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Up: Three-dimensional modelling of edge-on galaxies


Subsections

4 Results

4.1 Distribution of disk parameters

Table 2 contains the best fit model for each image. Together with the galaxy name (1), the filter (2), and the referring image (3), with integration time, and run ID, we list the inclination (4), the best fitting function for the z-distribution (5), the calibration index (6) (Ref. Sect. 2.4), and the central surface brightness of the model (7), without correcting for inclination. According to the distance tabulated in Table 1, the cut-off radius $R_{{\rm co}}$ (8) is given in kpc and arcsec as well as the scalelength h (9) and the vertical scaleheight z0 (10) which is normalised to the isothermal case, being two times an exponential scaleheight hz. For the seven galaxies with available images in more than one filter, we do not see any correlation of fitted parameters with different wavelength, although we find the same inclination angle for the best fitted disk within the range of the errors. Appendix A shows the best fitting model as an overlay to selected radial profiles for each image. The subsequent analysis of the distribution for the different parameters concerning the formation and evolution of galaxies will be given in forthcoming papers.

 

 
Table 2: Determined parameters set
galaxyfilterimageif(z)cali.$\mu_{0}$ $R_{{\rm co}}$hz0
   [ $\hbox{$^\circ$ }$]  mag $/\ifmmode\hbox{\rlap{$\sqcap$ }$\sqcup$ }\else{\unskip\nobreak\hfil
\penalty50\...
...$ }
\parfillskip=0pt\finalhyphendemerits=0\endgraf}\fi\hbox{$^{\prime\prime}$ }$[ $\;\hbox{$^{\prime\prime}$ }\;$][kpc][ $\;\hbox{$^{\prime\prime}$ }\;$][kpc][ $\;\hbox{$^{\prime\prime}$ }\;$][kpc]
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)
             
ESO 112-004r40E387.5sechl21.1445.0--22.7--2.8.-- 
ESO 150-014r20E390.0sechl22.0064.133.2723.412.145.52.85
NGC 585R20L188.0seche21.7670.024.4932.711.4410.63.71
ESO 244-048r15E387.0$\exp$l20.5445.419.1713.75.787.23.04
NGC 973R10L189.5$\exp$e20.83105.033.7451.516.5512.43.99
UGC 3326R30L188.0sech2e21.50101.528.4570.419.744.71.32
UGC 3425R30L187.0sech2e21.0180.522.2629.28.087.42.05
NGC 2424R15L186.5$\exp$e20.52112.023.4031.76.6211.92.49
IC 2207R10L286.5$\exp$e21.0456.017.6738.212.066.72.12
ESO 564-027r30E288.0sechl21.11140.418.3350.96.656.60.86
ESO 436-034g60E388.0sechl20.9682.118.3222.65.046.81.52
ESO 319-026g30E386.5seche21.5160.113.2114.23.123.10.68
ESO 319-026r30E286.0$\exp$i21.1463.013.8414.73.233.20.70
ESO 319-026r30E388.0sech2i21.9461.913.6013.22.902.80.62
ESO 319-026i30E388.0sechl21.0164.814.2414.33.143.40.75
ESO 321-010g30E388.0sechl20.1164.412.3419.93.814.70.90
ESO 321-010r30E388.0sechl19.5464.812.4221.64.145.00.96
NGC 4835Ar40E385.5$\exp$l20.9290.018.5640.48.337.91.63
ESO 575-059r15E287.0sech2l20.7860.517.5322.66.556.41.86
ESO 578-025g30E286.5$\exp$l21.5850.720.6317.06.927.53.05
ESO 578-025g30E386.5$\exp$l21.6950.720.6317.06.927.53.05
ESO 578-025r30E286.0$\exp$l21.0250.420.5114.96.067.43.01
ESO 578-025i30E286.0$\exp$l20.2747.519.3320.38.267.63.09
ESO 446-018r30E286.5sechl20.4575.622.7823.77.143.91.18
IC 4393r30E287.0$\exp$l20.3075.612.8829.85.085.80.99
ESO 581-006r30E386.5$\exp$l21.3355.811.0718.23.615.61.11
ESO 583-008r30E387.0$\exp$i20.8456.226.7014.06.653.81.81
UGC 10535r25E288.0sech2i21.5041.420.5711.45.664.32.14
NGC 6722r10E386.5sechl19.4786.030.7721.67.736.22.22
ESO 461-006r60E387.5$\exp$l20.9661.623.3721.18.013.81.44
IC 4937g20E188.5sech2l22.7374.910.4193.813.045.60.78
IC 4937r30E388.0sech2l21.2983.511.6132.14.465.90.82
IC 4937i20E188.5seche19.9278.110.8638.65.377.21.00
ESO 528-017g30E386.5$\exp$l21.9857.622.5420.58.023.51.37
ESO 528-017r60E386.5$\exp$l21.2855.121.5619.97.792.71.06
ESO 528-017i30E386.5sechl20.8950.019.5722.68.853.21.25
ESO 187-008r30E385.5$\exp$l20.7750.013.6515.14.124.41.20
ESO 466-001i40E387.0$\exp$e19.5252.623.7613.05.878.23.70
ESO 189-012g60E387.0$\exp$l21.5056.229.8126.814.213.82.02
ESO 189-012r30E386.5$\exp$l20.7156.529.9720.610.933.41.80
ESO 189-012i20E387.0sechl20.4454.729.0122.712.043.21.70
ESO 533-004r20E188.0$\exp$l20.3268.411.1233.15.387.71.25
IC 5199g30E386.5$\exp$l21.5564.120.4419.96.355.61.79
IC 5199i30E386.5$\exp$l19.9757.618.3719.96.354.91.56
ESO 604-006r30E390.0sechl21.2270.634.5427.913.653.81.86


