Re: GSC 00445-01993

[Chris Lloyd <cl@astro1.bnsc.rl.ac.uk>]

Period finding has the reputation of being something of a black art
because much of what the analysis tells you is open to interpretation. It
is rare to have enough data, of good quality where the light curve is well
enough sampled, to give a completely unambiguous result. For a start the
observations are usually made from one place and so they tend to be spaced
at one day intervals. This means that periods close to a fraction of a day
are poorly sampled but more generally it introduces an ambiguity of
1,2,...,n cycles per day in the frequency (period) of any variation
present in the data. These are usually referred to as the 1-day aliases
and can only be avoided by observing over a large range hour angles, or
from several widely spaced longitudes. Other regular spacings in the data
will also introduce their own aliases. The frequencies generated by the
data spacing are shown by the [spectral] window function and although it
is a good idea to generate this if you can, serious aliasing is obvious in
the periodogram as a series of regularly spaced features. This can happen
on all scales. Although people like periods it is important to realise
that periodograms work in frequency space, all the aliasing occurs at
equal frequency intervals.

All periodograms look at the data folded with a trial frequency (period)
and calculate some number that reflects the goodness of the light curve as
the program sees it. The methods basically fall into two camps - they
either look at the smoothness of the light curve, PDM, Jerzykiewicz,
Stellingwerf and the rest, or how sinusoidal it is, DFT, all Fourier based
methods, Lomb [-] Scargle, least-squares etc. Each has its own algorithm,
implementation, assumptions and approximations. Also real light curves are
rarely exactly sinusoidal and although smooth, not necessarily the kind of
smooth dictated by the periodogram.

On a good data set they will give the same answer but on real data with a
variety of gaps and data spacings the cracks begin to show. Each
periodogram measures what it measures, and usually that is pretty close to
you would call a good period, but it is not invariably the right period.
Ultimately it comes down to human interpretation. Is it a delta Scuti, or
a W UMa with twice the period, or twice the period of that alias that
won't go away?

The errors that come out of a periodogram, when they're provided at all,
are a pretty mixed bag. The only sensible way of estimating the error on
what you believe to be the right period is to fit the light curve with a
sine curve or Fourier series by least squares. If you can't do that,
purists look away now, then fold the data with periods slightly different
to the "best fit" and see where the light curve becomes obviously worse.
It ain't mathematical but it will give you a realistic idea.

The relatively long runs of TASS data make it an excellent instrument for
detecting interesting short-period variables. I know little about the
operation of the hardware but I believe it could significantly improve the
initial analysis if the data were taken over a range of hour angles, and
if the fields were repeated at irregular intervals.

Chris