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

3. Models with switching between condensation and evaporation

3.1. Experiences from full numerical solutions

This set of equations (Eqs. 1to 22), which includes the switching according to the criterion of Cowie et al. (1981), had been incorporated into the chemodynamical models to compute the evolution of a closed-box system (Hensler & Burkert 1991, Theis et al. 1992), and coupled with 1-dimensional hydrodynamics to study the evolution of spheroidal systems (Theis et al. 1992). It was found that star-formation occurs in two modes: in a continuous way as the consequence of self-regulation (cf. Köppen et al. 1995), but in a certain range of densities it would fluctuate very strongly. These fluctuations actually are regular, large amplitude, non-linear oscillations of the cloud temperature and hence of the star-formation rate (cf. Fig. 7 in Theis et al. 1992). The period of the order of 100 Myrs is longer than the cooling time scale of the cloud gas. This indicates that the strongly damped oscillations which may occur before star-formation reaches the self-regulated mode (Köppen et al. 1995) cannot be the origin. Further inspection reveals that the oscillations show up most strongly in the cloud temperature and the gas density, rather weakly in the gas temperature, and were almost absent in the cloud density.

In Fig. 1 we show the track in the ()-plane of a model entering the oscillatory mode and performing a few cycles. Starting from a point marked 'a', it quickly and rather directly enters a limit cycle. At point 'b' the system changes over from evaporation to condensation. While just a little amount of gas is condensed, the clouds heat up rapidly to a maximum, then cool along the dot-dashed curve, while much more gas condenses. This continues to point 'c' where the system changes over to evaporation. Cooling in the clouds follows, with initially little gas return which increases thereafter. Passing through a minimum temperature, the clouds reach a temperature marked by the horizontal dashed line, along which the system reaches again point 'b' and switches back to condensation. As will be shown below, the horizontal dashed line is identified as the self-regulated star-formation mode; the passage through the temperature minimum is merely the transient while the system settles into equilibrium (Köppen et al. 1995).

Fig. 1. Evolution of the full system from the initial state (marked 'a') until the completion of the first few oscillations (solid line). The dashed lines descending to the right are the loci where the system switches from evaporation and condensation (lower curve, at point 'b') and vice versa (at point 'c'). The horizontal dashed line is the locus of the evaporation funnel and the dot-dashed curve depicts the condensation funnel (see text).

The two dashed curves sloping down to the right delineate the border where the system changes between evaporation and condensation. One notices that points 'b' and 'c' do not lie on the same curve. This is because the gas temperature at point 'b' is about 10 percent lower than at point 'c'. Though this amplitude is rather modest, the strong dependence of the switch criterion on leads to a strong shift of the switch curve in the diagram. But these oscillations in are not essential for the behaviour of the model.

3.2. Origin of the oscillations

The fact that the numerical simulations show the appearance of the oscillations primarily in cloud temperature and gas density indicates that these two variables form an 'inner' system whose behaviour is governed by the values of the control parameters . A most convenient way to analyze this is plotting the streamlines of the equations in the ()-plane, with the other variables c, s, and either changing with time or held constant. We find that for the explanation of the fundamental structures, the exact dependences of the rate coefficients on density or temperature as given by McKee & Begelman (1990) or Cowie et al. (1981) are of secondary importance. It is completely sufficient to assume that the rate coefficients for condensation and evaporation are constant (). In the following, we regard this simplified description; for the criterion of switching between condensation and evaporation, the prescription to compute is kept as given above, but we shall consider the threshold value of 0.03 as another free constant .

The resulting streamlines are shown in Fig. 2, which corresponds to Fig. 1 except for the slightly different position of the limit cycle. One distinguishes the two different regions of evaporation and condensation. In the evaporation regime, all streamlines are always oriented towards increasing gas density, and bunch together to form a funnel centered at constant  K (corresponding to  K in Fig. 1). Along this feature, the clouds evaporate at constant temperature, until the gas density exceeds a critical value - corresponding to point 'b' in the full model - and condensation commences. Setting in Eqs. 1, 2, and 18and using , one gets the equation for the streamlines:

At all streamlines are horizontal, which is merely a feature of the logarithmic representation and gives the impression of the fan structure on the left as the lines bunch together into the funnel. The important horizontal streamline along the funnel is described by the balance between stellar heating and radiative cooling , i.e. the evolution proceeds with the clouds being essentially in thermal equilibrium. As these are the only features, it is evident that regardless of its initial state, the system will always end up in the funnel, along which it evolves at constant cloud temperature towards higher gas densities.

Fig. 2. The overall behaviour of the simplified system: streamlines for the combined equations. The dot-dashed curve is the locus where the clouds change from condensation to evaporation, the short dashed lines mark the funnels. All streamlines below the horizontal dashed line point upwards.

