A Modification to Allow Modelling of Dust

Due to the modularisation of the program it is easy to cut out and replace some of the individual elements of the program. One reason for doing this is to enable the modelling of other physical processes besides molecular line excitation whilst still using the same 2-D geometric structure. Other processes that might be of interest are line emission in other galaxies and processes associated with dust. Both of these would entail considerable work in replacing the present lambda iteration section with an appropriate new section. However, there is one fairly easy modification that enables dust continuum emission from star formation regions to be approximately modelled.  

A well known model for continuum dust emission from grains is that of Hildebrand [13]. This can be used to approximate the submillimetre dust emission in terms of the amount of hydrogen present in a cloud. The details of this model will not be presented here but the result is that optical depth is related to the amount of hydrogen present by

$${\rm N(H+H}_2)=1.2\times10^{25}\tau_\lambda\left( \frac{\lambda}{400}\right)^2 {\rm atom \: cm^{-2}}$$ (4.56)

where $\lambda$ is the wavelength (in $\mu$m) for which the optical depth is desired. Once the optical depth is known the emissivity coefficient can be calculated from

$$\eta_\lambda=\kappa_\lambda\frac{2h\nu^3}{c^2(e^{\frac{h\nu}{kT}}-1)}$$ (4.57)

provided $\tau \ll 1$ (this condition means the optical depth of each segment on a line of sight must be small). These two lines can be used to replace the lambda iteration section of the ASTRA program since once these values are known for every ring in the cloud the output section can calculate the intensity that a telescope would see if it were pointed at the cloud. This is in fact considerably easier than for molecular line emission since there is no velocity information required.

It should be noted that this is not a global solution of the radiative transfer equation in the way that the ASTRA program does for spectral lines. Instead it is simply a computation of the radiation intensity along a line of sight emitted by dust that has a specified density and temperature distribution.

The program that implements this (lesdust.f) is in fact considerably simpler than the ASTRA program. The entire lambda iteration routine is replaced by just two lines representing equations 4.56 & 4.57. Note that equation 4.56 is actually written as

$$\tau=2\frac{{\rm N(H}_2)}{1.2\times10^{25}}\left( \frac{\nu \times 1 \times 10^{-9}}{750} \right)^2$$

The initial factor of two is because the N(H$_2$) value counts the number of molecules of H$_2$ which each contain 2 atoms of hydrogen. The 750 term at the end is now in Gigahertz (ie. the 400$\mu$m in equation 4.56 is converted to 750GHz) whereas $\nu$ is in Hertz, hence the factor of $10^{-9}$. The value of $\nu$for which output is required is entered in a modified version of the MODELDATA.DAT input file. The program will calculate the emission for various positions on the cloud and thereby produce a dust emission map of the object. A grey scale map is then produced using PGPLOT which can be overlaid with contours if desired to enable easy comparison with observed dust maps (such as those produced by SCUBA on the JCMT).

Since only one line in the program is used to specify the emissivity law for the dust the program can easily be modified to run for dust emission laws other than that of Hildebrand.