3.1. Near-infrared profiles
In order to quantitatively distinguish between the various models, we tried to fit the extinction-corrected vertical light profiles in the K band with the extrapolated central surface brightness, the vertical scale parameter and the exponent 2/n as free parameters. To check the validity of the best-fitting 2/n thus obtained, we also tried to fit the vertical luminosity profiles with a continuous series of fixed values for the 2/n parameter. We compared the minimum in the distribution as a function of 2/n with the best-fitting n value resulting from the three-parameter fits. From this comparison, we concluded that resulting 2/n values from the three-parameter fits could reproduce the minima in the distribution with sufficient accuracy to be reliable indicators of the profiles' shapes.
To show the accuracy of our fits, in Fig. 5 we have compiled all our vertical K -band profiles. We show the best-fitting model luminosity distributions obtained with the three-parameter fitting routine for those profiles that are sufficiently symmetric. The vertical luminosity profiles for each galaxy are shown in two separate panels, representing either side of the galaxy with respect to the galaxy center; for comparison, the central profiles are shown in each panel. In Table 2 we have listed the positions along the major axis at which the vertical luminosity profiles were extracted.
Table 2. Positions along the major axis at which the vertical profiles of Fig. 5 were extracted. Columns: (1) Profile number (on either side of the galaxy center; 1 = central profile); (2) center position of radially averaged bin along the major axis (arcseconds); (3) bin width (arcseconds).
In Fig. 6 we present the distribution of the best-fitting values for the exponent of the family of luminosity laws ( 6) for our total sample of -band images.
We notice that, apart from the rather unpredictable behaviour in the central areas, where the disk luminosity is contaminated by a non-negligible bulge light contribution, the vertical luminosity profiles generally exhibit little variation of 2/n with position along the major axis. As has been shown by Andredakis et al. (1995), the early-type bulges are characterized by very steeply peaked luminosity profiles, whereas in the later-type galaxies the bulge profiles become closer to exponential. Therefore, we argue that the dominant bulge light contribution causes an increase of the cuspiness of the profiles, especially in the earlier-type galaxies.
In all panels of Fig. 6, indicated an estimated mean level of the best-fitting 2/n values by a dash-dotted line. These values are given in Table 3. In determining these mean levels, we have excluded both those fits that were substantially affected by bulge light contamination and those fits for which the resulting values of the minimization routine were excessively high, due to either a low signal-to-noise ratio or foreground stars superimposed on the galaxy. Especially in the (radially) outer parts of the galaxy disks our observations are rather noisy and therefore unreliable.
In general, we find that the mean levels for the sharpness of the band luminosity peaks indicate that the vertical luminosity distributions are more peaked than expected for the intermediate sech(z ) function, proposed by van der Kruit (1988), though less peaked than exponential.
To study the variation of the cuspiness of the vertical profiles as a function of position along the major axis, we have averaged the data points of our -band observations in radial bins of 0.5 scale length, both for our total sample and for subsamples in certain ranges of revised Hubble type, T. The results are shown in Fig. 7. The central values for 2/n, , were used for normalisation; the positions along the major axis are expressed in units of the K -band radial scale lengths, . We determined the scale lengths by fitting ellipses to the two-dimensional galaxy isophotes in the regions away from the central dust lane, roughly between 1 and 4 radial scale lengths, depending on the bulge influence.
We chose to look at the galaxies in the type range (4 galaxies) because of their clear bulge contribution, and those in the range (13 galaxies), since they are well-behaved late-type galaxies without prominent bulges. The intermediate type range, with , is shown as well. For these galaxies the bulge influence is generally small, though not negligible. In order to reduce the noise in the earliest-types bin, the radial binning was done in intervals of 1.0 scale length.
We notice that the distribution of the sharpness of the vertical profiles remains, within the errors, constant as a function of position along the major axis, irrespective of galaxy type. Some of the (radially) outermost profiles show a slight increase of 2/n, corresponding to a rounder vertical profile, but in those regions the number of useful profiles is small and hence the errors are large. The innermost profiles, especially those in the earliest-types bin, are affected by bulge light and should therefore be left out of the analysis.
