content
Log in

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

Programming and Computer Software Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.

REFERENCES

  1. 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.

    Article  Google Scholar 

  2. 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.

    Google Scholar 

  3. 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.

    Article  MathSciNet  MATH  Google Scholar 

  4. 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.

  5. 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.

  6. Buchberger, B., Theoretical basis for the reduction of polynomials to canonical forms, SIGSAM Bull., 1976, vol. 10, no. 3, pp. 19–29.

    MathSciNet  Google Scholar 

  7. 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.

  8. 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.

    Article  MathSciNet  MATH  Google Scholar 

  9. 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.

    Article  MathSciNet  MATH  Google Scholar 

  10. 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.

    Article  MathSciNet  MATH  Google Scholar 

  11. 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.

    Article  MathSciNet  MATH  Google Scholar 

  12. 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.

    Article  MathSciNet  MATH  Google Scholar 

  13. Wolfram Mathematica. http://www.wolfram.com/ mathematica.

  14. 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.

    Google Scholar 

  15. 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.

    Article  MathSciNet  MATH  Google Scholar 

  16. 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.

    Article  MathSciNet  MATH  Google Scholar 

  17. 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.

    Article  MathSciNet  MATH  Google Scholar 

  18. 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.

    Article  MathSciNet  MATH  Google Scholar 

  19. Sarychev, V.A., Problems of orientation of artificial satellites, Itogi Nauki Tekh., Ser.: Issled. Kosm. Prostranstva, 1978, vol. 11.

    Google Scholar 

  20. Batkhin, A.B., Parameterization of the discriminant set of a polynomial, Program. Comput. Software, 2016, vol. 42, pp. 65–76.

    Article  MathSciNet  MATH  Google Scholar 

  21. Batkhin, A.B., Parameterization of a set determined by the generalized discriminant of a polynomial, Program. Comput. Software, 2018, vol. 44, pp. 75–85.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding authors

Correspondence to S. A. Gutnik or V. A. Sarychev.

Ethics declarations

The authors declare that they have no conflicts of interest.

Additional information

Translated by Yu. Kornienko

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0361768823020093

Navigation