[Aavso-photometry] Uncertainty

[arne]

Richard Harvan wrote:
> Arne,
>         I took 12 images containing TT ARI, the 11.050 and 11.075 
> comparison stars, and performed the calibration with darks and flats. 
>         I then performed the aperature photometry, using the 11.050 star 
> as my reference.  When I obtained a measured magnitude of 11.050 of this 
> star, I then measured the magitudes of TT ARI and the 11.075 comp star.  
> This was done with all 12 images, giving 12 values for TT ARI and the 
> 11.075 comp star.  The value of the 11.050 comp star was always 11.050.  
> I obtained the standard deviation for the twelve TT ARI values from the 
> Excel function STDEV, however I calculated the standard deviation of the 
> twelve 11.075 comp star values manually using 11.075 as the mean in the 
> calculations.  Perhaps this was not the way to do this calculation for 
> the 11.075 comp star, however I thought the variance from this value was 
> more useful than the variance from the mean of the measurements.  I then 
> squared the calculated TT ARI and 11.075 standard deviation values, the 
> 11.050 error vaues from the VSP chart (0.010), and the 11.075 error 
> value from the VSP chart (0.049), summed these  four values, and took 
> the square root of the sum.  It is my intention to state the uncertainty 
> of the measurement as this value (square root of the sum). 
>        This is not what I entered into WebObs.  I intend on modifying 
> that observation, depending on the results of this discussion.
>  
Thanks, Rich - that helps.
You almost have it right.  Certainly adding every error source you
can identify in quadrature is the correct procedure.  There are three
areas where some confusion exists.

First, you use the 11.075 value instead of the instrumental mean
for the check star - this mixes and matches several error sources.
It includes the actual measurement error; the transformation error;
and the chart error (11.075 +/- 0.049) - the latter since your
instrumental error might actually be closer to the true magnitude
of the check star than what is reported on the chart.  So this
check-star procedure should end up with a slightly worse total
error for the check star than it really has.

Second and more important, you are taking the standard deviation of
the variable star as its "error".  This is composed of two parts: the
statistical uncertainty of the measurement, and the intrinsic variability
of the star.  TT Ari is known to flicker by 0.1-0.2mag on short timescales,
and taking the standard deviation of 12 measures incorporates this
actual variability.  The only error you can really determine for the
TT Ari measurement itself is its statistical (or Poisson) error, which
we all know underestimates the real error in a measurement.

In general, it is better to measure the error of a constant star in the
field that has the same magnitude as the variable, and assume that its
error will be identical to the variable.  That removes the intrinsic
variability issue.  Often you can use the check star as that constant star,
so the (K-C) difference is the crucial measure.

Finally, by fixing the magnitude of the comparison star and determining
the standard deviation of the check star with this fixed offset, means
that the statistical error of the comparison star measure is now transferred
to the check star - so the check star error is actually the total error
of the (K-C) measurement.

Bottom line:  I suggest that you take the standard deviation of the
(K-C) measurement (that is, the standard deviation of the check star
values from the instrumental mean, without the extra chart value offset)
as your base error.  That gives you the precision of the measurement,
which is what most people are interested in.  The additional offset
and error due to the lack of transformation and lack of accuracy in
the standardized chart values, is an extra "external" systematic error
that people would be concerned with if they were combining multiple
datasets without any kind of adjustment, which is rare since most
researchers will do zeropoint offsets of time series datasets before
analyzing them.  It is good to understand that it exists, and very
wise to apply if you are doing individual estimates rather than a
time series or multiple images on a night, but not necessary for
most time series work.
Arne