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3 Data analysis

3.1 General indications

The here used techniques of data reduction and colour measurements have been extensively described in Paper IV of this series (Michard 1999). The indications given there about the preliminary steps, the 1D and 2D colour reductions, the calibrations and usual corrections of the colours remain largely unchanged. Here is given a summary of a few important points:
The isophotal analysis according to Carter (1978) is a necessary step of our procedure. A set of isophotes, in parametric form, is used to locate the points to be measured to derive local magnitudes and colours, and define adequate averaging ranges to obtain a sufficient S/N without unwanted degradation of the resolution. The measured contours are then 0.1 mag distant, and they are symmetrized by simplifications in their Carter's representation: an invariant center is adopted, and odd harmonics are put to zero.

Radial 1D SuBr or colour distributions are obtained separately for each axis, and also for each opposite half of a given axis. To define data points for a given axis, pixels are averaged within a range of $\pm 22\hbox{$.\!\!^\circ$ }5$ around the nominal direction, this range being defined by the eccentric anomaly $\omega$ of the Carter Reference Ellipse. Note that a given range in $\omega$ corresponds to different ranges of polar angle at the majA and minA, depending upon the q ratio of the isophote. For instance, for a flat galaxy with q = .3 the adopted range of $\pm 22\hbox{$.\!\!^\circ$ }5$ in $\omega$ will correspond to $\pm7\hbox{$.\!\!^\circ$ }1$ in polar angle at the majA and to $\pm54\hbox{$^\circ$ }$ at the minA. Examples of such detailed radial profiles are shown below in Fig. 3.

Besides this we have obtained average radial colour profiles, giving the mean isophotal colour against the mean isophotal radius (see plots in the Appendix).

Azimuthal colour distributions have been systematically measured. These are obtained using all pixels within "measurements rings" limited by symmetrized isophotal contours, again defined by the Carter's representation and labelled by their inner and outer radii r. In azimuth there are 121 data points taken at equal steps in the eccentric anomaly $\omega$ along Carter's Reference Ellipse. These are usually plotted at the relevant polar angle, 0 and 180$^\circ $ for the majA, 90 and 270$^\circ $ for the minA. For strongly elongated objects, the points are thus much more closely packed near the major than the minor axis as may be noted in published diagrams (Figs. 1, 2).

2D colour maps have been produced from the best pairs of frames, using the usual definition of the colour as a difference of magnitudes. These have been used to produce the 12 maps assembled in Figs. A1 to A12, and the FITS images made available upon request to interested readers.

Again special care has been taken of correcting, at least approximately, the important errors in colours resulting from "differential seeing", that is the different PSF's of the two frames involved in measuring a colour. Some new features have been introduced in this part of the present work and will be now described.

3.2 PSF matching with improved resolution

As pointed out in Paper IV, the first step in attempting to equalize the PSF's of two different frames, is to determine a correcting function FC, such that the convolution of the sharper of the two PSF's with FC will reproduce the other one. Then one can convolve the "sharp" frame by FC to match its new PSF to the one of the "unsharp" one. Accordingly the colour is observed with the resolution of the worse frame. In principle, one could also deconvolve with FC the worse of the two frames, to match its PSF with the best one of the pair. In this case one has to be aware of the artefacts introduced by "ringing" of the solution, as experimented by Michard (1996) for the widely used Lucy's deconvolving routine. If FC is, however, much narrower than the original PSF such artefacts may remain negligible. A test is available, as the deconvolution by FC may be applied to a stellar image: in this case ringing will manifest itself by a darker ring around the deconvolved image.

It has been attempted to generalize the above considerations by determining the FC functions needed to bring the PSF's of a group of images of similar resolutions to a single narrower common PSF, indeed a Gaussian of suitable FWHM. Sometimes the best PSF in the group was used for matching instead of a Gaussian. The results of this exercize are given in Tables 1, 2, 3 under the label of "improved FWHM". It may be noted that we suceeded in matching the PSF's to very similar width's in most cases, and with noticeable improvements of between 20 to 33% of their width.

