A&A 394, 1023-1037 (2002)
DOI: 10.1051/0004-6361:20021221
S. Bagnulo ^{1} - M. Landi Degl'Innocenti ^{2} - M. Landolfi ^{3} - G. Mathys ^{1}
1 - European Southern Observatory,
Alonso de Cordova 3107, Vitacura,
Santiago, Chile
2 -
C.N.R., Istituto di Radioastronomia,
Sezione di Firenze,
Largo E. Fermi 5,
50125 Firenze, Italy
3 -
Istituto Nazionale di Astrofisica,
Osservatorio Astrofisico di Arcetri,
Largo E. Fermi 5,
50125 Firenze, Italy
Received 16 May 2002 / Accepted 6 August 2002
Abstract
We present the results of a statistical study of the magnetic structure of
upper main sequence chemically peculiar stars. We have modelled a sample
of 34 stars, assuming that the magnetic morphology is described by the
superposition of a dipole and a quadrupole field, arbitrarily oriented.
In order to interpret the modelling results, we have introduced a novel set
of angles that provides one with a convenient way to represent the mutual
orientation of the quadrupolar component, the dipolar component, and the
rotation axis. Some of our results are similar to what has already been
found in previous studies, e.g., that the inclination
of the dipole axis to the rotation axis is usually large for short-period stars
and small for long-period ones - see Landstreet & Mathys (2000). We also found
that for short-period stars (approximately
)
the plane
containing the two unit vectors that characterise the quadrupole is almost
coincident with the plane containing the stellar rotation axis and the dipole
axis. Long-period stars seem to be preferentially characterised by a
quadrupole orientation such that the planes just mentioned are perpendicular.
There is also some loose indication of a continuous transition between the two
classes of stars with increasing rotational period.
Key words: stars: magnetic fields - polarization - stars: chemically peculiar
The magnetic field of CP stars is relatively strong (up to a few tens of kG) and is characterised by a smooth geometry, thus it can be easily detected via spectropolarimetric techniques. At the same time, magnetic CP stars are not rare objects (they represent about 10% of all A and B stars), which makes it possible to observe how magnetic fields affect the stellar photospheres in many different situations (i.e., in stars of different age, mass, temperature, etc.). It appears for instance that magnetic fields tend to be stronger in hotter than in cooler stars (Landstreet 1992; Mathys et al. 1997; Hubrig et al. 2000). The presence of a magnetic field is generally associated with a non-homogeneous distribution of the chemical elements (both horizontally and vertically), and long rotational periods (up to several decades). Studying these phenomena, in particular the way various stellar features correlate with the magnetic field, may permit us to improve our knowledge of stellar astrophysics in general, as it is reasonable to hypothesise that those phenomena that are so pronounced (thus so well observed) in magnetic CP stars play also some role in "normal'' upper main sequence stars.
As a matter of fact, most of the physics that we could learn from the study of CP stars is not yet understood. One of the fundamental issues that still needs to be clarified is the origin of the magnetic field. Hints to the solution of this problem could come from more statistical information on the magnetic field geometries. In particular it would be important to know if there exist "typical'' magnetic structures, and if there are relationships between magnetic fields and rotational velocity. There is an obvious need for constraints from observations and constraints obtained from secure modelling results.
Some important steps have been accomplished in the last few years, and information about the magnetic strength of CP stars has been derived from simple attempts to correlate magnetic field measurements with various stellar features. For sharp-lined stars, Mathys et al. (1997) found that the mean field modulus, averaged through the rotational cycle, is typically in the interval between 3 and 9 kG, and that the 3 kG lower end represents in fact a sharp cutoff of the distribution. (However, it should be noted that, for mean field modulus values less than 1.5-2 kG, Zeeman splitting is typically smaller than the intrinsic line width, so that the mean field modulus cannot be measured. Therefore, it is more appropriate to say that the evidence found by Mathys et al. 1997 is for a shortage of sharp-lined stars with mean field modulus between about 1.5 and 3 kG.) Mathys et al. (1997) also discovered that the mean field modulus of stars with rotation period longer than 150 d is never stronger than 7.5 kG, which suggests a sort of anti-correlation between star's rotation period and mean field modulus. Finally, one of the most important findings about the magnetic structure of chemically peculiar stars was recently obtained from a simple modelling of the mean longitudinal field and the mean field modulus. Assuming an axisymmetric model described by the superposition of a dipole plus colinear quadrupole and octupole, Landstreet & Mathys (2000) found that slow rotating CP stars (i.e., stars with rotational period longer than one month) have a magnetic axis tilted at a small angle () with respect to the rotation axis. It is clear that these kinds of information are extremely valuable in order to explain the formation of the magnetic field (see Moss 2001), and also to understand all those phenomena that are most likely connected to the presence of a magnetic field, such as, e.g., the loss of angular momentum (Stepien & Landstreet 2002).
