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4 Results

4.1 Accuracy and comparison of the different decomposition methods

As Fig. 1 clearly shows, the elliptically-averaged profiles reproduce quite well the major-axis cut in almost all cases. In fact, our sample contains only moderately inclined systems ($i < 75^\circ$), whose isophotes are reasonably elliptical. However, a comparison between the 1-D and 2-D results from Table 2 shows that the 1-D decomposition is always less accurate, and sometimes inconsistent with the 2-D one. The inferior quality of the 1-D fits is attested to both by the larger errors on the parameters and by the larger values of $\chi^{2}$. A comparison between 1-D and 2-D results for the galaxies with significant non-axisymmetric structure suggests that 2-D fits manage to reproduce the whole distribution, whereas 1-D decompositions tend to match closely only the inner part of the profile, i.e. the less noisy one (see also Byun & Freeman 1995). Because of their superiority, when discussing parametric models, we will consider only the results of 2-D decompositions.

No systematic differences are found between the various fitted parameters: the 1-D bulge and disk scale lengths are, to within the scatter, the same as those of the 2-D fits, and the fitted inclination of the 2-D model corresponds, on average, to the fixed value determined from the outer isophotes (and used in the 1-D fits). However, disks are definitely better constrained than bulges: the rms difference of the 1- and 2-D $\mu_{\rm d}$'s (0.6 K-mag) is half that for $\mu_{\rm e}$ (1.1 K-mag), and the dispersion in the ratio of 1- and 2-D $r_{\rm e}$'s is 80%, while that for the $r_{\rm d}$'s is four times lower (21%), comparable to the discrepancies among disk scale lengths from different authors found by Knapen & van der Kruit (1991).

We have compared the results of our non-parametric (np) decompositions with the 2-D parametric ones, and with those of Kent (1988). There is no systematic difference between our 2-D and np bulge and disk luminosities, as the mean difference is $-0.4\,\pm\,0.6$ mag for $M_{\rm b}$, and $-0.1\,\pm\,0.4$ for the disk. Again, disk parameters are more consistent than those of the bulge. Parametric ellipticities are not significantly different from np ones: the mean ratio of the np $\epsilon_{\rm b}$'s and those fitted by the 2-D parametric method is 1.1 $\pm$ 0.5. System inclinations agree very well, with a mean ratio of $0.97\,\pm\,0.09$. Also, the bulge ellipticities determined by Kent (1988) are about the same as those used in our np decomposition, but with large scatter; the inclinations are in good agreement, with a mean ratio of $1.01\,\pm\,0.09$.

4.2 Dependence on n of parametric components  

In order to compare our results with those of other authors, we have investigated trends with n of the bulge and disk fitted parameters. Figure 2 shows the systematic variation in these, for a given galaxy, as a function of n. It can be seen from the figure that the derived bulge parameters strongly depend on the form of the fitting function. The same bulge, when fitted with small n, appears to be ``denser'' (that is to say brighter $\mu_{\rm e}$), more compact (smaller $r_{\rm e}$), and less luminous than when fitted with large n. Quantitatively, the changes are dramatic as n goes from 1 to 4 with a mean change in $\mu_{\rm e}$ of 3 mag arcsec-2, and in scale length of roughly a factor of 3. Moreover, the dispersion in the fitted values also increases with n; the spread of $\log\,r_{\rm e}$at n = 4 is 1.5 times larger than that at n = 1, and the spread of $\mu_{\rm e}$ is more than twice as large.

The derived disk parameters also change with the n of the bulge. $\mu_{\rm d}$ tends to be fainter for bulge n larger, and, as for the bulge, the dispersion in $\mu_{\rm d}$ increases with n. The only parameter that is stable with n is the disk scale length $r_{\rm d}$, although its dispersion does increase slightly with n.

