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

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3. Results

There are many parameters in our model that can be changed: IMF with its mass limits, ionization parameter, cloud mass [FORMULA], collapse time, star formation efficiency, and the number of particles. To obtain an estimate of the relative importance of these parameters, and of the sensitivity of our computations to these values, we tested them with a model for NGC6503 (see Bottema 1993 and Bottema & Gerritsen 1997 ). This is an Sc galaxy with a maximum rotation velocity of 120 km/s, star formation taking place all over the disk, and a global SFR of about [FORMULA] yr (Kennicutt 1983 ).

3.1. Model galaxy

We assume an exponential stellar disk with an isothermal z -distribution (e.g. Van der Kruit & Searle 1982 ):


with measured radial scale length [FORMULA] kpc and vertical scale height [FORMULA] kpc. The disk is truncated at 5 scale lengths (5.8 kpc).

The mass of the stellar disk is calculated from the measured stellar velocity dispersions (Bottema 1989 ) assuming that the z -dispersion, [FORMULA] is due to the surface density of the stellar disk ([FORMULA]):


This implies a stability parameter [FORMULA] throughout most of the disk, and yields [FORMULA].

Particles are distributed according to the density distribution (Eq. 11), and velocities are assigned according to the (gas) rotation curve (with a maximum of 120 km/s) corrected for asymmetric drift. Dispersions in [FORMULA] directions are drawn from gaussian distributions with dispersions of [FORMULA] respectively, using the relations


with [FORMULA] and [FORMULA] the orbital and epicyclic frequencies respectively.

The gas surface density distribution is modeled after the H I distribution,


with central surface density [FORMULA], and radial extension [FORMULA] kpc. In our calculations we truncate the gas distribution at 8 kpc, well outside the stellar disk, to save computing time. This yields a total gas mass [FORMULA]. Gas particle velocities are initiated according to the rotation curve. Dispersions are drawn from gaussian distributions, adopting an isotropic dispersion of 6 km/s.

A halo is included in the calculations as a rigid potential. This is justified since the galaxy evolves in isolation. As advantages we do not have to make assumptions about halo particle orbits, and we do not have to spend time in calculating the force of the halo particles on the galaxy. We assign an isothermal density distribution to the halo,


with central volume density [FORMULA] and core radius [FORMULA]  pc. This will correctly put the maximum rotation velocity at 120 km/s.

3.2. Parameter space

The influence of the model parameters on global parameters such as the star formation rate and cold gas fraction is tested by changing one parameter compared to a standard setting. The default setting is as follows:


We use a Salpeter initial mass function with [FORMULA] and [FORMULA].

Initially all SPH particles get a temperature of [FORMULA]  K, and all stars get an age of 300 Myr. At this age, the FUV flux of a stellar cluster has dropped below 1% of its maximum value. This assures an immediate cooling of the gas, but not too drastically. We let the simulations run for 1.5 Gyr. The results for the different runs are summarized in Table 2, where the first column gives the parameter that has been changed with respect to the default setting. The following columns give mean values for the interval [FORMULA] (thus starting after a period of 300 Myr, so that the simulations have reached approximate equilibrium). Column 2 gives the star formation rate in [FORMULA] for the whole galaxy, column 3 the total amount of cold gas, defined as having a temperature below 1000 K, and column 4 gives the amount of gas left at the end of the simulation.


Table 2. The results for various simulations of the model galaxy NGC6503. Only star formation parameters have been changed. The deviation from the standard setting is given in the first column.

3.2.1. Number of particles

Increasing the number of particles in steps of a factor 2 up to 40000/80000 SPH/star particles does not markedly influence the mean SFR and the total gas mass converted into stars. It has a small effect on the cold gas fraction due to the increased resolution: SPH particles can reach higher densities so that the cloud lifetimes become shorter, leading to less cold gas. However, since the free fall time goes with the square root of the density, this effect is not very large.

Given the consistency of SFR with number of particles we do not have to run very expensive calculations to test all parameters. Instead we feel safe in testing the recipe using 5000/10000 SPH/star particles, which takes about 60 CPU hours on a Sun SPARC20.

3.2.2. Cloud mass

[FORMULA] was chosen to be 10 times the mass of an individual SPH particle, with all SPH particles having the same mass initially. Eventually one would like to have a number for [FORMULA], typically the size of a small GMC, say [FORMULA]. But numerical problems can arise if [FORMULA] is chosen about equal to, or less than, the particle mass, since the smoothing over particles sets an upper limit to the maximum density than can be achieved.

