3 Cloud rotational properties

The information cited in Table 1 may be used to investigate a broad range of cloud rotational properties. In the following, we separately investigate variations in specific and total angular momentum, cloud orientation, angular velocity, stability and morphology, comparing the results with trends anticipated from theoretical analyses.

3.1 Orientations

Where cloud angular momentum derives from shear in the galactic disk, and there is no subsequent dynamical or magnetic interaction with other entities, then one would expect to find strong residual evidence for preferential orientation of the angular momentum vectors. In particular, the orientation of $\Omega$ depends upon the relative values of the Oort constants A and B, and the initial intrinsic cloud aspect ratio $\Gamma_{\rm i}$ (e.g. Field 1978); although in general, one would anticipate values $\Theta=90^\circ$ or $270^\circ$. Where, on the other hand, such rotation arises through turbulence, or is alternatively strongly disordered through (say) magnetic coupling between clumps, then one might expect that the orientations in $\Omega$ would be largely random.

What, from the present evidence, can one say regarding such trends in molecular clouds?

  

Figure 1: a) Distribution of molecular clouds as a function of angle $\Theta$ between the angular velocity vector $\Omega$ and galactic plane, where we have illustrated the subgroup of isolated clouds (i.e. cloud types MI, MC, etc.). b) As for Fig. 1a, but for disks and rings (i.e. subgroup D/R). c) As for Fig. 1a, but for clumps and condensations

The data is represented in the form of a series of histograms in Figs. 1a-c. It is clear, from these, that if we consider only the individual clouds (that is, we eliminate clumps, condensations, disks or rings), then there is a reasonably strong indication of preferential orientation, with approximately equal numbers of vectors $\Omega$ oriented towards the north and south galactic poles. The application of the Kolmogorov-Smirnov test to such a distribution yields a probability which depends, partially, upon which point in the angular sequence one chooses to commence with. If the lowest point in the data sequence is taken to be $\theta = 250^\circ$, then the probability of the double-peaked structure arising by chance is quite small, of order $2.8\ 10^{-2}$. If, on the other hand, the data sequence is folded about $\theta = 180^\circ$, so that the peaks approximately coincide, then the probability reduces to $\sim 5\ 10^{-10}$. Whatever statistical test one might choose to employ, therefore, it is likely that the probability of Fig. 1a arising by chance is going to be quite small.

Finally, we note that disks and rings also appear to share some evidence for preferential orientation (Fig. 1b), although clumps and condensations (which might be expected to possess a similar evolutionary pedigree) have a distribution which is essentially random (Fig. 1c).

Such trends are sufficiently disparate to confound any attempt at a simple explanation. Clouds, for instance, may acquire their momenta through shearing motions in the galactic disk, as noted above. Angular momentum in clumps, by contrast, is unlikely to derive through spin-up of gravitationally unstable sub- regions within larger cloud systems; or alternatively, if such spin-up did in the past occur, then the vector direction may have been randomised through dynamical or magnetic interactions. Given such a situation, it is then however unclear why disks may have retained some fossil memory of their original orientations - one would anticipate that these, too, should show random orientations.

  

Figure 2: Variation of projected cloud angular velocity $\Omega$ with radius for all of the clouds in the present study; the right hand key identifies various subgroups (see text for details). Solid lines connect régimes within differentially rotating disks, whilst the dashed lines represent expected trends for turbulent vorticity (A) and angular momentum conservation (B). Finally, the broad diagonal line (designated $\Delta D$) corresponds to the variation in source position resulting from a factor 2 change in distance

3.2 Variations in angular velocity

The variation of angular velocity with cloud radius ($\equiv L/2$) is represented in Fig. 2 for all of the cloud sample, where solid lines connect results determined for differing regimes of the same (differentially rotating) disks. A least squares analysis of these trends (Table 2) suggests that clouds and condensations taken separately are characterised by gradients dln$\Omega$/dln$R \cong -0.59$; a value which is similar to that determined for the sample as a whole. The disks and rings, on the other hand, display the tightest correlation between $\Omega$ and R (the correlation coefficient is r = 0.84), and are characterised by a gradient dln$\Omega$/dln$R \cong -1.04$ which is considerably steeper.

