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Astron. Astrophys. 347, 442-454 (1999)

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2. Basic equations for a three-component star forming disk

The physical processes in star forming galactic disks are too complicated for a detailed analysis from first principles. Among numerous dynamical and interchange processes one has to choose those which have a major influence on the evolution of the system. We perform our analysis using a simplified model which splits the disk into three components regulated by time-dependent mass transformations. We take into account a gas component containing all phases of the interstellar medium, and two stellar components, the massive and the low mass stars. The less massive stars are assumed to have no influence on the interstellar medium, and are simply accumulated as remnants, whereas massive stars are assumed to be responsible for gas heating.

The chain of mass transformations within the model includes spontaneous star formation and ejection of the stellar mass back into the gas phase. For the description of the star formation rate we use the approach developed by Köppen et al. (1995) which is a basic skeleton of the interaction scheme used in chemo-dynamical models (e.g. Theis et al. 1992, Samland et al. 1997). It describes the spontaneous stellar birth as a power law function of the gas density with an efficiency depending on the temperature of the gas. The latter just has to guarantee that the efficiency drops down with increasing temperature. In this scenario, the chain of mass transformation processes can be described by the following set of equations:

[EQUATION]

[EQUATION]

[EQUATION]

where the star formation rate [FORMULA] is given by a power law dependence on the gas density g and a function of the gas temperature T

[EQUATION]

s denotes the density of massive stars, whereas r corresponds to the low-mass stars and the stellar remnants. The parameters [FORMULA] and [FORMULA] determine the fraction of newly born massive stars, and the fraction of gas ejected by massive stars into the ISM, respectively. The parameter [FORMULA] corresponds to the mean stellar lifetime of a massive star.

In general, for the determination of the "efficiency" factor [FORMULA] which depends on the gas temperature, one has to integrate the energy balance equation. Köppen et al. (1995) noticed, however, that the model can be simplified, if the time-scales for heating and cooling of the gas are shorter than the characteristic dynamical time-scale of the system. In that case, the effective star formation rate depends only weakly on the exponent n and the efficiency function [FORMULA], and the star formation effectively follows the Schmidt law with the `true' star formation rate depending quadratically on the gas density. Additionally, due to the self-regulation inherent in this star formation prescription, the effective star formation rate is not strongly affected by substantial variations of f or [FORMULA]. Therefore, we can restrict our model to the simpler expression

[EQUATION]

We will use this expression in our analysis.

For the description of the spatial dynamics of the multi-component disk we will use a fluid dynamical approach. In this approach, gas, stars and remnants are considered as three fluids, coupled by nonlinear interchange processes and by the common gravity. While the application of fluid dynamics for the description of the gas is quite natural, it is not obvious that such an approach can be used for the collisionless components of the disk represented by stars and remnants. Kikuchi et al. (1997) analyzed this question and found that the stability properties of disks obtained in fluid approximation are in good qualitative, and to some extent in quantitative agreement with the stability properties of the collisionless models. We will use therefore a fluid approximation in the analysis of perturbations of a multi-component disk. The behavior of our model disk is described by the continuity equations, written for each component, a set of momentum equations, and the Poisson equation. In cylindrical coordinates they are:

[EQUATION]

[EQUATION]

[EQUATION]

where [FORMULA] are the corresponding substantial time derivatives written in cylindrical coordinates:

[EQUATION]

Using Eqs. (6)-(8) and definition (9) the momentum equations can be written as

[EQUATION]

[EQUATION]

[EQUATION]

Here [FORMULA] are the surface densities, and [FORMULA] and [FORMULA] are the radial and azimuthal components of the velocities [FORMULA] of gas, stars and remnants in the disk.

Without the exchange processes, i.e. when the right-hand sides of the Eqs. (10)-(12) are zero, the dynamics of these quantities is determined by the partial "pressures" of the components [FORMULA], the self-gravity of the disk [FORMULA] and the external gravity of the halo and bulge, [FORMULA] and [FORMULA]. Mass transformations between the components give an additional factor for the momentum balance and have to be taken explicitly into account in the numerical simulations.

The gravitational potential [FORMULA] is determined by the overall density of all components, and can be written in the form of a Poisson integral as

[EQUATION]

The equation of state closes the system of Eqs. (6)-(12). Throughout the simulations we use a polytropic equation of state applied to all three components:

[EQUATION]

Eqs. (6)-(14) are used for the analysis described in the subsequent sections.

In our simulations we will use a "galactic" system of units in which [FORMULA]kpc, [FORMULA] and the gravitational constant G is unity. With this choice we have a unit of time equal to [FORMULA] yr, and a unit of velocity equal to 147 k ms-1.

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© European Southern Observatory (ESO) 1999

Online publication: June 30, 1999
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