3 Disk models

3.1 Background

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 sech² 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]):

$$\hat{L}(R,z) = \hat{L}_0 \ \exp{\left(-\frac{R}{h}\right)} \ {\rm sech}^2{\left(\frac{z}{z_0}\right)} \qquad R < R_{{\rm co}}$$ (1)

$\hat{L}$ being the luminosity density in units of [ $L_{\hbox{$\odot$ }}$ pc^-3], $\hat{L}_0$ the central luminosity density, R and z are the radial resp. vertical axes in cylinder coordinates, h is the radial scalelength and z₀ the scaleheight, and $R_{{\rm co}}$ is the cut-off radius.

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 $z\!\approx\!0$ 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 $f(z)\!=\!\exp (-z/z_{0})$. 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 $f(z)\!=\!{\rm sech} (z/z_{0})$as an intermediate solution. De Grijs ([1997]) extended this to a family of density laws $g_m(z,z_0)\!=\! 2^{-2/m}\ g_0\ {\rm sech}^{2/m} \left( mz/2 z_0 \right) (m>0)$following van der Kruit ([1988]), where the isothermal (m=1), and the exponential ($m=\infty$) cases represent the two extremes. Therefore the luminosity density distribution can be written as:

$$\hat{L}(R,z) = \hat{L}_0 \ \exp{\left(-\frac{R}{h}\right)} \ f_n(z,z_0) \ H(R_{{\rm co}}-R)$$ (2)

with H(x₀-x) being the Heaviside function.

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 sech²). Due to the choice of our normalised isothermal case z₀ is equal to 2 h_z, where h_z is the usual exponential vertical scale height:

$$4\ \exp{\left(-2\ \frac{\mid z \mid}{z_0}\right)}$$ f₁(z) =

$$2\ {\rm sech} \left(\frac{2 z}{z_0} \right)$$ f₂(z) =

$${\rm sech}^2{\left(\frac{z}{z_0}\right)\cdot}$$ f₃(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.

3.2 Numerical realisation

The model of the two dimensional surface photometric intensity results from an integration along the line of sight of the three dimensional luminosity density distribution (2) with regard to the inclination i of the galaxy. Describing the luminosity density of the disk in a cartesian-coordinate grid $K(x\,-\,y\,-\,z)$ with $R=\sqrt{(x^2+y^2)}$ leads to the following coordinate transformation into the observed inclined system K'(x'-y'-z') with x' pointing towards the observer, whereas the rotation angle between the two systems corresponds to $90\hbox{$^\circ$ }-i$:

$$\begin{eqnarray*}x &=& x' \sin(i) - z'\cos(i)\\ y &=& y'\\ z &=& x' \cos(i) + z'\sin(i). \end{eqnarray*}$$

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 y' and z' on the CCD:

$$I(y',z') = \int\limits_{-\infty}^{+\infty} \hat{L}\bigl(x(x',z',i), y', z(x',z',i)\bigr) {\rm d}x'.$$ (3)

Together with Eq. (2) this gives:

 

$$I(y',z') = \qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$$

$$\hat{L}_0 \int\limits_{-\infty}^{+\infty}\!\! \exp{\left(-\frac{\sqrt{(x'\ \sin i - z'\ \cos i)^2+y'^2}}{h}\right)} f_n(z',z_0)$$

$$\ast\, H\left(R_{{\rm co}}-\sqrt{(x'\ \sin i - z'\ \cos i)^2+y'^2}\right) \:{\rm d}x'. \quad$$ (4)

Therefore six free parameters fit the observed surface intensity on the chip (y', z' plane) to the model:

$$I = I(y',\ z',\ i,\ n,\ \hat{L}_0,\ R_{{\rm co}},\ h,\ z_0).$$ (5)

Figure 1 shows a sequence of computed models with an isothermal z-distribution ($n\!=\!3$) for the exact edge-on case ( $i\!=\!90\hbox{$^\circ$ }$) with characteristic values for the ratio $R_{{\rm co}}/h$: 1.40,2.88,5.00 and for h/z₀: 2.0,4.0,7.3 keeping the cut-off radius $R_{{\rm co}}$ and the total luminosity $L_{{\rm tot}}\!\propto\!z h^2 \hat{L}_0$constant. The latter is causing a different central surface brightness $\mu_{0}$ for each model, ranging from 23.8 to 19.3 starting with $\mu_0\!=\!21.2$ mag $/\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil \penalty50\... ...$ } \parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\hbox{$^{\prime\prime}$ }$ for the reference model with $R_{{\rm co}}/h\!=\!2.88$and $h/{z_0}\!=\!4.0$. All contour lines falling within the interval $\mu\!=\!25.0 - 19.5$ are plotted with a spacing of 0.5.

