4 Isophotal shapes and azimuthally averaged brightness profiles

4.1 Isophotal shapes

The radial brightness variations were analysed by fitting ellipses to the isophotes of the surface brightnesses. For this purpose an IDL program was developed, based partially on the Fourier-technique as used by Kent ([1983]), Jedrzejewski ([1987]) and Rauscher ([1995]). The basic difference is that we first determined a good initial fit by choosing a fixed brightness level and finding the best ellipse to fit the image points at this level (method I), and only then proceeded with the Fourier-technique (method II).

I) Our IDL routine starts by choosing image points corresponding to a fixed brightness level. These points are fitted by an ellipse, defined by its center $(x_{\rm c},y_{\rm c})$, major and minor axis radii (a, b), and major axis position angle. The fitting is done with a general least squares iteration that minimizes the sum

$$\begin{array}{lp{0.8\linewidth}} g = \sum (x_{\rm p} - x_{\rm e})^2 - (y_{\rm p} - y_{\rm e})^2,\\ \end{array}$$

where $x_{\rm p}$ and $y_{\rm p}$ are the image points, and $x_{\rm e}$ and $y_{\rm e}$ are the corresponding nearest points in the fitting ellipse. The fit is further polished by accepting only those pixels within 3 standard deviations from the fitted ellipse. In the program two options to select the isophote pixels are available: automatic selection within the range of the tolerance (typically $2 \%$ of the specified brightness level) or manual selection by indicating the acceptable parts of the image with mouse. The manual option is valuable for example when fitting isophotes which are strongly affected by the spiral arms in the outer parts of the galaxy. Also, in the inner parts where fewer pixels fall into the tolerance interval, we use bilinear interpolation to create at least 50 points corresponding to the given surface brightness level.

II) Once a good initial fit is obtained as described above, the program proceeds with the intensity-fit technique of Jedrzejewski ([1987]). It thus fixes the semimajor-axis a and fits the intensity along the ellipse with the function

$$I=I_0 {+} A_1\sin(E) {+} B_1\cos(E) {+} A_2\sin(2E) {+} B_2\cos(2E),$$

where E is the eccentric anomaly used to parametrize the points of the ellipse. Intensity in these sample points is calculated with bi-linear interpolation. The ellipse parameters $x_{\rm c}$, $y_{\rm c}$, b and PA are iteratively corrected according to the coefficients A₁, A₂, B₁ and B₂ and the intensity gradient along the major axis, until a good fit to I = constant is found. The criterion to stop the iteration is that the largest harmonic amplitude is less than $4 \%$ of the rms residual intensity along the ellipse. If no convergence is found, as often happens in the outer portions of the image where the intensity gradient is small, the ellipse parameters of the initial fit are taken as final values.

The error estimate for the fit parameters is made in the same manner as in Rauscher ([1995]), namely from the errors in harmonic coefficients and the rms residual of the intensity. This error estimate is also used in the case the intensity-fit (method II) fails to converge, by applying Rauscher's error formulas for the initial fit obtained by method I.

We have tested our algorithm by measuring the position angle and ellipticity variations for NGC 4303 (using our high-resolution IR-data) and by comparing them with the published IR measurements by Rauscher ([1995]). We obtained reasonable ellipticities and position angles with small error bars for the whole image region (r < 30 arcsec), whereas Rauscher reports good measurements only for the inner 3 arcsec, where our measurements are identical to their values. The reason for the difference is that, in contrast to our combined methods, Rauscher's method fails when no good initial guess is available. Comparison to the IRAF program "ellipse'' was made using the B-band image of NGC 5905: similar results were obtained except that "ellipse'' gave slightly smaller error bars than our program. The program "ellipse'' is also based on the same Rauscher's method. We also tested our algorithm with the B and H-band data of the weakly barred galaxy IC 4214 (Buta et al. [1999]), obtaining practically identical results as compared to the program SPRITE developed at the University of Alabama. The isophotal parameters are shown in Fig. 2. Also marked are the PA and the ellipticity $\epsilon$ = 1-b/a, corresponding to our visual estimates of galaxy orientations. In most cases the visual estimations correspond to the averaged PA values in the outer parts of the disks, but there are a few exceptions for which the visually estimated value is different. For Kar 331 A this difference can be explained by a bright object in the galaxy area, whereas for Kar 296 A and NGC 5905 open spiral arms affect the automatic measurement.

