Dependency of Line Shape on Viewing Angle

Variation of Line Profiles with viewing angle.  

solid	45$^\circ$
dashed	30$^\circ$
dot-dashed	15$^\circ$
dotted	0$^\circ$

Variation of Line Profiles with viewing angle. 

solid	45$^\circ$
dashed	60$^\circ$
dot-dashed	75$^\circ$
dotted	89.9$^\circ$

Clearly a cloud that is rotating only around its central axis will have no component of velocity in the vertical direction. It might therefore at first be thought that when viewing the cloud from above, the line profile would revert to the same shape as for a completely stationary cloud. However, inspection of figures 6.2 & 6.3 shows this not to be the case. The figures show exactly the same model as used previously (ie. with the same parameters as given on page ). For the first 45$^\circ$ of rotation only relatively small changes in the line shape are visible, however, above 45$^\circ$ the line shape rapidly becomes narrower and since the emission is then less spread out in velocity it also becomes more intense. When viewing the cloud perpendicular to the plane of rotation^6.1 the line shape is indeed rather similar to the shape generated from a non-rotating cloud. Line emission at 89.9$^\circ$ for different rotation speeds  

solid	0
dashed	0.1 $r^\frac{1}{2}$
dot-dashed	0.2 $r^\frac{1}{2}$

[l] However, on closer inspection there are still small differences. These differences are more clearly visible in figure 6.4 where the solid line is the line profile for a stationary cloud. The dashed line is the rotating version which can be seen to have slightly less intense emission. The reason for this is that as the central core of the cloud is rotating its emission is distributed over a larger velocity range. Therefore at any particular velocity the optical depth of the central part of the cloud will appear to be less than for the equivalent non-rotating cloud. This enables radiation from the centre of the cloud to leak out to the outer portions of the cloud, thereby cooling the centre. If this explanation is correct then the effect ought to be stronger for faster rotating clouds. The dot-dashed line in figure 6.4 has twice the rotation velocity of the dashed line and it can be seen that the difference in intensity is indeed more pronounced.