In the past, it has been commonly assumed that magnetic fields of CP stars have a quasi-dipolar morphology that can be described in terms of a low-order and axisymmetric multipolar expansion. This assumption was based on the fact that in several cases an axisymmetric morphology was indeed found sufficient to interpret the available observations - mainly represented by determinations of mean longitudinal field. Recent developments of the observing techniques (including spectropolarimetry of all Stokes profiles, see Wade et al. 2000a), followed by attempts to interpret the newly obtained data, point to evidence for ubiquitous departures of the magnetic structures from axisymmetric geometries (e.g., Mathys et al. 1997; Bagnulo et al. 1999, hereafter Paper II; Bagnulo et al. 2001). From the theoretical point of view we note that - should the field be either a fossil relic or dynamo generated - there is no reason why stellar magnetic fields should have cylindrical symmetries. In fact, there is no reason why a fossil field should have an axisymmetric geometry, and it is theoretically predicted that dynamo generated fields do develop a non-axisymmetric geometry (Moss 2001). These considerations prompt for further searches for statistical properties of the magnetic structures of CP stars by making use of a modelling technique not limited by simplifying approximations on the magnetic geometry. A fully adequate modelling technique based on the inversion of Stokes profiles may exist: the Zeeman Doppler Imaging (ZDI; see Piskunov & Kochukhov 2002). However, only few stars have been modelled in detail through the inversion of Stokes profiles (e.g., CVn, see Kochukhov et al. 2002). We are lacking sufficient high quality data (in terms of spectral and phase coverage, spectral resolution and signal to noise ratio) to be able to perform a statistically meaningful analysis based on ZDI. On the other hand, we have at our disposal a large number of magnetic field observations obtained through measurements of Zeeman splitting and of the low-order moments of Stokes profiles I and V, such as the aforementioned mean field modulus and longitudinal field, but also the crossover and the mean quadratic field. These kinds of data (referred to as "magnetic observables'' - see, e.g., Bagnulo et al. 2000, hereafter Paper III) can be obtained from comparatively low-resolution, low signal-to-noise ratio spectra, even, in certain cases, from spectra recorded with photographic plates, making it possible to extract information on the stellar magnetic geometry from observational data that are not suitable for application of straight inversion techniques.
In this paper we present a statistical investigation of the magnetic structure of CP stars based on the interpretation of the magnetic observables, under the assumption that the stellar magnetic field is described by the superposition of a dipole and a quadrupole field, arbitrarily oriented. Although it seems unlikely that such a low-order multipolar expansion can accurately describe the real magnetic configurations of CP stars (see, e.g., Bagnulo et al. 2001), it is reasonable to hope that the analysis of a large sample of stars can still provide us with some insight into the problem of whether there exist statistical properties of the magnetic structures of CP stars other than those that have been already found.
A formalism for the description of the magnetic field in terms of dipole plus quadrupole (arbitrarily oriented) was presented by Bagnulo et al. (1996) and Landolfi et al. (1998) - hereafter Paper I - and modelling results for individual stars were presented, e.g., in Paper II and Bagnulo & Landolfi (1999). Preliminary statistical results were presented by Bagnulo (2001), who found for instance that in faster rotating stars, one of the unit vectors that define the quadrupolar component is generally aligned to the rotation axis. Then it was realised that a set of new angles could be adopted in order to obtain a clearer description of the statistical results, in particular by explicitly referring to the orientation of the plane defined by the quadrupolar component with respect to the dipole axis and the rotation axis. Such angles are introduced in Sect. 2, which also presents the guidelines for the statistical analysis performed in Sects. 3 and 4. In Sect. 5 we summarise and briefly discuss our results.
= | |
= | |
The four magnetic observables are denoted in this paper by the notations
The magnetic geometry of the dipole plus quadrupole model is illustrated in detail in Paper I. We simply recall that such model depends on the following 10 parameters:
As explained in Paper III, observations of
,
,
,
and
,
do not allow one to distinguish between two magnetic
configurations symmetrical about the plane containing the rotation axis and
the dipole axis. Such configurations are characterised by the same values of
,
,
,
,
,
f_{0}, while the remaining angles are related by
No. | HD | Other Identifier | P | R_{*} | g | p1 | p2 | ||||||
1 | 2453 | GR And | 33 | 6 | 6 | 14 | - | 2.97 | 2.08 | 2.25 | 1.57 | ||
2 | 12288 | V540 Cas | 20 | - | - | 28 | - | 3.28 | 1.75 | 1.75 | |||
3 | 14437 | AG+42 247 | 36 | - | - | 31 | - | 3.08 | 1.96 | 1.96 | |||
4 | 24712 | DO Eri | 82 | 7 | 5 | - | 5.6 | 1.83 | 3.56 | 3.56 | |||
5 | 51684 | CD-40 2796 | 7 | 4 | 7 | 10 | - | 3.58 | 0.93 | 0.59 | 0.68 | ||
6 | 61468 | CD-27 4341 | 5 | 5 | 5 | 15 | - | 4.16 | >8 | 6.29 | 1.57 | ||
7 | 65339 | 53 Cam | 66 | 6 | - | 36 | 13.0 | 2.56 | 4.21 | 4.21 | |||
8 | 70331 | CD-47 3803 | 16 | 16 | 16 | 38 | - | - | 1.46 | 1.58 | 1.34 | ||
