  By means of high resolution hydrodynamical, three-dimensional calculations
with nested-meshes, we evaluate the eccentricity reached by a low-mass,
luminous planet embedded in an inviscid disc with constant thermal diffusivity
and subjected to thermal forces. We find that a cell size of at most one tenth
of the size of the region heated by the planet is required to get converged
results. When the planet's luminosity is supercritical, we find that it reaches
an eccentricity of order $10^{-2}$--$10^{-1}$, which increases with the
luminosity and broadly scales with the disc's aspect ratio. Restricting our
study to the case of pebble accretion, we incorporate to our model the
dependence of the accretion rate of pebbles on the eccentricity. There is
therefore a feedback between eccentricity, which determines the accretion rate
and hence the planet's luminosity, and the luminosity, which yields the
eccentricity attained through thermal forces. We solve for the steady state
eccentricity and study how this quantity depends on the disc's turbulence
strength parameter $\alpha_z$, on the dimensionless stopping time of the
pebbles $\tau_s$, on the inward mass flux of pebbles and on the headwind (the
difference between the gas velocity and the Keplerian velocity). We find that
in general low-mass planets (up to a few Earth masses) reach eccentricities
comparable to the disc's aspect ratio, or a sizeable fraction of the latter.
Eccentric, low-mass protoplanets should therefore be the norm rather than the
exception, even if they orbit far from other planets or from large scale
disturbances in the disc.
