Orbital effects of Sun’s mass loss and the Earth’s fate

Author(s)
Lorenzo Iorio

ABSTRACT

I calculate the classical effects induced by 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. I perturbatively 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 undergo secular precession, like the longitude of the ascending node Ω and the inclination I. The anomalistic period is different from the Keplerian one, being larger than it. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion in Cartesian coordinates; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they only refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. By assuming for the Sun it turns out that the Earth's perihelion position is displaced outward by 1.3 cm along the fixed line of apsides after each revolution. By applying our results to the phase in which the radius of the Sun, already moved to the Red Giant Branch of the Hertzsprung-Russell Diagram, will become as large as 1.20 AU in about 1 Myr, I find that the Earth's perihelion position on the fixed line of the apsides will increase by AU (for ); other researchers point towards an increase of AU. Mercury will be destroyed already at the end of the Main Sequence, while Venus should be engulfed in the initial phase of the Red Giant Branch phase; the orbits of the outer planets will increase by AU. Simultaneous long-term numerical integrations of the equations of motion of all the major bodies of the solar system, with the inclusion of a mass-loss term in the dynamical force models as well, are required to check if the mutual N-body interactions may substantially change the picture analytically outlined here, especially in the Red Giant Branch phase in which Mercury and Venus may be removed from the integration.

I calculate the classical effects induced by 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. I perturbatively 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 undergo secular precession, like the longitude of the ascending node Ω and the inclination I. The anomalistic period is different from the Keplerian one, being larger than it. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion in Cartesian coordinates; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they only refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. By assuming for the Sun it turns out that the Earth's perihelion position is displaced outward by 1.3 cm along the fixed line of apsides after each revolution. By applying our results to the phase in which the radius of the Sun, already moved to the Red Giant Branch of the Hertzsprung-Russell Diagram, will become as large as 1.20 AU in about 1 Myr, I find that the Earth's perihelion position on the fixed line of the apsides will increase by AU (for ); other researchers point towards an increase of AU. Mercury will be destroyed already at the end of the Main Sequence, while Venus should be engulfed in the initial phase of the Red Giant Branch phase; the orbits of the outer planets will increase by AU. Simultaneous long-term numerical integrations of the equations of motion of all the major bodies of the solar system, with the inclusion of a mass-loss term in the dynamical force models as well, are required to check if the mutual N-body interactions may substantially change the picture analytically outlined here, especially in the Red Giant Branch phase in which Mercury and Venus may be removed from the integration.

Cite this paper

Iorio, L. (2010) Orbital effects of Sun’s mass loss and the Earth’s fate.*Natural Science*, **2**, 329-337. doi: 10.4236/ns.2010.24041.

Iorio, L. (2010) Orbital effects of Sun’s mass loss and the Earth’s fate.

References

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[24] Müller, J. and Biskupek, L. (2007) Variations of the gravitational constant from lunar laser ranging data. Classical and Quantum Gravity, 24, 4533-4538.

[25] Williams, J.G., Turyshev, S.G. and Boggs, D.H. (2007) Williams, Turyshev, and Boggs reply. Physical Review Letters, 98(55).

[26] Laskar, J. (1994) Large-scale chaos in the solar system. Astronomy and Astrophysics, 287, L9-L12.

[27] Ito, K. and Tanikawa, K. (2002) Long-term integra- tions and stability of planetary orbits in our solar system. Monthly Notices of the Royal Astronomy Society, 336, 483-500.

[28] Laskar, J. (2008) Chaotic diffusion in the solar system. Icarus, 196, 1-15.

[29] Duncan, M.J. and Lissauer, J.J. (1998) The effects of post-main-sequence solar mass loss on the stability of our planetary system. Icarus, 134, 303-310.

[1] Schröder, K.P. and Smith, R.C. (2008) Distant future of the Sun and Earth revisited. Monthly Notices of the Royal Astronomy Society, 386(1), 155-163.

[2] Krasinsky, G.A. and Brumberg, V.A. (2004) Secular increase of astronomical unit from analysis of the major planet motions, and its interpretation. Celestial Mecha- nics and Dynamical Astronomy, 90(3-4), 267-288.

[3] Standish, E.M. (2005) The astronomical unit now. Transits of Venus: New Views of the Solar System and Galaxy. Proceedings of the IAU Colloquium 196, Kurtz, D.W., Ed., Cambridge University Press, Cambridge, 2004, 163-179.