9 | 75445 | CD-38 4907 | 3 | 3 | 3 | 11 | - | 2.34 | 1.54 | 0.87 | |||
10 | 81009 | KU Hya | 13 | 13 | 13 | 42 | - | 2.61 | 1.95 | 1.32 | 1.38 | ||
11 | 83368 | HR 3831 | 12 | 12 | 11 | - | 32.6 | 2.23 | 2.08 | 2.08 | |||
12 | 93507 | CD-67 955 | 12 | 12 | 12 | 30 | - | 3.73 | 3.70 | 1.59 | 2.03 | ||
13 | 94660 | KQ Vel | 2764.0 | 12 | 12 | 12 | 25 | - | 3.15 | >8 | 1.60 | 1.39 | |
14 | 96446 | CD-59 3544 | 9 | 9 | 9 | - | 10.0 | 6.43 | 2.98 | 2.98 | |||
15 | 116114 | BD-17 3829 | 7 | 7 | 7 | 24 | - | 3.06 | 1.72 | 1.46 | 2.24 | ||
16 | 116458 | HR 5049 | 20 | 20 | 20 | 21 | - | 3.54 | 2.42 | 1.89 | 2.01 | ||
17 | 119419 | HR 5158 | 22 | 22 | 22 | - | 34.8 | 2.23 | 1.71 | 1.71 | |||
18 | 125248 | CS Vir | 31 | 21 | 21 | - | 8.0 | 1.97 | 1.85 | 1.85 | |||
19 | 126515 | Preston's star | 25 | 19 | 19 | 22 | - | 2.35 | 4.56 | 2.99 | 4.35 | ||
20 | 137509 | NN Aps | 14 | 14 | 14 | - | 28.1 | 3.59 | 1.20 | 1.20 | |||
21 | 137909 | CrB | 51 | 21 | 21 | 46 | 3.5 | 3.25 | 2.09 | 2.15 | 2.45 | ||
22 | 142070 | AG-00 2049 | 13 | 13 | 13 | 24 | 13.5 | 5.42 | >8 | 3.19 | 2.72 | ||
23 | 144897 | CD-40 10236 | 13 | 13 | 13 | 28 | - | 3.97 | 2.53 | 1.70 | 1.93 | ||
24 | 147010 | V933 Sco | 19 | 19 | 19 | - | 22.1 | 2.85 | 1.68 | 1.68 | |||
25 | 153882 | HR 6326 | 17 | 17 | 17 | - | 21.0 | 3.81 | 2.17 | 2.17 | |||
26 | 175362 | V686 CrA | 30 | 30 | 29 | - | 15.0 | 2.68 | 3.97 | 3.97 | |||
27 | 187474 | V3961 Sgr | 2364.0 | 20 | 20 | 20 | 40 | - | 2.93 | 3.82 | 3.59 | 3.36 | |
28 | 188041 | V1291 Aql | 77 | 8 | - | 18 | - | 3.14 | 1.41 | 1.41 | |||
29 | 192678 | V1372 Cyg | 14 | - | - | 34 | - | 3.05 | 2.17 | 2.17 | |||
30 | 200311 | V2200 Cyg | 24 | - | - | 35 | 9.0 | 2.18 | 2.37 | 2.37 | |||
31 | 208217 | BD Ind | 8 | 8 | 8 | 38 | 15.3 | 2.60 | 1.58 | 1.73 | 1.64 | ||
32 | 215441 | Babcock's star | 14 | - | - | 11 | 5.0 | 2.45 | 2.51 | 2.51 | |||
33 | 318107 | V970 Sco | 6 | 6 | 6 | 36 | - | 2.46 | 3.37 | 2.15 | 3.55 | ||
34 | 335238 | AG+29 2421 | 3 | 3 | 3 | 18 | - | 3.69 | 1.52 | 1.15 |
A convenient way to visualise the quadrupole orientation is to introduce
the plane containing the unit vectors ,
and its pole Q,
defined by
Figure 1: Geometry of the dipole plus quadrupole model. The rotation axis intersects the stellar surface at R, and makes an angle i (not shown in the figure) with the line of sight. D is the positive pole of the dipole. Q is the pole of the quadrupole plane defined by the unit vectors and . | |
Open with DEXTER |
The stars of our sample are listed in Table 1. The columns contain the following data (from left to right): a reference number used in this paper; the HD number of the object; a second identifier for the star; the stellar rotational period in days; the number of observations used in this work for each of the four magnetic observables; the observational value of the projected equatorial velocity; the estimated value of the stellar radius. The meaning of the last three columns (labelled with g, p1, and p2) will be explained later.
Most magnetic field measurements were taken from Mathys (1994) (mean longitudinal field), Mathys (1995a) (crossover), Mathys (1995b) (mean quadratic field, but the determinations were revised as explained in Mathys et al., in preparation), Mathys & Hubrig (1997) (longitudinal field, crossover, quadratic field), Mathys et al. (1997) (mean field modulus). We also considered additional measurements, to be published by Mathys et al. (in preparation), most of which were already used by Landstreet & Mathys (2000). Additional mean longitudinal field data were taken from Bagnulo et al. (2001), Borra & Landstreet (1977, 1978), Hildebrandt et al. (1997), Hill et al. (1998), Leone et al. (2000), Leone & Catanzaro (2001), Preston & Stepien (1968), Wade et al. (1996, 1997, 2000a, 2000b, 2000c). Further mean field modulus data were taken from Huchra (1972), and Preston (1969).
The adopted value of the rotational period and estimates of the projected equatorial velocity come mainly from the same references as for the magnetic data. However, in many cases, the rotational period was re-computed through a least-square technique based on a second-order Fourier expansion to the longitudinal field and mean field modulus measurements. Estimates for the stellar radius R_{*} were taken from North (1998) and Hubrig et al. (2000). Additional R_{*} estimates were kindly provided to us by North (2001, private communication).
The aim of this study is not so much to find the "best'' magnetic model for each individual star as to characterise the sample as a whole in order to see whether there are any "preferred'' magnetic geometries (in the framework of the assumed dipole plus quadrupole model) and/or correlations between magnetic model and the rotational period.
The main difficulty underlying an analysis of this kind is due to the fact that the inversion algorithm usually produces, for a given star, several different models, i.e., several sets of parameters, corresponding to local minima of the hypersurface. As discussed in Papers I, II, and III, this is primarily due to the relatively large observational errors. For some stars the values at the various minima are very different from each other, but in other cases these values are not so different. Therefore, it looks rather unsafe to restrict the statistical analysis to best-fit models (i.e., those corresponding to the absolute minimum). Furthermore, even if this restriction were applied, the magnetic models of the individual stars could not be considered equally reliable, being characterised by different values.
