When the study reported here started, there were in the literature about 80 published measurements of the mean magnetic field modulus of Ap stars. Although 12 stars with magnetically resolved lines were known, almost all of those 80 measurements pertained to only four of them: HD 65339, HD 126515, HD 137909, and HD 215441. As a result, these four stars were for a long time the Ap stars for which the best knowledge of the magnetic field geometry had been achieved. However, due to their small number, it was impossible to decide confidently whether their properties were representative of those of Ap stars in general, or if they rather were isolated unusual specimens. It was even less feasible to use the limited data available to derive information of statistical nature, such as on the distribution of the field strength among Ap stars, or on the relation between field strength and other stellar properties.
The more than 750 measurements of 40 stars reported here therefore open a wide range of new perspectives. A number of general results that can be inferred from these new data at the present stage (that is, in particular, prior to any modeling effort) are discussed hereafter.
On the spectra plotted in Figs. 2 (click here) to 4 (click here), one can see that
in virtually all studied stars in which the magnetic splitting of
is large enough, the profile of this line comes back to (or very
close to) the continuum between the two split components. There is no
significant absorption at the nominal wavelength (centre) of the line,
as should be the case if part of it was formed in regions devoid of a
magnetic field. Since there is no evidence for extreme
inhomogeneities in the distribution of iron over the surface of any of
the considered stars (nor in any Ap star known), the observed spectra
sample essentially the whole stellar hemisphere visible at the time of
observation. Therefore, it appears that the magnetic field of the
considered stars covers all (or most of) their surface or, in other
words, has a filling factor (nearly) equal to 1.
Moreover, it can also be seen in Figs. 2 (click here) to 4 (click here) that, in
most cases, the split components of 
are very sharp. This indicates
that the magnetic field prevailing in the line forming region has a rather
uniform strength. Indeed, if very different magnetic field intensities
were found at different locations on the star, locally the lines
originating from these regions of different magnetic fields would have
different splittings. The observed line, which is an average of them,
would as a result have broader components. We have actually mentioned
in the course of the discussion of the individual stars that a few
of them (HD 9996, HD 18078, HD 55719) appear from the profiles of the
split components of 
to have magnetic fields more inhomogeneous
than the bulk of the sample.
In a number of the studied stars (pointed out in Sect. 5), the
profiles of the split components of the line
appear distorted by
rotational Doppler effect. This distortion varies along the stellar
rotation period. Probably its most characteristic signature is that,
at some phases, the red component of 
is deeper than its blue
component, which is opposite to the behaviour due to partial
Paschen-Back effect alone. Rotational Doppler effect masks the
sharpness of the line components. When sufficiently large compared to
the line splitting, it may also result in partial overlap of the
components of 
, thus preventing them from reaching the continuum
at line centre.
Besides the cases when the magnetic field is too weak to yield full
splitting of sharp components and those when rotational
Doppler effect broadens the components so much that they are no longer
fully separated, there are two stars in which the split 
profile
fails to come back up to the continuum at line centre:
HD 166473 and HD 216018. These two cool Ap stars have very rich
spectra. It may be noted in Figs. 2 (click here) to 4 (click here) that they are
among the stars in which the unrecognized blue blend of 
is strongest,
and it seems probable that another blend affects the centre of the line.
Thus our observations are consistent with the view that the surface of the Ap stars is wholly covered by a magnetic field whose strength varies within a limited range from place to place. This result is not entirely new (see e.g. Preston 1971b). But thanks to the high quality of our spectra (as compared to the photographic ones), much more stringent constraints can be set on the filling factor and on the field distribution. Even more importantly, given the large number of stars that have been studied, the above-mentioned result can confidently be regarded as a generic property of Ap stars.
The much increased sample of stars with fairly to well studied magnetic field modulus (as compared with the data available in the literature so far) also allows one to derive statistically significant results, e.g. about the distribution of the strength and of the amplitude of variation of the fields.
