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ASTRONOMY
AND
ASTROPHYSICS
25.11.1996
Constraints on the magnetic configuration of Ap stars
from simple features of observed quantities
M. Landolfi 1 , S. Bagnulo 2 , M. Landi Degl'Innocenti 3 , E. Landi Degl'Innocenti 4 , and J.L. Leroy 5
1 Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I­50125 Firenze, Italy. E­mail: landolfi@arcetri.astro.it
2 Armagh Observatory, College Hill, Armagh BT61 9DG, Northern Ireland. E­mail: sba@star.arm.ac.uk
3 C.N.R., Gruppo Nazionale di Astronomia, Unit`a di Ricerca di Arcetri, Largo E. Fermi 5, I­50125 Firenze, Italy.
E­mail: mlandi@arcetri.astro.it
4 Dipartimento di Astronomia e Scienza dello Spazio, Universit`a degli Studi di Firenze, Largo E. Fermi 5, I­50125 Firenze, Italy.
E­mail: landie@arcetri.astro.it
5 Observatoire Midi­Pyr'en'ees, 14 Avenue Edouard Belin, F­31400 Toulouse, France. E­mail: leroy@srvdec.obs­mip.fr
Abstract. According to the oblique rotator model, the
time variations of the quantities usually employed to in­
vestigate the magnetic configuration of Ap stars (mean
longitudinal field, mean surface field, broad band linear
polarization) are described by simple laws. For each quan­
tity, certain typical features can easily be identified. We
show that these features set definite constraints on the
magnetic configuration.
Key words: polarization -- stars: magnetic fields -- stars:
chemically peculiar
1. Introduction
The study of the magnetic field of Ap stars is based on
various phenomena related to the Zeeman effect: the split­
ting of spectral lines (yielding the so­called mean magnetic
field modulus or mean surface field, B s ), the circular po­
larization in individual lines (yielding the so­called mean
longitudinal field, B l ), the broad band linear polarization
(BBLP) resulting from the differential saturation of oe and
ú components in all the magnetic lines contained in the
passband. The first measurements of B l and B s go back
to several years ago (Babcock 1947, 1960), while BBLP
measurements have been undertaken more recently (Kemp
& Wolstencroft 1974; Leroy 1995 and references therein),
probably because of the small BBLP degree -- usually, a
few parts in 10 \Gamma4 -- which requires sophisticated instru­
mentation.
B s measurements are mainly sensitive to the intensity
of the magnetic field, while they are not very sensitive to
its geometry. BBLP measurements, on the contrary, have
Send offprint requests to: M. Landolfi
a large sensitivity to the field geometry 1 and a small sen­
sitivity to the field intensity. B l measurements are some­
thing intermediate. Obviously, the maximum information
is obtained by using together, when available, all kinds
of observations. In particular, it has been shown (Leroy
et al. 1995, Appendix) that BBLP and B l measurements
are truly complementary in determining the magnetic field
geometry.
The combined use of observational data of different
kinds (and, in many cases, obtained at different times) re­
quires careful fitting techniques (see, e.g., Bagnulo et al.
1995). Here we just want to emphasize how it is possi­
ble, under certain assumptions, to establish a number of
constraints on the magnetic configuration of Ap stars on
the basis of a few features that can easily be recognized
in each kind of measurement. Some of the following ar­
guments are scattered in the literature (see, e.g., Landolfi
et al. 1993, hereafter referred to as Paper I; Leroy et al.
1993, 1995; Hensberge et al. 1977); only a unified treat­
ment leads, however, to the simple and significant picture
presented in this paper -- see Fig. 2 and Table 1.
Recent investigations have shown that the observations
of some Ap stars can adequately be explained in terms of
the dipolar oblique rotator model (Bagnulo et al. 1995),
while more complex models are required in other cases
(Leroy et al. 1994, 1996). The interpretative scheme de­
rived in this paper can only be applied to stars of the for­
mer group, and -- even in that case -- cannot replace the
rigorous fitting methods mentioned above. On the other
1 Incidentally, only BBLP measurements provide information
on the orientation of the stellar rotation axis projected on the
plane of the sky. Note that the dependence on the orientation
of particular directions, related to some kind of anisotropy of a
celestial body, is a quite general property of linear polarization;
see, e.g., Brown et al. (1978).

2 M. Landolfi et al.: Constraints on the magnetic configuration of Ap stars
hand, the scheme provides a useful tool to check the con­
sistency of a series of observations with the simple dipolar
oblique rotator model. When such consistency exists, the
scheme can be used to derive approximate values for the
geometrical parameters of the star even from low­accuracy
data, since the simple features considered here can directly
be deduced from a qualitative inspection of the observa­
tional material.
