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

 Stamatellos, Dimitrios, Whitworth, Anthony Peter, Bisbas, Thomas G. and Goodwin, S. 2007. Radiative transfer and the energy equation in SPH simulations of star formation. Astronomy and Astrophysics 475 (1) , pp. 37-49. 10.1051/0004-6361:20077373

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## Abstract

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 H$^{\rm o}$ 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 $1\,{M}_\odot$ cloud of initially uniform density and temperature. At first, the collapse proceeds almost isothermally ( $T\propto\rho^{0.08}$; cf. Larson 2005, MNRAS, 359, 211). The cloud starts heating fast when the optical depth to the cloud centre reaches unity ( $\rho_{_{\rm C}}\sim 7\times10^{-13}~{\rm g\ cm^{-3}}$). The first core forms at $\rho_{_{\rm C}}\sim 4\times10^{-9}~{\rm g\ cm^{-3}}$ and steadily increases in mass. When the temperature at the centre reaches $T_{_{\rm C}}\sim 2000\,{\rm K}$, 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 $1.2\,{M}_\odot$ cloud, which initially has uniform density and temperature, (ii) the collapse of a $1.2\,{M}_\odot$ 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 $0< \tau\la 10^{11}$.

Item Type: Article Publication Published Physics and Astronomy Q Science > QB Astronomy stars: formation methods: numerical radiative transfer hydrodynamics; ISM: clouds Pdf uploaded in accordance with publisher's policy at http://www.sherpa.ac.uk/romeo/issn/0004-6361/ (accessed 17/04/2014) EDP Sciences 0004-6361 30 March 2016 04 Jun 2017 04:54 http://orca-mwe.cf.ac.uk/id/eprint/46305