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None of the spatial modulations was able to reproduce the observations of the latter star Although satisfactory ts resulted for the other two stars, the magnitudes of the required inhomogeneities would also generate substantial photometric variations in the spectral lines, these not being observed It was therefore concluded that the peculiarities of the executed loci in the q u-plane mainly result from departures of the magnetic con guration from being simply dipolar Adopting the oblique rotator model, Landol , Bagnulo, Landi Degl Innocenti, et al (1997) have compared the various diagnostics whereby the magnetic con guration of an Ap star is determined Measurements of the splitting of lines leads to determination of the so-called mean magnetic eld modulus or mean surface eld, B s ; the circular polarization in individual lines leads to the so-called mean longitudinal eld, B l ; broadband linear polarization measurements provide information of the differential saturation effects in all of the magnetic lines contained in the passband Their study establishes the relationships between the three diagnostics in de ning the constraints on magnetic eld con gurations In a further paper, Leroy (1995) has brought together his broadband measurements of 55 Ap stars measured over a 4-year period, with data collected by other workers to provide a reference point catalogue for any future studies that might be undertaken An example of the behaviour of a well-studied star, CrB (= HD 137909), is depicted in Figure 123 Although the polarimetric behaviour looks very simple and readily interpretable, Leroy (1995) has noted that the tted q- and u-phased curves require third-order Fourier expansions to describe them, this indicating signi cant departures to a standard dipolar con guration In the nal paper of this series related to the behaviour of continuum polarization, Leroy, Landol & Landi Degl Innocenti (1996) addressed the problem again
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Fig 123 (a) displays the q and u variations of CrB (= HD 137909), phased on the star s rotational period of 18d 4868; (b) plots the same data in the qu-plane revealing a renal-shaped locus, the boxes carrying the phase values (Taken from Leroy, 1995)
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Stellar P larimetry
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of a dipole eld model, and the departures from the expected behaviour of the executed path that the data reveal when plotted in the q u-plane They were able to reconcile the issue by considering there to be inclination changes of the lines of force within their meridian plane Keeping the magnetic equator as a plane of symmetry, they have shown that it is suf cient to assume slightly expanded lines of force, over some parts of the magnetic equator, to explain the peculiarities in the polarization curves Such regions, where the lines of force expand outwards, seem to occur preferentially in the vicinity of the rotational poles for those stars having an angle of not far from 90 Magnetic eld determinations may be improved by combining the circular polarization signals within many spectral lines In a simulation exercise, Stift (1986) has shown that such statistical techniques are sensitive to macroscopic velocity elds Although multi-line combination methods may be useful for survey purposes, it was suggested that they do not provide a sound basis for detailed modelling of magnetic eld geometries of individual stars By applying spectropolarimetry, Donati, Howarth, Bouret, et al (2006) have recorded Zeeman signatures for the O f p spectrum variable star, HD 191612, suggesting that it displays a large-scale eld with a polar strength of about 15 kG The star is the second magnetic O star to have been discovered Its signature was essentially stable over four nights, indicating a slow rotation period They suggest that HD 191612 is an evolved version of the near-zero-age main-sequence magnetic O star, 1 Ori C, but with an even stronger eld; the rotation rate which is exceptionally slow by accepted O-star standards, could have been caused by angular momentum dissipation through a magnetically con ned wind A recent important observational development is the detection of patchy magnetic elds on rapidly rotating stars by the principle of Zeeman Doppler imaging (ZDI) By performing high spectral resolution polarimetry, features displaying polarization may be detected in rotationally broadened lines and followed, as they migrate across the pro le as the star rotates In the rst of a series of papers, Semel (1989) has described the principle and outlined the scheme whereby the average of the measurements of many lines may be used to improve the signalto-noise ratio of the determined magnetic elds, and their location on the stellar surface A numerical simulation of the model was undertaken by Donati, Semel & Praderie (1989), and some preliminary results were presented for the magnetic Ap star, 2 CVn Some of the problems associated with spurious signals which occur in the collection of data are discussed in Semel, Donati & Rees (1993), together with a listing of eld detections which have been achieved by instrumentation on three different telescopes Donati, Semel, Carter, et al (1997) have enhanced the sensitivity of ZDI by sophisticated reduction techniques to a selection of active stars, and have reported on the detection of magnetic elds for 14 objects The procedure, or least-squares deconvolution (LSD), involves the simultaneous analysis of many spectral lines LSD assumes that all spectral features in a given Stokes parameter fI, Q, U, V g spectrum have identical shape, but may differ in amplitude by known scaling factors
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