2. Basic features of the different kinds of measure­
ments
Let us consider the classical dipolar oblique rotator model:
the magnetic field at the stellar surface can be considered
as generated by a `frozen' magnetic dipole located at the
star center, whose axis is tilted with respect to the stellar
rotation axis. The magnetic configuration on the visible
hemisphere varies with the rotation phase, and this leads,
in general, to a variation of the intensity and polarization
of spectral lines, having the same period as the stellar
rotation. The fundamental parameters characterizing the
magnetic configuration are the inclination angle i between
the rotation axis and the line of sight and the angle fi
between the magnetic and rotation axes.
For a precise definition of the geometry of the oblique
rotator model, we refer the reader to Fig. 1 of Paper I.
Here we just point out that -- contrary to the conventions
that are often used in the literature -- the angles i and fi
are allowed to vary in the range
0 ffi Ÿ i Ÿ 180 ffi ; 0 ffi Ÿ fi Ÿ 180 ffi (1)
(i = 0 ffi means that the positive rotation pole lies at the
center of the visible hemisphere: the star is seen to rotate
in the counterclockwise direction; fi = 0 ffi means that the
positive magnetic pole coincides with the positive rotation
pole). The definition in Eqs. (1) allows one to fully specify
the geometry of the model, including the handedness of
the rotation and the sign of the magnetic poles.
First we consider the BBLP originated by this model.
A theory of the phenomenon has been presented in Pa­
per I, under the basic assumptions that the atmosphere
is chemically homogeneous and described by the Milne­
Eddington model, and that the spectral lines contained
in the passband are unblended and characterized by the
same (average) properties. As apparent from Figs. 2 and 4
of Paper I, the stellar rotation produces typical curves in
the plane Q­U (the frequency­integrated, disk­averaged
Stokes parameters) which strongly depend on the values
of the i and fi angles. The U vs. Q curves (or `polariza­
tion diagrams') just quoted were obtained by using the
simple analytical expressions for Q and U resulting from
the further assumption of weak magnetic field. However,
the thorough investigation of Bagnulo et al. (1995) has
shown that this assumption has negligible effects on the
shape of the polarization diagrams.
The first outstanding feature of these diagrams, which
can be deduced even from moderately accurate observa­
tions, is their direction (clockwise or counterclockwise).
From Fig. 2 and the symmetry properties (15) and (16)
-- or from Fig. 4 and the symmetry properties (21) and
(22) -- of Paper I, we see that the diagrams are described
in the counterclockwise direction for 0 ffi ! i ! 90 ffi and in
the clockwise direction for 90 ffi ! i ! 180 ffi . 2 This simple
result allows one to establish which rotation pole (positive
or negative) lies on the visible hemisphere; note that this
piece of information cannot be derived from B l or B s mea­
surements. In other words, the knowledge of the direction
of the polarization diagram is, by itself, sufficient to re­
strict to one half the parameter­space domain defined in
Eqs. (1) -- see Fig. 1a. It should be noticed that for i ú 90 ffi
the diagrams are `knotted' because of magneto­optical ef­
fects, so that their direction may be difficult to ascertain
(see Paper I, Fig. 4).
Another evident feature of the polarization diagrams
is that they can consist either of a single loop or of two
loops, depending on the values of i and fi. This entails
another partition in two regions of the domain defined in
Eqs. (1). According to Paper I (Sect. 5), the two regions
are those shown in Fig. 1b.
Let us now turn to the mean longitudinal field B l . It is
well­known that the dipolar oblique rotator model leads
to the simple relation
B l = k cos l = k (cos i cos fi + sin i sin fi cos f) ; (2)
where k is a positive constant (proportional to the po­
lar magnetic field intensity and dependent on the limb­
darkening coefficient), l is the (time­dependent) angle be­
tween the magnetic axis and the line of sight, and f is the
phase angle describing the stellar rotation (the reader is
again referred to Fig. 1 of Paper I for the precise definition
of l and f). Excluding the limiting values i = 0 ffi or 180 ffi ,
fi = 0 ffi or 180 ffi , Eqs. (1) imply sin i sin fi ? 0, so that the
curve B l (f) has a maximum for f = 0 ffi and a minimum
for f = 180 ffi , i.e.,
B max
l = B l (f = 0 ffi ) = k cos(i \Gamma fi) ;
B min
l = B l (f = 180 ffi ) = k cos(i + fi) :
(3)
According to the values of i and fi, the function B l (f)
can be everywhere positive, or everywhere negative, or
partly positive and partly negative. More precisely, we can
distinguish the four following cases, which in general can
easily be discriminated by inspection of the observational
data
I) B max
l ? B min
l ? 0 ;
II) B max
l ? 0 ? B min
l ;
fi
fi B max
l
fi
fi ?
