As mentioned in Sect. 2, one of the difficulties encountered in this study is to estimate the uncertainty of the mean field modulus measurements. In many previous studies of Ap stars, the magnetic field moment of interest (mean field modulus, longitudinal field, crossover, quadratic field) was derived through a least-squares fit of measurements of some quantity characterizing the profile of a sample of spectral lines by a function of some atomic parameters. The standard error of the magnetic quantity determined through this least-squares analysis could then be used as an estimate of the internal error affecting it (see e.g. Mathys 1994b, 1995a,b). This approach cannot be applied to the present case, where the mean field modulus is diagnosed from a single line. We cannot either use the kind of argument based on photon-counting statistics that allows one to estimate the error of longitudinal field measurements based on Balmer-line photopolarimetry (Landstreet 1982). In fact, we have no way to estimate independently the uncertainty of the field modulus determinations for each studied star.
However, we can take advantage of our knowledge that the field moments for Ap stars undergo smooth, periodical variations as a result of stellar rotation. For a significant fraction of the stars studied in this paper, these variations, or at least a clear trend of the evolution of the field with time, are well defined. If we fit them by an appropriate mathematical function, we can use the rms deviation of the measurements about this fit as an estimate of the internal measurement errors for the star under consideration. Actually, we may rather expect to derive in that way an upper limit of the errors, since the mathematical function that we fit can at best be an approximate representation of the actual variation of the field modulus. Once we have estimated the measurement errors in that way for a sufficient number of stars, we can get a feeling of how they are related to the appearance of the line profiles. We can in turn use this knowledge to estimate the uncertainty of the field modulus measurements in other stars.
The results of this approach are summarized in the last two columns of Table 3 (click here). In Col. 5 we give, for the 20 stars for which we judge that we can define sufficiently well the behaviour of the field modulus, the rms deviation of our measurements of this quantity with respect to the fit by the mathematical function that we find most appropriate to represent its variation.
For 7 of the stars (HD 2453, 12288, 81009, 93507, 144897, 188041, 192678), we used a simple sine function as an approximation to the variation curve. It must be stressed that this does not imply that we consider that the magnetic field modulus of these stars undergoes perfectly harmonic variations. Simply, the accuracy of our data does not allow us to distinguish whether there is any significant departure from harmonicity (even less so since we have no independent error estimate to compare with the scatter of the observations about a sine wave). This remark also applies to the other functions adopted to fit the variations of other stars. In the case of HD 192678, the AURELIE data have been shifted by -163 G before computing the fit (see Sect. 5.35), so that the scatter about the fit can be regarded as reflecting random measurement uncertainties, but not systematic instrumental errors.
For 3 stars, HD 65339, HD 137909 and HD 142070, our data are well represented by the superposition of a sine wave and of its first harmonic. The poor quality, discrepant measurement of HD 65399 that we have obtained at phase 0.098 (see Sect. 5.12) was not included in the fit, since it does not appear representative of the typical measurement uncertainty. In the case of HD 137909, we have pointed out in Sect. 5.24 the probable existence of a systematic difference between the OHP and the KPNO measurements: fitting the OHP measurements alone (by a sine wave and its first harmonic), the rms deviation was decreased to 28 G (compared to 43 G for the whole dataset).
For HD 126515, the superposition of a sine wave and of its first three harmonics gives the best fit (however, while the third harmonic appears significant, the second one at most marginally contributes).
On the other hand, there are several stars for which our observations cover only a fraction of the (long) rotation period, during which the field modulus variation is essentially linear. The rms deviation given in Col. 5 of Table 3 (click here) for those stars (HD 965, HD 150562, HD 201601) is computed with respect to a straight line fit. In the case of HD 201601, there is some evidence of systematic differences between measurements obtained at various sites (in particular, the AURELIE data seem systematically larger than those obtained with the CES and the KPNO coudé spectrograph). But this systematic difference is difficult to define reliably, especially because (as noted in Sect. 5.38) HD 201601 may be nearing its magnetic maximum. In any case, even more than for any other star, the rms deviation given in Table 3 (click here) for this star must be regarded as an upper limit of the measurement uncertainty. We also fitted a straight line to the last 19 measurements of HD 166473 (the first four show a flattening likely corresponding to the field maximum) and to the measurements of HD 50169 except for the first and the last one. The latter seems affected by an abnormally large error of unidentified origin (possibly an unrecognized cosmic ray hit on the CCD affecting the recorded line profile), while the former, which had been obtained two years before the following ones, lies above the extrapolation of the straight line fit to them. (This may well indicate that at that time, the star was around its magnetic minimum.)
