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

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 ∼ ρ Γ , T ∼ ρ Γ−1 . We use grid resolutions of 2048 3 cells and compare polytropic exponents Γ = 0.7 (soft EOS), Γ = 1 (isothermal EOS), and Γ = 5/3 (stiff EOS). We find a complex network of non-isothermal filaments with more small-scale fragmentation occurring for Γ < 1, while Γ > 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 Γ > 1. In contrast, the PDF becomes closer to a lognormal distribution for Γ 1. We derive and test a new density variance -Mach number relation that takes Γ 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 Γ and find that it decreases by a factor of ∼ 5 from Γ = 0.7 to Γ = 5/3.

The distribution of density in supersonic turbulence

“…Overall, model predictions seem to match with numerical results up to numerical uncertainties, although more detailed comparisons will be necessary to understand its successes and failures more completely. We finish with an extension to non-isothermal turbulence (illustrating reasonable qualitative agreement with the simulations of Federrath & Banerjee 2015) and a discussion of MHD, before concluding by reiterating the model's main predictions.…”

“…(iv) The assumption that the shock width is equal to the sonic scale-while useful as a phenomenological tool for deriving density variance-Mach number relations (Price et al 2011;Padoan & Nordlund 2011;Molina et al 2012;Federrath & Banerjee 2015)is inconsistent with log-normal density statistics. In particular, for isothermal shocks with a density contrast ∼ M 2 , such a model involves all of the mass being concentrated in a single shock, which leads to an unphysically intermittent density distribution.…”

“…These are crucial for star-formation applications and directly observable in the ISM. However, although there are certain wellestablished results-most importantly the density variance-Mach number relation: that the density distribution is approximately lognormal with a variance that increases with Mach number (Passot & Vázquez-Semadeni 1998;Price et al 2011;Padoan & Nordlund 2011;Molina et al 2012;Federrath & Banerjee 2015;Pan et al 2016)-we lack detailed understanding of many important issues. For example, the power spectrum of density and its variation with Mach number is not well understood (Kim & Ryu 2005;Kritsuk et al 2007;Kowal et al 2007;Konstandin et al 2016).…”

The Link Between Turbulence, Magnetic Fields, Filaments, and Star Formation in the Central Molecular Zone Cloud G0.253+0.016

“…It is supersonic, magnetized turbulence that determines the density PDF and, in particular, its standard deviation (Padoan et al 1997b;Federrath et al 2008b;Padoan & Nordlund 2011;Price et al 2011;Konstandin et al 2012;Burkhart & Lazarian 2012;Molina et al 2012;Federrath & Banerjee 2015;Nolan et al 2015). A high-density power-law tail can develop as a consequence of gravitational contraction of the dense cores in a cloud (Klessen 2000;Federrath et al 2008aFederrath et al , 2011bKritsuk et al 2011;Federrath & Klessen 2013;Girichidis et al 2014).…”

“…6. THE EFFECTIVE TURBULENT DRIVING Theoretical and numerical studies have shown that the density fluctuations (s r r 0 ) in a turbulent medium correlate with the Mach number () and the driving of the turbulence, which is controlled by the turbulence driving parameter b (Federrath et al 2008bPrice et al 2011;Konstandin et al 2012;Federrath & Banerjee 2015;Nolan et al 2015), . While Equation (8) lists the three special cases (solenoidal, mixed, compressive), the driving parameter can vary continuously in the range   b 1 3…”