Through three dimensional axisymmetric modelling of the stars and the dust in UGC 2048, we have been able to determine the parameters that describe the distribution of stars and dust in this galaxy. Fig. 3 shows the 2D image of
UGC 2048 in the I, V and B bands (top to bottom) with the model in the left half of the frame and the folded real galaxy in the right half of each panel. The model galaxy reproduces very well the real galaxy. We have verified this by taking the residuals between the observed galaxy and the model galaxy that we produced. Fig. 4 shows how the residuals in the I band are distributed throughout the galaxy's image, in terms of the absolute value of the percentage error. Different colours indicate areas with different error values. Red colour corresponds to areas with error less than , yellow colour corresponds to areas with error between and , and black colour corresponds to areas with error between and . White circles indicate the positions of the foreground stars. In the upper panel, a de Vaucouleurs profile was used for the bulge, while a Hubble profile was used in the bottom panel. The distribution of residuals in the B and V bands is similar to the one in the I band.
Fig. 4 has a lot more information than discussed above. First of all it gives us an idea of the quality of the fit that was done to the observed image. For the law bulge (top panel), the red colour (error less than in absolute value) covers of the total image of the galaxy, while the red and the yellow colours (error less than ) cover of the total image. Only of the total image has errors grater than . For the Hubble bulge (bottom panel), of the total galaxy image has error less than , while in of the galaxy's image the error is less than . In this case, of the total image has errors grater than . Considering the three-dimensional clumpiness and the spiral structure that the real galaxy may have, these numbers show on the one hand the goodness of the fit and on the other the validity of the 3D stellar and dust distributions of the model galaxy that we have used.
Another thing that these residual maps show us is a hint at the detailed structure and clumpiness of the real galaxy. The clumpiness in the 3D distribution of stars and dust is recognized by the regions of the galaxy where the observed surface brightness has a large difference from a smooth distribution. In other words, regions in the residual maps (between the observed galaxy and the smooth model) with high errors indicate that along the line of sight the 3D distribution of stars and dust has significant departures from the assumed smooth distribution. These departures could be due to spiral structure or just inhomogeneities. For example, one can see in Fig. 4 small black regions near the major axis of the galaxy that are caused probably by inhomogeneities, i.e., by local departures from the smooth 3D distribution. Similarly, relatively large black and yellow areas in the galactic disk away from the major axis may indicate the existence of spiral arms. Also, one can easily spot four "lobes" distributed symmetrically around the center of the galaxy in the bulge region. These features come from the fact that the projection of the bulge in the plane of the sky has a characteristic "box/peanut" shape and differs from the commonly used or Hubble profiles.
Yet another thing that comes from Fig. 4 is a direct comparison of the two types of bulge that we have used. It is evident that the bulge of the galaxy is described somewhat better with the law (top panel) than with the Hubble law (bottom panel) especially at the outer regions of the bulge.
We also want to stress from Fig. 4 the fact that the model stellar disk and the model dust distribution characteristics are not affected significantly by the different types of bulge that we have used. This is also evident from Tables 1 and 2, where the various derived parameters are shown. Apart from the different bulge characteristics, because two completely different functions are used, only small differences are seen for the parameters that describe the exponential distributions of the dust and the stars in the disk. This gives us the freedom to use any of these two types of bulges when the aim is to derive only stellar disk and dust distribution characteristics.
One may claim that the structure and the inhomogeneities in the 3D distribution of stars and dust in the galaxy make it practically impossible to find smooth distributions to describe the galaxy. This would certainly be the case if one tried to fit only a small part of the galaxy. Fortunately, by doing a global fit and taking into account the whole image of the galaxy, the model tries to cancel out these inhomogeneites and a mean distribution of the light emissivity is evaluated. As a result, there are regions in the image of the galaxy where the residuals are positive and regions where they are negative, distributed in such a way as to accomplish the highest possible symmetry between positive and negative values. This is shown in Fig. 5, where a histogram of the percentage relative errors between the folded observed image (in which the fit was done) and the model image is shown for the case where the de Vaucouleurs bulge is used.
Once the distribution of light in the galaxy was determined, we wanted to investigate how dust behaves if for the moment we neglect scattering and assume that only absorption takes place. To do this we fitted the model image to the galaxy image again, but now only the optical depth of the dust was left as a free parameter. It was found that the central optical depth of the dust dropped by about in all three bands. This indicates that scattering plays an important role in the determination of the galaxy characteristics and a serious underestimate of the opacity can be made if only absorption is considered. This conclusion was also reached by other authors (see, e.g. Bruzual et al. 1988, Di Bartolomeo et al. 1995, Corradi et al. 1996).
From the face-on central optical depth of the dust , and given a mean value of the scaleheight for all the filters, one can calculate the absorption coefficient in each band. We have found that and for the case where the Hubble () law is used. These values can be directly compared with the ratio of the extinction and are plotted as a function of the effective wavelength in Fig. 6. Solid triangles (stars) correspond to the values derived with our model for the case where the Hubble () law is used. The open circles are the values given by Rieke & Lebofsky (1985) for our Galaxy.
Finally, we would like to point out that all the characteristic lengths of the stellar and dust distribution have been found to be larger by about a factor of two compared to mean values for these lengths derived for a number of galaxies (see, e.g. Byun 1992). Thus, we caution the reader that the reported distance of 63 Mpc might be an overestimate by a factor of 2 from the true distance. In this case, where the distance is half of that reported above, the dust and the gas masses become and , leaving the gas to dust ratio unchanged and equal to 121.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998