  We analyse the size evolution of pebbles accreted into the gaseous envelope
of a protoplanet growing in a protoplanetary disc, taking into account
collisions driven by the relative sedimentation speed as well as the convective
gas motion. Using a simple estimate of the convective gas speed based on the
pebble accretion luminosity, we find that the speed of the convective gas is
higher than the sedimentation speed for all particles smaller than 1 mm. This
implies that both pebbles and pebble fragments are strongly affected by the
convective gas motion and will be transported by large-scale convection cells
both towards and away from the protoplanet's surface. We present a simple
scheme for evolving the characteristic size of the pebbles, taking into account
the effects of erosion, mass transfer and fragmentation. Including the
downwards motion of convective cells for the transport of pebbles with an
initial radius of 1 millimeter, we find pebble sizes between 100 microns and 1
millimeter near the surface of the protoplanet. These sizes are generally
amenable to accretion at the base of the convection flow. Small protoplanets
far from the star (>30 AU) nevertheless erode their pebbles to sizes below 10
microns; future hydrodynamical simulations will be needed to determine whether
such small fragments can detach from the convection flow and become accreted by
the protoplanet.