We have also included in Fig. 2 the trends which would be anticipated were angular momentum conserved, and those predicted through the turbulent vorticity model of Fleck & Clark (1981). Neither represents observed trends particularly well. Thus, the observed variation of $\Omega$ with R is significantly less steep than would be envisaged were angular momentum conserved - a disparity which might be regarded as favouring models in which J is leached through a process of magnetic braking. If this were the case, then it must be concluded that such a mechanism must operate over all cloud sizes $R\leq 4$ pc. Whilst the variation in $\Omega$ for larger cloud sizes is rather less clear, it appears on current evidence to be relatively modest.

  

Figure 3: Variation of angular velocity $\Omega$ with cloud mass M (where both here and the proceeding figures, M is given in units of the solar mass)

The variation of $\Omega$ with mass is particularly interesting, since the correlation between these two parameters appears in all cases to be very low (Fig. 3); indeed, it would appear that the relevant figures constitute a species of scatter diagram, with correlation coefficients ranging between 0.03 and 0.41 (the latter value corresponding to the disks; the only subgroup to display evidence for systematic trends (Table 2)). As we shall see later, this lack of correlation is in marked contrast to the corresponding variation in angular momentum J(M).

Finally, two final points are worth making. Although the contribution of uncertainties in distance D is difficult to quantify precisely, it seems unlikely that estimates can be in error by greater than a factor 2. Thus, whilst certain of the larger values D may be open to appreciable revision, the mean influence of such uncertainties is likely to be modest. This is illustrated in Figs. 2 and 3 by means of broad diagonal bars (labelled $\Delta D$) wherein is indicated the variation in cloud location arising from a factor 2 change in distance. Similar loci are illustrated in the remaining figures. This question is also further analysed in the appendix, where we conclude (from an analysis of 10³ model clouds) that even order of magnitude errors in distance would be unlikely to reproduce observed gradients and correlation coefficients.

The second point concerns the role of cloud projection in modifying deduced rotational properties. In particular, where the angular velocity vector $\Omega$ is inclined at some angle $\gt 0^\circ$ to the plane of the sky, then intrinsic angular momentum J, angular velocity $\Omega$, and a raft of other parameters (including J/M (Sect. 3.3), $\alpha$ and $\beta$ (Sect. 3.4) and $\Gamma$ (Sect. 3.5)) are likely to be undervalued. Similarly, since the projected mean values of such parameters depend, in part, upon cloud morphology, then any systematic trend of morphology with radius may lead to changes in the gradients dlnJ/dlnR, dln$\Omega$/dlnR and so forth. Such a possibility must in fact be taken quite seriously when considering the clouds as a whole, since many larger clouds appear to be spindle shaped, whilst disks are concentrated towards the lower end of the radial range.

Although such effects may be far from negligible, they are also extremely difficult to quantify precisely. Taken as a whole, however, it seems unlikely that the (statistical mean) trends and conclusions of this analysis would be greatly modified by such corrections.

  

Figure 4: Variation of specific angular momentum J/M as a function of cloud radius R, where we have included all of the cloud subgroups in the present study (see Fig. 2 for key to symbols)

  

Figure 5: As in Fig. 3, but for the variation of J/M with mass M. The diagonal line corresponds to the trend expected for isothermal, non-magnetic rotating clouds

  

Figure 6: Variation of angular momentum J with cloud mass M. The diagonal line represents the trend anticipated for simple clump merger models (Benz 1984)

  

Figure 7: Variation of angular momentum J with cloud radius R

3.3 Specific and total angular momentum

The variation of specific angular momentum with radius is illustrated for the complete data set in Fig. 4, where $J/M\equiv \phi\Omega R^2$, and the parameter $\phi$ varies from 0.5 in the case of disks through to 0.33 for prolate structures; we shall adopt an intermediate value $\phi = 0.4$ appropriate for spheres. It is clear, from this, that J/M declines systematically with decreasing cloud radius, and that the gradient is of order dln(J/M)/dln$R \cong 1.43$ for both isolated clouds and condensations. On the other hand, disks again display a trend which is disparate from that of other regions in this study, yielding a gradient dln(J/M)/dln$R \cong 0.96$.

