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A&A manuscript no.
(will be inserted by hand later)
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02(02.13.1; 02.16.2; 08.03.2)
ASTRONOMY
AND
ASTROPHYSICS
5.2.1996
Multipolar magnetic fields in rotating Ap stars:
modeling of observable quantities
S. Bagnulo 1 , M. Landi Degl'Innocenti 2 , and E. Landi Degl'Innocenti 3
1 Armagh Observatory, College Hill, Armagh BT61 9DG, N. Ireland
2 C.N.R., Gruppo Nazionale di Astronomia, Unit`a di Ricerca di Arcetri, Largo E. Fermi 5, I­50125 Firenze, Italy
3 Dipartimento di Astronomia e Scienza dello Spazio, Universit`a di Firenze, Largo E. Fermi 5, I­50125 Firenze, Italy
Received 11 May, accepted 28 August 1995
Abstract. Magnetic field is present in most of the chem­
ically peculiar stars of the upper main sequence. For a
long time, the component of the magnetic field along the
line of sight and the magnetic field modulus, averaged on
the stellar disk, were the only diagnostic contents of the
observational techniques. Recent developments in the ap­
plication of the theory of line formation, joined to the in­
creased sensitivity of instrumental techniques, augmented
the number of observable quantities related to the mag­
netic fields of such a class of stars. On the other hand, the
analytical modeling techniques, that is, the capability to
reproduce the observable quantities -- predicted by a given
magnetic configuration -- through simple analytical formu­
lae, is still limited only to some of the quantities nowadays
measurable and for the simplest magnetic configurations.
This work gives a contribution in order to fill this gap.
Through the formalism of the spherical tensorial cal­
culus, analytical formulae for most of the observable quan­
tities are provided for a magnetic field of arbitrary com­
plexity.
Namely, the magnetic field is thought of as origi­
nated by a multipolar expansion of arbitrary order (dipole,
plus quadrupole, plus octupole, . . . ) and it is expressed
through spherical tensors and spherical harmonics. By us­
ing such a formalism, it is straightforward to carry on the
integration over the stellar disk of the particular combi­
nations of the magnetic field components which represent
the observable quantities. The mean longitudinal mag­
netic field, the so--called mean asymmetry of the longi­
tudinal magnetic field and the mean quadratic magnetic
field, together with the quantities related to the observed
broadband linear polarization, are expressed through suit­
able sums of terms including the spherical components of
tensors describing the magnetic configuration. These sums
can numerically be performed, or they can be further de­
veloped in order to provide simpler analytical expressions
in the case of a defined multipolar expansion: in partic­
Send offprint requests to: S. Bagnulo
ular, such simpler analytical expressions are given in the
case of a magnetic field due to the superposition of a dipole
with a quadrupole. Finally, few diagrams of the observable
quantities predicted by such a configuration are shown.
Key words: Magnetic fields -- Polarization -- Stars:
Chemically peculiar
1. Introduction
Magnetic field is present in the atmosphere of a variety of
non--degenerate stars. While there are evidences that in
late--type stars the magnetic field -- when detected -- shows
a complex and fragmented structure, in early--type stars it
usually has a simpler, smoothed morphology, whose scale
length is comparable with the stellar radius. In fact, in
magnetic stars of early spectral type, the magnetic field
plays an important role, having deep consequences on the
global properties of the whole stellar surface, in partic­
ular on the distribution of the elements: indeed, among
the stars belonging to the upper main sequence, strong
spectrum variability is found only in the magnetic ones
(Landstreet 1993).
Due to its smoothed topology, the magnetic field of
early--type stars is easily detected (with respect to the
case of late--type stars) through the analysis of the Zeeman
effect in spectral lines, and it has been studied for almost
half a century, since Babcock detected for the first time a
magnetic field in a star other than the Sun, the Ap star
78 Virginis (Babcock 1947).
The main technique for the detection of the magnetic
field in early--type stars consists in measuring the wave­
length shifts of a large sample of spectral lines observed
in left and right circular polarization; from the measure­
ment of such shifts, one can derive the mean longitudi­
nal magnetic field, that is, the component of the mag­
netic field along the line of sight, averaged on the stellar

2 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
disk. The mean longitudinal magnetic field can also be
detected through the analysis of the wings of hydrogen
Balmer lines, observed via photopolarimetric techniques
(see Borra & Landstreet 1980).
Extensive application of these techniques, carried out
by several authors during the last five decades, allowed
one to know that most of the chemically peculiar stars
of the upper main sequence have a magnetic field, with
strength ranging from 0.3 to 30 kG. The mean longitu­
dinal magnetic field is time--dependent, and its observed
variation has the same periodicity as the stellar rotation
(deduced via photometric measurements). The interpreta­
tion is given in terms of the oblique rotator model: the mag­
netic field has an approximately dipolar structure, with
the symmetry axis not parallel to the rotation axis, so
that the observer sees a magnetic configuration changing
with time.
Besides this fundamental technique, another one, with
a different diagnostic content, has been applied to a lim­
ited sample of Ap magnetic stars: the method is based on
the analysis of the Zeeman splitting in spectral lines ob­
served in non--polarized light (I Stokes parameter), and
the quantity derived is the mean magnetic field modulus.
Unfortunately, only the stars with small projected rota­
tional velocity and strong magnetic field show resolved
Zeeman split lines, and, up to date, the mean magnetic
field modulus has been measured in only 21 stars (Mathys
& Lanz 1992).
Recently a number of new techniques have been intro­
duced, leading one to increase the number of the diagnos­
tic tools for the study of magnetic fields in Ap stars.
i) The well--known cross--over effect, observed for the
first time by Babcock (1951) in the magnetic Ap star
HD 125248, can be measured by evaluating the second
order moment of the V Stokes parameter profiles of sev­
eral spectral lines about their symmetry centers. From
such a measurement, one can derive the mean asymme­
try of the longitudinal magnetic field, which is defined as
the first moment of the component of the magnetic field
along the line of sight, about the plane defined by the
line of sight and the stellar rotation axis (Mathys 1995a).
Mathys (1995a) detected the cross--over effect in 10 stars,
among a sample of 29 observed stars.
ii) The measurement of the second order moment of
the I Stokes parameter profiles of the stellar spectral lines
about their symmetry centers, allows one the determina­
tion of the mean quadratic magnetic field, which is defined
as the square root of the sum of the mean square magnetic
field modulus with the mean square longitudinal magnetic
field. Measurements of mean quadratic magnetic field were
performed by Mathys (1995b) on a sample of 29 stars, such
a quantity having been detected in 22 of them.
iii) A small percentage (generally some units of 10 \Gamma4 )
of the radiation arising from some Ap stars is linearly po­
larized: this fact is well--known since Kemp & Wolstencroft
(1974) observed this phenomenon for the first time on the
star 53 Camelopardalis. The presence of broadband linear
polarization is explained by the mechanism of differential
saturation (Leroy 1962, Calamai, Landi Degl'Innocenti &
Landi Degl'Innocenti 1975), which has been known for a
long time to be the main agent of a similar phenomenon
observed in sunspots. The Zeeman effect is the origin of
the linear polarization in spectral lines, and the transfer
of radiation through the stellar atmosphere is responsible
for a differential saturation of the oe and ú components
of the Zeeman multiplet. The combination of these two
mechanisms gives rise to broadband linear polarization,
which is basically sensitive to the transverse component
of the magnetic field, that is, the component perpendicu­
lar to the line of sight (Landi Degl'Innocenti et al. 1981,
Landolfi et al. 1993). Systematic observations were per­
formed by Leroy (1995) (see also Leroy, Landolfi & Landi
Degl'Innocenti 1993): 16 stars, among a sample of 55 ob­
served, show a conspicuous time--dependent signal of lin­
ear polarization.
The observational data supplied by the new techniques
quoted above yield a large number of information and con­
straints not available before, and in principle they permit a
further refiniment of our knowledge of Ap magnetic stars.
In several cases, the observational data show to be not
consistent with the classic oblique rotator model with a
pure dipolar field. Neglecting abundance anomalies -- as
often done in models -- could be thought of as the ex­
planation of the observed discrepancies, but taking them
into account is not sufficient to smooth the disagreement.
Indeed, Leroy et al. (1995) showed that inhomogeneus dis­
tributions of the elements abundances cannot account for
the deviations of the observed linear polarization from the
one expected by a pure dipolar configuration. The inad­
equacy of the classical picture of a magnetic star as an
oblique rotator with a pure dipolar magnetic structure is
evident for a number of stars, and one is forced to consider
magnetic configurations more complex than the dipolar
one.
As a first step, the superposition of a dipole and a
linear quadrupole, aligned to the dipole, can be consid­
ered. However, the presence of such a co--linear quadrupole
does not introduce substantial differences from the simple
dipole. Namely, the resulting mean longitudinal magnetic
curves, as well as the linear polarization diagrams, are not
remarkably modified with respect to those predicted by
the pure dipolar field (Schwarzshild 1950, Landolfi et al.
1993).
Several authors considered more sophisticated config­
urations, like the decentered dipole (e.g. Stift 1975, Hens­
berge et al. 1977, Landi Degl'Innocenti et al. 1981), but
such works were finalized to find out suitable expressions
only for some of the quantities which are nowadays cur­
rently observed, or to model -- through numerical tech­
niques -- some particular stars.
We decided to approach the problem in its generality
by representing the magnetic field as originated by a se­

