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Astron. Astrophys. 357, 920-930 (2000)

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3. Variability of the mean longitudinal magnetic field

The analysis of the Ap stellar spectra is usually complicated, due to the overabundances of some chemical elements, which are probably connected with the magnetic field structure. As it has been noted above, HD 83368 was very carefully studied by photometry (Kurtz, 1990; Kurtz et al., 1997) and also a large number of magnetic field measurements were carried out. Mathys (1995) has determined the values of the quadratic magnetic field, which is a function of the effective field [FORMULA] and of the surface one [FORMULA].

One of the problems of this work is the necessity to take into account the Doppler shift effects due to the non-homogeneous stellar surface distribution of the chemical elements and the effects of the magnetic intensification of some spectral lines. Special attention was given to the lithium line 6708 Å, and also to the lines of other elements with high values of Landé factor. The magnetic field geometry in Ap stars is often characterized by the parameter [FORMULA], where [FORMULA] (Hensberge et al., 1977). For most of the Ap stars, the variation of the mean longitudinal magnetic field observed throughout the rotation cycle appears nearly sinusoidal:

[EQUATION]

where [FORMULA] is the angle between the line of sight and the axis of the magnetic field. Spherical geometry gives for [FORMULA] the following exspression:

[EQUATION]

where [FORMULA] and i are the angle of inclination of the rotational axis to the axis of the magnetic field and to the line of sight respectively, and [FORMULA] is the rotational phase. When [FORMULA], we have the minimum value of angle [FORMULA] and consequently the maximum value of longitudinal magnetic field. It is easy to see that

[EQUATION]

For HD 83368, supposing a pure dipolar configuration of the magnetic field, Mathys (1991) has performed a least-squares fit of the values of [FORMULA] previously observed by Thompson (1983) through [FORMULA] photopolarimetry, and measurements of [FORMULA], obtained by Mathys in 1985, 1987 and 1988, with the Eq. (1). The fitted parameters are [FORMULA] and [FORMULA], which give us [FORMULA]. In the case of an oblique rotating magnetic dipole, the relation of the mean longitudinal magnetic field to the polar field strength [FORMULA] is proportional to [FORMULA] (Hensberge et al., 1977)

[EQUATION]

where u is the limb-darkening coefficient. Matthews et al. (1996) have derived the limb-darkening coefficient for HD 83368 directly from the observed ratio of pulsation amplitudes in the optical (Strömgren vby and Cousins RI) and IR (JHK) bandpasses supposing [FORMULA]. For the spectral range, with central wavelength [FORMULA] 6200 Å (this wavelength corresponds to Mathys' observations), a value of [FORMULA] has been derived, that was used in our calculations (see Eq. (4) and Eq. (5)). By varying the parameters i and [FORMULA], which define the angle [FORMULA] at a given rotational phase (see Eq. (2)), we have performed a least-squares fit of the available values of [FORMULA] (excluding the two measurements published by Mathys (1991) for one night in 1986 near [FORMULA] with huge estimated errors) to the Eq. (4) with minimization of the discrepancies between the calculated and observed values of [FORMULA] (see Fig. 1). By this method we have obtained an estimate of the polar magnetic field strength [FORMULA] with fit goodness [FORMULA] and parameters [FORMULA], that are in good agreement with the data of Mathys (1991) and Kurtz (1990). Last parameters were used in order to construct the variability of the mean surface magnetic field modulus, averaged over the full visible stellar disk versus the rotational phase for a pure dipolar configuration (Hensberge et al., 1977):

[EQUATION]

where u is the same as in Eq. (4). The variation of [FORMULA] with the phase [FORMULA] is shown in Fig. 2.

It must be emphasized, that the mean surface magnetic field modulus for HD 83368 determined in such a way does not correspond to the recent estimate of the mean quadratic magnetic field [FORMULA] = 11 kG by Mathys (Mathys, 1995, Mathys & Hubrig, 1997). The mean quadratic magnetic field is diagnosed from the second-order moment of the unpolarized line profile (about the line centre), [FORMULA], where the integration was carried out over the whole width of the observed line. Here [FORMULA] is the equivalent width and [FORMULA] is the unpolarized line profile. This moment characterizes the unpolarized line width, that includes the constant part (natural width, rotational and thermal Doppler broadening, instrumental profile, etc.) and the part variable with [FORMULA], which is proportional to the mean quadratic magnetic field. But, if we suppose that some lines under analysis are generated in the "abundance spots" on the stellar surface, then the line broadening due to the stellar rotation also would vary with [FORMULA] and will not be constant as supposed by Mathys (1995). Consequently this evaluation of the mean quadratic magnetic field could be significantly greater than our estimation of the mean surface magnetic field modulus.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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