Light Curves

An important application of photometry is the observation of objects variable in brightness. The plot of brightness versus time for such an object is its light curve.

Ephemerides of a Variable Star

A regular variable star can be characterized by its light elements; viz. its period and a reference epoch (which is typically the center of the primary minimum for an eclipsing system). The epoch will be given as a Julian Date (JD) and the period will be in days in most cases. The phase is the fraction (ranging from 0.0 to 1.0) of the period at the time of the observation. Thus phase can be calculated as where T is the JD of the observation, T0 is the reference epoch and P is the period. The epoch (time and date) of the occurrence of a particular phase is calculated from where n is the number of full periods since T0.

Finding the period

The raw light curve may not look like anything other than a random scattering of points. It is necessary to plot the points as a function of the phase in order to see the true nature of the variability. This requires the determination of the period of variability which may not always be obvious. The early researchers did period finding essentially by trial and error. A period was guessed and then phases were calculated by hand or with a mechanical calculator and a light curve was plotted.

Observations of the Cepheid variable star R Cru plotted against the last four digits of Julian date

Observations of the Cepheid variable star R Cru plotted against phase; P=5.8257d.

One of the first systematic methods of period determination was described by Lafler and Kinman (1965, ApJ Supp, 11, 216-222. "The calculation of RR Lyrae periods by electronic computer"). This method provides a systematic approach to testing a series of trial periods and looking for the period that results in the "smoothest" light curve. The observed points are sorted by phase and the sum of the squares of the difference in magnitude of successive pairs of points is used to rank the trial period. The smallest value of the figure of merit should be the "best" period since this represents the smallest successive changes in the light curve.

trial period	5.8	5.81	5.82	5.825	5.8257
L-K "Theta"	0.94	0.81	0.44	0.016	0.020

Since the square-root of this figure of merit is a length, this method is called a "string" method. Dworetsky (1983, Mon. Not. R. Astr. Soc. 203, 917-924 "A period-finding method for sparse randomly spaced observations or "How long is a piece of string?"") discusses and extends a method introduced by earlier researchers that is literally a "string" method. Here the total distance between points in a phased light curve is calculated. The smaller the total distance, the smoother the light curve.

More recently the PDM method present by Stellingwerf, R.F., 1978, ApJ 224, 953-960. "Period determination using phase dispersion minimization" has been adopted by many researchers and the IRAF data analysis routines include a PDM period finder. In this method a smoothed light curve is calculated (typically between 5 and 10 points) and the dispersion of each point is calculated. The trial period that minimizes this dispersion is the "best" period.

While it would appear that fourier analysis of the light curve might represent the most natural and direct approach to period finding there are difficulties with this approach for light curve analysis. Observations are frequently not uniformly spaced in time and there are usually gaps in the data set (which may be large when the weather turns bad).

Spurious Periods

Astronomers, of necessity, have periodicities imposed on their observations by outside forces. For ground-based observatories sunrise and sunset impose limits except at the South Pole (where the lunar cycle may introduce other periodicities). Orbiting telescopes have their own limitations, usually related to their orbital period. The habit of observing always near meridian passage (transit) can introduce alias periods related to the sidereal day so that one may find false periods related to the true period by .

An example of a binary system where a problem could arise is V471 Tau (BD +15° 516) whose period is 0.52118346 hours (12.5084 days). This presented a problem when first finding the period and it also presents a continued problem in planning observations. Each night one has almost the same portion of the light curve. It takes several weeks to get the complete curve.

Interesting and useful links:

Some variable stars observed by the Hipparcos satellite are displayed at http://astro.estec.esa.nl/Hipparcos/education_lcA.html in an interactive web page where different periods can be tried.

References on Period Finding:

http://www.umanitoba.ca/faculties/science/astronomy/gao/aids/biblios/

A class lecture by Michael Richmond at the Rochester Institute of Technology at http://spiff.rit.edu/classes/phys445/lectures/period/period.html

In the literature

(these papers can be found at http://adsabs.harvard.edu by searching on authors and titles)