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

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.