Figure 47: Histogram showing the distribution of the phase-averaged
mean magnetic field moduli of the 42 Ap stars with resolved
magnetically split lines presently known. The shaded part of the
histogram corresponds to those stars that have been observed
throughout their rotation cycle (see text)
An histogram illustrating the distribution of the average 
of the mean magnetic field modulus is shown in Fig. 47 (click here). It is
plotted from the values of 
appearing in Table 2 (click here),
with this quantity defined as explained in Sect. 4
. The shaded part of the
histogram corresponds to the stars that have been observed throughout
their whole rotation cycle; the unshaded area accounts for the stars
for which only partial phase coverage has been achieved. The highest
bin of the histogram is ``open'' on the high value side: it contains
all stars for which 
kG.
The vast majority of the studied stars have average magnetic field
moduli in the range 
. There are a few, more or less isolated
stars, with considerably stronger fields. Babcock's star, HD 215441,
more than 35 years after its discovery (Babcock 1960), still
stands as the record holder, with its mean field modulus close to 34 kG. The
next strongest fields hardly reach 17 kG.
But while the high field
strength tail of the distribution seems fairly compatible with the
combination of a ``standard'' statistical distribution (analogous to
e.g. Poisson's law) and of small number statistics, the low field end
shows a puzzling, rather sharp cutoff. Indeed, while more than 40%
of the stars of the sample have a value of 
between 3
and 5 kG, for only 2 stars (less than 5%) is this quantity less than
3 kG. 
in these 2 stars, HD 29578 and HD 75445, is resp.
2782 G and 2985 G. Although it is not clear whether a full rotation cycle
of those two stars has been sampled so far, it seems unlikely that their
mean field modulus may at any phase go much below its lowest value
found until now (resp. 2.7 kG and 2.9 kG: see Sects. 5.7 and 5.14).
As a matter of fact, there is only one star, the SB2 HD 59435, for
which some of our measurements
of the mean field modulus yield values significantly lower (down to
approx. 2.2 kG). But this star has one of the largest relative
amplitudes of field modulus variation, and since our observations
likely cover most of its rotation period (and in particular its mean
field modulus minimum: see Sect. 5.10 and Fig. 19 (click here)), it can be
regarded as well established that the average over its rotation cycle of its
mean field modulus is not smaller than 3 kG. As mentioned in
Sect. 5.6, the profile of 
in HD 18078 at phases where no
splitting is observed is still consistent with a field modulus of the
order of 3 kG.
Thus all the data gathered so far are consistent with the view that
the average of the mean field modulus of Ap stars over their rotation
cycle is never less than 
. However, we expect to be able to
observe resolved magnetically split lines for significantly weaker fields.
Indeed, taking 
K as a representative temperature for an Ap
star, the thermal broadening for iron is 1.7 
. Microturbulence is
not thought to be large in Ap stars: it is unlikely to exceed 2 
.
For the
configuration most used in this study, the long camera of the ESO CES,
the full width at half maximum of the instrumental profile is 3 
.
Combining these three contributions quadratically, it appears possible
two resolve two lines (or two line components) separated by 4.0 
.
For the split components of 
, this corresponds to a limit of
resolution of 1.7 kG, in the absence of any broadening source other
than those already considered. This latter restriction is realistic,
as can be seen from the consideration of the stars with resolved lines
studied here. The uniformity of the field of most of them (and the
resulting sharpness of the split components of 
) has been stressed
above. Furthermore, rotational Doppler broadening is negligible in
most cases: for a stellar radius of 5 
and a rotation period
of 1 yr (a value reached or exceeded by about half the stars with
resolved lines known), 
is smaller than 0.7 
. These
arguments indicate that we should be able to detect magnetic line
splitting in a non-negligible number of
stars with a mean field modulus of significantly less
than 3 kG. This view is also supported by the consideration of the
splitting of 
in the spectrum of HD 75445 shown in Fig. 2 (click here)
(corresponding to a mean field modulus of approx. 3 kG):
clearly the components of the line could still be (partly) resolved
even if the star had a considerably weaker field.
