Calculating the final output

Line of sight velocities   [l] In order to correctly generate the combined line profile that all the lines of sight in the beam grid produce it is necessary to consider the velocity field in the cloud. The geometry2 subroutine returns the velocity of all the intersections on a line of sight relative to the starting point on that line of sight. What is actually required are the velocities relative to the cloud centre since the receiver on the telescope will be tuned to the rest frequency of the emission line being observed with a correction for the motion of the cloud relative to the telescope. This motion will effectively be the velocity of the centre of the cloud. Earlier versions of the STENHOLM program assumed that the beam size was small relative to the radius of the cloud and therefore all lines of sight in the beam grid will have the same velocity relative to the centre of the cloud. It is therefore possible to simply calculate the emission from each line of sight, add them all together (with an appropriate weighting) and then subtract the velocity difference between the end of the lines of sight and the cloud centre. This simply shifts the velocity (frequency) scale but has no effect on the line shape. Considering figure 4.26 shows that if the beam size is not small compared to the radius of the cloud this method may no longer work well for a radially collapsing cloud and will definitely not work at all for a rotating cloud as the velocities of the ends of the lines of sight will not be the same. Consider the three lines of sight shown. If the outer edge of the cloud is rotating with a velocity of $v$ then the component of this velocity at the end point of the line of sight (marked by a blob) will be $v \sin \theta$ for line 1, zero for line 2 and $-v \sin \theta$ for line 3 (in each case this is relative to the central point in the cloud - marked by a blob). So the way to deal with this is to adjust the velocities for each line of sight so that all velocities are relative to the central velocity of the cloud. If this is done then the emission from each line of sight can simply be added together (with the appropriate weighting) to give the total emission.

For each line of sight the emission is calculated in the same way as described in section 4.7. This time however, the velocity range for which the output from each line of sight must be calculated is determined as described in the previous section. The number of velocity steps within this range is given in the MODELDATA.DAT file as $outpts$. Depending on the size of the grid chosen to calculate the beam with this section can take a long time as there are $outpts$ velocity points for which each line of sight must be calculated (eg. for a 51 $\times$ 51 grid if $outpts=50$ there will be 130,050 lines of sight for which emission must be calculated - and these are by no means extreme numbers).