Radiative transfer and the energy equation in SPH simulations of star formation

Aims. We introduce and test a new and highly e fficient method for treating the thermal and radiative e ffects influencing the energy equation in SPH simulations of star formation. Methods. The method uses the density, temperature and gravitational potential of each particle to estimate a mean optical depth, which then regulates the particle’s heating and cooling. The method captures – at minimal computational cost – the e ffects of (i) the rotational and vibrational degrees of freedom of H 2; (ii) H 2 dissociation and H o ionisation; (iii) opacity changes due to ice mantle melting, sublimation of dust, molecular lines, H − , bound-free and free-free processes and electron scattering; (iv) external irradiation; and (v) thermal inertia. Results. We use the new method to simulate the collapse of a 1 M ⊙ cloud of initially uniform density and temperature. At first, the collapse proceeds almost isothermally ( T ∝ ρ 0 .08; cf. Larson 2005). The cloud starts heating fast when the optical depth to the cloud centre reaches unity (ρ C ∼ 7 × 10 −13 g cm − 3 ). The first core forms at ρ C ∼ 4 × 10 − 9 g cm − 3 and steadily increases in mass. When the temperature at the centre reaches TC ∼ 2000 K, molecular hydrogen starts to dissociate and the second collapse begins, leading to the formation of the second (protostellar) core. The results mimic closely the detailed calculations of Masunaga & Inutsuka (2000). We also simulate (i) the collapse of a 1 .2 M ⊙ cloud, which initially has uniform density and temperature, (ii) the collapse of a 1 .2 M ⊙ rotating cloud, with an m = 2 density perturbation and uniform initial temperature, and (iii) the smoothing of temperature fluctuations in a static, uniform density sphere. In all these tests the new algorithm reproduces the results of previous authors and/or known analytic solutions. The computational cost is comparable to a standard SPH simulation with a simple barotropic equation of state. The method is easy to implement, can be applied to both particle- and grid-based codes, and handles optical depths 0 < τ <∼ 1011 .