Thus one is led to the intriguing conclusion that the absence among
the known stars with resolved magnetically split lines of any
star with a phase-averaged field modulus lower than 
is not
due to observational limitations but rather reflects an intrinsic
stellar property. This is somewhat unexpected, since the
distribution of the mean longitudinal field (the field moment that has
been most determined) is continuous, strongly skewed toward
small field values, down to the limit of detectability (Landstreet
1992).
Beside the stars discussed here, we have in the course of this project observed stars in which we failed to detect magnetically resolved lines. The detailed study of those (much more numerous) stars is beyond the scope of the present work. From a preliminary inspection of their spectra, it appears that their lines either are sufficiently broadened by rotational Doppler effect to smear out a possible splitting by a field
Figure 48: Observed average of the mean magnetic field modulus
against the Strömgren photometric index (b-y). Dots:
stars with field modulus determination throughout the rotation
cycle; triangles: stars with incomplete phase coverage of
the field modulus measurements. Open symbols are used to
distinguish those stars for which the photometric index is
affected by the contribution of a companion
of the order of 3 kG, or (like HD 133792, shown in Fig. 1 (click here)) have profiles so sharp and ``clean'' that the considered star must have quite a weak field (probably not exceeding 1 kG), or perhaps even no field at all. In any case, there is no flagrant inconsistency with the existence of a sharp discontinuity in the distribution of the mean field moduli around 3 kG.
We shall not, for the time being, attempt to interpret this discontinuity, which is reported for the first time here. To address this question properly, we consider that it is necessary in a first stage to exploit more fully the information contained in our observations of stars with unresolved lines (in particular, by deriving more quantitative constraints on their magnetic fields). This will be the subject of a future work.
We shall now discuss possible correlations between the mean magnetic field modulus and other stellar properties.
Figure 49: Observed average of the mean magnetic field modulus
against the Geneva photometric index Z. Dots:
stars with field modulus determination throughout the rotation
cycle; triangles: stars with incomplete phase coverage of
the field modulus measurements. Open symbols are used to
distinguish those stars for which the photometric index is
affected by the contribution of a companion
In Fig. 48 (click here), the average 
of the field modulus is
plotted against the Strömgren photometric index (b-y).
In the figure, different symbols are
used to distinguish the stars for which full phase coverage is achieved
(dots) from those where our data only sample part of the rotation cycle
(triangles). Open symbols are used to single out three stars whose
colour is distorted by the contribution of a companion: the SB2s HD 55719
and HD 59435, and the visual binary HD 81009 (a system
discussed in some detail in Paper I, whose components of
almost equal visual magnitude are separated by less than 02).
(b-y) is primarily a temperature indicator, in spite of the facts
that it is affected by interstellar reddening and that
the chemical peculiarities of Ap stars confer them anomalous
colours. Therefore Fig. 48 (click here) essentially illustrates the relation
between the mean magnetic field modulus and the stellar temperature.
From its consideration, it appears that while the 
kG
lower limit of the
field strength distribution is roughly independent of the temperature,
hotter stars may have stronger fields than cooler stars. In other words,
the scatter of the possible intensities of the field modulus for a given
temperature increases with the temperature. The inconsistency between
this conclusion and the location of HD 215441 (the star with the
strongest magnetic field known) in Fig. 48 (click here) is only apparent and due
to our neglecting the fairly strong reddening of this star (see
e.g. Stepien 1968).
It is actually one
of the very hottest stars of the whole sample, if not the hottest one:
if its intrinsic (and not apparent) colour was used for the plot, it
would appear very close to the right edge.
The indications found here that hot Ap stars may achieve higher
magnetic field moduli than cool Ap stars may bear some relation
with the result that the longitudinal fields of hot He weak
stars and of He strong stars are in the average stronger than those
of Ap stars and of cool He weak stars (Landstreet 1992).
However, it must be stressed that our sample is limited to stars of the
latter groups, and does not encompass any hot He weak or He strong star
(which, as a matter of fact, are too hot to show the line 
).
