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Astron. Astrophys. 325, 972-986 (1997)

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2. Numerical technique

We model the evolution of galaxies using a hybrid N- body/hydrodynamics code (TREESPH; Hernquist & Katz 1989). A brief summary of the techniques pioneered by these authors follows. A tree algorithm (Barnes & Hut 1986 , Hernquist 1987 ) determines the gravitational forces on the collisionless and gaseous components of the galaxies. The hydrodynamic properties of the gas are modeled using SPH. The gas evolves according to hydrodynamic conservation laws, including an artificial viscosity for an accurate treatment of shocks. Each particle is assigned an individual smoothing length, [FORMULA], which determines the local resolution and an individual time step. Estimates of the gas properties are found by smoothing over 32 neighbours within [FORMULA]. We adopt the equation of state


where P is the pressure, [FORMULA] the density, u the thermal energy density, and [FORMULA] for an ideal gas. In the Lagrangian formulation of SPH energy conservation can be expressed as


where [FORMULA] is the velocity, and [FORMULA] represents the energy sources and sinks.

The cooling term [FORMULA] in Eq.(2) describes radiative cooling of the gas and is sensitive to the chemical abundances. We do not alter this composition, but instead we assume a hydrogen gas mix with a helium mass fraction of 0.25. For this "standard" gas, cooling functions can be found in the literature (e.g. Dalgarno & McCray 1972 ). We parametrize the cooling function as




where T is the temperature in Kelvin. The second term on the right hand side of Eq.(3) determines the cooling below [FORMULA]  K. This part of the cooling function is strongly dependent on the ionization parameter [FORMULA]. We parametrize this part of the cooling function with a and b. Values can be found in Table 1. Since we do not solve the ionization balance, we choose the ionization parameter a priori.


Table 1. Cooling function parameters a, b for various ionization parameters x.

The largest contributor to the gas heating [FORMULA] is photoelectric heating of small grains and PAHs (see e.g. Wolfire et al. 1995 ). The heating rate is given by


where [FORMULA] is the heating efficiency and [FORMULA] is the incident far-ultraviolet (FUV) field (91.2 nm to 210 nm) normalized to Habing's (1968) estimate of the local interstellar value ([FORMULA]). The heating efficiency [FORMULA] for temperatures below 10000 K. Since at about [FORMULA]  K the cooling increases more than a factor [FORMULA], this temperature is effectively an upper limit to the gas temperature, virtually independent of the heat input.

During the simulations the cooling time in some regions can become much shorter than the SPH time step, which is determined by the Courant-Friedrichs-Levy (CFL) condition. If the cooling time is very short the gas will reach thermal equilibrium within the SPH time step. Therefore, in those regions where [FORMULA] we enforce thermal equilibrium by requiring [FORMULA] and choose as time step [FORMULA]. In other regions we choose as particle time step [FORMULA]. For these latter particles we do not allow a particle to lose more than half its thermal energy in one time step (Katz & Gunn 1991 ).

Since the gas is heated mainly due to FUV photons, we must calculate the stellar radiation field. Due to the limited resolution of our simulations each star represents a stellar association. Such an association will be called a star particle, which we assume to be formed instantaneously. That means that all star particles have an age, but these ages differ from particle to particle. We can attribute to each particle an FUV flux according to its age. For this we constructed look-up tables with FUV fluxes using the population synthesis models of Bruzual & Charlot (1993). The FUV flux is dependent on the Initial Mass Function (IMF) of the stellar cluster and on the lower and upper mass limits of the IMF. Once we have adopted a specific IMF with mass limits, we can calculate the FUV radiation field by summing the FUV fluxes from all stars, corrected for geometrical dilution but not for extinction. The stellar FUV fluxes inside the smoothing length are softened using the spline-kernel which is also used for softening the acceleration (Hernquist & Katz 1989 ).

A single dynamical particle thus represents an entire stellar association. We suppose that this does not invalidate our models as long as the dynamical disintegration of the association takes more time than the main sequence lifetime of its massive members, which contribute most to the heating flux. This condition is normally fulfilled for the usual stellar mass functions.