This difference largely disappears when we consider the variation of J/M with mass M, on the other hand (Fig. 5; Table 2), whence it is clear that all of the cloud components share comparable gradients dln(J/M)/dln$M \cong 0.7$. It is interesting, in this case, to note that models of isothermal, non-magnetic rotating clouds would require J/M to scale as M (i.e. dln(J/M)/dlnR takes a value unity; Bodenheimer & Black 1978). Indeed Boss (1987), in his discussion of model rotating disks, concludes that some such scaling law is consistent with observed trends in disk-like structures.

The gradient indicated by the present observations appears, in reality, to be somewhat shallower, suggesting that such analyses may be inapplicable to real cloud structures.

The variation of angular momentum J with mass (Fig. 6) constitutes one of the tightest correlations in the present data set ($r \cong 0.98$); a feature which is all the more interesting given that J depends upon a total of three independent parameters (R, M & $\Omega$) and their associated errors. Although the scale range in J is rather larger than that of other functions investigated here - and this, in turn, can lead to an apparent enhancement in correlation - we shall find below that this is by no means the entire story.

As might be expected from the narrowness of the distribution in Fig. 6, clumps, disks and clouds share closely similar gradients dlnJ/dln$M \cong 1.7$; a value which is closely similar the model gradient dlnJ/dln$M \cong 5/3$ noted for clumps in non-magnetic rotating clouds (Benz 1984).

If one now determines the variation of J with R, as in Fig. 7, we obtain a distribution which is qualitatively somewhat different; the scatter in results appears to be broader, and correlation coefficients ($r \sim 0.88$) are correspondingly lower. Most of the gradients for individual cloud sub-groups are also rather similar (of order dlnJ/dln$R \sim 3.3$), although it appears that disks are again different in having a gradient some 30% smaller.

It follows, from this, that the decrease in angular momentum appears to be directly related to the decrease in mass, and less so to changes in radius; it is difficult to avoid the conclusion that these various distributions are telling us something rather important about the way angular momentum is transferred from large to small scales. At the very least, one might infer that rotational braking processes must be strongly mass dependent, whilst rotational mechanisms which depend primarily upon size scale (cf. turbulent vorticity) would be unlikely to reproduce the very tight correlations noted above. Similarly, we note that where cloud angular velocities derive from galactic shear, then the value of |J| will depend upon the morphology and orientation of the initial cloud structure; it is possible for J to possess a comparatively broad range of values for any single cloud mass M (see, for instance, the discussion in Blitz 1993). Under these circumstances, it follows that the tight correlation noted in Fig. 6 would imply closely similar initial collapse configurations.

  

Figure 8: Variation of the stability parameter $\beta(R)$ for isolated clouds and disks (see text for details). $\beta$ appears in most cases to be significantly less than unity, implying that turbulent and thermal virial expressions greatly exceed the contribution due to rotation

  

Figure 9: Variation of the stability parameter $\beta(R)$ for clumps and condensations. Comparison with Fig. 8 suggests that rotation may be even less important for these sources than is the case for disks and larger cloud structures

  

Figure 10: Variation of the stability parameter $\alpha(R)$ for the complete data set (see text for details); where the lower dotted line corresponds to the parametric limit at which rotation becomes marginally significant. The upper dashed line represents the limiting value of $\alpha$ for highly flattened rotating disks

3.4 Cloud stability

There are various means whereby one might investigate the importance of rotation for overall cloud stability. In particular, it is possible (for instance) to define a ratio $\beta$ between the rotational and combined turbulent and thermal virial terms through

$$\beta=7.04\ 10^{-2}\left[\frac{R}{\rm pc}\right]^2\left[\fra... ... s}}\right]^2 \left[\frac{\Delta V}{\rm km\ s^{-1}}\right]^{-2}$$ (1)

where $\Delta V$ is the observed line FWHM. The influence of rotation and (primarily) turbulence are comparable when $\beta\cong 1$.