In the condensation regime, all streamlines are oriented towards lower gas densities. Here, one finds a funnel, marked by the short-dashed curve sloping down from upper right to lower left. All streamlines, whether coming from lower or higher temperatures, merge into this funnel which meets the switch line at and  K, and the system is changed back to evaporation corresponding to point 'c' in Fig. 1. Setting yields the equation for the streamlines

Fig. 3 depicts the whole condensation regime in the ()-plane. To show the behaviour at large densities, a lower value of  K is assumed. Apart from a region of horizontal streamlines at , as for the evaporation region, one finds these structures:

• at low gas densities, a horizontal streamline exists where , the same condition as for the evaporation funnel. Its position agrees closely with the upper short dashed curve in Fig. 3, which is the locus where one has thermal balance in the clouds .
• at large gas densities, a horizontal streamline appears where . The dot-dashed curve (labeled 'A') shows where the cloud temperature does not change with time. The clouds are not in thermal equilibrium, because the condensation occurs faster than the cooling time, and the clouds are kept at the condensates' temperature.
• both two parts are connected with a curve, along which the evolution proceeds towards thermal equilibrium in the clouds. Where the transition occurs, is essentially determined by the ratio of and .
• there is an additional feature, showing up as a clump of streamlines: the intersection of the condensation funnel with the condition at and  K (in Fig. 3) forms an attracting node. The evolution of the clouds proceeds in thermal equilibrium and in balance between stellar gas return and condensation.

Fig. 3. Similar to Fig. 2, but showing the case for pure condensation, and assuming a temperature of the condensate  K. On the dashed line thermal equilibrium in the clouds exists, the dot-dashed line shows where the cloud temperature is constant.

Independent of the initial conditions, the system always reaches the condensation funnel, where the clouds cool and the gas density decreases. However, under certain conditions, the system may enter an equilibrium evolution at constant gas density.

The overall evolution can thus be pieced together: The two regimes of evaporation and condensation are separated by the curve where the criterion of Cowie et al. (1981) is met exactly (), depicted in Fig. 2. One may work out how the position of this switching line depends on the parameters, using the prescriptions (Eqs. 6to 9). For a cloud filling factor much smaller than unity, which is the common case, one obtains:

The curve's location is independent of the cloud density c, and it is shifted towards lower gas densities, if one raises the threshold or lowers the gas temperature , which is the most sensitive parameter.

The existence of the funnels makes it relatively easy to predict the conditions under which oscillations occur. From the pattern of the streamlines it is clear that a condition for a closed track in the plane is the intersection of evaporation funnel and switching line should happen at a lower cloud temperature (and higher gas density) than that of condensation funnel and switch curve. The conditions for the funnels and the switch line constitute a set of simultaneous non-linear equations which can be solved numerically to yield the curve in the -plane, and the positions of the intersections. The stellar density can be estimated from the low density limit (cf. Köppen et al. 1995). The geometry of the funnels thus allows a classification of the behaviour, as a function of the 'outer' parameters c and , as shown in Fig. 4. Furthermore, the difference and ratio of the densities g can be used together with the two forms of Eq. 1to estimate the times spent in each phase, the oscillation period and the keying ratio. Because these considerations neglect the changing gas temperature, they predict the types of behaviour of the complete network only in a qualitative way.

Fig. 4. The behaviour of the simplified system as a function of the control parameters c and , as estimated with the assumption of constant stellar density: Circles denote where oscillations occur (filled circles: the gas temperature is within a factor of 2 of the value for thermal equilibrium of the gas). In the white area, oscillations also occur, but with a period of over 5 Gyrs. Plus-signs indicate where the streamlines in both regimes are too flat, and the system comes to rest at an intermediate point on the switching line. Small dots show when the funnels meet very closely, and the asterisks indicate that the system enters the equilibrium.

All these investigations - together with numerical solutions of the complete system - show that oscillations are restricted to a finite range in cloud density, which may be quite narrow (less than a decade) and which is rather sensitive to some of the parameters and constants in the model.

During a cycle, the following sequence of physical processes takes place: When condensation starts, the clouds are rapidly heated up by the gas condensates, until the condensation funnel is reached. There, the clouds cool radiatively and evolve towards thermal equilibrium. When the switching criterion is met again, evaporation commences. Lacking the heating by the condensates, the clouds cool until the new thermal equilibrium is reached. Then the on-going evaporation of the clouds increases the gas density, until the condensation criterion is again fulfilled.

The cycle is thus essentially driven by the deposition of thermal energy of the condensates into the clouds, which radiate the energy away in the form of line emission. Thus the source which makes the cycle possible is the deposition of sufficiently hot material onto the clouds. The temperature of the condensates is an extremely sensitive parameter: oscillations can only be found if the condensing gas remains hot . Assumption of a lower temperature greatly reduces the region for the occurrence of oscillations, or can completely suppress them. This is quite easily understood in terms of the condensation funnel (Fig. 3): the position of its high-density branch is determined by the condensate temperature. Any cooling reduces the range in cloud temperatures that the funnel covers and thus the possibility for having a cyclic path between the two regimes is reduced. Thus, the prescription of the transfer of energy during the condensation phase is a most critical part of the network.

© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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