We conclude that, at least statistically, bulge structural parameters are strongly influenced by the form of the function used to derive them. Not only are the parameters themselves altered by constraining the form of the bulge, but also the dispersion in the parameters is changed. Independently of the best-fit n, requiring a de Vaucouleurs law to fit the bulge yields more tenuous, extended, and luminous spheroids, together with wider distributions of the parameters (larger dispersion), than does using a simple exponential.

4.3 Bulges  

In terms of the ``quality indicators'' mentioned in Sect. 3.1, n = 3 gave superior results for the majority of galaxies, while n = 2 was the second-best choice. In only two cases did n = 4 give the highest quality fit, and in one case (NGC 3593) n = 1. These values are in general agreement with the trend noted by Andredakis et al. (1995) who fitted the n of non-parametric bulge profiles and found that early-type spirals tend to have bulges with $n\sim 2-3$.We also note that, in our experience, it is usually difficult to determine the best n with a precision much better than 1.

The distribution of the best-n bulge parameters is illustrated in the left panels of Fig. 3. The median bulge parameters are $\mu_{\rm e}^{\rm c}(K)=16.8$mag arcsec-2, and $r_{\rm e}=1.6$ kpc, as reported in Table 4. Our bulges are more tenuous, larger, and more luminous than those of similar type in de Jong (1996a), as expected given the fixed n = 1 used by him. The median apparent $\epsilon_{\rm b}$ is 0.24 from parametric decompositions and 0.25 in np decompositions. This translates into a median intrinsic ellipticity of 0.36 or 0.33 from np values. Notably, 0.33 is also the commonest intrinsic ellipticity found in elliptical galaxies, if they are assumed to be rotational ellipsoids, either oblate or prolate (Mihalas & Binney 1981). In any case, bulges are rarely spherical and the results of studies assuming so should be treated with caution.

\resizebox {17cm}{!}{\rotatebox{90}{\includegraphics{fig3.eps}}}\end{figure*} Figure 3: Distributions of best-n bulge and disk parameters. Surface brightnesses are corrected to face-on and in units of mag arcsec-2. Scale lengths are in kpc

Table 4: Statistics of parametric components

& & \m...
 ...25.4$--$\, -22.5$ 

Correlations between (average) surface brightness and scale length have been found for spiral bulges (Kent 1985; Kodaira et al. 1986) and for ellipticals (Kormendy 1977; Hoessel & Schneider 1985; Djorgovski & Davis 1987), and these two observables constitute an almost face-on view of the ``fundamental plane'' (FP) (e.g., Kormendy & Djorgovski 1989, and references therein). The left panels of Fig. 4 show scatter plots of bulge $\mu_{\rm e}^{\rm c}(K)$ and $<\mu^{\rm c}(K)\gt _{\rm e}$ vs. $r_{\rm e}$, $<\mu^{\rm c}(K)\gt _{\rm e}$ is the average surface brightness within the half-light isophote commonly used in FP studies. The upper panel shows results for all values of n, and the lower one only best-n values. Our results are consistent with the slope within the FP found for bulges by Andredakis et al. (1995), shown as a dashed line in the figure. Although we have transformed their regression line to our distance scale and to the K-band according to their precepts, we note a slight offset: our best-n bulges are generally dimmer ($\sim 0.5$ mag), for a given $r_{\rm e}$, than theirs.


\resizebox {12cm}{!}{\includegraphics{fig4.eps}}

\parbox[b]{55mm}{}\end{figure*} Figure 4: Scatter plots of bulge and disk $\mu^{\rm c}(K)$ vs. $\log r$.Scale lengths are in kpc. The left panels show bulge parameters, and the right those of disks. The upper panels show points for all values of n, with n = 1, 2, 3, 4 shown by $\times$'s, open circles, filled triangles, and filled squares, respectively. The lower panels show only the best-n points, with n denoted as in the upper panel, together with the non-parametric results shown as crosses. In the lower panels, $<\mu^{\rm c}(K)\gt _{\rm e}$ is the average surface brightness within the effective isophote; note that also for disks the abscissa is $r_{\rm e}$.In the lower panels, the dashed line shows the regression found by Andredakis et al. (1995) converted to our distance scale and to the K-band according to their precepts. The inclined error bars represent the average shift related to an uncertainty of $\pm 1$ in the bulge index n

