Classical and relativistic orbital effects of Sun’s mass loss and the Earth’s fate

We calculate the classical and general relativistic effects induced by an isotropic mass loss Ṁ/M of a body on the orbital motion of a test particle around it; the present analysis is valid also for a variation Ġ/G of the Newtonian constant of gravitation. Concerning the Newtonian case, we perturbatively obtain negative secular rates for the osculating semimajor axis a, the eccentricity e and the mean anomaly M, while the argument of pericenter ω does not undergo secular precession; the longitude of the ascending node Ω and the inclination i are left unaffected. The anomalistic period is different from the Keplerian one and is 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 decreasing of a and e because they refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. 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. By assuming for the Sun Ṁ/M = −9× 10−14 yr−1 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 − 1.5 Myr, we find that the Earth’s perihelion position on the fixed line of the apsides will increase by ≈ 0.29 AU (for Ṁ/M = −2 × 10−7 yr−1); other researchers point towards an increase of 0.37− 0.63 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 10 − 1000 AU. Simultaneous long-term numerical integrations of the equations of motion – 3 – 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. Subject headings: gravitation – stars: mass-loss – celestial mechanics – relativity