SRX Physics

SRX Physics / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 261249 | 10 pages | https://doi.org/10.3814/2010/261249

Classical and Relativistic Orbital Motions around a Mass-Varying Body

Received22 Nov 2009
Revised29 Dec 2009
Accepted31 Dec 2009
Published09 Mar 2010

Abstract

I work out the Newtonian and general relativistic effects due to an isotropic mass loss M˙/M of a body on the orbital motion of a test particle around it; the present analysis is also valid for a variation G˙/G of the Newtonian constant of gravitation. Concerning the Newtonian case, I use the Gauss equations for the variation of the elements and obtain negative secular rates for the osculating semimajor axis a, the eccentricity e, and the mean anomaly , while the argument of pericenter ω does not experience secular precession; the longitude of the ascending node Ω and the inclination i remain unchanged as well. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion with MATHEMATICA; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. A comparison with the results obtained with different approaches by other researchers is made. General relativity induces positive secular rates of the semimajor axis and the eccentricity completely negligible in the present and future evolution of the solar system.

References

  1. K.-P. Schröder and R. C. Smith, “Distant future of the Sun and Earth revisited,” Monthly Notices of the Royal Astronomical Society, vol. 386, no. 1, pp. 155–163, 2008. View at: Publisher Site | Google Scholar
  2. G. A. Krasinsky and V. A. Brumberg, “Secular increase of astronomical unit from analysis of the major planet motions, and its interpretation,” Celestial Mechanics and Dynamical Astronomy, vol. 90, no. 3-4, pp. 267–288, 2004. View at: Publisher Site | Google Scholar
  3. E. M. Standish, “The Astronomical Unit now,” in Transits of Venus: New Views of the Solar System and Galaxy, D. W. Kurtz, Ed., Proceedings of the IAU Colloquium, no. 196, pp. 163–179, Cambridge University Press, Cambridge, UK, 2005. View at: Google Scholar
  4. P. D. Noerdlinger, “Solar mass loss, the Astronomical Unit, and the scale of the solar system,” http://arxiv.org/abs/0801.3807. View at: Google Scholar
  5. S. A. Klioner, “Relativistic scaling of astronomical quantities and the system of astronomical units,” Astronomy and Astrophysics, vol. 478, no. 3, pp. 951–958, 2008. View at: Publisher Site | Google Scholar
  6. W. Jin, P. Imants, and M. Perryman, Eds., A Giant Step: From Milli-to Micro-Arcsecond Astrometry, Proceedings of the IAU Symposium 248, Cambridge University Press, Cambridge, UK, 2008.
  7. W. H. Oskay, S. A. Diddams, E. A. Donley et al., “Single-atom optical clock with high accuracy,” Physical Review Letters, vol. 97, no. 2, Article ID 020801, 2006. View at: Publisher Site | Google Scholar
  8. D. P. Rubincam, “On the secular decrease in the semimajor axis of Lageos's orbit,” Celestial Mechanics, vol. 26, no. 4, pp. 361–382, 1982. View at: Publisher Site | Google Scholar
  9. J. D. Anderson and M. M. Nieto, “Astrometric solar-system anomalies,” in Relativity in Fundamental Astronomy: Dynamics, Reference Frames, and Data Analysis, S. Klioner, P. K. Seidelmann, and M. Soffel, Eds., Proceedings of the IAU Symposium, no. 261, 2009. View at: Google Scholar
  10. E. Strömgren, “ber die Bedeutung kleiner Massennderungen fr die Newtonsche Centralbewegung,” Astronomische Nachrichten, vol. 163, no. 9, pp. 129–136, 1903. View at: Google Scholar
  11. J. H. Jeans, “Cosmogonic problems associated with a secular decrease of mass,” Monthly Notices of the Royal Astronomy Society, vol. 85, pp. 2–11, 1924. View at: Google Scholar
  12. J. H. Jeans, Astronomy and Cosmogony, Dover, New York, NY, USA, 1961.
