How to predict the polarimetric measurement error from the Tabulated NEFD.

• The basic method for calculating NEFD is given in Tom's Table of SPARO characteristics. Here we explain how to derive polarimetric errors from the NEFD numbers given by Tom's Table:
• Photometry errors:
• First note that the NEFD calculated by Tom is the NEFD that would apply if doing photometry with just one of SPARO's two arrays. Using both arrays and taking an average improves things by a factor of root-2.
• Second, note that Tom's calculation does not account for the need to "chop", which introduces a factor of two loss in signal-to-noise.
• Putting the two points above together, one finds that the NEFD of SPARO, if used as a photometer (averaging the results for the two arrays), would be equal to the NEFD given in Tom's table multiplied by the square root of two.
• Note that it is straightforward to calculate the expected rms measurement error in the averaged value of a signal when that signal consists low-pass filtered white noise. This error is given by the rms noise of the signal divided by the square root of (2.0 * bandwidth of signal * integration time). This is given in Appendix B of Lena, Lebrun and Mignard, "Astrophysical Observations", second edition, page 452. Thus {sigma-F} = {NEFD} / {(root-2)(root-t)}
• Polarimetric errors:
• Measurement errors in Q and U are in general bigger than measurement errors in I by a factor of root-2. To see this, imagine a polarimeter that measures only Q, by splitting the signal into two photon streams, I_H and I_V. Clearly, the error in I_H plus I_V is the same as the error in I_H minus I_V. So Q can be measured just as accurately as I. But polarimetrists spend half their time on Q and half on U so the errors in Q and U end up being bigger by root-2 from what one would get if one spent the same amount of time just measuring I alone.
• Note that sigma-P equals sigma-u, which is sigma-U divided by the total intensity, I.
• Note that for imperfect polarimeters, polarimetric errors should increase in inverse proportion to the polarimetric efficiency.
• Thus sigma-P equals {(root-2)(sigma-I)} / {(I)(eta)} , where eta is the polarimetric efficiency of the polarimeter, I is the total intensity, and sigma-I is the measurement error one would get if one used the polarimeter as a photometer and measured "I" with it, instead of Q and U.
• Putting together all of the above results, we see that sigma-P equals (NEFD)/{(F)(eta)(root-t)} , where NEFD is the photometric NEFD. The photometric NEFD is given in Tom's table, but must be multiplied by root-2 as explained above, to correct for some effects that were ignored. (We now use F for the total Flux from the source instead of I.)

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Last updated June 5, 2002.
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