4 Photometric decomposition: The method

4.1 Decomposition of close to edge-on systems

In this case, the analysis procedure follows two modes, depending on whether the disk is completely embedded in the bulge or whether it is dominant in the outer parts. As already mentioned, the decomposition method applied here is a modified version of the method described in Scorza & Bender (1995). In the latter work it was found that the modeling of twelve early-type galaxies required disk models with profiles departing from the exponential behaviour. These were constructed by adding several exponential functions, most frequently two exponentials were required. In the present work we have developed an automatized method to determine the optimal flux intensities of the disk models without constraints on the shapes of the disk profiles. As a first step, the initial values of an exponential disk model were guessed as described in Scorza & Bender (1995). The models are described by a central surface brightness SB₀, a scale length $r_{\rm d}$ and an inclination angle $\it i$. After subtraction of this input disk model (convolved to the appropriate seeing) from the galaxy image, an isophote analysis of the remaining bulge was carried out. This analysis yielded, among several other parameters, the a₄/a amplitudes along the major axis. The radial flux of the input disk model was then corrected according to the amplitude of the remaining a₄/a signature of the bulge via:

$$F_{\rm b} = F_{\rm a}(r).(1.0+a_4/a(r)/{\rm const})$$ (1)

where $F_{\rm a}$ is the initial flux, $F_{\rm b}$ the corrected one and r the radius. With this corrected flux profile, a new disk model was constructed and subtracted from the galaxy image. The procedure was repeated until the isophotes of the bulge show a₄/a $\sim 0$. If it was found that the iteration did fail to converge properly, an input disk model with either a higher or a lower surface brightness was used.

Simulations of the decomposition procedure with perfect galaxy models showed that both input disks with too high or too low surface brightness did converge to the true disk. However, the convergence from brighter input disk models was in general better than from fainter input disks.

In the present version of the method the surface brightness profile of the disk is left free and can take any shape, provided that after disk subtraction the remaining bulge shows elliptical isophotes. The method described so far gave satisfactory results for objects with embedded disks (case I in Sect. 3). In case of a boxy bulge (case I.b) the disk profile was modified in such way that after disk subtraction, the remaining bulge showed also boxy isophotes and therefore negative a₄/4 values (e.g. see the case of NGC 3818, Fig. A15 in the Appendix).

If the disk was found to dominate at large radii (case II), the iteration procedure was similar, however, the initial disk parameters had to be estimated more carefully. This was done in two steps. First, a disk model was constructed following the above procedure. This model failed to produce a convincing solution in the outer disk dominated parts (specially the surface brightness profile) but allowed to estimate the bulge and disk profile in the inner parts, where the bulge still dominates. The bulge profile was then fitted by an r^1/4 model and extrapolated out to radii where the disk dominates. The such constructed bulge model was subtracted from the galaxy and the remaining disk was used as input disk for a further iteration. In this way, a satisfactory decomposition was achieved.

Figure 1 illustrates this procedure for NGC 4564. In Fig. 1a we show the surface brightness profile (uppermost panel) and the a₄ and ellipticity profiles (middle and lower panel) before (filled circles) and after (crosses) subtraction of a first (too small) disk model. Although the subtraction of this disk model yields a bulge with vanishing a₄/a, the brightness profile of the bulge still shows signatures of the disk: mainly a hump at radii > 23 arcsec (2.5 in the a^1/4 scale) and ellipticity values which continue increasing at large radii. After having modified the bulge profile and having constructed and subtracted an r^1/4 bulge model, a second input disk was obtained. With the latter, a better solution (shown in Fig. 1b) was achieved. The change in the ellipticity profile is noticeable (lower panel in Fig. 1b): the bulge has lower ellipticity when compared with the previous solution.

  

Figure 1: Surface brightness, a4/a*100 and ellipticity profiles of NGC 4564 a) before (filled circles) and after (crosses) the subtraction of a too small disk and b) before and after subtraction of a brighter and larger disk (see text)

4.2 Decomposition of close to face-on barred-systems

As already mentioned in Sect. 3, a detailed decomposition of these systems is not possible by means of the present method. However, the bar hypothesis infered from the behaviour of the Fourier coefficients can be tested after subtraction of a bulge model and examination of the residual structures. We follow here the same procedure applied in Scorza & Bender (1995) to recover the embedded bar in NGC 4660. As a first step, the decomposition method described above was applied. This yielded an approximated bulge profile from which a bulge model having constant position angle and ellipticity was constructed. This bulge model was then subtracted from the galaxy image and the residual structures were analysed. Notice that in this case it is assumed that the observed twists and abrupt ellipticity changes are due to the embedded bar component and therefore the constructed bulge-model is free from these features. Although triaxiality can also lead to projected twists of the isophotes (see e.g. Benacchio & Galleta 1980) the bar hypothesis is more likely for isotropic oblate rotators, which is the case of the galaxies examined here. In all galaxies showing the typical behaviour of the Fourier coefficients described in Sect. 3 (case III), elongated barred-like structures could be recovered after the bulge-model subtraction. Examples are shown in Sect. 5.2.3.

4.3 Calibration and determination of characteristic parameters

Similar to Scorza & Bender (1995), the zero point calibration was done here by comparing integrated fluxes within apertures with values given in the literature (Poulain 1993). After calibration, we derived characteristic parameters from growth-curve-fitting: the half light radius $r_{\rm e}$, the surface brightness $SB_{\rm e}$ at $r_{\rm e}$and the total apparent magnitude $m_{\rm tot}$.All these parameters were derived for galaxies, bulges and disks separately. For the latter, both $r_{\rm e}$ and $SB_{\rm e}$ were corrected to face-on.

Because the disks analysed here often have surface brightness profiles which deviate from exponentials and central surface brightnesses are ill-defined (see Scorza & Bender 1995), the best way to characterize the disks is then by their mean effective surface brightness $\langle{SB}\rangle_{\rm e}$:

$$\langle{SB}\rangle_{\rm e} = m_{\rm tot} + 5.\log(r_{\rm e})+1.995.$$ (2)

However, given that throughout the literature the disks of late-type galaxies are parameterized by the extrapolated central surface brightness SB₀ and scale length $r_{\rm d}$, we have determined also for comparison purposes the SB₀ for the disks and a mean $r_{\rm d}$ value obtained from the fit of exponential growth curves to the disks (for details see Scorza & Bender 1995). In this case, the mean $r_{\rm d}$is representative of the dominant exponential part of the disk profiles. The SB₀ values were obtained directly from the central intensities of the disk models and corrected to face-on. Due to systematic effects, assumptions concerning bulge shapes, non-exponentiality of the disks and seeing, the errors for these parameters could not be estimated reliably for each case. In general, however, from exploring a range of models we have estimated the following rough errors: $\Delta SB_0$ $\simeq$ 0.2 and $\Delta r_{\rm d}$ $\simeq$ 20% to 30%.

The distances to the galaxies were determined from the radial velocities given in the literature (RC3 catalogue) and a Hubble constant of H₀=75 km s^-1 Mpc^-1). With these quantities, the absolute magnitudes and scale lengths of the disks in kpc were derived. The absolute magnitudes and the $\langle{SB}\rangle_{\rm e}$ were corrected for dust absorption in the galactic plane according to Burstein (1984). The magnitudes and surface brightness of the galaxy images obtained only in the R band were transformed to the V band following Peletier (1989):

$$V=R-(B-V)+\frac{(B-V)+0.013}{0.620}$$ (3)

where the (B-V) were taken from the RC3 catalogue.