Hertz-mode Data Analysis for SHARP Giles ... Sept. 21, 2005 This deals with the possibility of developing a data analysis mode for SHARP that is analogous to what we used for Hertz. In this approach, we treat SHARP as two 12 x 12 arrays. The idea would be to develop Hertz-mode in time for the January run. SHARP alignment adjustments can produce very good alignment (0.1 pixel) for the center pixels, but somewhat worse (0.5 pixel) near the edges. As discussed below, the misalignment near the edges will make it problematic to do any project that is based on the idea of taking small amounts of data (e.g. one polarimetric cycle) at a large number of positions. In such cases, pointing errors will dominate and will preclude achieving background limited signal-to-noise. For this reason, the basic Hertz-mode will not work well for a GMC survey, and we would have to simply put off the GMC survey until a later run. This problem and possible solutions to this problem are discussed in (3a) below. As for Hertz, the analysis will be done using pairs of pixels. (When one of the detectors in a pair is bad, the other pixel will not be used.) The basic idea is the same as is discussed in papers by Dowell and Kirby. The only difference is that we have a rotating field, but if we assume that the i.p. can be described by two fixed contributions, one from the telescope and one from SHARP, then it is fairly straightforward to adapt Kirby's method to SHARP: (1) ANALYSIS OF ONE CYCLE: Start with raw data files for theta = 0, 22.5, 45, 67.5 (a) for each pixel-pair compute PF_theta,i and TF_theta,i this step is from Dowell et al. need f (e.g., use Dowell et al. method) (b) compute PF_theta and TF_theta this step is from Dowell et al. errors in PF_theta and TF_theta are computed as in Dowell et al. (c) compute Q_inst, U_inst, and I Q_inst = PF_45 - PF_0 ... also propagate errors U_inst = PF_67.5 - PF_22.5 ... also propagate errors I = SUM-OVER-THETA (TF_theta) sigma_I derived from scatter in TF_theta (ignore sky rotation) (2) COMBINING CYCLES: (a) combine all of the measurements of I and sigma_I into one map this map will be on a 2" grid we need to know the RGM to ~5% accuracy we need to know how to correct for tau to ~5% accuracy we need to know how to correct for loading gain compression (~5% acc.) techniques for doing this step are available (e.g., Harper used "COMB") techniques for propagating errors to final map should also be available (b) compute q and u from each cycle first correct Q_inst, U_inst for RGM, tau, loading gain compress. (~5%) interpolate I from above map correct for IP(instr) and IP(tel) rotate to q_sky, u_sky (c) make q, u maps, using the same algorithm used for I maps will usually want to downgrade resolution to 4" or 8" grid (e.g., bin) (3) OPEN QUESTIONS: (a) effect of pixel-pair mis-alignment To first order, ~2.5" pixel misalignment will merely degrade resolution (perhaps we go from 9" -> 10") at edges. But there is a bigger problem. Flux gradients, combined with pixel-pair misalignment, combined with pointing drifts within a cycle, will cause a fake polarization. It will integrate down as it will be uncorrelated from file to file. It will probably be of order 10% per cycle. For faint sources this will not be a big problem, as the errors will be 10% per cycle anyway. For these sources we require 100 cycles (~ 10 hours) for a good measurement. For the proposed survey of Orion, however, this is a killer. It degrades our signal-to-noise by a factor of 5. Here are some possible solutions to the edge pixel-pair misalignment, which would probably have to wait until after the January run: Finish development of a sharpsolve-type method Hardware fix: the biggest problem is relative rotation between H and V subarrays. This might be correctable by tipping the combiner. But we have not studied this carefully. Correct for this effect: For some sources, the TF_theta map can tell us the pointing for that particular value of theta, and how it might be different from the pointing at other values of theta. Then we can correct for the fake polarization caused by this pointing drift. Improve the pointing: Martin says it may be possible to obtain extremely stable pointing using a new pointing system being developed at CSO (b) calculating f f may need to be known to high accuracy, just as in past (Hertz). One possibility is to use the method of Dowell et al. to calculate f; another is to incorporate a variation of John's (sharpsolve/sharpstokes) method to calculate f in to Hertz mode analysis. (c) determining RGM One method is outlined in Dowell et al. It works to 10%, which is almost good enough. There may be alternatives based on short sharpsolve integrations.