Calculating the Optimum Velocity Window

Selecting the velocity window   [r] Failure to select the correct velocity window would lead to the output either not covering the entire width of the line or the resolution of the line being very poor. Inherent in finding this window is deciding where the line begins and ends - this is defined as being when the line temperature drops below 1% of its peak value. Even with a very powerful computer it would still make sense to correctly establish the velocity window as it is much better to use the computing power to work out the line shape in more detail than to waste it on calculating lots of zeros. The program splits the problem into three stages. First the extremes of velocity that occur in the cloud are established simply by checking all points in a 100 by 100 evenly spaced grid that covers the entire cloud and noting the maximum velocity. To this maximum value is added twice the velocity width of the line ($\Delta V$, as calculated in equation 4.20). This is then taken as the initial velocity window. The second stage then refines this first estimate by splitting this range into 35 velocity steps and then for each position on the grid that will be used for the convolution (see section 4.8.3) an approximation for the emission at that velocity is calculated. The key here is that the approximation is very much faster than the final detailed calculation so it pays to spend some time refining the window further at this stage. The approximation used is simply the same method as used in the final detailed calculation but without the interpolation between the rings^4.20. This drastically reduces the number of segments that can lie on a line of sight and thus enables the emission from each line of sight to be rapidly calculated. Taking each velocity position in turn this section locates the velocity where the emission rises above a certain predetermined level and flags this point. It then finds where it drops back below this level and flags that point. These two points then define the final velocity window within which the detailed calculations will be made. This final window is split into $outpts$^4.21 velocity steps for each of which the emission and optical depth will be calculated. An example of this is shown in figure 4.25. The velocity field in the cloud lies between the two extremes of $V_{min}$ and $V_{max}$. The approximate line emission is calculated for each velocity position marked by a spot. The final velocity window is that region where the approximation yielded an emission greater than $T_{min}$. Within this final velocity window the detailed line emission calculation is done for each velocity position marked by the unfilled circles thus yielding a much more detailed line shape than if the same number of unfilled circles had been spread over the initial velocity window.