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 3
ries of multipoles of increasing order. This enables us to
describe with a convenient accuracy, magnetic configura­
tions of arbitrary complexity.
For all the observable quantities -- except for the mean
magnetic modulus -- we performed the calculations analyt­
ically. To reach this aim, we followed the approach, exotic
in this context, of making an extensive use of the powerful
formalism of the spherical tensor calculus.
This paper is organized as follows. In Sect. 2 we de­
fine the geometry of the oblique rotator model and the
quantities we are interested in calculating. In Sect. 3 we
introduce the formalism which we have adopted in this
work and we give the general expression for the magnetic
field due to a multipolar expansion of arbitrary order. In
Sect. 4 we give the expressions that we have to integrate
over the stellar disk, and the integration is carried out in
Sect. 5. Then, in Sect. 6, the results are given by specifying
the magnetic configuration in the reference frame of the
star. The final expressions for the observable quantities
are given in Sect. 7. As an application of the formalism of
this work, in Sect. 8 we consider the particular case of a
magnetic field originated by the superposition of a dipole
and a quadrupole (not necessarily linear neither aligned
with the dipole) and few diagrams of the observable mag­
netic quantities are shown.
2. Formulation
2.1. Definition the geometry of the oblique rotator model
The relevant quantities concerning the geometry of the
oblique rotator are specified in Fig. 1.
On the left of Fig. 1 a star with centre C and radius
R is represented. CR is the stellar rotation axis, defined
in such a way that an observer standing in R sees the star
rotating counterclockwise under himself.
If the magnetic configuration includes the contribu­
tion of a dipole, CM identifies the direction of the dipole
axis, and M is the positive magnetic pole. Otherwise the
geometry of the oblique rotator model is referred to the
first non--zero multipole. Namely, CM is aligned with the
vector s (kM )
1 = s (kM )
1 u (kM )
1 , kM denoting the first non--zero
order of the multipolar expansion of the magnetic field:
such quantities will be defined in Sect. 3.3. CM forms an
angle fi (kM )
1 (0 ffi Ÿ fi (kM )
1 Ÿ 180 ffi ) with CR. Note that, if
the magnetic field includes the dipolar contribution, the
angle fi (kM )
1 j fi (1)
1 coincides with the angle fi defined in
previous works (e.g. Landolfi et al. 1993).
The right--handed coordinate system ~
K(~x; ~
y; ~ z) is fixed
with the star: the ~
z--axis is directed as CR, and the ~
x--axis
lies on the plane identified by CR and CM, in such a way
that M stands on the positive half--plane (~x – 0; ~
z); finally,
the ~
y--axis is chosen in order to make ~
K a right--handed
reference frame.
The phase angle f , formed by the half--planes RCV
and RCM (j ~
x – 0; ~ z), V being the apex, grows linearly
with time t (f = !t, ! being the stellar angular ve­
locity). During a rotation period, M intersects twice the
plane RCV: the instant at which the positive half--plane
(~x – 0; ~ z) crosses the apex V defines the zero--point phase.
The right--handed coordinate system K(x; y; z) on the
right of Fig. 1 has its z--axis directed to the observer and
its x--axis lying along the celestial meridian and pointing
to the North celestial pole. In such an observer's reference
frame K, the North rotational pole R is identified by the
polar angle i (0 ffi Ÿ i Ÿ 180 ffi ) and by the azimuth angle \Theta
(0 ffi Ÿ \Theta ! 360 ffi ).
A given point P is specified, in the observer's reference
frame K, by the vector r, which has modulus r, polar
angle ` (0 ffi Ÿ ` Ÿ 180 ffi ), azimuth angle ü (0 ffi Ÿ ü ! 360 ffi ).
In Fig. 1, the point P is drawn on the surface of the star.
2.2. Definition of the observable quantities
The various measurable quantities quoted in Sect. 1 are
the mean magnetic field modulus and suitable linear, bi­
linear and trilinear combinations of the components of the
magnetic field in the observer's reference frame K.
Since the stellar disk is not resolved by the observa­
tions, such quantities are averaged on the stellar disk,
weighted with the local emergent line intensity. Unfortu­
nately, the limb--darkening law of magnetic Ap stars is
not well established, so that one is forced to introduce
an approximation. The Milne--Eddington atmosphere im­
plies a center to limb dependence of the intensity as cos `.
Such a picture is not fully realistic since it predicts that
lines disappear at the stellar limb, a phenomenon which
is not observed in the Sun. An empirical improvement of
the Milne--Eddington model consists in assuming for the
limb--darkening a polymonial law in cos `:
X
n–0
an (cos `) n : (1)
For simplicity, in the following we will carry on our calcula­
tions considering only the first two terms of the expansion
of Eq. (1), writing the limb--darkening law as
1 \Gamma u + u cos ` (0 Ÿ u Ÿ 1) ; (2)
u being the so--called limb--darkening coefficient. The
Milne--Eddington model is recovered assuming u = 1.
The observable quantities are given by averages of the
form
hf(`; ü)i = (3)
g(u)
Z 2ú
0
dü
Z ú
2
0
d` sin ` cos `(1 \Gamma u + u cos `)f(`; ü) ;
where
g(u) =
'' Z 2ú
0
dü
Z ú
2
0
d` sin ` cos `(1 \Gamma u + u cos `)
# \Gamma1
= 3
ú(3 \Gamma u) : (4)

4 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
Fig. 1. The geometry of the oblique rotator model. The figure is explained in the text
For the mean longitudinal magnetic field we simply have
hB z i : (5)
Defining d as the distance of a given point on the stellar
surface from the plane defined by the line of sight and the
stellar rotation axis (that is, the RCV plane), the mean
asymmetry of the longitudinal magnetic field is
hd B z i : (6)
The mean magnetic field modulus is
h
q
B 2
x +B 2
y +B 2
z i j hBi ; (7)
and the mean quadratic magnetic field is the square root
of the expression
hB 2 + B 2
z i : (8)
A detailed description of the phenomenon of broad­
band linear polarization requires the calculation of numer­
ical integrals (see Eqs. (4) of Landolfi et al. 1993) which
are not really handy, especially in order to interpret ob­
servational data. However, Landolfi et al. (1993) showed
that under the weak--field approximation, the observed lin­
ear polarization can be obtained through the calculation
of suitable bilinear and trilinear forms built up with the
components of the magnetic field along the axes of the
reference frame K of Fig. 1. Namely, the observed broad­
band linear polarization, PQ and PU , can be written
PQ = P (0)
Q + P (F )
Q
PU = P (0)
U + P (F )
U
where P (0)
Q and P (0)
U are proportional to bilinear forms:
P (0)
Q / hB 2
x \Gamma B 2
y i ; (9)
P (0)
U / h2B x B y i ; (10)
while P (F )
Q and P (F )
U , which account for magneto--optical
effects, are proportional to trilinear forms:
P (F )
Q / h\Gamma2B x B y B z i ; (11)
P (F )
U / h
\Gamma
B 2
x \Gamma B 2
y
\Delta
B z i : (12)
It should be remarked that the weak--field approximation
is not commonly verified in Ap stars. However Landolfi et

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 5
al. (1993) and Bagnulo et al. (1995) demostrated that, at
least for the simplest cases of a dipole or a dipole with a
linear quadrupole aligned with the dipole, the shape of the
polarization diagrams is well reproduced even under such
a drastic approximation.
3. Multipolar expansion for the magnetic field
The magnetic field B(r) of a star, which we suppose for
simplicity to be a sphere of radius R, can be thought of
as the superposition of the magnetic fields due to a conve­
nient series of multipoles placed at the center of the star 1 :
B(r) =
X
k–1
B (k) (r) ; (13)
where r is the position vector with its origin at the cen­
ter of the star, and where B (k) (r) is the magnetic field
due to the multipole of order k; the multipoles of order
1, 2, 3, . . . represent the dipole, the quadrupole, the oc­
tupole, . . . respectively.
3.1. Defining the multipole of order k
The multipole of order k is identified by k displacement
vectors whose moduli have to be much smaller than the
stellar radius:
s (k)
1 ; s (k)
2 ; : : : ; s (k)
k ;
and by the scalar quantity M (k) (magnetic mass, sup­
posed – 0). The magnetic field due to such a multipole,
B (k) (r), is given, for r AE s (k)
i (i = 1, 2, . . . , k), by
B (k) (r) = (14)
\Gammas (k)
k \Delta r
i
\Gammas (k)
k\Gamma1 \Delta r
i
\Delta \Delta \Delta
i
\Gammas (k)
1 \Delta r
`
kGM (k) “ r
r 2
''
\Delta \Delta \Delta
''
;
where r is the modulus of r, “ r is a unit vector having the
same origin and direction as r, s (k)
i are the moduli of the
vectors s (k)
i , kG is the Gilbert constant, and r is the dif­
ferential operator. Equation (14) expresses the fact that
the multipolar field of order k is obtained by the following
iterative process. Let us consider a multipole of order k \Gamma 1
identified by the quantities M (k) ; s (k)
1 ; s (k)
2 ; : : : ; s (k)
k\Gamma1 ; the
multipole of order k is obtained by displacing such multi­
pole of order k \Gamma 1 by a vector s (k)
k from the center of the
star and by placing another identical but opposite multi­
pole at the center of the star, where opposite means that
the multipole has all its magnetic masses changed in sign
(see Fig. 2).
1 In fact the considerations developed here -- with minor ad­
justments -- have a general validity, and they are not necessarily
confined to the study of stellar magnetic fields
Fig. 2. The multipole of order k is obtained by displacing the
multipole of order k \Gamma 1 by a vector s (k)
k from the center of
the star and placing another identical but opposite multipole
at the center of the star; see text for further details
3.2. Introducing spherical components for vectors and dif­
ferential operator
The expression in the r.h.s. of Eq. (14) can be conveniently
transformed by introducing the spherical components of
the different vectors. These components are defined in an
assigned, but arbitrary, reference frame.
The spherical components a q , with q = +1; 0; \Gamma1 , of
a vector a, characterized by a modulus a, and by a direc­
tion identified by the polar angle ` and the azimuth angle
ü, and having components a x , a y , a z in a right--handed
orthogonal frame of reference, are defined by
a 0 = a z = a cos ` = aC 1
0 (`; ü)
a \Sigma1 = \Upsilon1 p
2 (a x \Sigma ia y ) = \Upsilon1
p
2 a sin `e \Sigmaiü = aC 1
\Sigma1 (`; ü) ;
(15)
where the quantities C j
m (`; ü) are proportional to the
spherical harmonics and are defined as in Brink and Satch­
ler (1971, hereafter referred to as BS).
The scalar product of two vectors a and b, expressed
by their spherical components, is given by
a \Delta b =
1
X
q=\Gamma1
(\Gamma1) q a q b \Gammaq ; (16)
the spherical components of the operator r are given by
r 0 = @
@z

