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3. Magnetic field diagnosis

Stellar magnetic fields are diagnosed from the observational material described in the previous section through application of the moment technique (Mathys 1988), in essentially the same manner as in Papers III to V.

The following moments of the magnetic field have been determined in the present study:

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the mean longitudinal magnetic field tex2html_wrap_inline2667,
-
the crossover, that is, the product of the projected equatorial velocity tex2html_wrap_inline2777 and of the mean asymmetry of the longitudinal magnetic field tex2html_wrap_inline2779,
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and the mean quadratic magnetic field tex2html_wrap_inline2781.

For the sake of clarity, let us briefly recall the meaning of each of these quantities, and how they are derived. The reader is referred to Mathys (1988, 1989) and to Papers II to V for a discussion of the approximations underlying these derivations.

 

tex2html_wrap2869
 
Table 3: Mean longitudinal magnetic field, crossover, and mean quadratic magnetic field

The mean longitudinal magnetic field (in short, the longitudinal field) is the average over the stellar hemisphere visible at the time of observation of the component of the magnetic field parallel to the line of sight, weighted by the local emergent spectral line intensity. It is diagnosed from the first order moment of the line profiles in the Stokes parameter V (characterizing the circular polarization), that is, from the wavelength shift of the spectral lines between RCP and LCP. The latter can be written under the form:
equation1345
tex2html_wrap_inline2785 (tex2html_wrap_inline2787) is the wavelength of the centre of gravity of the line as recorded in RCP (LCP). tex2html_wrap_inline2789 is the Lorentz unit (tex2html_wrap_inline2791 is the laboratory wavelength of the line and tex2html_wrap_inline2793 Åtex2html_wrap_inline2795G-1). tex2html_wrap_inline2799 is the effective Landé factor of the considered transition, which characterizes the displacement of the centre of gravity of its tex2html_wrap_inline2801 components with respect to the line centre.

In practice, the wavelength shifts tex2html_wrap_inline2803 are measured for a sample of spectral lines. tex2html_wrap_inline2667 is determined by performing a linear regression of these measurements as a function of tex2html_wrap_inline2807, applying the least-squares method. This linear regression is forced through the origin and weighted by the uncertainties of the measurements of tex2html_wrap_inline2809. The evaluation of the latter is described in detail in Paper III. The standard error tex2html_wrap_inline2811, obtained from the least-squares analysis, is used as an estimate of the uncertainty of the derived longitudinal field.

The mean asymmetry of the longitudinal magnetic field tex2html_wrap_inline2779 (more briefly, the asymmetry of the longitudinal field) is the first-order moment about the plane defined by the line of sight and the stellar rotation axis of the component of the magnetic vector parallel to the line of sight. This moment is computed over the visible stellar hemisphere and weighted by the local emergent spectral line intensity. The quantity which is directly obtained from the observations is the product of the asymmetry of the longitudinal field and of the projected equatorial velocity tex2html_wrap_inline2777. We call this product the crossover. It is derived from the second-order moment about the line centre of the line profile in the Stokes V parameter, tex2html_wrap_inline2819. This moment characterizes the difference of the widths of the spectral lines between RCP and LCP (for more about the intuitive interpretation of the low-order moments of the line profiles, see Mathys 1993). The moment tex2html_wrap_inline2819 can be expressed as:
equation1359
where tex2html_wrap_inline2823 (as usual, v is the stellar equatorial velocity and i the angle between the rotation axis and the line of sight).

In practice, the crossover is derived like the longitudinal field, by performing a weighted regression analysis over a set of measurements of tex2html_wrap_inline2819 for a selected sample of spectral lines; this regression is forced through the origin (for details, see Paper IV).

The mean quadratic magnetic field (or quadratic field) is the square root of the mean square magnetic field. The latter is the sum of two terms:

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the mean square magnetic field modulus tex2html_wrap_inline2831, that is, the average over the visible stellar hemisphere of the square of the modulus of the magnetic vector, weighted by the local emergent line intensity;
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the mean square longitudinal magnetic field tex2html_wrap_inline2833, that is, the average over the visible stellar hemisphere of the square of the component of the magnetic vector parallel to the line of sight, weighted by the local emergent line intensity.

The quadratic field is at least equal and generally greater than the mean magnetic field modulus tex2html_wrap_inline2691gif. Indeed, even in the (hypothetical) case of zero contribution of the longitudinal field term tex2html_wrap_inline2833, the mean quadratic field modulus tex2html_wrap_inline2841 still exceeds tex2html_wrap_inline2691 by an amount characterizing the scatter of the local values of the field modulus around tex2html_wrap_inline2691 across the stellar surface (see Paper V and Mathys 1993 for details).

The quadratic field is diagnosed from the second-order moment about the line centre of the unpolarized line profile, tex2html_wrap_inline2847. This moment characterizes the unpolarized line width. The quadratic field determination rests on the relation:
equation1374
S2 and D2 are atomic parameters obtained from linear combinations of the effective Landé factor and of the second-order moments of the tex2html_wrap_inline2853 and tex2html_wrap_inline2855 components of the transition about their respective centre. They characterize the total spread of the Zeeman pattern and its anomalous character. Their expression in terms of the Landé factors and of the total angular momentum quantum numbers of the atomic levels between which the transition takes place has been established by Mathys (1988) and Mathys & Stenflo (1987a,b). C is a constant accounting for the non-magnetic part of the unpolarized line width (including contributions such as natural width, rotational and thermal Doppler broadening, instrumental profile, etc.).

Again, the actual determination of the quadratic field is carried out through measuring tex2html_wrap_inline2847 for a sample of lines and performing a weighted linear regression of the form given in Eq. (3) over this set of measurements. Contrary to the cases of the longitudinal field and of the crossover, this regression is not forced through the origin. Instead, two free parameters are determined from it: the quadratic field and the constant C. In Paper V, for stars repeatedly observed at various rotation phases, it was assumed that C is phase-independent: accordingly, a single value of this constant was derived from the simultaneous consideration of all the observations of the considered star, which allowed a better accuracy to be achieved. Here, this approach can no longer be used, because of the inhomogeneity of the data (resulting from the use of different instrumental configurations). As a result, the accuracy of the present determinations of both C and tex2html_wrap_inline2781 is less good than in Paper V, even when these determinations rely on spectra of better resolution.


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