Variable stars are stars for which the intensity of the emitted energy changes over time; for periodic variable stars the change of intensity is periodic over time. Common types of periodic variable stars include eclipsing binaries, RR Lyraes, and Cepheids.
Eclipsing binaries consist of two stars orbiting each other in a conformation relative to the observer such that brightness variability occurs as one star passes in front of the other in turn; as the stars may be of different brightness, the drop in light flux depends on which star is in the front. These stars have periods of between 3 hours and 24 years, although 0.5 to 10 days is the most common range. The light curve from an eclipsing binary is shown in the next figure:
The brightness changes in the remaining classes of periodic variables is caused by periodic pulsation (contraction and expansion) of the stars and their outer layers.
RR Lyrae stars are the second most common class of known periodic variable stars; they have periods in the range 0.2 to 0.9 days. There are two common types, RRab stars which have an asymmetric signal and RRc stars which have a symmetric signal. This is an example of the light curve of an RRab with a period of .53 days::
Cepheids are very bright stars with periods of 1-70 days. The light curve has an asymmetric shape, and rises more rapidly than it falls. Cepheids with periods of about 1 week tend to have a bump in the descending part of the curve. For periods of about 10 days, the bump is at the peak of the curve, and for longer periods it is on the rising part of the curve. Here is a Cepheid with a period of 4 days:
The most common class of known periodic variable stars are the Long Period Variables. These are red giant stars with periods in the range 30-1000 days. The period can vary by about 10% with accompanying changes in average flux and amplitude of the harmonic component, and so these are more accurately semiperiodic stars. Here is an example:
There are numerous additional types of variable stars, and each of the categories above contains subcategories. For example, Beat Cepheids and Beat RR Lyrae oscillate at more than one frequency. More details are given in the references below. Different classes of variable stars can be located in different regions of a plot of magnitude versus temperature or spectral type. For example, RR Lyrae and Cepheids lie on a strip called the "instability strip." Different types of variable curves are classified also on the basis of the shapes of their light curves and the relationships of shapes to period, for example. As well as being important for studies of stellar structure and evolution, these classes are used to determine distances on a cosmic scale by means of the relationship of their periods to their magnitudes.
Observations of these stars are typically made at rather irregular times, depending on observation schedules and sky conditions. Different observations have differing errors. All this makes determination of the periods and the shapes of the corresponding phased light curves an interesting statistical problem. Here is a plot of the intensity versus time for a particular star:
Any structure in this time series is quite obscure. Next is a plot when the data are folded with a period of 1.4 days---now the pattern of this eclipsing binary is quite clear!
How was this period found? That was the subject of James Reimann's Ph.D. thesis (Berkeley Department of Statistics, 1994). The basic idea is that the adequacy of a candidate period is judged by the magnitude of the variation about a smooth curve passed through the data folded at that period.
The MACHO project monitors millions of stars every night with a dedicated telescope at Mount Stromlo Observatory in Australia (home of the kookaburra). The collaboration is probing the halo of our galaxy in order to detect dark matter in the form of Massive Compact Halo Objects---MACHOs. These are astronomical bodies that emit negligible visible light, such as dwarf or neutron stars, large planets, and black holes. Detection of a MACHO is achieved by observing its gravitational lensing effect on a chance background star as the MACHO crosses near the line of sight between the observer and this star. In order to detect a sufficiently large number of MACHOs, the project collects observations on an large number of distant stars over an extended period of time. Data are being collected daily over a 4-year period (weather permitting), on approximately 8 million stars in the Large Magellanic Cloud (LMC) and the bulge of the Milky Way.
This database is a valuable resource for many other types of astronomical research. It is the most comprehensive catalog of stars in the LMC and contains stars much dimmer than those covered by previous surveys. Temporal coverage is unusually long compared to most star surveys, which permits a comprehensive study of star variability, including long periods and transient phenomena. About 40,000 variable stars have been observed in the LMC and a similar number in the galactic bulge.
A complete database of variable stars will available at some time in the near future. For the purposes of this workshop, data from a small number of stars can be downloaded as a tar file of about 9 megabytes.. It will unpack into directories containing Cepheids, RR Lyrae, and eclipsing binaries. Data is recorded in two color bands, a blue band from 4500--6300 Angstroms and a red band from 6300--7600 Angstroms. Files containing the character "r" relate to the red band and those containing "b" relate to the blue band. Files with the suffix .dat are the data files. After a header, the first column gives the time, the second gives the (differential) value of the magnitude at that time, and the third gives the SD of the measurement. A value of -99 means that the data point is missing and any other negative value means that the data point is suspect and should not be used. You can ignore the remaining columns. Files with the suffix .per contain some results of the fitting program. Postcript files show phased plots of the light curves. These data were kindly provided for purposes of this workshop by Kem Cook of the Institute of Geophysics and Planetary Physics at Lawrence Livermore National Laboratories.
C. Hoffmeister, G. Richter, and W. Wenzel (1985). Variable Stars. Springer-Verlag.
R. Ferlet, J.P. Maillard, and B. Raban (1996). Variable Stars and the Astrophysical Returns of Microlensing Surveys. Editions Frontieres.
James Reimann's Ph.D. thesis [postscript pdf] contains more information about the MACHO data. It compares several methods for estimating periods on both theoretical and empirical grounds.
The MACHO Project
The OGLE Project is also searching for microlensing and maintains an online database of variable stars.
T. Axelrod et al. (1994). Statistical issues in the MACHO project. In Statistical Challenges in Modern Astronomy II. (eds. E. Feigelson and G.J. Babu). Springer Verlag.