[4] Noerdlinger, P.D. (2008) Solar mass loss, the astronomical unit, and the scale of the solar sys- tem. http://arxiv.org/abs/0801.3807

[5] Klioner, S.A. (2008) Relativistic scaling of astrono- mical quantities and the system of astronomical units. Astronomy and Astrophysics, 478, 951-958.

[6] Jin, W., Imants, P. and Perryman, M. (2008) A giant step: From milli- to micro-arcsecond astrometry. Proceedings of the IAU Symposium 248, Cambridge University Press, Cambridge, 2007.

[7] Oskay, W.H., Diddams, S.A., Donley, E.A., Fortier, T. M., Heavner, T.P., Hollberg, L., Itano, W.M., Jefferts, S.R., Delaney, M.J., Kim, K., Levi, F., Parker, T.E. and Bergquist, J.C. (2006) Single-atom optical clock with high accuracy. Physical Review Letters, 97(2), 1-4.

[8] Rubincam, D.P. (1982) On the secular decrease in the semimajor axis of Lageos’s orbit. Celestial Mechanics and Dynamical Astronomy, 26(4), 361-382.

[9] Williams, J.G. and Boggs, D.H. (2008) DE421 lunar orbit, physical librations, and surface coordinates. 16th International Workshop on Laser Ranging, Poznań, 13-17 October 2008.

[10] Strömgren, E. (1903) Über die bedeutung kleiner massenänderungen für die newtonsche centralbewe- gung. Astronomische Nachrichten, 163, 129-136.

[11] Jeans, J.H. (1924) Cosmogonic problems associated with a secular decrease of mass. Monthly Notices of the Royal Astronomy Society, 85, 2-11.

[12] Jeans, J.H. (1961) Astronomy and cosmogony. Dover, New York.

[13] Armellini, G. (1935) The variation of the eccentricity in a binary system of decrasing mass. The Observatory, 58, 158-159.

[14] Hadjidemetriou, J.D. (1963) Two-body problem with variable mass: A new approach. Icarus, 2, 440-451.

[15] Hadjidemetriou, J.D. (1966) Analytic solutions of the two-body problem with variable mass. Icarus, 5, 34-46.

[16] Kholshevnikov, K.V. and Fracassini, M. (1968) Le problème des deux corps avec G variable selon l'hypothèse de dirac. Conferenze dell' Osservatorio Astronomico di Milano-Merate, 1(9), 5-50.

[17] Deprit, A. (1983) The secular accelerations in Gylden’s problem. Celestial Mechanics and Dynamical Astrono- nomy, 31(1), 1-22.

[18] Kevorkian, J. and Cole, J.D. (1966) Multiple scale and singular perturbation methods. Springer, New York and Berlin.

[19] Bertotti, B. Farinella, P. and Vokrouhlický, D. (2003) Phyiscs of the solar system. Kluwer, Dordrecht.

[20] Roy, A.E. (2005) Orbital motion. 4th Edition, Institute of Physics, Bristol.

[21] Casotto, S. (1993) Position and velocity perturbations in the orbital frame in terms of classical element pertu- rbations. Celestial Mechanics and Dynamical Astronomy, 55, 209-221.

[22] Pitjeva, E.V. (2005) The astronomical unit now. Transits of Venus: New Views of the Solar System and Galaxy. Proceedings of the IAU Colloquium 196, 2004, Kurtz, D. W., Ed., Cambridge University Press, Cambridge, 177.

[23] Pitjeva, E.V. (2008) Personal communication to P. Noerdlinger.

[24] Müller, J. and Biskupek, L. (2007) Variations of the gravitational constant from lunar laser ranging data. Classical and Quantum Gravity, 24, 4533-4538.

[25] Williams, J.G., Turyshev, S.G. and Boggs, D.H. (2007) Williams, Turyshev, and Boggs reply. Physical Review Letters, 98(55).

[26] Laskar, J. (1994) Large-scale chaos in the solar system. Astronomy and Astrophysics, 287, L9-L12.

[27] Ito, K. and Tanikawa, K. (2002) Long-term integra- tions and stability of planetary orbits in our solar system. Monthly Notices of the Royal Astronomy Society, 336, 483-500.

[28] Laskar, J. (2008) Chaotic diffusion in the solar system. Icarus, 196, 1-15.

[29] Duncan, M.J. and Lissauer, J.J. (1998) The effects of post-main-sequence solar mass loss on the stability of our planetary system. Icarus, 134, 303-310.