6 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
r \Sigma1 = \Upsilon 1
p
2
` @
@x
\Sigma i @
@y
'
:
The evaluation of the magnetic field of the multipole of
order k needs the repeated application of the formula (BS
p. 150)
r q
\Theta
f(r)C j
m (`; ü)
\Lambda =
(\Gamma1) j+m+q+1
h (j+1)(2j+3)
2j+1
i 1
2
`
1 j j + 1
q m \Gammam \Gamma q
'
\Theta
\Gamma d
dr f(r) \Gamma j
r f(r)
\Delta
C j+1
m+q (`; ü)
+ (\Gamma1) j+m+q
h j(2j \Gamma1)
2j+1
i 1
2
` 1 j j \Gamma 1
q m \Gammam \Gamma q
'
\Theta
\Gamma d
dr f(r) + j+1
r f(r)
\Delta
C j \Gamma1
m+q (`; ü) :
(17)
Note that the second term of the r.h.s. of Eq. (17) gives
no contribution, since in our case the term
` d
dr f(r) + j + 1
r
f(r)
'
is always null.
3.3. Deriving the expression for the magnetic field
By tedious but straightforward calculations, we obtain for
the q--component of the magnetic field due to a multipole
of order k (with k – 1 ) the following expression:
B (k)
q (r) = (18)
kGM (k) (\Gamma1) k(k\Gamma1)
2
Ÿ (k + 1)!
3(2k + 3)
– 1
2
(2k + 3)!! 1
r k+2
\Theta
X
q1 q2 :::q k
X
p1p2 :::p k
(\Gamma1) q+q1+\Delta\Delta\Delta+q k\Gamma1
\Theta
`
1 1 2
p 1 \Gammaq q 1
'`
1 2 3
p 2 \Gammaq 1 q 2
'
\Delta \Delta \Delta
`
1 k k + 1
p k \Gammaq k\Gamma1 q k
'
\Theta
i
s (k)
1
j
p1
i
s (k)
2
j
p2
\Delta \Delta \Delta
i
s (k)
k
j
pk
C k+1
qk (`; ü) ;
where the indices p 1 , p 2 , . . . , p k , which label the spheri­
cal components of the vectors s (k)
i , can assume the val­
ues +1, 0, \Gamma1, while the indices q 1 , q 2 , . . . , q k span the
integer values included in the ranges [\Gamma2; 2], [\Gamma3; 3], . . . ,
[\Gammak \Gamma 1; k + 1] respectively.
Let us write each vector s (k)
i in the form
s (k)
i = s (k)
i u (k)
i (i = 1; 2; : : :; k) ; (19)
where s (k)
i is the modulus of s (k)
i and u (k)
i is a unit vector.
The product of the spherical components of the k vec­
tors u (k)
i , which implicitly appears in Eq. (18), can be ex­
pressed as a linear combination of the components of suit­
able spherical tensors whose rank ranges between 0 and k,
by repeatedly applying the formula
R k
q S k 0
q 0 =
X
KQ
(\Gamma1) \Gammak+k 0 \GammaQ (2K + 1) 1
2
`
k k 0 K
q q 0 \GammaQ
'
T K
Q ;
whose inverse relation
T K
Q = (20)
X
qq 0
(\Gamma1) k\Gammak 0 +Q (2K + 1) 1
2
`
k k 0 K
q q 0 \GammaQ
'
R k
q S k 0
q 0
is directly derived from BS (p. 52, Eq. 4.6) by explicitily
introducing the 3­j symbols. We obtain
i
u (k)
1
j
p1
i
u (k)
2
j
p2
\Delta \Delta \Delta
i
u (k)
k
j
pk
=
X
j1 j2 \Delta\Delta\Deltaj k\Gamma1
X
m1m2 \Delta\Delta\Deltam k\Gamma1
(\Gamma1) j1 +m1+j2+m2+\Delta\Delta\Delta+j k\Gamma1 +mk\Gamma1
\Theta (2j 1 + 1) 1
2 (2j 2 + 1) 1
2 \Delta \Delta \Delta (2j k\Gamma1 + 1) 1
2
\Theta
` 1 1 j 1
\Gammap 1 \Gammap 2 m 1
'`
j 1 1 j 2
\Gammam 1 \Gammap 3 m 2
'
(21)
\Theta
`
j 2 1 j 3
\Gammam 2 \Gammap 4 m 3
'
\Delta \Delta \Delta
`
j k\Gamma2 1 j k\Gamma1
\Gammam k\Gamma2 \Gammap k m k\Gamma1
'
\Theta T jk\Gamma1
mk\Gamma1 (j 1 ; j 2 ; : : : ; j k\Gamma2 ; u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) ;
where the sums must be extended to those values of the
indices for which the 3­j symbols do not vanish; each of
the tensors T j k\Gamma1 , which appear in this formula, is singled
out by the ordered chain j 1 , j 2 , . . . , j k\Gamma2 , which specifies
the rank of its `progenitor' tensors.
By introducing Eqs. (19) and (21) in Eq. (18), and by
repeatedly using the identity (BS pp. 141 and 142)
X
fifflffi
(\Gamma1) \Gammaffi
` a b e
ff fi \Gammaffl
'` d c e
ffi fl ffl
'` b d f
fi ffi \GammaOE
'
=
(\Gamma1) \Gammaa\Gammab\Gammac\Gammad+f \Gammae\Gammaff
`
c a f
fl ff OE
' ae
a b e
d c f
oe
;
the component B (k)
q of the magnetic field due to a multi­
pole of order k assumes the form
B (k)
q (r) =
kG M (k) s (k)
1 s (k)
2 \Delta \Delta \Delta s (k)
k
\Theta (\Gamma1) q+ k(k\Gamma1)
2
Ÿ (k + 1)!
3(2k + 3)
– 1
2
(2k + 3)!! 1
r k+2
\Theta
X
j1 j2 :::j k\Gamma1
X
mn
(\Gamma1) j1 +j2+\Delta\Delta\Delta+j k\Gamma1
\Theta (2j 1 + 1) 1
2 (2j 2 + 1) 1
2 \Delta \Delta \Delta (2j k\Gamma1 + 1) 1
2 (22)
\Theta
ae 1 j 1 1
1 2 3
oe ae 1 j 2 j 1
1 3 4
oe