   
4.2 Comparison of different methods

The different fitting methods were independently developed within two diploma theses (Lütticke 1996; Schwarzkopf 1996). The quality of the data basis for each project was the same. From the sample presented here there were five objects in common. These are used to compare the two methods and determine the quantitative difference of the derived parameters.

 

 
Table 3: Comparison of the different determined parameter sets for the same galaxy images
galaxyf(z)izh $R_{{\rm co}}$
  [ $\;\hbox{$^\circ$ }\;$] [ $\;\hbox{$^{\prime\prime}$ }\;$] [ $\;\hbox{$^{\prime\prime}$ }\;$] [ $\;\hbox{$^{\prime\prime}$ }\;$]
ESO 321-010 r { sech88.05.021.664.8
$\exp$ 88.05.225.964.1
ESO 461-006 r { $\exp$87.53.821.161.6
$\exp$87.53.616.261.2
IC 4937 r { $\exp$86.56.126.883.5
sech89.57.027.475.6
ESO 189-012 r { $\exp$86.53.219.356.2
$\exp$88.03.621.356.9
ESO 604-006 r { sech88.54.139.867.7
sech289.53.224.573.4


Table 3 shows the results for the five images. The mean deviation in the determined inclination is $\approx \! 1 \hbox{$^\circ$ }$ and 12.4% for the scaleheight (ranging from $5.0\%-26.6\%$) whereas for three images different functions for the z distribution were used. The mean difference for the radial scalelength is 20.6% $(2.1\%-47.2\%)$ and 4.2% for the determination of the cut-off radius.

A subsequent analysis shows that it is not possible to ascribe the sometimes quite large discrepancies to the quality of the individual method. It turns out, that the main problem is the non-uniform determination of the fitting area. The intrinsic asymmetric variations of a real galactic disk compared to the model enforce a more subjective restriction of the galaxy image to the fitting region, whereby for example one has to exclude the bulge area and the dust lane.

