  The late phases of the orbital evolution of an Earth-like planet around a
Sun-like star are revisited considering the effect of the density fluctuations
associated with convective motions inside the star. Such fluctuations produce a
random perturbation of the stellar outer gravitational field that excites a
small residual eccentricity in the orbit of the planet counteracting the
effects of tides that tend to circularize the orbit. We compute the power
spectrum of the outer gravitational field fluctuations of the star in the
quadrupole approximation and study 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. Adopting a model of the
stellar evolution of our Sun computed with MESA, we find that the Earth will be
engulfed close to the tip of the red giant branch evolution phase. We find a
maximum mean value of the residual eccentricity of about 0.026 immediately
before the 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 up to about 1.4 au. The engulfment of the Earth by the red giant Sun
is found to be a stochastic process, in contrast to the deterministic character
assumed in previous studies. If an Earth-like planet escapes engulfment, its
orbit around its remnant white dwarf star will be moderately eccentric. Such a
residual eccentricity on the order of a few hundredths can play a relevant role
in sustaining the pollution of the white dwarf atmosphere by asteroids and
comets as observed in several objects.
