- 3.1 Orientations
- 3.2 Variations in angular velocity
- 3.3 Specific and total angular momentum
- 3.4 Cloud stability
- 3.5 Cloud aspect ratios

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.

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 depends upon the relative values of the Oort constants A and B, and the initial intrinsic cloud aspect ratio (e.g. Field 1978); although in general, one would anticipate values or . 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 would be largely random.

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

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 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 , then the probability of the double-peaked structure arising by chance is quite small, of order . If, on the other hand, the data sequence is folded about , so that the peaks approximately coincide, then the probability reduces to . 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.

The variation of angular velocity with cloud radius () 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/dln; 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 and *R* (the correlation
coefficient is *r* = 0.84), and are characterised by a gradient dln/dln
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 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 pc. Whilst the variation
in for larger cloud sizes is rather less clear, it appears on current evidence to
be relatively modest.

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

The variation of 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 ) 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 10^{3} 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 is
inclined at some angle to the plane of the sky, then intrinsic angular
momentum *J*, angular velocity , and a raft of other parameters (including *J*/*M*
(Sect. 3.3), and (Sect. 3.4) and (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 dln*J*/dln*R*, dln/dln*R* 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) |

The variation of specific angular momentum with radius is illustrated for the
complete data set in Fig. 4, where , and the parameter varies from
0.5 in the case of disks through to 0.33 for prolate structures; we shall adopt an
intermediate value 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 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.

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.
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*)/dln*R* 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 (); a feature which is all the
more interesting given that *J* depends upon a total of three independent
parameters (*R*, *M* & ) 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 dln*J*/dln; a value which
is closely similar the model gradient dln*J*/dln 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 () are correspondingly
lower. Most of the gradients for individual cloud sub-groups are also rather
similar (of order dln*J*/dln), 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.

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 between the rotational and combined turbulent and thermal virial
terms through

(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 d*v*/d*r* we are in fact determining a
parameter , where *i* is the inclination of vector to the line of sight.
Values of will therefore tend to be depressed by a factor . A second
qualification concerns the values of line width 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 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

(2) |

The distribution of values for the present sample of nebulae is indicated in
Fig. 10. We have also indicated (by a dashed line) an upper limit 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 , then the gravitational acceleration at either end of the filament would
be . Rotational stability at the limits of the filament would then require
values . It would appear, therefore, that 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 depends upon cloud inclination through
, 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 of cases ( where clouds alone are considered), whilst the clumps possess in only of cases. As was found for , 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 and 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 also appears to be prone to systematic errors, and is likely (on average) to be appreciably undervalued.

Given that , we can further investigate the reasons for the depressed values of through an analysis of virial mass . 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 are larger by a factor ; 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 by dex. The parameter would, under these circumstances, represent a rather more credible indicator of cloud stability.

Figure 12:
Distribution of cloud aspect ratios for the stability parameters
a) and b). Note the broader distribution in
for larger values of |

Figure 13:
Distribution of cloud aspect ratios for the stability parameters
a) and b).
Note the broader distribution in for larger values of |

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

(3) |

(4) |

(5) |

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 oriented close to the minor axis in of cases (Table 1, Col. 13); a result which is certainly consistent with rotational deformation.

Similarly, if we take separately the sources having and , and plot the corresponding distributions of aspect ratio (Fig. 12), then it seems that is larger for high values of than is the case for clouds having lower values of . In particular, Gaussian fits to these distributions would suggest a width for the values some 53% greater than for sources having lower values of , whilst very similar results are found for sources having and (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 and .

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