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
2
N
N

1 
 
 
I ( )   yk cos 2  tk    yk sin 2 tk  
N  k 1
  k 1
 
1 
 

  yk cos  tk    yk sin  tk 
N  k 1
  k 1

N
2
N
2



Regular time spacing
• Frequency range
• Frequency spacing
0   0.5 / t
 j  j / N ; j  1, 2, ...,  N2 
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
2
N
N

1 


 
I ( )   yk cos  tk    yk sin  tk  
N  k 1

 k 1
 
I (*   )  I (*   )
 sin * (t k  t j )  0 for all k , j
N 1
S ( )  
 sin  (t
j 1 k  j 1

2
N
k
tj)  0
Solutions for Nyquist frequency
Time spacing between exposures (IRSF)
Top: IRSF exposures
Bottom: Hipparcos
Frequency spacing
• Frequency resolution is
~ 1/ T
(Loumos & Deeming 1978, Kovacs 1981)
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
Y (t )  C (t ) cos (0 t   (t ))  e(t )
 cos 0 t
 A(t )
sin 0 t 
 e(t )

 B(t )
 A(t )  A(t  1)  (t )
 B (t )   B (t  1)   (t )

 
 

 ,  Gaussian
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:
n j  number of cycles elapsed between T j and T j 1
j
N j   ni cumulative cycle count
i 1
(O  C ) j  T j  (T*  N j P0 )
SZ Lyn (Delta Scuti pulsator in a binary orbit)
The Light-time Effect
 1  e2

T (t )  A
sin [ (t )   ]  e sin  
1  e cos  (t )

 1 e
E (t ) 
 (t )  2 arctan 
tan

2 
 1 e
2
E (t )  e sin E (t ) 
(t  t0 )
Pb
TX Her (P = 1.03 d)
SV Cam (P = 0.59 d)
A stochastic period-change model
Pj   j   j
j
Nj
i 1
k 1
T j  T*   ni i   k  ek


(O  C ) j   ni i  N j P0    k  ek
 i 1
 k 1
2
 j   j 1   j  j ~ (0,   )
j
Nj
State Space Formulation:

Tj  1
nj
U 
1
    0
 j 

U 
    T*  e j
 j
n j 1 
1 
U 
  
  j 1
 
 
 
 ,  Gaussian
j 1
Nj
j 1
j
i 1
k 1
i 1
k 1
U j   ni i   k   ni i    k
TX Her (units of 10
5
d)
SV Cam (units of 10
6
d)
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?