Transcript Probability Distribution Function

太陽雑誌会
2005.02.07
T.T.Ishii
The Astrophysical Journal, 619:1160–1166, 2005
DISTRIBUTION OF THE MAGNETIC FLUX IN
ELEMENTS OF THE MAGNETIC FIELD IN
ACTIVE REGIONS
V. I. Abramenko1,2 and D. W. Longcope3
Received 2004 August 17; accepted 2004 October 12
1 Big
Bear Solar Observatory, New Jersey Institute of Technology
2 Crimean Astrophysical Observatory, Nauchny, Crimea, Ukraine
3 Department of Physics, Montana State University
Abstract
The unsigned magnetic flux content in the flux
concentrations of two active regions is calculated by using a
set of 248 high-resolution SOHO/MDI magnetograms for
each active region.
Data for flaring active region NOAA 9077 (2000 July 14)
and nonflaring active region NOAA 0061 (2002 August 9)
were analyzed.
We present an algorithm to automatically select and
quantify magnetic flux concentrations above a threshold p.
Each active region is analyzed using four different values of
the threshold p ( p = 25, 50, 75, and 100 G).
Abstract (cont.)
Probability distribution functions and cumulative
distribution functions of the magnetic flux were calculated
and approximated by the lognormal, exponential, and
power-law functions in the range of flux > 1019 Mx.
The Kolmogorov-Smirnov test, applied to each of the
approximations, showed that the observed distributions are
consistent with the lognormal approximation only.
Neither exponential nor power-law functions can
satisfactorily approximate the observed distributions.
lognormal distribution: 対数正規分布
ln(X)が正規分布に従う分布
log-log表示だと放物線
Abstract (cont.)
The parameters of the lognormal distribution do not depend
on the threshold value; however, they are different for the
two active regions.
For flaring active region 9077, the expectation value of the magnetic
flux content is m = 28.1×1018 Mx, and the standard deviation of the
lognormal distribution is s = 79.0×1018 Mx.
For nonflaring active region NOAA 0061, these values are
m = 23.8×1018 Mx and s = 29.6×1018 Mx.
The lognormal character of the observed distribution
functions suggests that the process of fragmentation
dominates over the process of concentration in the
formation of the magnetic structure in an active region.
Introduction
Magnetic fields in the solar atmosphere are thought to be
concentrated in thin flux tubes anchored in the photosphere,
where their footpoints form concentrated clusters of
magnetic flux.
Information on the dynamics and statistical characteristics
of the photospheric magnetic field is necessary when
analyzing processes in the corona because of the magnetic
coupling between the photosphere and the corona.
Modern observational techniques allow us to calculate the
distribution function of flux concentrations of the magnetic
field only at the photospheric level.
Introduction (cont.)
Wang et al. (1995) studied the dynamics and statistics of
the network and intranetwork magnetic fields using
BBSO videomagnetograph data. The authors argued that
the distribution function follows a power law.
They found a power index of -1.68 for areas in which the
flux was in the range (0.2 －1)×1018 Mx (intranetwork
fields) and -1.27 for areas in which the flux was in the
range (2 －10)×1018 Mx (network elements).
Wang et al. 1995
Introduction (cont.)
Schrijver et al. (1997) used high-resolution data of a
quiet network area from the SOHO/MDI.
They reported that the flux distribution function follows an
exponential law
with a slope of
approximately
1×1018 Mx-1
in areas in which
the flux ranges from
1 to 5 ×1018 Mx.
Introduction (cont.)
In this study, we calculate and analyze the distribution of
magnetic flux concentrations in the two well-developed
active regions in the range of flux >1019 Mx.
We pay special attention to the analytical approximation of
the observed distribution.
