  Context: Planet formation by pebble accretion is an alternative to
planetesimal-driven core accretion. In this scenario, planets grow by accreting
cm-to-m-sized pebbles instead of km-sized planetesimals. One of the main
differences with planetesimal-driven core accretion is the increased thermal
ablation experienced by pebbles. This provides early enrichment to the planet's
envelope, which changes the process of core growth.
  Aims: We aim to predict core masses and envelope compositions of planets that
form by pebble accretion and compare mass deposition of pebbles to
planetesimals.
  Methods: We model the early growth of a proto-planet by calculating the
structure of its envelope, taking into account the fate of impacting pebbles or
planetesimals. The region where high-Z material can exist in vapor form is
determined by the vapor pressure. We include enrichment effects by locally
modifying the mean molecular weight.
  Results: In the pebble case, three phases of core growth can be identified.
In the first phase, pebbles impact the core without significant ablation.
During the second phase, ablation becomes increasingly severe. A layer of
high-Z vapor starts to form around the core that absorbs a small fraction of
the ablated mass. The rest either rains out to the core or mixes outwards
instead, slowing core growth. In the third phase, the vapor inner region
expands outwards, absorbing an increasing fraction of the ablated material as
vapor. Rainout ends before the core mass reaches 0.6 M_Earth, terminating
direct core growth. In the case of icy H2O pebbles, this happens before 0.1
M_Earth.
  Conclusions: Our results indicate that pebble accretion can directly form
rocky cores up to only 0.6 M_Earth, and is unable to form similarly sized icy
cores. Subsequent core growth can proceed indirectly when the planet cools,
provided it is able to retain its high-Z material.