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 7
\Theta
ae 1 j 3 j 2
1 4 5
oe
\Delta \Delta \Delta
ae 1 j k\Gamma1 j k\Gamma2
1 k k + 1
oe
\Theta
` 1 j k\Gamma1 k + 1
\Gammaq m n
'
C k+1
n (`; ü)
\Theta T jk\Gamma1
m (j 1 ; j 2 ; : : : ; j k\Gamma2 ; u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
It is simple to see that the triangular conditions of the 6­j
symbols (BS p. 142) impose
j 1 = 2; j 2 = 3; : : : ; j k\Gamma1 = k :
The 6­j symbols can now be explicitly evaluated by mak­
ing use of the relation
ae 1 a a \Gamma 1
1 a a + 1
oe
= (\Gamma1) 2a 1
2a + 1
which can easily be derived from Varshalovich, Moskalev
& Khersonskii (1988, p. 300). Equation (22) assumes a
much more compact form:
B (k)
q (r) =
kG M (k) s (k)
1 s (k)
2 \Delta \Delta \Delta s (k)
k
\Theta (\Gamma1) 1+k+q [(k + 1)! (2k + 3)!!] 1
2
1
r k+2 (23)
\Theta
X
mn
`
1 k k + 1
\Gammaq m n
'
U k
m (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta C k+1
n (`; ü) ;
where
U k
m (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) =
T k
m (2; 3; : : :; k \Gamma 1; u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
The explicit expression of the tensor U k as a function of
the spherical components of the vectors u (k)
i is the follow­
ing:
U k
q (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) =
X
q1 +q2+\Delta\Delta\Delta+q k =q
Ÿ (k \Gamma q)!(k + q)!
k!(2k \Gamma 1)!!
– 1
2
(24)
\Theta
Ÿ 1
(1 + q 2
1 )(1 + q 2
2 ) \Delta \Delta \Delta (1 + q 2
k )
– 1
2
\Theta (u (k)
1 ) q1 (u (k)
2 ) q2 \Delta \Delta \Delta (u (k)
k ) qk ;
where the indices q 1 , q 2 , . . . , q k can assume the values +1,
0, \Gamma1, and the sum is extended to all the solutions of the
equation q 1 + q 2 + : : : + q k = q. This expression has been
obtained by making use of the analytical expression
`
a b a + b
ff fi fl
'
= (\Gamma1) a\Gammab\Gammafl (25)
\Theta
Ÿ (2a)! (2b)! (a + b + fl)! (a + b \Gamma fl)!
(2a + 2b + 1)! (a + ff)! (a \Gamma ff)! (b + fi)! (b \Gamma fi)!
– 1
2
given by BS (p. 138). Note that the tensor U k -- and con­
sequently the expression of the magnetic field too -- is in­
dependent of the order in which the k unitary vectors
u (k)
1 ; u (k)
2 ; : : : ; u (k)
k are chosen.
Equation (23) expresses the magnetic field B(r) pro­
duced in r by a multipole of order k characterized by the
quantities
M (k) ; s (k)
1 ; s (k)
2 ; : : : ; s (k)
k ; u (k)
1 ; u (k)
2 ; : : : ; u (k)
k : (26)
Equation (23) is valid in whatever reference frame, pro­
vided that (`; ü) represent the polar and azimuth angles
of the position vector in that frame; in particular such an
expression is true in the observer's frame K and in the
frame ~
K fixed with the star.
3.4. Defining the magnetic configuration
We introduce now a suitable set of angles which allows
one to define the magnetic configuration (see Fig. 3).
The angles fi (k)
i and fl (k)
i (i = 1, 2, . . . , k) represent
the polar and azimuth angles of the unit vectors u (k)
i in
the star's reference frame ~
K. Since the star's reference
frame ~
K is identified by the first non--zero multipole kM ,
the azimuth angle of the unit vector u (kM )
1 , that is fl (kM )
1 ,
turns to be identically equal to zero. In particular the
azimuth angle fl (1)
1 of the dipole, whenever it is present, is
always null.
Since the star is rotating, in the observer's reference
frame K the vectors u (k)
i are time--dependent. They can
be identified by the polar angles l (k)
i (t) and by the azimuth
angles \Lambda (k)
i (t) (which hereafter will be shortly denoted by
l (k)
i and \Lambda (k)
i ). Note that in previous works (e.g. Landolfi
et al. 1993), the polar angle of the dipolar axis was denoted
by l(t), and the azimuth angle by \Lambda(t). In the notation in­
troduced in the present work, the same angles -- whenever
a dipole is present -- are denoted by l (1)
1 and \Lambda (1)
1 .
From these definitions, and from Eq. (15), it is trivial
to see that in the star's reference frame ~
K
(u (k)
i ) 0 = cos fi (k)
i
(u (k)
i ) \Sigma1 = \Upsilon 1
p
2 sin fi (k)
i e \Sigmaifl (k)
i ; (27)
while in the observer's reference frame K
(u (k)
i ) 0 = cos l (k)
i
(u (k)
i ) \Sigma1 = \Upsilon 1
p
2 sin l (k)
i e \Sigmai\Lambda (k)
i :
(28)
For the applications (see Sect. 8) it is also useful to
introduce the (time--dependent) cartesian components of
the unit vectors u (k)
i in the observer's reference frame K:
¸ (k)
i j (u (k)
i ) x = sin l (k)
i cos \Lambda (k)
i
j (k)
i j (u (k)
i ) y = sin l (k)
i sin \Lambda (k)
i
i (k)
i j (u (k)
i ) z = cos l (k)
i :
(29)

8 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
Fig. 3. The relevant angles which define the magnetic configuration in the reference frame fixed with the star ~
K (that is, fi (k)
i
and fl (k)
i
), and in the observer's reference frame K (that is, l (k)
i
and \Lambda (k)
i
). See text and Fig. 1
By making use of elementary trigonometric formulae of
spherical geometry it is easy to relate the cartesian com­
ponents of the unit vectors u (k)
i in the observer's reference
frame K to the angles fi (k)
i and fl (k)
i :
¸ (k)
i = sin i cos fi (k)
i cos \Theta
\Gamma cos i sin fi (k)
i cos \Theta cos
i
f + fl (k)
i
j
+ sin fi (k)
i sin \Theta sin
i
f + fl (k)
i
j
j (k)
i = sin i cos fi (k)
i sin \Theta (30)
\Gamma cos i sin fi (k)
i sin \Theta cos
i
f + fl (k)
i
j
\Gamma sin fi (k)
i cos \Theta sin
i
f + fl (k)
i
j
i (k)
i = cos i cos fi (k)
i + sin i sin fi (k)
i cos
i
f + fl (k)
i
j
:
3.5. The k--pole magnetic field intensity B (k)
Equation (23) shows that the k--pole contribution to the
spherical components of the magnetic field is proportional
to the factor kG M (k) s (k)
1 s (k)
2 \Delta \Delta \Delta s (k)
k , which does not have
an immediate physical interpretation. It is convenient to
relate this quantity to the k--pole magnetic field intensity
B (k) that we are going to introduce in the present subsec­
tion. As it will become clear from the definition, for a star
having a pure dipolar configuration, the quantity B (1) co­

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 9
incides with the intensity of the magnetic field at the pole
of the star.
Indeed, the concept of magnetic pole of a star does not
have an obvious meaning when the magnetic configuration
is more complex than the dipolar one. For instance, in the
pure quadrupolar configuration, two poles can be defined.
In general, the concept of magnetic pole looses any mean­
ing, except when the vectors u (k)
i are all parallel.
Let us then consider a fictitious star having the same
radius of the one we are considering. Let us suppose that
the magnetic field of this star is due only to the k--pole
contribution, and that such k--pole is the same as the one
that is present in our star, except for the fact that all the
unit vectors
u (k)
1 ; u (k)
2 ; : : : ; u (k)
k
are pointing to the same direction (instead of being ar­
bitrarily oriented). For this fictitious star the concept of
magnetic pole is meaningful. We then define the k--pole
magnetic field intensity B (k) of the original star as the
modulus of the magnetic field that would be present at
the pole of the fictitious star.
Since the k--pole magnetic intensity is a scalar quan­
tity, it can be calculated working in whatever reference
frame. By choosing to work in the star's reference frame
~
K of the fictitious star, we can suppose, without loosing in
generality, that all the unit vectors defining the k--pole are
aligned with the positive direction of the ~ z­axis. The mag­
netic pole of the fictitious star is specified by the vector
r, having polar cohordinates (R; 0; ~
ü), and the magnetic
field at this point can be derived by means of Eqs. (23)
and (24). Taking into account that
(u (k)
1 ) 0 = (u (k)
2 ) 0 = \Delta \Delta \Delta = (u (k)
k ) 0 = 1
(u (k)
1 ) \Sigma1 = (u (k)
2 ) \Sigma1 = \Delta \Delta \Delta = (u (k)
k ) \Sigma1 = 0 ;
we get, substituting in Eq. (24),
U k
m (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) = ffi m0
Ÿ k!
(2k \Gamma 1)!!
– 1
2
; (31)
where ffi m0 is the Kronecker symbol; moreover we have
(BS, Eq. 2.11, p. 19)
C k+1
n (0; ~
ü) = ffi n0 : (32)
Substituting Eqs. (31) and (32) in Eq. (23), and evaluat­
ing the 3­j symbol, we get the explicit value of B (k) :
B (k) = kG M (k) s (k)
1 s (k)
2 \Delta \Delta \Delta s (k)
k (k + 1)! 1
R k+2 :
3.6. The final expression for the magnetic field
The magnetic field B(r) produced in r j (r; `; ü) by a
multipole of order k, characterized by the set given by
Eq. (26), is
B (k)
q (r; `; ü) =
(\Gamma1) 1+k+q
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
B (k)
` R
r
' k+2
(33)
\Theta
X
mn
` 1 k k + 1
\Gammaq m n
'
U k
m (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta C k+1
n (`; ü) :
In the following we will use expression (33); however
it is worthwhile to note the Eq. (33) can be written in a
different way by making use of the analytical expression
of the 3­j simbols of the form (25):
B (k)
q (r; `; ü) =
Ÿ (2k \Gamma 1)!!
(k + 1)! (k + 1)
– 1
2
Ÿ 1
1 + q 2
– 1
2
B (k)
`
R
r
' k+2
(34)
\Theta
k
X
m=\Gammak
(\Gamma1) m
Ÿ (1 + k + q \Gamma m)! (1 + k \Gamma q + m)!
(k + m)! (k \Gamma m)!
– 1
2
\Theta U k
m (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) C k+1
q\Gammam (`; ü) :
By substituting Eq. (33) or (34) in Eq. (13), we get
the most general expression for the stellar magnetic field
as the sum of different multipolar magnetic fields, in an
arbitrary cartesian reference frame:
B q (r; `; ü) =
X
k–1
B (k)
q (r; `; ü) : (35)
4. Linear, bilinear and trilinear forms in the mag­
netic field components
In order to calculate the quantities described by Eqs. (5)--
(12), we introduce a number of linear, bilinear and trilin­
ear forms in the magnetic fields components:
B z = B 0 (a)
d B z = i
p
2
\Gamma “ r 1 e \Gammai\Theta + “ r \Gamma1 e +i\Theta
\Delta
B 0 (b)
B 2
x +B 2
y +B 2
z = (B 0 ) 2 \Gamma 2B 1 B \Gamma1 (c)
B 2
x +B 2
y + 2B 2
z = 2
h
(B 0 ) 2 \Gamma B 1 B \Gamma1
i
(d)
B 2
x \Gamma B 2
y = (B 1 ) 2 + (B \Gamma1 ) 2 (e)
2B x B y = \Gammai
h
(B 1 ) 2 \Gamma (B \Gamma1 ) 2
i
(f)
\Gamma
B 2
x \Gamma B 2
y
\Delta
B z =
h
(B 1 ) 2 + (B \Gamma1 ) 2
i
B 0 (g)
\Gamma2B x B y B z = i
h
(B 1 ) 2 \Gamma (B \Gamma1 ) 2
i
B 0 (h)
(36)

