3 Data analysis

3.1 Outline of the operations

The analysis of the frames entails the following steps:

1.

The usual corrections for offset, the "flat-fielding'', and the interpolation of bad columns. As explained below we tried to improve the flat-fields by ad hoc "superflats'';

2.

The registration of the 5 frames in each passband to a common geometry, based on a set of measured coordinates for 6 to 12 stars. This is intended to simplify the derivation of colour maps if needed;

3.

The preparation of each frame for measurement, involving a final attempt to measure and correct residual large scale background trends, corrections for parasitic objects, a treatment against cosmic rays peaks, and the calibration against the available results of aperture photometry (see Sect. 3.2.3);

4.

Large errors in colour measurements may result from small differences between the widths of the PSFs of the two frames involved (see for instance Michard 1999, and the previous literature quoted therein). When the FWHMs of measured PSFs in the colour set for a given object differed by more than 10%, it was our practice to modify the frame PSFs and try to make them equal to that of the best frame of the group (see Sect. 3.2.4);

5.

The isophotal analysis of the V frame was performed according to Carter (1978), as implemented in the Nice technique described in Michard & Marchal (1994) (MM94);

6.

The correction procedures for the red halo effect in V-I, and eventually for the effects of different PSF far wings in other colours, were performed (see Sect. 3.2.5). The correction necessitates the crossed correlation of the V frame by the I PSF and conversely. This operation cancels out the errors in the colour distribution induced by the red halo, but degrades the resolution. A correction to the calibrations performed before the convolutions is needed;

7.

Finally, colour measurements were performed along the previously found isophotal contours, and the average isophotal colours tabulated. Our routine at this stage involves corrections to the adopted values of the sky backgrounds, in order to eliminate the obvious effects of inaccuracies in these (see Sect. 3.2.6);

8.

The V surface brightness and colours have been collected in ad hoc files, and corrected for galactic absorption (or reddening) and the K effect, according to the precepts and constants given in the Third Reference Catalogue of Bright Galaxies (RC3, de Vaucouleurs et al. 1991). The usual linear representation of colours against $\log r$ have been calculated in selected ranges, avoiding on the one hand the central regions affected by imperfect seeing corrections and/or by important dust patterns; and on the other the outer range presumably affected by poor background corrections and residual noise.

 

Table 5: Measurements of residual fluctuations in background before and after the final "rectification''. Unit: % of sky background.

Colour	U	B	V	R	i
Flats+superflats 1st run	1.84	1.19	0.72	0.68	0.95
id. 2nd run	1.90	0.77	0.45	0.54	0.79
id. 3rd run	0.69	0.79	0.56	0.60	0.64
Final treatment 1st run	0.36	0.34	0.30	0.29	0.28
id. 2nd run	0.40	0.27	0.26	0.31	0.30
id. 3rd run	0.28	0.28	0.24	0.20	0.32

3.2 Detail of operations

Some important details of the above summarized procedures will now be discussed.

3.2.1 The background

The sky background of flat-fielded frames showed disappointingly large scale trends, specially in the U band. To improve upon this situation, it was tried to derive corrections by mapping the background of such frames, at least those which were not "filled'' by a large galaxy. Such maps were found to be correlated, although less so than expected, and their average used as a "superflat''. The quality of the background was then generally improved: if not, the superflats were not used. Note that, in the observing runs of May 2000 and January 2001, a number of frames of "empty'' fields were obtained (sometimes in moonlight hours) to contribute to the derivation of superflats.

A final improvement was obtained by measuring, during the treatment of each frame, a number of background patches, and subtracting a linearly interpolated map of these, instead of a constant. The background residual large scale fluctuations were often measured at the three steps of the procedure, that is, after the application of the flats, of the superflats, and after the final treatment. Table 5 summarizes the results. It may be noted that the combination of flats and superflats left large errors in our first and second runs, specially in the U colour. The final background linear "rectification'' allowed quite significant improvements, as seen by comparing the upper and lower halves of the table.