  13. G. Armellini, “The variation of the eccentricity in a binary system of decrasing mass,” The Observatory, vol. 58, pp. 158–159, 1935. View at: Google Scholar
  14. J. D. Hadjidemetriou, “Two-body problem with variable mass: a new approach,” Icarus, vol. 2, pp. 440–451, 1963. View at: Google Scholar
  15. J. D. Hadjidemetriou, “Analytic solutions of the two-body problem with variable mass,” Icarus, vol. 5, no. 1–6, pp. 34–46, 1966. View at: Google Scholar
  16. K. V. Kholshevnikov and M. Fracassini, “Le Problème des deux corps avec G variable selon l'hypothse de Dirac,” Conferenze dell' Osservatorio Astronomico di Milano-Merate, Serie I, no. 9, pp. 5–50, 1968. View at: Google Scholar
  17. A. Deprit, “The secular acceleratons in Gylden's problem,” Celestial Mechanics, vol. 31, no. 1, pp. 1–22, 1983. View at: Publisher Site | Google Scholar
  18. J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, New York, NY, USA, 1966.
  19. B. Bertotti, P. Farinella, and D. Vokrouhlický, Physics of the Solar System, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
  20. A. E. Roy, Orbital Motion, Institute of Physics, Bristol, UK, 4th edition, 2005.
  21. S. Casotto, “Position and velocity perturbations in the orbital frame in terms of classical element perturbations,” Celestial Mechanics & Dynamical Astronomy, vol. 55, no. 3, pp. 209–221, 1993. View at: Publisher Site | Google Scholar
  22. M. Beech, “Tidal circularization in massive binaries,” Astrophysics and Space Science, vol. 132, no. 2, pp. 269–276, 1987. View at: Publisher Site | Google Scholar
  23. J. P. Zahn, “Tidal friction in close binary stars,” Astronomy and Astrophysics, vol. 57, pp. 383–394, 1977. View at: Google Scholar
  24. J. P. Zahn, “Erratum; tidal friction in close binary stars,” Astronomy and Astrophysics, vol. 67, p. 162, 1978. View at: Google Scholar
  25. B. Mashhoon, “Gravitoelectromagnetism,” in Reference Frames and Gravitomagnetism, J. F. Pascual-Sánchez, L. Floriá, A. San Miguel, and F. Vicente, Eds., pp. 121–132, World Scientific, Singapore, 2001. View at: Google Scholar
  26. B. Mashhoon, “Gravitoelectromagnetism: a brief review,” in Measuring Gravitomagnetism: A Challenging Enterprise, L. Iorio, Ed., pp. 29–39, NOVA, Hauppauge, NY, USA, 2007. View at: Google Scholar
  27. D. Bini, C. Cherubini, C. Chicone, and B. Mashhoon, “Gravitational induction,” Classical and Quantum Gravity, vol. 25, no. 22, Article ID 225014, 2008. View at: Publisher Site | Google Scholar
  28. L. Iorio, “A gravitomagnetic effect on the orbit of a test body due to the Earth's variable angular momentum,” International Journal of Modern Physics D, vol. 11, no. 5, pp. 781–787, 2002. View at: Publisher Site | Google Scholar
  29. E. V. Pitjeva and E. M. Standish, “The Astronomical Unit now,” in Transits of Venus: New Views of the Solar System and Galaxy, D. W. Kurtz, Ed., Proceedings of the IAU Colloquium, no. 196, p. 177, Cambridge University Press, Cambridge, UK, 2005. View at: Google Scholar
  30. E. V. Pitjeva, “private communication to Noerdlinger P.,” 2008. View at: Google Scholar
  31. J. Müller and L. Biskupek, “Variations of the gravitational constant from lunar laser ranging data,” Classical and Quantum Gravity, vol. 24, no. 17, pp. 4533–4538, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  32. J. G. Williams, S. G. Turyshev, and D. H. Boggs, “Williams, Turyshev, and Boggs reply,” Physical Review Letters, vol. 98, no. 5, Article ID 059002, 2007. View at: Publisher Site | Google Scholar

Copyright © 2010 L. Iorio. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

130 Views | 0 Downloads | 0 Citations
 PDF  Download Citation  Citation
 Download other formatsMore
 Order printed copiesOrder