fi
fi B min
l
fi
fi ;
III) B max
l ? 0 ? B min
l ;
fi
fi B max
l
fi
fi !
fi
fi B min
l
fi
fi ;
IV) 0 ? B max
l ? B min
l :
(4)
2 Obviously, the direction of the observed polarization di­
agrams depends crucially on the definition of the Q and U
Stokes parameters. We adopt the operational definition given
by Shurcliff (1962).

M. Landolfi et al.: Constraints on the magnetic configuration of Ap stars 3
b
b
b
b b
b
Fig. 1. Limitations on the possible values of the angles i and fi imposed by certain features of magnetic field measurements
b
Fig. 2. Partition of the domain (i; fi) produced by the set of
features considered
These four cases yield a partition in four regions of the
domain defined in Eqs. (1). Using Eqs. (3) and (4), it can
be seen that the regions are those illustrated in Fig. 1c.
Next, consider the mean surface field B s . For the dipo­
lar oblique rotator model, a good approximation to the ex­
act expression of B s is given by the simple formula (Hens­
berge et al. 1977, Eqs. (6), (8), and (21))
B s = k 0 cos 2 l + k 00 sin 2 l ; (5)
where k 0 and k 00 are positive constants (proportional to
the polar magnetic field intensity and dependent on the
limb­darkening coefficient), and l is the angle appearing
in Eq. (2). Substitution of the expression of cos l given by
Eq. (2) yields
B s = k 00 (1 + h cos 2 i cos 2 fi + h sin 2 i sin 2 fi cos 2 f
+ 1
2 h sin 2i sin 2fi cos f) ;
(6)
where h = (k 0 \Gamma k 00 )=k 00 is a positive number ranging from
0.20 to 0.33 according to the value of the limb­darkening
coefficient. Thus the function B s (f), everywhere positive,
has a simple dependence on the rotation phase. In a ro­
tation period it has (excluding the limiting values i = 0 ffi

4 M. Landolfi et al.: Constraints on the magnetic configuration of Ap stars
Table 1. The six asterisks in each column characterize the properties of the region marked in Fig. 2 by the letter in the header
a b c d e f g h a 0 b 0 c 0 d 0 e 0 f 0 g 0 h 0
[BBLP] counterclockwise \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[BBLP] clockwise \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[BBLP] 1­loop \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[BBLP] 2­loop \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[B l ] I \Lambda \Lambda \Lambda \Lambda
[B l ] II \Lambda \Lambda \Lambda \Lambda
[B l ] III \Lambda \Lambda \Lambda \Lambda
[B l ] IV \Lambda \Lambda \Lambda \Lambda
[B s ] I A \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[B s ] II A \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[B s ] I B \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[B s ] II B \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[BBLP] internal \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
[BBLP] external \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda \Lambda
or 180 ffi , fi = 0 ffi or 180 ffi ) either one maximum and one
minimum or two maxima and two minima, so that we can
distinguish between the two cases
I A) B s (f) has 2 extrema ;
II A) B s (f) has 4 extrema :
(7)
In both cases, there is an extremum at f = 0 ffi and an­
other one at f = 180 ffi ; in case II A, the two additional
extrema are symmetrical about f = 0 ffi (or f = 180 ffi ) and
correspond to the same value of B s . It is easily seen that
cases I A and II A take place when j tan i tan fi j is less than
unity or greater than unity, respectively. We thus have a
partition of the (i; fi) plane as illustrated in Fig. 1d.
The preceding arguments show that certain simple fea­
tures of BBLP, B l , and B s measurements contain different
and complementary information on the magnetic geome­
try of the oblique rotator. In principle, the three features
summarized in Figs. 1a, 1b, and 1c allow one to restrict
the possible values of the fundamental angles i and fi to
a region whose size is 1=16 of the complete domain (see
Fig. 2; note that two stars characterized by (i 0 ; fi 0 ) and
(180 ffi \Gamma i 0 ; 180 ffi \Gamma fi 0 ) are identical except for the rotation
direction). On the other hand, the three features summa­
rized in Figs. 1a, 1b, and 1d restrict the values of i and fi
to a region whose size is 1=8 of the complete domain.