Figure 4 gives further insights as to what happens when the form of the bulge is constrained to one value of n. As discussed in the previous section, when for example n is 4, the resulting bulge parameters lie in the tenuous, extended portion of the parameter space; when n is 1, resulting bulges are smaller and denser (see also Fig. 2). From Fig. 4, it appears that different n values, that is to say different bulge shapes, occupy different regions of the FP. Such a behavior is evident in both the upper- and lower-left graphs. An analysis of the correlation coefficients shows that while the best-n set of points is significantly correlated, as is the global set of points for all values of n, each individual set with fixed n is not. Moreover, the slopes of each fixed n group increase with n, although even the slope of n = 4 is not as large as the global one. The appearance of the top-left graph is determined mainly by a ``geometrical" effect, that is a constant luminosity relation although, see Fig. 2, higher n's produce slightly more luminous bulges. Independently of the details, it is clear that the large scatter related to the uncertainty on the decomposition has a high incidence on the position of a bulge within the FP, as shown by the error bar in the lower-left panel.

It has been suggested that residuals relative to the FP are correlated with shape parameters (Hjorth & Madsen 1995; Prugniel & Simien 1997). Although this projection of the FP is not appropriate for such considerations, the lower-left graph (bulge best n) in Fig. 4 suggests that even the distribution within the FP is at least partially generated by form variations. The distribution of the bulge np parameters does not reveal any dramatically different behavior; if anything, the distribution is tighter and situated in the low-n region of the plot.

4.3.1 Bulge colors  

The median $(J-K)_{\rm b}$ of 1.06 (1.04 for the np decomposition) agrees with the colors measured by Giovanardi & Hunt (1996), and is redder by about 0.1-0.2 mag than those measured in later types (Frogel 1985; Giovanardi & Hunt 1988). The scatter is large, 0.3 mag, with some bulges having J-K as high as 1.5 (NGC 4845). We find that $(J-K)_{\rm b}$ correlates (98% significance) with $\mu_{\rm e}^{\rm c}(K)$, in the sense that redder colors are associated with ``denser'' bulges. In contrast, $(J-K)_{\rm b}$ is independent of $M_{\rm b}(K)$, and of total galaxy luminosity.

The four objects (NGC 3593, 4419, 4845, and IC 724) with $(J-K)_{\rm b}\, \gt\,1.3$also show red extended circumnuclear structure in the color images of Fig. 1, inflections or bumps in their surface brightness profiles, and red gradients in the inner color profiles. Such features have been observed in starburst galaxies (Hunt et al. 1997), and we would argue that the red J-K bulge color is revealing star formation in progress. The most clear-cut case is NGC 3593 which, besides a high mid-infrared 12$\mu$msurface brightness (Soifer et al. 1989) and high molecular gas content (Sage 1993), hosts two counterrotating stellar disks and a disk of ionized gas (Bertola et al. 1996). Moreover, ${\rm H}\alpha$ images reveal an HII-region ring (Pogge & Eskridge 1993) whose structure closely resembles that seen in our J-K image. NGC 4419, the only barred galaxy in our sample, is a LINER (Huchra & Burg 1992) with mid-infrared properties (Soifer et al. 1989; Devereux 1987) and CO content (Young et al. 1995) typical of starbursts. NGC 4845 was defined as a starburst by David et al. (1992) on the basis of its FIR-to-blue luminosity ratio and X-ray excess. IC 724, one of the most distant in our sample, harbors more than 109$M_\odot$ of HI (Eder et al. 1991), but we have found no evidence in the literature for star formation activity. The J-K image may be just revealing a normal bulge, partially obscured by a dusty disk.

