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Astron. Astrophys. 351, 140-146 (1999)

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3. Radiative transfer in galactic nuclei

Radiative transfer models for the disks and nuclei of galaxies have been carried out in various approximations by a number of authors (Krügel & Tutukov 1978, Efstathiou & Rowan-Robinson 1990, Spagna et al. 1991, Pier & Krolik 1992, Granato & Danese 1994, Krügel & Siebenmorgen 1994, henceforth KS94, Efstathiou & Siebenmorgen, 1995, Silva et al. 1998).

3.1. The source function

We compute the radiative transfer for the nucleus of the galaxy. The maps of nuclei generally appear at all wavelengths to first order round and therefore we feel justified to use for our crude endeavor models of spherical symmetry. The radiation transport code which we employ is detailed in KS94. Its potential was demonstrated by successfully modeling all relevant infrared data for the nucleus of M 82.

The crucial parameter in the transfer equation is the source function. For a dust cloud without distributed luminosity sources (like stars in a galactic nucleus) it reads

[EQUATION]

The dust emission coefficient is given by

[EQUATION]

The absorption, scattering and extinction coefficient are denoted by [FORMULA], [FORMULA], [FORMULA], and the mean intensity of the interstellar radiation field by [FORMULA]. The distribution function [FORMULA] in Eq. (2) equals the [FORMULA]-function in the case of large grains because their temperature is constant. Then this equation just reads [FORMULA]. For very small grains (such as PAHs) the function [FORMULA] has to be evaluated separately (see e.g. Siebenmorgen et al., 1992).

The model nuclei contain two kinds of stellar populations:

  • Giants with 4000 K effective temperature, number density [FORMULA] and spectral luminosity [FORMULA]. They are very numerous and smoothly distributed and introduce into the source function the term [FORMULA] which enters in the numerator of the right side of Eq. (1).

  • OB stars from the starburst with an effective temperature of 30 000 K, number density [FORMULA] and luminosity [FORMULA][FORMULA]. Because they are so luminous, each immediate environment of an OB star presents a hot spot where the dust temperature has a local peak.

As was shown in KS94, when the luminosity of the galactic nucleus is dominated by OB stars, the hot spots have a great influence on the mid IR spectrum. The size of a hot spot follows from the condition that inside it, heating of the dust is dominated by the OB star, whereas outside heating by the interstellar radiation field [FORMULA] is more important. The emission from a hot spot, denoted by [FORMULA], has to be computed in a special routine (for the original description see Krügel & Tutokov 1978) and is taken into account in the source function by the term [FORMULA]. Altogether, the source function of a galactic nucleus filled with stars is

[EQUATION]

3.2. Dust model

We employ a dust model similar to that by Siebenmorgen & Krügel (1992). The large spherical grains have radii from 300 to 2400 Å with a size distribution exponent 3.5. They consist of silicate and amorphous carbon particles with a volume ratio [FORMULA]. A fraction of 25% of the solid carbon is graphitic in the form of very small grains with radii from 10 to 20 Å and distribution exponent 4. Another fraction [FORMULA] of the solid carbon is in PAHs. All PAHs have a skeleton of 25 C-atoms and 14 H-atoms at the periphery, corresponding to the maximum hydrogenation parameter for circular PAHs. Optical constants for silicate are from Laor & Draine (1993), for amorphous carbon (type BE) from Zubko et al. (1996), for graphite from Draine (1985), for the PAHs from Schutte et al. (1993). To minimize the number of free parameters in the fits, the dust model is kept constant and we only vary the abundance [FORMULA] of the PAHs.

The dust-to-gas ratio is not required for computing the radiative transfer models. The total amount of dust in the galactic nuclei, [FORMULA], follows from the visual extinction [FORMULA] and the dust radius [FORMULA], as given in Table 2:

[EQUATION]

Here it is assumed that the dust density [FORMULA] is constant, so [FORMULA]. The exact value of [FORMULA] depends on the visual extinction coefficient [FORMULA] and thus on the particular dust model, but [FORMULA] cm2 per gram of dust (or [FORMULA] cm2 per gram of gas) is a value that will not arouse much controversy.


[TABLE]

Table 2. Parameters of the radiative transfer models.


3.3. Photodissociation of PAHs

In a strong UV radiation field, PAHs may evaporate. Although evaporation is not important for our objects, it was taken into account and treated by the following formalism details of which are found in Zota (1998). The photo-dissociation rate [FORMULA] of the PAHs is described only by its internal energy U and the degrees of freedom s (see e.g. Forst 1973),

[EQUATION]

Despite this simplification, Jochims et al. (1996) achieved good agreement between dissociation measurements in the lab and a linearized version of Eq. (5) for small PAHs ([FORMULA]) by properly choosing the fit-parameters [FORMULA] and [FORMULA]. After Jochims et al. (1994), [FORMULA] eV and [FORMULA] s-1 for the dissociation channels H, H2 and C2H2. As typical emission rates for IR-photons are of order [FORMULA] s-1, they suggest [FORMULA] s-1 as the critical value for the onset of PAH dissociation. This corresponds after Eq. (5) to some internal energy [FORMULA] and the latter is converted into an evaporation temperature [FORMULA] by the canonical representation of the internal energy,

[EQUATION]

The [FORMULA] are the fundamental vibrations of the molecule; for small PAHs, these frequencies have been measured (e.g. Cyvin et al. 1979, Whitmer et al. 1978) or can be calculated. The effect of photo-ionization, which is the only competing reaction path for PAHs of high internal energy, stabilizes the PAHs against evaporation. This effect is accounted for in our models following Verstraete et al. (1990).

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© European Southern Observatory (ESO) 1999

Online publication: November 2, 1999
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