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Astron. Astrophys. 331, 347-360 (1998)

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Appendix A: treatment of the diffuse field

The ionized gas also contributes to the radiation field through the capture of free electrons, hydrogen and helium ions. For a nebula which is optically thick in the Lyman continuum a good approximation is to suppose that the photons are absorbed in a local environment near the place where the recombination has occurred, i.e. the OTS approximation. This allows the use of the local variables, x, y, z, T and [FORMULA] in order to calculate the diffuse radiation and so it is not necessary to solve the transport equation. Recombinations to the ground levels of the H0 produce an emission centred at frequencies [FORMULA] and, as a consequence, it is entirely absorbed by hydrogen. In the OTS approximation this emission can be easily accounted for by considering only recombinations to the excited levels of hydrogen ([FORMULA]) in both the ionization and the energy equations. The inclusion of the diffuse photons that ionize helium is not as direct as with hydrogen. The recombinations to the ground levels of He0 and [FORMULA] can ionize both the hydrogen and the helium because these photons are emitted with energies greater than [FORMULA] eV and [FORMULA] eV, respectively. The photoionization terms for these diffuse photons can be written as

[EQUATION]

where [FORMULA] and [FORMULA].

The recombinations to the excited levels of He0 and [FORMULA] (recombinations to the [FORMULA] level) can also produce photons able to ionize the H0 and He0. The photoionization terms can be written as in Eq. (A1) substituting [FORMULA] by [FORMULA], where [FORMULA] is the efficiency of photoionization, i.e. the probability that each transition [FORMULA] could produce an ionizing photon, where now [FORMULA] and [FORMULA].

The OTS approximation is not as precise in the thermal balance as it is in the ionization balance because it does not take into account the fact that photons emitted after recombination may have different energies so that different penetration depths into the nebula, and thus some of these cannot be absorbed locally. For this reason we have solved for the transport of the diffuse field using the outward approximation (Williams 1967), i.e. each cell may receive ionizing photons from all inner cells. To calculate the radiation intensity [FORMULA] at each point of the nebula we have built a grid of rays with impact parameters ([FORMULA]) at different radius. Then, given that [FORMULA] is equivalent to know [FORMULA] as in spherical symmetry, [FORMULA]. The radiative transport along each ray with impact parameter [FORMULA] is then solved:

[EQUATION]

for [FORMULA], and [FORMULA], and with

[EQUATION]

where [FORMULA] is the coordinate along the ray ([FORMULA]) and [FORMULA] is the gas source function (we have omitted the dependence on [FORMULA] for clarity) and [FORMULA] and [FORMULA] are correspondly the emissivity and the gas opacity. The mean intensity is then calculated through a weighted summing over the rays:

[EQUATION]

Appendix B: line transfer

As output, the hydrodynamical models produce a set of magnitudes: density, temperature, velocity, etc, at each spatial location, but these can only be related to observable quantities if they are used to calculate either surface brightness or line profiles.

To solve the radiative transport along a line of sight we used the method described by Yorke (1982). The line intensity is given by

[EQUATION]

where [FORMULA], assuming that the continuum contribution to the line is negligible. If the source function [FORMULA] varies linearly with optical depth, Eq. (B1) can be integrated to yield

[EQUATION]

The emissivity and the opacity are [FORMULA] and [FORMULA] where [FORMULA] and [FORMULA] are the values at the line centre. [FORMULA] is the frequency in the ion reference system, which is displaced from the frequency at rest [FORMULA] by the Doppler effect, i.e. [FORMULA], assuming [FORMULA], where [FORMULA] is the velocity component along the line of sight. After complete redistribution [FORMULA] one obtains

[EQUATION]

where [FORMULA] is the line thermal width. As the emissivity and the opacity have the same frequency dependence the source function [FORMULA] is independent from the frequency, and in the Eq. (B2) only the optical depth remains as a function of the frequency:

[EQUATION]

where

[EQUATION]

[FORMULA] is the error function and we have supposed that the velocity component [FORMULA] varies linearly in the interval [FORMULA], i.e. [FORMULA]. [FORMULA] is the coordinate along the line of sight. Finally the surface brightness can be determined from the integration over all frequencies of [FORMULA].

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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