## Optical Spectropolarimetry With Incomplete Data Sets

##### Abstract

Context. Linear spectropolarimetry is a >photon-hungry> observing technique, requiring a specific sequence of observations to determine the Stokes Q and U parameters. For dual-beam spectropolarimeters, the Q and U Stokes parameters can be ideally determined using observations at N = 2 retarder plate positions. The additional polarization signal introduced by instrumental effects requires the redundancy of N = 4 observations to correct for these effects and to accurately measure the linear polarization of astronomical objects. Aims. We wish to determine if the >instrumental signature corrections> for the Stokes Q and U parameters, is an element of(Q) and is an element of(U), are identical for observations with dual-beam spectropolarimeters. For instances when observations were only acquired at N = 3 retarder plate angles, we wish to determine if the complete measurement of one Stokes parameter and the associated instrumental signature correction can be used to determine the other Stokes parameter. Methods. We constructed analytical and Monte Carlo models of a general dual-beam spectropolarimeter to study the factors affecting the assumption is an element of(Q) = is an element of(U) and the uncertainty thereon. We compared these models with VLT FORS1 linear spectropolarimetry observations. Results. We find that, in general, is an element of(Q) - is an element of(U) approximate to 0, with the variance around zero (Delta(is an element of(Q) - is an element of(U))) being directly related to the signal-to-noise ratio of the observations. Observations of a polarized standard star, observed under identical instrumental conditions over the period of 2002-2007, show that the assumption of is an element of(Q) - is an element of(U) = 0 is generally true over a long period, although the absolute values of is an element of(Q) and is an element of(U) vary between observational epochs. While the variance of is an element of(Q) - is an element of(U) is not dependent on the polarization angle, significant deviations from is an element of(Q) - is an element of(U) = 0 arise when p greater than or similar to 20%. Conclusions. Incomplete VLT FORS1 spectropolarimetry datasets, for which observations at only N = 3 retarder plate position angles have been acquired, can be analyzed under the assumption that is an element of(Q) approximate to is an element of(U). The uncertainty associated with this assumption is directly related to the signal-to-noise ratio of the observations. This property of the analysis of spectropolarimetry, with dual beam spectropolarimeters, can be used to test for the presence of artifacts affecting individual observations and to assess the quality of the data reduction, when observations at all four retarder plate angles have been acquired.