10 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
where B x , B y , B z and B 1 , B 0 , B \Gamma1 are, respectively, the
cartesian and spherical components of the surface mag­
netic field of the star in the observer's reference frame K,
and d is the distance, in units of the stellar radius, of a
point r of the stellar surface from the plane containing
the line of sight and the rotation axis of the star (plane
RCV). The unit vector “ r (with spherical components “
r 1 ,
“ r 0 , “ r \Gamma1 ) has the same origin and direction as r.
4.1. Linear forms
Equation (35) for r = R gives the explicit expression of
the spherical components of the magnetic field at the sur­
face of a star, and in particular the z--component (q = 0),
which is necessary to evaluate the mean longitudinal mag­
netic field.
As far as the quantity (36 b) is concerned, note that
from Eq. (15) we have
“ r q = C 1
q (`; ü) ; (37)
therefore, by multiplying Eq. (37) by Eq. (35), evaluated
in r = R, and by the use of the formula (BS p. 146)
C a
ff (`; ü)C b
fi (`; ü) = (38)
X
cfl
(2c + 1)(\Gamma1) fl
`
a b c
0 0 0
'`
a b c
ff fi \Gammafl
'
C c
fl (`; ü) ;
we get
“ r q B q 0 (R; `; ü) =
X
k–1
X
h`
X
j
X
m
(\Gamma1) 1+k+q 0 +m (2j + 1)
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
B (k) (39)
\Theta
`
1 k + 1 j
0 0 0
'`
1 k + 1 j
q ` \Gammam
'`
1 k k + 1
\Gammaq 0 h `
'
\Theta U k
h (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) C j
m (`; ü) :
By Eq. (39), the evaluation of the quantity (36 b) is direct.
4.2. Bilinear forms
The evaluation of the quantities (36 c--f) directly results
from the evaluation of the diadic forms B q B q 0 . However it
is convenient to follow a more simple and elegant way. By
the use of Eq. (20), it is possible to build up two tensors
from the diadic forms B q B q 0 , T 0 and T 2 , of rank 0 and 2
respectively (the rank 1 tensor is identically zero), whose
components have the following expressions:
T 0
0 = \Gamma
q
1
3
h
(B 0 ) 2 \Gamma 2B 1 B \Gamma1
i
T 2
0 =
q
2
3
h
(B 0 ) 2 +B 1 B \Gamma1
i
T 2
\Sigma1 =
p
2 B 0 B \Sigma1
T 2
\Sigma2 = (B \Sigma1 ) 2 :
(40)
By inverting Eq. (40), we obtain that the quantities (36
c--f) can be expressed through the tensors T K in the fol­
lowing forms, respectively
B 2
x +B 2
y +B 2
z = \Gamma p
3 T 0
0
B 2
x +B 2
y + 2B 2
z =
q 2
3
i
T 2
0 \Gamma 2
p
2 T 0
0
j
B 2
x \Gamma B 2
y = T 2
2 + T 2
\Gamma2
2B x B y = \Gammai
i
T 2
2 \Gamma T 2
\Gamma2
j
:
(41)
Therefore, by substituting in Eq. (20) the explicit expres­
sions of the magnetic field components given by Eq. (35),
we can get the following general expression for the com­
ponents of the tensors T K :
T K
Q (R; `; ü) =
X
kk 0 –1
X
qq 0
X
hh 0
X
`` 0
(\Gamma1) k+k 0
(2K + 1) 1
2
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!
(k + 1)!(k 0 + 1)!
– 1
2
B (k) B (k 0 ) (42)
\Theta
` 1 1 K
q q 0 \GammaQ
'` 1 k k + 1
\Gammaq h `
'` 1 k 0 k 0 + 1
\Gammaq 0 h 0 ` 0
'
\Theta U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) C k+1
` (`; ü)
\Theta U k 0
h 0 (u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k ) C k 0 +1
` 0 (`; ü) :
Expressing the product C k+1
` (`; ü)C k 0 +1
` 0 (`; ü) by
Eq. (38), and using the relation (BS p. 144)
X
OEš ffiae
`
c f i
fl OE š
'`
a d g
ff ffi ae
'`
d e f
ffi ffl OE
'`
g h i
ae j š
'
=
X
b
(2b + 1)
`
a b c
ff fi fl
'`
b e h
fi ffl j
' 8 !
:
a b c
d e f
g h i
9 =
; ; (43)
Eq. (42) can be compacted in the form
T K
Q (R; `; ü) =
X
kk 0 –1
X
hh 0
X
jj 0
X
mm 0
(\Gamma1) j 0 +m (2K + 1) 1
2
\Theta (2j + 1)(2j 0 + 1)
Ÿ (2k + 3)!!(2k 0 + 3)!!
(k + 1)!(k 0 + 1)!
– 1
2
B (k) B (k 0 )
\Theta
`
K j j 0
Q \Gammam m 0
'`
k k 0 j 0
h h 0 m 0
'
(44)
\Theta
`
k + 1 k 0 + 1 j
0 0 0
' 8 !
:
1 k k + 1
1 k 0 k 0 + 1
K j 0 j
9 =
;
\Theta U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta U k 0
h 0 (u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k ) C j
m (`; ü) :
Specifying the convenient values of K and Q, and substi­
tuting in the r.h.s. of Eqs. (41), it is straightforward to
evaluate the bilinear forms (36 c--f).

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 11
4.3. Trilinear forms
The evaluation of the trilinear quantities (36 g,h) can be
obtained directly by the triadic forms B q B q 0 B q 00 , which
can be achieved by Eq. (35). However it is again conve­
nient to proceed analogously with what we have done for
the bilinear quantities.
Combining three tensors of rank 1 (i.e. three vectors)
by the iterative use of Eq. (20), seven tensors are built
up: one of rank 0, three of rank 1, two of rank 2, and one
of rank 3. It can be easily shown that the trilinear forms,
which we are interested in, are related only to the tensor of
rank 3. The expression of this tensor, which we will denote
by the symbol S 3
Q , is the following:
S 3
0 =
q 2
5 B 0
h
(B 0 ) 2 + 3B 1 B \Gamma1
i
S 3
\Sigma1 =
q
3
5 B \Sigma1
h
2 (B 0 ) 2 + B 1 B \Gamma1
i
S 3
\Sigma2 =
p
3 B 0 (B \Sigma1 ) 2
S 3
\Sigma3 = (B \Sigma1 ) 3 :
(45)
By inverting Eqs. (45), we get the following expressions
for the quantities (36 g,h):
\Gamma
B 2
x \Gamma B 2
y
\Delta
B z =
q 1
3
i
S 3
2 + S 3
\Gamma2
j
\Gamma2B x B y B z = i
q
1
3
i
S 3
2 \Gamma S 3
\Gamma2
j : (46)
The tensor S 3
Q can be obtained, through Eq. (20), by com­
bining the tensor T 2
Q , given by Eq. (44), with a further
component of the magnetic field B q 00 given by Eq. (35).
Performing the calculations in analogy with what we did
in order to get Eq. (44) (we again use Eqs. (38) and (43)),
we obtain the expression
S 3
Q (R; `; ü) =
X
kk 0 k 00 –1
X
hh 0 h 00
X
jj 0 j 00 j 000
X
mm 0 m 00
(\Gamma1) 1+j+m+m 00
\Theta
p
35 (2j + 1)(2j 0 + 1)(2j 00 + 1)(2j 000 + 1)
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!(2k 00 + 3)!!
(k + 1)!(k 0 + 1)!(k 00 + 1)!
– 1
2
B (k) B (k 0 ) B (k 00 )
\Theta
` k + 1 k 0 + 1 j 000
0 0 0
'` k 00 + 1 j j 000
0 0 0
'
(47)
\Theta
`
k k 0 j 00
h h 0 m 00
'`
j 0 3 j
m 0 Q \Gammam
'`
k 00 j 0 j 00
h 00 m 0 \Gammam 00
'
\Theta
8 !
:
1 k k + 1
1 k 0 k 0 + 1
2 j 00 j 000
9 =
;
8 !
:
1 k 00 k 00 + 1
2 j 00 j 000
3 j 0 j
9 =
;
\Theta U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta U k 0
h 0 ( u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k )
\Theta U k 00
h 00 (u (k 00 )
1 ; u (k 00 )
2 ; : : : ; u (k 00 )
k ) C j
m (`; ü) :
Specifying the convenient values of Q, and substituting in
the r.h.s. of Eqs. (46) it is straightforward to evaluate the
trilinear quantities (36 g,h).
5. Integrals over the stellar disk
In the following we will have to evaluate some integrals
over the stellar disk of the form
In [f(R; `; ü)] = (48)
Z 2ú
0
dü
Z ú
2
0
d` f(R; `; ü) (cos `) n sin ` ;
where f(R; `; ü) stands for any of the quantities in
Eqs. (36), and where the angles ` and ü specify the point
over the stellar disk in the reference frame K.
The evaluation of the integrals expressed by Eq. (48)
is brought back, through Eqs. (35), (39), (41) and (46), to
the evaluation of the following integrals:
In [B q (R; `; ü)] (a)
In [“r q B q 0 (R; `; ü)] (b)
In
h p \GammaT 0
0 (R; `; ü)
i
(c)
In
\Theta T K
Q (R; `; ü)
\Lambda (d)
In
\Theta
S 3
Q (R; `; ü)
\Lambda
: (e)
(49)
The integrals (49 a,b,d,e) can be performed analytically as
the quantities B q , “ r q B q 0 , T K
Q , S 3
Q depend on the integra­
tion variables ` and ü only through the spherical harmon­
ics C j
m (`; ü), which appear linearly in their expressions.
The same is not true for the integral (49 c), that we
will have to integrate numerically (see Sect. 7).
The spherical harmonics C j
m (`; ü) depend upon the
angle ü only through the exponential factor e imü (see e.g.
BS p. 145); therefore the integral in dü can be easily eval­
uated, giving the result 2úffi m0 . Since the spherical har­
monics C j
0 (`; ü) coincide with the Legendre polynomials
P j (cos `) (see e.g. BS p. 145), making use of the formula
(Abramowitz & Stegun 1970, p. 338, Eq. (8.14.15))
Z 1
0
d cos ` P j (cos `) (cos `) n = Fn (j) ;
where
Fn (j) =
p
ú \Gamma(1 + n)
2 n+1 \Gamma(1 + n
2 \Gamma j
2 )\Gamma( 3
2 + n
2 + j
2 ) ;
and \Gamma is the Euler function, we get the relation
In [C j
m (`; ü)] = ffi m0 2ú Fn (j) : (50)
5.1. Linear forms
Through Eq. (50) and the expression of the component
B q of the magnetic field (Eq. (35)), the calculation of the
integral (49 a) is straightforward and yields
In [B q (R; `; ü)] = (51)