If we consider an E galaxy observed under the typical conditions of the present series (see above for a tabulation of sky background values), the final residual fluctuations quoted here represent local errors of less than 0.1 mag near the isophote $\mu_B=25$. We will return later to the question of errors resulting from background uncertainties.

3.2.2 Parasitic objects

In galaxy photometry it is necessary to remove parasitic objects, stars and galaxies, that overlap the measured object. In the present work we used concurrently the following techniques:

• replacement of pixels in a circle enclosing the "parasite'' by a circle symmetric about the center of the galaxy;

• replacement of the circle by another one chosen in a nearby area;

• marking of the pixels to be discarded in such a way that they are later left aside in the measuring programmes.

3.2.3 Calibrations

The frames were calibrated by comparisons with the results of aperture photometry. Our first choice was to use the UBVRI data of Poulain (1988) (PP88), Poulain & Nieto (1994) (PN94) that are available for 26 objects of our survey, and are in Cousins's system, notably for R and I. In a few cases the data collected in the HYPERCAT catalogues were used. These contain both primary data (those with an independent photometric calibration), and secondary data (actually calibrated with part of the primary data). Only primary data were used, selected according to our previous experience or prejudices. A few completely missing calibrations in R and I were replaced by values calculated from the tight correlations of V-R and V-I with B-V, derived from Poulain's data for E galaxies.

Although the I filter in the camera is of Gunn's type, our photometry is transfered to Cousins's system through the calibration. It is assumed that the difference of pass-bands has no significant effect on colour gradients.

3.2.4 Equalization of PSF $\mathsfsl{ FWHM}$'s

Many studies of colour distribution in galaxies are affected by errors resulting from the difference in the PSFs of the two frames involved in a colour measurement. Franx et al. (1989), Peletier et al. (1990), Goudfrooij et al. (1994) calculated the radial range where the errors due to "differential seeing'' are larger than some accepted threshold, and discarded the corresponding colours. Our policy, for instance in RM99, was to correct for this effect by adjusting the two frames to have PSFs with a common FWHM. In the present study we tried to equalize the PSFs of the 5 frames in a given colour set. This is feasible if the 5 frames are taken in rapid succession, so that the PSFs have similar widths. The frame with the narrowest PSF is selected, and we find by trial and error a narrow Gaussian (or sum of two Gaussians) which can restore another frame to the same quality, or rather the same FWHM, by deconvolution. The parameters at hand are the $\sigma$ of the Gaussian and the number of iterations in the deconvolution. In Tables 1 to 4 we give the original FWHM $W_{\rm o}$ of each frame, and the improved $W_{\rm i}$ after the procedure described here. Obviously this cannot lead to perfect results, and sometimes we find in our data the signature of "differential seeing'', in the form of large colour variations, peaks or dips, within the seeing disk: these defects were edited out unless there was some good reason to suspect a genuine central colour anomaly, such as large dust patterns, or the jet of NGC 4486.

The reader may notice that two observations of NGC 4406 are listed in Table 2, one of May 30, the other of May 31 2000. The first one was taken through fog and with average seeing, while for the other the "mistral'' brought a clearer sky and very poor seeing. A special treatment was then applied: the central peak of the sharp images was "grafted'' on the corresponding regions of the unsharp but deeper images. This explains why the $W_{\rm i}$ is so much narrower than the original $W_{\rm o}$ for the frames of May 31.

 

Table 6: Comparisons of average color gradients for different subsamples of E galaxies, before or after tentative corrections for the effects of PSF far wings according to RM01. No corrections are applied if the average observed gradients are close to the adopted reference. N, number of objects. $\Delta _{U-B}$, etc. mean gradients.

