  Thermodynamics play an important role in determining the way a protostellar
disc fragments to form planets, brown dwarfs and low-mass stars. We explore the
effect that different treatments of radiative transfer have in simulations of
fragmenting discs. Three prescriptions for the radiative transfer are used, (i)
the diffusion approximation of Stamatellos et al., (ii) the barotropic equation
of state (EOS) of Goodwin et al., and (iii) the barotropic EOS of Bate et al.
The barotropic approximations capture the general evolution of the density and
temperature at the centre of each proto-fragment but (i) they do not make any
adjustments the particular circumstances of a proto-fragment forming in the
disc, and (ii) they do not take into account thermal inertia effects that are
important for fast-forming proto-fragments in the outer disc region. As a
result, the number of fragments formed in the disc and their properties are
different, when a barotropic EOS is used. This is important not only for disc
studies but also for simulations of collapsing turbulent clouds, as in many
cases in such simulations stars form with discs that subsequently fragment. We
also examine the difference in the way proto-fragments condense out in the disc
at different distances from the central star using the diffusion approximation
and following the collapse of each proto-fragment until the formation of the
second core (~10^{-3} g/cm3). We find that proto-fragments forming closer to
the central star tend to form earlier and evolve faster from the first to the
second core than proto-fragments forming in the outer disc region. The former
have a large pool of material in the inner disc region that they can accrete
from and grow in mass. The latter accrete more slowly and they are hotter
because they generally form in a quick abrupt event.
