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Astron. Astrophys. 331, 524-534 (1998)

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4. The physically consistent formulation

McKee & Cowie (1977) and McKee & Begelman (1990) give expressions for the mass transfer rate, but not for the rate of energy exchange between gas and clouds. Therefore one considered in the previous chemodynamical models (e.g. Theis et al. 1992) limiting cases, such as that the condensing gas does not cool or cools down to some fixed temperature. Forming only one of the many processes of the ISM model network, this rather simple recipe was preferred. Of course, it is not a physically entirely satisfactory description, even more so since the oscillations were found to be rather sensitive to the exact recipe.

A physically consistent prescription can be formulated from noting that McKee & Cowie (1977) as well as McKee & Begelman (1990) derive their evaporation and condensation rates on the basis of a stationary solution for the gas flow between cloud and intercloud matter, with a fixed temperature profile. This means that any parcel of condensing gas follows the local temperature from the intercloud gas to the cloud interior, and that it arrives in the cloud with the cloud's temperature: [FORMULA]. As discussed above, cooling of the condensating material results in a lowering of the high-density part of the condensation funnel (cf. Figs. 2 and 3). If now the condensates cool down to the cloud temperature, the condensation funnel is merely a horizontal line, at the same height of the evaporation funnel. Because of the opposite directions of the streamlines in the two regimes, the intersection of the funnels - which are characterized by thermal balance in the clouds - with the switching curve is an attracting node, in which the system will always end up. Here, the system evolves with the cloud gas in thermal equilibrium, and without any oscillations.

The switching condition implies that [FORMULA] (Eq. 15) and so the evaporation and condensation terms in Eqs. 1, 2, 16, and 18vanish. Usually one finds that [FORMULA], so that one has [FORMULA], and the overall evolution is described by


which is nothing but the single-gas-phase system which has been shown to evolve almost exclusively in an equilibrium between star-formation and its inhibition due to the heating of the ambient gas by the massive stars (Köppen et al. 1995).

Note that in this more general view, the conversion of gaseous matter into remnants is independent of the rate coefficients for evaporation and condensation, and even on the actual prescription for the switching. But how the gas is divided among clouds and intercloud gas, depends on these specifications.

The switching of the system between condensation and evaporation gives rise to non-continuous terms in the differential equations. However, since in both regimes the system evolves towards the switching condition, this apparent mathematical difficulty makes the solution instead more simple: It forces the system to evolve along the switch line.

This permits to estimate the timescales of the equations: For conditions appropriate to a galactic disk ([FORMULA] [FORMULA] /pc3) one has [FORMULA]  K from the figures shown. The gas density is typically [FORMULA]. The cloud filling factor is usually quite small [FORMULA]. From Eq. 23, one gets for the temperature of the intercloud gas


which is quite insensitive to the actual gas density. The rate coefficients are obtained from Eqs. 6to 14


This makes these processes much slower than the cooling in the clouds [FORMULA]  Myr, but faster than star-formation [FORMULA]  Myr. For larger heating coefficients than we used here, the star-formation is even slower (few Gyr), and the two equilibria influence each other even less.

Thus, after some transient phase, the evolution proceeds in a hierarchy of equilibria: the fast cooling in the clouds will establish the self-regulated mode of star-formation; on a longer timescale the condensation and evaporation will balance the distribution of the gaseous matter among cloud and intercloud gas; and even more slowly, all gas is consumed to be turned into stellar remnants.

4.1. Balance of condensation and evaporation

How the gas is distributed among the gas and the cloud component, depends on the condensation and evaporation. As stated above, the system evolves along the switching line, which can be formulated as:


Thermal equilibrium in the clouds implies constant temperature, so the last term vanishes, and with Eq. 16one gets


The changes of the densities can be written in the low density limit of self-regulated star-formation [FORMULA]


where [FORMULA] is taken from Eq. 24, and using the abbreviations


One notes that the equations have a critical point which obeys the condition [FORMULA], or


Thus the ratio [FORMULA] due to the balance of condensation and evaporation increases only very slightly as the cloud gas is consumed. It is independent of the actual rate coefficients, and only weakly dependent on the switching threshold. These properties are found in the numerical solutions, shown in Fig. 5. After the transient evolution subsides - within 1 Gyr, which depends on the rate coefficients - the system enters the equilibrium, where the above derived dependences are obeyed. This shows that, apart from transient phases, the results of the chemo-dynamical models are quite insensitive to the details of the recipe of Cowie et al. (1981), for example the correction factor [FORMULA] for the influence of magnetic fields.

[FIGURE] Fig. 5. The mass ratio of gas and clouds, as a function of cloud density, for different values of the switching threshold [FORMULA].

The stability of the system against perturbations of the gas and cloud densities and temperatures are investigated by numerical experiments, adding or removing a substantial (50 percent) portion of the gas present in the system or changing strongly the temperatures. After a transitory phase lasting some 100 Myrs, a new equilibrium solution is reached which has the same character as before, i.e. without oscillations. In the transient there may be a few bumps and wiggles, but the system never breaks into oscillations. This is as expected, since the disturbance moves the system in e.g. the [FORMULA] -plane to another position, from where it always evolves towards one of the funnels to end up in the intersection with the switch line. Since the character of the streamlines cannot be altered into the pattern responsible for oscillations, the system is also very robust against inflows, outflows, and external heating.

4.2. Metallicity

In Fig. 6 we depict the relation of abundance in the cloud component of a primary element with the gas fraction [FORMULA]. In the Simple Model (closed-box with instantaneous recycling) the gas metallicity follows [FORMULA] with the true yield y, shown as the dashed line. For simplicity, we divide the abundances by the yield. The abundance rises more steeply at early times than the Simple Model, because we allow for a finite life-time of the massive stars which are responsible for the metal enrichment.

[FIGURE] Fig. 6. The abundance of a primary element (oxygen) in the clouds as a function of the gas fraction [FORMULA]. Shown are models with constant evaporation/condensation rate coefficients, assuming that the condensates retain the gas temperature (stepped curve) or cool down to the cloud temperature. The behaviour of the Simple Model is shown as a dashed line.

For comparison, we also show the model where the condensing gas does not cool: the metallicity remains constant during the evaporative phases, separated by a steep rise when the metal-rich gas is mixed to the clouds at the beginning of the condensation phase. If one allows the gas to cool down to cloud temperature before condensing, after a single evaporative intervall, the evolution proceeds in what seems to be the average evolution of the model with oscillations. It is worth emphasizing that for [FORMULA], the evolution of either model is quite close (within 0.1 dex) to that of the Simple Model, i.e. the effective yield is nearly equal to the true yield.

In Fig. 7 we show the behaviour of the abundance of a secondary element such as nitrogen, by the N/O vs. O/H plot. The Simple Model predicts a strictly linear relationship, viz. a straight line with slope 1 for the logarithmic values. Apart from a systematically lower yield for nitrogen (by 0.1 dex), the numerical models show a quite similar slope. The oscillations are less noticeable here than in the abundances themselves, and in the overall evolution, they make the slope of the N/O-O/H relation steeper. If one allows for cooling of the condensing gas, the slope is very close to unity.

[FIGURE] Fig. 7. The evolution of the abundances of a secondary (nitrogen) and a primary element (oxygen), for the same models shown in Fig. 6.

Thus, in both aspects, the chemical enrichment of the clouds from which the stars are born occurs in the very much same way as in the Simple Model. This should not be too surprising, because we do consider a closed-box model. Moreover, the time-scales for mixing of the freshly-produced metals into the cloud gas are determined by the condensation/evaporation times, which are still short compared to the overall evolution, viz. the gas consumption.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998