4.1.2 Fitting of Ellipses

The fitting of ellipses to the galaxy images is done in a three step procedure. First a harmonic expansion along concentric circles is performed. Second the residuals from this expansion are used to flag additional pixels. Third the actual ellipse fit is performed, using another harmonic expansion to calculate an initial guess.

The harmonic expansion is done as follows. The user manually determines the coordinates of the center of the galaxy. Along concentric circles with this center a 6 term harmonic series is fitted to the intensities I. The series is given by

$$I(r,\varphi) = I_0^{({\rm c})}(r) + \sum_{n=1}^6 \left[ A_n^... ...varphi) + B_n^{({\rm c})}(r) \cos(n\varphi) \right] \enspace ,$$ (4.1)

where r is the radius and $\varphi$ is the position angle (measured from the CCD x-axis counterclockwise), and where the superscript c denotes that we are fitting along circles. Actually, discrete radii r_i are used; they are calculated as $r_i = r_{\rm min} \cdot s^{i-1}, \, i = 1,\ldots,N_{\rm max}$. In this way, the radii are equally spaced in $\log r$. $r_{\rm min}$ was set to 0.3 pixels, and the scaling factor s was usually set to 1.1, giving the radius sequence 0.3 pixels, 0.33 pixels, 0.363 pixels, .... $r_{\rm min}$ has to be as small as 0.3 pixels in order to get a good fit of the central pixels of the galaxy. The maximum radius number $N_{\rm max}$ is basically determined from the condition that 60% of the circle needs to be within the image. For the pixels outside the maximum radius only the intensity is fitted. The pixels contaminated by other objects are excluded from the fit, cf. above.

A residual image is calculated by simply subtracting the fit from the original image. This residual image will normally be flat, since the only thing left is what the harmonical terms of order higher than 6 would account for. In the cases where the galaxy has a very strong disk, some residual can be seen, though.

This residual image is used in the second step to flag all pixels that deviate by more than 5 sigma. This gives an additional list of pixels to exclude from future fits. The pixels flagged in this process could be due to cosmic-ray-events that extend over several pixels, unremoved remanence or overflow stripes, and a very strong disk in the galaxy. Of course one wants to keep the disk pixels, so in these cases it is necessary to specify a region where the 5 sigma flagging should not be done.

In the third step, the above harmonic expansion along concentric circles is done again, this time also excluding the additional pixels found in the above second step. From the resulting Fourier coefficients initial guesses on the center of the ellipses $x_{\rm c}(r),y_{\rm c}(r)$, the ellipticity $\varepsilon(r)$, and the position angle PA(r) are calculated. Then at each equivalent radius $r_i = \sqrt{a_i b_i}$ an ellipse is fitted to the image. The same values of r_i as above are used. For the pixels outside the maximum radius, the mean of the last three ellipses is used to define the center and shape of the ellipses with larger radii, and only the intensity is fitted. The fit is iterated 20 times. In each iteration step there is a limit on how much the center, ellipticity, and position angle can change from the values they had in the previous step. These limits are imposed to safeguard the iteration from running wild, and most of the time they work fine. However, if the galaxy has a large change of for example center position (such as R269) or position angle (also known as isophote twist), the user will need to increase these limits.

A residual image is calculated by subtracting the ellipse fit from the original image. The structure seen in this image is per definition how the galaxy deviates from elliptical isophotes. This is quantified by fitting a 6 term harmonic series along the fitted ellipses,

$$I(r,\varphi) = I_0^{({\rm e})}(r) + \sum_{n=1}^6 \left[ A_n^{... ...varphi) + B_n^{({\rm e})}(r) \cos(n\varphi) \right] \enspace .$$ (4.2)

From $A_n^{({\rm e})}(r)$ and $B_n^{({\rm e})}(r)$ we calculate `normalized' coefficients s_n(r) and c_n(r) as

$$s_n(r) = \frac{ A_n^{({\rm e})}(r) }{ r \cdot \frac{dI(r)}{dr... ... \cdot \frac{dI(r)}{dr} } \, , \quad n = 1,\ldots,6 \enspace .$$ (4.3)

s_n(r) and c_n(r) have the advantage of measuring the relative radial deviation of the isophotes from ellipses. The names of s_n(r) and c_n(r) reflects the relation to the sine and cosine terms of the Fourier expansion, respectively. One of the most interesting of these coefficients is c₄(r), since it is an indicator of whether the galaxy is disky (c₄(r)>0) or boxy (c₄(r)<0) (Carter 1987; Bender et al. 1989; Peletier et al. 1990). The first and second order coefficients will in general be zero since the expansion is done in the residual image where the best fitting ellipses have been subtracted.

