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

Aims.We introduce and test a new and highly efficient method for treating the thermal and radiative effects 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 effects of (i) the rotational and vibrational degrees of freedom of H2; (ii) H2 dissociation and Hᵒ 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∝ρ0.08; cf. Larson 2005, MNRAS, 359, 211). 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 T_C~ 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, ApJ, 531, 350). 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.