3. Photo-ionizing radiation flash as trigger of efficient cooling in free metal gas clouds
In the context of formation of galaxies and globular clusters, the non-equilibrium formation of has been identified as a key process for achieving a rapid cooling bellow (Palla & Stahler 1983; Izotov & Kolesnik 1984; Shapiro & Kang 1987; Palla & Zinnecker 1987; Kang et al. 1990; Anninos et al. 1996; Padoan et al. 1996; Tegmark et al. 1996). On the other hand, the ionization and dissociation of primordial gas by UV background radiation have been taken into account as an important parameter in many studies of the thermo-chemical evolution of pregalactic and intergalactic structures (Kang et al. 1990; Donahue & Shull 1991; Ferrara & Giallongo 1996; Haardt & Madau 1996; Navarro & Steinmetz 1996; Mucket & Kates 1997). In general, the UV background radiation acts as an inhibitor of formation. However, there are exceptions to this rule. Recently Haiman et al. (1996) have shown that UV background radiation can enhance the formation of in primordial gas at high densities () and low temperatures (); but for densities lower than the above value, the effect of a constant UV background radiation is to inhibit the formation. As it will be shown in the present section, rapid variations of the UV background radiation can also enhance the non-equilibrium formation of . In particular, we focus our attention on flash like variations capable of heating and increasing the ionization fraction of an initially warm-neutral gas cloud near thermo-chemical equilibrium. After the passing of the radiation pulse, the enhanced ion fraction makes the gas phase formation of molecules possible by the creation of the intermediaries and . The presence of small quantities of molecules then makes possible further radiative cooling to temperatures as low as . The rapid cooling ( free-fall times) abruptly reduces the Jeans mass by a factor , permitting the fragmentation of clouds initially marginally stable. Izotov (1989) has considered the homogeneous contraction approach of a gravitational unstable cloud, but here we are interested in delimiting the necessary conditions that provoke the rapid cooling of the cloud without invoking the gravitational collapse. UV radiation pulses have similar effects than shock waves because the post-shock flow also recombines out of equilibrium.
The spectrum, amplitude and duration of the radiation pulse are free parameters in our model. However, it should be noticed that, in order to induce efficient cooling in the cloud, the detailed form of the flash is not important as long as that, during the flash, the ionization of the cloud increases appreciably and after this, the ionizing flux decreases to background values in a short time compared to the recombination time.
The initial conditions of the cloud and the characteristics of the hypothetic ionizing pulse depend on the chosen scenery. For example in the Fall and Rees (1985) scenery, during the proto-galactic collapse, if the gas is assumed to be lumpy, the overdense regions will cool more rapidly than the underdense regions producing a two-phase medium. But the developing rate of this overdense regions depends on the initial conditions (i.e. the denser regions develop faster; Murray and Lin 1990) and on their interaction with the system. Thus, a dependence of the density contrast and of the mean cloud masses on the galactocentric distance are expected. The assumption of an ionizing flash produced in the galactic nucleus implicitly assumes a radial increment of the delay in the evolution of the clouds relative to the center. The dilution and attenuation of the ionizing and dissociating radiation produced in the flash also introduce a radial dependence. In fact, the possibility that proto-galactic structures are exposed to UV radiation emitted by massive young stars or an active galactic nucleus has been considered previously (Kang et al. 1990). Therefore, it would be useful to know the dependence of the cloud evolution on the flash characteristics, and on the cloud initial state (i.e cloud mass (), temperature (T), number density (n), and the relative number density of the species ().
To follow the thermal and chemical evolution of a metal-free cloud
in presence of a variable radiation flux, we consider an idealized
uniform cloud. The gas model adopted for this application considers
the following 9 species: H, , e,
, , ,
, , and
. Assuming a mass fraction y
() of Helium relative to Hydrogen, then
, where is the ratio of
the number density of species i to the total number density of
Hydrogen nuclei n. The processes of formation and destruction
of these nine species are assumed to be the 24 reactions in Table 1 of
Rosenzweig et al. 1994. The adopted cooling rates, include: 1) the
collisional ionization of H by electron impact; 2) the
free-free transitions of (Izotov (1989) and
references therein); 3) the collisional ionization by electron impact
of and ; 4) the
free-free transitions of and
; 5) the total dielectronic cooling rate of
(Shapiro & Kang 1987); 6) the
cooling rates due to rotational and vibrational
transitions excited by () and
() collisions, calculated according to Lepp and
Shull (1983). These cooling rates must be
multiplied by a factor and
to take into account the escape probability of
the vibrational and rotational photons, respectively. Finally, 7) the
excitation of the vibrational levels
() by low energy electron-impact was taken into
account (Klonover & Kaldor, 1979).
