Our disk model is based upon the fundamental work of van der Kruit & Searle
([1981a], [1981b], [1982a], [1982b]).
They tried to find a fitting function for the light distribution in
disks of edge-on galaxies.
These galaxies are, compared to the face-on view, preferred for studying
galactic disks due to the fact, that in this geometry it is possible to
disentangle the radial and vertical stellar distribution.
Their model include an exponential radial light distribution found for
face-on galaxies (de Vaucouleurs [1959]; Freeman [1970]), a
sech^{2} behaviour in *z*, which is expected for an isothermal population in a
plan-parallel system (Camm [1950]; Spitzer [1942]), and a
sharp edge of the disk, first observed by van der Kruit ([1979]) in
radial profiles of edge-on galaxies. The resulting luminosity density
distribution for this symmetric disk model is
(van der Kruit & Searle [1981a]):

(1) |

being the luminosity density in units of [ pc

The empirically found exponential radial light distribution is now well
accepted and it is proposed that viscous dissipation could be responsible
(Firmani et al. [1996]; Struck-Marcel [1991];
Saio & Yoshii [1990]; Lin & Pringle [1987]), although there is so
far no unique explanation for the disk being exponential.
An alternative description of the form 1/*R* proposed by Seiden et al.
([1984]) did not get much attention, although it emphasizes the
empirical nature of the exponential fitting function.

To avoid the strong dust lane and to follow the light distribution down to
the region
Wainscoat ([1986]) and Wainscoat et al.
([1989]) carried out NIR observations using the much lower
extinction in this wavelength regime compared to the optical.
They found a clear excess over the
isothermal distribution and proposed the *z*-distribution to be better fitted by
an exponential function
.
According to van der Kruit ([1988]) such a distribution would led to
a sharp minimum of the velocity dispersion in the plane, which is not observed
(Fuchs & Wielen [1987]). Therefore he proposed
as an intermediate solution. De Grijs ([1997]) extended this to a family
of density laws
following van der Kruit ([1988]), where the isothermal (*m*=1), and
the exponential ()
cases represent the two extremes.
Therefore the luminosity density distribution can be written as:

with

In order to limit the choice of parameters we restrict our models to the
three main density laws for the *z*-distribution (exponential, sech, and
sech^{2}). Due to the choice of our normalised isothermal case *z*_{0} is equal
to 2 *h*_{z}, where *h*_{z} is the usual exponential vertical scale height:

f_{1}(z) |
= | ||

f_{2}(z) |
= | ||

f_{3}(z) |
= |

In contrast to Paper I and Barteldrees & Dettmar ([1989]) we define the cut-off radius at the position where the radial profiles become nearly vertical, corresponding to the mathematical description. They tried to avoid any confusion due to the lower signal-to-noise in the outer parts, by fixing the cut-off radius where the measured radial profile begin to deviate significantly from the pure exponential fit.

Taking into account this transformation we have to integrate Eq. (2), obtaining an equation for the intensity of the model disk depending on the observed radial and vertical axes

Together with Eq. (2) this gives:

Therefore six free parameters fit the observed surface intensity on the chip (

Figure 1 shows a sequence of computed models with an isothermal

Figure 1:
Different galaxy models computed according to Eq. (4)
with constant values for the ratio
within the columns, or
within the rows, whereas
and
are
constant for all models using an isothermal description for the exact edge-on
case |

We are aware of the problem that these are rather rough definitions difficult to reproduce without quoting the exact values for . However, the final choice of the fitting area is a complex and subjective procedure depending on the intrinsic shape of each individual galaxy, the influence of their environment, and the quality of the image itself. Therefore it is not possible to quote exact general selection criteria and to derive the structural parameters straight forward. One solution is to do it in a consistent way for a large sample, leaving the problem of comparing results from different methods (cf. Sect. 4.2).

The numerical realisation of the fitting procedure minimizes the difference
(*SQ*) between the averaged quadrant and a
modelled quadrant based in Eq. (4).

is the intensity within the average quadrant (observed intensity) and of the modelled intensity. In contrast to Shaw & Gilmore ([1989], [1990]) and Shaw ([1993]) using a similar approach for their models we do not weight individual pixels. They weight the difference by the error in the surface brightness measure, which Shaw ([1993]) derives from the averaging of the quadrants. However, this method implies an absolute symmetry for the disk in

Figure 2:
Example of the fitting procedure: Set of typical radial
(left panel) and vertical (right panel) profiles used for the fitting process
of the galaxy ESO 461-006 together with the model |

As a first step, the inclination *i* is determined by using the axis
ratio of the dust lane. Depending on the shape of the dust lane it is
possible to restrict the inclination to
.
The central
luminosity density
is calculated automatically for each new
parameter set by using a number of preselected reference points along the
disk. Given a sufficient signal-to-noise ratio, the cut-off radius
can
be determined from the major axis profile. Thus, the remaining fitting
parameters are the disk scale length and height, *h* and *z*_{0}, as well as
the set of 3 functions *f*(*z*).
The scale length *h* is fitted to a number (usually between 4 and 6) of major
axes profiles (left panel of Fig. 2).
The quality of the fit for the vertical profiles along the disk can be used
as a cross-check for the estimated scalelengths (right panel of
Fig. 2).
The disk scaleheight *z*_{0} is estimated by fitting the *z*-profiles, outside
a possible bulge or bar contamination.
The vertical disk profiles of most of the galaxies investigated enable
a reliable choice of the quantitatively best fitting function *f*(*z*).
This is due to the fact that the deviations between different functions
become visible at vertical distances larger than that of the most
sharply-peaked dust regions.
The first raw fitting steps are usually carried out by using a reduced
number of both major and minor axis profiles simultaneously. Afterwards,
when a good first fitting quality is reached, a complete set of major and
minor calculated and observed axes profiles are investigated in detail.
At the end of this procedure a final, complete disk model is calculated
using all previously estimated disk parameters.

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