3 Quantitative morphology

3.1 The parametric decomposition  

For the 2-D parametric fits, we have modeled the surface brightness distribution with a generalized exponential bulge (Sèrsic 1982; Sparks 1988):

$$I_{\rm b}\,(x,y) = I_{\rm e} \, \exp \left\{-\alpha_n \lef... ...^2} {(1-\epsilon_{\rm b})^2}}\, \right)^{1/n}-1\right]\right\},$$ (1)

plus an exponential thin disk:

$$I_{\rm d}\,(x,y)=I_{\rm d}(0)\exp\left[-\frac{1}{r_{\rm d}}\sqrt{x^2+ \frac{y^2}{\cos^2 \,i}}\right].$$ (2)

$I_{\rm e}$ and $r_{\rm e}$ are effective (half-light) values, $\epsilon_{\rm b}$ is the apparent bulge ellipticity, $\alpha_n$ is a constant relating the effective brightness and radius to the exponential values (see Appendix). x and y are in arbitrary units, with origin at the galaxy's center, and x along the major axis. The bulge is assumed to be an oblate rotational ellipsoid, coaxial with the disk, and its apparent eccentricity $e_{\rm b}$ is related to the intrinsic eccentricity $e_{\rm b}'$ by:  

$$\protect e_{\rm b} = e_{\rm b}'\sin i \;\; .$$ (3)

The structural parameters have been determined by fitting the photometric data to the model, convolved with a circular Gaussian seeing disk of appropriate FWHM, using a $\chi^{2}$ minimization. The fitted parameters are: the two surface brightnesses $I_{\rm e}$ and $I_{\rm d}(0)$ (or $\mu_{\rm e}$ and $\mu_{\rm d}$ when given in magnitudes), the two scale lengths $r_{\rm e}$ and $r_{\rm d}$, the bulge ellipticity $\epsilon_{\rm b}$, and the system inclination i. Because of the difficulty inherent in estimating a luminosity distribution of unknown form (bulge) which is partially embedded in another (disk), we did not attempt to fit the bulge index n explicitly (cf. Andredakis et al. 1995). Instead, for each galaxy the bulge was modeled with four different values of the exponent n: 1, 2, 3, and 4 ($n\,=\,4$ for a de Vaucouleurs bulge). We then assessed the ``quality'' of the results for each set of 2-D fits, and determined the value of n that gave the best all-round fit, thus assigning, in effect, a shape index to the bulge. Three quality indicators were considered in the assessment: the successful convergence of the process, the value of $\chi^{2}$, and the mean absolute residual.

When both J and K data were available, tests showed that bulge and disk decompositions and colors of the components were more stable when the two bands were fitted simultaneously. We also found no evidence for systematic scale length changes with wavelength of either component (cf., Evans 1994). Hence, when two bands were available, we performed a single fit keeping scale lengths, inclination, and bulge ellipticity the same for both bands, and letting the surface brightnesses of both components vary independently, but with constant color indexes $(J-K)_{\rm b}$ and $(J-K)_{\rm d}$.

The decomposition technique was extensively tested on a set of synthetic maps of axisymmetric bulge+disk distributions, covering a range of values for i, $\epsilon_{\rm b}$, and B/D ratio similar to ours. We adopted an exponential thin disk, and generalized exponential laws for the bulge. An appropriate amount of noise was added to simulate the typical signal-to-noise levels we achieve, and finally the models were convolved with a range of PSF's to simulate the effects of seeing. The 2-D decomposition method was always able to recover the true parameter set, within the estimated errors, independently of the starting point in parameter space.

We also performed a 1-D fit to the radial profile, obtained by averaging along elliptical annuli; center and position angle of the annuli were held fixed, while the ellipticity was allowed to vary with semi-major axis. In the 1-D case, the convolution of the model with the PSF requires i and $\epsilon_{\rm b}$ to be assigned a priori: as customary in 1-D techniques, $\epsilon_{\rm b}$ was fixed at 0 and i was determined from the outer J-band isophotes. The best-fit bulge index n was determined in the same way as for the 2-D models. 1- and 2-D techniques are compared in Sect. 4.1.