4.4 Disks  

The distribution of the disk parameters is shown in the right panels of Fig. 3. Similar to the results of de Jong (1996a) for early spiral types, the median disk has a $\mu_{\rm d}^{\rm c}(K)$ of 17.1mag arcsec-2, $r_{\rm d}=4.6$ kpc; with $M_{\rm d}(K)=$ -24.3 mag it is slightly more luminous than the median bulge. The median ratio of $r_{\rm d}$ and isophotal (optical) radius R25 is 0.24 (shown in Fig. 5 as a dotted line in the upper left panel), comparable to what is found in late-type spirals (Giovanardi & Hunt 1988; Giovanelli et al. 1995). Although similar in size, these early-type disks are more than 1 K-mag arcsec-2 brighter than those in late-type spirals (Giovanardi & Hunt 1988).

The right panels of Fig. 4 show scatter plots of disk $\mu_{\rm d}(K)$ vs. $r_{\rm d}$(upper panel) and of disk $<\mu^{\rm c}(K)\gt _{\rm e}$ vs. $r_{\rm e}$ (lower panel). As for the bulge, correlations of disk $\mu_{\rm d}^{\rm c}(K)$ with $r_{\rm d}$ have been noted for some time (e.g., Kent 1985). It is interesting to note that, when plotted in terms of the photometric observables commonly used in FP studies (lower-right panel), disks dwell in a region of this FP projection which is contiguous and similar in shape and extent to that of bulges. The disks appear to extend the bulge relation to larger radii and fainter surface brightnesses. Also evident, in the lower-right panel, is the rough consistency with the slope for bulges found by Andredakis et al. (1995), although with a large offset. It is clear from Fig. 4 that, unlike the bulge, the relation between disk parameters does not vary substantially with bulge n.

4.4.1 Disk colors  

The median $(J-K)_{\rm d}$ of 0.94 (0.91 for the np decomposition) is similar to the central colors of late-type spirals (Frogel 1985; Giovanardi & Hunt 1988). The scatter about the mean is 0.08 mag, smaller than for $(J-K)_{\rm b}$.The median disk is 0.12 mag bluer than the bulge, an effect not noted by Terndrup et al. (1994) whose sample was dominated by later types. As for bulges, redder disks tend to be ``denser'' (98% significance ), and $(J-K)_{\rm d}$ is independent of $M_{\rm d}$ and inclination.

\resizebox {12cm}{!}{\includegraphics{fig5.eps}}

\parbox[b]{55mm}{}\end{figure*} Figure 5: Disk scale lengths, ratio of bulge and disk scale lengths, and B/D vs. optical isophotal radius R25. Best-n values are shown in the left panels, and values for all n are shown on the right; symbols are as in Fig. 4. The upper panels show $\log r_{\rm d}$ vs. $\log R_{25}$;the dotted line in the upper left panel illustrates a linear relationship between $r_{\rm d}$ and R25 with a mean ratio $r_{\rm d}/R_{25}$ of 0.24. The middle panels show the ratio $\log r_{\rm e}/r_{\rm d}$ vs. $\log R_{25}$;the dotted line on the left gives the best-n median constant value $r_{\rm e}/r_{\rm d}$ = 0.3, while the pair of lines on the right show the n = 1 median $r_{\rm e}/r_{\rm d}$ = 0.2, and the n = 4 median $r_{\rm e}/r_{\rm d}$ = 0.7. The lower panels show $\log(B/D)$ vs. $\log R_{25}$

4.4.2 Opacity of the disks  

Although the diagnostics of dust content in galaxies have been extensively revised in recent years (e.g. Byun et al. 1994 - BFK; Bianchi et al. 1996), such studies have made clear that disk opacities are not easy to determine. In the following we gather the indications about disk opacity obtained here; none of the tests we adopt is particularly stringent, due mainly to the small number statistics, but all converge on conservative estimates for the opacity of early type disks: $\tau_V(0)$ ranges from 2 to 4, where $\tau_V(0)$ is the central optical depth in the V band (face-on). This is essentially the same result reached by Peletier & Willner (1992) and Giovanardi & Hunt (1996).

