A prolate model

Looking at figure 7.3 it is clear from the general shape of the cloud that the most obvious simple cloud models to consider are prolate or oblate spheroids. A prolate spheroid (a rugby ball shaped object) would have its axis of symmetry aligned in the SE-NW direction. In order to simulate such a shape the hydrogen distribution is described by

$$n_{\rm H_2}=c \times r^{a-b \left\vert \sin \theta \right\vert}$$ (7.1)

where $\theta$ is given by $\theta=\tan^{-1}\frac{r_d}{r_c}$ where $r_{c}$ is the radial distance from the axis of symmetry and $r_d$ is the distance along the axis of symmetry. In figure 4.3 this is the angle formed by the dashed line and the line labeled $r$. This enables two different power laws to be specified, one along the major axis and one along the minor axis of the cloud. The $\sin \theta$ function then varies smoothly between the two. For example with $a=-1.5$ and $b=0.5$ the hydrogen density drops off as $r^{-1.5}$ along the plane in the centre of the cloud that is perpendicular to the axis of symmetry and with $r^{-2}$ along the axis of symmetry. Lines radiating from the centre of the cloud but pointing elsewhere will drop off with some power law between these two values. It is possible to get quite easily an approximate fit to the line intensities with a cloud with a radius of order 0.1pc and a distribution of H$_2$ given by

$$n_{\rm H_2} \propto r^{-1.5+0.5 \left\vert \sin \theta \right\vert}$$

However, although this fits the peak intensities the modelled spectra to the NW and SE are clearly offset from the telescope spectra. This offset is probably caused by rotation and can be corrected by making the cloud rotate as a solid body with a speed of order 100 m s$^{-1}$. The problem with this is that it is rotation around the cloud's minor axis. In other words this is end over end rotation of the major axis which is a probably an unrealistic situation. It therefore seems clear that the cloud cannot be prolate.