Up to now BBLP, B l , and B s measurements have es­
sentially been regarded as independent. However, a reli­
able interpretation in terms of the dipolar oblique rotator
model requires their phase­consistency. As apparent from
Eq. (3), the epoch corresponding to f = 0 ffi is univocally
determined by the condition B l = B max
l . According to
the discussion following Eqs. (7), one of the extrema of
the function B s (f) should occur at that epoch. 3 The exis­
tence of a significant phase shift should be regarded as an
indication of inconsistency of the measurements with the
oblique rotator dipolar model.
Under the assumption of phase­consistency, it can eas­
ily be seen that the behavior of the B s (f) curve yields
another partition of the (i; fi) domain. In fact, we can dis­
tinguish the following cases
I B) B s (f = 0 ffi ) ? B s (f = 180 ffi ) ;
II B) B s (f = 0 ffi ) ! B s (f = 180 ffi ) :
(8)
As apparent from Eq. (6), the corresponding regions are
those shown in Fig. 1e.
Finally, the position of the zero­phase point (still de­
termined from the condition B l = B max
l ) on the polar­
ization diagrams leads to a further partition of the (i; fi)
3 Accurate knowledge of the rotation period is of course re­
quired when considering observations taken at distant times.

M. Landolfi et al.: Constraints on the magnetic configuration of Ap stars 5
domain. As apparent from Figs. 2 and 4 of Paper I, when
0 ffi ! i ! 90 ffi and 0 ffi ! fi ! 90 ffi the zero­phase point
lies on the apex of the internal loop for 2­loops diagrams;
for 1­loop diagrams, where the internal loop reduces to
a `cusp' or a `maximum concavity' point, it lies on that
cusp. Denoting by internal both these situations, and us­
ing again the symmetry properties (15) and (16) -- or (21)
and (22) -- of Paper I, we obtain a partition of the (i; fi)
plane as shown in Fig. 1f. In the regions labelled exter­
nal , the zero­phase point lies on the apex of the external
loop or on the point `opposite to the cusp' for 2­loop and
1­loop diagrams, respectively. As already noticed for the
direction of the diagrams, the position of the zero­phase
point may be difficult to ascertain for i ú 90 ffi . Obviously,
the above argument assumes phase­consistency of BBLP
and B l measurements.
Clearly the three last properties, illustrated in
Figs. 1d, 1e, and 1f, are redundant and do not lead to
a partition of the (i; fi) domain finer than in Fig. 2. It
should also be noticed that, owing to the small value of
the parameter h in Eq. (6), it may be difficult in practice
to distinguish between cases I A and II A of Eqs. (7).
Table 1 summarizes the characteristics of the various
kinds of measurements associated with each of the 16 re­
gions shown in Fig. 2. It provides a quick way to test
the consistency of a set of observations with the dipolar
oblique rotator model, and -- if consistency is found -- to
get an approximate estimate of the values of the funda­
mental angles i and fi.
3. Conclusion
Measurements of the time variations of the mean longi­
tudinal field, mean surface field, and broad band linear
polarization contain complementary information on the
magnetic configuration of Ap stars. This appears clearly
if appropriate features, easily deduced from the observa­
tions, are considered (see Fig. 1). Figure 2 and Table 1
show the considerable amount of information that can be
obtained from these features alone, when measurements
of different kinds are available for the same star.
Acknowledgements. S. Bagnulo was supported in this work by
the UK PPARC grant GR/L21259.
References
Babcock H.W., 1947, ApJ 105, 105
Babcock H.W., 1960, ApJ 132, 521
Bagnulo S., Landi Degl'Innocenti E., Landolfi M., Leroy J.L.,
1995, A&A 295, 459
Brown J.C., McLean I.S., Emslie A.G., 1978, A&A 68, 415
Hensberge H., van Rensbergen W., Goossens M., Deridder G.,
1977, A&A 61, 235
Kemp J.C., Wolstencroft R.D., 1974, MNRAS 166, 1
Landolfi M., Landi Degl'Innocenti E., Landi Degl'Innocenti
M., Leroy J.L., 1993, A&A 272, 285
Leroy J.L., 1995, A&AS 114, 79
Leroy J.L., Landolfi M., Landi Degl'Innocenti E., 1993, A&A
270, 335
Leroy J.L., Landstreet J.D., Bagnulo S., 1994, A&A 284, 491
Leroy J.L., Landolfi M., Landi Degl'Innocenti M., et al., 1995,
A&A 301, 797
Leroy J.L., Landolfi M., Landi Degl'Innocenti E., 1996, A&A
311, 513
Shurcliff W.A., 1962, Polarized light, Harvard University
Press, Cambridge
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