In the ellipse fit we keep the center ( $x_{\rm c}(r)$ and $y_{\rm c}(r)$) and the shape ( $\varepsilon(r)$ and PA(r)) as free parameters, since these quantities are not constant with radius in real galaxies. However, at some point in the profile the signal-to-noise becomes too low to keep the center and the shape as free parameters. The last radius where the center is free, $r_{\rm free-center}$, is determined as the point where the uncertainty on the first order Fourier coefficients from the harmonic expansion along concentric circles is below 0.02 for Gunn r and Johnson B, and 0.04 for Johnson U. Likewise, $r_{\rm free-shape}$ is determined using the uncertainty on the second order coefficients. The condition $r_{\rm free-center} \le r_{\rm free-shape}$ is imposed, since otherwise one could easily get overlapping ellipses. In general, however, one gets $r_{\rm free-center} = r_{\rm free-shape}$, and one can therefore speak of just one radius, $r_{\rm free}$. Beyond the last free radius, the parameters are fixed at the mean value of the last three free radii.

The actual fitting of ellipses is done as described above by starting with suitable default values for the different parameters that control the fit. The residual image from the ellipse fit is then inspected as well as the text output from the fitting task (it might for example report overlapping ellipses at some radii). The parameters and the object flagging is then `tuned' until a good fit is obtained. This is described in more detail in Sect.  (p. ).

The method of simply excluding from the fit the pixels that are contaminated by signal from other objects does not work if too large a fraction of a given radius is excluded by this. This is for example the case where a neighbor galaxy is sufficiently close to the galaxy we want to fit. In these cases we fitted the two galaxies iteratively, cf. Sect. . An example is the central field (field 00, see the image on p. ), where the two bright galaxies R256 and R269 were fitted iteratively. When these two galaxies had been successfully fitted, models of the two were subtracted from the original image, and the remaining 8 program galaxies in the central field were fitted in the normal way using this image. The same models of the R256 and R269 were subtracted from the neighboring fields. In this way, the galaxies in the overlap region between field 00 and a neighboring field were in any case fitted to an image where models of R256 and R269 had been subtracted. An example of this is R255 and R273, which are located both in field 00 and in field 12 (see image on p. ). Note, that the paper reproduction of the field 00 image (p. ) might not convey the impression that there is signal from R256 and especially the cD galaxy R269 all over the image. However, that is easily seen when one views the image on screen, and that is also seen in the derived surface photometry. Another example of galaxies that needed this kind of iterative fitting is R336/R337 (p. or ).

The above method of iteratively fitting ellipses to two objects was also used when a bright star was close to the center of the program galaxy in question. It worked well. In the cases where this method was used, the separation between the centers of the star and the galaxy was typically 20 pixels (10''). An example is the galaxy R194 which has a star 21 pixels (11'') from the center - see the image on p. . In our images, the star had a peak intensity of about four times larger that of the galaxy. Another example is R308 (p. ).

The output from the ellipse fit is the radial profiles of intensity (in ADU) $I_0^{({\rm e})}(r)$, center $x_{\rm c}(r),y_{\rm c}(r)$, ellipticity $\varepsilon(r)$, position angle PA(r), and the normalized Fourier coefficient s_n(r) and c_n(r) ( $n = 1,\ldots,6$); as well as the corresponding uncertainties, also as function of radius. The position angles are afterwards transformed so they are measured exactly as north through east, which is the standard. In addition the PA-profiles that cross $0^\circ$ or $180^\circ$ are made look `continuous' by adding or subtracting $180^\circ$ at certain points in the profile. These corrections to the PA-profiles are described in Sect.  (p. ).