Periodicities in variable stars: a few issues
description
Transcript of Periodicities in variable stars: a few issues
![Page 1: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/1.jpg)
Periodicities in variable stars: a few issues
Chris Koen
Dept. Statistics
University of the Western Cape
![Page 2: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/2.jpg)
Summary
• Variable stars
• The periodogram
• Quasi-periodic variations
• Periodic period changes
![Page 3: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/3.jpg)
Some Example Lightcurves
• Lightcurve:
brightness plotted against time (or
sometimes phase)
![Page 4: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/4.jpg)
An eclipsing double star (P=7.6 h)
![Page 5: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/5.jpg)
A pulsating star (P=1.4 h)
![Page 6: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/6.jpg)
![Page 7: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/7.jpg)
Residual sums of squares after fitting sinusoids with different frequencies
![Page 8: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/8.jpg)
Phased lightcurve, adjusted for changing mean values
![Page 9: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/9.jpg)
![Page 10: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/10.jpg)
![Page 11: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/11.jpg)
![Page 12: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/12.jpg)
![Page 13: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/13.jpg)
![Page 14: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/14.jpg)
The Periodogram
2
1
2
1
2
1
2
1
sincos1
2sin2cos1
)(
N
kkk
N
kkk
N
kkk
N
kkk
tytyN
tytyN
I
![Page 15: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/15.jpg)
Regular time spacing
• Frequency range
• Frequency spacing
t /5.00
2...,,2,1;/ Nj jNj
![Page 16: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/16.jpg)
Periodogram of sinusoid (f=0.3) with superimposed noise: regularly spaced data
![Page 17: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/17.jpg)
Periodogram of sinusoid (f=0.3) with superimposed noise: irregularly spaced data
![Page 18: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/18.jpg)
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
![Page 19: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/19.jpg)
Solutions for Nyquist frequency
![Page 20: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/20.jpg)
![Page 21: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/21.jpg)
Time spacing between exposures (IRSF)
![Page 22: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/22.jpg)
Top: IRSF exposuresBottom: Hipparcos
![Page 23: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/23.jpg)
Frequency spacing
• Frequency resolution is
(Loumos & Deeming 1978, Kovacs 1981)
T/1~
![Page 24: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/24.jpg)
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
![Page 25: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/25.jpg)
Correlation between periodogram ordinates for increasing separation between frequencies
(irregularly spaced data)
![Page 26: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/26.jpg)
Horne & Baliunas (1986): “independent frequencies”
![Page 27: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/27.jpg)
Quasi-periodicities (QPOs)
• Sinusoidal variations with changing amplitude, period and/or phase
![Page 28: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/28.jpg)
A 32 minute segment of fast photometry of VV Puppis
![Page 29: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/29.jpg)
Periodogram of the differenced data
![Page 30: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/30.jpg)
Periodograms of first and second quarters of the data
![Page 31: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/31.jpg)
Wavelet plot of the first quarter of the data
![Page 32: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/32.jpg)
Complex Demodulation
• Transform data so that frequency of interest is near zero
• Apply a low pass filter to the transformed data
![Page 33: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/33.jpg)
Complex demodulation of the first quarter of the data
![Page 34: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/34.jpg)
Time Domain Modelling
Gaussiant
t
tB
tA
tB
tA
tetB
tAtt
tetttCtY
,)(
)(
)1(
)1(
)(
)(
)()(
)(sincos
)())((cos)()(
00
0
![Page 35: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/35.jpg)
Amplitude and phase variations from Kalman filtering
![Page 36: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/36.jpg)
The results of filtering the second quarter of the data
![Page 37: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/37.jpg)
Periodic period changes
• Apsidal motion
• Light-time effect
• Stochastic trends?
![Page 38: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/38.jpg)
O-C (Observed – Calculated)
• Equivalent to CUSUMS• Sparsely observed process:
)()( 0*
1
1
PNTTCO
countcyclecumulativenN
TandTbetweenelapsedcyclesofnumbern
jjj
j
iij
jjj
![Page 39: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/39.jpg)
SZ Lyn (Delta Scuti pulsator in a binary orbit)
![Page 40: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/40.jpg)
The Light-time Effect
)(2
)(sin)(
2
)(tan
1
1arctan2)(
sin])([sin)(cos1
1)(
0
2
ttP
tEetE
tE
e
et
ette
eAtT
b
![Page 41: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/41.jpg)
TX Her (P = 1.03 d)
![Page 42: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/42.jpg)
SV Cam (P = 0.59 d)
![Page 43: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/43.jpg)
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
![Page 44: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/44.jpg)
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
![Page 45: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/45.jpg)
)10( 5 dofunitsHerTX
![Page 46: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/46.jpg)
)10( 6 dofunitsCamSV
![Page 47: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/47.jpg)
General form of Information Criteria:
IC = -2 log(likelihood)+penalty(K)
• Akaike : penalty=2K
• Bayes: penalty=K log(N)
• Model with minimum IC preferred
![Page 48: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/48.jpg)
Models:
• Polynomial + noise
• Random walk + noise
• Integrated random walk + noise
![Page 49: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/49.jpg)
![Page 50: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/50.jpg)
![Page 51: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/51.jpg)
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
![Page 52: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/52.jpg)
![Page 53: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/53.jpg)
![Page 54: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/54.jpg)
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
![Page 55: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/55.jpg)
![Page 56: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/56.jpg)
![Page 57: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/57.jpg)
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*
![Page 58: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/58.jpg)
A brief mention…
Transient deterministic oscillation or purely stochastic variability?
![Page 59: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/59.jpg)
![Page 60: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/60.jpg)
![Page 61: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/61.jpg)
![Page 62: Periodicities in variable stars: a few issues](https://reader036.fdocuments.in/reader036/viewer/2022062409/568145bd550346895db2c71e/html5/thumbnails/62.jpg)