A semi-analytic model of the turbulent multi-phase interstellar medium

Mon. Not. R. Astron. Soc. 421, 1838–1860 (2012) doi:10.1111/j.1365-2966.2011.19889.x A semi-analytic model of the turbulent multi-phase interstellar medium H. Braun and W. Schmidt Institut für Astrophysik, Universität Göttingen, Friedrich-Hund Platz 1, D-37077 Göttingen, Germany Accepted 2011 September 26. Received 2011 August 12; in original form 2011 April 29 ABSTRACT Key words: turbulence – methods: numerical – stars: formation – ISM: structure – galaxies: ISM. 1 I N T RO D U C T I O N The capabilities of contemporary supercomputing enable us to model the evolution of the baryonic gas in the universe with unprecedented sophistication. Adaptive methods such as smoothed particle hydrodynamics (SPH) and adaptive mesh refinement (AMR) in Eulerian grid codes allow us to cover a huge dynamic range such  E-mail: that simulations of the formation and evolution of galaxies from cosmological initial conditions at high resolution (∼100 pc) are within reach (Gnedin & Kravtsov 2010; Agertz, Teyssier & Moore 2011). In simulations of isolated disc galaxies, it is feasible to resolve length-scales down to ∼10 pc (Agertz et al. 2009; Tasker & Tan 2009). Computations on these length-scales entail the problem to account for various physical processes in the multi-phase interstellar medium (ISM; Mayer, Governato & Kaufmann 2008). Notwithstanding the high numerical resolution that can be achieved, several important processes cannot be fully resolved and have to be described by means of a sub-grid scale (SGS) model.  C 2012 The Authors C 2012 RAS Monthly Notices of the Royal Astronomical Society  We present a semi-analytic model for the interstellar medium that considers local processes and structures of turbulent star-forming gas. A volume element of the interstellar medium is described as a multi-phase system, comprising a cold and a warm gas phase in effective (thermal plus turbulent) pressure equilibrium and a stellar component. The cooling instability of the warm gas feeds the cold phase, while various heating processes transfer cold gas to the warm phase. The cold phase consists of clumps embedded in diffuse warm gas, where only the molecular fraction of the cold gas may be converted into stars. The fraction of molecular gas is approximately calculated, using a Strömgren-like approach and the efficiency of star formation is determined by the state of the cold gas and the turbulent velocity dispersion on the clump length-scale. Gas can be heated by supernovae and ultraviolet emission of massive stars, according to the evolutionary stages of the stellar populations and the initial mass function. Since turbulence has a critical impact on the shape of the gaseous phases, on the production of molecular hydrogen and on the formation of stars, the consistent treatment of turbulent energy – the kinetic energy of unresolved motions – is an important new feature of our model. Besides turbulence production by supernovae and the cooling instability, we also take into account the forcing by large-scale motions. We formulate a set of ordinary differential equations, which statistically describes star formation and the exchange between the different budgets of mass and energy in a region of the interstellar medium with given mean density, size, metallicity and external turbulence forcing. By exploring the behaviour of the solutions, we find equilibrium states, in which the star formation efficiencies are consistent with observations. Kennicutt–Schmidt-like relations naturally arise from the equilibrium solutions, while conventional star formation models in numerical simulations impose such relations with observed efficiency parameters as phenomenological calibrations. Beyond the semi-analytic approach, a potential application is a complete sub-grid scale model of the unresolved multi-phase structure, star formation and turbulence in simulations of galaxies or in cosmological simulations. The formulation presented in this article combines various models focusing on particular processes and yet can be adopted to specific applications, depending on the range of resolved length-scales. A model of the turbulent multi-phase ISM  C 2012 The Authors, MNRAS 421, 1838–1860 C 2012 RAS Monthly Notices of the Royal Astronomical Society  Padoan & Nordlund (2011, hereafter PN11) parametrize the star formation rate per free-fall time as a function of the virial parameter, i.e. the turbulent velocity dispersion relative to the specific gravitational energy, by using data from forced isothermal magnetohydrodynamic (MHD) turbulence simulations. Following Krumholz & McKee (2005, hereafter KM05), the star formation rate is calculated by integrating density fluctuations beyond a critical density that is given by the virial parameter and the Mach number of the turbulent cold neutral medium. However, as pointed out by Krumholz et al. (2009, hereafter KMT09), new observations reveal a tight correlation between the molecular hydrogen surface density and the star formation rate. They present an analytic model that includes approximate calculations of molecular hydrogen fraction from a spherical-cloud model and the star formation efficiency per freefall time on the basis of the numerical parametrization in KM05. This model reproduces the Kennicutt–Schmidt relation between the star formation rate and the surface density on length-scales of the order of a kiloparsec in recent surveys. By assuming a constant star formation efficiency, the formation of molecular hydrogen in cosmological simulations is modelled by an approximate treatment of shielding and photodissociation in Gnedin et al. (2009, hereafter GTK09). As in KMT09, the star formation rate is assumed to be proportional to the molecular hydrogen density rather than the density of the cold neutral medium. The unresolved density structure of the gas is parametrized by a clumping factor, and the efficiency of star formation per free-fall time in molecular clouds is set to 1 per cent. Using this model, Gnedin & Kravtsov (2010) investigate the Kennicutt–Schmidt relation in galaxies at high redshifts. For simulations of isolated discs with molecular hydrogen chemistry, see Dobbs et al. (2008) and Robertson & Kravtsov (2008). The KMT09 and GTK09 models focus on molecular hydrogen to predict the star formation rate, whereas the multi-phase structure and the turbulent dynamics of the ISM are not addressed explicitly. In contrast, Koppen, Theis & Hensler (1998) formulate a dynamical model for the evolution of a massive and a low-mass star component and clouds embedded in hot gas, with various interaction processes. In a similar way, the model of Springel & Hernquist (2003) considers interacting cold and warm phases and stars. A simple multi-phase SGS model of star formation and supernova feedback is proposed by Murante et al. (2010). By assuming that the amount of molecular hydrogen is controlled by the pressure of the ISM, rate equations for the mass (...truncated)