A&A 385, 896899 (2002)
DOI: 10.1051/00046361:20020177
V. R. Khalack ^{1,2}
1  Main Astronomical Observatory,
27 Zabolotnogo Str., 03650 Kyiv, Ukraine
2 
Isaac Newton Institute of Chile, Kiev Branch
Received 12 October 2001 / Accepted 8 January 2002
Abstract
Reconstruction of phasedependent crossover effect
variations is performed for a differentially rotating star
assuming that pointlike magnetic charges are distrubuted under
the stellar surface. Models adopting the solar differential rotation law
and an exponential rotation law are considered and compared to the case
of a rigidly rotating star.
It is shown that neglect of differential stellar rotation
leads to underestimation of the equatorial rotational velocity
by 25%.
Key words: stars: magnetic fields  stars: rotation  Sun: rotation
Variations of the surface magnetic field with stellar rotation are observed for the majority of chemically peculiar (CP) stars (Romanyuk 1997). These variations are thought to be caused by nonuniformity of the magnetic fields on the surface of CP stars. Probably, the location of patches of chemical elements (spots or rings) on the stellar surface is related to the distribution of the magnetic fields (Shavrina et al. 2001). Therefore, it is evident that precise reconstruction of the surface magnetic field can help provide a comprehensive element abundance analysis and anomaly mapping.
The last two decades have seen rapid progress in the measurement of the surface magnetic fields (Mathys 1988, 1991, 1995a, 1995b; Mathys & Hubrig 1997; Leone & Catanzaro 2001). First introduced by Mathys (1988), the moment technique relates the line profile asymmetries (while measuring the Stokes V parameter) to the mean values of the longitudinal magnetic field (first moment) and the crossover effect (second moment) for a given phase of stellar rotation.
Special attention should be paid to the crossover effect, since it reflects the result of a joint action of the Zeeman and Doppler effects on the investigated line profile of the Stokes V parameter (Mathys 1995a). Such an effect was first detected in HD 125248 and was named a crossover effect by Babcock (1951). This term comes from the fact that the effect usually achieves its maximum value when the mean longitudinal magnetic field reverses its sign, i.e. it crosses zero from one polarity to another. For example, in the oblique rotator model (when the magnetic and rotational axes do not coincide) with a dipolar magnetic field for a rigidly rotating star (Stibbs 1950), the longitudinal component of the local magnetic field is different in the two halves of the stellar disk at the surface points with equal radial velocity of rotation. Apparently, this difference is maximal at the particular moment when the two magnetic poles approach the stellar limb, while the mean longitudinal magnetic field is close to zero.
Recently, Mathys (1995a) proposed a new term, asymmetry of the longitudinal field, emphasizing the relationship of the crossover to the asymmetry in the distribution of the longitudinal component of the magnetic field, when considering the two halves of the visible stellar disk separated by the plane including the line of sight and the stellar rotational axis. In general, for a differentially rotating star with a complicated structure of its surface magnetic field it is not evident that the maximum of the effect coincides with the moment of sign reversion of the mean longitudinal magnetic field. Nevertheless, Babcock's term crossover effect is usually used to refer to the value that is actually derived from measurements of the secondorder moment of the V line profile (Mathys 1995a).
As mentioned above, the observable value of the crossover effect depends on the combined contribution of the Zeeman and Doppler effects to the line profile asymmetry of the Stokes V parameter averaged over the visible stellar disk for a given rotation phase. Therefore, at first it is necessary to model the structure of the surface magnetic field together with the velocity law for stellar rotation.
Among the known models the most appropriate one is the oblique rotator model proposed by
Gerth et al. (1997, 1998), especially its
modification (Khalack et al. 2001) that allows one
to reconstruct with high accuracy the geometry of the surface magnetic field
under the given distribution of the point field sources
embedded in the stellar body. In order to simplify the calculations,
let us consider the case of centered symmetric magnetic dipole
with two magnetic charges located close to the stellar centre.
Then the modeled mean longitudinal magnetic field
(directed along Zaxis in Fig. 1)
varies with phase of the stellar rotation in the following way
(Khalack et al. 2001):
Figure 1: Geometry of the oblique rotator model with one pointlike magnetic field source.  
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Here
and
specify an arbitrary point M
on the stellar surface in the spherical reference frame
related to the observer.
The variables =

and
determine
orientation of the dipolar magnetic axis
(defined by location of the magnetic charges)
in the spherical frame related to the star (see Fig. 1).
Angles i and
define the inclination of the rotational
axis to the line of sight and its planeofsky orientation
relative to the north celestial pole, respectively.
