4. Test cases
We have implemented the methods described in the previous sections into a program where we can choose between methods A and B. We can also use the reference field to suppress random noise and choose between two random number generators i.e. use either pseudorandom or quasirandom numbers.
The program has been tested by comparing its results to some model calculations published in the past. We present here the results from the comparison with the microturbulent models of Liszt & Leung (1977). The model cloud of Liszt & Leung has a radius of 2.5 pc, a constant kinetic temperature of 40 K, and an intrinsic linewidth of 3.0 km s-1. The hydrogen density and the fractional abundance of CS are varied so that the column density of CS stays constant (see Fig. 2a in their article). In the calculations we used a value of µ=1.95 D for the permanent dipole moment of the CS molecule and the collision rate constants published by Green and Chapman (1978). These are presumably the same constants as used by Liszt & Leung for collisions between CS and H2. The collisions between CS and He are neglected.
We made the calculations using both the one-dimensional and the full three-dimensional models. The one-dimensional spherical model cloud consisted of 30 shells. The radii of the shells were selected so that all the spherical cells had the same volume. The three-dimensional model cloud consisted of 31 31 31 cells from which the corners were rounded off. Since the radius of the cloud is only some 15 cells, the discretization of the three-dimensional cloud is more coarse. The line profiles calculated with the three-dimensional model cloud were almost identical and the differences could hardly be seen by eye.
We also repeated the calculations with and without the aid of the reference field (using only positive reference temperatures although some transitions may have negative excitation temperatures) and also using traditional Monte Carlo simulation (method A) and our new simulation method (method B). The results remain the same irrespective of the method of calculation. The profiles were compared with the results of Liszt & Leung and the differences in the peak temperatures were less than 0.2 K for all transitions and usually much smaller.
The number of photon packages used in the calculation of the one-dimensional models was also varied. When method B was used and the random number generators were reset on each iteration the changes in the calculated spectra were observable only when the number of photon packages was 70 per one iteration. This illustrates the very low number of separate model photons needed with method B. In three-dimensional models 10000 photons per iteration were enough to get results similar to those of the one-dimensional model. Lower photon numbers were not tested.
We reproduced also the results of Park & Hong (1995) for three-dimensional, clumpy clouds with a volume filling factor f =0.12 (Fig. 5 in their article). Using simulation method B we made 15 iterations with 10 000 model photon packages per iteration. This means that on each iteration there are approximately 10 photon packages going through each cell and each package represents the total radiation field in one direction. The spectra were calculated by using a convolving gaussian beam with a FWHM corresponding to four cells. Since the number of cells within the beam is very large, the random errors in the calculated excitation temperatures of individual cells do not show up in the line profiles. All line intensities agree within 2% with the results of Park & Hong (1995), and the differences become smaller if more than 15 iterations are made.
In Fig. 1 b we show the radial excitation temperature distributions of two 12 CO transitions in a cloud with f =1.0. The cloud is similar to the one in Fig. 1 of Park & Hong (1995). The calculations were made using the method B with the aid of a reference field. We used 20 000 model photons per iteration and the figure presents the situation after 15 iterations. In spite of the relatively low number of model photons the result is still quite comparable with the results of Park & Hong (1995).
These results can also be compared with the calculations made by Bernes (1979) using a spherically symmetric model cloud. Although our three-dimensional cloud has a somewhat lower kinetic temperature and no infall velocity the models are still rather similar. For the calculation of the one-dimensional model Bernes (1979) used 200 model photons per iteration and the absorption event counters contained in the end the average over 20 iterations. Our photon numbers cannot be compared directly with this, however, since each of our model photons contains information of the whole line profile and is therefore more expensive to calculate. Even if this is taken into account it seems that the cost of modelling three-dimensional clouds with method B is not excessively high compared with the cost of modelling one-dimensional clouds. We calculated the absorption counters on each step as the average of the values collected during the current iteration and the average from past iterations weighted with 2.0. Increasing this weight (to e.g. 4.0) would considerably lower the noise in the excitation temperatures but would also slow down the convergence. Another possibility would be to use a larger weight to decrease the noise and reset the counters after some iterations to enhance the convergence. In the case of Fig. 1 b the line was divided into 30 velocity channels and the calculations took 35 minutes on a Pentium-166MHz computer. Using only 10000 model photons per iteration would give higher noise in the excitation temperatures but the computed line profiles would remain much the same.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998