Figure 50: Observed average of the mean magnetic field modulus
against rotation period.
Dots:
stars with well determined periods (open dots are used to
distinguish those stars for which our field modulus measurements do
not cover the whole rotation cycle); triangles:
stars for which only a lower limit of the period is known
Another photometric quantity of interest is the reddening-free
index Z of Geneva photometry. Cramer & Maeder (1981) had
suggested the existence of a linear relation between Z and
for stars earlier than A5 (that is, with the Geneva
index Y>-0.110) with a field modulus not exceeding 5 kG.
This result has been questioned by various authors (see in
particular Oetken 1984). The most recent discussion of the
issue (Paper II) did not support its validity. However, the
discussion relied to some extent on indirect determinations of the
field modulus, from the differential magnetic intensification of the
lines 
and 
. The relation
between the latter and 
appears at present more ambiguous than
it seemed from the consideration of the much smaller number of data
available when Paper II was written (see also Takeda 1991), so
that it will have to be rediscussed. Because this discussion calls on a
number of considerations that go beyond the scope of the present
paper, and because it is especially relevant to the study of the
stars whose lines are not magnetically resolved, we leave it
for a future work in which those aspects will be addressed.
Therefore, the results reported in this paper are insufficient to
settle the question of the possible relation between Z and 
,
since a large fraction of the stars for which new data have been
obtained are either too cool or too strongly magnetic to fall within
the purported range of validity of that relation. For the sake of
completeness, we show in Fig. 49 (click here) a plot of the mean magnetic field
modulus against the Z parameter. As for the dependence on
(b-y), but somewhat more marginally, there is some indication that
the field strengths can span a wider range (or have a higher upper
limit) for higher absolute value of Z. Also the absence of any
star with |Z|<0.030 is noticeable. But these two results must
be taken cautiously: the sample of stars in which we have looked for
resolved lines is likely significantly biased toward high absolute
values of Z, since in a first stage of our study, such a high value
was one of the criteria used to select some of
the candidates to be observed.
There also seems to be some anticorrelation between the mean
magnetic field modulus and the stellar rotation period. This is
illustrated in Fig. 50 (click here), where the observed average of the field
modulus, 
, is plotted against the rotation period
P. For stars for which only a lower limit of the latter has been
derived so far, this lower limit is shown. 35 stars appear in
Fig. 50 (click here) (not enough is known about the period of the 7 remaining
stars), of which 16 have a rotation period shorter than 150 days. Of these
16 stars, 10 have 
kG, while the 19 stars with a period
longer than 150 d all have 
kG. In order to check
whether this might result from a possible correlation between
the effective temperature and the rotation period, we have in
Fig. 51 (click here) plotted P against (b-y) for the stars of our sample.
From this figure, period and temperature appear essentially independent from
each other.
Let us now discuss the variations of the mean field modulus, and their meaning in terms of geometrical structure of the studied fields.
Of all the stars for which the observations obtained until now
sample adequately the variations of the mean magnetic field
modulus, only one (HD 208217) has a double-waved variation
curve, while all others only have one minimum and one maximum per
rotation cycle. This indicates that the field of these stars is
generally not a centred dipole. Indeed for such a dipole, the field
would be weakest at the magnetic equator and strongest at the
rotation poles, and would have the same intensity at
both of those poles. Provided that the geometry of the observation is
such that both poles come alternatively into view, there should
be two (identical) maxima and two (identical) minima of the field
modulus per rotation period. Statistically, in a sample of the size
studied here, there should be several stars
fulfilling the condition 
: that the magnetic field
of Ap stars is not a centred dipole, a result already clear for a
number of stars, is established here much more generally. This is
even more so because in HD 208217, the two field maxima appear
different: the field is not a centred dipole either, in spite
of the double-wave variation.
As a matter of fact, many of the 
variation curves show some
anharmonicity. This indicates that the fields
Figure 51: Rotation period of the stars with magnetically
resolved lines against the Strömgren photometric index (b-y).
