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

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5. Models with simultaneous evaporation and condensation

The models that treat the clouds as being identical are certainly a simplification. For a more realistic approach, one would consider a cloud population with a distribution in e.g. mass, as is done in the models of Samland et al. (1997). With such a description of the cloud phase, the Cowie et al. criterion implies that all clouds smaller than a certain radius will evaporate, while those larger will grow by condensation. Thus, one has at the same time both processes taking place, and for the overall rates one integrates over the cloud spectrum. How this is done and which rates one gets, depends on any further assumptions on the evolution of the cloud population. Samland et al. obtain for the solar neighbourhood, coefficients [FORMULA] Myr-1 and [FORMULA] Myr-1 which remain quite constant throughout the evolution.

Such a system settles quickly, within a few condensation time-scales [FORMULA] into the balance between condensation and evaporation


which is time-independent. The overall gas consumption takes place much more slowly, with [FORMULA]. For a linear SFR, this can easily be shown by solving the equations analytically. Since one normally has [FORMULA], most of the gaseous matter is in the cloud component, which is slowly converted into stellar remnants.

In Fig. 8 we show the ratio [FORMULA] as a function of cloud density for models with [FORMULA], starting with an initial density [FORMULA]. The self-regulated star-formation is quickly established as soon as the star-formation time-scale has become larger than the mean life-time of the massive stars ([FORMULA] or [FORMULA]). As long as the condensation occurs faster than the stellar gas return ([FORMULA] Myr-1), the system settles into the condensation/evaporation equilibrium. In models with very low condensation coefficients, the ejecta from the stars formed in the initial period of rather high star-formation are first accumulated in the hot gas, whose mass may remain constant while the clouds continue to be consumed - [FORMULA]. After a few condensation time-scales the gas condenses following the steady-state condition (Eq. 25). The peak in the curve for the rather extreme model with [FORMULA] at [FORMULA] marks where the gas and cloud phases are in equilibrium. In this model, about 10 percent of the initial mass is present in the gas component for about a few Gyr.

[FIGURE] Fig. 8. The mass ratio of gas and clouds, as a function of cloud density, for different values of the condensation rate coefficient K (in Myr-1), and with the same ratio [FORMULA].

In Fig. 9 we show the ratio of the metallicities in gas and cloud components, to study the efficiency of the mixing between the two components. Comparing with the evolution of the density ratio, one notices that the metallicities become equal as soon as the model reaches the steady-state solution (Eq. 25).

[FIGURE] Fig. 9. Like Fig. 8, but for the ratio of the oxygen metallicities. The dashed line is a model with [FORMULA] and [FORMULA].

The oxygen abundance in gas and clouds as a function of the gas fraction is depicted in Fig. 10, compared to the expectation from the Simple Model. If condensation and evaporation occur faster than star-formation, gas and clouds remain well-mixed, and the model follows the Simple Model very closely. The metallicity of the clouds is essentially determined by the amount of mixing due to condensation, it does not depend on the value of E. On the other hand, the gas metallicity is determined by how much the stellar ejecta are mixed with the low metal material evaporated from the clouds. For low values of E the mixing is poor, and the metallicity is as large as the metal abundance of the fresh stellar ejecta which is as large as [FORMULA] (Eq. 19). Eventually, all models evolve with the true yield.

[FIGURE] Fig. 10. The oxygen metallicity as a function of the gas mass fraction, for the intercloud gas (and for different values of the evaporation rate coefficient E) and the clouds (independent of E). The condensation coefficient is [FORMULA], the initial cloud density is [FORMULA] [FORMULA] /pc3. The behaviour of the Simple Model with a unity yield is shown as a dashed line, the short dashed lines for half and double yield.

In systems with a large initial density [FORMULA]   [FORMULA] /pc3 (shown in Fig. 11) the initial SFR [FORMULA] is greater than the condensation rate [FORMULA]. As a consequence, metals accumulate in the gas, and because of the small but existing reprocessing of enriched cloud matter the metallicity even rises. The condensation of the metal-rich gas occurs only at a more evolved stage, and then the large gas metallicity causes the clouds to have metallicities larger than expected from the Simple Model.

[FIGURE] Fig. 11. Same as Fig. 10, but with a higher initial cloud density [FORMULA] [FORMULA] /pc3.

5.1. Distribution of metal mass among the components

In a galaxy, the total mass of a primary element such as oxygen is proportional to the total number of type II supernovae that have exploded until the present time, independent of when and where this happened. This makes the total metal mass an essential indicator for the global state of a galaxy, though it is not easy to obtain, as it requires the determination of not only the mean abundances of gas in the various forms and stars, but their mass fractions as well. This would involve different wavelength regions and abundance analysis techniques with the problems associated combining the data.

Is it necessary to measure all gaseous components, or would the observation of a single phase, e.g. the cloud components being observable with H II regions, suffice? What is the magnitude of the errors involved? In the following, we use our closed-box chemodynamical models to provide some rough answers. Since the freshly produced metals are first injected into the warm or hot intercloud gas, observations of the dense cloud medium might miss an important fraction of the metals.

In Figs. 12 to 14 we show how the distribution of oxygen mass over gas, clouds, and remnants depends on the gas mass fraction [FORMULA], which indicates the chemical age of a system. The metals in chemically very young systems are still mainly in the hot gas. At more advanced stages, the metal rich gas has condensed into the clouds, which then contain most metal mass. In highly evolved systems, the metals are found in the remnants. The shape of the curves in intermediately evolved systems, i.e. around the peak of the contribution from the clouds, is mainly determined by the ratio of the inverse time-scales for condensation and initial star formation [FORMULA]. Because in the closed-box model most of the star formation occurs during the initial phase, this time-scale determines how much metals are produced and which are released from the intercloud medium by condensation K. For [FORMULA] [FORMULA] pc-3 Myr-1 (Fig. 12) the metals are hardly to be found in the gas, because of the rapid condensation into clouds. For values smaller than about 0.1 (Fig. 14) the metals remain in the gas until rather late, and because of the fast conversion of clouds into stars, they are swiftly locked up into remnants. As the clouds may contain less than a third of all the metals, measurements of the metallicity of the cloud gas only could be rather misleading for the determination of the total metal mass in a galaxy. Such a situation arises for low condensation rates or for high initial gas densities, i.e. a centrally rather concentrated proto-galactic cloud.

[FIGURE] Fig. 12. The fraction of the oxygen mass in the gas, clouds, and remnants, as a function of the gas mass fraction [FORMULA]. Model parameters are [FORMULA] [FORMULA] /pc3, [FORMULA]  Myr-1, and [FORMULA].

[FIGURE] Fig. 13. Same as Fig. 12 but with [FORMULA]  Myr-1.

[FIGURE] Fig. 14. Same as Fig. 13 but with an initial density [FORMULA] [FORMULA] /pc3.

The shapes of the curves are much less strongly influenced by the ratio [FORMULA] which we kept constant in the figures: It determines the ratio of metal mass contained in gas and clouds in the late evolution; thus because one has [FORMULA], the clouds keep a larger portion of the metal mass than the gas.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998