![Page 1: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/1.jpg)
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 Technology2 Crimean Astrophysical Observatory, Nauchny, Crimea, Ukraine3 Department of Physics, Montana State University
太陽雑誌会2005.02.07
T.T.Ishii
![Page 2: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/2.jpg)
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).
![Page 3: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/3.jpg)
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
![Page 4: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/4.jpg)
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 = 28.1×1018 Mx, and the standard deviation of the lognormal distribution is = 79.0×1018 Mx. For nonflaring active region NOAA 0061, these values are = 23.8×1018 Mx and = 29.6×1018 Mx.
The lognormal character of the observed distribution functions suggests that the process of fragmentationdominates over the process of concentration in the formation of the magnetic structure in an active region.
![Page 5: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/5.jpg)
Introduction
Magnetic fields in the solar atmosphere are thought to beconcentrated in thin flux tubes anchored in the photosphere,where their footpoints form concentrated clusters of magnetic flux.
Information on the dynamics and statistical characteristicsof the photospheric magnetic field is necessary whenanalyzing processes in the corona because of the magnetic coupling between the photosphere and the corona.
Modern observational techniques allow us to calculate thedistribution function of flux concentrations of the magnetic field only at the photospheric level.
![Page 6: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/6.jpg)
Introduction (cont.)
Wang et al. (1995) studied the dynamics and statistics of the network and intranetwork magnetic fields usingBBSO 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).
![Page 7: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/7.jpg)
Wang et al. 1995
![Page 8: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/8.jpg)
Introduction (cont.)
Schrijver et al. (1997) used high-resolution data of aquiet 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.
![Page 9: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/9.jpg)
Introduction (cont.)
In this study, we calculate and analyze the distribution ofmagnetic 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) : X5.7NOAA 0061 (2002 Aug. 9) : several C-class flares
![Page 10: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/10.jpg)
NOAA 9077
145 ’’
2000 July 14, 06:26 UT
![Page 11: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/11.jpg)
NOAA 0061 2002 Aug. 9, 11:00 UT
220 ’’
116 ’’
![Page 12: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/12.jpg)
Selection of magnetic flux concentrations
0. set the threshold1. determine local peak2. outline the flux concentrations3. calculate their flux content
![Page 13: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/13.jpg)
Check the completeness
OK
<Bz>
![Page 14: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/14.jpg)
Probability Distribution Function
観測結果
![Page 15: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/15.jpg)
Cumulative Distribution Function
1 -CDF
Power law
Kolmogorov-Smirnov test : ×
![Page 16: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/16.jpg)
Power law
![Page 17: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/17.jpg)
Probability Distribution Function
![Page 18: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/18.jpg)
Cumulative Distribution Function
1 -CDF
exponential
Kolmogorov-Smirnov test : ×
![Page 19: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/19.jpg)
exponential
![Page 20: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/20.jpg)
Probability Distribution Function
![Page 21: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/21.jpg)
Cumulative Distribution Function
1 -CDF
lognormal
Kolmogorov-Smirnov test : ○
![Page 22: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/22.jpg)
lognormal
![Page 23: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/23.jpg)
Conclusions and Discussion
We have presented the results of fitting the probability distribution function, PDF of the magnetic flux concentrations of two active regions.
Lognormal distributions are consistent with eachdata set; however, the two active regions are fitted by distributions with different parameters.
The lognormal distribution of the flux content in magneticflux elements of an active region suggests that the processof fragmentation dominates the process of flux concentration.
![Page 24: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/24.jpg)
Conclusions and Discussion (cont.)
Assuming that the lognormality of the concentration fluxresults from repeated, random fragmentation, we may attribute meaning to the distribution parameter.
The variance of ln , 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.
![Page 25: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/25.jpg)
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).
![Page 26: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/26.jpg)
Liu and Zhang 2001
論文の内容はここまで以下つっこみ
![Page 27: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/27.jpg)
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
![Page 28: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/28.jpg)
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)
赤が lognormal 、青が exponential 、緑が power law
黄色が観測結果
![Page 29: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/29.jpg)
そこで適当にずらして表示
赤が lognormal 、青が exponential 、緑が power law
黄色が観測結果
Fitting は > 1019 Mx
![Page 30: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/30.jpg)
Fitting した範囲のみ表示
赤が lognormal 、青が exponential 、緑が power law
黄色が観測結果
![Page 31: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/31.jpg)
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
![Page 32: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/32.jpg)
Double power law でも小さい側は再現できない
![Page 33: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/33.jpg)
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
![Page 34: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/34.jpg)
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
![Page 35: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/35.jpg)
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
![Page 36: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/36.jpg)
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
![Page 37: The Astrophysical Journal, 619:1160–1166, 2005](https://reader035.vdocuments.mx/reader035/viewer/2022062717/56812b1f550346895d8f1aac/html5/thumbnails/37.jpg)
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