ON THE THERMAL INSTABILITY IN NUMERICAL MODELS OF THEINTERSTELLAR MEDIUM

SAMI DIB, ANDREAS BURKERT and AHMAD HUJEIRATMax-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany

E-mail: dib@mpia.de

Abstract. We investigate with 3D hydrodynamical simulations the role played by thermal processesin the dynamical evolution of the interstellar medium (ISM). A parametric approach of the coolingprocess shows that the observed mass fraction of the cold (< 300 K) and unstable gas (300K < T <

6000K) can not be produced by turbulent compression or background heating of the medium alone.An analysis of the properties of the clouds that are formed by the combined effect of the thermal andgravitational instability shows that the cloud’s scaling relations imprinted by the thermal instability(TI) are in good agreement with observational values.

Keywords: thermal instability, ISM, molecular clouds

1. Introduction

Recent studies confirm the important role played by thermal processes in the in-terstellar medium. On the observational side, there are evidences that many of theshell- and void-like structures observed in the ISM of different galaxies, especiallyin the outer parts of their galactic disk are unlikely to be the result of supernovaexplosions (Kim et al., 1999). Furthermore, a careful study by Rhode et al. (1999)of the holes of the galaxy Holmberg II shows no sign of a remnant stellar clusterinside the voids as would be expected from the supernova scenario. From a theor-etical point of view, Kolesnik (1991) has shown that TI plays an important role inthe formation of GMC’s and Burkert and Lin (2000) showed, via 1D simulations,that TI leads to the build up of significant overdensities. Vázquez-Semadeni et al.(2000), Gazol et al. (2001) carried out a number of 2D simulations that hint to thefact that thermal processes of cooling and heating, coupled to energy feedback inthe form of stellar winds can reasonably reproduce the observed mass fraction ofthe unstable gas. However it is still not clear from their simulations what is thetime evolution of this mass fraction and what is explicitly the origin of the unstablegas, see Gazol et al. (2001) for more discussion (also Gazol et al., this volume).In §2, we use a parametric study to constrain what could be the mass fractionof the unstable gas of non-stellar origin and in §3, we present scaling relationsderived from a typical model presented in §2. Calculations were performed usingthe ZEUS-3D code (Stone and Norman, 1992) at the Rechnenzentrum Garching ofthe Max-Planck-Gesellschaft.

Astrophysics and Space Science 289: 465–468, 2004.© 2004 Kluwer Academic Publishers. Printed in the Netherlands.

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2. Parametric Study

We solve the equations of ideal gas dynamics for a one component fluid on amedium resolution grid (1283 grid) which represents a 1 Kpc3 volume. We assumea polytropic equation of state with a specific heat ratio of 1.4 and impose periodicboundary conditions in the three directions. The initial conditions of all runs areidentical and taken from a fully driven isothermal (T=104K) simulation evolvedover one dynamical timescale τdyn = L/Vrms0 ≈ 108 years, where L = 1 Kpcand Vrms0 = 10 km sec−1 is the rms turbulent velocity. Thus, the initial densitydistribution is inhomogeneous with (δρ/ρ)max = 2.8, and the velocity distributioncorresponds to a Mach number Ma0 = 1 turbulent medium. For a parametric studywe assume the cooling and heating terms in the energy equation to be respectively�(ρ, T ) = �0ρ

αT β and L(ρ) = q�0ρε . The latter is responsible for keeping a

fraction of the gas in the unstable regime. Shielding of the densest condensations istaken into account, by nulling the heating inside dense clouds and by applyinga shape-dependent heating at the surface cells of each condensation. Shieldingapplies whenever ρ ≥ ρsh = 25ρ̄ where ρ̄ = 1.53 × 10−27 g cm−3 is the meandensity. �0 can be related to the cooling timescale τcool by the relation displayedin Eq. 1.

�0 = 3

2

(Ma0Vrms0)1−β

ρ̄α−1τcool

(1)

To avoid the task of fixing an arbitrary cooling timescale, we parametrize the lat-ter as a function of the dynamical timescale by introducing the parameter η =τcool/τdyn. In this study α and ε have been fixed in all the simulations at the valuesof 2 and 1 respectively and minimum and maximum temperatures of 10 K and12000 K are assumed. We varied the three other parameters η (with β = 0.25,q = 0.1), β (with η = 0.3, q = 0.1) and q (with η = 0.3, β = 0.25) and analysedthe mass fractions of the cold (Fc) and unstable gas (Fu). Figure 1 shows that Fu

can reach high values of the order of 0.3 close to the value of 0.4–0.5 derivedfrom observations (Heiles et al., 2001), but at a cost of an unrealistic small valuefor Fc. We ran simulations with other permutations of the parameters leading toFu ≈0.30–0.40, but it turned out that the outmost of what Fc can reach is a valueof 0.1. We conclude by stating that the contribution of thermal processes of non-stellar origin to the mass fraction of the unstable gas is in the range of 10–20%and the location of most of the unstable gas should be directly correlated to starforming regions.

3. Scaling Relations

We use a clump-finding algorithm based on a density threshold and a friend-of-friend search method for neighbor cells to extract the scaling relations of the con-densations which are formed following the TI. We use a model for which η = 0.3,

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Figure 1. Mass fraction of the cold gas (Fc) and of the unstable gas (Fu) at t=2 τdyn as a function ofthe cooling-heating parameters.

β = 0.25 and q = 1, where the self-gravity of the gas has also been accountedfor. Keeping in mind the conclusion of Sec. 1, we present the scaling relationsat a time t≈ 0.6τdyn ≈ 56 Myrs where star formation is believed not to haveplayed an important role yet and before a runaway gravitational collapse can takeplace (at this time the numerical density in the densest clouds has reached 300–400cm−3). The medium shows a rather filamentary structure at this stage as displayedin Figure 2a, with the densest clouds sitting at the intersection of filaments. Figure2b displays the mean density-size relation as extracted from the 3 dimensionaldata cube. The coefficient m in the relation n̄ ≈ Rm is found to be equal to –0.9 and –0.7 when density thresholds of 8 and 10 times the average density areused, respectively. In comparaison, observations converge towards a value of ≈ –1, (Larson, 1981). We should stress that the relations we obtain are derived fromthe three dimensional physical data. A comparison of those relations to relationsderived from the integration of the data on given line of sights as presented recentlyin Ballesteros-Paredes and Mac-Low (2002) (molecular cloud simulation s on ascale of 0.1 pc), will be presented in detail in a subsequent paper.

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Figure 2. a) (Left) Density structure at t=0.6 τdyn. Iso-density contours correspond to n≥ n̄/2, where

n̄=0.5 cm−3 is the average numerical density. b) (Right) Mean density-size relation for two densitythresholds in the clump-finding algorithm. 8 n̄ (triangles), 10 n̄ (diamonds).

Acknowledgements

S.D. acknowleges the financial support of the EAS and would like to thank J.Ballesteros-Paredes, A. Gazol and Fabian Heitsch for stimulating discussions.

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