Subsample	N	$\Delta _{U-B}$	$\sigma$	$\Delta_{B-V}$	$\sigma$	$\Delta_{V-R}$	$\sigma$	$\Delta_{V-I}$	$\sigma$
RM00	29	-0.152	0.048	-0.061	0.025	-0.018	0.030	-0.053	0.022
2000 Observ.	23	-0.138	0.037	-0.064	0.018	+0.018	0.015	+0.093	0.047
2000 Correc.	23	-0.116	0.038	-	-	-0.016	0.013	-0.048	0.026
2001 Observ.	14	-0.174	0.045	-0.080	0.022	-0.017	0.013	+0.040	0.037
2001 Correc.	14	-0.140	0.036	-	-	-	-	-0.062	0.025

3.2.5 "Red halo'' and PSFs far wings

Before the start of this survey, the CCD camera on the telescope used was known to be affected by the "red halo'', an unfortunate property of thinned CCDs. The aureoles surrounding stellar images are obviously brighter and more extended in the I band than in B or V. Not only the red halo, but more generally the outermost wings of PSFs, were measured during our observing runs in 2000-1. The techniques and results are described in Michard (2001) (RM01). The choice of appropriate star fields allowed us to extend the measurements up to a radius of nearly 3$^\prime$, and down to a level of about $0.5\times 10^{-6}$ of the central peak. Due to the red halo effect, PSF wings in I may be a factor of 3 brighter in an extended radius range than the V ones. Much smaller but still significant differences may also occur between the PSFs of various spectral bands, the V PSF wings always being fainter. The V PSF wings however, and all the others at the same time, were reinforced between our observing runs of spring 2000 and winter 2001, probably an effect of 10 months ageing of mirrors coatings. The final output of the measurements are average "synthetic'' PSFs in the format 512 $\times$ 512 pixels, or $5.8\times 5.8$$^\prime$, for each run and pass-band.

Figure 1: Example of the "correction'' of a colour profile through changes in the sky background constants. Abscissae: V surface brightness in mag. Ordinates: B-V. Uncorrected: open circles. Corrected: filled circles. The changes of the background amount here to -0.25% in B and 0.10% in V, that is more than average (see Table 7).

A set of numerical experiments on model galaxies is also presented in RM01, to illustrate the effects of these far wings on the observed surface brightness and colour distributions. The most striking effect occurs for the gradient  $\Delta _{VI}$, which appears strongly positive, while it is negative according to the classical results of Bender & Möllenhof (1987), or Goudfrooij et al. (1994). More subtle effects are found for other colours, with relatively small but definite changes in gradients.

To correct for the consequences of the red halo, or other similar effects upon the colour distribution in the index C1-C2, frame C1 is convolved with the PSF of frame C2 and conversely. After this operation, the resulting images have been submitted to the same set of convolutions, one in the atmosphere plus instrument, the other in the computer: they lead therefore to correct colour distributions, but with a significant loss of resolution. As the convolutions attenuate the central regions of the galaxy, and much more so for the V frame convolved by the I PSF, the mean colours are biased: a correction to the calibrations performed before the convolutions is needed. This has been done by a comparison of simulated aperture photometry to the observed one, an operation also used to estimate the errors in calibration (see below).

Since the extended PSFs are found with limited accuracy, it is necessary to discuss the validity of the corresponding corrections obtained through crossed convolutions, the more so because of the obvious changes of the PSF far wings between run 1 and run 3. The mean values of the colour gradients for subsamples of E galaxies have been used for these checks, with the results of Table 6. For a subsample of 12 or more E galaxies the mean colour gradients and their dispersions cannot differ much, so that their values may be used as checks of the need for a correction and its eventual success. The reference for these comparisons is the subsample in Michard (2000) (RM00), mostly a rediscussion of the "classical'' data by Peletier et al. (1990), Goudfrooij et al. (1994) and others.

Looking at the Table 6, it is clear that the red halo introduces enormous errors in the V-I gradients, but that the corrections are remarkably successful in restoring the agreement of the results with the accepted reference, both as regards the mean values and the dispersion. The same may be said about the V-R gradients. The wings of the V PSF were strongly reinforced between our run 3 and run 1 or 2, but much less so for the I and R PSF wings. As a result the red halo effect is less in V-I for the frames of run 3 and disappears in V-R. The situation is less clear for the U-B distributions. Our mean uncorrected gradients are in good agreement with the "classical'' data, essentially from Peletier et al. (1990), as rediscussed in RM00. On the other hand, the U PSF wings are consistently above the V ones in all our runs, so that the true slopes of the U-B variations may be a bit smaller than the observed ones. This error might well be present in the classical observations. Incidentally, the data of Peletier et al. were obtained in U-R, and it is impossible to be sure that the far PSF wings of the used telescope were the same in both pass-bands!