As can be seen from Table 2, changing [FORMULA] up or down by a factor 4 does not have a significant effect on the SFR. This results from the two-phase structure of the ISM, which allows for stable warm and cold phases of the gas, but not for stable lukewarm gas. As long as the value of [FORMULA] implies an instability temperature well below that of the warm phase Eq.(7) essentially selects the same particles. This is demonstrated in Fig. 1, where the straight lines correspond to a Jeans mass equal to [FORMULA]. The two-phase structure of the gas is easily visible in this figure. For [FORMULA] (Fig. 1e), problems with limited resolution start to appear.

[FIGURE] Fig. 1. Diagnostic plots of temperature versus density in various simulations. Each dot indicates one SPH particle. The fully drawn line marks the Jeans mass corresponding to [FORMULA], which is stated at the top at each panel together with the collapse time. The dashed line gives the collapse time versus density (the right axis shows the scale). The cold SPH particles have densities corresponding to the amount of heating: lowest density particles have lowest heating, highest density particles have highest heating (see also Fig. 11a where SPH particles are colour-coded with the heating).

3.2.3. Collapse time

The most influential model parameter is the collapse time for the cold gas particles, although its influence is still smaller than that of the IMF. If the collapse time is taken very small, essentially all regions that cool will produce stars rapidly, thus inhibiting the growth of the cold phase. If the collapse time is very large, it may effectively inhibit star formation. The dashed lines in Fig. 1 show the collapse time as function of density (time scale is plotted on the right axis). For the default recipe the collapse time is 20-100 Myr for unstable particles, which corresponds roughly to the lifetimes of GMCs.

Fig. 1f shows a simulation with an extremely long collapse time: over 1 Gyr. This leads to a high fraction of cold gas, almost no star formation (the simulation lasted only 1.5 Gyr) and a severe resolution problem. Such a choice for the collapse time is not realistic.

3.2.4. Star formation efficiency

Remarkably, the star formation efficiency [FORMULA] does not influence the star formation rate, but it has some influence on the amount of cold gas. The formation of huge stellar clusters destroys the cold clouds efficiently, whereas smaller ones leave the clouds more nearly intact. In the standard simulation a new star particle has a mass of [FORMULA], which is a bit large.

3.2.5. Ionization fraction

The recipe is very sensitive to the ionization fraction x assumed for the gas. Decreasing x from 0.1 to 0.01 depresses the SFR from 0.27 to 0.06 [FORMULA] /yr. This is due to the cooling properties of the gas, which are very sensitive to the ionization fraction. The cooling function at [FORMULA]  K drops by a factor about 7 when x changes from 0.1 to 0.01. This means that the gas density must be higher by a factor 7 before it can start to cool, or, alternatively, less stellar heat input is required to keep the gas warm.

The ionization fraction is determined mainly by cosmic rays and soft X-rays, heating sources that we do not include in the calculations. Thus we have to adopt a value for x. A value of [FORMULA] seems reasonable for the solar neighbourhood (Cox 1990 ).

3.2.6. Initial mass function

The IMF directly controls the energy input from the stellar disk. Therefore our recipe should be very sensitive to it, and it is. In Fig. 2 we show the various IMFs used in our simulation, together with the FUV flux for each IMF (as given by Bruzual & Charlot 1993 ). If one integrates the FUV flux in time the total heat input from the stellar cluster is found. For mass cutoffs of [FORMULA] one finds [FORMULA] ergs/ [FORMULA] for Salpeter, Scalo and Miller-Scalo IMFs respectively, where this value is reached within about [FORMULA]  yr. These differences in heat input are reflected directly in the SFR, which is lowest for the Miller-Scalo IMF (with highest energy input) and highest for the Scalo IMF (with lowest energy input). As is the case with the ionization fraction, the SFR does not scale linearly with the heat input, but has a weaker dependence. A Salpeter IMF with [FORMULA] gives a higher SFR, since the high mass stars contribute much to the FUV heating.