Dots:
stars with well determined periods; triangles:
stars for which only a lower limit of the period is known.
Open symbols are used to
distinguish those stars for which the photometric index is
affected by the contribution of a companion
of the corresponding
stars have significant non-dipolar
components (e.g., higher-order
multipoles). Several of the 
curves (HD 137909, HD 142070,
probably HD 94660 and HD 187474) are not even symmetric about the
phases of their extrema. This means that their fields do not have
the property of cylindrical symmetry with respect to an axis
passing through the centre of the star. This conclusion is the
same as reached by Mathys (1993) from the consideration of
other field moments in other, (generally) faster-rotating Ap
stars: again, the
present study confirms that a result (the lack of symmetry of the
magnetic field) that could in the past be suspected to belong
only to some isolated, possibly exceptional object, is in fact
widespread among magnetic Ap stars.
Figure 52: Histogram showing the distribution of the relative
amplitudes of variation 
of the mean magnetic field modulus of the 42 Ap stars with resolved
magnetically split lines presently known. The shaded part of the
histogram corresponds to those stars that have been observed
throughout their rotation cycle. For stars for which observations
well distributed in phase have not been obtained yet (unshaded
part of the histogram), a lower limit of q is used
The distribution of the ratio 
of the maximum
to the minimum of the mean magnetic field modulus (hereafter
called the relative amplitude of the variation) is shown in
Fig. 52 (click here). Again, the
shaded part of the histogram corresponds to ``real''
extrema (that is, to stars observed throughout their whole
rotation cycle). Both the complete histogram and the shaded part
are strongly skewed toward small values of the ratio: this consistency
suggests that few of the stars for which phase coverage is still
incomplete will be found to have a large relative amplitude of
variation of 
. This, however, makes sense only provided that
there is no relation between the rotation period and the variation
amplitude (since the stars with incomplete phase coverage have periods
longer in the average than the periods of the stars
for which our data sample well the rotation
cycle). The possible existence of such a correlation, although
it does not seem very likely (cf. the large amplitude of variation
of the short-period star HD 65339), cannot be
definitely ruled out at present. On the other hand, it can be
noted that for 7 at least of the stars with resolved magnetically
split lines currently known, q is greater than 1.25, that is,
the upper limit for a centred dipole.
Knowing the mean field modulus alone is insufficient to characterize
the geometrical structure of the stellar magnetic field. The latter
can be better (although not uniquely) constrained from the simultaneous
consideration of the field modulus and of the longitudinal field, and
of their variation along the star's rotation cycle. As can be seen
in Table 2 (click here), measurements of the longitudinal field throughout the
rotation period are available for 12 of the stars with magnetically
resolved lines currently known. For all of them but two (HD 9996
and HD 187474), enough determinations of the field modulus
have been performed to define its variations. In Fig. 53 (click here),
we have plotted the ratio q of the extrema of the field
modulus against r, the quantity commonly used to characterize
the relative amplitude of variation of the longitudinal field.
The latter is defined as 
, where
is the smaller
Figure 53: Relative amplitude of variation q of the mean magnetic
field modulus against the ratio of the extrema of the mean longitudinal
magnetic field. Open symbols are used to distinguish two stars,
HD 9996 and HD 187474, for which only incomplete phase coverage is achieved
by our measurements of the field modulus
and 
the larger (in absolute
value) of the observed
extrema 
and 
of 
. r is negative when the longitudinal field reverses its
sign, that is (in the simple picture of dipolar field geometry)
when both magnetic poles come into view during a rotation period.
r gets close to -1 when the maximum and the minimum of 
are symmetric with respect to 0, which in the dipole model occurs
when the angle 
between the dipole axis and the stellar rotation
axis is close to 90
. Conversely, for positive values of r,
the same magnetic pole remains within view throughout the whole
rotation cycle. r tends towards 1 whenever the line of sight or
the magnetic axis (or both) become parallel to the rotation axis.