Observational Data
SOHO / MDI, high resolution magnetograms
NOAA 9077 (2000 July 14) : X5.7
NOAA 0061 (2002 Aug. 9) : several C-class flares
NOAA 9077
2000 July 14, 06:26 UT
145 ’’
2002 Aug. 9, 11:00 UT
NOAA 0061
116 ’’
220 ’’
Selection of magnetic flux concentrations
0. set the threshold
1. determine
local peak
2. outline
the flux
concentrations
3. calculate
their flux content
Check the completeness
<Bz>
OK
Probability Distribution Function
観測結果
Cumulative Distribution Function
1－CDF
Kolmogorov-Smirnov test : ×
Power law
Power law
Probability Distribution Function
Cumulative Distribution Function
1－CDF
Kolmogorov-Smirnov test : ×
exponential
exponential
Probability Distribution Function
Cumulative Distribution Function
1－CDF
Kolmogorov-Smirnov test : ○
lognormal
lognormal
Conclusions and Discussion
We have presented the results of fitting the probability
distribution function, PDF (F) of the magnetic flux
concentrations of two active regions.
Lognormal distributions are consistent with each
data set; however, the two active regions are fitted by
distributions with different parameters.
The lognormal distribution of the flux content in magnetic
flux elements of an active region suggests that the process
of fragmentation dominates the process of flux
concentration.
Conclusions and Discussion (cont.)
Assuming that the lognormality of the concentration flux
results from repeated, random fragmentation, we may
attribute meaning to the distribution parameter.
The variance of ln F, s2, is proportional to the number of
independent fragmentations that produced a given
concentration from a single initial concentration.
If the basic fragmentation process is similar in all active
regions, then the value of s2 is proportional to the time over
which fragmentation has occurred.
Conclusions and Discussion (cont.)
Since the value of s2 for AR 9077 is larger than that of
AR 0061 by a factor of 2.3, AR 9077 may be older than
AR 0061 by approximately that factor.
Alternatively, AR 9077 may have undergone more
vigorous fragmentation over a comparable lifetime.
This explanation may also account for their very
different levels of flaring activity.
Note that a very intense fragmentation of sunspots during
several days before the Bastille Day flare in AR 9077
was reported by Liu & Zhang (2001).
Liu and Zhang 2001
論文の内容はここまで
以下つっこみ
分布関数のどっちがより観測を説明するかの
議論にKS test を使うのは良くない
lognormal以外合わないっていう言い方なので
比較してるわけではないのかもしれないが
適合度検定(test)は、仮定したモデルが
合っているかどうかを評価するもので
モデルの優劣を評価するものではない
モデルの優劣は、分布間の距離の指標
例えばAIC (Akaike’s information criterion)などで
評価する
論文のパラメータでグラフかいたら
normalizationがあわなかった
（PDFを積分して 1になってるか心配）
黄色が観測結果
赤がlognormal、青がexponential、緑がpower law
そこで適当にずらして表示
黄色が観測結果
Fitting は > 1019 Mx
赤がlognormal、青がexponential、緑がpower law
Fittingした範囲のみ表示
黄色が観測結果
赤がlognormal、青がexponential、緑がpower law
Power law のベキを変えてみる
緑が論文の(-1.45)、青が -2.5、
赤はdouble power law (-1.45と-2.5)
黄色が観測結果
Double power law でも小さい側は再現できない
Exponential では大きい側が再現できない
青が論文の(beta 0.05)、緑が 0.1、赤が 0.01
Lognormal のパラメータを変えてみる
ｍ(平均)をいじると横方向にシフト
赤が論文の(ｍ 2.2)、青が 1.0、緑が ３.0
Lognormal のパラメータを変えてみる
ｓ(分散)をいじると幅が変わる
赤が論文の(s 1.49)、青が 1.0、緑が 2.0
Lognormal は小さい側で減る
Completeでない観測結果より本来は数は多いはず
赤がlognormal、青がexponential、緑がpower law
小さい側で減らないで大きい方も合いそうな関数形
Saunders’ Luminosity Function (LF) for IRAS galaxies
どうしてこういう形になるかの物理的解釈はまだない