The most natural way to deal with these problems is to "weight'' the
parameter sets associated with the
minima. In this work we
introduced a weight
for each star, defined by
An additional difficulty concerns the type of observations to be used for the modelling. In particular, as noticed already in previous works (e.g., Paper III; Landstreet & Mathys 2000), for many stars, and measurements are not consistent among themselves. For this reason we performed separate investigations, by using all observations together (Sect. 3), and by disregarding either or observations (Sect. 4). The former kind of analysis, i.e., based on all observations together, has the obvious advantage that all the information available for a given star is used in the modelling; moreover, all the 34 stars listed in Table 1 can be considered. The latter kind of analysis may help to reveal if some results are likely biased by systematic measurement errors.
Figure 2: Left: distribution of the minimum values for the magnetic models of the 34 stars of the sample (see Col. g of Table 1). Right: same for the models corresponding to all (absolute and relative) minima. Further absolute and relative minima with larger values are not shown in the figure. See text for details. | |
Open with DEXTER |
Figure 2 (right) shows the distribution of the reduced values for all the fits (corresponding either to the absolute minimum or to relative minima of the hypersurface). Altogether there are 168 minima, 12 of which with . The relevant distribution is quite similar to that of the left panel, with a main group around and a secondary group at ; for larger values, the number of minima decreases rapidly. We decided to confine our study to models with : this leads to 147 models, associated with the 31 stars of the sample.
Consider first the magnetic field strength. Figure 3 (left) shows
the distribution of the dipole amplitude, .
It is a rather
regular, bell-shaped distribution peaking at something less than 10 kG; most
models are characterised by
values ranging from 3 kG to 20 kG.
We recall that the histogram represents the
distribution
weighted according to Eqs. (5)-(7): explicitly, the height of
the histogram in the ith interval is
Figure 3: Left: distribution of the dipole field amplitude. is expressed in kG. The isolated tooth on the far right is associated with Babcock's star (HD 215441, ). Right: distribution of the ratio of quadrupole to dipole field amplitude. | |
Open with DEXTER |
Figure 4: Distribution of the phase-averaged mean field modulus (expressed in kG). The tooth on the right is Babcock's star. | |
Open with DEXTER |
Next we consider the distributions of the different angles. These are plotted
in Fig. 5, together with the curves describing the relevant random
distributions (sinusoidal variation for i, ,
,
;
flat distribution for
and ). It should
be noticed that the apparent symmetry characteristics of several histograms
is just a consequence of the properties in Eqs. (2)-(4). The
distributions of i and
are symmetrical about
because of
the degeneracy of Eq. (2), the distribution of
because of Eq. (3); similarly, the distributions of
and
are invariant under a
shift owing to Eqs. (2)
and (4), respectively. By contrast, the distribution of
is
unaffected by those properties (no symmetry characteristics).
Figure 5: Distributions of the angles specifying the magnetic geometry, compared with the relevant random distributions. Angles are expressed in degrees. | |
Open with DEXTER |
It is obvious from Fig. 5 that most distributions, and especially
that of
,
are very different from the relevant random
distributions. In order to quantify such difference, we decided to apply the
Kolmogorov-Smirnov test (see, e.g., Press et al. 1986) to
each distribution. As is well known, the test is based on the measurement of the
maximum distance, D, between the cumulative distribution of the
observational data and the corresponding theoretical distribution. Such
distributions are shown in Fig. 6.
Figure 6: Cumulative distributions of the angles in Fig. 5, and the corresponding random distributions. | |
Open with DEXTER |
Strictly speaking, the Kolmogorov-Smirnov test in not applicable in our case, because it presupposes a set of "unweighted'' measurements of the observational quantity, whereas our determinations are weighted as described in Sect. 2. For a set of N unweighted (different) measurements, all the "steps'' of the cumulative distribution are the same (=1/N). The probability that the distribution of the measurements is consistent with an assigned theoretical distribution obviously depends both on Dand on N, and, for a given D value, decreases strongly with increasing N.
In our case, we are forced to introduce an "effective number'' of
measurements,
.
The choice
(the number
of stars in the sample) would be correct if we had just one model (with the
same
value) for each star: as this is not the case, the value
is certainly an underestimate. The choice
(the overall number of models) would be correct if all
the determinations had the same weight: as many models are characterised by
rather large
values (hence, rather small weights),
is certainly an overestimate. The most reasonable choice
is something intermediate: for instance, one could take the quadratic mean
of the steps' heights ,
Table 2 shows the test results for the distributions of the six
angles, according to different choices of
.
It is clear that
all the distributions are non-random, and that the largest discrepancies
concern
,
,
and i.
angle | D | p_{1} | p_{2} | p_{3} | |
i | 0.24 | 0.05 | |||
0.29 | 0.01 | ||||
0.33 | |||||
0.16 | 0.35 | 0.01 | |||
0.12 | 0.69 | 0.09 | 0.02 | ||
0.12 | 0.72 | 0.13 | 0.03 |
One might suspect that deviations from the random distribution like those of Figs. 5 and 6 could be an artifact due to the inversion algorithm. Because of the large number of parameters and the ensuing complexity of the hypersurface, noisy observational data might generate some unexpected systematic effect. To check this point we performed extensive numerical simulations, by presenting to the fitting code synthetic and noisy data for a variety of dipole plus quadrupole, randomly oriented models. Such calculations showed that non-random distributions like those of Figs. 5 and 6 can by no means be ascribed to systematic effects of the inversion code.
Let us consider the inclination angle i - the only "non-magnetic'' angle.
Its distribution clearly shows an excess of small values. It should be noticed,
however, that for most of the stars (22 out of 31) the mean field modulus
was actually detected: this means that the sample is certainly
affected by a selection effect which tends to favour small i values, since
positive detections of that quantity require small equatorial velocity,
.
In fact, the distribution of i for the 9
remaining stars (Fig. 7 left) is very different from that
in Fig. 5, and much closer to the random distribution: for
,
14, and 16 (corresponding to the criteria adopted in
Table 2 for p_{1}, p_{2}, and p_{3}, respectively) the probability
p resulting from the Kolmogorov-Smirnov test is 0.91, 0.73, and 0.65,
respectively.
