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Astron. Astrophys. 355, 333-346 (2000)

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5. Dense inclusions in diffuse clouds?

5.1. Profile variations in HCO+ absorption lines

Profile variations in molecular gas over time have been discussed by (Moore and Marscher 1995) and (Marscher et al. 1993); (Diamond et al. 1989) and (Faison et al. 1998) present VLBA maps of strong ([FORMULA]) variations in HI optical depth. In both cases, it is suggested that AU-sized inclusions are needed to reproduce such high degrees of variability on yearly or milli-arcsecond scales. More or less the same considerations apply here, except that the column densities and column density differences are typically much greater in molecular gas compared to HI.

The existence of AU-sized structure in pc-sized clouds, producing 10% - 100% changes in optical depth, seems to imply the presence of some kind of inclusion having a local number density 0.1 - 1.0 x 206265 higher than in the ambient material. Since most profiles seem subject to variability (at least from the references cited), these extreme conditions are common. Since the profile variations occur as fluctuations, rather than disruptions, of a template profile, even this extreme material somehow partakes of normal cloud kinematics.

(Heiles 1997) recently discussed the (in retrospect) rather mild fluctuations which are seen in HI absorption spectra toward pulsars (Frail et al. 1994) where 2 km s-1-wide changes with [FORMULA] commonly occur. This implies a column density variation of only [FORMULA][FORMULA] [FORMULA]: by careful tuning, and by considering a very dense underlying atomic cloud ([FORMULA]), he was able to lower the required over-densities and pressures considerably. Heiles noted that if his model inclusions were to be very cold (needed to lower the column density associated with a given level of fluctuation in the optical depth) they would cool by CO emission with line brightnesses just at/above the nominal 0.3K sensitivity limits of surveys like those used to define our sample of background objects (Liszt and Wilson 1993; Liszt 1994). In suggesting a more sensitive search for these lines in emission, he neglected the fact that the CO in any such features would readily have been manifested in absorption, and in related molecules as well, and is not seen (ibid, (Liszt and Lucas 1996)).

Stellar extinction measurements conducted by Thoraval and her collaborators (Thoraval et al. 1996, 1997) put interesting constraints on the small-scale structure of extinguishing material of moderate column density. In time series, it is found that stars do not vary in brightness as dark masses temporarily occult them. In spatial extinction maps, the fluctuations are such that the column density is smoothly distributed, without extremes. This means that any internal cloud structure responsible for variation in spectral line profiles cannot contain a large fraction of the total amount of material through a typical line of sight.

Here we show that our measurements, although exhibiting the same profile variations, also bear strong direct evidence that the usual dense molecular concentrations within clouds are not the cause. These arguments could probably have been adduced earlier and are not terribly subtle. Shown in Fig. 9 are HCO+ emission profiles from our earlier survey (Lucas and Liszt 1996), to make the point that HCO+ emission is very weak toward our sources, typically below 0.05K. This is consonant with the low thermal pressures deduced from CO absorption (Liszt and Lucas 1998). In Fig. 10, we show the result of a very straightforward HCO+ excitation calculation for a bit of gas bearing a column [FORMULA] over a 1 km s-1 velocity interval. The calculation is parametrized by the temperature and density; we have assumed that a piece of molecular gas is exposed to the interstellar radiation field and determined the electron density with carbon depleted in abundance by a factor of 2.5 from the Solar value; the standard free-space carbon photoionization rate [FORMULA]; and a cosmic-ray ionization rate of [FORMULA]. Knowing n([FORMULA]), n(e), [FORMULA] , and N(HCO+) we can calculate the line brightness and optical depth, which we have plotted.

[FIGURE] Fig. 9. HCO+ emission spectra toward three sources, from (Lucas and Liszt 1996)

[FIGURE] Fig. 10. HCO+ J=1-0 optical depth and brightness temperature for a column [FORMULA] and linewidth [FORMULA]km s-1 at various densities and kinetic temperatures

At densities of [FORMULA], the observed line emission brightnesses and optical depths are produced naturally by the conditions which prevail in diffuse clouds, without extra assumptions. At such low density, all the curves for the optical depth coincide when the excitation is minimal: they separate substantially at higher density, depending on the temperature, as population is shifted into the upper levels. At high density, n[FORMULA], the predicted brightness temperature varies relatively little with either kinetic temperature or number density.

Consider the case that in a typical HCO+ line with [FORMULA] = 1.17 km s-1 (like that of the -17 km s-1 component toward B0355+508) equivalent to N(HCO+) = [FORMULA] over a 1 km s-1 linewidth, we would try to induce an optical depth change like that which is shown in Fig. 3, with [FORMULA] km s-1. For gas at [FORMULA] = 30 K, n([FORMULA]) = [FORMULA], we infer from Fig. 10 that the column density of HCO+ required would be [FORMULA], which would produce HCO+ emission of 1.4 K. Since that is more than 100 times higher than what is seen, less than 1% of the HCO+ can exist at such conditions.

