*Astron. Astrophys. 325, 135-143 (1997)*
## 4. Model
In our model we assume that the stellar distribution consists of
two components, the disk and the bulge. For the light distribution in
spiral disks, two laws seem to be generally accepted so far. The
sech^{2} z law (van der Kruit & Searle 1981) and the
exponential law (Freeman 1970; Wainscoat et al. 1989). Both of these
laws provide a good representation of the vertical distribution of
light in the galactic disks. For the radial distribution, an
exponential law is used. In the present work, we use the exponential
law in both the radial and vertical direction.
For the bulge several profiles have been introduced. The well known
law (de Vaucouleurs 1953; Young 1976), the
Hubble-Reynolds law (Reynolds 1913;
Hubble 1930), the Hernquist law
(Hernquist 1990) and the exponential law (Andredakis & Sanders
1994) are good representations of bulges. For our model both the
law and the Hubble profile are used and a
comparison is made. A good description of the luminosity densities is
given by Christensen (1990) for the law and by
Binney & Tremaine (1987) for the Hubble law.
The stellar emissivity is then described by
where *R* and *z* are the cylindrical coordinates,
is the stellar emissivity at the center of the
disk and and are the
scalelength and scaleheight respectively of the stars in the disk. The
second term in this equation gives the two different types of bulge
luminosity density profiles, the first being the Hubble law and the
second the law, with a
normalization constant and
where is the effective radius of the bulge
and *a* and *b* are the semi-major and semi-minor axis
respectively of the bulge. Because the profile
has an infinite luminosity density at the center, and in order to
avoid computational problems, the luminosity density in a small sphere
of radius 0.2 kpc around the center was given a constant value and
equal to that at 0.2 kpc. To be consistent, this region was excluded
when the fitting procedure was done.
For the extinction coefficient we also use a double exponential
law, namely
where is the extinction coefficient at
wavelength at the center of the disk and
and are the scalelength
and scaleheight respectively of the dust.
If the above model galaxy is seen edge-on, and for the moment we
ignore completely the effects of dust, the surface photometry due to
the disk alone, after integration of the first term on the right hand
side of Eq. (1), is
where is the modified Bessel function of the
second kind, first order (Abramowitz & Stegun 1965), with a
central value of
The central value of the bulge surface brightness is
for the modified Hubble profile and
for the de Vaucouleurs profile (Christensen
1990).
The optical depth through the disk, in directions parallel to the
plane of the disk is
Thus, the central optical depth of an edge-on galaxy is
and the central optical depth of the same
galaxy seen face-on is .
The radiative transfer is done in the way described by Kylafis
& Bahcall (1987). The intensity *I* reaching a pixel is
thought of as the sum of , where
consists of the photons that suffered no
scattering between their point of emission in the galaxy and the
pixel, consists of the photons that suffered
one scattering between their point of emission and the pixel,
consists of the photons that suffered two
scatterings between their point of emission and the pixel, and so on.
The term is proportional to the albedo
of the dust, the term is
proportional to and so on. Since
(say , see below), the
contribution to the intensity *I* of the terms
, with , is generally
small compared to . Thus, to save computer time,
we compute and very
accurately and approximate the sum with
, where (see Eq. 19 of
Kylafis & Bahcall 1987). The error that this approximation
introduces to the intensity *I* is typically less than 1%.
For specific values of the parameters in Eqs. (1) - (3), a 2D image
of a model galaxy is produced. The goal is to find those values of the
parameters which create an image of the model galaxy as close as
possible to the image of the observed galaxy. A >Henyey-Greenstein
phase function has been used for the scattering of the dust (Henyey
& Greenstein 1941). A mean value 0.4 was used for the anisotropy
parameter *g*, while the value 0.6 was used as an average albedo
for all B, V, I bands. The effects of changing
these parameters within the limits given by (Bruzual et. al. 1988)
have been explored. It has been found that varying these parameters
within the above limits has no important changes in the intensity
because of the low optical thickness that has been found for this
galaxy.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998
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