12 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
X
k–1
(\Gamma1) 1+k+q 2ú
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
Fn (k + 1) B (k)
\Theta
` 1 k k + 1
\Gammaq q 0
'
U k
q (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
In a completely similar way we get for the inte­
gral (49 b) the expression
In [“r q B q 0 (R; `; ü)] =
X
k–1
X
h
X
j
(\Gamma1) 1+k+q 0
\Theta 2ú(2j + 1)
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
Fn (j) B (k) (52)
\Theta
` 1 k + 1 j
0 0 0
'` 1 k + 1 j
q \Gammaq 0
'` 1 k k + 1
q 0 \Gammah q
'
\Theta U k
h (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
5.2. Bilinear forms
Analogously, for the the integral (49 d) we get
In
\Theta
T K
Q (R; `; ü)
\Lambda
=
X
kk 0 –1
X
hh 0
X
jj 0
(\Gamma1) j
\Theta 2ú (2K + 1) 1
2 (2j + 1)(2j 0 + 1)
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!
(k + 1)!(k 0 + 1)!
– 1
2
Fn (j 0 ) B (k) B (k 0 )
\Theta
`
k k 0 j
h h 0 \GammaQ
'`
j j 0 K
Q 0 \GammaQ
'
(53)
\Theta
`
k + 1 k 0 + 1 j 0
0 0 0
' 8 !
:
1 k k + 1
1 k 0 k 0 + 1
K j j 0
9 =
;
\Theta U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta U k 0
h 0 (u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k ) :
5.3. Trilinear forms
Finally, for the integral (49 e), we obtain the expression
In
\Theta S 3
Q (R; `; ü)
\Lambda =
X
kk 0 k 00 –1
X
hh 0 h 00
X
jj 0 j 00 j 000
X
m
(\Gamma1) 1+j 000 +m
\Theta 2ú
p
35 (2j + 1)(2j 0 + 1)(2j 00 + 1)(2j 000 + 1)
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!(2k 00 + 3)!!
(k + 1)!(k 0 + 1)!(k 00 + 1)!
– 1
2
\Theta Fn (j 000 ) B (k) B (k 0 ) B (k 00 )
\Theta
`
k + 1 k 0 + 1 j 00
0 0 0
'`
k 00 + 1 j 00 j 000
0 0 0
'
(54)
\Theta
` k k 0 j
h h 0 m
'` j k 00 j 0
\Gammam h 00 \GammaQ
'` j 0 j 000 3
Q 0 \GammaQ
'
\Theta
8 !
:
1 k k + 1
1 k 0 k 0 + 1
2 j j 00
9 =
;
8 !
:
1 k 00 k 00 + 1
2 j j 00
3 j 0 j 000
9 =
;
\Theta U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta U k 0
h 0 ( u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k )
\Theta U k 00
h 00 (u (k 00 )
1 ; u (k 00 )
2 ; : : : ; u (k 00 )
k ) :
6. Expressing the tensor U k in the star's reference
frame
Since the beginning of Sect. 4 we have worked in the ob­
server's reference frame K. Consequently, the tensorial
components U k
q in Eqs. (51)--(54) are the components of
the tensor U k in the system K. Since the tensors U k de­
scribe the geometry of the magnetic multipoles of the star,
and since the star is rotating, their spherical components
are implicitly time--dependent. It is then convenient to ex­
press the tensor U k by its components ~
U k
q in the reference
system ~
K, fixed with the star; in this way, the quanti­
ties ~
U k
q will result to be time--independent and will have
an intrinsic meaning strictly connected with the magnetic
geometry (see Sect. 3.4).
The components of the tensor U k in the two systems K
and ~
K, are connected by the relation (BS, p. 51, Eq. (4.5))
U k
q =
X
p
~
U k
p D k
pq (ff 1 ; ff 2 ; ff 3 ) ; (55)
where D k
pq are the rotation matrices and ff 1 , ff 2 , ff 3 are the
Euler angles of the rotation ! which brings the system ~
K
to coincide with the system K . It can be easily seen that
! is specified by the following Euler angles
! !
8 !
:
ff 1 = \Gammaf
ff 2 = i
ff 3 = ú \Gamma \Theta :
6.1. Linear forms
Substituting Eq. (55) in Eqs. (51) we obtain
In [B q (R; `; ü)] = (56)
X
k–1
X
h
(\Gamma1) 1+k+q
\Theta 2ú
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
Fn (k + 1) B (k)
`
1 k k + 1
q \Gammaq 0
'
\Theta D k
hq (ff 1 ; ff 2 ; ff 3 ) ~
U k
h (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
Substituting Eq. (55) in Eq. (52) we obtain
In [“r q B q 0 (R; `; ü)] = (57)
X
k–1
X
hh 0
X
j
(\Gamma1) 1+k+q 0
\Theta 2ú(2j + 1)
Ÿ (2k + 3)!!
(k + 1)!
– 1
2
Fn (j) B (k)

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 13
\Theta
` 1 k + 1 j
0 0 0
'` 1 k + 1 j
q \Gammaq 0
'` 1 k k + 1
q 0 \Gammah 0 q
'
\Theta D k
hh 0 (ff 1 ; ff 2 ; ff 3 ) ~
U k
h (u (k)
1 ; u (k)
2 ; : : : ; u (k)
k ) :
6.2. Bilinear forms
Substituting Eq. (55) in Eq. (53), and making use of the
relations (BS p. 147)
D A
aa 0 (ff; fi; fl)D B
bb 0 (ff; fi; fl) =
X
Ccc 0
(2C + 1)
`
A B C
a b c
'`
A B C
a 0 b 0 c 0
' \Gamma D C
cc 0 (ff; fi; fl)
\Delta \Lambda
;
\Gamma D A
aa 0 (ff; fi; fl)
\Delta \Lambda
= (\Gamma1) a\Gammaa 0
D A
\Gammaa\Gammaa 0 (ff; fi; fl) ;
together with the closure relation for the 3­j symbols, we
obtain
In[T K
Q (R; `; ü)] =
X
kk 0 –1
X
hh 0
X
jj 0
X
m
(\Gamma1) Q+j+m
\Theta 2ú (2K + 1) 1
2 (2j + 1)(2j 0 + 1)
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!
(k + 1)!(k 0 + 1)!
– 1
2
Fn (j 0 ) B (k) B (k 0 )
\Theta
`
k k 0 j
h h 0 \Gammam
'`
j j 0 K
Q 0 \GammaQ
'
(58)
\Theta
`
k + 1 k 0 + 1 j 0
0 0 0
' 8 !
:
1 k k + 1
1 k 0 k 0 + 1
K j j 0
9 =
;
\Theta D j
mQ (ff 1 ; ff 2 ; ff 3 ) ~
U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta ~
U k 0
h 0 (u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k ) :
6.3. Trilinear forms
Finally, substituting Eq. (55) in Eq. (54), and making an
iterative use of the same relations which led us to Eq. (58),
we obtain
In [S 3
Q (R; `; ü)] =
X
kk 0 k 00 –1
X
hh 0 h 00
X
jj 0 j 00 j 000
X
mm 0
(\Gamma1) 1+Q+h 00 +j 000
\Theta 2ú
p
35 (2j + 1)(2j 0 + 1)(2j 00 + 1)(2j 000 + 1)
\Theta
Ÿ (2k + 3)!!(2k 0 + 3)!!(2k 00 + 3)!!
(k + 1)!(k 0 + 1)!(k 00 + 1)!
– 1
2
\Theta Fn (j 000 ) B (k) B (k 0 ) B (k 00 ) (59)
\Theta
`
k + 1 k 0 + 1 j 00
0 0 0
'`
k 00 + 1 j 00 j 000
0 0 0
'
\Theta
`
k k 0 j 0
h h 0 m 0
'`
k 00 j j 0
h 00 \Gammam \Gammam 0
'`
j j 000 3
Q 0 \GammaQ
'
\Theta
8 !
:
1 k k + 1
1 k 0 k 0 + 1
2 j 0 j 00
9 =
;
8 !
:
1 k 00 k 00 + 1
2 j 0 j 00
3 j j 000
9 =
;
\Theta D j
mQ (ff 1 ; ff 2 ; ff 3 ) ~
U k
h ( u (k)
1 ; u (k)
2 ; : : : ; u (k)
k )
\Theta ~
U k 0
h 0 ( u (k 0 )
1 ; u (k 0 )
2 ; : : : ; u (k 0 )
k )
\Theta ~
U k 00
h 00 (u (k 00 )
1 ; u (k 00 )
2 ; : : : ; u (k 00 )
k ) :
7. General expressions for the observable quanti­
ties
From Eqs. (3) and (48) it is direct to obtain 2
hf(R; `; ü)i = (60)
g(u)
n
(1 \Gamma u)I 1 [f(R; `; ü)] + uI 2 [f(R; `; ü)]
o
:
The observable quantities given by Eqs. (5), (6) and (8)--
(12) can be derived by using Eqs. (36 a,b), (41) and (46),
thus obtaining, from Eq. (60):
hB z i = g(u)
(
(1 \Gamma u)I 1 [B 0 ] + uI 2 [B 0 ]
)
(61)
hd B z i = (62)
g(u)
i
p
2
(
(1 \Gamma u)
i
I 1 [“r 1 B 0 ] e \Gammai\Theta + I 1 [“r \Gamma1 B 0 ] e +i\Theta
j
+ u
i
I 2 [“r 1 B 0 ] e \Gammai\Theta + I 2 [“r \Gamma1 B 0 ] e +i\Theta
j )
hB 2 +B 2
z i =
g(u)
r
2
3
(
(1 \Gamma u)
i
I 1
\Theta
T 2
0
\Lambda \Gamma 2
p
2I 1
\Theta
T 0
0
\Lambda j
(63)
+ u
i
I 2
\Theta T 2
0
\Lambda \Gamma 2
p
2I 2
\Theta T 0
0
\Lambda j )
P (0)
Q / hB 2
x \Gamma B 2
y i =
g(u)
(
(1 \Gamma u)
i
I 1
\Theta T 2
2
\Lambda + I 1
\Theta T 2
\Gamma2
\Lambda j
(64)
+ u
i
I 2
\Theta T 2
2
\Lambda + I 2
\Theta T 2
\Gamma2
\Lambda j )
P (0)
U / h2B x B y i =
g(u)(\Gammai)
(
(1 \Gamma u)
i
I 1
\Theta
T 2
2
\Lambda \Gamma I 1
\Theta
T 2
\Gamma2
\Lambda j
(65)
+ u
i
I 2
\Theta T 2
2
\Lambda \Gamma I 2
\Theta T 2
\Gamma2
\Lambda j )
2 The results of this work are easily generalized to the case of
a limb--darkening law of the form given by Eq. (1) by express­
ing the averages hf(R; `; ü)i as G
P
n–0 anIn+1 [f(R; `; ü)],
where G is a suitable normalization constant which general­
izes Eq. (4)

