Astrophysics and Space Science

, Volume 11, Issue 2, pp 191–221 | Cite as

The relative motion of two spheroidal rigid bodies, II

  • Paolo Lanzano
Article

Abstract

This study constitutes the second phase of an effort devoted to the relative motion of two spheroidal rigid bodies.

An isolated binary system was considered whose components are bodies: (1) of comparable size; (2) of constant density; and (3) having the shape of an oblate ellipsoid of revolution with small meridional eccentricity.

The equations that determine the relative motion of the centroids and the angular motion for the two sets of body axes constitute a simultaneous system of seven nonlinear, second-order differential equations, for which solutions were obtained as power series in the two meridional eccentricities.

A recurrent procedure was formulated to ascertain the various approximations in terms of lower order terms; it gave rise to linear differential equations with constant coefficients for the angular variables and to differential equations of the Hill type for the other coordinates. The zero-order approximation for the motion of the centroids was assumed to be a Kepler elliptic orbit of small eccentricity.

The following contributions were made:
  1. (1)

    The general solution to the zero-order approximation of the rotational motion was obtained in terms of elementary functions;

     
  2. (2)

    Certain functionals, related to the Kepler motion and depending on two parameters, were expressed in terms of the mean anomaly up to the sixth power of the orbital eccentricity in order to evaluate the lower order terms of the various approximations;

     
  3. (3)

    The secular terms were eliminated from the first-order approximation;

     
  4. (4)

    The second-order approximation was also obtained; and

     
  5. (5)

    An alternate procedure was suggested that might be more appropriate for achieving higher order approximations.

     

Keywords

Relative Motion Linear Differential Equation Elliptic Orbit Angular Motion Lower Order Term 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Gröbner, W. and Hofreiter, N.: 1949,Integraltafel, Unbestimmte Integrale, Springer Verlag, Vienna.Google Scholar
  2. Kondurar, V. T.: 1961, ‘The General Case of the Translational-Rotational Motion of a Spheroid Attracted by a Sphere’,Soviet Astron. 5, 232–241.Google Scholar
  3. Kondurar, V. T.: 1962, ‘On the Perturbations in the Translational-Rotational Motion of a Satellite and Planet Caused by Their Oblateness’,Soviet Astron. 6, 405–411.Google Scholar
  4. Kopal, Z.: 1968, 1969a, b, ‘The Precession and Nutation of Deformable Bodies, I’,Astrophys. Space Sci. 1, 74–91; II,op. cit. 4, 330–364; III,op. cit. 4, 427–458.Google Scholar
  5. Lanzano, P.: 1962, ‘Theoretical Considerations on the Translational-Rotational Motion of Two Attracting Spheroidal Bodies’, Proceedings 13th International Astronautical Congress, pp. 379–410, Springer Verlag, New York.Google Scholar
  6. Lanzano, P.: 1967, ‘Stability of a Class of Periodic Orbits in the Restricted Three-Body Problem’,Icarus 7, 105–113.Google Scholar
  7. Lanzano, P.: 1969, ‘The Relative Motion of Two Spheroidal Rigid Bodies’,Astrophys. Space Sci. 5, 300–322.Google Scholar
  8. Plummer, H. C.: 1960,An Introductory Treatise on Dynamical Astronomy, Dover Publications, New York.Google Scholar
  9. Smart, W. M.: 1953,Celestial Mechanics, Longmans, Green and Company, London.Google Scholar

Copyright information

© D. Reidel Publishing Company 1971

Authors and Affiliations

  • Paolo Lanzano
    • 1
  1. 1.North American Rockwell CorporationDowneyU.S.A.

Personalised recommendations