Research Article | Open Access
Classical and Relativistic Orbital Motions around a Mass-Varying Body
I work out the Newtonian and general relativistic effects due to an isotropic mass loss of a body on the orbital motion of a test particle around it; the present analysis is also valid for a variation 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 , the eccentricity , and the mean anomaly , while the argument of pericenter does not experience secular precession; the longitude of the ascending node and the inclination 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 and 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.
- 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.
- 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.
- 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.
- P. D. Noerdlinger, “Solar mass loss, the Astronomical Unit, and the scale of the solar system,” http://arxiv.org/abs/0801.3807.
- 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.
- 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.
- 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.
- 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.
- 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.
- E. Strömgren, “ber die Bedeutung kleiner Massennderungen fr die Newtonsche Centralbewegung,” Astronomische Nachrichten, vol. 163, no. 9, pp. 129–136, 1903.
- 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.
- J. H. Jeans, Astronomy and Cosmogony, Dover, New York, NY, USA, 1961.
- G. Armellini, “The variation of the eccentricity in a binary system of decrasing mass,” The Observatory, vol. 58, pp. 158–159, 1935.
- J. D. Hadjidemetriou, “Two-body problem with variable mass: a new approach,” Icarus, vol. 2, pp. 440–451, 1963.
- J. D. Hadjidemetriou, “Analytic solutions of the two-body problem with variable mass,” Icarus, vol. 5, no. 1–6, pp. 34–46, 1966.
- 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.
- A. Deprit, “The secular acceleratons in Gylden's problem,” Celestial Mechanics, vol. 31, no. 1, pp. 1–22, 1983.
- J. Kevorkian and J. D. Cole, Multiple Scale and Singular Perturbation Methods, Springer, New York, NY, USA, 1966.
- B. Bertotti, P. Farinella, and D. Vokrouhlický, Physics of the Solar System, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003.
- A. E. Roy, Orbital Motion, Institute of Physics, Bristol, UK, 4th edition, 2005.
- 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.
- M. Beech, “Tidal circularization in massive binaries,” Astrophysics and Space Science, vol. 132, no. 2, pp. 269–276, 1987.
- J. P. Zahn, “Tidal friction in close binary stars,” Astronomy and Astrophysics, vol. 57, pp. 383–394, 1977.
- J. P. Zahn, “Erratum; tidal friction in close binary stars,” Astronomy and Astrophysics, vol. 67, p. 162, 1978.
- 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.
- B. Mashhoon, “Gravitoelectromagnetism: a brief review,” in Measuring Gravitomagnetism: A Challenging Enterprise, L. Iorio, Ed., pp. 29–39, NOVA, Hauppauge, NY, USA, 2007.
- D. Bini, C. Cherubini, C. Chicone, and B. Mashhoon, “Gravitational induction,” Classical and Quantum Gravity, vol. 25, no. 22, Article ID 225014, 2008.
- 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.
- 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.
- E. V. Pitjeva, “private communication to Noerdlinger P.,” 2008.
- 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.
- 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.
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