Error estimation in differential photometry

It is possible, if one knows in detail many of
the electronic characteristics of one's CCD camera, to estimate *a
priori *the sorts of errors that one may expect from
differential photometry. This is by no means a small task, and
may well be beyond the technical ability (and interest) and time
constraints of many observers. This is in no way meant as a
put-down of this method; rather, that it is involved and perhaps
more so than many would wish to accept.

As mentioned on the main Photometry page, one
may use a check star (and preferably more than one) not only to
assess the stability of the comparison star, but also as a means
of estimating the error of the data on the variable star. If the
check star, *K*, is of similar brightness to the variable
star, *V*, then the statistics of *C-K* will be
similar to the statistics of *C-V*.

If *K* is fainter than *V*, the *C-K*
error will over-estimate the *C-V* error.

If* K* is brighter than *V*, the *C-K*
error will underestimate the *C-V* error.

If the only possible option is to measure a
single *K* star only, try to choose one that is around the
average brightness of *V*. This way, *C-K* errors
will provide an "average" indication of the expected
errors in *C-V*.

Now, if one can choose and measure a whole
swarm of *K* stars, which range from fainter than, to much
tha same as, and to brighter than *V*, one may estimate
the errors for any value of *C-V*. The more stars the
merrier; perhaps around six as a minimum, and a dozen or so would
be good.

For a data run on a particular field, obtain
the differential magnitudes for the variable star and for all
your selected check stars. You will then have a set of
differential magnitudes vs. time (i.e. time series) for *C-V*
and *C-K1*, *C-K2*, *C-K3,* etc. For each of
these time series, calculate the standard deviation of the time
series, and its average magnitude. If, as an example, you have
ten check stars and one variable, you will end up with 11 pairs
of numbers of the form

(ave. magnitude, standard deviation)

to be considered as (x,y) pairs for plotting on
a graph. Plot them out (using a spreadsheet or other suitable
software), and fit an exponential curve to the (x,y)-pairs for
the **C-K****
values only** (do not
include the *C-V* pair in this step- treat it strictly as
a separate series on the graph). The software should be able to
tell you the equation that was used to fit the curve to the
points.

This equation will allow you to estimate a
y-value (i.e. scatter, to be interpreted as expected error) for
any x-value (i.e. differential magnitude) for the conditions that
prevailed at the time of that particular observing run. Use the
software to generate an expected error for each *C-V*
point, and then use this as the size of the y-error-bars on a
plot of the *C-V* time series. Each of these errors will
be appropriate to the magnitude of *V* at that time.

*Diff. mag. vs. time
(minutes)* : about 9 hours of *C-V* and *C-K*
data for the field of EUVE J2115-58.6 on the night of
27-28/7/1997. Check stars are crosses of various colours, and the
variable is denoted by a circle. It is evident that the scatter
in the data increases as one goes to fainter stars. It also is
clear that the *V* star varies considerably in overall
brightness, so that a single error estimate would not be accurate
for all the *C-V* values shown.

*Scatter vs. ave.
diff. magnitude* : a plot of standard deviation (scatter) vs.
average differential magnitude for the time series shown in the
first diagram. An exponential fit to the *C-K* pairs is
shown, along with the equation of the fit. *K* star value
are shown as crosses; the variable is denoted by a circle. Note
that the *V* star is highly variable- it lies well above
the trend for expected errors due to measurement errors alone,
and is therefore significantly intrinsically variable.

*Diff. mag. vs. time
(minutes)* : *C-V *data with error bars added. Note
that the size of the error changes with the brightness of the *V*
star; larger when *V* is fainter, smaller when it is
brighter.

This method makes no assumptions about the
characteristics of the CCD camera, and as it uses the actual data
to estimate the error, it will reflect the actual observing
conditions prevailing during the observing run. Nights which are
photometrically poor (even for the relavtively forgiving nature
of differential photometry), will show larger scatter than those
nights which are of high quality. This will show up in the size
of the errors allocated to the *C-V* points, and will
alert the observer to data which may be too poor to use. Also,
changing photometric quality during a night will, quite likely,
be readily apparent as an increse or decrease in scatter of the *C-K*
data during the night.

It may by chance happen that one of the *K*
stars will be variable. This may be shown by a continued habit of
it appearing above the general trend of your scatter points- in
which case, you may have discovered a new (but possibly quite
faint) variable star.

A photometric reduction package which allows
the choosing of multiple* K* stars will allow this sort of
reduction to be undertaken with minimal extra effort on the part
of the observer. If such a method is employed, one will have a
thorough idea of the errors in your data for each data run, a
good long-term check of the stability of *C*, a sample of
stars to check for new variables, and, over time, a good guide to
the sort of accuracy attainable with your equipment and the
general stability of your observing site with regard to
photometric quality.