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:
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.
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:
(
) is the wavelength of the centre of gravity of
the line as recorded in RCP (LCP).
is the
Lorentz unit (
is the laboratory wavelength of the line
and
Å
G-1).
is the effective
Landé factor of the considered transition, which characterizes the
displacement of the centre of gravity of its
components with
respect to the line centre.
In practice, the wavelength shifts are measured
for a sample of spectral lines.
is determined by performing a
linear regression of these measurements as a function of
, applying the least-squares method.
This linear regression is forced through the
origin and weighted by the uncertainties of the measurements of
. The evaluation of the latter is described in
detail in Paper III. The standard error
, 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
(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
. 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,
.
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
can be expressed
as:
where (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 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:
The quadratic field is at least equal and generally greater
than the mean magnetic field modulus . Indeed, even in the (hypothetical) case
of zero contribution of the longitudinal field term
, the mean
quadratic field modulus
still exceeds
by an amount
characterizing the scatter of the local values of the field modulus
around
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, . This moment
characterizes the unpolarized line width. The quadratic field
determination rests on the relation:
S2 and D2 are atomic parameters obtained from linear combinations
of the effective Landé factor and of the second-order moments of the
and
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 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
is less good than in Paper V,
even when these determinations rely on spectra of better resolution.