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Abstract:
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Study on variable stars is an important topic of modern astrophysics. After
the invention of powerful telescopes and high resolving powered CCD’s, the
variable star data is accumulating in the order of peta-bytes. The huge amount
of data need lot of automated methods as well as human experts. This thesis
is devoted to the data analysis on variable star’s astronomical time series data
and hence belong to the inter-disciplinary topic, Astrostatistics.
For an observer on earth, stars that have a change in apparent brightness
over time are called variable stars. The variation in brightness may be regular
(periodic), quasi periodic (semi-periodic) or irregular manner (aperiodic) and
are caused by various reasons. In some cases, the variation is due to some
internal thermo-nuclear processes, which are generally known as intrinsic vari-
ables and in some other cases, it is due to some external processes, like eclipse
or rotation, which are known as extrinsic variables. Intrinsic variables can
be further grouped into pulsating variables, eruptive variables and flare stars.
Extrinsic variables are grouped into eclipsing binary stars and chromospheri-
cal stars. Pulsating variables can again classified into Cepheid, RR Lyrae, RV
Tauri, Delta Scuti, Mira etc. The eruptive or cataclysmic variables are novae,
supernovae, etc., which rarely occurs and are not periodic phenomena. Most
of the other variations are periodic in nature.
Variable stars can be observed through many ways such as photometry,
spectrophotometry and spectroscopy. The sequence of photometric observa-
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tions on variable stars produces time series data, which contains time, magni-
tude and error. The plot between variable star’s apparent magnitude and time
are known as light curve. If the time series data is folded on a period, the plot
between apparent magnitude and phase is known as phased light curve. The
unique shape of phased light curve is a characteristic of each type of variable
star. One way to identify the type of variable star and to classify them is by
visually looking at the phased light curve by an expert. For last several years,
automated algorithms are used to classify a group of variable stars, with the
help of computers.
Research on variable stars can be divided into different stages like observa-
tion, data reduction, data analysis, modeling and classification. The modeling
on variable stars helps to determine the short-term and long-term behaviour
and to construct theoretical models (for eg:- Wilson-Devinney model for eclips-
ing binaries) and to derive stellar properties like mass, radius, luminosity, tem-
perature, internal and external structure, chemical composition and evolution.
The classification requires the determination of the basic parameters like pe-
riod, amplitude and phase and also some other derived parameters. Out of
these, period is the most important parameter since the wrong periods can
lead to sparse light curves and misleading information.
Time series analysis is a method of applying mathematical and statistical
tests to data, to quantify the variation, understand the nature of time-varying
phenomena, to gain physical understanding of the system and to predict future
behavior of the system. Astronomical time series usually suffer from unevenly
spaced time instants, varying error conditions and possibility of big gaps. This
is due to daily varying daylight and the weather conditions for ground based
observations and observations from space may suffer from the impact of cosmic
ray particles.
Many large scale astronomical surveys such as MACHO, OGLE, EROS,
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ROTSE, PLANET, Hipparcos, MISAO, NSVS, ASAS, Pan-STARRS, Ke-
pler,ESA, Gaia, LSST, CRTS provide variable star’s time series data, even
though their primary intention is not variable star observation. Center for
Astrostatistics, Pennsylvania State University is established to help the astro-
nomical community with the aid of statistical tools for harvesting and analysing
archival data. Most of these surveys releases the data to the public for further
analysis.
There exist many period search algorithms through astronomical time se-
ries analysis, which can be classified into parametric (assume some underlying
distribution for data) and non-parametric (do not assume any statistical model
like Gaussian etc.,) methods. Many of the parametric methods are based on
variations of discrete Fourier transforms like Generalised Lomb-Scargle peri-
odogram (GLSP) by Zechmeister(2009), Significant Spectrum (SigSpec) by
Reegen(2007) etc. Non-parametric methods include Phase Dispersion Minimi-
sation (PDM) by Stellingwerf(1978) and Cubic spline method by Akerlof(1994)
etc.
Even though most of the methods can be brought under automation, any of
the method stated above could not fully recover the true periods. The wrong
detection of period can be due to several reasons such as power leakage to
other frequencies which is due to finite total interval, finite sampling interval
and finite amount of data. Another problem is aliasing, which is due to the
influence of regular sampling. Also spurious periods appear due to long gaps
and power flow to harmonic frequencies is an inherent problem of Fourier
methods. Hence obtaining the exact period of variable star from it’s time
series data is still a difficult problem, in case of huge databases, when subjected
to automation. As Matthew Templeton, AAVSO, states “Variable star data
analysis is not always straightforward; large-scale, automated analysis design
is non-trivial”. Derekas et al. 2007, Deb et.al. 2010 states “The processing of
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huge amount of data in these databases is quite challenging, even when looking
at seemingly small issues such as period determination and classification”.
It will be beneficial for the variable star astronomical community, if basic
parameters, such as period, amplitude and phase are obtained more accurately,
when huge time series databases are subjected to automation. In the present
thesis work, the theories of four popular period search methods are studied, the
strength and weakness of these methods are evaluated by applying it on two
survey databases and finally a modified form of cubic spline method is intro-
duced to confirm the exact period of variable star. For the classification of new
variable stars discovered and entering them in the “General Catalogue of Vari-
able Stars” or other databases like “Variable Star Index“, the characteristics
of the variability has to be quantified in term of variable star parameters. |