スーパーナイキスト周波数事象の解析
柴 橋 博 資
(東大理)
!
Simon Murphy and Don Kurtz !
(UCLan)
2013
年度宇宙科学情報解析シンポジウム2014.2.14
Question: !
!
Are the Kepler LC (29.4-min sampling) data useless or still usefull for short period
pulsators (roAp, delta Sct, sdB, WD) ?
Nyquist frequency of the LC mode = 24.5 d -1
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-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10
signal
time
Observational data = Discrete function of time
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-1.5 -1 -0.5 0 0.5 1 1.5
0 2 4 6 8 10
signal
time
-1.5 -1 -0.5 0 0.5 1 1.5
0 0.2 0.4 0.6 0.8 1
signal
time
Sinusoidal fitting of discrete function
a multiply ambiguous fitting!
4
Fourier transform of discrete function
These are indistinguishable!
sampling freq.
Nyquist freq.
5
6
F ( ) := f (t) exp(i t) dt
F
obs( ) := f
obs(t) exp(i t) dt
Fourier transform of discrete functions :
f
obs(t) :=
n=
f (t) (t t
n) dt
Fourier transform of continuous functions :
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W ( ) :=
n=
exp(i t
n)
=
n=
exp(i t) (t t
n) dt F
obs( ) = (F W ) ( )
Window spectrum : Convolution of !
the continuous function and the window spectrum
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t
n:= n t
=
n=
exp(i t) (t n t) dt W ( )
F
obs( )
s
:=
2 t=
n=
( n
s)
=
n=
F ( n
s)
Uniform cadence :
sampling interval
sampling angular frequency
0.0 0.2 0.4 0.6 0.8 1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5
W N()
( t/2)/
N=100 N=10
̲
̲
| W
N( ) | = (N + 1)
1 sin{sin((N+1)t)/2t/2}Window spectrum in the case of uniform cadence with ∆t
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10
“light-time effect”
Romer’s measurement!
of the light speed !
by using Io’s eclipse timing
Binary pulsar !
(cf. Taylor & Hulse)
1 au = 500 light seconds
equinox Earth a
star
Kepler data:!
1. taken with regular time interval according to the clock on-board!
!
!
2. the time stamps converted to the barycentre arrival times t
,n= t
n+ cos( t
n)
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t
n= t
0+ n t
= (a/c) cos
“Romer delay”
𝛽 Ω t
n- 𝛌
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t
n= n t + sin t
exp(ix sin ) =
k=
Jk(x) exp(ik )
exp { i (n t + sin t) }
F
obs( )
Modulated sampling :
W ( )
?
Bessel function :
=
n= k=
J
k( )F ( n
sk )
=
n= k=
J
k( ) ( n
sk )
| W
N( ) | = (N + 1)
1k=
J
k( ) sin { (N + 1)( + k ) t/2 } sin { ( + k ) t/2 }
Window spectrum for the Kepler LC mode
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-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0
0.0 1.0 2.0 3.0 4.0 5.0
J n() S 2 S
S 2 S 3 S 4 S 5 S 6 S 7 S 8 S
S 2 S 3 S 4 S 5 S 6 S 7 S 8 S
J0( ) J1( ) J2( ) J3( ) J4( ) J5( ) J6( ) J7( ) J8( )
Long Cadence (30-min sampling)
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triplet quintuplet
septuplet nonuplet
singlet
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Example: KIC 6861400 (δSct star)
true freq. triplet quintuplet
septuplet nonuplet
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Beyond the Nyquist frequency: KIC 10195926 (roAp star)
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0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.99996 1.00000 1.00004
| W
N( ) |
/
SShort Cadence (1-min sampling) !
window spectrum near the sampling frequency
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KIC 10139564 (sdB star)
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結論
サンプリングに周波数変調を導入すれば、ナイキスト周波 数に関わらずに、唯一的に事象を解析可能。
スーパーナイキスト事象も問題なく解析可能。
Kepler データに応用、成果大。 !
時間についてのフーリエ変換（周波数解析）のみならず、
空間についてのフーリエ解析（波長解析）についても同様。
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