Figure 7: Distribution of the inclination angle i for the 9 stars with no observations (left), and for the 10 stars with rotational period larger than 40 days (right). | |
Open with DEXTER |
Following another approach, we might consider the distribution of i for all stars with sufficiently small rotational velocity (period P>P_{0}): such sample should likely be unaffected by selection effects. Setting days - actually, any number of days between 35 and 48 - we get a sub-sample of 10 stars (none of which pertaining to the former 9-star sample) leading to the distribution of Fig. 7, right. For , 30, and 37, corresponding to the same criteria as above, we have p=0.82, 0.21, and 0.12, respectively. This probability decreases strongly with decreasing P_{0}.
On the whole, the difference from the random distribution looks rather reduced, though some excess of small i values remains. This is not fully understood. It could be due either to inadequacy of the dipole plus quadrupole model, or to a weak, but real intrinsic non-randomness of the star sample, although it should be noted that this latter phenomenon does not have an obvious physical justification. We recall that - contrary to axisymmetric models - our magnetic model involves no degeneracy between the angles i and , the only "physical'' degeneracy being that of Eq. (2). Thus a randomization - so to say - of the i distribution at expense of the distribution is not allowed in our case (see Landstreet & Mathys 2000).
Let us now turn our attention to the "magnetic'' parameters of the model. Since the magnetic structure may plausibly be connected with the star's rotation, we looked for possible correlations between the magnetic parameters and the rotational period P.
Figure 8 shows the dipole amplitude
(upper panel)
and the ratio of quadrupole to dipole amplitude
(lower panel) as
functions of P. The figure refers to all the 147 models associated with the
31 stars of our sample. Both quantities, and particularly the latter, decrease
on average with increasing P: faster rotators seem to be characterised by
stronger and predominantly quadrupolar fields.
Figure 8: Dipole amplitude (upper panel) and ratio of quadrupole to dipole amplitude (lower panel) versus rotational period. P is expressed in days, in kG. The dots' area is proportional to the weight of Eq. (6). | |
Open with DEXTER |
Similarly, Fig. 9 shows the phase-averaged mean field modulus as a function of P. The number of dots is much smaller than in Fig. 8 because - as already noted - the different models of a given star yield almost the same value. Consistently with the data in Fig. 8, tends to decrease with increasing P.
As far as the magnetic geometry is concerned, consider first the
quadrupolar component. The first step is to find the location of the
point Q on the stellar surface (cf. Fig. 1). The plot in
Fig. 10 shows the location of Q in the
plane for all models^{}. It turns out that the
dots tend to cluster round
,
and, to a smaller
extent, round
and
:
such characteristics were already apparent
from the histograms of Fig. 5. But the most interesting aspect arises
when connecting the dots corresponding to all the models of a given star: one
finds that, for a number of stars, the location of the point Q lies within a
small region around the point
.
There are
seven stars of this kind, shown in the lower panel of Fig. 10. For
three of them - HD 96446, HD 119419, and HD 147010, labelled in the figure as
No. 14, 17, and 24, respectively - the location of the point Q is univocal
(one minimum in the
hypersurface), while for the others - HD 70331,
HD 137509, HD 208217 and HD 318107 (labelled in the figure with 8, 20, 31, and
33, respectively) - there are several possible locations. In any case, for all
of them the point Q lies within a small spherical cap, centred at the point A
and having a width
(which amounts to about 7% of the spherical
surface) - see Fig. 11. These stars will be referred to in the
following as "class I''.
Figure 9: The phase-averaged mean field modulus (in kG) against the rotational period (in days). | |
Open with DEXTER |
Figure 10: Upper panel: the position of the point Q in the plane for each of the fitting models. Angles are expressed in degrees. The axis spans a interval centred at in order to show clearly the regions of interest (cf. Fig. 11): two spherical caps with widths and centred at the point . The dots' area is proportional to defined in Eq. (6). Full dots indicate "class I'' and "class II'' stars, labelled according to their reference number in Table 1. Class II stars HD 2453, HD 12288, and HD 187474 (labelled as 1, 2, and 27) have five, one, and two models, respectively. Lower panel: an enlargement of the class I region, showing the connections between models of the same star. Class I stars are HD 70331, HD 96446, HD 119419, HD 137509, HD 147010, HD 208217, and HD 318107 (indicated in the figure as No. 8, 14, 17, 20, 24, 31, and 33). | |
Open with DEXTER |
Figure 11: Location of the quadrupole pole Q on the stellar surface for "class I'' and "class II'' stars. | |
Open with DEXTER |
Similarly, we found that for 3 other stars - HD 2453, HD 12288, and HD 187474 (No. 1, 2, and 27, respectively) - the point Q lies within a small "ring'' between and , i.e., a ring around the meridian containing the stellar rotation axis and the dipole axis (about 10% of the spherical surface). These stars, denoted as "class II'' in the following, are shown in the upper panel of Fig. 10.
For the remaining stars, the location of the point Q is usually scattered throughout the plane according to the different models, so that no precise characterisation is possible.
It should be noticed that, in terms of the angles , , , , class I stars are characterised by the property or and or (the unit vectors and lie in the plane of the rotation axis and dipole axis). It follows that the degeneracy of Eq. (2) does not apply to such stars: this can be formally seen by combining Eqs. (2) and (4).
Let us now examine whether any correlation exists between the location of the
point Q on the stellar surface and the rotational period. If we restrict
attention to class I and class II stars (for which such location is more
precise), we obtain the remarkable result shown in Fig. 12: there
is striking evidence that class I stars are short-period, and
class II stars are long-period. A similar result was anticipated by
Bagnulo (2001).
