Astronomical research at VMI has developed in two directions:
- Observations of pulsating variable stars using the 20-inch reflector telescope.
- Theoretical investigations involving time-series analysis of variable star data
The photograph at right was obtained by Dr. DuPuy, using one of the telescopes at the VMI Observatory (see Astrophotography).
Observations of pulsating variable stars: At the end of a star's lifetime, the star becomes unstable and begins to pulsate, literally becoming first larger, then smaller in a cyclical fashion, with a corresponding change in the temperature of its atmosphere. By measuring the changes in the star's brightness over a period of time, we obtain a light curve showing how that particular star pulsates. Different classes of stars exhibit different pulsation characteristics. Over the years, astronomers have learned to use pulsating stars to indicate cosmic distances, map out the spiral arms in the Milky Way galaxy, and probe the interior structure of stars.
The VMI Observatory includes a 20-inch telescope which is ideal for obtaining professional-quality observations of pulsating variable stars. We use an astronomical CCD electronic imaging system, which is cooled to -40 degrees C to permit time exposures. The procedure is to obtain images of a small field around the variable star and then to use a few surrounding stars as comparison stars. Images are obtained in several standard filter bandpasses to obtain color information as well as the brightness changes of the variable star.
A sample light curve is shown at left for the variable star HZ Persei, a Cepheid variable which completes a pulsation cycle in 11.28 days and undergoes a change in apparent magnitude of approximately 0.6 mag (a factor of 1.7 in brightness). The data points were obtained over roughly 10 pulsation cycles and collapsed onto one cycle to show the average curve (a standard technique).
The graph shows the change in HZ Persei's brightness compared to an average brightness of four comparison stars (top graph). The bottom graph shows the difference in brightness of two of the comparison stars vs. that of the other two as an indication of the precision of the observations.
This particular Cepheid variable star shows very little scatter in the light curve, which is typical for this type of variable star. Note the asymmetry in the light curve: there is a relatively rapid rise to maximum light and a slower fall to minimum light. The line through the points is an average drawn by hand.
Time-Series Analysis of Pulsating Variable Star Data: Some variable stars exhibit very simple light curves that are easily interpreted, while others show very complicated changes in brightness. Other pulsating stars pulsate simultaneously at two or three different frequencies, making it very difficult or impossible to interpret the observations directly.
Various techniques have been developed over the years at many observatories to help decipher pulsating star observations. One of our areas of research has been to explore mathematical techniques that assist in identifying the pulsation frequencies in variable star pulsations. These techniques are related to Fourier transforms, and the data are examined in the frequency domain, rather than in the time domain, where the observations were obtained.
Example of the Technique: The following shows an example of one of the techniques we have used on variable star data to determine the pulsation periods of variable stars.
Some stars pulsate simultaneously at two frequencies. The following graph
shows a hypothetical light curve of a star pulsating once every 20 days and simultaneously pulsating (with a smaller amplitude) at 11 days. Time (in days) is shown on the x-axis, and the star's brightness is shown on the y-axis. This example is an ideal case, but even so, the two pulsations interact with each other so that the two pulsation periods cannot simply be measured directly. Real observations are also greatly complicated by the effects of breaks in the data from day/night and from cloudy nights. The hypothetical data shown here are continuous, with no noise, in order to offer a simple example.
One algorithm we have explored for variable star data is a power spectrum technique called the Scargle algorithm. It is similar to Fourier transforms, but more powerful for variable star data. It is an integral smoothing algorithm, and the data (brightness vs. time of observation) are used to calculate a power spectrum, showing the amount of "power" in the data at various frequencies.
We have programmed this type of work in various computer languages, and an example of a portion of the program in MathCAD is shown at below. (We require physics majors to obtain experience in two computer languages appropriate for physics and astronomy, including MathCAD.)
Although this appears to be just a set of equations, this is an extract from the actual program in MathCAD, a visually oriented programming language.
The results of the calculation are displayed in a graph shown below. The two peaks represent the two frequencies present in the data. The peak at 0.05 corresponds to the 20-day periodicity (0.05 times per day), the major pulsation in our hypothetical data.
The smaller peak at a frequency of 0.09 times/day represents the pulsation component at the 11-day periodicity in the data. Hence the two pulsation periods are clearly shown. Compare this with the graph of the data in the time domain above, where the two pulsation periodicities interact, and the pulsation periods are not readily identifiable.
Cadets Working on this Research: Working with me on this investigation were Robert Youngren ('97), Kevin Ryan ('98), and Peter Hugger ('03). Col. D. L. DuPuy