Similar remarks might be made about our B-V data. Since the mean measured gradient is the same for our data of the year 2000 and the adopted reference (and as a good B PSF is not available) we take as correct this set of results. For our data of 2001, there is evidence that the wings of the B PSF were slightly above those of the V one, so that the B-V gradients might also be biased upwards.

It appears that a significant source of error in the measurement of the small colour gradients in E galaxies has hitherto been overlooked. It might be that small systematic errors, of the order of some 15-20%, are still present in the U-B or U-V gradients published here. Although such errors would not have significant astrophysical implications, control observations are planned.

3.2.6 Measurements of isophotal colours

Our procedure uses Carter's isophotal representation of the V frame. The successive contours at 0.1 mag intervals are fitted to each of the two frames to be compared, the mean surface brigtnesses calculated and the corresponding magnitudes and colours derived. A sliding mean smoothing is applied to the data for the outer contours and a graph of the colour against $\log r$ displayed. This might hopefully be linear, or nearly so, in the studied range. If it is not, it is our practice to introduce corrections to the provisional values of the sky background for one or both of the frames, in order to get rid of the "breaks'' in the colour-radius relation typically associated with a poor choice of the background constant. The reader is referred to the graphs published in Goudfrooij et al. (1994) for examples of such features. In Fig. 1 we show a color profile with a rather important defect due to poor backgrounds, and its adopted correction.

The introduction of such "aesthetic'' corrections to the raw data might be criticized, since it assumes a regular behaviour of the colours at large $\log r$. This is however a reasonable hypothesis: the introduced corrections remain small, as shown by the statistics of Table 7. It should be noted that the mean sky background values derived for our large field frames are more precise than in previous works based on small field frames, where the sky was not reached at all. The problem lies in the presence of residual large-scale background fluctuations (see above): their effects are similar to those resulting from the poor evaluation of a constant background, and can be approximately corrected by the introduction of an ad hoc constant, or rather a set of constants, for the 5 frames in the colour set.

The linear fit was finally performed on a range of $\log r$ selected so as to avoid the central regions affected by known dust patterns or possible residual seeing-induced errors, and the outermost regions visibly affected by deviations from the expected straight line.

 

Table 7: Corrections to provisional sky background values applied to cancel out "breaks'' in the run of colours aginst $\log r$. The table gives the mean absolute values of the applied corrections. Unit: % of sky background.

Colour	U	B	V	R	I
1st and 2nd run	0.22	0.14	0.12	0.18	0.20
3rd run	0.14	0.07	0.02	0.08	0.14

Figure 2: Example of a set of "regular'' colour profiles for NGC 4473. In this case, the colours are nearly linear in $\log r$ through the observed range. In this particular case a linear fit was made in the range 5-80 $^{\prime \prime }$ to provide the tabulated data.

Figure 3: Example of a set of colour profiles for NGC 4125, a galaxy with a central dust pattern of importance index 3. In this case, the colours show a hump for r < 10 $^{\prime \prime }$, small in U-B but much larger in other colours. In this case, the linear fit was restricted to the range 10-80 $^{\prime \prime }$.

Figure 4: Example of a set of colour profiles for NGC 3377, a galaxy with a central red hump in U-B or B-V but not in V-I. This suggests a metallicity effect. The fit was obtained in the range 8-80 $^{\prime \prime }$.