The lower left panels in Fig. 1 show the results of the 2-D decompositions in terms of cuts along the semi-major axis (K band). Table 2 gives the best-fit parameters, including the shape index n: the first line reports the values from the 2-D fit, the second reports the 1-D ones, and the third gives the values from the non-parametric decompositions (Sect. 3.2). Bulge and disk surface brightnesses are corrected to face-on values assuming optical transparency with C=1:  

$$\mu_{\rm d}^{\rm c} = \mu_{\rm d} -2.5C\log\, (\cos\, i) ,$$ (4)

$$\mu_{\rm e}^{\rm c} = \mu_{\rm e} -2.5C\log\, (1-\epsilon_{\rm b}),$$

and are denoted by $\mu_{\rm e}^{\rm c}(K)$ and $\mu_{\rm d}^{\rm c}(K)$ respectively. Table 3 lists the derived luminosities and bulge-to-disk ratios (B/D); the first line reports the 2-D parametric fit values, and the second gives the non-parametric values (see Sect. 3.2). The 1-D fixed-position-angle profiles together with Tables 2 and 3 are available in electronic form on the ftp node: sisifo.arcetri.astro.it in directory  ftp/pub/nir.

  

Table 2: Decomposition results: structural parameters

Line 1: 2-D parametric decompositions.
Line 2: 1-D parametric decompositions.
Line 3: K-band 2-D non-parametric decompositions.

3.2 The non-parametric decomposition  

Kent (1986) was the first to introduce a non-parametric (np) method to separate bulge and disk; recently this has been extended to the profile analysis of 2-D images by Andredakis et al. (1995). We propose here a new, completely 2-D technique. Assuming the apparent ellipticities of the two components, $\epsilon_{\rm b}$ and $\epsilon_{\rm d}=1-\cos\, i$, are known, let us consider the average surface brightness of an annulus of ellipticity, say, $\epsilon_{\rm d}$.For a sufficiently narrow annulus, A, the disk contribution will be $I_{\rm d}(a)$, where a is the semi-major axis. The total average brightness will be:  

$$\protect <I(a)\gt _{\rm d}=I_{\rm d}(a)+\frac{1}{S_{\rm A}}\int_{A}I_{\rm b} \,{\rm d}S$$ (5)

where $S_{\rm A}$ is the annular area. An analogous equation can be written for an annulus of ellipticity $\epsilon_{\rm b}$. The integral in Eq. (5) can be written as:  

$$\protect \frac{2S_{\rm A}}{\pi}\int_0^{\pi/2} I_{\rm b} \lef... ...2_{\rm d}}{\rho^2_{\rm b}}\sin^2\theta}\right) \,{\rm d}\theta$$ (6)

where $\rho_{\rm b}$ and $\rho_{\rm d}$ are the axial ratios of the components. We then obtain for any a, a system of integral equations:  

$$\protect \left\{ \begin{array} {lll} {\displaystyle I_{\rm ... ...\sin^2\theta} \right)\, {\rm d}\theta}. & \end{array}\right.$$ (7)

To solve the system we first compute $<I(a)\gt _{\rm b}$ and $<I(a)\gt _{\rm d}$ for two sets of annuli of ellipticities $\epsilon_{\rm b}$ and $\epsilon_{\rm d}$, respectively, each set sampling the entire light distribution. The two equations can then be solved iteratively for $I_{\rm b}$ and $I_{\rm d}$at each a, starting with $<I\gt _{\rm b}$ and $<I\gt _{\rm d}$ as initial guesses to evaluate the integrals.

When defining the two sets of annuli, a proper sampling of the semi-major axis is of crucial importance. A reliable estimate of the integrals in Eq. (7)  requires that each annulus of either set intersects with a sufficient number of annuli of the other set. If we constrain this number to be the same for every ellipse, the sampling is linear in $\log a$, i.e. the annuli become wider with increasing radius.

Results of the np decomposition technique are given in the third line of Table 2, and in the second line of Table 3. The non-parametric parameters are defined as follows. The bulge $r_{\rm e}$ is the semimajor axis of the ellipse enclosing half light; $\mu_{\rm e}$ is the bulge surface brightness at $r_{\rm e}$; $r_{\rm d}$ is the semimajor axis of the ellipse enclosing 0.264 of the disk light, as in the exponential case. Since the central disk is not tightly constrained by the np fits, $\mu_{\rm d}$ is the central brightness of an exponential disk having scale length $r_{\rm d}$ and the same total luminosity as the np disk. The np colors of the components are derived by the total component luminosities in the two bands.

3.2.1 Testing of non-parametric decomposition

The np decomposition algorithm was tested on the same set of synthetic maps used for the parametric tests. Since the method applies only to components with different apparent ellipticity, we only tested models with $\epsilon_{\rm b}$ at least 10% smaller than $\epsilon_{\rm d}$. Under these conditions, to within the noise level, the method always proved to be able to recover the true bulge and disk distributions given the correct values of the ellipticities.