Values deriving from the present data are plotted in Figs. 8 and 9, where we have represented, separately, the distributions for clouds+disks and clumps.

It would appear, in both of these figures, that turbulence is by far the most critical factor in determining cloud stability; all but a handful of the results are appreciably less than unity (although the influence of turbulence appears rather stronger in the case of clumps).

Certain caveats should be expressed at this point, however. The first is that in measuring projected velocity gradients dv/dr we are in fact determining a parameter $\Omega\sin(i) \leq\Omega$, where i is the inclination of vector $\Omega$ to the line of sight. Values of $\beta$ will therefore tend to be depressed by a factor $\sin^2(i)$. A second qualification concerns the values of line width $\Delta V$ used to determine combined thermal and turbulent support. We shall discuss this in rather more detail later this section; suffice to say that observed line widths $\Delta V$ may also lead to an overestimate of kinematic broadening.

Taken as a whole, therefore, it would appear that such an analysis is likely to provide a somewhat pessimistic impression of the importance of cloud rotation for overall cloud stability.

A differing measure of the influence of cloud rotation upon cloud stability is provided through the parameter

$$\alpha=22.0\left[\frac{\Omega}{10^{-4}{\rm s}}\right]^2\left[\frac{R}{{\rm pc}}\right] ^3\left[\frac{M}{M_\odot}\right]$$ (2)

which represents a ratio between rotational and gravitational virial terms. Where $\alpha$ is less than 0.5, then the influence of rotation may be regarded as small (e.g. McKee et al. 1993).

The distribution of values $\alpha$ for the present sample of nebulae is indicated in Fig. 10. We have also indicated (by a dashed line) an upper limit $\alpha = 3\pi/4$ relevant for highly flattened, rotationally stabilised disks. On the other hand, if one represents filamentary clouds in terms of a cylinder of diameter d, length L, and density $\rho$, then the gravitational acceleration at either end of the filament would be $\pi G\rho d$. Rotational stability at the limits of the filament would then require values $\alpha = L/d = \Gamma$. It would appear, therefore, that $\alpha$ may approach values >4, based on the observed aspect ratios of filamentary clouds in the present study.

The results in Fig. 10 can be very broadly separated into three radial ranges. For sources having R > 2 pc, or R < 0.06 pc (primarily disks), it appears that rotation is frequently important in maintaining overall stability. For intermediate radii, on the other hand, significantly fewer sources (approximately 1/3) appear to be influenced by rotation; although since $\alpha$ depends upon cloud inclination through $\sin^2(i)$, it follows that the actual proportion of rotationally stabilised clouds may be significantly greater.

This data may also be sliced in a slightly different way. Thus, and taken as a whole, the individual clouds and disks appear to be strongly influenced by rotation in $\sim 50\%$ of cases ($\sim 64\%$ where clouds alone are considered), whilst the clumps possess $\alpha \gt 0.5$ in only $\sim 31\%$ of cases. As was found for $\beta$, therefore, it appears that clumps and condensations are less susceptible to the influence of rotation than is the case for other subgroups in this study.

In summary, it would appear that $\beta$ and $\alpha$ give somewhat differing impressions concerning the importance of cloud rotation. This, in part, may arise from the differing natures of the parameters we are evaluating, although $\beta$ also appears to be prone to systematic errors, and is likely (on average) to be appreciably undervalued.

Given that $\beta\propto \Delta V^{-2}\propto M_{\rm vir}^{-1}$, we can further investigate the reasons for the depressed values of $\beta$ through an analysis of virial mass $M_{\rm vir}$. Thus, for instance, Fig. 11 illustrates the variation of virial mass (evaluated using linewidths quoted in Table 1) against corresponding column density mass estimates derived from the literature. It seems clear, from this, that there is a systematic disparity between the two comparative mass estimates, in the sense that values $M_{\rm vir}$ are larger by a factor $\sim 4.8$; similar trends have also been noted by Zimmerman & Ungerechts (1990), Keto & Myers (1986), Stutzki & Gusten (1990), Zhou et al. (1994) and Magnani et al. (1985) for other cloud samples. Various physical explanations might be sought for this difference, including pressure stabilisation by the ambient medium; the possibility that certain of these features are transient; or finally, that apparent line widths are somewhat broadened by line saturation - a problem which is by no means completely eliminated through our use of more optically thin transitions.