The representative points of all the stars for which the relevant
information exists are found in the lower left half of Fig. 53 (click here).
This is emphasized on the figure by the dashed line crossing it
diagonally. Thus, when the same magnetic pole is seen
at all phases, the relative amplitude of variation of the
mean field modulus is small. It seems that large variations of
can be observed only provided that both magnetic poles
come within view at some phase. This is not trivial, as
for a centred dipole, the field on the surface of the star is
maximum at both poles and minimum at the equator. Even though
this result is based upon a rather small number of stars, it
points towards the rather general existence of a quite significant
difference between the two magnetic poles of Ap stars.
From the simultaneous consideration of the curves of variation
of the longitudinal field and of the mean field modulus,
one can in principle derive uniquely a simple model of
the magnetic field structure (e.g., dipole offset along its axis
or superposition of collinear dipole and quadrupole),
provided that the extrema of 
and of 
occur at the same phases, and that the curves
of variations of both quantities are symmetric with respect to
the phases of these extrema. We have shown above that those two
conditions are generally not fulfilled. Thus more complex field
models are required: we postpone their discussion to a future
paper.
As a by-product of the present study, we have been able to perform or to improve the determination of the rotation period of a number of stars, and to set lower limits on the period of several others. Consequently, the number of Ap stars known to rotate with periods longer than 30 days has almost doubled, rising from 16 to 31 (assuming that the fact that no variations of the magnetic field of HD 177765 have been observed so far is due to its having a long period). This allows a better characterization of the slow-rotation tail of the period distribution of those stars, as shown in Fig. 54 (click here). The shaded part of the histogram corresponds to those stars whose period is known, the unshaded one to stars for which only a lower limit is available so far. The upper bin of the histogram is ``open'' towards the large values of the period. Note however that there is currently no star showing definite evidence of having a period that would exceed its upper boundary, if it had the same width as the other bins (that is, if its upper boundary was 84 years).
Figure 54: Histogram showing the distribution of the rotation
periods of Ap stars above 30 days. The shaded part of the
histogram corresponds to stars whose actual rotation period is
known; the unshaded part to stars for which only a lower limit
of the period has been set so far
Taking into account the fact that the stars for which only lower limits
of the period have been obtained so far may, when they have been studied
long enough, shift to higher bins in the histogram, the distribution
in Fig. 54 (click here) is consistent with an equipartition of the long periods
(say, over 1 year) of Ap stars on a logarithmic period scale,
or in other words, with a distribution
decreasing exponentially with the period. Nevertheless, this result at
present still has a limited statistical significance, and it will be
very interesting (but it will require quite a long time) to see if it
remains valid as new long periods are determined. But it is already
clear that periods in excess of one year, albeit infrequent, are not
exceptional among Ap stars. It has sometimes been questioned in the
past whether these periods do indeed correspond to stellar rotation,
or if they rather belong to intrinsic, secular field variations.
Leroy et al.'s (1994)
study of 
Equ (with HD 137949 the Ap star with the largest
lower limit of the period until now) in linear polarization strongly
favours the former interpretation. It should also be stressed that
extremely slowly rotating Ap stars do not seem to differ systematically
from their faster counterparts in any other respect (note in particular
the absence of temperature dependence for the stars of the present
sample in Fig. 51 (click here)). Accordingly, the mechanisms by which stellar
rotation can be slowed down to the extent observed in some Ap stars
remain enigmatic.
At the opposite extreme, a few of the Ap stars with magnetically resolved lines have periods of the order of one week or less. Their interest comes not from their period by itself (the vast majority of Ap stars have period in the 1-7 day range), but rather from its conjunction with the appearance of resolved magnetically split lines. Indeed, the fact that rotational Doppler effect is small with respect to the magnetic splitting implies that the inclination of the rotation axis with respect to the line of sight must be (very) small. In other words, these fast rotating stars with magnetically resolved lines are unique examples of stars definitely seen (rotation) pole on.