Figure 12: The angular distance d (in degrees) of the point Q from the point as a function of the rotational period (in days), for class I (lower left) and class II (upper right) stars. Dots' area is proportional to of Eq. (6). | |
Open with DEXTER |
The idea of a general "law'', valid for all the stars in our sample,
according to which the angular distance d of the point Q from the point
is an increasing function of the
rotational period, looks really tempting. However, we just observed that for
a star other than class I or II, the locations of the point Q associated with
the various models are generally far apart in the
plane - which entails scattered values (i.e., both large
and small) for d. In fact, when d is plotted against P for all models
of all the stars in the sample, we obtain the results of Fig. 13,
which look rather ambiguous. In particular, some long-period stars (especially
HD 116458, HD 188041, HD 51684, and HD 93507 - No. 16, 28, 5, and 12,
respectively) have both large-d and small-d models. (Note that in the case
of HD 93507, the determination of the magnetic observables - hence the
modelling - is hampered by the fact that the distribution of Fe over the
stellar surface is rather strongly inhomogeneous.)
Figure 13: Same as Fig. 12 for all models of all the stars in the sample. The vertical lines connect the dots corresponding to the different models of stars HD 116458, HD 188041, HD 51684, and HD 93507, that are labelled in the figure as 16, 28, 5, and 12, respectively. | |
Open with DEXTER |
However, if we restrict attention to the best model (associated with the absolute minimum) of each of the 31 stars, we obtain the plot in Fig. 14. It really seems that the data of the whole sample support (approximately) the "law'' mentioned above.
Finally, one could suspect that a relationship exists between d and , since both quantities are somewhat related to the rotation period. However, inspection to the plot of d vs. did not reveal any obvious correlation.
Figure 14: Same as Fig. 13 for the best models of all the stars in the sample. Full dots are class I and class II stars. Dots' area is proportional to the weight of Eq. (5), which is more appropriate in this context. | |
Open with DEXTER |
In order to characterise the quadrupole orientation, one needs to know
- beside the location of the point Q - the directions of the unit vectors
and
in their plane, i.e., the angles
and
or, equivalently,
and .
Contrary to Q, those
directions proved to be basically uncorrelated with the rotational period: this
is obvious even if we restrict the analysis to the best models, as shown in
Fig. 15. The only "anomalies'' of
and
are those
appearing in Fig. 5: some excess of
values around
,
and of
values around
and
.
Though the discrepancies
from the relevant random distributions are less marked than for the other
angles (cf. Table 2), it is interesting to observe that the distribution tends to rule out linear quadrupole models
(parallel or antiparallel to ,
which entails
or
).
Figure 15: The angles (upper panel) and (lower panel) - expressed in degrees - against the rotational period in days. Only the best-fit values are shown. Full/open dots and dots' area have the same meaning as in Fig. 14. | |
Open with DEXTER |
Next we consider the dipole orientation. Figure 16 shows the angle as a function of the rotational period. The upper panel refers to all models of all the 31 stars, and shows a deficiency of large values for long period stars. If we confine attention to the best-fit models (lower panel), we better see that tends to decrease with increasing rotational period. This is in agreement with the main result of Landstreet & Mathys (2000): is usually small for slow rotators and large for fast rotators. Such agreement is especially significant for long-period stars, where the quadrupolar component is less important (see Fig. 8, lower panel), so that the orientation of the magnetic structure is dominated by the dipole orientation. A notable exception to this general trend is represented by HD 187474. This star has a rotation period of 2364 d (the longest in the g sample), and the best-model predicts a tilt angle between dipole axis and rotation axis of 80. A detailed study of its chemical peculiarity has been recently presented by Strasser et al. (2001).
Figure 16: The angle (in degrees) versus the rotational period (in days), for all models (upper panel) and for best-fit models (lower panel). Dot's area is proportional to and to in the upper and lower panel, respectively. Full dots are class I and class II stars. | |
Open with DEXTER |
On the whole, the results above show the existence of some correlation between the magnetic structure of the stars of our sample and the rotational period.
In fact, in Paper III we determined the minimum combinations of magnetic observables which allow a dipole plus (non linear) quadrupole configuration to be recovered. In particular, Table 1 of that paper shows that the configuration can be recovered from (a sufficient number of) and observations, as well as from and observations; whereas it cannot be recovered either from alone, or from and . Bearing in mind these results, we see that the sub-samples for our partial analyses contain 25 stars each.
These are shown in the last two columns of Table 1, where the label "p1'' means that the fit is based on all available observations except , while the label "p2'' means that it is based on all available observations except . It should be noticed that stars HD 75445 and HD 335238, which in principle pertain to sub-sample p2 as for both of them there are measurements of , , and , are actually excluded from p2 because of the small number of observations (9 in all), which renders the fit impossible. On the other hand, stars HD 61468, HD 94660, and HD 142070, which in the "global'' investigation give rise to low-quality fits ( ) so that they were excluded from the statistical analysis, in the "partial'' investigations give rise to much better fits, thus they have been included in the analysis.
If we restricted the study to the stars common to all three samples g, p1, and p2, we would be left with only 13 objects - too small a number for a statistical analysis. On the other hand, a direct comparison of samples gand p1, or g and p2, is not very meaningful because the fit of several of the common stars is identical by definition: for instance, this is the case for the g-fit and the p1-fit of star HD 12288, because for that star there are no observations of , hence either fit is in fact the same fit (based on and observations).
For these reasons, the correct attitude is to forget comparisons and to look at the "global'' and "partial'' analyses as separate investigations. The final goal is to see whether the overall picture of magnetic CP stars arising from analyses of different kind (in the sense specified above) is the same or not; where by "overall picture'' we intend the presence of recurrent features and/or correlations among the parameters of the magnetic model.
