The density structure and star formation rate of non-isothermal polytropic turbulence

Federrath, C. & Banerjee, S., 2015

Monthly Notices of the Royal Astronomical Society, 448, 3297
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Abstract

The interstellar medium of galaxies is governed by supersonic turbulence, which likely controls the star formation rate (SFR) and the initial mass function (IMF). Interstellar turbulence is non-universal, with a wide range of Mach numbers, magnetic fields strengths, and driving mechanisms. Although some of these parameters were explored, most previous works assumed that the gas is isothermal. However, we know that cold molecular clouds form out of the warm atomic medium, with the gas passing through chemical and thermodynamic phases that are not isothermal. Here we determine the role of temperature variations by modelling non-isothermal turbulence with a polytropic equation of state (EOS), where pressure and temperature are functions of gas density, P ~ rho^Gamma, T ~ rho^(Gamma-1). We use grid resolutions of 2048^3 cells and compare polytropic exponents Gamma = 0.7 (soft EOS), Gamma = 1 (isothermal EOS), and Gamma = 5/3 (stiff EOS). We find a complex network of non-isothermal filaments with more small-scale fragmentation occurring for Gamma < 1, while Gamma > 1 smoothes out density contrasts. The density probability distribution function (PDF) is significantly affected by temperature variations, with a power-law tail developing at low densities for Gamma > 1. In contrast, the PDF becomes closer to a lognormal distribution for Gamma <= 1. We derive and test a new density variance - Mach number relation that takes Gamma into account. This new relation is relevant for theoretical models of the SFR and IMF, because it determines the dense gas mass fraction of a cloud, from which stars form. We derive the SFR as a function of Gamma and find that it decreases by a factor of ~5 from Gamma = 0.7 to Gamma = 5/3.

Simulation movies

The animation shows the gas density projections (top panels) and temperature projections (bottom
panels) of supersonic, non-isothermal turbulence with polytropic
exponents Gamma = 0.7 (left-hand panels) and Gamma = 5/3 (right-hand
panels). Gas with Gamma < 1 cools when it is compressed in dense shocks, while gas
with Gamma > 1 heats up. Real molecular clouds are in the Gamma < 1 regime over a wide range of gas densities
and only when dense cores form do they turn into the Gamma > 1 regime as the gas becomes optically thick.
We also see that lower Gamma results in a more fragmented density cloud on small scales, while Gamma > 1 smoothes out density contrasts.
These simulations use unprecedented resolutions of 2048^3 grid cells.

The next movie shows slices of the gas density (top) and sound speed (bottom) through our 3D simulations with 2048^3 grid cells; polytropic Gamma = 0.7 (left-hand panels)
and Gamma = 5/3 (right-hand panels), as in the first movie above.

We thank P. Hennebelle and E. Vaquez-Semadeni for stim-lating discussions on polytropic turbulence, and we thank the anonymous referee for a constructive and useful reportt.
C.F. acknowledges funding provided by the Australian Research Council's Discovery Projects (grants DP130102078 and DP150104329).
We gratefully acknowledge the Jülich Supercomputing Centre (grant hhd20), the Leibniz Rechenzentrum and the Gauss Centre for Supercomputing (grant pr32lo),
the Partnership for Advanced Computing in Europe (PRACE grant pr89mu), and the Australian National Computing Infrastructure (grant ek9).
The software used in this work was in part developed by the DOE-supported Flash Center for Computational Science at the University of Chicago.