Ring parameter calculations

Intersection/Ring relationship     The parameter files store the values of the parameters for each intersection of the cylinders and disks in the cloud, however, the program requires most of these values for each ring, not the intersections. Therefore the program needs to convert the data from the files. In a 1-D model there is a fairly simple relationship between the shell boundaries and the shells themselves since if there are $n$ shells then there are also $n$ boundaries (or $n+1$ if the centre point of the cloud is counted). With a 2-D model the connection is more difficult. Consider the example in figure 4.11: this shows a 3 cylinder by 6 disk model. The centre line of the cloud is the thicker line. If the rest of the diagram is rotated around this centre line the cylinders and disks are formed (ie. this is a cross-section of half the cloud). The convention used in the program is that the intersections and rings are labelled from the bottom centre of the cloud outwards and upwards as shown. The relationship between the ring numbers and the intersection numbers can be stated as^4.7:

Any ring $x$ in a cloud with $n_{cyls}$ cylinders has four surrounding vertices given by:

$$x+1+n_{cyls}+\left[\frac{(x-1)}{n_{cyls}}\right] \hspace{2cm} x+2+n_{cyls}+\left[\frac{(x-1)}{n_{cyls}}\right]$$

$$x+\left[\frac{(x-1)}{n_{cyls}}\right] \hspace{3cm} x+1+\left[\frac{(x-1)}{n_{cyls}}\right]$$

The parameter for a ring is then defined as the values at these 4 positions averaged together. This explains the rather long statements describing each ring parameter in the program.

First the temperature in each ring is worked out. Once this is known the collision coefficients for each transition can be worked out in each ring. This uses simple linear interpolation of the figures provided for various temperatures in the COEFDATA.DAT file. At present the program can only interpolate for temperatures that lie below the smallest temperature provided (ie. if a ring has a kinetic temperature higher than the largest value listed in the COEFDATA.DAT file the program will stop). The interpolation is a simple linear interpolation between the two surrounding temperatures for temperatures that lie in the range for which collision coefficients are provided. For temperatures below the lowest provided a linear decline between the two lowest temperatures provided is assumed and continued down to the required temperature. The collision rate file provides only the downward collision rates so the program has to calculate the upward rates using the equation of detailed balance:

$$Rate_{\rm i \rightarrow j}=Rate_{\rm j \rightarrow i} \times ... ...on coefficients} \frac{g_i}{g_j} \times e^{\frac{h \nu}{k T_k}}$$

where i and j are the lower and upper levels respectively, the $g_i$ and $g_j$ are the level degeneracies (given by $2j-1$ where $j$ is the quantum number for that level), $h$ is Planck's constant, $k$ is Boltzman's constant, $\nu$ is the transition frequency and $T_{\rm k}$ is the kinetic temperature.

For each ring the full width at half maximum is calculated using the parameters for the turbulent velocity given in the input file. From this the ring velocity width is calculated from

$$\Delta V=\frac{1}{10} \times \frac{\sqrt{v^2+V_{\rm turb}^2}}{2\sqrt{\ln2}}$$ (4.20)

and where $v^2=\frac{8 k T_k \times \ln2}{M}$(with $M$ being the mass of the molecule). The factor of $2\sqrt{\ln2}$ is due to the different definitions of FWHM as explained next to figure 4.24.

The program next calculates how many molecules there are in each ring and then makes a first guess at how many of those molecules are in each excitation level. The radiation intensity is set equal to that caused by the cosmic background radiation - ie. $BPL=\frac{2h \nu^3}{c^2\left(e^{h \nu /kT_B}-1\right)}$ and $T_B$ is set to 3 K (ie. the cosmic background temperature). The level populations can then be calculated using the levels subroutine (see section 4.7.3). With the population levels calculated the initial excitation temperature can be calculated for each ring. This is done by re-arranging the definition of excitation temperature $\frac{n_i}{n_j}=\frac{g_i}{g_j}e^{\rm\frac{-h \nu}{kT_{ex}}}$ to give:

$$T_{\rm ex}=\frac{h \nu}{k}\frac{1}{\ln\left[\frac{n_jg_i}{n_ig_j}\right]} \index{temperature!excitation}$$ (4.21)

Column density   The column density is calculated simply by adding up the individual segments in each ring for a line of sight that runs through the centre of the cloud out perpendicularly to the centre line of the cloud (figure 4.12). Each ring's column density is easily calculated as the density in the ring and the length of the line of sight segment is known. The figure returned is:

$${\rm column\ density}=2 \times \sum_{i=1}^{n_{cyls}} N_{i} \times \left( r_i - r_{i-1} \right)$$

where $n_{cyls}$ is the number of cylinders, $N_i$ is the molecular density in the ${\rm i^{th}}$ ring and $r_i$ is the radius of the ${\rm i^{th}}$ cylinder (with $r_0=0$). Note that this is the column density right through the cloud (ie. not just to the centre but effectively the total optical depth between the telescope and the edge of the universe).