  It is well known that asteroids and comets fall into the Sun. Metal pollution
of white dwarfs and transient spectroscopic signatures of young stars like
$\beta$-Pic provide growing evidence that extra solar planetesimals can attain
extreme orbital eccentricities and fall onto their parent stars. We aim to
develop a general, practically implementable, semi-analytical theory of secular
eccentricity excitation of small bodies in mean motion resonances with an
eccentric planet valid for arbitrary values of the eccentricities and including
the short-range force due to General Relativity. Our semi-analytic model for
the restricted planar three-body problem does not make use of any series
expansion and therefore is valid for any values of eccentricities and
semi-major axes ratios. The model is based on the application of the adiabatic
principle, which is valid when the precession period of the longitude of
pericenter of the planetesimal is much longer than the libration period in the
mean motion resonance. This holds down to vanishingly small eccentricities in
resonances of order larger than 1. We provide a Mathematica notebook with the
implementation of the model allowing direct use to the interested reader. We
confirm that the 4:1 mean motion resonance with a moderately eccentric planet
is the most powerful one to lift the eccentricity of planetesimals from nearly
circular orbits to star-grazing ones. However, if the planet is too eccentric,
we find that this resonances becomes unable to pump the planetesimal's
eccentricity to very high value. The inclusion of the General Relativity effect
imposes a condition on the mass of the planet to drive the planetsimals into
star-grazing orbits. For a planetesimal at $\sim$1 AU around a solar-mass star
(or white dwarf), we find a threshold planetary mass of about 17 Earth masses.
We finally derive an analytical formula for this critical mass.