As argued in Sect. 5.20, the field modulus of HD 116458 has been regarded as not undergoing any significant variation and the standard deviation of our measurements (30 G) is taken as a good estimate of their uncertainties. We dealt in the same manner with HD 137949 and HD 177765, since we judge that we have not observed (yet?) any variation of their field modulus.
Finally, the distribution in phase of our observations of HD 200311 is
not sufficiently uniform to allow us to ascertain the shape of the
variation curve of its mean field modulus. However, since this star
appears representative of a category of observations (somewhat noisy
spectra of stars with a fairly weak line 
, possibly affected by
some blend or distortion) for which we have no
better possibility to estimate the measurement uncertainties, we
found important to try to get at least a rough indication of what the
errors on the magnetic field modulus determinations may be for
it. Therefore, we fitted our data for this star with a sine wave
alone, and with superpositions of a sine wave and of up to its
first three harmonics. None of these fits represents the observations
significantly better or worse than the others, but the standard
deviations of the data with respect to all of them fall between 320 and
355 G. Since due to its relative faintness,
our KPNO spectra of this star are rather noisy (which
shows up in the scatter of the measurements), we repeated the same
procedure with our AURELIE data alone: the rms deviation was then down
to around 260 G. For inclusion in Col. 5 of Table 3 (click here), we adopt an
intermediate approximate value of 300 G for this star.
As mentioned in the beginning of this section, we regard the standard deviation of the measurements of the field modulus of a given star with respect to a fit of its variation by a suitable mathematical function as representative of the uncertainty affecting these measurements. Admittedly, in some cases (e.g., HD 93507: see Sect. 3.16), the measurement error shows some dependence on the rotation phase. However, this should only occur for a very small number of the stars under consideration, and even for these stars, the adoption of an average value of the uncertainty appears as the most reasonable (or even only acceptable) approach.
The smallest values of the standard deviations appearing in Col. 5 of
Table 3 (click here) are of the order of 
. They
are obtained for stars in which the components of the line
are well resolved, sharp and mostly symmetric (not showing any
distortion by rotational Doppler effect), and not too heavily affected
by blends (especially, the unknown blending line(s)
frequently affecting the blue side of 
should not be too strong
with respect to the latter). In such cases the accuracy of the
measurements appears to be limited primarily by the accuracy of the
wavelength calibration. Indeed, the latter can be estimated from the
rms deviation of the wavelengths of the lines of the calibration arc
spectrum with respect to the fitted dispersion curve. This deviation
is typically of the order of 1/20 to 1/30 of a pixel. Taking as an
example a spectrum recorded with the long camera of the ESO CES, this
corresponds in wavelength units to between 0.85 and 1.3 mÅ, or in
terms of magnetic field (see Eq. (1)), to between 18 and 27 G.
As the next step, we evaluated the uncertainty of the determination
of the magnetic field modulus in the stars for which the variation
of this quantity is not sufficiently well defined to fit it by a
mathematical function. For this, we compared the profile of 
in
those stars with its profile in the stars discussed earlier in this
section. The resulting error estimates appear in Col. 6 of Table 3 (click here).
We are confident that these estimates should be correct to within
%.
HD 59435 is the only star for which there is no entry either in Col. 5
or in Col. 6 of Table 3 (click here). On the one hand, estimating the
measurement error from the aspect of the profile of 
may not be safe,
since in many cases, this profile could only be recovered after removing the
contribution of the other component of this SB2 system: possible
additional errors may be introduced by this operation. On the other
hand, as mentioned in
Sect. 5.10, while the field modulus of this star undergoes large
variations, our measurements probably do not cover a full rotation
cycle yet, so that their rms deviation with respect to a mathematical
fit may not be quite representative of the measurement uncertainties.