[FIGURE] Fig. 2. The frame on the left shows the mass spectrum for three different IMFs: Salpeter, Miller-Scalo and Scalo (straight, dotted, and dashed lines respectively), with mass limits of 0.1 and 125 [FORMULA] as given by Bruzual & Charlot (1993). The corresponding FUV fluxes versus evolution time are shown in the frame on the right.

3.2.7. Absorption

One of the most difficult processes to include is the absorption of the FUV photons, since the gas column density between emitter and receiver of the photons has to be known: this is not easily solved in simulations of the type conducted here. Therefore we treat our galaxies as optically thin for FUV photons, thus setting an upper limit to the amount of dust heating of the ISM. In this respect the calculated SFR is a lower limit to the actual SFR.

To mimic the effects of absorption we run a simulation with an absorption coefficient of [FORMULA]  kpc. Thus the flux an SPH particle receives from a star particle declines not as [FORMULA] but rather as [FORMULA]. This results in a much higher SFR, since stars contribute only locally to the heating.

3.2.8. Summary

In summary we can say that the global star formation parameters resulting from our recipe are strongly dependent on physical input parameters, while only mildly dependent on model parameters. The cloud mass [FORMULA] has virtually no influence on the SFR or on the cold mass fraction. The collapse time has little influence on the SFR and cold mass fraction. The star formation efficiency has little influence on the cold mass fraction. The primary influence on the outcome of the simulations is the physical process of heating and cooling of the ISM. In our simplified model the heat input is controlled by the IMF (and the absorption), while the cooling is controlled by the ionization fraction.

3.3. Detailed analysis

In the previous section we discussed the global star formation rate and the cold gas fraction as a function of the input parameters. In this section we discuss one run in detail: the highest resolution simulation, which consists initially of 40,000 SPH particles and 80,000 star particles. The model is evolved using a time step of [FORMULA], using a tolerance parameter [FORMULA] and quadrupole moments to calculate the gravitational forces. The gravitational softening length for the particles is: [FORMULA]. The hydrodynamic properties are calculated using variable smoothing lengths, such that each SPH particle has 32 neighbours within 2 smoothing lengths.

During the simulation, the energy and angular momentum of the galaxy were conserved to better than 0.2%. The simulation was performed on a Cray J32, and took 260 CPU hours.

For this simulation [FORMULA], [FORMULA], so newly formed star particles have a mass of [FORMULA]. Therefore this simulation has about the highest resolution one can obtain with our star formation recipe: increasing the resolution must be accompanied with the input of new physics.

3.3.1. Evolution

First we describe the global evolution of the galaxy. Recall that we do not start from scratch with a gaseous protogalaxy, but our initial conditions resemble a present-day galaxy. We focus on the SFR, the evolution of the various gas components, and show the correlation between the total gas mass and SFR.

In Fig. 3 the SFR is plotted against time. The SFR immediately rises to [FORMULA] and then remains constant at [FORMULA]. After [FORMULA]  Myr the SFR declines slowly. At this time all influence from the old stellar population, now 600 Myr old, has disappeared, and the SFR is completely determined by new stars. The episodic changes in SFR are probably real and due to the discrete nature of star formation. Only if the major mode of star formation would be in single stars, the global SFR could smear out to a single value.

[FIGURE] Fig. 3. The star formation rate during the simulation. After settling of the system there is a slow decline of the SFR. The episodic changes in SRF are real and not due to limited resolution.

Fig. 4 shows the gas mass as function of time. Prominent in this figure is the approximately constant decline of the total gas mass, which reflects the rather constant SFR. The warm gas matches this decline perfectly, while the cold and lukewarm gas masses stay constant (here cold means below 1000 K). At the end of our simulation the gas mass contributes about 20% to the total galaxy mass, with 17% being cold. If we extrapolate this figure to a total gas mass of [FORMULA], which is 10% of the galaxy mass, we would find that roughly half of the gas is in the cold phase.

[FIGURE] Fig. 4. The global evolution of the gas. The dash-dot line shows the total gas, the fully drawn line the cold gas ([FORMULA]  K), the dotted line the lukewarm gas ([FORMULA]  K), and the dashed line shows the warm gas ([FORMULA]  K).

Fig. 5 shows the ratio of the SFR to the total gas mass versus time. This ratio is remarkably constant during the simulation, which suggests that the total gas mass controls the global SFR of an isolated galaxy. This explains the surprising observational result that the SFR correlates better with the total [FORMULA] masses than with the individual gas components (Ryder & Dopita 1994 , Kennicutt 1989 ).