3.2.7 Errors

The noise is not a significant source of error in this type of work, because averages can be performed upon thousands of pixels in the galaxy regions of low S/N ratio. Residual noise effects at large r can be easily recognized in sample plots of the data (see Figs. 1-4). The main sources of errors lie:

• in the calibration, giving a probable error $\sigma_{\rm C}$, affecting equally all parts of a given object, but dependent upon the quality of the available aperture photometry in each colour;

• in the "differential seeing'', i.e. the small differences in PSF between the frames in a colour set, also after the adjustments described above. This error $\sigma_0$ occurs only near the center of the objects, say in a diameter of twice the seeing FWHM (it would be a much larger range without the performed adjustments). There is no obvious reason for it to be colour dependent;

• in the sky background residual large-scale fluctuations, or equivalently, errors in the sky background estimates. This error $\sigma_{\rm S}$ is strongly dependent upon colour and the studied radius, or rather the corresponding surface brightness.

These three components of the total error will now be considered, together with other relevant topics.

1.

Errors in colours from calibration inaccuracies.

When a number of measurements are available for a given object, for instance 5 apertures in PP88 and PN94, a probable error of the resulting calibration is readily derived from the dispersion of these measured values about the results of simulated aperture photometry, calculated from our data, i.e. V magnitudes, isophotal parameters and colours. For the preferred calibrations with Poulain's data, the computed error is often less than 0.01 in B-V or V-R but may rise to 0.02 in U-B or V-I. Still larger calibration errors, up to 0.04, have been estimated for objects with very scanty or uncertain aperture photometry. These probable errors apply to the zero point of the colour regression as shown in Table 9, and to all colour data from the same object.

2.

Residual errors from PSF equalization.

The residual errors after this step in our data treatment can be objectively ascertained by comparing the "central'' B-R colours in our survey with the equivalent data from RM99, derived from high-resolution CFHT frames. For the present survey the "central colours'' are the integrated colours within a radius of 3 $^{\prime \prime }$. From RM99, Table 6, we find the colours at the isophote of 1.5 $^{\prime \prime }$, which are likely similar. The statistics of the difference B-R(new)-B-R(RM99) are for 31 objects in common: $\rm mean=0.01$ $\sigma=0.038$. Assuming then that the errors are equal in the two surveys, the probable error associated with poor PSF adjustements is $\sigma_0=0.027$ in B-R. This source of error has no reason to vary significantly from one colour to another. It is independent of the error of calibration previously discussed.

The comparison between the two surveys is made possible because the same set of calibrations has been used, although the field of the CFHT frames was often not sufficient to use all calibrations apertures. A minor part of the above differences may come from this source. In RM99, the B-R of NGC 2768 is quoted too red by 0.08 and that of NGC 3610 too red by 0.10.

3.

Errors from sky background inaccuracies

Given $\epsilon_A$ and $\epsilon_C$, the relative errors in the sky background evaluation for frames A and C, A and C beeing a pair among UBVRI, the magnitude error in the colour A-C may be expressed as $\delta_{A-C}=-2.5 \log (1+\epsilon_A 10^{0.4\Delta \mu _A})+ 2.5 \log (1+\epsilon_C 10^{0.4\Delta \mu _C})$. Here $\Delta \mu _A$ is the magnitude contrast between the object and the sky in colour A. Using average values for the colours of E-galaxies and for the sky brightnesses, $\Delta \mu _A$ for any colour may be expressed in terms of $\Delta \mu _V$. Then, in the range of small $\delta_{A-C}$, the expression reduces to $\delta_{A-C}= 1.0857(K_A \epsilon_A-K_C \epsilon_C) 10^{0.4\Delta \mu _V}$where $K_A=10^{0.4\Delta \mu _A}/10^{0.4\Delta \mu _V}$. K_V=1 by definition; we find K_U=3.1 and K_I=1.55, while K_B and K_R are slightly below 1.

