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

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2. The model for the ISM

2.1. The description used in CDE models

We shall consider a somewhat simplified version of the present CDE models which captures their characteristic behaviour. As in the full models, there are four components: the hot intercloud gas (named hereafter 'gas', with a mass density g), the gas in the form of clouds ('clouds' c), as well as massive stars (s), and low mass stars and remnants (r). Between the components the following interactions are taken into account: star-formation, gas return from dying stars, evaporation of clouds, condensation of gas onto clouds, (radiative or mechanical) heating of the gas by massive stars, radiative cooling of the gas. The full network also includes other processes, such as the formation of clouds by compression in supernova shells, dissipation by cloud-cloud collisions. These will not be included in our investigation, because comparison with the results of the complete network showed that they do not essentially determine the type of the system's behaviour. Then the time evolution of the mass densities of the components is described by the following equations:

[EQUATION]

Throughout the paper, we shall use the units parsec, [FORMULA] years, and solar masses.

Star-formation is described by the stellar birth function used in the form of Köppen et al. (1995)

[EQUATION]

Normally we use a quadratic dependence on density ([FORMULA] and [FORMULA]). The exponential factor involving the temperature [FORMULA] of the cloud gas describes what fraction of a cloud is in the form of star forming molecular clumps.

The mass returned to the interstellar gas by dying massive stars (with a mean life-time [FORMULA] Myr) is taken to be the fraction [FORMULA] of the stellar mass. Of all stars born, the fraction [FORMULA] is in the form of massive stars.

The remaining terms pertain to evaporation of clouds, whose rate coefficient E can be a function of densities and temperatures, and condensation of gas onto clouds (coefficient K).

In the formulations of Hensler & Burkert (1991) and Theis et al. (1992) the cloudy medium is composed of clouds which have identical properties (radius [FORMULA], mass [FORMULA], density [FORMULA]) and which are embedded in the (hot) intercloud gas. One assumes pressure equilibrium ([FORMULA]), which gives for the volume filling factor of the cloudy medium:

[EQUATION]

with the mean molecular masses [FORMULA], [FORMULA] in the two gas phases. For simplicity, we set [FORMULA] equal to one proton mass, in all what follows. The true mean density [FORMULA] within a single cloud is given by

[EQUATION]

Assuming that the clouds are just unstable to the Jeans criterion, the mass is determined by

[EQUATION]

with Boltzmann's constant [FORMULA] and gravitational constant G. Hence the radius of the cloud is

[EQUATION]

Depending on its properties, such a cloud evaporates into the surrounding medium due to thermal conduction from the ambient hot gas or the gas condenses onto it. Cowie et al. (1981) give a criterion for this behaviour: If the quantity defined as

[EQUATION]

is smaller than 0.03, condensation occurs, otherwise the clouds evaporate. Note that Cowie et al. use a slightly different notation: [FORMULA]. Since the clouds have identical properties, the whole system switches between condensation and evaporation, depending on the criterion. In a further development, Samland et al. (1996) consider a cloud population with a spectrum of masses, and therefore one has at any time clouds that evaporate as well as those condensing. We discuss the implications in Sect. 5.

The mass loss rate per cloud due to evaporation is from Cowie et al.:

[EQUATION]

For [FORMULA] the mass loss rate decreases with increasing [FORMULA]. In what follows, we do not consider this additional detail, as it alters the behaviour of the models only slightly. This applies also to the modifications introduced by McKee & Begelman (1990). From the number density of the clouds

[EQUATION]

one gets the rates for evaporation and condensation

[EQUATION]

We note that with [FORMULA], on the border between evaporation and condensation behaviour, the rates are equal:

[EQUATION]

The changes in the internal energy densities are

[EQUATION]

with the abbreviation [FORMULA] in our units. The first term is the heating of the gas by massive stars. Because in our treatment of the stars the rate from continuous heating during the life-time of the stars [FORMULA] has the same dependence on s as that from an explosive stellar death e.g. [FORMULA], processes of both types are included in the heating term: photoionization (with all stellar ionizing photons being absorbed by the gas), deposition of mechanical energy by stellar winds and supernova explosions. All three processes give rate coefficients of the same order [FORMULA]. To permit a direct comparison with the model of Köppen et al. (1995) we keep [FORMULA] (for photoionization with an efficiency of [FORMULA] for the conversion into thermal energy). A higher value gives a smaller self-regulated SFR and thus a longer star-formation time-scale. The second term is radiative cooling with a general cooling function [FORMULA] (cf. Theis et al. 1992). In our units, and by noting the use of mass densities instead of number densities:

[EQUATION]

The third and fourth terms are the energy gained by the gas through the addition of the evaporated cloud material - whose temperature shall be [FORMULA] - and the energy lost by the gas which is condensed onto the clouds. The (usually negligibly small) fifth term is that part of the gain if the stellar ejecta had gas temperature. We include this merely for convenience of analytical considerations. Likewise, the energy density of the cloudy medium is changed

[EQUATION]

by gains from heating by massive stars and losses by radiative cooling, by the losses due to matter locked up into stars (usually a minor term), and by the losses due to evaporating material or by gains from incorporating the condensing gas. The latter temperature is designated as [FORMULA]. For simplicity, we consider [FORMULA].

2.2. Metallicity

The metal masses in the gas [FORMULA], where [FORMULA] is the metallicity, and the other components evolve like:

[EQUATION]

The first term in the equation for [FORMULA] describes the supply to the ISM of metals from the massive stars. It is composed of two parts: the metals that where incorporated into the stars at their birth and are unchanged, and the metals freshly synthesized in the stars. For a whole stellar generation the latter is [FORMULA], with the fraction [FORMULA] of the mass locked up into remnants which is here [FORMULA]. Since the yield y (e.g. Köppen & Arimoto 1991) refers to the entire stellar mass spectrum, the massive stars alone contribute [FORMULA]. For a primary element such as oxygen, the yield is constant, except for the (negligible) factor [FORMULA] which takes into account that the primordial elements are used up with metal enrichment. For a secondary element (nitrogen) one has [FORMULA]. The abundances of every metal can be scaled to convenient reference values. In this paper, we divide the abundances by the yields, which are taken to be solar.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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