The external radiation flux affects the photoionization of the Hydrogen and Helium, and the photo-dissociation (and its associated heating rates; see Rosenzweig et al. 1994). The external photon flux is assumed to have the quasar-like distribution (Shapiro and Kang 1987)
where is the H Lyman-edge frequency. The function is introduced to represent the changes of the flux level outside the cloud, and is assumed to behave as in the Lyman continuum, and as for the photo-dissociating photons. Since a homogeneous cloud is assumed in this application, the optical depth at its center, for the ionizing radiation, can be written as:
where M is the mass of the cloud in units of and are the absorption cross sections of species i at frequency . As a first approximation, the attenuation of the external radiation field by the factor is adopted to schematize the opacity effects. For the dissociating radiation field a similar approximation is adopted (Rosenzweig et al. 1994). Obviously, by using this first approximation, the effect of the ionizing radiation is minimized, and the study of the internal structure of the cloud is not possible.
For the time dependence of the flux level we assume a flash like variation. More precisely, during a first period of integration the external radiation flux remains at the assumed background value (i.e. ). The duration of this initial period of integration is large enough (i.e. yr) to ensure that the cloud is near its thermo-chemical equilibrium state. After this first period of stabilization, it is assumed that both, and , increase linearly to the value in a time , and remains at this value during a time . Then, the flux level is assumed to decrease exponentially with a characteristic time to the background value for the ionizing flux, and to for the dissociating flux. This kind of flux variations are expected in the case when the radiation pulse is assumed to be produced by an intense but short event of stellar formation or by the radiation coming from a front shock. In a few million years after the end of the stellar formation process, the ionizing flux is expected to decrease to background values. On the other hand, due to the contribution of intermediate and low mass stars to the dissociating flux, it is expected that its level remain higher than the background value () long after the death of massive stars.
For a given initial state of the cloud, there is a critical background flux level , bellow which the cloud spontaneously cools because the self-shielding allows the formation of at relatively high concentrations. This critical background flux level is mainly sensible to the initial ion fraction due to its strong influence on the formation rate. In order to describe this initial state dependence, it is useful to look at the thermo-chemical equilibrium (TCE) curve (i.e. , and ). Fig. 1a shows the TCE curves for two different values of the background flux level (lower curve , and upper curve ) when the mass of the cloud is .
Note the three phase structure characterized by the presence of three stable branches (solid lines) denoted in Fig. 1a as W-I (i.e. warm-Ionized; ), W-N (i.e. warm-Neutral; ), and C-N (i.e. cool-Neutral; ). Note also the qualitative difference of the TCE curve for low () and high () background flux level. That is, for the left-hand maximum (denoted as ) is over the right-hand maximum (denoted as ), whereas, for . This difference is important because if the cloud is initially in TCE in the W-I branch, and the pressure is progressively increased from to a transition to the W-N phase occurs in the case , but in the case the C-N phase is reached. For , Fig. 1b shows the typical isobaric evolution tracks for initially ionized states (arrows A, B, C and D) and for initially neutral states (arrows E, F and G). For cases A, B, and D, as expected the cloud evolves toward the stable branch W-I, W-N, and C-N, respectively. However, in case C, even when the track reaches the C-N branch transversing the W-N branch due to the inertia of the ionization fraction. More precisely, the excess of the ionization fraction shifts the maximum pressure of the corresponding thermal equilibrium curve to a value bellow the pressure corresponding to case C. On the other hand, for the initially neutral states the evolution is substantially different. In case E the track stops in the W-N branch; note the difference with case C. In case F the cooling rate is so small that in practice the cloud reaches a quasi-stationary state similar to that in the W-N branch. The small cooling efficiency is due to the very small formation rate for the low ion fraction in the quasi-stationary W-N state. Finally, for high enough gas pressure as in case G, the cloud reaches the C-N branch. The isobaric evolution tracks for a high background flux level (i.e. ) are similar to those in Fig. 1b except that the W-N branch can not be reached from an ionized initial state as in case B in Fig. 1b. It is interesting to note that when the cloud is in the quasi-stationary state schematized by track F, a large enough ionizing flash is able to increase formation rate and to stimulate the rapid condensation to reach the C-N branch.