This finding is in agreement with the study of Knapen & van der Kruit ([1991]) who compared published values of the scalelength and find an average value of 23% for the discrepancy between different sources. As already mentioned by Schombert & Bothun ([1987]) the limiting factor for accuracy of the decomposition is not the typical S/N from the CCD-telescope combination nor the errors in the determination of the sky background, but the deviation of real galaxies from the standard model.

4.3 Comparison with the literature

In our former study (Paper I) with an earlier method to adapt Eq. (4), 20 of our 45 galaxy images have already been used. We decided to re-use them in this study to get models for as many galaxies as possible in a homogeneous way. Additionally, Paper I only presents the best fit values for the isothermal model, and uses a different definition of the cut-off radius.

Only three galaxies are in common with the sample of de Grijs ([1998]): ESO 564-027, ESO 321-010, and ESO 446-018. The mean difference for the scalelength is 10.4% (ranging from $1.2\%~-~20.3\%$) and for the scale height (normalised to the isothermal case) 4.0% $(0.0\%-8.2\%)$. For the remaining galaxies there are no models in the literature.

4.4 Model limitations

Our model only represents a rather simple axisymmetric three-dimensional model for a galactic disk, consisting of an one component radial exponential disk with three different laws for the density distribution in the z-direction and a sharp outer truncation. Therefore it does not include additional components, such as bulges, bars, thick disks, or rings, and cannot deal with any asymmetries. Features like spiral structure or warps are not included, whereas Reshetnikov & Combes ([1998]) multiply their exponential disk by a spiral function introducing an expression to characterise an intrinsic warp depending on the position angle outside of a critical radius.

The choice of our fitting area tries to avoid the dust lane, possible only for almost edge-on galaxies, as a first step to account for the dust influence (cf. Sect. 4.4.2). Examples of models including a radiative transfer with an extinction coefficient $\kappa_{\lambda}(R,z)$ can be found in Xilouris et al. ([1999]). However, introducing more and more new components and features automatically increases the amount of free parameters. Therefore we restricted our model to the described six parameters, to obtain statistically meaningful characteristics for galactic disks.

In the following we demonstrate that a simple disk model omitting the bulge component and the dust lane give indeed reasonable parameters.

4.4.1 The influence of the bulge component

We have studied the influence of the bulge for some of our objects including the earliest type galaxy in our sample (ESO 575-059) presented here. We have subtracted our derived disk model from the galaxy and then tried to find the best representation for the remaining bulge by a de Vaucouleurs r1/4 or an exponential model. Taking the slope of the vertical profile at $R\!=\!0$ and a fixed axis ratio, we have constructed the 2-dimensional model of the bulge. In agreement with Andredakis et al. ([1995]) we find, that bulges of early type galaxies are better fitted by an exponential profile than by a r1/4. Figure 3 shows the resulting vertical and radial cuts for ESO 575-059 together with the models.

  \begin{figure}\par\psfig{figure=ds1758f3.eps,width=8cm,angle=0}\end{figure} Figure 3: Minor and major axes profile for ESO575059r15E2 (solid lines) together with the best disk model (dotted), the best bulge model (dashed), and the resulting combined profile (dotted-dashed)

Despite the deviation between $R\!=\!10''$ and $R\!=\!20''$ which could be attributed to an additional component (inner disk or bar), we do not find any evidence for changing our disk model due to the influence of the bulge.

Therefore we conclude, that it is possible to nearly avoid any influence of the central component by fitting outside the clearly visible bulge region.

   
4.4.2 The influence of dust

Dust disturbs the light profile by a combination of absorption and extinction and the net effect has to be calculated by radiation transfer models. Therefore it is not obvious that outside the "visible'' dust lane, which is excluded for the fitting area as a first step, the dust will not play a major role in shaping the light distribution. Xilouris et al. ([1999]), Bianchi et al. ([1996]), de Jong ([1996a]), and Byun et al. ([1994]) have recently addressed this problem in more detail. Although they investigated the influence of the dust on the light distribution by quoting best fit structural parameters for the star-disk as well as the dust-disk, they did not quantify the influence on the star-disk parameters derived by standard fitting methods without dust. Even Kylafis & Bahcall ([1987]) state within their fundamental paper on finding the dust distribution for NGC 891 that "in order to avoid duplication of previous work, we will take... (the values for the star-distribution estimated by standard fitting methods)''.