The eventual effect of deconvolution artefacts was checked by performing the isophotal analysis upon both the original and "improved" frames: the parameters q and e4 provide very sensitive tests. Other tests are feasible by comparing the colours measured from pairs of frames matched by either convolution or deconvolution: if new colour features occur with the "improved" frames, one should be cautious!

3.3 Errors in the study of near edge-on disk objects

The errors induced by seeing in colour studies of E-type objects have been discussed at some length in our Paper IV, and in the previous papers by several authors quoted there. The errors may be very large in the central part of such objects, involving a strong light gradient, but at radial distances larger than a few times the seeing disk, the measurements are hopefully unbiased. This will not be the case however for objects embodying a thin disk at nearly edge-on projection. The "transverse" light gradient in such objects, that is parallel to the minA, is then large, and the colour badly affected through most of the minA length. We have made a number of numerical experiments relevant to this question, specially in connection with the use of our OHP data to supplement the frames of much better resolution obtained at the CFHT and the TBL. The "OHP seeing" leads to reduced transverse visiblity of the disk in edge-on objects, and reduced contrast of the disk colour (if different from the bulge one). Since part of our OHP frames have elongated and asymetric PSF's in the $\alpha$ direction, due to poor performance of the autoguider, the effects may be dramatic for edge-on objects of unfavourable PA! A list of the main causes of errors encountered in this work is offered here, besides the trivial ones of photometric noise (negligible except far from galaxian centers), and calibrations errors (of little importance for our purposes):

- Improper PSF matching leaves important systematic errors in observed colours both near the galaxian center and along the minA of near edge-on disk galaxies;

- PSF asymmetries may also give important errors in minA light and colour profiles, unless they are very similar in both the frames used. Such asymmetries, specially in the outer wings of the PSF, are not readily corrected by our techniques of PSF matching;

- The background will lead to large errors in colours at low light levels: these will come in part from uncertainties in the adopted sky background constants, but also from residual background fluctuations and trends. These are no doubt present after the flat-fielding and other correction techniques, and they were large with one of the used OHP CCD's, at least in the I band. A poor choice of the average background level also results in calibration errors;

- The unsufficient resolution of the frames, even if properly matched, leads to important errors if sharp colour features are present: this is certainly the case near the galaxian center, and for many dust patterns.

Table 5: Estimates of mean errors in radial colour profiles from duplicate frames. The columns refer to ranges of measurement, i.e. C0, the central region inside the approximate FWHM, 2W, a small range near a radius r about twice the FWHM, and the $\mu _V$ of V magnitude ranges, roughly 1 mag broad. The first two values are intended to describe the errors associated with resolution problems (after PSF matching), the other to errors associated with background problems (choice of mean level and residual fluctuations). The succesive lines are $\sigma _{13}$, $\sigma _{24}$ the mean errors for a data point in one of the semi-majA and -minA respectively as in Fig. 3; $\sigma _{\rm l}$, $\sigma _{\rm s}$, the mean errors for the average colours along the majA or minA respectively; and finally $\sigma _{\rm r}$ for the data points of the radial colour profiles of Figs. A1 to A12. A colon stands for values of lower weight
Obs. C0 2W 17 18 19 20 21 22 23 24
CFH $\sigma _{13}$ - .012 .010 .016 .042 .055: - - - -
CFH $\sigma _{24}$ - .023 .018 .015 .030 .043: - - - -
CFH $\sigma _{\rm l}$ - .009 .008 .011 .032 .040: - - - -
CFH $\sigma _{\rm s}$ - .022 .018 .013 .028 .030: - - - -
CFH $\sigma _{\rm r}$ .026 .015 .012 .012 .028 .036: - - - -
TBL $\sigma _{13}$ - - - .020 .015 .015 .015 .035 .070 -
TBL $\sigma _{24}$ - - - .035 .030 .030 .030 .040 .070 -
TBL $\sigma _{\rm l}$ - - - .015 .010 .010 .015 .025 .055 -
TBL $\sigma _{\rm s}$ - - - .035 .030 .020 .020 .035 .055 -
TBL $\sigma _{\rm r}$ .035 - - .020 .015 .010 .015 .030 .055 -
OHP $\sigma _{13}$ - .022 - - .020 .020 .025 .030 .055 .090
OHP $\sigma _{24}$ - .043 - - .050 .040 .045 .060 .060 .085
OHP $\sigma _{\rm l}$ - .013 - - .010 .015 .015 .020 .040 .075
OHP $\sigma _{\rm s}$ - .035 - - .035 .025 .030 .035 .045 .070
OHP $\sigma _{\rm r}$ .050 .019 - - .020 .010 .010 .020 .030 .055

