Periodicities in variable stars: a few issues

Chris Koen

Dept. Statistics

University of the Western Cape

Summary

• Variable stars

• The periodogram

• Quasi-periodic variations

• Periodic period changes

Some Example Lightcurves

• Lightcurve:

brightness plotted against time (or

sometimes phase)

An eclipsing double star (P=7.6 h)

A pulsating star (P=1.4 h)

Residual sums of squares after fitting sinusoids with different frequencies

Phased lightcurve, adjusted for changing mean values

The Periodogram

2

1

2

1

2

1

2

1

sincos1

2sin2cos1

)(

N

kkk

N

kkk

N

kkk

N

kkk

tytyN

tytyN

I

Regular time spacing

• Frequency range

• Frequency spacing

t /5.00

2...,,2,1;/ Nj jNj

Periodogram of sinusoid (f=0.3) with superimposed noise: regularly spaced data

Periodogram of sinusoid (f=0.3) with superimposed noise: irregularly spaced data

2

1

2

1

sincos1

)(N

kkk

N

kkk tyty

NI

jkallforttII

jk ,0)(sin)()(

*

**

0)(sin)(21

1 1

N

j

N

jkjk ttS

Solutions for Nyquist frequency

Time spacing between exposures (IRSF)

Top: IRSF exposuresBottom: Hipparcos

Frequency spacing

• Frequency resolution is

(Loumos & Deeming 1978, Kovacs 1981)

T/1~

Significance testing of the largest peak

• For regularly spaced data:

- statistical distribution of ordinates known

- ordinates independent in Fourier frequencies

• For irregularly spaced data:

- ordinates can be transformed to known distribution – ordinates not independent

Correlation between periodogram ordinates for increasing separation between frequencies

(irregularly spaced data)

Horne & Baliunas (1986): “independent frequencies”

Quasi-periodicities (QPOs)

• Sinusoidal variations with changing amplitude, period and/or phase

A 32 minute segment of fast photometry of VV Puppis

Periodogram of the differenced data

Periodograms of first and second quarters of the data

Wavelet plot of the first quarter of the data

Complex Demodulation

• Transform data so that frequency of interest is near zero

• Apply a low pass filter to the transformed data

Complex demodulation of the first quarter of the data

Time Domain Modelling

Gaussiant

t

tB

tA

tB

tA

tetB

tAtt

tetttCtY

,)(

)(

)1(

)1(

)(

)(

)()(

)(sincos

)())((cos)()(

00

0

Amplitude and phase variations from Kalman filtering

The results of filtering the second quarter of the data

Periodic period changes

• Apsidal motion

• Light-time effect

• Stochastic trends?

O-C (Observed – Calculated)

• Equivalent to CUSUMS• Sparsely observed process:

)()( 0*

1

1

PNTTCO

countcyclecumulativenN

TandTbetweenelapsedcyclesofnumbern

jjj

j

iij

jjj

SZ Lyn (Delta Scuti pulsator in a binary orbit)

The Light-time Effect

)(2

)(sin)(

2

)(tan

1

1arctan2)(

sin])([sin)(cos1

1)(

0

2

ttP

tEetE

tE

e

et

ette

eAtT

b

TX Her (P = 1.03 d)

SV Cam (P = 0.59 d)

A stochastic period-change model

),0(~

)(

21

110

1 1*

jjjj

N

kkk

j

ijiij

j

i

N

kkkiij

jjj

j

j

ePNnCO

enTT

P

State Space Formulation:

jN

k

j

kk

j

iiik

j

iiij

j

j

j

j

j

jj

nnU

GaussianUnU

eTU

nT

1 1

1

1

1

1

1

1

*

,10

1

1

)10( 5 dofunitsHerTX

)10( 6 dofunitsCamSV

General form of Information Criteria:

IC = -2 log(likelihood)+penalty(K)

• Akaike : penalty=2K

• Bayes: penalty=K log(N)

• Model with minimum IC preferred

Models:

• Polynomial + noise

• Random walk + noise

• Integrated random walk + noise

Order Sigma_error BIC

3 1.1921 153.57

4 1.1036 142.74

#5 0.51673 -4.4166*

6 0.51335 -1.1247

7 0.51519 4.1961

RW 0.43166 41.661

IRW 0.51412 55.247

Order sigma_error BIC 1 0.24656 -170.82

4 0.23132 -169.76

5 0.21551 -179.32

6 0.21558 -174.65

7 0.21589 -169.76

# RW 0.19477 -185.97*

IRW 0.21756 -171.33

Order sigma_error BIC

4 0.29048 -124.22

5 0.27773 -128.59

6 0.24941 -145.5

7 0.24809 -141.95

8 0.24678 -138.41

RW 0.17886 -119.37

#IRW 0.2194 -149.06*

A brief mention…

Transient deterministic oscillation or purely stochastic variability?