When errors on the ellipticities are introduced, the shape of the resulting components is affected in a systematic way: the value of $\epsilon_{\rm b}$ affects mainly the inner part of the disk, while $\epsilon_{\rm d}$ determines the shape of the outer bulge. In particular, if $\epsilon_{\rm b}$ is underestimated the inner disk is too steep, whereas if the bulge is too elliptical the inner disk develops a hole which gets deeper as $\epsilon_{\rm b}$increases. If $\epsilon_{\rm d}$ is too small the outer bulge is too steep and viceversa. Although qualitatively these are expected trends, they are difficult to quantify in practice, since they depend on the shapes of the distributions, on the ellipticities, and on the seeing. Especially when the two components have similar ellipticities, even small errors on $\epsilon_{\rm b}$ and $\epsilon_{\rm d}$ (<0.1) perceptibly affect parameters such as the luminosity of the components or the B/D ratio, as well as the shape of the profiles.

3.2.2 Seeing corrections

Simulations show that, if no correction is applied, seeing significantly affects np decompositions. The effects are stronger in the inner parts, the main feature being a central hole in the disk, whose extension is proportional to the seeing width. In fact, the seeing makes inner isophotes rounder, causing the component of higher ellipticity to be depressed. The disk profile, however, can be corrected by extrapolating the outer points to the center.

To recover the intrinsic bulge profile, we introduced a second iterative algorithm which proved to be quite effective on synthetic maps. The effect of seeing is accounted for by defining a correction coefficient $k_{\rm b}(a)$, for each point of the bulge profile, as the ratio of the convolved bulge distribution and an estimate of the true one. Each decomposition is performed using $k_{\rm b}(a)$to degrade the assumed true distribution. After decomposition, the coefficients are recomputed by convolving the extracted bulge profile with the PSF, and a new decomposition is performed until both the coefficients and the profiles converge. The initial estimate of $k_{\rm b}(a)$ is provided by the convolution of the bulge profile extracted without any correction.

3.2.3 Estimate of the ellipticities

To determine the apparent ellipticity of the components, we extracted from each image a radial profile of the ellipticity by fitting isophotes of variable ellipticity and fixed position angle. We then selected the lowest value beyond one seeing disk from the center as $\epsilon_{\rm b}$, and the average value in the outer parts as $\epsilon_{\rm d}$(see also Andredakis et al. 1995). The values for the J and K bands were then averaged to obtain a single value of $\epsilon_{\rm b}$ and $\epsilon_{\rm d}$ for each galaxy. The accuracy of these estimates is difficult to assess: bright bulges can affect the isophotes even at large radii, leading to an underestimate of $\epsilon_{\rm d}$, while bars and spiral arms may often lead to overestimates. In turn, $\epsilon_{\rm b}$ can be easily underestimated because of the seeing. From simulations, it seems that a reasonable estimate for the uncertainty in $\epsilon_{\rm d}$ is about 0.1; it is certainly worse in $\epsilon_{\rm b}$, 0.2 or more in the worst cases.

We note that comparing the observed image to the extracted bulge+disk distribution, for example in terms of residuals, is of little use to test the ellipticities, since it is often possible to find excellent agreement for a wide range of $\epsilon_{\rm b}$ and $\epsilon_{\rm d}$.A more stringent constraint is provided by the color profiles of bulge and disk, which turn out to be quite sensitive to the values of $\epsilon_{\rm b}$ and $\epsilon_{\rm d}$. This has ultimately allowed us to verify the ellipticities a posteriori (in particular $\epsilon_{\rm b}$) by letting them vary within the estimated errors and checking the plausibility of the resulting color profiles. The final accuracy is estimated to be better than 0.05.

  

Table 3: Decomposition results: K-band integrated luminosities and bulge-to- disk ratios

Line 1: $\\ gt$ 2-D parametric decompositions.
Line 2: $\\ gt$ 2-D non-parametric decompositions.

  

Figure 2: Bulge and disk fitted parameter variation with n. Bulge parameters are shown in the left panels: from top to bottom, absolute K-magnitude $M_{\rm b}(K)$, effective surface brightness $\mu_{\rm e}^{\rm c}(K)$,and effective scale length $r_{\rm e}$.Analogous disk parameters are shown in the right panels. The curves show, for a given galaxy, fit parameters obtained with n going from 1 to 4. $\mu_{\rm e}^{\rm c}(K)$ and $\mu_{\rm d}^{\rm c}(K)$ are corrected to face-on and in units of mag arcsec-2. Scale lengths are in arcsec