The $\epsilon_A$, ... are unknown, but it is feasible to get statistics of the linear combinations $K_A \epsilon_A-K_C \epsilon_C$. Indeed, as explained above, we have adopted ad hoc corrections to provisional sky background values, in order to regularise the colour-$\log r$ relations. It is reasonable to assume that the errors left after these corrections are proportional to the adopted corrections themselves. We take $\delta_{A-C}=\eta K_{AC}=\eta (K_A \epsilon_A-K_C \epsilon_C)$, where $\eta$ is a small constant and the K_AC may be derived from the statistics of the adopted corrections given in Table 7. In practice somewhat different statistics have been calculated, to take into account the fact that our corrections for two colours are not necessarily uncorrelated. The constant $\eta$, different for our observing runs of 2000 and 2001, was chosen so as to get a system of errors compatible with the appearance of the data and also with the errors found for the slopes of the colour-$\log r$ relation. The finally adopted errors $\sigma _S$ from sky background inaccuracies are given in Table 8. Note that this source of error is negligible for $\mu_V=22$ or smaller. Predicted errors are reduced for our run 3 as compared to the two others.

4.

Total errors

The above estimated errors are independent and should be added quadratically. In the central region with r<6 $^{\prime \prime }$, the total error will be $\sigma_{\rm T} = (\sigma_C^2+\sigma_0^2)^{1/2}$. In the mean region with $18 < \mu_V < 22$ the total error equals the calibration error. Finally, in the outer range of $\mu_V > 23$ one can use $\sigma_{\rm T} = (\sigma_C^2+\sigma_S^2)^{1/2}$.

5.

Errors from the corrections for red halo and other far wings effects

To our knowledge, the crossed convolutions used to correct for the red halo and similar effects do not give rise to random errors, but rather to systematic errors due to inaccuracies in the adopted PSFs. Such problems may be detected from the study of the distributions of the slopes of the colour-$\log r$ relations discussed below.

 

Table 8: Probable errors $\sigma _S$ associated with sky background inaccuracies. Units: magnitude. Estimated errors are the same in V-R as in B-V.

Colour	$\mu _V$	U-B	U-V	B-V	V-I
1st and 2nd run	23	0.023	0.026	0.008	0.012
id	24	0.058	0.064	0.020	0.030
id	24.5	0.093	0.102	0.032	0.048
3rd run	23	0.017	0.019	0.006	0.010
id	24	0.042	0.048	0.015	0.026
id	24.5	0.067	0.077	0.024	0.041

Figure 5: Correlation between the colour gradients, with the signs changed. Abscissae: $\Delta _{BV}$. Ordinates: $\Delta _{UB}$. Compare with the similar diagram in RM00, or with the original correlation diagram between $\Delta _{BR}$ and $\Delta _{UR}$ in Peletier et al. (1990). The dispersion is clearly reduced here, which can only be attributed to an improved accuracy.

6.

Errors in the slopes of the colour-$\log r$ relations

These have been calculated by two complementary methods. On the one hand, we can look for the correlations between the slopes derived here and those from the literature, notably the data collected and discussed in RM00. Assuming then that the errors are the same in both sources, we get an estimate of our slope errors, hopefully an upper limit. On the other hand we can consider the internal correlations between the slopes of the various colour-$\log r$ relations, specifically $\Delta _{UB}$ and others with $\Delta _{BV}$. Figures 5 and 6 show the correlations of the U-B and V-I colour gradients with that in B-V. The coefficients of correlation are respectively 0.73 and 0.40. A weighted mean Gm4 of the slopes in the 4 colours has also been used as reference instead of $\Delta _{BV}$with analogous results. From the dispersions of such correlations the slope errors can be estimated, if the error for the reference $\mathrm{d}{B-V}/d\log r$ or Gm4 is "guessed''. The two techniques give results in very good agreement, the internal correlations indicating somewhat smaller errors.

The probable errors of the slope estimates are then 0.03 in U-B or U-V, 0.01 in B-V, 0.015 in V-R or B-R, 0.02 in V-I.

Figure 6: Correlation between the colour gradients, with the signs changed. Abscissae: $\Delta _{BV}$. Ordinates: $\Delta _{VI}$. Here the comparison with similar diagrams in RM00 does not show much increase in the accuracy of the new data.