Columns p1 and p2 of Table 1 show the value of the reduced for the best-fit model of the 25 stars pertaining to either sample. Such data
is also shown in the left panels of Fig. 17. The two distributions
are similar to that of the g-case (cf. Fig. 2 left), though somewhat
displaced towards smaller values - which reflects the difficulties encountered
in fitting together
and
observations.
In the right panels of Fig. 17 we show the distributions of the
reduced
for all models (absolute and relative minima): on the whole, there are 140 models for the p1-case (7 of which with
)
and 60 models for the p2-case (20 of which with ).
These figures show that the average number of models per star - i.e., the
number of minima in the
hypersurface - increases strongly when
observations are included: from 2.4 for the p2-case to 4.9
for the g-case and 5.6 for the p1-case. Thus inclusion of
observations renders the
hypersurface much more complex: this is
possibly due to a stronger degeneracy of
compared to the
other magnetic observables (different dipole plus quadrupole
configurations leading to similar
curves). As in
Sect. 3, we restricted the statistical analysis to the models with
.
Figure 17: Same as Fig. 2 for the stars of samples p1 (upper panels) and p2 (lower panels). Either sample contains 25 stars. | |
Open with DEXTER |
Figure 18 shows the distributions of the dipole amplitude, and of the ratio between quadrupole and dipole amplitude, for the two "partial'' samples, while Fig. 19 refers to the phase-averaged mean field modulus. Such distributions are in fact quite similar to those of Sect. 3 (cf. Figs. 3 and 4). Note that Babcock's star does not belong to the p2-sample (see Table 1).
It is interesting to compare our Figs. 4 and 19 with the histogram showing the
observed phase-averaged magnetic field modulus presented in Fig. 47 of
Mathys et al. (1997). We recall that Fig. 4 is based on the modelling results
of all available magnetic observables; the left panel of Fig. 19 is obtained
from
the results of the analysis of all quantities except the mean quadratic field;
the right panel of Fig. 19 from results of the analysis of all quantities except
the mean field modulus; Fig. 47 of Mathys et al. (1997) is purely based on
the observations, and includes stars that have not been monitored throughout
the rotation cycle. A similar overall picture emerges from all these figures.
A certain degree of consistency is to be expected for the
following reason: the mean field modulus is generally a quantity that does
not change much over the rotation phase, and even few observed points, or a
very rough model are sufficient to characterise the
curve
with good approximation. We note in particular that all figures show
a definite lower limit in the
distribution at about 3 kG
- this characteristic has been already discussed in Mathys et al. (1997),
and in Hubrig et al. (2000). It is interesting to note that the same result
is here recovered also for stars with no magnetically split lines,
though still with relatively low
values. Notably the same sharp
cut-off is found from the modelling of
,
,
and
(see right panel
of Fig. 19).
Figure 18: Same as Fig. 3 for samples p1 (upper panels) and p2 (lower panels). | |
Open with DEXTER |
Figure 19: Same as Fig. 4 for p1 (left) and p2 (right). | |
Open with DEXTER |
The distributions of the angles i, ,
,
,
,
for the p1 and p2 cases are shown in
Figs. 20 and 21, respectively. It appears that the
deviations from the relevant random distributions are basically the same as
in the g-case (cf. Fig. 5). These include an excess of small values and large
values, and, to a smaller extent, of
and
values around
.
The peaks of the
"barycentric angle''
around
and
are still visible,
though the latter is somewhat shifted in the p2-case.
Figure 20: Same as Fig. 5 for the p1-case. | |
Open with DEXTER |
Figure 21: Same as Fig. 5 for the p2-case. | |
Open with DEXTER |
As far as the inclination angle i is concerned, it should be borne in mind that - contrary to the g-case - was detected for all stars of the p1-sample. The selection effect is therefore larger than in the g-case, so that the excess of small i values is even more prominent (see Fig. 20). On the other hand, the p2-sample contains a larger percentage of long-period stars (say, with ) with respect to the g-sample, which explains why the excess of small ivalues is less prominent (see Fig. 21).
Consider next the correlations with the rotational period. Concerning the
field strength, the behaviour of the dipole and quadrupole amplitudes
is quite similar to the g-case; we show only the phase-averaged mean field
modulus (Fig. 22), which tends to decrease with increasing P as in
Fig. 9.
Figure 22: Same as Fig. 9 for cases p1 (upper panel) and p2 (lower panel). The larger dots' area in the lower panel is just a consequence of the smaller average number of models per star in the p2-case. | |
Open with DEXTER |
As to the quadrupole orientation, we repeated for the p1 and p2 cases the
same analysis on the location of the point Q on the spherical surface described
in Sect. 3 (see Fig. 10). For both cases, we found a number
of "class I'' and "class II'' stars (cf. Fig. 11), and we plotted
the angular distance d between points Q and A against the rotational period
(cf. Figs. 12 to 14). The final results (based on the
best fit of each star) are shown in Fig. 23.
Figure 23: Same as Fig. 14 for cases p1 (upper panel) and p2 (lower panel). Full dots represent class I and class II stars, defined as in Sect. 3. Some stars are labelled by their reference number of Table 1. | |
Open with DEXTER |
Altogether, the p1-case yields a picture similar to the g-case (see Fig. 14), with the exception of star HD 94660 (absent in the g-sample) - which is the longest-period star of all the 34 objects and which is characterised by a small d value - and possibly of HD 116458.