[FIGURE] Fig. 5. The evolution of the ratio of the star formation rate to the total gas mass.

3.3.2. Spatial distribution

We now focus on the spatial distribution of the star formation and analyze the situation at [FORMULA]  Myr. The initial stellar component has an age of 1.2 Gyr and will no longer influence the star formation other than gravitational.

Fig. 6 shows the face-on and edge-on distributions of several components of the galaxy: (a) the old stellar component (the initial stellar disk), (b) the youngest stellar component (age under 150 Myr), (c) the warm gas, [FORMULA]  K, (d) the cold gas, [FORMULA]  K. There are two striking features in this plot. First there is the flocculent spiral structure in the cold gas and young stars and the absence of it in the warm gas and old stellar disk. Can we believe this spiral structure? In Fig. 7 the face-on cold gas is plotted again, now with four regions encircled. The characteristics of these regions are written in Table 3. From this table it is clear that the smallest group falls below our resolution of 32 particles, but the other groups are resolved and are candidates for real structures. Moreover, the same spirals already arise in the simulations presented in the previous section, with the same global appearance, so that they seem to be resolution independent. Still, our evolution movies show that these are transient structures, temporarily amplified by local processes, and are certainly not 'grand design' spirals. We expect to find those in our forthcoming work on interacting galaxies.

[FIGURE] Fig. 6. The particle distribution at [FORMULA]  Myr. The left column shows the face-on distribution, the right column the edge-on distribution. From top to bottom are shown: the old stars, the young stars (younger than 150 Myr), the warm gas and the cold gas. The size of each box is [FORMULA]  kpc. A random sample of at most 10000 particles is plotted.
[FIGURE] Fig. 7. The cold SPH particle distribution at [FORMULA]  Myr shown face-on. The characteristics of the four encircled regions are given in Table 3. The box has a size of [FORMULA]  kpc.


Table 3. Cloud mass and extreme temperatures for the four cold cloud complexes encircled in Fig. 7. These are listed clockwise, starting with the lower one.

The second striking feature is that the cold gas and the young stars are confined to the plane of the stellar disk. It means that the gas can only cool sufficiently in the plane. In regions outside, the density is never high enough to allow cooling, or, conversely, the FUV radiation is strong enough to prevent cooling. This is in accordance with the observation that the scale height of young stars in the Milky Way is much smaller than for the old stars.

The radial distribution of the three gas components is shown in Fig. 8, where (a) shows the gas surface density against radius and (b) shows the relative gas distribution with radius. As can be seen from these figures most of the cold gas is in the centre of the galaxy (over 70% of the gas there is cold), while there is no cold gas outside 8 kpc.

[FIGURE] Fig. 8. The spatial distribution of the gas. The panel on the left shows the surface density of the various components; the panel on the right shows the fractions occupied by these components. The dash-dot line represents the total gas, the fully drawn line the cold gas, the dotted line the lukewarm gas, and the dashed line the warm gas.

The interpretation of the truncation of the cold disk, and consequently the truncation of the stellar disk, requires some care. Poor resolution effects at the edge of the galaxy may cause spurious results. Since the gas properties are always estimated by smoothing over 32 particles, SPH particles far out may have very large kernel lengths. Clumpy structures are improperly modeled in this regime, but may exist in reality and form stars. Another point of concern is that the observed H I disk for NGC6503 extends to 14.55 kpc, while we truncated it at 8 kpc. The outer gas thus lacks the pressure of the gas that should be surrounding it, so that it moves outward. The outermost gas particle at [FORMULA]  Myr is 13.6 kpc away from the centre so the effect is indeed observed in the simulations. This leads to lower gas densities, which might inhibit cooling and contraction.

The final subject we discuss is the radial distribution of the new stars. In Fig. 9 we plot the surface density distributions of the young stars (fully drawn line) and the old stars (dotted line) in arbitrary units, scaled to fit in one plot. The decline of the surface density is slower for the young stars, which corresponds with the observed trend for spiral galaxies that the scale lengths are longer at blue wavelengths than in the red (de Jong 1996 ).

