Periodicities in variable stars: a few issues

1
Periodicities in variable stars a few issues

• Chris Koen
• Dept. Statistics
• University of the Western Cape

2
Summary

• Variable stars
• The periodogram
• Quasi-periodic variations
• Periodic period changes

3
Some Example Lightcurves

• Lightcurve
• brightness plotted against time (or
• sometimes phase)

4
An eclipsing double star (P7.6 h)
5
A pulsating star (P1.4 h)
6
(No Transcript)
7
Residual sums of squares after fitting sinusoids
with different frequencies
8
Phased lightcurve, adjusted for changing mean
values
9
(No Transcript)
10
(No Transcript)
11
(No Transcript)
12
(No Transcript)
13
(No Transcript)
14
The Periodogram
15
Regular time spacing

• Frequency range
• Frequency spacing

16
Periodogram of sinusoid (f0.3) with superimposed
noise regularly spaced data
17
Periodogram of sinusoid (f0.3) with superimposed
noise irregularly spaced data
18
(No Transcript)
19
Solutions for Nyquist frequency
20
(No Transcript)
21
Time spacing between exposures (IRSF)
22
Top IRSF exposuresBottom Hipparcos
23
Frequency spacing

• Frequency resolution is
• (Loumos Deeming 1978, Kovacs 1981)

24
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

25
Correlation between periodogram ordinates for
increasing separation between frequencies(irregul
arly spaced data)
26
Horne Baliunas (1986) independent frequencies
27
Quasi-periodicities (QPOs)

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

28
A 32 minute segment of fast photometry of VV
Puppis
29
Periodogram of the differenced data
30
Periodograms of first and second quarters of the
data
31
Wavelet plot of the first quarter of the data
32
Complex Demodulation

• Transform data so that frequency of interest is
  near zero
• Apply a low pass filter to the transformed data

33
Complex demodulation of the first quarter of the
data
34
Time Domain Modelling
35
Amplitude and phase variations from Kalman
filtering
36
The results of filtering the second quarter of
the data
37
Periodic period changes

• Apsidal motion
• Light-time effect
• Stochastic trends?

38
O-C (Observed Calculated)

• Equivalent to CUSUMS
• Sparsely observed process

39
SZ Lyn (Delta Scuti pulsator in a binary orbit)
40
The Light-time Effect
41
TX Her (P 1.03 d)
42
SV Cam (P 0.59 d)
43
A stochastic period-change model
44
State Space Formulation
45
(No Transcript)
46
(No Transcript)
47
General form of Information Criteria

• IC -2 log(likelihood)penalty(K)
• Akaike penalty2K
• Bayes penaltyK log(N)
• Model with minimum IC preferred

48
Models

• Polynomial noise
• Random walk noise
• Integrated random walk noise

49
(No Transcript)
50
(No Transcript)
51

• 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

52

53

54

• 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

55

56

57

• 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

58
A brief mention

• Transient deterministic oscillation or purely
  stochastic variability?

59
(No Transcript)
60
(No Transcript)
61
(No Transcript)
62
(No Transcript)