We checked the influence of the dust on our determined parameter set, by studying simulated galaxy images with three different dust distributions. These dusty galaxies were kindly provided by Simone Bianchi who calculated images with known input parameters for the star and dust distribution with his Monte Carlo radiative transfer method (Bianchi et al. [1996]). We have defined a worst, best, and transparent case, according to the dust distributions presented by Xilouris et al. ([1999]), of our mean stellar disk ( $R_{{\rm co}}\!=\!2.9$, $h/z\!=\!4$, and f1(z)). The worst case is calculated with $\tau_{\rm R}=0.51$, $h_{{\rm d}}/h_{\rm *}=1.55$, and $z_{{\rm d}}/z_{\rm *}=0.75$, the best case with $\tau_{{\rm R}}=0.20$, $h_{{\rm d}}/h_{\rm *}=1.08$, and $z_{{\rm d}}/z_{\rm *}=0.32$, and a transparent case without dust. To be comparable we used the same method for selecting the fitting region by masking the "visible'' dust lane and reserving a typical area for a possible bulge component using mean values for the transparent case. In contrast to our standard procedure we do not restrict the inclination range from the appearance of the dust lane, but specify the best SQ model in the range $i=80\hbox{$^\circ$ }\!-\!90\hbox{$^\circ$ }$ in Table 4. For the models marked with a "$\star$'' we pretend the correct input inclination. Table 4 demonstrates, that even for the worst case we are able to reproduce the input parameters within the range of the typical 20% error discussed in Sect. 4.2. It should be mentioned that for each case we overestimate the input scalelength and -height, whereas the determination of $R_{{\rm co}}$ does not depend on the dust distribution. The implication on the distribution of the ratio $R_{{\rm co}}/h$ will be discussed in a forthcoming paper.

 

 
Table 4: Comparison of the results for the stellar disk (input: $z_{\rm *}\!=\!8.4$, $h_{\rm *}\!=\!33.7$, and f(z)= exp) from our fitting procedure for the three different dust distributions
case $i_{\rm in}$fn $i_{\rm out}$$z_{\rm *}$$h_{\rm *}$ $\Delta z_{\rm *}$ $\Delta h_{\rm *}$
 [ $\;\hbox{$^\circ$ }\;$] [ $\;\hbox{$^\circ$ }\;$] [ $\;\hbox{$^{\prime\prime}$ }\;$] [ $\;\hbox{$^{\prime\prime}$ }\;$] [%] [%]
trans.87.5187.58.534.01.20.9
best 85.0184.58.535.01.23.9
best 87.5186.58.635.52.45.3
best 90.0187.08.736.03.66.8
best$\star$90.0190.08.835.04.83.9
worst85.0285.08.939.96.018.4
worst 87.5286.08.839.14.816.0
worst90.0288.09.739.415.516.9
worst$\star$90.0290.09.839.116.716.0


4.5 Comments on individual galaxies

Trying to adapt a simple, perfect, and exact symmetric model to real galaxies always implies a compromise between the degree of any deviation and the final model (Sect. 4.6). The following list will provide some typical caveats found during the fit procedure which will characterise the quality of the specified model for individual galaxies.

ESO 112-004: warped, asymmetric, central part slightly tilted compared to disk, after fitting still remaining residuals.

ESO 150-014: slightly warped, minor flatfield problems.

NGC 585: remaining residuals.

ESO 244-048: possible two component system, slope of inner radial profile significantly higher than of an outer one, final model fits the inner parts.

NGC 973: one side disturbed by stray light of nearby star, seems to be radially asymmetric, remaining residuals.

UGC 3425: superimposed star on one edge.

NGC 2424: model does not fit very well without obvious reason.

