Stellar P larimetry in Visual Studio .NET

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the oscillatory components, although the behaviour of determined amplitudes, using other values of period for the t, was not investigated as the number of data points was considered to be too small Again, in respect of an observational study of Ori E, Kemp & Herman (1977) used the photometric period to make a Fourier analysis to obtain the amplitudes of the polarimetric variations The same data were later explored to provide a periodogram by Clarke & McGale (1988b) using the F-statistic with [4;N] degrees of freedom, based on tting the four free parameters for the cosine and sine amplitudes of the rst and second harmonics, with N data values Statistical signi cance levels were placed across the periodogram according to the standard tables for F-tests, making it easy to highlight the period of the binary The determined Fourier coef cients can be translated into geometric parameters describing the stellar system according to the BMcLE model They can be related to i, , 1 and 2 , and, according to the nomenclature here, may be written as Fundamental: 1 C cos i 2 (b c C a s )2 C (a c b s )2 (1124) D , 1 cos i (b a s )2 C (a c C b s )2 c bc as bc C as 1 tan 1 C tan 1 , (1125) D 4 bs C ac ac bs bc as bc C as 1 tan 1 tan 1 (1126) 1 D 2 bs C ac ac bs Harmonic: 1 C cos i 4 (Bc C A s )2 C (A c Bs )2 D , 1 cos i (B A s )2 C (A c C Bs )2 c Bc A s Bc C A s 1 1 1 tan C tan , D 4 Bs C A c A c Bs Bc A s Bc C A s 1 tan 1 tan 1 2 2 D 2 Bs C A c A c Bs
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(1127) (1128) (1129)
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It may be noted that the alternative expressions associated with the fundamental and harmonic serve as a consistency check on determined values of i and Drissen, Lamontagne, Moffat, et al (1986) provide an excellent example of how data may be investigated for the determination of orbital parameters In later papers, the tting of the canonical model above to data have been investigated by Simmons, Aspin & Brown (1980) and Aspin, Simmons & Brown (1981) in terms of the 2 statistic; these works show that normal experimental noise produces uncertainties to the model parameters which are larger than those predicted by the formal error treatment and, in addition, the determined system values carry bias In a paper concerned on the accuracy of polarimetric determinations of inclination, Aspin & Simmons (1982) re-examined the values obtained by Kemp and colleagues for 7 binaries For ve systems, they found the uncertainties in the orbital inclination were signi cantly larger than the previously quoted formal values,
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but with the optimum values of i coinciding with those given earlier For Algol and Ori E (B- lter), the model was found to be unacceptable carrying only a 10% signi cance
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115 AM Her Stars Polars
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The star AM Her drew attention to itself as a possible optical counterpart of a variable soft X-ray source catalogued as 3U 1809+50 As part of its investigation, Tapia (1977) undertook both linear and circular polarization observations and discovered remarkable changes which proved to be cyclic, with a 3h 094 period Using the Vband, on the rising section, p V increased from about 1% to a maximum, 53%, in about 30 min, followed by a near symmetrical fall The circular polarization, vV , displayed a more complicated behaviour over the period, with swings in level from C3% to 9% Krzemi ski & Serkowski (1977) also detected an extremely high circular polarizan tion for AN UMa, with changes over a 1h 9 period from 9% to 35% in the blue spectral region (see Figure 112) Pronounced variations in p were also recorded achieving a peak of 9%, near the time of minimum v, and maximum brightness They proposed that, with the similarity of behaviour with AM Her, a distinct set of objects had been discovered with the polarimetry providing strong evidence of cyclotron emission from hot electrons in a magnetic eld 108 G Direct evidence of the strength of the magnetic eld of AN UMa was obtained from the polarimetric spectra of Schmidt, Stockman & Margon (1981) The behaviour could not be explained by the canonical models available for close X-ray binary stars, and gave rise to a new genre of cataclysmic variables emitting strongly polarized light, and proposed by Krzemi ski & Serkowski (1977) to be called polars, although this term n is not universally applied Early reports on the polarimetric behaviour of AN UMa and HZ Her were given by Shakhovskoi (1978) Michalsky, Stokes & Stokes (1977) con rmed the polarimetric behaviour of AM Her and presented a preliminary geometrical model The basic model for a polar involves accretion from a low-mass secondary star onto a highly magnetic (B 20 50 MG) white dwarf The strong eld prevents the formation of an accretion disc with the stream of material being channelled along the eld lines High amounts of polarized light are generated by cyclotron emission The polarimetric behaviour is orbitally phase-locked, and, as a starting point, a centred dipole eld may be considered The polarization waveforms depend on the angle of the magnetic axis relative to the rotation axis of the white dwarf, and on the inclination, i, of the binary system to the line of sight Monitoring their form leads to determinations of magnetic and binary geometries With this simple model geometry, the angle, , between the magnetic eld and the line of sight is given by cos D cos i cos sin i sin cos[2 ( s)] , (1130)
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