14 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
P (F )
Q / h\Gamma2B x B y B z i =
g(u)
i
p
3
(
(1 \Gamma u)
i
I 1
\Theta S 3
2
\Lambda \Gamma I 1
\Theta S 3
\Gamma2
\Lambda j
(66)
+ u
i
I 2
\Theta S 3
2
\Lambda \Gamma I 2
\Theta S 3
\Gamma2
\Lambda j )
P (F )
U / h(B 2
x \Gamma B 2
y )B z i =
g(u)
r
1
3
(
(1 \Gamma u)
i
I 1
\Theta
S 3
2
\Lambda
+ I 1
\Theta
S 3
\Gamma2
\Lambda j
(67)
+ u
i
I 2
\Theta
S 3
2
\Lambda
+ I 2
\Theta
S 3
\Gamma2
\Lambda j )
;
where all the quantities of the form In [\Delta \Delta \Delta] are given by
Eqs. (51), (52), (53) and (54), if the magnetic configura­
tion is described by the tensor U k (observer's reference
frame), or by Eqs. (56), (57), (58) and (59), if the same
configuration is described by the tensor ~
U k (star's refer­
ence frame).
These equations are very general and they allow one
to consider a large variety of analytical investigations.
As an example, the depencence of the observable quan­
tities on the geometrical configuration of a given multipole
of order k, can be easily studied by taking the derivative
(with respect to the suitable variable) of the same equa­
tions.
As a further example, let us consider the case where the
observable quantites are expressed in terms of the tensor
~
U k (star's reference frame). From Eqs. (56)--(59), we can
see that the dependence of the observable quantities on the
phase f is contained only in the rotation matrices which
are of the form
D k
pq (\Gammaf; i; ú \Gamma \Theta) = e ipf d k
pq (i) e \Gammaiq(ú\Gamma\Theta) ;
where d k
pq (i) are the reduced rotation matrices (see BS
p. 22). When using the tensor ~
U k , the observable quanti­
ties are thus naturally expanded as a Fourier series of the
phase f . This may allow one to get a direct insight into the
star's multipolar configuration by comparison with obser­
vational data -- when Fourier analyzed.
For more direct applications -- to which we limit our­
selves in the present work -- we found simpler to work
in the observer's reference frame K, that is, building up
the tensors U k by substituting in Eq. (24) the spherical
components of the unit vectors u (k)
i given by Eqs. (28).
The angles l (k)
i and \Lambda (k)
i -- which represent the polar and
azimuth angles of the unit vectors u (k)
i in the observer's
frame K -- are related to the stellar rotation phase and
to the angles fi (k)
i and fl (k)
i -- which define the magnetic
configuration intrinsec to the star -- through the rela­
tions (30).
In order to obtain the behavior with phase of the ob­
servable quantities, one can proceed along two different
ways:
i) to develop a suitable FORTRAN code for the more
general case (multipoles of arbitrary order);
ii) to derive simpler analytical expressions for particular
magnetic configurations.
This latter application has been considered in detail in
Sect. 8 for the case of a dipole plus a quadrupole.
As pointed out in Sect. 5, the mean magnetic field
modulus has to be numerically calculated. This can be
done by evaluating the spherical components of the mag­
netic field through Eq. (33) or (34) for r = R, as a function
of the polar and azimuth angles ` and ü which identify
the position vector over the stellar disk in the observer's
reference frame K . Then the magnetic field modulus is
calculated by making use of Eq. (16) and the integration
is performed numerically.
It should be remarked that the mean longitudinal mag­
netic field, the mean asymmetry of the longitudinal mag­
netic field, the mean quadratic magnetic field and the
mean modulus of the magnetic field are quantities which
are by definition independent of the position of the stel­
lar rotation axis in the plane of the sky. Consequently,
Eqs. (61)--(63), and the numerical integral over the stellar
disk of the magnetic field modulus, are independent of the
angle \Theta.
8. Applications: the magnetic field due to the su­
perposition of a dipole with a quadrupole
As a particular application, we give the expressions of
Eqs. (61)--(65) further developed in the case of a mag­
netic field due to the superposition of a dipole with an
arbitrary quadrupole. Similar equations (with somewhat
different notations) have been presented, for the quantity
hB z i by Schwarzschild (1950), and, for the quantities P (0)
Q
and P (0)
U (and for P (F )
Q and P (F )
U too), by Landolfi et al.
(1993). However, these previous derivations were limited
to the case of a dipole plus a linear quadrupole aligned
with the dipole.
Posing, for simplicity,
U 1
q j U 1
q (u (1)
1 )
U 2
q j U 2
q (u (2)
1 ; u (2)
2 ) ;
the observable quantities are given by
hB z i = úg(u) (68)
\Theta
Ÿ
B (1)
` 15 + u
60 U 1
0
'
+ B (2)
i u
24
p
6 U 2
0
j –
hd B z i = úg(u)
\Theta
Ÿ
B (1) 8 \Gamma 3u
40
i
p
2
\Gamma U 1
1 e \Gammai\Theta + U 1
\Gamma1 e +i\Theta
\Delta (69)

