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Techniques of Observational

Astronomy

AST3722C

Light Curves

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

Ephemeredes of a Variable StarA regular variable star can be characterized by its light elements viz its period and areference epoch (which is typically the center of the primary minimum for an eclipsingsystem) 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 00 to 10) of the period at the time of

the observation Thus phase can be calculated as whereT is the JD of the observation T 0 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 T 0

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

Observations of the Ce heid variable star R Cru Observations of the Ce heid variable star R Cr

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plotted against the last four digits of Julian date plotted against phase P=58257d

One of the first systematic methods of period determination was described by Lafler andKinman (1965 ApJ Supp 11 216-222 The calculation of RR Lyrae periods byelectronic 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 Theobserved points are sorted by phase and the sum of the squares of the difference inmagnitude of successive pairs of points is used to rank the trial period The smallest value

of the figure of merit should be thebest period since this represents the smallest successive changes in the light curve

trial period

58 581 582L-K Theta 094 081 044

trial period 5825 58257

L-K Theta 0016 0020

Since the square-root of this figure of merit is a length this method is called a stringmethod Dworetsky (1983 Mon Not R Astr Soc 203 917-924 A period-findingmethod 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 astring method Here the total distance between points in a phased light curve iscalculated The smaller the total distance the smoother the light curveMore recently the PDM method present by Stellingwerf RF 1978 ApJ 224 953-960Period determination using phase dispersion minimization has been adopted by manyresearchers 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 thedispersion of each point is calculated The trial period that minimizes this dispersion isthe best periodWhile it would appear that fourier analysis of the light curve might represent the mostnatural 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 thereare usually gaps in the data set (which may be large when the weather turns bad)

Spurious Periods

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Astronomers of necessity have periodicities imposed on their observations by outsideforces For ground-based observatories sunrise and sunset impose limits except at theSouth Pole (where the lunar cycle may introduce other periodicities) Orbiting telescopeshave their own limitations usually related to their orbital period The habit of observingalways near meridian passage (transit) can introduce alias periods related to the siderealday 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 +15deg 516)whose period is 052118346 hours (125084 days) This presented a problem when firstfinding the period and it also presents a continued problem in planning observationsEach night one has almost the same portion of the light curve It takes several weeks toget the complete curve

Interesting and useful linksSome variable stars observed by the Hipparcos satellite are displayed athttpastroestecesanlHipparcoseducation_lcAhtml in an interactive web page wheredifferent periods can be tried

References on Period Finding

httpwwwumanitobacafacultiesscienceastronomygaoaidsbiblios

A class lecture by Michael Richmond at the Rochester Institute of Technology athttpspiffriteduclassesphys445lecturesperiodperiodhtml

In the literature

(these papers can be found at httpadsabsharvardedu by searching on authors and titles)- Lafler J and Kinman TD 1965 ApJ Supp 11 216-222 The calculation of RR Lyrae periods by electronic computer - Dworetsky MM 1983 Mon Not R Astr Soc 203 917-924 A period-findingmethod for sparse randomly spaced observations or How long is a piece of string

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- Renson P 1978 AampA 63 125-129 (In french + english abstract) Meacutethode deRecherche des Peacuteriodes des Eacutetoiles variables- Stellingwerf RF 1978 ApJ 224 953-960 Period determination using phasedispersion minimization- Horne J and Baliunas S 1986 ApJ 302 757-763 A Prescription for Period Analysisof Unevenly Sampled Time SeriesThere is a comparison of many methods in- Heck A et al 1985 AampA 59 63-72 On period determination methods