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)

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

8. Phased lightcurve, adjusted for changing mean values

14. The Periodogram

15. Regular time spacing Frequency range Frequency spacing

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

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

19. Solutions for Nyquist frequency

21. Time spacing between exposures (IRSF)

22. Top: IRSF exposures Bottom: 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 (irregularly 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:

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

48. Models: Polynomial + noise Random walk + noise Integrated random walk + noise

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

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

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?