3.2. Effects of seeing
Since we are particularly interested in the behaviour of the vertical light distributions in edge-on disk galaxies at small z, we have to be careful to account for the effects of atmospheric smearing. No matter what the exact light distribution is, atmospheric seeing causes the profiles to show a flat-topped distribution in the galaxy planes. In order to circumvent this problem, we have to convolve our models with the seeing profile. Assuming that the actual point spread function can be adequately described by a Gaussian, the seeing convolved model profile is:
where and K(z) are the corrected and intrinsic model intensity profiles, is the dispersion of the seeing Gaussian and is the zeroth-order modified Bessel function of the first kind (Pritchet & Kline 1981; Andredakis & Sanders 1994).
The importance of atmospheric smearing on a particular vertical profile depends on the ratio of the seeing FWHM and the vertical scale height of the profile. The effects are most noticeable for (nearly) exponential profiles. In the case of an exponential light distribution, the seeing convolution becomes non-negligible for those observations with seeing FWHM , see Fig. 8a. In the present study of near-infrared observations, the seeing FWHM varies between 0.12 and 0.57 . Therefore we conclude that the (variation of the) cuspiness of the observed vertical profiles is real rather than an artifact of the method applied. This conclusion is supported by the fact that, in some of the earliest-type galaxies, we see a variation of the 2/n parameter along the major axis due to bulge light contamination, causing more sharply peaked vertical profiles, which is likely to be real.
3.3. Comparison with I- band observations
To be able to compare our results with those published previously, we applied this method to our total sample of I -band observations, thereby using the same fitting ranges as for the -band data. It is immediately obvious that the I -band results are much noisier than those obtained in the band, largely due to the presence of clear dust lanes along the galaxies' major axes. In those galaxies that show a well-behaved dependence of the 2/n value as a function of position along the galaxy's major axis, we cannot distinguish statistically between the best-fitting 2/n values in the I band and those obtained in the band, within the errors. This is made clear in Fig. 9, where we compare the distributions of best-fitting 2/n levels between the two passbands. If we approximate the distributions of 2/n in the I and bands by a Gaussian, we find 2/n and 2/n . However, -band observations are clearly preferable to determine the sharpness of the peak of the vertical luminosity profiles unambiguously, because of their relative insensitivity to contamination by galactic dust. In the presence of a central dust lane, i.e. a deep trough in the vertical light distribution, one has to remove the central regions from the fit to get meaningful answers. While doing this, one often ends up only with the exponential outer parts of the vertical distribution, forcing the fitting routine to yield a more sharply peaked solution. Although we tried to fit the vertical profiles over the same range in z for both the I and -band observations, this was not always possible due to the disturbing effects of dust in the I -band data. Therefore the best-fitting I -band 2/n values are not very reliable. In the best-fitting 2/n value is hardly affected by a different fitting range, since the fit is dominated by the inner data points, which represent the luminosity distribution in the galaxy plane.
From Fig. 9 it can be seen that the median of the best-fitting 2/n value lies in between the exponential (2/n = 0.0) and the intermediate sech(z ) model (2/n = 1.0). It is therefore not surprising to note that in previously published papers by, e.g., Wainscoat et al. (1989) for the large southern edge-on galaxy IC 2531, and Gilmore & Reid (1983) for the Galaxy, the exponential and the sech(z ) models were found to fit the data equally well (van der Kruit 1988).
When studying the relation between the sharpness of the peaks and various global parameters for our sample galaxies, little correlation was detected, although it seems that the smallest galaxies generally show a more sharply peaked vertical distribution than the larger galaxies. In other words, for those galaxies with the faintest absolute magnitudes ( or ), we find a lack of "rounder" profiles ( ) compared with those galaxies with brighter absolute magnitudes, as can be seen in Fig. 8b.
Although we do not find clear correlations with global galaxy parameters, we argue that the radial distribution of 2/n is universal (i.e., we do not find any significant variations as a function of galaxy type), exhibiting only local variations due to, e.g., residual dust contamination (see, e.g., Wainscoat et al. 1989; Aoki et al. 1991).
3.4. Extinction analysis
In Fig. 10 we show the central profiles of all sample galaxies in both I and band, and their corresponding I-K colour, to give an indication of the importance of contamination by dust in the I -band observations.
The vertical colour profiles seem to indicate strong extinction in the plane of the galaxy, and no extinction in the outer parts. We therefore define the constant colour in the outer parts to be the colour of the stars in the galaxy, and the maximum I-K difference to be the I-K colour excess, .
