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

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4. Discussion

The most prominent outcome of our simulations is the self-regulating character of the star formation. It is almost impossible to change the SFR unless one changes the gross physical properties of the parent galaxy. The explanation for the self-regulation is due to the star formation coupling back into the ISM, and can be viewed as thermal equilibrium of the ISM. If many stars form, the ISM heats up, the amount of cold gas declines, and the SFR must also decline. If only few stars form, the ISM cools down, the amount of cold gas increases and the SFR goes up. This effect has been discussed previously, e.g. in the chemo-dynamical models of Burkert et al. (1992).

Without additional constraints the simulations produce a power law dependence of the SFR on the gas density, flocculent spiral arms and truncation of the stellar disk. The physical reasons for these processes are the following.

4.1. Schmidt Law

Due to the self-regulation of the star formation the two major players in determining the SFR are the gas volume density and the heating rate. In a stellar disk that is much heavier than the gas disk the scale height of the gas (and hence its density) is completely determined by the stellar disk. Descriptions of the gas density for a gas layer embedded in an isothermal stellar disk can be found in e.g. Dopita & Ryder (1994) and Bottema (1996). If the z -density distribution is described by an exponential [FORMULA] the mid-plane velocity dispersion [FORMULA] of the gas is given by

[EQUATION]

where [FORMULA] is the total matter surface density (Dopita & Ryder 1994). In those regions where [FORMULA] and [FORMULA] (both conditions are true for our model galaxy within about 4 scale lengths), the gas density in the plane is approximated by

[EQUATION]

Observations of the velocity dispersion in external galaxies indicate that [FORMULA] is roughly constant among galaxies (Van der Kruit & Shostak 1984 ). Neutral hydrogen observations further show a variety of shapes of the H I surface density, but in general this is almost constant compared to the stellar surface density (see e.g. Cayatte et al. 1994 ). Together with the constant stellar scale height this implies [FORMULA].

The other component we have to know is the heating rate as function of radius. In the case of an optically thin ISM the heating rate is found by summing the contributions of all stars

[EQUATION]

where [FORMULA] is the FUV-flux/mass ratio. This integral cannot be solved analytically, but numerical integration shows that out to about 5 scale lengths the heating rate in the plane of the galaxy, resulting from a thin stellar disk, is well represented by

[EQUATION]

Accordingly, the heating rate declines as [FORMULA].

If there are no large gradients in the abundance of elements heavier than helium, the cooling properties of the gas are the same everywhere. Then in order to maintain thermal equilibrium, the radial dependence of the heating rate must equal the radial dependence of the gas density: [FORMULA], hence [FORMULA]. For the SFR this implies that

[EQUATION]

in perfect agreement with the results of Ryder & Dopita (1994) and Kennicutt (1989).

4.2. Spiral structure

Now return to Fig. 6, and consider once more the flocculent spirals that are visible in the cold gas and young stellar disk. The lack of spiral structure in the old stellar disk means that the spiral structure in the cold gas is not due to a density wave. The structure must be due to the dissipational nature of the gas, which allows for clustering of the gas in large assemblies. The shear of the disk then causes the (trailing) spiral shape. The same conclusion about the origin of flocculent spiral structure has been reached by Elmegreen & Thomasson (1993) using 2D simulations.

The young stars also show this spiral structure since they form out of these cold spiral filaments. This is, in effect, the stochastic star formation mechanism (Seiden & Schulman 1990), with the physical driver being the molecular cloud complexes. Apparently there is no need for supernovae to stimulate star formation, although it might help to make the spiral structure more pronounced. Note, by the way, that this finding by no means excludes the usual swing amplification or density wave mechanisms for 'grand design' spirals, as we hope to show in our subsequent work.

4.3. Truncation

An interesting effect seen in the simulations is the sharp truncation of the cold gas disk and consequently of the young stellar disk. Truncation of galaxies is seen in edge-on galaxies, where the cut-off radii occur at about 4 radial scale lengths (Van der Kruit & Searle 1982 , Bosma & Freeman 1993 ), while in our simulations this truncation occurs at approximately 6 scale lengths. The traditional explanation for truncation of a spiral galaxies invokes the angular momentum distribution of a collapsing protogalaxy, where the material with the highest specific angular momentum settles at a radius of about 4.5 scale lengths (Van der Kruit 1987 ).

Here we propose a thermal mechanism for the truncation. Although the interpretation in our simulations is complicated, it is interesting to see that the FUV radiation of the young stellar disk is capable of heating the entire gas disk. We can make a rough estimate of the conditions under which this effect could indeed prohibit star formation outside the stellar disk, using the radiation field and the gas density.

The radiation field outside the stellar disk declines with radius as [FORMULA] due to geometrical dilution, where absorption is very unlikely to be important. The gas scale height [FORMULA] outside the stellar disk is determined by the halo and increases linearly with radius. If the gas is always warm, the density [FORMULA] must decrease at least as fast as the radiation field. Since [FORMULA] this implies that the gas surface density [FORMULA] should decline at least as [FORMULA].

Note that we do not include any extragalactic radiation field in our simulations. Inclusion of such an external field requires less heat input from the stellar disk for heating the gas. Consequently the truncation will occur closer to the centre.

Abundance gradients may alter the situation greatly. If the gas outside the stellar disk is less enriched with heavy elements than the disk gas, it might be more difficult to heat the outer gas. On the other hand, this gas cools less efficiently. If the latter effect dominates, the requirement of a gas surface density which declines as [FORMULA] can be relaxed, making truncation due to heating more likely.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998

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