Extinction is probably not very important below a hydrogen column density [FORMULA] (Wolfire et al. 1995 ). As an approximation we can model the extinction as exponential decay with distance ([FORMULA]), where [FORMULA] is an absorption coefficient. Unless otherwise stated we will take [FORMULA] in our calculations, which produces a transparent ISM.

2.1. From ISM to stars and back

Star formation is governed by the delicate symbiosis between stars and the interstellar medium. Stars influence the ISM by heating it, due to FUV-radiation, stellar winds and supernovae, while the ISM in return provides the necessary material to form stars. In this section we give a rough description of star formation and the (local) ISM.

The local ISM consists of several components: hot intercloud medium ([FORMULA]  K), H II ([FORMULA]  K), warm intercloud medium (8000 K), warm H I (8000 K), H I clouds (10-100 K) and molecular clouds (5-30 K). The hot and warm intercloud medium occupy almost 100% of the volume, equally divided. On the other hand, the H I and H2 clouds contain about 90% of the mass, with the other 10% in the warm intercloud medium (Knapp 1990 ). The hot gas is probably heated by supernova shocks, whereas the warm and cold H I and H2 clouds are heated mainly by the photoelectric ejection of electrons from dust grains by the interstellar radiation field (Wolfire et al. 1995 ).

Star formation occurs in giant molecular clouds (GMCs). These typically have masses of a few [FORMULA], temperatures around 10 K and number densities in excess of [FORMULA] (e.g. Shu et al. 1987 , Bodenheimer 1992 ). GMCs do not collapse as a whole, but fritter away as smaller subclouds inside contract to form stars. Most stars form in associations containing on the order of [FORMULA] of stars. The collapse time is some 10 times the free-fall time, because the clouds cannot dissipate their internal energy fast enough to collapse at their free-fall rates (Elmegreen 1992 ). The lifetimes of molecular clouds range from [FORMULA]  yr for GMCs to over [FORMULA]  yr for dwarf molecular clouds (Shu et al. 1987 ).

The formation of stellar associations inside a GMC destroys the cloud within 10 Myr, due to FUV radiation, stellar winds and supernovae (Blaauw 1991 ). This behaviour is reflected in the star formation efficiency. On the scale of stellar associations the efficiency may be as high as 50%, while for GMCs it is only a few percent (Bodenheimer 1992 ).

2.2. From ISM to stars numerically

The amount of physics in our simulations is limited by the number of particles our computers can handle, which is far smaller than the number of stars and gas clouds in galaxies. We think that the most interesting gas phases for star formation are the warm and cold phases ([FORMULA]  K), while the hot phase is of less importance for controlling the star-gas life cycle. The cooling is described by a standard cooling function between [FORMULA]  K (Dalgarno & McCray 1972 ). We do not solve the ionization balance in detail, so we cannot distinguish between cold neutral or molecular gas, and warm neutral and fully ionized gas.

Since photoelectric emission from dust is the most important heating source for the cold and warm gas, we have to calculate the stellar FUV radiation field. A star particle in our simulations corresponds to a stellar cluster, with a given IMF and age. Thus we do not form individual stars, but only associations. If the stellar FUV radiation is included, a direct feedback from new star particles upon the ambient gas particles is assured.

Another important feedback of star formation on the ISM is the energy injection into the ISM by supernovae and stellar winds from massive stars. Numerical studies conducted by others have so far concentrated on the feedback from supernovae (and ignored radiative energy input from stars). This supernova energy has been modeled as thermal or mechanical energy input. If it is modeled (partly) as thermal energy, all studies agree that there is little effect on the evolution of the system since the energy is rapidly radiated away (Katz 1992 , Navarro & White 1993 , Friedli & Benz 1995 ). This is due to the cooling properties of the gas, which limit the gas to [FORMULA]  K. If no cooling below this temperature is allowed, then the thermal energy input indeed cannot have a large effect on the evolution. Modeling the supernova energy as mechanical energy poses other problems: only a small fraction can be converted into kinetic energy of surrounding gas particles, otherwise even normal galaxies would expel much of the gas (Mihos & Hernquist 1994 , Friedli & Benz 1995 ). Given the problems involved with incorporating supernova energy, and since we are not trying to model the hot gas and do not return mass from stars to the surrounding ISM, we do not include supernova energy in the present study, although it will be included in future work. In this article however, we focus on the effects of the stellar heating on star formation.