The analysis of the various evolution tracks in Fig. 1b have been made for a fixed background flux level. Taken into account that an increase of the background flux level shift the TCE curve upward, a similar analysis can be made for a fixed pressure but varying the background flux level. Fig. 2a shows, in the constant pressure approximation, the time dependence of the gas temperature for a cloud subject to the three labeled values of the background flux level . The initial condition is the same in the three cases and corresponds to an out of equilibrium warm-ionized state at a gas pressure , and , , , , , , , , , , and . The temperature evolution in the three plotted cases differs because for the system stabilizes in the W-I branch (i.e. as in track A of Fig. 1b), for the system stabilizes in the W-N branch (i.e. as in track B of Fig. 1b), and for the system stabilizes in the C-N branch (i.e. as in track C of Fig. 1b). On the other hand, Fig. 2b shows the temperature evolution for the same pressure as in Fig. 2a, but for an initially neutral warm state, (i.e. , , , , , , , , , , and ). In this case, the evolution for corresponds to the situation schematized by track F in Fig. 1b; that is, the system enters in a quasi-stationary state characterized by a very slow decrease of the temperature. However, for the system evolves rapidly to the C-N branch, a situation that corresponds to that schematized by track G in Fig. 1b. The results in Figs. 2a and 2b illustrate how the initial cloud state affects the critical value of below which the cloud evolves toward the C-N branch (i.e. the cloud condenses). In order to show the effect of the constant density approximation, Fig. 2b also shows the temperature evolution for when the constant pressure approximation is switched to the constant density approximation at the time when the rate of decrement of the cloud radius equals the sound speed.
For initially ionized states at pressure P, the critical value roughly corresponds to the value of that produces a maximum in the TCE curve at pressure P. For a high background flux level, the left-hand maximum (denoted as in Fig. 1a) is the higher maximum and must be equaled to P. For a low background flux level, the right-hand maximum must be equaled to P. On the other hand, for initially neutral states, the critical value must be calculated by finding the value of bellow which the evolution does not fall in the quasi-stationary warm-neutral state. The dependence of the critical value of on the initial cloud state is summarized in Fig. 3 for a cloud. The results are showed as functions of the initial cloud column density (). The left-hand side curve corresponds to the critical background flux level for initially ionized states. The upper segment (labeled I-branch) corresponds to background flux levels for which , and therefore, the condition is used. The lower segment (labeled N-branch) corresponds to the case when the condition is used because for these values of . The initial column density is calculated assuming that initially the cloud is at and at a pressure ; the initial concentrations are assumed to be the same as in Fig. 2a. For values of () above the left-hand curve, the initially ionized cloud reaches TCE in the stable warm phase, but below this curve the cool-neutral branch is reached. On the other hand, the right-hand side curve corresponds to the critical background flux level for initially neutral states. In this case, the initial column density is calculated assuming that initially the cloud is at and the initial concentrations are assumed to be the used in Fig. 2b; the initial density is varied in order to cover the plotted range of . For values of () above the right-hand curve, the initially neutral cloud attains a quasi-stationary warm-neutral state, but below this curve the cool-Neutral branch is reached. It is to be noticed that as the initial electron concentration increase, the right-hand curve approaches the left-hand curve. In any case, there exits a set of initial conditions for which the cloud reaches a quasi-stationary warm-neutral state. The point to be emphasized is that these quasi-stationary states are susceptible to be induced to condensate if the cloud is exposed to an intense enough ionizing flash.
In order to illustrate the effect of the ionizing flash on a cloud that has fallen in the quasi-stationary warm-neutral state, Figs. 4a-c show respectively the time dependence of the temperature, the relative number density of electrons (), and of hydrogen molecules (), for a cloud. The initial condition corresponds to an out of equilibrium warm-neutral state at a gas pressure (i.e. , , and the initial concentrations used in Fig. 2b). The curves labeled (a) in Figs. 4 are plotted for reference, and correspond to the case when the cloud is subjet to a constant background flux level of . As expected for , the cloud evolves toward a quasi-equilibrium warm neutral state. The curves labeled (b) and (c) in Figs. 4 correspond to the evolution of the cloud when it is subjected at yr to a flash. In case (b), the flash characteristics are: yr, yr, yr, , and . In case (c) the flash characteristics are the same as in case (b) but with . Notice in Figs. 4 that during the increase of the external radiation flux from to , the electron density and the temperature increase, whereas the and densities decrease. During the time when , the electron density and the temperature continue to increase because the variation of during the time is rapid enough to leave the gas far from equilibrium. At the end of the lapse of decrement of , if is short enough, the gas has an excess of electrons and thermal energy compared with the equilibrium values corresponding to and . After the lapse of UV flux decrement, recombination continues, but at a lower rate than the cooling. The excess of electrons at these relatively low temperatures results in an enhancement of the rate formation. Even when , the abundance of may reach a large enough value to produce considerable self-shielding. If the cloud reaches a critical value () for the optical depth at dissociating frequencies, the abundance grows very fast, allowing the cooling of the cloud to temperatures of the order of . This is the situation for case (b) in Figs. 4, where the flash induce rapid cooling even when . On the other hand, for case (c) a post flash dissociating level with is enough to inhibit the formation of , and then, the cloud remains warm.
A detailed study of the dependence of the cloud evolution on the phase space of free parameters (cloud mass, initial conditions, and flash characteristics) is out of the scope of this simple application. However, such detailed study may reveal that the stimulating condensation process studied here can be effective in a restricted region of the free parameters space, and therefore, may act as a selective effect that contributes to the formation of dense structures at certain scales.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998