Analysis of the data from photoelectric gas polarimeters
We review the tools and procedures for the analysis of the data collected by X-ray photoelectric gas polarimeters, like the ones on-board the Imaging X-ray Polarimetry Explorer (IXPE). Although many of such tools are in principle common with polarimeters working at other energy bands, the peculiar characteristics and performance of these devices require a specific approach. We will start from the analysis of the raw data read-out from this kind of instruments, that is, the image of the track of the photoelectron. We will briefly present how such images are processed with highly-specialized algorithms to extract all the information collected by the instrument. These include energy, time of arrival and, possibly, absorption point of the photon, in addition to the initial direction of emission of the photoelectron. The last is the quantity relevant for polarimetry, and we will present different methods to obtain the polarization degree and angle from it. A simple method, used extensively especially during the development phase of X-ray photoelectric gas polarimeters, is based on the construction and fitting of the azimuthal distribution of the photoelectrons. We will discuss that there are several reasons to prefer an analysis based on Stokes parameters, especially when one wants to analyze measurements of real, i.e., not laboratory, sources. These are quantities commonly used at all wavelengths because they are additive, and then operations like background subtraction or the application of calibration are trivial to apply. We will summarize how Stokes parameters can be used to adapt current spectroscopy software based on forward folding fitting to perform spectro-polarimetry. Moreover, we will derive how to properly associate the statistical uncertainty on a polarimetry measurement and the relation with another statistical indicator, which is in the minimum detectable polarization.