Period Criteria

Actual stars are expected to be more complicated than these simulations. For example, if a star is inclined and has a nearly polar spot, the variation will be far more complicated. The spot may be entirely visible most of the time (one would observe a prolonged flux minimum low), and then part of it may disappear over the horizon briefly and then return (slow rise and then fall back to constant level). In this case, the observed modulation is actually a brightening from the base signal. Flares on stars are even more troublesome, since they usually occur near spotted regions. When the photosphere is becoming dimmer, there can be a very bright group of data points. In these realistic cases, it would be very difficult to determine the correct period without high S/N and additional information. With the set of observations taken here, I have information available from several comparison stars and color information. I use four consistency criteria to mitigate against noise, and two additional criteria to fight aliasing.

1. The same period must be found using both period finding techniques. The rule that was used here was that a peak in the power distribution of the periodogram had to coincide with a local minimum of the PDM code.

2. The same period must be found using multiple comparison stars. This is an obvious criterion which is used to remove variable comparison stars.

3. The same period must be found in the V, R and I filters. If the periodic behavior does not have the same period in the three filters, it is probably not due to rotation. Five percent deviation is considered acceptable to allow for measurement errors in the time of observation and in the sampling grid used by the period search routines. As is shown in the Figures 3.1, 3.2, 3.3 and 3.5. The full width half maximum of the peaks in the periodogram functions are quite wide. This width is caused by the low number of sample used in the simulation, and the actual data sets.

4. The same phase must be found in the three filters. This criterion is similar to the previous one. Again five percent errors are considered acceptable.

5. The period found must be consistent with the data from any given night. One way to test whether a true period or an alias has been found is to compare the fit to the period with the data for a given night. For example, if the fit to the data predict that on a given night the target star should be getting fainter, yet the data for that night show that star is getting significantly brighter, the period is discarded.

6. The color changes observed are consistent with the starspot hypothesis. All the period measurements rest on the initial hypothesis that the observed modulations are induced by the rotation of spotted regions of the star onto and off of the side of the star facing the Earth. This hypothesis has fundamental predictions for the observed color changes that should be observed. For a star with very cold spots, the spots appear black relative to the star and the same color variation is observed in all colors. For a star with warmer spots, 300K cooler than the photosphere, more flux is removed from the V band than the I band, therefore the observed variations are greater in V than in R, and greater in R than in I. If the behavior differs from this prediction by more than the observed noise in the signal, the period is discarded.

Before discussing the results, a few cautionary notes should be made clear. Even with these criteria, there is no method available to completely prevent false period detections. The statistical FAP calculation should be looked upon as a relative measure for comparing different results and not an absolute measure. It is also very difficult to distinguish between periods that are aliases of each other and are also close to one day, which is a fairly common occurrence. In the following section, I present the most probable periods that are consistent with the criteria that have been discussed. The data are separated into two main sections, those with FAPs below 1% which are expected to be very secure, and those with FAPs between 1% and 20% which are far less certain.