By contrast, the p2-case yields a very different picture, which tends to deny the existence of a correlation between d and P. In particular, it is worth noticing the behaviour of stars HD 116458, HD 93507, and HD 51684. As shown in Fig. 13, in the g-case they have both large-d and small-d models, but the best-fit ones are the former (see Fig. 14). In the p2-case, on the contrary, the best-fit models of all three stars correspond to small d values. On the other side, it should also be noticed that the short-period star HD 142070 (absent in the g-case) proves to be a class I object both in the p1 and the p2 case, and that the long-period star HD 61468 (also absent in the g-case) proves to be of class II in the p1-case and nearly of class II in the p2-case, in accordance with the tentative "law'' of Sect. 3.
On the whole, the existence of some correlation between d and P (small-dmodels for short-period stars, large-d models for long-period ones) seems somewhat related to the reliability of observations, being more likely if they were not fully reliable, and vice versa. In any case, there is a considerable excess of models with and/or : this comes out clearly from all three samples (see Figs. 5, 20, and 21).
As for the angles
and ,
they seem to be uncorrelated - as in
the g-case - with the rotational period. For the angle ,
characterising
the dipole orientation, we found the results presented in Fig. 24.
They roughly confirm, though with a larger scattering, the behaviour deduced
from the g-case (cf. Fig. 16): large
values for short-period
stars, small
values for long-period ones.
Figure 24: Same as the lower panel of Fig. 16 for cases p1 (upper panel) and p2 (lower panel). | |
Open with DEXTER |
In most of the cases we could not obtain a unique model, even for those stars whose magnetic geometry seems strongly constrained by the observations. This is related to the relatively large observational errors, and possibly to the fact that the actual magnetic geometry of CP stars is too complex to be described in terms of a second-order multipolar expansion. We thus decided to follow an approach that gives a limited importance to the best-model of the individual stars, searching instead for characteristics that emerge from the whole set of models (corresponding to the absolute and relative minima of the hypersurface), and weighting the model parameters with the value.
Furthermore, we had to face the problem of the inconsistency of mean field modulus and mean quadratic field measurements: in fact, evidence exists that for many stars, either the former are systematically slightly overestimated, or the latter are systematically slightly underestimated. To deal with this problem, we repeated our analysis considering different observational datasets. We first performed a modelling considering all the kinds of measurement that were available for each individual star. Then we repeated the same analysis neglecting the observations of mean quadratic field; finally, we performed the analysis considering all available data but the observations of mean field modulus. Numerical simulations performed by Bagnulo, Mathys, & Stift (in preparation) suggest that the technique used to determine the mean quadratic field tends to underestimate the measurement. In addition to these numerical simulations, another argument supports the view that measurements of the mean field modulus are likely more reliable than determinations of the mean quadratic field. Field modulus measurements rely directly on the basic physics of the Zeeman effect and, for simple lines such as the Fe II magnetic doublet, are for all practical purposes independent of radiative transfer effects (Mathys 1989, Sect. 4.2.1), in contrast to quadratic field determinations, which involve fairly gross approximations of the effects of the radiative transfer in the stellar atmosphere. Accordingly, one could be tempted to give more weight to the results obtained neglecting quadratic field measurements. We nevertheless decided to accept as reliable only those results that were recovered in all three kinds of analysis. These are as follows.
i) The dipolar strength is usually comprised in the interval 3-20 kG, with a peak at something less than 10 kG. The strength of the quadrupolar component is the same as, or larger than the dipole strength.
ii) The dipole and quadrupole amplitude, as well as the ratio of quadrupole to dipole amplitude, tends to decrease with increasing rotational period. Consistently, the phase-averaged mean field modulus, an indicator of the typical field strength, also tends to decrease with increasing P.
iii) The inclination of the dipole axis to the rotation axis is usually large for short-period stars, and small for long-period ones. This is in accordance with a recent result obtained by Landstreet & Mathys (2000), who found a similar result for the magnetic axis of the axisymmetric model that they adopted in their best-fit technique.
iv) For several stars, the plane containing the two unit vectors that characterise the quadrupole is almost coincident with the plane containing the stellar rotation axis and the dipole axis. These are basically short-period stars (approximately, ). Long-period stars seem to be preferentially characterised by a quadrupole orientation such that the planes just mentioned are perpendicular. There is also some indication of a continuous transition between the two classes of stars with increasing rotational period, but the existence of such a "law'' is partly related to the question of the reliability of mean quadratic field observations. This result is not recovered by the analysis performed neglecting the observations of mean field modulus (thus when quadratic field measurements have a larger weight).
It should be noted that the results of this analysis pertain to a sample of relatively slow rotating stars; many of them are in fact very long-period stars.
It is an unanswered question whether the magnetic field of CP stars is the relic of an interstellar field that became stronger and stronger as the primordial nebula collapsed into a star ("freezing'' the field flux in a smaller and smaller plasma surface), or whether the field is at least in part dynamo generated. The results of our and previous studies show that there is some correlation between magnetic structure and rotational period, which encourages one to think in terms of the dynamo theory. In reality, it is not obvious to determine what is the consequence of what. For instance, Stepien & Landstreet (2002) have argued that the interaction of the magnetic field and a circumstellar disk before the star reaches the Zero-Age-Main-Sequence tends to slow-down preferentially those stars that have magnetic field vectors tilted at a small angle with respect to the rotation axis (thus presumably with the field lines perpendicular to the disk). Therefore, the long rotational period would be a consequence of the magnetic structure, and the magnetic field itself could well be a fossil relic. On the other hand, finding within the framework of a fossil field an explanation for the statistical properties of the quadrupolar component that were found in this work may be less easy. In any case we still need firmer conclusions about the geometrical structures of magnetic CP stars, that could come for instance from extensive applications of Zeeman Doppler Imaging. At the same time, there is a strong need for a theory explaining the origin of the magnetic fields in A and B type stars capable of leading to predictions about the geometrical structures to be compared with the modelling results.
Acknowledgements
Many thanks to Pierre North for providing us with (unpublished) estimates of the radius for some of the stars of Table 1.