4.2 Azimuthally averaged profiles

For computing the intensity profiles the standard approach was followed. Instead of fitting ellipses to the isophotes, fixed inclination and position angle were used and the mean intensities were calculated in elliptical annuli evenly distributed in radius. The algorithm included automatic elimination of foreground stars. Identification of the data points contaminated by star light was based on pixel values exceeding the mean value in the annulus at least by 2-4 standard deviations, depending on the image.

Interacting galaxies have the additional problem that they are sometimes superimposed by the nearby galaxies. Therefore the fitting routine includes the possibility to compensate the affected data points with the mean values of the good pixels in the same annulus. The excluded area is specified by radius and position angle intervals.

Figure 3 shows the measured azimuthally averaged profiles. No corrections for Galactic or internal extinction or seeing were applied. The sources of error in the surface brightnesses were taken to be the background noise and the global variations in the sky level, taken to be the difference between the minimum and maximum sky brightnesses divided by four. The latter is somewhat smaller than used for example by de Jong ([1996b]), but on the basis of our tests with synthetic data it is still probably an overestimation for the true uncertainties due to sky variations.

In the literature deep intensity profiles were found for three galaxies common with our sample: NGC 5905, Arp 87 A and Kar 302 A. For NGC 5905 our profiles were identical with the B, V and R profiles by Wozniak et al. ([1995]), whereas in the I-band there was a half magnitude shift, which may be related to the large amount of processing of the I-image by them: Wozniack et al. had strong interference fringes in the I-frame so that additional correction frames had to be used. The comparison was made to the surface brightness level of $24 \ \rm mag \ arcsec^{-2}$ in the B-band. For Arp 87 A our V-profile was identical with that by Gavazzi & Randone ([1994]) who had the same limiting surface brightness as we did. Also, for Kar 302 A our I-profile was identical with the profile by Heraudeau & Simien ([1996]) to the distance of about 80 arcsec, after which there was a half magnitude shift. This shift is most probably caused by uncertainties in the sky subtraction. We note that both Arp 87 A and Kar 302 A were observed during the run in Calar Alto in 1992, for which campaign an additional correction of 0.6 mag to the zero points was applied in Paper I. The correction was done by comparing with the magnitudes in RC3, which are generally based on lower quality data than the above CCD- observations by Wozniak et al. and Gavazzi and Randone. Therefore the correction applied in Paper I should NOT be applied. In the literature intensity profiles can be found also for Arp 298 A, Kar 125 A and B and NGC 5908 (Kotilainen et al. [1993]; Marquez & Moles [1996]; de Robertis et al. [1998]), but as their data occupy only the high surface brightness parts of the profiles no comparisons were done.

Table 2 provides information similar to that generally given in galaxy catalogues. R₂₅ is the radius and m₂₅ the integrated apparent magnitude measured within the isophote having a surface brightness of $\mu_B$ = 25 mag arcsec^-2. The parameters $r_{\rm e}$ and $\mu_{\rm e}$ are the effective radius and surface brightness in units of arcsec and mag arcsec^-2, respectively. Also shown are the total integrated apparent magnitudes and the B-band absolute magnitudes after correcting for redshift and Galactic extinction. The standard correction terms are from RC3. Internal extinction corrections are much more difficult to assess and were not considered. The total magnitudes were in most cases derived by extrapolating the exponential profiles to infinity. However, as some of the galaxies had flat regions outside the exponential parts of the profiles this extrapolation was not done. Note that the total magnitudes in Table 1 are not necessarily the same as in Paper I (its Table 4), where the magnitudes were measured by single apertures without correcting for the superpositions of the nearby galaxies or extrapolation to infinity.

The error bars for $\mu_{\rm e}$ were estimated from the global variations in the sky level and by taking into account the background noise, whereas for R₂₅ and $r_{\rm e}$ they were estimated from the uncertainties in the surface brightnesses. The error limits for integrated magnitudes include the magnitude zero-point errors and the background noise.

The R₂₅ radii in the B-band were compared with the values given in RC3. Our measurements did not show any systematic shift as compared to the RC3 values, but measurements for individual galaxies in many cases disagreed, typically by $5-30\%$. One of the cases with large difference was NGC 5905, for which we obtained R₂₅ = 89'' whereas RC3 gives R₂₅ = 119''. The value we obtained is reasonable, because our profile was identical with that measured by Wozniak et al. ([1995]) in the region common with their measurement. In general CCD-images are a more accurate way of measuring the bulge and disk parameters than the aperture photometry often used in RC3.