The variables Q and
determine the modulus of
the magnetic charge and its distance
from the centre of the star, expressed in the units of stellar radius
.
From the spherical triangle ZPL_{j}
with the known angles
and i, the relation between the two
mentioned spherical frames is derived as (omitting the subscript
j):
Stars are expected to rotate differentially
when energy transfer to the surface is dominated by
convection. Let us consider a spherically symmetric star
with a decrease in the angular rotation rate
from equator to pole. Our Sun (if
we neglect its oblation) has the differential rotation law
of the form (Gadun et al. 1985)
,
where
is the
latitude,
specifies the equatorial velocity
and parameter
(the case of b=0 corresponds to a rigidly
rotating star). Here we neglect the contribution of turbulence and
meridional circulation to the surface velocity at a given latitude.
Correspondingly, the lineofsight projection of the
velocity vector at a surface point M(
) is
Figure 2: Shape of the function f(b=1,u,i)defined by the Eq. (10) for the solar differential rotation law.  
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In order to simplify the analysis of the obtained results it is useful
to introduce the function f(b,u,i) defined as ratio of
the crossover magnitude in the model of rigidly rotating star
to the crossover magnitude in the differentially rotating model:
The plot of function f(b=1,u,i) (see Eq. (10)) is given in Fig. 2. As follows from this figure, maximal underestimation (that corresponds to the minimum of function f(b,u,i)) of the real equatorial velocity in the model of a rigidly rotating star is about 50% of the velocity estimate for and or . For the main sequence stars with and (spectral type B, A and F) the limb darkening coefficient u is about (DiazCordoves et al. 1995). Accepting the mean inclination of the rotational axis to the line of sight to be (Landstreet & Mathys 2000) and assuming , as for the case of the Sun (Gadun et al. 1985), Eq. (10) leads to underestimation of the order of 25% (see. Fig. 4) for the real equatorial velocity of the differentially rotating star. It is of the order of the typical error bar in modeling estimates of the rotational velocity on the stellar equator.
Figure 3: The same as Fig. 2, but for the differential rotation with an exponential law (14, 15).  
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In the stability analysis of a magnetized, differentially rotating star
researchers frequently use an exponential law (Uprin 1996)
that significantly simplifies the analytical calculations.
In such a case the rotational velocity depends on the
stellar latitude
as
,
where
specifies the equatorial velocity, and
the parameter b is restricted as
(the case of b=0 corresponds to a rigidly rotating
star). At an arbitrary surface point
M(
), projection of the rotational
velocity onto the line of sight is (see Fig. 1)
Figure 4: Dependence of the analyzed function f on the parameter bin the model with the solar differential rotation law (solid line) and in the model with an exponential law (dotted line).  
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This paper is devoted to modeling of the mean crossover effect in a differentially rotating, spherically symmetric star with a centered magnetic dipole. Geometry of the surface magnetic field is derived for the point field sources with the magnetic charges embedded in the stellar body. This provides high accuracy for the field reconstruction (Khalack et al. 2001).
Since the crossover effect is defined by the joint action of the Zeeman and Doppler effects, the corresponding theoretical expression (7) for its mean (averaged over the visible stellar hemisphere) value is obtained here for the differentially rotating star with a solarlike rotation law (5). It is shown that neglect of the differential character of the stellar rotation leads to underestimation of the real equatorial velocity up to 25%, comparable to the errors in model estimates of the stellar rotation. Comparatively smaller errors appear in the model considering the differentially rotating star with an exponential rotation law for the same value of the parameter b.
Nevertheless, the value (10) (related to underestimation of the equatorial velocity) is very sensitive to the parameter b (see Fig. 4) that specifies the rate of decrease in the angular rotation of the star from equator to pole. Thus, in a general case, for correct modeling of the observable variability of the values related to the surface magnetic field (the mean longitudinal magnetic field, the crossover effect, etc.), the possibility of differential rotation of the star should be comprehensively considered and then taken into account, if necessary.
Strong differential rotation characterized by sufficiently large values of bwould induce major distortions in the line profiles, which are not observed in the majority of CP stars (Mathys 2001). Therefore, in the case of differential rotation of these stars, the probable value of b is not very high and the respective underestimation error of the real equatorial velocity is of the order of the model uncertainty due to other effects. Consequently, application of the model with a rigidly rotating star remains valid.
Acknowledgements
Author is grateful to Prof. Y. Geroyanis, Prof. G. Mathys and Dr. S. Marchenko for useful advice and fruitful discussion.