Astron. Astrophys. 332, 703-713 (1998)

Appendix A: the Monte Carlo method of radiative transfer

For clarity, the equations below are presented for a two energy level molecule, but the generalization to the multilevel case of the code is quite straightforward.

Model photons with a weight W in proportion to the number of real photons emitted by the source are sent out in random directions and at random frequencies. To save time the model photons represent every transition simultaneously, and thus they have a separate weight for each transition. In addition to photons from the molecular gas, photons which represent background radiation, dust radiation and radiation from a central black body (star) can also be emitted. The cloud is composed of a series of concentric spherical shells, each having its own values of the physical parameters such as density, temperature, etc. The model photons are followed across the cloud, causing excitations and de-excitations along their paths, which affect the level populations in the different shells. The total number of induced de-excitations, , as a photon of weight W travels from to is given by

, (A1)

where m is the index denoting the shell and the other symbols have their usual meaning. The total number of molecules in the upper energy state in shell m is , where is the volume of the shell. With , we can write the number of radiative de-excitations per molecule in the upper state in shell m as

, (A2)

where is the optical depth along the distance . The optical depth is calculated as

. (A3)

is accumulated in sums . When all the model photons have been emitted, new level populations are calculated from the equation of statistical equilibrium,

, (A4)

which in the Monte-Carlo formulation is written as

. (A5)

After adjustment of the level populations, new model photons are sent out, and the process is repeated. In order to reduce the random noise, the counters are not emptied every iteration, and is then replaced by in the statistical equlibrium equation. The iteration is continued either for a fixed number of iterations, or until some pre-specified criterion for convergence is fulfilled.

Appendix B: Monte Carlo method with core saturation

B.1. Implementation of the core saturation method

The core saturation method is only applied to photons emitted by the molecular gas (i.e not to those from the dust or the background radiation). When a model photon is emitted, the optical depth to the nearest boundary is calculated, and it is decided, whether the photon is in the core of the line. As the distinction between core and wing is not precise, we follow Flannery et al. (1980) by introducing an exponentially tapered weighting function, , which tends to unity in the wing and to zero in the core, but also allows intermediate cases, viz.

(B1)

is the customary adjustable parameter, which roughly represents the optical depth separating the core and the wing. A smaller will result in faster convergence, but if it is made too small, the code will not converge. Note that even though the core photons are eliminated in the calculations, we must continue to follow every model photon, as each model photon represents all the transitions simultaneously.

The ratio of photons in the wing, , to the total number of photons emitted by the gas is given by

. (B2)

In the core saturation method, the core photons are eliminated from the statistical equilibrium equation, implicitly making use of the approximation in the core. This leads to much faster convergence and is possible, because these photons contribute so little to the energy transport. The statistical equilibrium equation is modified to

(B3)

and, in Monte-Carlo formulation, one arrives at

. (B4)

For simplicity, the core saturation procedure is only executed in the shell where the photon in question was created, since the core photons will in any case never travel far.

B.2. Correcting the core saturation for lost photons

The core saturation method simply neglects the energy transport in the core. This will cause an error, which becomes larger as is decreased, i.e. as the frequency space defining the core grows larger. We introduce a procedure, where a correction term, , is added to the statistical equilibrium equation. represents the difference between the correct number of radiative de-excitations per molecule in the upper state, , and the implicit aproximation in the core saturation method, . The equation of statistical equilibrium with the correction term included reads

. (B5)

Every photon contributes a certain amount to the de-excitations. For a certain gas photon in the core, this is aproximated by , where the subscript 0 refers to the initial value of the photon weight. Using the excitation temperatures from the previous iteration, can then be calculated as

. (B6)

The total weight, , of the model photons emitted in shell m is equal to the total number of real photons emitted, . From Eqs. (A2) and (A3) and using the relations

one finds

(B7)

and

(B8)

so that one finally can write

. (B9)

which is the expression actually used in the code.

© European Southern Observatory (ESO) 1998

Online publication: March 23, 1998
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