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Astron. Astrophys. 333, 399-410 (1998)

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2. Simulations

The simulations presented here follow the evolution of two fluids: a collision-less component that comprises [FORMULA] of the mass and represents the dark matter, and a second hydrodynamical gas component. Gravitational forces are calculated using a tree code and the hydrodynamical equations are solved for the gas by use of the smooth particle hydrodynamics (SPH) method. Details of the implementation can be found in Hultman & Källander (1997).

The gas consists of hydrogen and a [FORMULA] mass fraction of helium, typical of estimated primordial values. Furthermore, we assume that the gas is optically thin and in ionization equilibrium at all times. The most important cooling processes are radiative cooling by bound-bound and bound-free transitions. Compton cooling by electron scattering against the Cosmic Microwave Background (CMB) can also have a significant effect at high redshifts, but at low redshifts, [FORMULA], the photon density of the CMB is too low to lead to any significant Compton cooling. In some of the simulations an external strong UV radiation background field is present. The time evolution of this field is fixed already at the start of the simulations, and does not depend on the state of the gas in the simulated region. This field changes the gas cooling rate, and adds a redshift dependence to the cooling function. The gas is also heated by this field.

Radiative molecular cooling is not included, and this limits us when choosing objects to study. Below [FORMULA] K, atomic radiative cooling is inefficient and molecular radiative cooling may be important. By neglecting molecular cooling we cannot with any certainty simulate systems where the gas pressure at temperatures below [FORMULA] K can be dynamically important. This implies a lower mass limit on the applicability of these simulations.

Neutral gas of primordial abundance is stable against collapse in a dark matter halo if the halo mass, [FORMULA], is (Rees 1986, Quinn et al. 1996)


T, [FORMULA] are respectively the gas temperature, the proton mass, the gas mean molecular weight, the formation redshift and the Hubble constant in units of km [FORMULA] [FORMULA]. For a neutral gas, [FORMULA] K, and a formation redshift of [FORMULA] this corresponds to a mass of [FORMULA]. A further complication at these mass scales is that galactic halos with as low masses as [FORMULA] are very susceptible to disruption by supernovae (Dekel & Silk 1986). The least total mass in the sequence of simulations, was for these reasons chosen to have a total mass of [FORMULA].

Each object was simulated both with and without a photo-ionizing background, using the same initial density field. In this way it is possible to investigate the effects of the photo-ionizing background, without making a statistical study that would require a large number of simulations. The assumed background radiation field is of the form


where [FORMULA] is the Lyman limit frequency, and the time dependence of the field is contained in the function [FORMULA].

Observational determinations of [FORMULA] in recent years include the following (Haardt and Madau 1996): [FORMULA] [FORMULA] 0.5 at [FORMULA] (Bechtold 1994), [FORMULA] at [FORMULA] (Williger et al. 1994), [FORMULA] at [FORMULA] (Bajtlik et al. 1988, Lu et al. 1991), and [FORMULA] at [FORMULA] (Espey 1993). A decline in the intensity is expected at high redshifts due to a decline in the number density of quasars and increased absorption by Lyman-alpha clouds. Here [FORMULA] is taken to have the form


Proto-galactic gas is enriched with metals ejected from heavy stars, through stellar winds and supernovae. This leads to an significant increase in the gas cooling rate at the temperatures that are relevant for galaxy formation, as can be seen from cooling curve by Sutherland & Dopita (1993), in Fig. 1. To take cooling effects of metals into account, a description of the star formation rate is necessary. Star formation is, however, not well understood. Since the only purpose here is to roughly estimate the amount of heavy metals ejected, a simple description is adequate.

[FIGURE] Fig. 1. The normalized cooling function as given by Sutherland & Dopita. Different curves correspond to different gas metallicity, zero metallicity ([Fe/H] = - 3, solid curve), [Fe/H] = - 2 (dashed curve), [Fe/H] = - 1 (dotted curve) and [Fe/H] = 0 (dot-dashed curve). ([Fe/H] being the value of the logarithm of the metal content normalized to the solar value.)

All gas with a density contrast above 200 is taken to be inside "collapsed structures", and gas inside such collapsed structures is assumed to be transformed completely into stars. This is only made for the purpose of estimating the average gas metallicity in the proto-galaxy; dynamically, baryonic mass remains gaseous everywhere. Instantaneous recycling is also assumed (heavy stars eject their metals immediately after being formed) and complete mixing of the metals throughout the proto-galaxy (making the gas metallicity homogeneous) which simplifies the model further. The gas metallicity is then given by (Tinsley 1980)


where Z, y, M, [FORMULA] and t are, respectively, gas metallicity, yield, total baryonic mass, total mass in stars and time. That is, the average gas metallicity in the proto-galaxy, is only dependent of the average gas fraction in the proto-galaxy, (the proto-galaxy being the entire simulation volume in this context). Here a yield [FORMULA] is adopted, which was found to give a gas metallicity in accordance with observed values.

Assuming a time evolution and spectral shape for the background field, a four dimensional table in redshift, metallicity, density and temperature was calculated for both the cooling and heating. We used the publicly available program CLOUDY (Ferland 1993) to calculate cooling and heating tables. The tables were stored in a file and subsequently read in at the beginning of each simulation. Linear interpolation was then used to calculate heating and cooling rates during the simulation.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998