Solid Body Rotation vs Keplerian Rotation

$\includegraphics[scale=0.9]{buksk1.eps}$ $\includegraphics[scale=0.9]{buksk2.eps}$ $\includegraphics[scale=0.9]{buksk3.eps}$ $\includegraphics[scale=0.9]{buksk4.eps}$ $\includegraphics[scale=0.9]{buksk5.eps}$

$\includegraphics[scale=0.9]{bukbsk1.eps}$ $\includegraphics[scale=0.9]{bukbsk2.eps}$ $\includegraphics[scale=0.9]{bukbsk3.eps}$ $\includegraphics[scale=0.9]{bukbsk4.eps}$ $\includegraphics[scale=0.9]{bukbsk5.eps}$

Line profiles for HCO+ $4\rightarrow 3$ (left column) and CS $5 \rightarrow 4$ (right column) for a cloud rotating as a solid body (dashed lines) at a speed of $\Omega =30$ km s $^{-1}$ pc $^{-1}$ and for a cloud rotating with a Keplerian velocity distribution of $0.1 \times r^{-\frac {1}{2}}$ km s $^{-1}$ where

is the distance from the axis of rotation in parsecs. The rows represent output at offsets of (from the top down) -20 $^{\prime \prime}$ , -10 $^{\prime \prime}$ , 0 $^{\prime \prime}$ , 10 $^{\prime \prime}$ & 20 $^{\prime \prime}$ for a JCMT type telescope (ie. beamsize of 13.6 $^{\prime \prime}$ for HCO+ and 20.0 $^{\prime \prime}$ for CS.

The problem with the solid body rotation that was modelled in the previous chapter is that it is not a physically realistic simulation for a non-solid cloud. This was acknowledged by Buckley, however, it might be hoped that solid body rotation at least provides a first approximation to a more realistic model. This can be checked with the ASTRA program by running a model with Keplerian rotation. This is presented in figure 6.1. This shows the output from the solid body rotation model (dashed lines) against a model where the cloud rotates with a $r^{-\frac{1}{2}}$ velocity distribution (the dashed lines are the same model as the solid lines in figure 5.31). The distribution chosen was $\frac{1}{10}r^{-\frac{1}{2}}$ where

is the distance in parsecs of any given point from the centre line of the cloud. For the Buckley model this yields a velocity on the outer edge of the cloud of 0.46 km s $^{-1}$ , rising to a maximum of 8 km s $^{-1}$ at the

pc cylinder. The velocity is capped at 8 km s $^{-1}$ . This is to stop extremely high velocities occurring in the centre of the cloud which, whilst they may be realistic, make it very difficult for the program to deal with. The emission from the innermost ring or two will not have a significant effect on the line profiles. This model then simulates a sphere with differential rotation which, although this is also not a particularly realistic model (since in reality such a system would collapse to a rotating disk very quickly), is a better approximation than solid body rotation. It can be seen from figure 6.1 that for the line profiles that are taken well off the centre of the cloud the solid body rotation is indeed a rough approximation to the differential rotation model. This is as might be expected since for neither model do the velocities change very rapidly in the outer part of the cloud. However, closer to the centre large differences occur. This too follows from the change in velocity distribution since for the solid body rotation there is very little velocity at the centre of the cloud whereas the Keplerian velocity distribution reaches its peak at the centre of the cloud. Note also that the apparent similarity in the line shapes at the central position is a coincidence as the double peaked shapes have completely different origins. For the solid body rotation the twin peaks are caused by there being an absorption dip in the centre. For the Keplerian rotation the dip is caused by there being very little material in the beam that has a low velocity component along the line of sight. Only the material that lies directly on the line of sight through the centre of the cloud has all its velocity perpendicular to the line of sight. Slightly off centre the material will either be moving rapidly away or towards the telescope thus causing the extended line wings. Only lines of sight further away from the centre will have large amounts of material all moving at the same velocity. On one side of the cloud this material will be moving away from the telescope and on the other side it will be moving towards the telescope, each side producing its own peak on the line profile. A related point to notice is the difference in the size of the line wing at the 10 $^{\prime \prime}$ position for the HCO+ and CS line profiles. This is only partly due to the differences in the type of molecule but is mainly due to the differences in the emission frequency and hence telescope beam size. The HCO+ beam does not reach all the way to the centre being only 6.8 $^{\prime \prime}$ in radius (at the half maximum) and hence does not see all the high speed material in the central, more sensitive, part of the beam, thus restricting the size of its line wing. In contrast the CS beamsize is 10 $^{\prime \prime}$ in radius and therefore reaches to the centre thereby giving a much more powerful line wing.

It would thus seem to be a reasonable conclusion that although the solid body rotation modification to the STEN program is a useful exercise it is not capable of even approximately reproducing the effects of Keplerian rotation. There are, however, some clouds that appear to rotate very slowly as solid bodies (see the next chapter) in which case provided spherical symmetry is also present this version of STEN would run very much faster than any 2-D program could hope to achieve. It is also very useful as a test of more sophisticated models since its inherent simplicity makes it easy to understand and it does not need many changes to the program thereby reducing the possibility of errors being introduced.