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 root2.

Second, note that Tom's calculation does not account for the need
to "chop", which introduces a factor of two loss in signaltonoise.

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 lowpass 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
{sigmaF} = {NEFD} / {(root2)(roott)}

Polarimetric errors:

Measurement errors in Q and U are in general bigger
than measurement errors in I by a factor of root2. 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 root2 from
what one would get if one spent the same amount of time just measuring I
alone.

Note that sigmaP equals sigmau, which is sigmaU divided
by the total intensity, I.

Note that for imperfect polarimeters, polarimetric errors
should increase in inverse proportion to the polarimetric
efficiency.

Thus sigmaP equals {(root2)(sigmaI)} / {(I)(eta)} ,
where eta is
the polarimetric efficiency of the polarimeter, I is
the total intensity, and sigmaI 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 sigmaP equals (NEFD)/{(F)(eta)(roott)} , where NEFD
is the photometric NEFD. The photometric NEFD is given in Tom's
table, but must be multiplied by root2 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.)
Last updated June 5, 2002.
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