ESO 436-034: strong bulge component, possibly barred, hard to pinpoint final model, remaining residuals.

ESO 319-026: outer parts show u-shaped behaviour, remaining residuals, therefore large ( $\pm 2 \hbox{$^\circ$ }$) difference in inclination angle.

ESO 321-010: u-shaped, no clear major axis visible, therefore uncertain rotation angle, bar visible, bulge rotated against disk.

NGC 4835A: strong residuals.

ESO 446-018: the different sides of the disk are asymmetric visible in radial profiles and on the contour plot.

IC 4393: similar to NGC 4835A.

ESO 581-006: galaxy shows typical late type profile, $R_{{\rm co}}$ questionable, but nevertheless final model seems to fit well.

ESO 583-008: disturbed by superimposed star, shows warp feature and a bar structure, $R_{{\rm co}}$ questionable, remaining residuals.

UGC 10535: one side slightly extended.

NGC 6722: only one side observed, bulge rotated against disk, barred, strongly disturbed by dust absorption, radial extension visible, therefore $R_{{\rm co}}$ should be treated with caution.

ESO 461-006: minor flatfield problem seems to cause asymmetry, although model looks fine.

IC 4937: similar to NGC 6722, dominating bulge, small disk, model significantly different compared to the i and r image, model possibly hampered by strong dust lane.

ESO 578-025: bar visible.

ESO 466-001: maybe two components, final model represents only inner part, outer part clearly different from normal disk component.

ESO 189-012: slightly warped.

ESO 533-004: similar to NGC 4835A, model fits the whole galaxy, leaving more or less no bulge component.

IC 5199: slightly radial asymmetric.

ESO 604-006: only one side observed, bar structure visible.

   
4.6 Comments on some rejected galaxies

The model limitations described above constrain the application of our fitting process. Therefore we had to exclude about 20 galaxies from our original sample. They all show significant deviations from the simple geometry and an inclusion of their parameters obtained by forcing the model to fit the data will spoil the resulting parameter distribution. One larger group classified mainly as S0 galaxies (e.g. NGC 2549, ESO 376-009, NGC 7332, ESO 383-085, ESO 506-033) shows a completely different behaviour of the luminosity distribution in the outer parts compared to the other galaxies. They all show an additional component, mainly characterised as an elliptical envelope. This is already visible in the contour plot, but becomes even more evident in a radial cut parallel to the major axis. In these cases the usual common curved decline of the profile (e.g. ESO 578-025) is missing, and is replaced by a more or less straight decline into the noise level, sometimes even by an upwards curved profile. Fitting these luminosity distribution by our one component exponential disk with cut-off, will therefore naturally provide parameters qualitatively different compared to late type disks. This will be discussed in detail in a forthcoming paper.

Another group consists of galaxies dominated mainly by their bulges, whereas the disk is only an underlying component, partly characterized as having thick boxy bulges (Dettmar & Lütticke [1999]), e.g. IC 4745, ESO 383-005, although there are also pure elliptical bulges (e.g. ESO 445-049, NGC 6948).

In the case of ESO 383-048 and ESO 510-074 the radial profiles clearly indicate that a more complex model will be needed to fit these kind of multicomponent galaxies. Galaxies like UGC 7170 or ESO 113-006 were excluded due to their strong warps, which made it impossible to fit the model in a consistent way. Mainly late type galaxies such as ESO 385-008, IC 4871, UGC 1281, or ESO 376-023 show a too patchy and asymmetric light distribution, that any attempt to fit the profiles will give only very crude, low quality parameters. UGC 11859 and UGC 12423 were rejected due to their thin faint disks, which will maybe overcome by taking new images with longer integration times to get a higher signal-to-noise ratio, whereas NGC 5193A is completely embedded into the surface brightness distribution of its near companion.

Acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft, DFG. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. We have made use of the LEDA database (www-obs.univ-lyon1.fr). The authors wish to thank Simone Bianchi, who kindly provided us dusty-galaxy images produced with his radiative transfer code.


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