Residual eccentricity of an Earth-like planet orbiting a red giant Sun

Abstract

Context. The late phases of the orbital evolution of an Earth-like planet around a Sun-like star are revisited in order to consider the effect of density fluctuations associated with convective motions inside the star. Aims. Such fluctuations produce a random perturbation of the stellar outer gravitational field that excites a small residual eccentricity in the orbit of the planet. This counteracts the effects of tides, which tend to circularize the orbit. Methods. We computed the power spectrum of the outer gravitational field fluctuations of the star in the quadrupole approximation and studied their effects on the orbit of the planet using a perturbative approach. The residual eccentricity is found to be a stochastic variable showing a Gaussian distribution. Results. Adopting a model of the stellar evolution of our Sun computed with Modules for Experiments in Stellar Astrophysics (MESA), we find that the Earth will be engulfed by the Sun when it is close to the tip of the red giant branch phase of evolution. We find a maximum mean value of the residual eccentricity of ~0.026 immediately before engulfment. Considering an Earth-mass planet with an initial orbital semimajor axis sufficiently large to escape engulfment, we find that the mean value of the residual eccentricity is greater than 0.01 for an initial separation of up to ~l.4 au. Conclusions. The engulfment of the Earth by the red giant Sun is found to be a stochastic process instead of being deterministic as assumed in previous studies. If an Earth-like planet escapes engulfment, its orbit around its remnant white dwarf (WD) star will be moderately eccentric. Such a residual eccentricity of on the order of a few hundredths can play a relevant role in sustaining the pollution of the WD atmosphere by asteroids and comets, as observed in several objects.