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 15
+ B (2) 35 \Gamma 3u
420 i
\Gamma U 2
1 e \Gammai\Theta + U 2
\Gamma1 e +i\Theta
\Delta –
hB 2 + B 2
z i = úg(u)
\Theta
Ÿ
B (1) B (1)
` 255 \Gamma 79u
360
+ 105 + 13u
1260
i
3
\Gamma U 1
0
\Delta 2
\Gamma 1
j '
+ B (1) B (2)
` \Gamma1392 + 377u
1680
p
2 (U 1
1 U 2
\Gamma1 +U 1
\Gamma1 U 2
1 )
+ 192 \Gamma 31u
336 U 1
0
p
6 U 2
0
'
(70)
+ B (2) B (2)
` 9765 \Gamma 4517u
40320 4 U 2
2 U 2
\Gamma2
+ 819 \Gamma 227u
2016
\Gamma \Gamma4 U 2
1 U 2
\Gamma1
\Delta
+ 3045 \Gamma 677u
20160 (
p
6 U 2
0 ) 2
'–
P (0)
Q = hB 2
x \Gamma B 2
y i = úg(u) (71)
\Theta
Ÿ
B (1) B (1) 105 \Gamma 17u
1680
i \Gamma
U 1
1
\Delta 2
+
\Gamma
U 1
\Gamma1
\Delta 2
j
+ B (1) B (2)
` 288 \Gamma 43u
3360
p
2 (U 1
1 U 2
1 +U 1
\Gamma1 U 2
\Gamma1 )
+
\Gamma128 + 23u
1120 U 1
0
\Gamma
U 2
2 + U 2
\Gamma2
\Delta '
+ B (2) B (2)
` 945 \Gamma 209u
10080
i \Gamma
U 2
1
\Delta 2
+
\Gamma
U 2
\Gamma1
\Delta 2
j
+
\Gamma165 + 37u
2880
p
6 U 2
0
\Gamma
U 2
2 +U 2
\Gamma2
\Delta '–
P (0)
U = h2B x B y i = úg(u) (72)
\Theta
Ÿ
B (1) B (1) 105 \Gamma 17u
1680 (\Gammai)
i \Gamma
U 1
1
\Delta 2 \Gamma
\Gamma
U 1
\Gamma1
\Delta 2
j
+ B (1) B (2)
` 288 \Gamma 43u
3360 (\Gammai)
p
2
\Gamma U 1
1 U 2
1 \Gamma U 1
\Gamma1 U 2
\Gamma1
\Delta
+ \Gamma128 + 23u
1120 (\Gammai)U 1
0
\Gamma U 2
2 \Gamma U 2
\Gamma2
\Delta '
+ B (2) B (2)
` 945 \Gamma 209u
10080 (\Gammai)
i \Gamma U 2
1
\Delta 2
\Gamma
\Gamma U 2
\Gamma1
\Delta 2
j
+ \Gamma165 + 37u
2880
p
6 U 2
0 (\Gammai)
\Gamma U 2
2 \Gamma U 2
\Gamma2
\Delta '–
:
The particular combinations U k
q and U k
q U k 0
q 0 which appear
in Eqs. (68)--(71) are given by
U 1
0 = i (1)
1 (73)
i
p
2
\Gamma U 1
1 e \Gammai\Theta +U 1
\Gamma1 e +i\Theta
\Delta = j (1)
1 cos \Theta \Gamma ¸ (1)
1 sin \Theta (74)
\Gamma U 1
1
\Delta 2 +
\Gamma U 1
\Gamma1
\Delta 2 =
Ÿ i
¸ (1)
1
j 2
\Gamma
i
j (1)
1
j 2
–
(75)
\Gammai
h \Gamma U 1
1
\Delta 2 \Gamma
\Gamma U 1
\Gamma1
\Delta 2
i
= 2
i
¸ (1)
1 j (1)
1
j
(76)
p
6U 2
0 = 2i (2)
1 i (2)
2 \Gamma ¸ (2)
1 ¸ (2)
2 \Gamma j (2)
1 j (2)
2 (77)
i
\Gamma
U 2
1 e \Gammai\Theta +U 2
\Gamma1 e +i\Theta
\Delta
=
i
j (2)
1 i (2)
2 + j (2)
2 i (2)
1
j
cos \Theta (78)
\Gamma
i
¸ (2)
1 i (2)
2 + ¸ (2)
2 i (2)
1
j
sin \Theta
U 2
2 +U 2
\Gamma2 = ¸ (2)
1 ¸ (2)
2 \Gamma j (2)
1 j (2)
2 (79)
\Gammai \Gamma U 2
2 \Gamma U 2
\Gamma2
\Delta = ¸ (2)
1 j (2)
2 + ¸ (2)
2 j (2)
1 (80)
\Gamma U 2
1
\Delta 2 +
\Gamma U 2
\Gamma1
\Delta 2 =
1
2
Ÿ i
i (2)
1
j 2
` i
¸ (2)
2
j 2
\Gamma
i
j (2)
2
j 2
'–
(81)
+ 1
2
Ÿ i
i (2)
2
j 2
` i
¸ (2)
1
j 2
\Gamma
i
j (2)
1
j 2
'–
+ i (2)
1 i (2)
2
i
¸ (2)
1 ¸ (2)
2 \Gamma j (2)
1 j (2)
2
j
\Gammai
i \Gamma
U 2
1
\Delta 2 \Gamma
\Gamma
U 2
\Gamma1
\Delta 2
j
=
i
i (2)
1
j 2
¸ (2)
2 j (2)
2 +
i
i (2)
2
j 2
¸ (2)
1 j (2)
1 (82)
+ i (2)
1 i (2)
2
i
¸ (2)
1 j (2)
2 + ¸ (2)
2 j (2)
1
j
4 U 2
2 U 2
\Gamma2 =
Ÿ
1 \Gamma
i
i (2)
1
j 2
– Ÿ
1 \Gamma
i
i (2)
2
j 2
–
(83)
\Gamma4 U 2
1 U 2
\Gamma1 =
i
i (2)
1
j 2
Ÿ
1 \Gamma
i
i (2)
2
j 2
–
+
i
i (2)
2
j 2
Ÿ
1 \Gamma
i
i (2)
1
j 2
–
(84)
+ 2i (2)
1 i (2)
2
i
¸ (2)
1 ¸ (2)
2 + j (2)
1 j (2)
2
j
p
2
\Gamma
U 1
1 U 2
\Upsilon1 +U 1
\Gamma1 U 2
\Sigma1
\Delta
=
\Upsilon¸ (1)
1
i
¸ (2)
1 i (2)
2 + ¸ (2)
2 i (2)
1
j
(85)
\Gammaj (1)
1
i
j (2)
1 i (2)
2 + j (2)
2 i (2)
1
j
\Gammai
p
2
\Gamma U 1
1 U 2
1 \Gamma U 1
\Gamma1 U 2
\Gamma1
\Delta =
¸ (1)
1
i
j (2)
1 i (2)
2 + j (2)
2 i (2)
1
j
(86)
+j (1)
1
i
¸ (2)
1 i (2)
2 + ¸ (2)
2 i (2)
1
j
;

16 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
where ¸ (k)
i , j (k)
i , i (k)
i are expressed as functions of the pa­
rameters of the oblique rotator model and of the magnetic
configuration through Eqs. (30).
In Figs. 4--5 we show a number of diagrams of the ob­
servable quantities defined by Eqs. (5)--(10). Such plots
have been obtained -- a part from the mean magnetic field
modulus -- through Eqs. (68)--(72); as far as the mean mag­
netic field modulus is concerned, numerical integrals have
been performed as suggested in Sect. 7.
We have considered a limited sample of magnetic con­
figurations for different values of the inclination angle i
(20 ffi , 50 ffi , 80 ffi ) and for the fixed value of the azimuth an­
gle \Theta = 0 ffi .
The mean longitudinal magnetic field, the mean
asymmetry of the longitudinal magnetic field, the mean
quadratic magnetic field, and the mean magnetic field
modulus, expressed in kG, are plotted versus rotational
phase (normalized to 100), while h2B x B y i is plotted ver­
sus hB 2
x \Gamma B 2
y i (in arbitray units).
Continuous lines in Figs. 4 and 5 show the observ­
able quantities for a magnetic configuration composed of a
dipole plus a quadrupole specified by the following param­
eters: B (1) = 5 kG fi (1)
1 = 25 ffi , 45 ffi , 65 ffi , 85 ffi ; B (2) = 7:5 kG,
fi (2)
1 = fi (2)
2 = 90 ffi , fl (2)
1 = 0 ffi and fl (2)
2 = 90 ffi . Note that in
any case such parameters identify a quadrupole lying on
a plane perpendicular to the stellar rotation axis having
the unit vectors u (2)
1 and u (2)
2 in quadrature. In all cases,
the limb--darkening coefficient is u = 0:5.
Dotted lines show the observable quantities corre­
sponding to the magnetic configuration including the
dipole alone (that is, posing B (2) = 0). The linear polar­
ization loci predicted by the simple dipolar configuration
(dotted lines) are plotted with a different scale with re­
spect to the cases which include also the contribution of
the quadrupole (continuous lines). This makes easier to
compare the shapes of the linear polarization diagrams
obtained with different magnetic configurations. Filled cir­
cles represent the points with phase f = 0 ffi , empty circles
those with phase f = 90 ffi .
9. Conclusions
We have expressed the magnetic field due to a multipolar
expansion of arbitrary order as a sum of terms involving
the components of spherical tensors which describe the
magnetic configuration. Such a representation allowed us
to perform analytically the integrals over the stellar disk
of a number of linear, bilinear and trilinear forms built
up with the components of the magnetic field in an ar­
bitrary reference system. Then we have provided analyt­
ical expressions for most of the quantities related to the
magnetic field which are commonly measured in Ap stars,
namely, the mean longitudinal magnetic field, the mean
asymmetry of the longitudinal magnetic field, the mean
quadratic magnetic field, and the observable linear polar­
ization as predicted under the weak--field approximation,
including magneto--optical effects too. The mean magnetic
field modulus has to be calculated by performing numeri­
cal integrals.
Such expressions have been further developed for the
case of a magnetic field due to a dipole with a superposed
quadrupole.
Acknowledgements. Research at Armagh Observatory is grant­
aided by the Dept. of Education for N. Ireland, while support is
provided in terms of both software and hardware by the STAR­
LINK Project which is funded by the UK PPARC. S. Bagnulo
thanks Armagh Observatory for a studentship. The authors
are deeply indebted to R. Casini for Figs. 1, 2 and 3.
E­mail addresses
S. Bagnulo Internet sba@star.arm.ac.uk
E. Landi Degl'Innocenti Internet landie@arcetri.astro.it
M. Landi Degl'Innocenti Internet mlandi@arcetri.astro.it
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18 S. Bagnulo et al.: Multipolar magnetic fields in Ap stars
Fig. 4. hBz i (top), hd Bzi (middle), (hB 2 + B 2
z i)
1
2 (bottom) as a function of the phase (normalized to 100) for different values
of the inclination angle i and for the sample of magnetic configurations described in the text: continuous lines are referred to the
magnetic configurations including a dipole plus a quadrupole lying in a plane perpendicular to the rotation axis and identified
by the angles fi (2)
1 = 90 ffi , fl (2)
1 = 0 ffi , fi (2)
2 = 90 ffi , fl (2)
2 = 90 ffi . Dotted lines are referred to the dipole alone (see text)

S. Bagnulo et al.: Multipolar magnetic fields in Ap stars 19
Fig. 5. hBi is plotted versus rotational phase (top), h2BxByi is plotted versus hB 2
x \Gamma B 2
y i (bottom). The magnetic configurations
are the same as in Fig. 4. Continuous lines are referred to the magnetic configurations including dipole and quadrupole, dotted
lines to those including the dipole alone. The linear polarization loci given by the simple dipolar configurations are plotted in
a different scale with respect to the cases including the quadrupole (see text)