The Influence of Radiative Transfer on SPH Simulations of Star Formation
Stuart C. Whitehouse and Matthew R. Bate
E-mail: scw@astro.ex.ac.uk, mbate@astro.ex.ac.uk
School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL
We have performed a series of calculations of star formation using a smoothed particle hydrodynamics (SPH) code which includes radiative transfer in the flux-limited
diffusion approximation. Previous calculations of star formation using SPH have shown that a large proportion of binary systems are formed by disc fragmentation (e.g. the
large simulations of Bate et al (2002)). However, these simulations used a simple barotropic equation of state, and a study performed by Boss et al (2000) shows that the
amount of fragmentation is sensitive to the equation of state. We have performed SPH calculations including radiative transfer to investigate how the barotropic equation of
state has affected previous results.
Introduction
Simulations of star formation play an important role in determining how
protostars form and evolve. Over the past few years large simulations have
given us insight into the chaotic nature of star formation (Bate et al 2002).
However, such simulations have used an equation of state based on an
approximation of grid-based radiative transfer codes. We show the
evolution of a collapsing sphere of gas using an SPH code including
radiative transfer.
Method
We have developed a three-dimensional smoothed particle hydrodynamics
code with radiative transfer in the flux-limited diffusion approximation
based on the one-dimensional code of Whitehouse & Bate (2004). The
results of a supercritical shock with the one-dimensional code can be seen
in figure 1.
Figure 3: The evolution of collapsing clouds in the log-density log-temperature plane. The
red and green crosses are the evolution of the Boss & Bodenheimer (1979) initial conditions
in the spherically symmetric (red) and binary (green) case. The blue crosses are the Boss &
Myhill (1992) collapse at a low resolution (5000 particles), and the cyan the same conditions
with a higher resolution (50,000 particles). The long-dashed line is the equation of state
used by Bate et al (2003), and the short-dashed line is the equation of state used by Bate,
Bonnell & Price (1995) as a comparison with previous work.
Results
Figure 3 shows the evolution of the collapsing cloud simulations in the logdensity log-temperature plane, together with two barotropic equations of state
for comparison.
Figures 1 & 1b: A supercritical shock. On the left is the result from our SPH code with radiative
transfer. On the right is the result from a comparable grid-based code (Turner & Stone 2001). In
the left graph the dotted line is the gas temperature and the solid the radiation temperature. On the
right the dashed line is the analytic solution.
Using the United Kingdom Astrophysical Fluids Facility, we have
performed simulations on of symmetric cloud collapses similar to those of
Boss & Myhill (1992) and Boss & Bodenheimer (1979) using our threedimensional code. Boss & Myhill start with a 1.2 Msol sphere of density
1.7 x 10-19 g cm-3, and Boss & Bodenheimer use a 1.0 Msol sphere of
density 1.44 x 10-17 g cm-3. All collapses were performed with the
opacities of Alexander (1975) at high temperatures and Pollack et al
(1985) at low temperatures. We use the specific heat capacity from Black
& Bodenheimer (1975) for both cases.
As can be seen from the figure, the Boss & Bodenheimer initial conditions
follow a greatly different evolutionary path than the Boss & Myhill conditions,
even though the opacities used are the same and the only major difference is the
starting density. The Boss & Bodenheimer conditions are consistently much
hotter than the Boss & Myhill conditions for a given density.
The spherically symmetric Boss & Bodenheimer collapse (red) starts to heat at a
density of around 10-14, but applying the same initial conditions with the m=2
density perturbation superimposed upon them (green, which should produce a
binary star) leads to the collapse still being isothermal at much higher densities.
The significantly different evolutionary tracks from the different simulations
imply that the temperature evolution of a collapse is dependent on the initial
density distribution. This has significant implications for previous simulations
that have used a barotropic equation of state.
Summary
• The temperature evolution of simulations of collapsing clouds depends upon
the initial conditions.
• There is no “one-size-fits-all” temperature-density relation.
• Previous simulations using simple barotropic equations of state may be
inadequate.
• Radiative transfer is necessary to properly simulate these conditions.
References
Figures 2a & 2b: The density (left) and temperature (right) profiles of the collapsing lowresolution Boss & Myhill collapses. The red crosses are at a time of 1.3127 free-fall times
(maximum density of approximately 10-13 g cm-3), the green at 1.3164 tff (10-11 g cm-3) and the
blue at 1.3281tff (10-9 g cm-3). The pressure-supported core can clearly be seen at the later times
in the inner regions of the density profile, while the outer part of the cloud remains isothermal.
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