On the Statistical Analysis of X-ray Polarization Measurements
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 of the form alpha plus beta cosine (exp 2)(phi - phi(sub 0) (0 (is) less than phi is less than pi). We explore the statistics of such polarization measurements using both Monte Carlo simulations as well as analytic calculations based on the appropriate probability distributions. 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 alpha by itself (single interesting parameter case) or jointly with the position angle phi (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 3sigma level. For the latter case, when one requires a joint measurement at the same confidence level, then more counts are needed, by a factor of approximately equal to 2.2, than that required to achieve the MDP level. We find that the position angle uncertainty at 1sigma confidence is well described by the relation sigma(sub pi) equals 28.5(degrees) divided by beta.