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5 Polarization Statistics and Data Treatment
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51 Introduction
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The levels of polarization that are generally observed from stellar sources are usually very small, and it is important to know how experimental noise affects measurements and their interpretation This is particularly important in the assessment of detectivity, and in ascribing con dence levels to any data Polarimetry is sometimes referred to as being a photon-intensive pursuit in that many detection counts need to be accumulated before the polarimetric signal emerges In the limit, polarimetric detectivity and accuracy depend only on the statistical behaviour of the detection of photons Most instruments claim to overcome all other generated noises and to achieve the fundamental accuracies imposed by the limits set by the number of photons recorded For the purpose of discussing the statistics associated with polarimetry, it will be assumed that no sources of noise other than photon shot noise affect the measurements
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52 Basic Statistics
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The formulation of the way that photon shot noise affects determinations of a normalized Stokes parameter, or p, depends on the modulation technique Any measurement of an NSP is essentially one of differential photometry comprising the determination of intensities of a pair of orthogonal polarizations The smaller the Stokes parameter, the smaller is this intensity difference, and the larger is the total photon count required to tease out the value of the parameter with statistical signi cance Because of the nature of the observations, it is a normal practice to break down the total integration time needed to count the required number of photons in smaller intervals, with a series of measurements being undertaken Thus, for example, these might correspond to a polarizer being set rst parallel to some reference frame, say Q, and then perpendicular to the same frame, with the procedure repeated many times The signals may comprise direct photon counts, say from a photomultiplier detector, or analogue-to-digital units (ADUs) from a CCD camera,
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Stellar Polarimetry David Clarke Copyright 2010 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim ISBN: 978-3-527-40895-5
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Stellar P larimetry
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with each unit carrying a calibrated photon count conversion factor If n ki and n i correspond to the recorded signals (photon counts) from a single measurement, then the NSP values are determined from qi D n ki n i n ki C n i (51)
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0, and assuming that n ki For a situation with small polarization such that q i and n i are independent variables, with 1 values determined by photon counting Poisson statistics, and given by the square root of the signals themselves, manipulation of (51) provides a simple conservative estimate for the 1 uncertainty of each NSP measurement as s 1 qi D , (52) no where n o D n ok C n o , these identities corresponding to the true underlying photon counts Comparisons of papers in the literature suggest a minor confup sion with respect to (52) in that it sometimes carries an additional factor of 1/ 2 Whether or not this numerical term occurs is purely a matter of de nition in that some authors de ne n o D n ok D n o , rather than as above A similar formula also applies to the determination of the NSPs corresponding to u i It should be noted that ascribed uncertainties calculated by (52) are conservative estimates as the exact underlying true value of n o is unknown, and is an estimate itself, based on the same measurements made to determine q i For calculations of a mean value for q, Clarke, Stewart, Schwarz, et al (1983) have commented on the various estimators that are used For example, if the total observation time is divided so that N repeated samples represented by n ki , n i are recorded, the individual values of q i may be used to evaluate a mean given by N N 1 X 1 X n ki n i qD Q qi D N N n ki C n i
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