Figure 1: Different galaxy models computed according to Eq. (4) with constant values for the ratio $\frac{R_{{\rm co}}}{h}$ within the columns, or $\frac{h}{z_0}$ within the rows, whereas $\hat{L}_{{\rm tot}}$ and $R_{{\rm co}}$ are constant for all models using an isothermal description for the exact edge-on case

3.3 Method 1

3.3.1 Determination of fitting area

The first step is to divide the galaxy into its quadrants. The four images are then averaged according to their orientation, following van der Kruit & Searle ([1981a]) and Shaw ([1993]). Thereby larger foreground stars and asymmetrical perturbations in the intensity distribution are eliminated by omitting this region during averaging. Smaller foreground stars are removed by median filtering. The average quadrant should result at least from three quadrants to get a representative image of the galaxy. This averaging additionally increases the signal-to-noise ratio. In order to determine the fitting area for modelling the disk component on the final quadrant, one has to avoid the disturbing influence of the bulge component and the dust lane. The region dominated by the bulge is fixed following Wyse et al. ([1997]) defining the bulge component by "light that is in excess of an inward extrapolation of a constant scale-length exponential disk''. Therefore the clear increase of the intensity towards the center which can be seen in radial cuts determines an inner fitting boundary $R_{\rm min}$. We tried to minimize the dust influence (cf. Sect. 4.4.2) by placing a lower limit $z_{\rm min}$ in the vertical direction by visual inspection. Additionally we restricted the remaining image by a limiting contour line $\mu_{\rm lim}$, where the intensity drops below a limit of 3$\sigma$ on the background.

We are aware of the problem that these are rather rough definitions difficult to reproduce without quoting the exact values for $R_{\rm min}, z_{\rm min}, \mu_{\rm lim}$. 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).

3.3.2 Numerical fitting

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

$$\nonumber SQ = \sum_{j}\Bigl({\rm log}\left(I_{{\rm O}_j}(y_j... ...)\right)-{\rm log}\left(I_{{\rm M}_j}(y_j,z_j)\right)\Bigr)^2$$

$I_{{\rm O}_j}$ is the intensity within the average quadrant (observed intensity) and $I_{{\rm M}_j}$ 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 z and y, which is not the case for real galaxies. These kind of errors only reflect the asymmetry of the galaxy. Using the observed errors in the surface brightness for weighting individual pixels does not result in a considerable advantage because they are nearly the same after smoothing. The minimal SQ is found by varying five of the six free parameters of the model (cf. Eq. 5), whereas the parameter $R_{{\rm co}}$ is determinated by cuts parallel to the major axis. A significant decrease of the intensity extrapolated to I = 0 gives the value of $R_{{\rm co}}$ (van der Kruit & Searle [1981a]), therefore it is important that the intensity at the cut-off radius is well above the noise limit. The other five parameters are determined by fixing the smallest SQ. For the three different functions f(z) and every possible i ( $\Delta i = 0.5\hbox{$^\circ$ }$) the remaining parameters ($\hat{L}_0$, h, and z₀) are varied with the "downhill simplex-method'' (Nelder & Mead [1965]; Press et al. [1988]) until the global minimum of SQ is found. $I_{{\rm M}_j}$ is calculated by a numerical Gaussian integration of Eq. (4). The possible inclination angles can be restricted from the dust lane of the galaxy (Paper I). Tests of model disks with additional noise show that the "downhill simplex-method'' found the input inclination i, the used model f(z) and the other disk parameters within errors of $\delta h, \delta z_0$, and $\delta \hat{L}_0 <$ 1%. An estimation of the errors of the parameters for the best model disk can be made by inspection of the parameter space for SQ around the smallest SQ, with slightly different fitting areas, and different values of $R_{{\rm co}}$. f(z) is in almost all cases the same and the variation of i is only small ( $\pm 1\hbox{$^\circ$ }$). The differences in h and z₀ are about 15% (in some cases up to 25%), and $\hat{L}_0$ varies about a factor of 2.

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

3.4 Method 2

Within this method all disk parameters as well as the choice of the optimum function f(z) are estimated by a direct comparison of calculated and observed disk profiles by eye.

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 $\pm 1.5 \hbox{$^\circ$ }$. The central luminosity density $\hat{L}_0$ 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 $R_{{\rm co}}$ can be determined from the major axis profile. Thus, the remaining fitting parameters are the disk scale length and height, h and z₀, 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 z₀ 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.