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Astron. Astrophys. 342, 655-664 (1999)

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2. Numerical technique

We model the evolution of galaxies using a hybrid N- body/hydrodynamics code (TREESPH; Hernquist & Katz 1989). An extensive description is given in Gerritsen & Icke (1997, 1998). A tree algorithm (Barnes & Hut 1986; Hernquist 1987) determines the gravitational forces on the collision-less and gaseous components of the galaxies. The hydrodynamic properties of the gas are modeled using smoothed particle hydrodynamics (SPH) (see Lucy 1977; Gingold & Monaghan 1977). 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, h, which determines the local resolution and an individual time step. Estimates of the gas properties are found by smoothing over 32 neighbors within [FORMULA]. We adopt an equation of state for an ideal gas.

We allow radiative cooling of the gas according to the cooling function for a standard hydrogen gas mix with a helium mass fraction of 0.25 (Dalgarno & McCray 1972). Radiative heating is modeled as photo-electric heating of small grains and PAHs by the FUV field (Wolfire et al. 1995), produced by the stellar distribution.

All our simulations are advanced in time steps of [FORMULA] yr for star particles, while the time steps for SPH particles can be 8 times shorter ([FORMULA] yr). The gravitational softening length is 150 pc. A tolerance parameter [FORMULA] is used for the force calculation, which includes quadrupole moments.

2.1. Star formation and feedback

Gerritsen & Icke (1997, 1998) extensively describe the recipe for transforming gas into stars and the method for supplying feedback onto the gas. The recipe works well for normal HSB galaxies, with the energy budget of the ISM as prime driver for the star formation. The simulations allow for a multi-phase ISM with temperature between [FORMULA] K. This allows us to consider cold [FORMULA] K regions as places for star formation (Giant Molecular Clouds in real life). Below we summarize the important ingredients of the method.

From our SPH particle distribution we select conglomerates where the Jeans mass is below the mass of a typical Giant Molecular Cloud. In the simulations performed here we use [FORMULA] for this aggregate mass (the method is not very sensitive to the exact value for this mass). The Jeans mass depends on the local gas properties and is calculated from the SPH estimates of the density [FORMULA] and sound speed s,


with G the constant of gravity. During the simulations the maximum number density achieved is of the order 1 cm-3. It follows that only regions below [FORMULA] K are unstable and may form stars.

We follow unstable regions during their dynamical and thermal evolution and if an SPH particle resides in such a region longer than the collapse time,


half of its mass is converted into a star particle. Experiments with a different number of particles and different star formation efficiencies show no dependence on these parameters, as already shown in Gerritsen & Icke (1997).

Important for our calculations is that we consider star particles as stellar clusters with an age. Thus for each individual star particle we can attribute quantities like the SN-rate, the mass loss, and the FUV-flux, according to its age. We use the spectral synthesis models of Bruzual & Charlot (1993) to determine these quantities, where we adopt a Salpeter Initial Mass Function (IMF) with slope 1.35 and with lower and upper mass limits of [FORMULA] and [FORMULA] respectively.

The radiative heating for a gas particle is calculated by adding the FUV-flux contributions from all stars, which is done together with the force calculation. The mechanical luminosity from a star particle is determined by both the SN-rate and the mass loss rate. We assume that each SN injects [FORMULA] ergs of energy and that the energy injected by stellar winds is [FORMULA], with [FORMULA] the wind terminal velocity and [FORMULA] the stellar mass loss. For massive stars [FORMULA] depends critically on the stellar mass, luminosity, effective temperature, and metallicity (e.g. Leitherer et al. 1992; Lamers & Leitherer 1993). For simplicity we adopt [FORMULA] km s-1. After [FORMULA] yr the last [FORMULA] stars explode and no more mechanical energy is supplied to the gas. Thus we return mechanical energy from massive stars into the ISM, and ignore the mechanical energy from low mass stars.

In the simulations the parent SPH particle is the carrier of the mechanical energy from the new star particle. This SPH particle ("SN particle") mimics a hot bubble interior. Radiative cooling is temporarily switched off (the resolution does not allow the creation of a low-density, hot bubble), the temperature of the SN particle is set to the mechanical energy density (of a few [FORMULA] K), and the particle evolves adiabatically. The first [FORMULA] yr, the position and velocity of the SN particle are equal to the position and velocity of the associated star particle. Afterwards, the SN particle evolves freely. After [FORMULA] yr radiative cooling is switched on again, and the particle behaves like an ordinary SPH particle.

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999