Table 6: Estimates of mean errors in azimuthal colour profiles from duplicate frames. The columns refer to the successive "measurements rings" introduced in 3.1.3. In succesive lines are given the mean errors $\sigma _{\rm a}$ and $\sigma _{\rm c}$ for data points in near majA and near minA regions respectively, in graphs like Figs. 11, 12, or 14. The $\sigma _{\rm D}$ are mean errors for such data as Table 9, i.e. colour differences between majA and minA at a given isophote
Obs. 2.-3.2 3.2-5. 5.-7.9 7.9-12.6 12.6-20. 20.-31.6 31.6-50.
CFH $\sigma _{\rm a}$ .018 .017 .021 .043 .062 - -
CFH $\sigma _{\rm c}$ .025 .018 .023 .036 .061 - -
CFH $\sigma _{\rm D}$ .018 .011 .012 .016 .026 - -
TBL $\sigma _{\rm a}$ - .02 .02 .03 .045 .06 -
TBL $\sigma _{\rm c}$ - .03 .03 .04 .055 .07 -
TBL $\sigma _{\rm D}$ - .02 .02 .02 .03 .04 -
OHP $\sigma _{\rm a}$ - - - .017 .026 .041 .103
OHP $\sigma _{\rm c}$ - - - .056 .076 .105 .087
OHP $\sigma _{\rm D}$ - - - .033 .033 .036 .051

3.4 Evaluation of various errors from multiple observations

If two or more frames of a given object in a given band C are available, one may evaluate the pseudo-colour C-C from such pairs. These are controlled by the measurement errors of various origins listed above. The 4th one, i.e. errors due to unsufficient resolution, do not enter however if the two frames are from the same observatory and therefore similar resolution (after PSF matching).

Errors in CFHT observations have been discussed in our previous Paper IV, from 8 pairs of duplicate observations of mostly elliptical objects, plus NGC 3115. This study has been repeated with emphasis upon objects containing a thin disk, such as NGC 3115, 3377, 3610. The analysis of duplicate frames was conducted in such a way as to derive errors upon the quantities to be considered in the present study.

A number of duplicate observations also occur in our collection of OHP frames. Because our sample mostly contains very flat galaxies, with very large transverse light gradients, the seeing effects are quite large. The experiments with duplicate OHP frames led to the rejection of part of the results from this material, except where it could supplement the higher resolution data without introducing unwanted bias.

Another technique is needed to estimate errors in the TBL data. It should be noted that there are 6 galaxies in common between the TBL and OHP series: the comparison of measurements from these two sources gives indications upon errors in both, and, together with the above noted experiments upon duplicate frames in the OHP material, opens a way to estimate a set of errors for both sources. It should be emphasized however that errors are mostly systematic, leaving no hope to estimate these by straightforward statistics. The two applied techniques, that is pseudo-colours C-C from duplicate frames on the one hand, and comparisons of colours from different sources on the other, are not equally sensitive to the various systematic errors listed above. For instance, near centre colour features are much sharper at 1 arcsec resolution than with the OHP seeing: this introduces large local differences between the colour distributions of the TBL and OHP series.

Probable errors derived by the above techniques are summarized in Tables 5 and 6, separately for the three sources of material and various quantities of interest. In Tables 5 are given the errors for radial colour profiles. First appear the errors for the separate semi- majA an minA as in Fig. 3 (noted $\sigma _{13}$, $\sigma _{24}$). For OHP material $\sigma _{24}$ is much larger than $\sigma _{13}$ as expected from poor PSF matching. Then come the errors $\sigma _{\rm l}$, $\sigma _{\rm s}$ for mean colours along the majA and minA respectively: they are smaller mainly because the effects of PSF asymmetries cancel out. Finally is given the estimated mean errors $\sigma _{\rm r}$ for the isophotal radial colour profiles.