(a) Correlation between apparent disk brightness and inclination. We find a slight trend in both bands, with slopes $C_J = 0.66\,\pm\,0.23$ and $C_{K} = 0.73\,\pm\,0.23$, both compatible with a fully transparent disk (with C=1, see Eq. (4)). Taken at face value, a C=0.7 corresponds to a $\tau_V(0)$ of $\sim$ 1.1 if measured in the J band, and to $\tau_V(0) \simeq 1.8$ if in K[*]. These moderate values for the central opacity imply that the spread observed in the NIR $\mu_{\rm d}^{\rm c}$ is intrinsic and not due to extinction; de Jong (1996b) reached a similar solution on the basis of a larger sample.

(b) Variation of disk scale length with wavelength and inclination (Evans 1994; Peletier et al. 1994). Five of the sample galaxies have been parametrically decomposed in the optical: either r (Kent 1985; NGC 2639), V (Kodaira et al. 1986; NGC 3898, 4698), or B (Boroson 1981; NGC 2775, 2841, 3898). We find a trend in the ratios of our to their $r_{\rm d}$: for the only measurement in r the ratio is exactly 1, it decreases to 0.85 in V, and to 0.70 in B. In addition, these ratios depend on the inclination, thus providing a test which is largely free from the influence of intrinsic color gradients. The correlation, in the sense of smaller ratios for higher i, implies $\tau_V (0) \leq 3$.These results are consistent with Peletier et al. (1994), who find that B and K scale length ratios vary from 1.2 to 2.0, and with inclination.

(c) Colors. As noted in the previous section, $(J-K)_{\rm d}$ does not depend on i. We estimate the maximum (3 $\sigma$) slope of $(J-K)_{\rm d}$ vs. $\sec\, i$ which is still compatible with our data to be 0.075. For a Triplex model (Disney et al. 1989) with $\zeta = 0.5$ (Peletier & Willner 1992), this implies a $\tau_V (0) \leq 3$.We noted in Sect. 4.4.1 that red disks were associated with bright $\mu_{\rm d}^{\rm c}$, which again points to moderate opacities. Indeed, since $\mu_{\rm d}^{\rm c}$ is corrected for inclination assuming transparency, a high opacity would translate into faint brightnesses for reddened disks.

4.5 Relationship between bulge and disk  

The best-n median B/D ratio is 0.8 with values ranging from 0.2 to 2; even in this early-type sample, more than two thirds of the galaxies have disks more luminous than bulges. With the exception of NGC 1024, B/D ratios obtained from the np decomposition are always less than 1 (as can be seen from the lower left panel in Fig. 5). The two methods yield B/D values which differ by almost a factor of $\sim 2$, but with large scatter. We have verified that this is mainly imputable to the choice of $\epsilon_{\rm b}$ and i, the values adopted in the np case being lower. Our parametric B/D ratios are comparable, although somewhat larger, to those (parametric) found by Kent (1985) in the r band. Also, our B/D's (both parametric and np) in the K band are 10 $\sim$ 15% larger than in J. That the K-band B/D is larger than in the optical was also noted by de Jong (1996a), but with values smaller than ours due to his choice of n = 1 bulges. It is evident that the B/D ratio is rather model dependent, and different decomposition methods provide estimates differing by factors of 2 or more, as illustrated in Fig. 5 where $\log r_{\rm d}$, $\log\,r_{\rm e}/r_{\rm d}$, and $\log\,B/D$are shown as a function of optical (isophotal) radius[*]. Inspection of this figure also shows that the derived bulge and disk parameters, including best n, are not appreciably affected by biases associated with galaxy apparent size.