Of course the density and temperature do not vary independently but this naive approach makes it clear that any dense inclusions would have to be composed of gas which is very cold, even though it is not strongly shielded. To see how cold such dense gas might be in equilibrium, we went through the exercise of calculating the temperature in a gas of moderate column density at various number densities. The results are as shown in Fig. 11 at top; the gas cannot be made arbitrarily cold as long as it is exposed to the interstellar radiation field. (Heiles 1997) calculated that (atomic) gas would not get colder than 13.5 K and we find 10-12 K as well. This is still warm enough to produce quite strong HCO+ emission at high density, so that material at densities above (say) [FORMULA] cannot include more than a few percent of the HCO+ in any cloud.

[FIGURE] Fig. 11. Top: Variation of computed kinetic temperature across model gas spheres of various densities, for N(H) = [FORMULA]. Bottom: CO and C+ column densities for the model gas spheres shown at top

As the gas becomes denser, it invariably makes the C+-C-CO transition as well (Fig. 11 at bottom) and what begins as an attempt to induce relatively mild fluctuations in the HCO+ optical depth produces unacceptably large amounts of CO. The abundance of CO increases by about a factor of 100 at high density, even at low extinction; thus, again, only about 1% of the gas can exist under such conditions.

5.2. HCO+ profile variations are not caused
just by extremely dense inclusions

The existence of a limit on the fraction of HCO+ which can exist under extreme conditions is sufficient to eliminate the usual models of dense clumps, at least in the context of the present discussion of equilibrium conditions.

We want these inclusions to be commonly observable, seen of order half the time; this implies that the total amount of surface area in clumps is comparable to that of the cloud in which they are embedded. So let the ratio of total clump to cloud area be denoted by f[FORMULA]: since the probability of encountering a clump along a random line of sight is 1-exp(-f[FORMULA]), f[FORMULA] [FORMULA] 0.7 to give an even chance of encountering one clump per cloud. Next, denote the ratio of the HCO+ column density in a clump to that through the cloud by fcl: for the dense 30 K gas considered as an example in Sect. 5.1, fcl was unity and would have been about 0.4 for gas at the same density but 10 K.

Then, the constraint from the emission, that only a fraction fem of the HCO+ can be at extreme conditions, may be expressed as [FORMULA] and this is a problem. If f[FORMULA] is fixed at a relatively high value of order unity to make the clumps commonly detected, and if fcl is of order unity, there is no way to make their product be as small as it must be. Much smaller fluctuations or much less frequent variations can be explained, but only because they approach the limit of no variations.

These arguments are hardly unique to HCO+ but apply to older observations of OH and as well.

5.3. Chemical and other fluctuations as the cause of profile variations in molecular absorption line profiles

The usual assertion (Moore and Marscher 1995) that molecular absorption line profile variations require small substructures of correspondingly high hydrogen volume density is really based on the implicit constancy of the molecular abundance, or, for that matter, all the other quantities affecting the optical depth: for OH in material of modest density, these are manifested in its poorly-understood excitation temperature. If the molecular abundance is allowed to vary widely, other, contradictory aspects of the problem largely disappear (for instance, the extinction need not vary). Especially in diffuse gas, where the chemistry is hardly driven to saturation, it would seem quite possible that large fluctuations in chemical abundance could easily be driven by local changes in ambient physical circumstances, for instance, due to the temporary dissipation of turbulent energy, the onset of bistability, and so forth.

As an example, consider the case where a fluctuation of size l (typically AU) in a cloud of size L (typically pc; L [FORMULA] l) is required to harbor a column N[FORMULA] of some species X whose relative abundance varies as [FORMULA]. It follows that the density over the l-sized region must be [FORMULA]. In the usual case, p = 0 and [FORMULA]. But if p = 1, the required density increase is only [FORMULA], which is smaller by a factor [FORMULA] 100, and the column of hydrogen required over the l-sized fluctuation is reduced by the same factor. Clearly this sort of (chemical) inhomogeneity can go a long way toward alleviating difficulties in explaining the rapid variability of some molecular line profiles.

The previous discussion relies on the assumption that a significant change in the molecular column density occurring while the line of sight is displaced by a small distance l across a cloud must originate in a relatedly small part [FORMULA] along that line of sight. This assumption might conceivably fail for some models of cloud structure: for instance in a cloud with density enhancements occuring in two-dimensional sheets, the main fluctuations in the column density as a function of impact parameter would occur when the line of sight tangentially hits one of these sheets, in which case the need for high density could be relaxed by a large amount, somewhat reminiscent of the geometrical tuning described by (Heiles 1997). Fractal models of diffuse clouds would actually be needed to investigate this point in detail.

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Online publication: March 17, 2000