Table 6 contains estimated mean errors for azimuthal colour profiles. Again errors $\sigma _{\rm a}$ for the tips of the majA and $\sigma _{\rm c}$ for the arcs in the minA regions are distinguished: such errors are expected to apply to such graphs as in Figs. 1, 2. We also give the error $\sigma _{\rm D}$ expected for the colour differences between majA and minA such as tabulated in Table 10. The above estimates were mainly obtained for B-V and B-R colours, while errors are probably larger for U-V and V-I. The contrast of E-S0 galaxies against sky background is reduced by roughly 1 mag in U and 0.5 mag in I (greatly varying with zenith distance!). Besides this, the S/N ratio is poor for part of our U OHP frames, and the background unclean for part of the I frames of the same source. Larger mean OHP $\sigma _{\rm r}$ errors for U-V and V-I were "estimated" and are entered in the relevant plots.

Finally the standard error upon the colour gradients in Tables 7 and 8 has been evaluated to 0.03 from a comparison between the various data series.

3.5 Calibrations and calibration errors

The frames were calibrated from the UBV aperture photometry results compiled in the catalogues by Longo & de Vaucouleurs (1983, 1985), and the RI results in de Vaucouleurs and Longo (1988). For NGC 3115 and 4350 the more recent data by Poulain (1988) was prefered. Since aperture photometry data in R and I are missing for many of the studied objects, they were replaced by (very tight) correlations between V-R and V-I against B-V, derived from Poulain's photometry of early-type galaxies.

Errors in calibrations should be added to the errors analysed above. They are of course closely bound to the availability and accuracy of calibration data. These are abundant for NGC 2549, 3115, 4111, 4350, 7332, 7457, but rather scanty for NGC 3098, 4036, 5308, 5422. The data are best in B-V, and less good in U-V (due to scarcer data and poorer S/N in U), or V-I (due to the use of an ancillary correlation with B-V). It is believed that calibration errors reduce to 0.01 in the best cases and may reach 0.04 in the worse.

Remark: The survey in BP94 includes minA colour profiles for 4 objects of the present sample. These authors obtained their own photometry by observing standard stars (although in rather unsatisfactory conditions). To quote their paper, their "photometric accuracy is estimated to be better than 0.05 mag in R and I and 0.10 mag in B and U". We could compare results for 7 B-R or R-I colours measured at the same place, i.e. s=5 arcsec along the minA. We find for Them-Us a mean of .02 and a $\sigma$ of 0.05, well in line with estimated calibrations errors in BP94 and above.

Incidentally we also compared minA gradients (measured in identical ranges). These are in good agreement for NGC 5422 and 7457, but not for 5308 and 7332: in these two objects BP94 gradients (their Table 2) are anomalous, B-R increasing outwards instead of decreasing as usual. This might be an effect of "differential seeing" if the R frame was distinctly sharper than the B one.

3.6 Corrections to observed colours

The total colour corrections CC for galactic extinction and K-effect, are based upon the galactic extinction coefficient $A_{\rm g}$ and the radial velocities $V_{\rm opt}$ given in the RC3 (de Vaucouleurs et al. 1991). The following corrections have been adopted, in the 4 used colours:

. in B-V, $CC=A_{\rm g}(1/4.3)+0.099\,10^{-4}~V_{\rm opt}$

. in V-I, $CC=A_{\rm g}(1.40/4.3)+0.045\,10^{-4}~V_{\rm opt}$

. in B-R, $CC=A_{\rm g}(1.75/4.3)+0.126\,10^{-4}~V_{\rm opt}$

. in U-V, $CC=A_{\rm g}(1.70/4.3)+0.084\,10^{-4}~V_{\rm opt}$.

The coefficients for galactic reddening take into account the approximate passbands used in our observations, that is Johnson's U, B and V, Cousins's R and I. The corrections have been applied only to tabulated results, not to graphical data dealing with individual galaxies.

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