A linear correlation between $r_{\rm e}$ and $r_{\rm d}$ over all spiral types has been recently found by de Jong (1996a) and Courteau et al. (1996). They interpret the correlation as an indication that the Hubble sequence is scale-free since the relative size of bulge and disk does not depend on morphological type. It can be seen from the middle panels that our data are also consistent with constant $r_{\rm e}/r_{\rm d}$;the sample median best-n $r_{\rm e}/r_{\rm d}$ of 0.3 is shown as a dotted line. This value is a factor of 2 larger than that found by Courteau et al. in the r band, $\langle r_{\rm e}/r_{\rm d} \rangle\,=\,0.13$ and by de Jong for the $K^\prime$ data alone, $\langle r_{\rm e}/r_{\rm d} \rangle\,=\,0.14$[*]. However, $r_{\rm e}/r_{\rm d}$ appears to be strongly influenced by the bulge parameterization: the dashed lines shown in the middle right panel of Fig. 5 illustrate the values obtained from our n = 1 fits (sample median $r_{\rm e}/r_{\rm d}$ = 0.2), and for the n = 4 fits (median $r_{\rm e}/r_{\rm d}$ = 0.7). According to whether bulges are fit with a simple exponential or with the de Vaucouleurs law, $r_{\rm e}/r_{\rm d}$ changes by more than a factor of 3. Hence, if the best-fit n changes with morphological type as suggested by Andredakis et al. (1995), the claims made by Courteau et al. for a scale-free Hubble sequence may be premature.

4.6 Color gradients  

Since we adopt the same scale length in J and K, our parametric decompositions yield bulges and disks with uniform color. On average, the resulting bulges are redder than the disks by more than 0.1 mag, and we should detect a significant color gradient at the transition between bulge and disk. Such gradients are clearly evident in NGC 3593, 3898, 4419, 4845, 6314 and IC 724, all objects whose bulge and disk colors differ greatly. When such gradients are present, they also appear, enhanced, in the r-K profiles.

Regarding the gradients within the single components, we give no estimate of bulge color gradients[*]. For the disk, following Terndrup et al. (1994), we estimated outer (> 3 kpc) color gradients, computed by fitting J-K and r-K versus $\log r$ (in kpc); they will be denoted with $\delta(J-K)$ and $\delta(r-K)$ respectively[*]. We detect J-K gradients at 3$\sigma$ in only one case, NGC 6314, and in six if we consider a 2 $\sigma$ limit. All galaxies with J-K gradients for which we have r data, namely NGC 3593, 3898, 4378, 4419 and 6314, also show significant $\delta(r-K)$.Significant $\delta(r-K)$ is also found in NGC 2639, 4845 and IC 724. In agreement with de Jong (1996b), we only find negative gradients, ranging from -0.24 to -0.47 mag per decade in J-K, and from -0.26 to -1.03 in r-K. There is no correlation between $\delta(J-K)$ and inclination, but inclined galaxies have steeper r-K profiles (see Fig. 6).

Such color gradients provide a last assessment of the disk opacity, already discussed in Sect. 4.4.2. While NIR colors are stable across the disk, we find a prevalence of negative trends in r-K, especially in inclined galaxies (see Fig. 6). A weighted fit of $\delta(r-K)$ vs. $-2.5\log\, (\cos\, i)$ yields a slope of $-0.30\,\pm\,0.08$. In B-I, a common feature of the models (e.g. BFK) is that gradients tend to steepen with increasing i only in rather transparent disks; for $\tau_V(0) \geq 3$ the trend is reversed due to a saturation effect. It has also been shown by Bianchi et al. (1996) that disk color gradients are not greatly influenced by the dust scattering properties. In r-K the reddening will be larger by a factor of 2 for the same $\tau_V(0)$, so the observed slope is roughly indicative of a $\tau_V(0)\simeq 1.5$ (BFK; Bianchi 1995).

\resizebox {12cm}{!}{\includegraphics{fig6.eps}}
}\end{figure} Figure 6: Colors and color gradients vs. inclination. The upper panels show the mean of J-K and r-K colors beyond 3 kpc from the center. Color gradients evaluated in the same region are shown in the lower panels

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