[FIGURE] Fig. 9. The spatial distribution of the young stars compared with that of the old. The thick line gives the young stellar surface density against radius, the dotted line shows the old stellar surface density, the dot-dash line shows Kennicutt's (1989) Schmidt law ([FORMULA]), and the dashed line gives the Ryder & Dopita (1994) fit for the star formation ([FORMULA]). The (surface) densities are scaled to fit in one plot.

Recently Ryder & Dopita (1994) obtained photometric CCD imaging data for 34 spiral galaxies. From their I, V and H [FORMULA] images they find a correlation between the surface brightness in the I -band (which measures the surface density of the old, low-mass stars, and this component determines the mass of a galaxy) and the H [FORMULA] band: SFR  [FORMULA]. In a similar study Kennicutt (1989) found a correlation between the gas density and the H [FORMULA] emission. Written as a Schmidt law the fit for his sample is: SFR  [FORMULA].

The dashed line in Fig. 9 represents the Ryder & Dopita results for the stellar surface density, and the dot-dashed line shows the Schmidt law with exponent 1.3. The correlation of these two lines with the surface density of the young stars is striking. The physical reason of this correlation is be discussed in Sect. 4.1.

3.3.3. Two-phase ISM

The first detailed description of the galactic ISM was the two-phase model of Field et al. (1969). Their model accounts for what we call the warm and cold gas, but not for the hot gas. Since our simulations do not include supernovae we do not get the three-phase model of McKee & Ostriker (1972), but the two phases are reproduced faithfully in our simulations.

In the standard two-phase model there is a limited range of thermal pressures for which there are two solutions for the gas density. Very low density gas is always warm, while very high density gas is always cold. In between the gas may be either. Starting with a warm, low density gas, one may increase the density until a certain fixed point. If the density is increased above that critical value the gas cools rapidly and will keep that low temperature when the density is increased further (see e.g. Wolfire et al. 1995 ), as long as the heating rate is constant.

In our simulations the heating rate depends on position and time. This implies that the critical density is similarly dependent. This behaviour is shown in the two phase diagrams of Fig. 11. Fig. 11a displays the density versus temperature, where the colour coding denotes the heating rate. To one colour, meaning one heating rate, corresponds one critical density. If the density is lower than that, the gas is warm, while if the density is higher the gas is cool. The spread in heating rates thus explains the spread in density for the cold gas component.

Fig. 11b shows the heating rate plotted against the pressure, where the colour coding now denotes density. The gas particles that are plotted are all in the plane of the disk. The motion of the particles is illustrated in Fig. 10 and explained below.

[FIGURE] Fig. 10. The motions of SPH particles in the phase diagram of Fig. 11b. The dotted line shows the trajectory of an SPH particle in the inner part of the galaxy. For particles farther away from the centre, the curve moves to the lower left in the diagram.
[FIGURE] Fig. 11. Phase diagrams for the SPH particles at [FORMULA]  Myr. The upper diagram a shows temperature against density, while the colour denotes the heating, blue particles receive lowest heating, red particles receive most heating; the lower diagram b shows heating against pressure, where the colour denotes the density, red particles have lowest density, blue particles have highest density. The two-phase structure of the ISM is especially clear. The diagram only shows the particles in the plane of the galaxy. Those above above and below the plane occupy the region of [FORMULA], and these particles are all warm.

Consider a particle at, say, [FORMULA] and [FORMULA] having a temperature of [FORMULA]  K. If the density does not change, the particle only moves upward in the diagram if the heating increases, and downward otherwise. The latter happens if no stars form in the neighbourhood. The particle moves down (dotted line in Fig. 10) until it reaches the sharp transition at [FORMULA] (indicated by the straight line). There the particle reaches its critical density (or critical heating). If the heating continues to decrease the particle cools rapidly to about 100 K and moves to the left in the diagram to [FORMULA]. As long as no new stars form nearby the heating decreases steadily and the gas particle contracts with its neighbours to higher density and lower temperature, thus moving down and left. This process will stop if a new star is formed, which heats the gas, letting the gas particle jump to its original position (dashed line).

Interesting in this phase diagram is that we can immediately see why particles outside the galactic plane cannot cool. Particles outside the plane only occupy the region left of [FORMULA]. These mostly move up and down in the diagram. The crucial point is that such particles cannot reach the line giving the critical heating (there is a gap between the particles and the line), so they will never be able to cool and collapse: no star formation can occur.

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© European Southern Observatory (ESO) 1997

Online publication: April 28, 1998