In SPH simulations the particles sample gas properties such as density and temperature. These particles cannot be correctly interpreted as gas clouds: only groups of particles can be considered as clouds. Bearing this in mind we adopted the following recipe for star formation. For each particle, we calculate the Jeans mass


(e.g. Binney & Tremaine 1987 ) where s is the sound speed of the gas and G the gravitational constant. If this quantity is smaller than the mass of a GMC the particle is in a cold, dense environment, resembling a GMC and we declare the particle to be part of a (gravitationally) unstable cloud. Once a region is labeled unstable, it is allowed to form star clusters. The first condition for an SPH particle to form stars can thus be summarized as:


where the critical cloud mass [FORMULA] is set a priori and does not change during the simulations (see also Sect. 3.2.2).

Once a region is unstable, it takes a collapse time to form a stellar cluster. This collapse time is the most influential model parameter in our recipe and its value is uncertain. The problem is mainly numerical. We know that subunits inside GMCs exist, where densities reach values exceeding a few 100 cm-3 and temperatures fall below a few times 10 K. Especially these high densities are beyond the dynamical range of the present simulations. Moreover new physics is involved in forming these high densities, in particular the self-shielding of molecular clouds (e.g.  Van Dishoeck & Black 1986 ). In these subunits the collapse time is governed by the dissipation time scale which we cannot estimate due to the problems mentioned above. Given all these uncertainties we couple the collapse time simply to the free-fall time. In practice this leads to collapse times of [FORMULA]  yr, which corresponds to the lifetimes for molecular clouds. In order to have some handle on the collapse time [FORMULA], we introduce a scale parameter [FORMULA],


The second condition for an SPH particle to form stars can now be stated as:


where [FORMULA] is the time that the particle remains in an unstable cloud.

As soon as an SPH particle has fulfilled both conditions (Eq.(7) and Eq.(9)) part of its mass is converted into a star particle of mass


where [FORMULA] is the star formation efficiency (SFE). We fix this efficiency at the beginning of the simulation, so that all new star particles have the same mass since [FORMULA] does not change. The star particle gets the same IMF as all other star particles, and age zero. New star particles are then included in the calculation of the radiation field; their emission evolves in time according to standard stellar evolution.

To prohibit infinitely low-mass particles we put a lower limit to the minimum mass of a particle, [FORMULA], where [FORMULA] is the initial mass of an SPH particle. If the mass of a gas particle falls below this limit when forming a star particle, it is totally converted into stars.

The new star particle gets initial velocity and acceleration equal to that of the parent gas particle; its position is offset in a random direction by an amount [FORMULA].

In summary, we devised the following recipe. From our SPH particle distribution we select conglomerates where the Jeans mass is sufficiently below the aggregate mass. We take this to mean that such regions resemble GMCs. We follow them during their dynamical and thermal evolution and if an SPH particle resides in such a region longer than the collapse time, part of its mass is converted into a star particle. This star particle then heats the surrounding gas as its stellar population evolves, possibly inhibiting further star formation.

In this way we have tried to minimise the damage due to the inevitable lack of resolution by choosing star formation criteria that are based on global properties. Our recipe uses a Jeans criterion for the gas as a whole, supplemented with an estimate of the cloud collapse timescale. Thus we forge a connection between the parent galaxy and its unresolved constituents. In the same vein we calculate cloud heating due to the global radiation field. Accordingly, even though an SPH 'neighbourhood' consists of 32 gas particles, we can still pick a much smaller value (such as 10 SPH masses) as the critical cloud mass. We do not thereby claim that this mass scale is resolved (it isn't), but merely maintain that only a small fraction forms stars. This is completely in keeping with the prevailing view that star formation is a stochastic process when seen from the large-scale perspective. All the same it remains true that these assertions must be verified numerically by charting the influence of the choices of various parameters, which indeed we try to do.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998