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

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 = 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.

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.

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).

Wang et al. 1995

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.

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

NOAA 9077

145 ’’

2000 July 14, 06:26 UT

NOAA 0061 2002 Aug. 9, 11:00 UT

220 ’’

116 ’’

Selection of magnetic flux concentrations

0. set the threshold1. determine local peak2. outline the flux concentrations3. calculate their flux content

Check the completeness

OK

<Bz>

Probability Distribution Function

観測結果

Cumulative Distribution Function

1 －CDF

Power law

Kolmogorov-Smirnov test : ×

Power law

Probability Distribution Function

Cumulative Distribution Function

1 －CDF

exponential

Kolmogorov-Smirnov test : ×

exponential

Probability Distribution Function

Cumulative Distribution Function

1 －CDF

lognormal

Kolmogorov-Smirnov test : ○

lognormal

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.

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.

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）

赤が lognormal 、青が exponential 、緑が power law

黄色が観測結果

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

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

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