However, it appears e.g. from the consideration of Fig. 19 (click here), that
the latter should not exceed much 50 G.
The discussion so far has been focussed onto the random measurement errors. Our field modulus determinations may plausibly also be affected by systematic errors. The latter might introduce e.g. a constant offset of the field values, or an incorrect scaling of the measurements. Such effects affect all the determinations of the mean magnetic field modulus of a given star in essentially the same way: they do not show through in the dispersion of the measurements.
As a matter of fact, two sources of systematic errors can be readily identified in our measurements.
First, as mentioned in Sect. 2, the line 
is formed in a regime
of partial Paschen-Back effect, and the interpretation of its
splitting in terms of the mean magnetic field modulus through Eq. (1),
which corresponds to pure Zeeman effect, is only a good approximation.
The magnitude of the error that is introduced in that way depends on:
and 
line components, which behave differently from each
other in the partial Paschen-Back regime);
(since the degree of
saturation also affects the relative intensities of the 
and
components).
(resp.
) components is 1.021 (resp. 0.979) times the pure Zeeman splitting.
Accordingly, in the stars discussed in this paper, the systematic
error introduced by neglecting the departures from the Zeeman effect
should never be much larger than 2%
.
The second type of systematic errors that is definitely present is of instrumental origin. Indeed, it has been seen in Sect. 5 that for several stars (in particular, HD 137909, HD 192678, and HD 201601), the field modulus measurements obtained from OHP AURELIE spectra are systematically different from those obtained with other instruments, in particular with the ESO CES and the KPNO coudé spectrograph. As a matter of fact, all the instrumental configurations used but AURELIE yield field determinations that are quite consistent, at the achieved level of accuracy.
Because the effect varies from star to star, as well as, for HD 137909, with rotation phase (see Sect. 5.24), we are led to think that it results from the modification of the linear polarization of the stellar light by the optical train. The resulting effect may be rather complex, since the polarization of the light emitted by the star varies across the line profile in a manner which depends on the structure of the magnetic field and on the geometry of the observation. The interpretation of the observed systematic measurement differences in terms of instrumental polarization receives further support from the fact that HD 137909, HD 192678, and HD 201601 are among the stars for which Leroy (1995b) reports the definite detection of broad-band linear polarization. A more definite characterization of the effect would require the knowledge of the geometric structure of the magnetic field of the considered stars and is therefore beyond the scope of this paper.
The vast majority of the observations discussed in this paper, including those performed with AURELIE, were obtained through coudé systems. We do not believe, however, that oblique reflections on the plane coudé mirrors can account for the modification of the polarization discussed above. Otherwise, the effect would also depend on the position of the star in the sky: thus, it would show through (at least, in part) as additional scatter about the field variation curve of a given star. Such additional scatter does not appear to contribute significantly to the rms deviation of the HD 137909 AURELIE measurements about a smooth variation curve, which as reported earlier in this section, is fully consistent with the accuracy achievable in the wavelength calibration. In addition, there is no reason to suspect that the 4-mirror coudé train of the OHP 152 cm telescope would affect the light polarization more badly than the other coudé (or Nasmyth) systems used in this study. Some of them actually might be expected to be much more harmful to the stellar light polarization: for instance, the CFHT with its train of 7 mirrors with optimized coatings, or the alt-alt mounted ESO CAT with its 3 mirror system whose flat tertiary is frequently used at very large incidences. However, there is no indication in our data of major polarizing effect from any of these coudé systems.
It seems much more plausible that the main source of instrumental polarization rather is the spectrograph grating. With respect to this, it is noteworthy that AURELIE was equipped for our observations with a ``normal'' grating used in the second order, while all the other spectrographs that we employed had echelle gratings used in higher orders (that is, at high angles).
As already stressed in Sect. 5.24, the kind of systematic effect discussed here may set a rather severe limitation to our ability to determine the geometrical structure of the stellar magnetic fields, since it may modify the shape of the curve of variation of the mean field modulus (and in particular, there is some hint that it may shift the apparent phases of field extrema). Fortunately, the good consistency of the data that we obtained with all instrumental configurations but AURELIE seems to indicate that only the latter may be significantly affected by instrumental polarization.
Our data do not show any evidence of systematic errors other than those discussed above.