Stellar P larimetry

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114 Binary Orbit Theory

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The polarimetry of close binary systems has been modelled by Buerger & Collins II (1970) by considering the distortions on the distribution of scattering material by axial rotation and gravitational interactions The method allows calculation of intrinsic polarization in and out of eclipse Predictions are for polarization variations of several tenths of a per cent By investigating the effects of Thomson scattering in optically thin stellar envelopes, Brown, McLean & Emslie (1978) established the fundamental polarimetric behaviour associated with binary systems with co-rotating steady state electron clouds This paper is frequently cited, and the model will be referred to here as the BMcLE model The form of the variation, as plotted in the q u-plane, is dependent on the inclination, i, of the system and may be expressed in terms of the fundamental period and its rst harmonic It was demonstrated how the formulation may be formally inverted to provide a systematic diagnostic for inferring the characteristics of a binary and its envelope This seminal work has provided the basis of studies of a variety of binary types Simmons (1983) later extended the BMcLE model by considering arbitrary scattering mechanisms other than Thomson scattering, with the results illustrated by Mie scattering An alternative approach to modelling of the effects of scattering material in binary systems has been made by Dolan (1984) using regularized Monte Carlo calculations Aspin, Simmons & Brown (1981) performed an analysis of the required accuracy of q, u measurements to allow their variation to provide useful estimates of i The procedure comprised evaluation of the con dence interval of i for a model involving single Thomson scattering in a co-rotating envelope which yields an acceptable 2 t to simulated data when optimized over the free parameters involved in the model The required accuracy of polarimetric observations was found to be significantly higher for low values of inclination A later paper by Simmons, Aspin & Brown (1982) showed that tting the canonical model to binary star data will tend to yield values of inclination greater than the true value The statistical bias is most pronounced for data with higher noise values when the inferred value, and the formal linear error, have no bearing on the actual value As the noise increases, the inferred inclination approaches 90 Errors for inclination which are established by formal techniques are seriously over optimistic, except for data with high signal-tonoise ratios It was also demonstrated by Brown, Aspin, Simmons, et al (1982) that if the binary orbits are eccentric, third harmonics are generated in the polarimetric variation, making a clear diagnosis of noisy data more problematic The polarimetric behaviour of binary systems embedded in electron shells of various geometries has been explored by Manset & Bastien (2000) It was shown that the polarimetric variations should be more apparent for systems with the low inclination, a high optical depth, a at envelope, a small cavity, or an orbit that brings the stars close to the inner edge of the cavity They also demonstrated that the BMcLE model remains powerful in nding orbital inclinations & 45 for a variety of distribution forms for optically thin Thomson scattering clouds; the atness of the

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envelope, the size of any central cavity and the size of the orbit have no signi cant in uence on the inclination deduced by the BMcLE model Returning to the BMcLE model, the polarimetric effect of Thomsom scattering by material between a pair of stars can be considered by applying the simple approach as summarized in (934) and (935), i e q D e q 0 C q 1 cos(2 t) C q 2 cos(4 t) , and u D e u 1 sin(2 t) C u 2 sin(4 t) (111) (112)

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with the coef cients in the expression describing q and u given by ! 3 sin2 0 n 2 1 I e D 2 , q 0 D sin i 2 r 1 (sin 2 sin 2i) I q1 D u 1 D sin 2 sin i , 2 1 sin2 (1 C cos2 i) I q2 D u 2 D sin2 cos i 2

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(113) (114) (115)

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With consideration to the theorem of optical equivalence, the effect of the light from the two stars, scattered by a complicated geometric optically thin distribution of electrons, will have the same form as the equations above The coef cients, however, result from the sum or integrations of the scattering of the light from the pair of stars by the complex geometry associated with distribution of the scattering centres, and, in the rst place, it is assumed that the stars move in circular orbits and that the envelope material co-rotates in a xed fashion The scattering integrals may be expressed as products in the form of 0 j , where the term 0 is the effective Thomson scattering optical depth integrated over all directions The ve terms, given by 0 0 to 0 4 , correspond to optical depths integrated over solid angle with various weightings associated with direction, and averaged for the two stars according to their brightnesses These expressions summarize the scattering and polarigenic geometry In keeping with an extension to (111) and (112), the expressions for the intensity and the NSPs, may be written, according to Brown, McLean & Emslie (1978), as I D I0 1 C 0 2(1 C 0 ) C (1 3 0 ) sin2 i C 0 sin 2i( 1 cos 2 sin ) C sin2 i( 3 cos 2 (1 2 sin ) (1 C cos2 i)( 3 cos 2 4 sin 2 ) ,

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4 sin 2 )

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