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Periodicities in variable stars: a few issues

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

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Periodicities in variable stars: a few issues

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  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 (P=7.6 h)

  5. A pulsating star (P=1.4 h)

  6. Residual sums of squares after fitting sinusoids with different frequencies

  7. Phased lightcurve, adjusted for changing mean values

  8. The Periodogram

  9. Regular time spacing • Frequency range • Frequency spacing

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

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

  12. Solutions for Nyquist frequency

  13. Time spacing between exposures (IRSF)

  14. Top: IRSF exposuresBottom: Hipparcos

  15. Frequency spacing • Frequency resolution is (Loumos & Deeming 1978, Kovacs 1981)

  16. 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

  17. Correlation between periodogram ordinates for increasing separation between frequencies(irregularly spaced data)

  18. Horne & Baliunas (1986): “independent frequencies”

  19. Quasi-periodicities (QPOs) • Sinusoidal variations with changing amplitude, period and/or phase

  20. A 32 minute segment of fast photometry of VV Puppis

  21. Periodogram of the differenced data

  22. Periodograms of first and second quarters of the data

  23. Wavelet plot of the first quarter of the data

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

  25. Complex demodulation of the first quarter of the data

  26. Time Domain Modelling

  27. Amplitude and phase variations from Kalman filtering

  28. The results of filtering the second quarter of the data

  29. Periodic period changes • Apsidal motion • Light-time effect • Stochastic trends?

  30. O-C (Observed – Calculated) • Equivalent to CUSUMS • Sparsely observed process:

  31. SZ Lyn (Delta Scuti pulsator in a binary orbit)

  32. The Light-time Effect

  33. TX Her (P = 1.03 d)

  34. SV Cam (P = 0.59 d)

  35. A stochastic period-change model

  36. State Space Formulation:

  37. General form of Information Criteria: IC = -2 log(likelihood)+penalty(K) • Akaike : penalty=2K • Bayes: penalty=K log(N) • Model with minimum IC preferred

  38. Models: • Polynomial + noise • Random walk + noise • Integrated random walk + noise

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