Whatever the reasons, it seems clear that observed line widths are rather larger than would be consistent with virial equilibrium, and this leads to a reduction in $\beta$ by $\sim0.7$ dex. The parameter $\alpha$ would, under these circumstances, represent a rather more credible indicator of cloud stability.

  

Figure 11: Comparison of virial masses $M_{\rm vir}$ deduced using line widths from Table 1 with cloud masses M derived from molecular column densities. The diagonal dashed line represents the relationship which would be anticipated were clouds stable, line widths unaffected by saturation, and turbulent and thermal velocities constituted the dominant virial components

  

Figure 12: Distribution of cloud aspect ratios $\Gamma$ for the stability parameters $\alpha\leq 0.3$ a) and $\alpha \gt 0.3$ b). Note the broader distribution in $\Gamma$ for larger values of $\alpha$

  

Figure 13: Distribution of cloud aspect ratios $\Gamma$ for the stability parameters $\beta\leq 0.1$ a) and $\beta \gt 0.1$ b). Note the broader distribution in $\Gamma$ for larger values of $\beta$

3.5 Cloud aspect ratios

An interesting and related question is that of how strongly rotation affects the apparent aspect ratios of molecular clouds. Put at its most basic, one might anticipate that rapidly rotating structures would show typically higher aspect ratios than slowly rotating clouds. Thus, Miyama et al. (1984) find that aspect ratios for isothermal rotating disks are of order

$$\Gamma=k\left[\frac{GM}{(J/M)c}\right]^2$$ (3)

providing rotational forces at the cloud boundary exceed thermal support; where c is the velocity of sound, and k is a constant of order unity. This expression can also be generalised to include the combined turbulent and thermal pressure terms, whence for observed FWHM linewidths $\Delta V$ we determine

$$\Gamma =1.03\ 10^{-4}\left[\frac{M}{M_\odot}\right]^2\left[\... ...\right]^{-2} \left[\frac{\Delta V}{ \rm km\ s^{-1}}\right]^{-2}$$ (4)

$$\approx 4.31 \left[\frac{M}{M_\odot}\right]^{0.6}\left[\frac{\Delta V}{{\rm km\ s}^{-1}}\right]^{-2}$$ (5)

providing the stability parameter $\beta\gt 1$, where we have substituted the trend for J/M(M) from Table 2. For reasonable mean line widths $\Delta V \sim 2$ km s^-1 it is therefore apparent that disks, and indeed clouds taken generally may show appreciable departures from sphericity. On the other hand, numerical simulations of comparable isothermal structures indicate that whilst $\Gamma$ is likely to increase with increasing J/M, the change overall may be comparatively modest (Boss 1987). Similarly, Myers et al. (1991) have noted that observed aspect ratios in dark clouds imply levels of rotational gradient which are not observed.

Do the present results offer any help in resolving this question? The answer, on the whole, is unclear. Thus, we note that the present sample of clouds appear to have values $\Omega$ oriented close to the minor axis in $\sim75\%$ of cases (Table 1, Col. 13); a result which is certainly consistent with rotational deformation.

Similarly, if we take separately the sources having $\alpha\leq 0.3$ and $\alpha \gt 0.3$, and plot the corresponding distributions of aspect ratio $\Gamma$ (Fig. 12), then it seems that $\Gamma$ is larger for high values of $\alpha$ than is the case for clouds having lower values of $\alpha$. In particular, Gaussian fits to these distributions would suggest a width for the $\alpha \gt 0.3$ values some 53% greater than for sources having lower values of $\alpha$, whilst very similar results are found for sources having $\beta\geq 0.1$ and $\beta < 0.1$ (Fig. 13). On the other hand, the sample numbers are somewhat restricted, and it is not possible to claim a very high level of significance for these trends.

It would therefore seem that the present analysis offers some possible hint of rotational deformation, although further analyses (using larger samples) are necessary to establish any clear relationship between $\Omega$ and $\Gamma$.