In Table 4 we give, on the first line, crude estimates for the maximum (edge-on) I-K colour excess, , in the galaxy planes, at various positions along the galaxies' major axes. We obtained these estimates by averaging our vertical colour profiles on both sides of the galaxy centers. In deriving these colour excesses we assume that there is little or no dust mixed in with the stellar population outside the dust lane region, as seems likely from the smooth vertical colour profiles. From a comparison with published colours of moderately inclined Sc galaxies, Kuchinski & Terndrup (1996) have shown that there is indeed little or no reddening away from the dust lane. Secondly, we have neglected any change in intrinsic colour due to a stellar population gradient by assuming that the intrinsic colours of the dominant stellar population in the dust lane region are identical to those away from the dust lane. Kuchinski & Terndrup (1996) argue that a population gradient would result in a very small colour change compared to the total colour change across the dust lane and that we can therefore assume that the colour exces is largely due to dust extinction.
Table 4. I -band extinction estimates in the galaxy planes Columns: (1) Galaxy name; (2)-(4) first line: Extinction estimates at the galaxy center, at 1 and at 2 radial ( -band) scale lengths. The accuracy of the numbers is indicative of the quality of the colour profile; second line: Inferred maximum I -band optical depths, , assuming a uniform mixture of dust and stars (edge-on and face-on values).
By using the Galactic extinction law we derive that , the maximum (edge-on) I -band extinction. From this, one can derive the optical depth, and the optial depth in V: .
In all our sample galaxies, remains constant or decreases as a function of projected galactocentric distance, which could be a characteristic of an exponentially decreasing dust contribution. Jansen et al. (1994) find, for a small sample of highly inclined spiral galaxies, that the maximim extinction in the dust lane ( ) decreases rapidly with increasing galactocentric distance.
For each position for which we give the maximum I-K colour excess, we also tabulate the inferred maximum I -band optical depths, , assuming a uniform mixture of dust and stars in the galaxy planes, taken from Walterbos & Kennicutt (1988):
For the same optical depth, the uniform mixture of dust and stars causes less I -band extinction, , than the classical foreground screen model, because part of the extinction lies behind the source. In Table 4 we have also listed the inferred face-on optical depth estimates. For face-on galaxies we have assumed the galaxies to be 9 times less opaque than in the highly-inclined case, which reflects the difference in path length through the disk at both positions. Of course, the face-on optical depth can only be estimated in a rough way; in reality the optical depth depends on the geometry of the dust and stars (which we have assumed to be uniformly mixed), the filling factor of the disk components (in our approximation we have neglected the possibility of a patchy dust distribution and the presence of spiral arms), and the major-to-minor axis ratio, which does not need to be 9 for all our sample galaxies.
In Fig. 11a we present the mean values for the maximum I -band edge-on optical depth as a function of Hubble type and for different positions. The scatter in this figure is large due to the small number of data points available, but also because of the uncertainties in the galaxy classification inherent to edge-on galaxies. Therefore, we can only draw qualitative inferences from the behaviour of the optical depth as a function of galaxy type. This behaviour suggests an increasingly important dust contribution from the lenticular and early spiral galaxies towards later types, although for the latest galaxy types the dust content seems to diminish relative to the intermediate types.
This result is in qualitative agreement with that of Peletier et al. (1994), and Peletier & Balcells (1997) (see also Peletier & Balcells 1996), who studied the type dependence of the R and I vs. radial scale length ratios of a sample of some 70 disk galaxies, both face-on and edge-on. From Fig. 11b it is clear that the radial colour gradients, indicated by deviations from unity of this ratio, are smallest for the early-type sample galaxies, whereas in the later types they vary considerably.
Peletier et al. (1994) have shown that scale length ratios due to stellar population changes are of order 1.1-1.2 in the blue - near-infrared range; in the I vs. range this contribution is likely to be less. Therefore, the observed scale length ratios largely represent the galaxies' dust content.
Although the scale length ratios indicate a more or less constant dust content for galaxy types later than about (Sb), the data suggests that the ratios decrease towards later types, as is also found from the optical depth measurements.
© European Southern Observatory (ESO) 1997
Online publication: April 6, 1998