Accurate radial velocity determinations can be performed from the high-resolution spectra obtained for the present study of the magnetic fields. Thanks to this, and to the fact that we had to observe repeatedly the same stars over long time spans, we could report in this paper the occurrence of radial velocity variations in 8 stars for which such variability was not mentioned in Renson's et al. (1991) General Catalogue of Ap and Am stars. Our observations also confirmed the indication given in that catalogue that the radial velocity of HD 94660 is variable. In addition, 7 of the stars considered here belong to well studied spectroscopic binaries, HD 81009 is a visual binary system (discussed to some extent in Paper I), and HD 165474 is the secondary of a visual binary whose primary (7'' apart) is the fast-rotating, normal A star HD 165475. The detailed presentation of our radial velocity data and the derivation of the orbital elements of the binary systems are beyond the scope of this paper. For the time being, we shall restrict ourselves to a brief discussion of some intriguing facts that are unveiled through a first inspection of the measurements.
Of the 41 stars with resolved magnetically split lines that have been repeatedly observed (that is, excluding HD 47103), 18 appear to belong to binaries. This rate of occurrence of binarity (44%) is between the frequencies found by Gerbaldi et al. (1985) for SrCrEu stars (46%) and for hot Ap stars (37%). However, 16 of the 18 Ap binaries considered here have an orbital period longer than 100 d, and for at least 10 of them, the orbital period must even exceed 1000 d. Thus the distribution of the orbital periods of the binary systems containing an Ap star with magnetically resolved lines is much more skewed towards long periods than that of Ap binaries in general (see Fig. 8 (click here) of Gerbaldi et al. 1985).
The interpretation of this observational result is not straightforward. Components of wide binaries like those of interest here must have evolved in an essentially independent manner. In particular, the slow rotation of the Ap star cannot be attributed to an interaction with its companion. As a matter of fact, there is no relation between the rotation and the orbital periods of the stars of interest: for 6 of them the rotation period is shorter than the orbital period, while for another 6, the orbital period is longer (for the 6 remaining systems, the data available so far are inconclusive).
An alternative explanation is that, rather than an excess
of long orbital periods among binaries containing Ap stars
with magnetically resolved lines, what we observe in fact is a
deficiency of short orbital periods among such
binaries. In support of this idea, one may note that the binary nature
of several of the stars with resolved lines has been established from
the observation of very small radial velocity changes (of a few 
)
which have taken place over quite long timescales (several years). The
detection of such minute variations was possible only because over a
long period we repeatedly obtained high-resolution spectra of very
slowly rotating stars (which allowed us to derive very accurate radial
velocities). For the vast majority of Ap stars, observations have not
been performed in such favourable conditions: therefore we conjecture
that a significant number of Ap binaries with very long orbital
periods may have been omitted from Gerbaldi et al.'s (1985)
study. If this assumption is correct, the frequencies of binary systems
containing Ap stars obtained by Gerbaldi et al. (1985) are
underestimated. Then the occurrence of binarity among
Ap stars with magnetically resolved lines would be lower than in Ap
stars in general, due to a lack of short orbital periods for stars
with resolved lines. This would seem to indicate that Ap stars cannot
achieve slow enough rotation to show resolved magnetically split lines
if they belong to (fairly) close binaries.
Another remarkable result of the present study is that 6 of the 42 Ap stars with magnetically resolved lines belong to the group of the rapidly oscillating Ap (roAp) stars (e.g., Kurtz 1990). To some extent, this results from a selection bias, since due to the interest of some of us in roAp stars (and especially in their magnetic field), we have systematically checked all of them for possible resolution of magnetically split lines. Nevertheless, the fraction of stars showing magnetically resolved lines appears much higher among roAp stars (6 out of 28 roAp stars presently known) than among Ap stars in general. There is no obvious explanation for this. Let us just note that one may possibly draw some parallel between the low frequency of binaries among Ap stars with resolved lines (according to the hypothesis developed above) and the absence of spectroscopic binaries among roAp stars (Hubrig & Mathys 1996).