6 Appendix

The parameters J, $\Omega$, J/M, M and so forth depend to varying degrees upon distance D, such that (for instance) $\Omega\propto D^{-1}$, and $J/M \propto D$. It follows that errors in distance may give rise to correlations between rotational and physical parameters over and above those attributable to intrinsic cloud properties. How important are such effects for the sample investigated here?

To assess this, we have investigated a model containing 10³ clouds having intrinsic values log J, log $\Omega$ etc. distributed randomly within pre-set limits. These limits are in turn defined such that the final "observed" ranges in these parameters (including the effects of distance uncertainties) are comparable to those noted in Table 1, and Figs. 2-7. In particular, if the intrinsic logarithmic range in J is given by $\Delta\log J_{\rm int}$, and the range of values J which would be determined through observation (and in the presence of distance errors) is $\Delta\log J_{\rm obs}$, then $\Delta\log J_{\rm int}$ is tailored to make $\Delta\log J_{\rm obs }\sim 11$; comparable to the range noted in Figs. 6 and 7.

Errors in estimated distance are taken to be of a form:

$$\log D_{\rm obs} = \log D_{\rm int} + \Sigma(R - 0.5)$$

where again $D_{\rm obs}$ is the "observed" or estimated value of distance, $D_{\rm int}$ is the intrinsic value of D, $\Sigma$ is a constant, and R is a random number generator varying between 0 and 1. The clouds, in brief, have estimated distances which vary randomly between limits $D_{\rm int}10^{\pm 0.5\Sigma}$ on either side of the actual distances. Similarly, the intrinsic range in cloud distances is restricted to $\Delta\log D_{\rm int} = 2-\Sigma$. Where $\Sigma = 0$ then errors in estimated distance are negligible, and the range in log $D_{\rm obs}$ is comparable to the intrinsic range (i.e. $\Delta \log D_{\rm int} = \Delta \log D_{\rm obs} = 2$). Where $\Sigma = 2$ then the range in $D_{\rm obs}$ is due almost entirely to errors.

Under these circumstances, it is possible to investigate how important such errors would be in creating spurious correlations, for a sample in which there is no initial correlation between the various cloud parameters (i.e. in which intrinsic values of $\Omega$, say, are uncorrelated with M, J or any other parameter). A summary of the results is provided in Table 3, wherein we indicate correlation coefficients for various values of $\Sigma$, and relative parametric trends. The simulated variation between J and M is also indicated in Fig. 14 for two values of $\Sigma$.

 

Table 3: Correlation coefficients and gradients for simulated cloud trends

 

  

Figure 14: Theoretical trends between J and M for a 103 cloud sample, and two values of the distance error parameter $\Sigma$ (see appendix for details). The diagonal lines correspond to linear least-squares fits

It is clear, from these, that appreciable correlations may indeed result, but only where $\Sigma$ is large; and even for $\Sigma = 1.5$, which would correspond to a factor 5.6 uncertainty in distance, it is apparent that values for r are very much less than observed here (Table 2). Similarly, the gradients b for these distributions depend upon the intrinsic ranges in the parameters, and error factor $\Sigma$. Even for the most extreme case investigated here (the gradients b in Table 3 correspond to $\Sigma = 1.5$) it is apparent that gradients are significantly different from what is observed (Table 2).

Given that errors in D are almost certainly appreciably smaller, and $\Sigma$ is no more than $\sim 0.3\Rightarrow 0.5$, we conclude that the influence of such uncertainties upon observed correlations is likely to be small.