On the Statistical Analysis of X-ray Polarization Measurements

Abstract: In many polarimetry applications, including observations in the X-ray band, the measurement of a polarization signal can be reduced to the detection and quantification of a deviation from uniformity of a distribution of measured angles. We explore the statistics of such polarization measurements using Monte Carlo simulations and chi-squared fitting methods. We compare our results to those derived using the traditional probability density used to characterize polarization measurements and quantify how they deviate as the intrinsic modulation amplitude grows. We derive relations for the number of counts required to reach a given detection level (parameterized by beta, the "number of sigma's" of the measurement) appropriate for measuring the modulation amplitude by itself (single interesting parameter case) or jointly with the position angle (two interesting parameters case). We show that for the former case when the intrinsic amplitude is equal to the well known minimum detectable polarization (MDP) it is, on average, detected at the 3-sigma level. For the latter case, when one requires a joint measurement at the same confidence level, then more counts are needed than that required to achieve the MDP level. This additional factor is amplitude-dependent, but is approximately 2.2 for intrinsic amplitudes less than about 20%. It decreases slowly with amplitude and is 1.8 when the amplitude is 50%. We find that the position angle uncertainty at 1-sigma confidence is well described by the relation 28.5 (deg) / beta.