A TOPOLOGICAL MODEL OF THE LARGE X-RAY FLARE
ON 03.11.2003
I. V. ORESHINA and B. V. SOMOV
Sternberg Astronomical Institute
Moscow State University
Universitetskii pr., 13, Moscow, 119992, Russia
O. G. DEN
IZMIRAN, Troitsk, Russia
Abstract. We present the results of modelling of the X3.9 flare occurred on 2003, November 3 in the Active Region (AR) 0488. In contrast to many other large flares, this one
shows a very simple morphology with well defined loop-top and foot-point hard X-ray
sources. These data obtained by RHESSI allow us to use the flare as an excellent test for
the magnetic-reconnection and topological models. We reconstruct the coronal magnetic
field on the base of line-of-sight magnetograms from SOHO/MDI, find the separator, and
explain the appearance of all hard X-ray sources (the chromospheric pair as well as the
coronal one). Then, we estimate the energy of the potential magnetic field in the AR
above the photosphere and show that it is an order of magnitude greater than the nonpotential part of energy. So, the potential approximation can be used for investigation of
the coronal field topology in ARs on the Sun. Furthermore, the topological model allows
us to deduce a simple analytical formula to calculate the magnetic field value in any
point of the corona. Note that the advantages of our model are its simplicity and clarity
in describing topology of the magnetic field in ARs.
1. Introduction
The question on the best approximation for coronal magnetic field is still
open. The potential reconstruction is used as well as various non-potential
models (Gorbachev and Somov, 1988; Longcope, 2002; Somov, Oreshina,
Lubimov, 2004; Burnette, Canfield, Pevtsov, 2004; Schmieder, Rust, Georgoulis et al., 2004). Some authors believe that there are ARs with potential
configuration while there are others ARs with non-potential configuration
(Burnette, Canfield, Pevtsov, 2004). Other authors think that there may
be potential and non-potential loops in the same AR (Schmieder, Rust,
Georgoulis et al., 2004). We investigate the plausibility of a simple topological model of an AR (Gorbachev and Somov, 1988). In the previous
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I. V. ORESHINA, B. V. SOMOV, O.G. DEN
paper (Somov, Oreshina, Lubimov, 2004) the model was compared with
the photospheric/chromospheric observations during the Bastille Day Flare:
two-ribbon nature of the flare was shown to be a consequence of the field
topology; a place and a form of bright chromospheric ribbons and the
appearance of bright kernels on the ribbons’ ends were explained. Now,
we intend to compare our model with direct coronal observations obtained
by RHESSI (Liu,Jiang,Liu and Petrosian, 2004). For that, we present the
result of modelling of the X3.9 flare occurred on 2003, November 3 in the
AR 0488 in the limb vicinity.
The aims of our study are (1) to estimate the availability of the topological model, i.e. to explain the existence and the location of all X-ray
sources in the corona as well as on the chromospheric level (on the base
of the X-ray data from RHESSI); (2) to deduce an analytical formula to
estimate the magnetic field value above the photosphere; (3) to answer the
question: “Can the energy of the potential magnetic field be enough to play
the key role in the topology of the solar ARs ?”
2. Model
Our model consists in the following (Gorbachev and Somov, 1988). Magnetic field is created by effective charges, situated under the photosphere:
B(x, y, z) =
N
X
i=1
ei
r − ri
,
|r − ri | |r − ri |
(1)
where N is the number of charges, ei denote their values, ri - their radiusvectors. The photosphere is the plane z = 0.
Note, since the flare 03.11.2003 took place in the vicinity of the limb
(its precise coordinates are N 09◦ , W 77◦ ), its real proportion is different
from that we can see on the SOHO magnetogram (Figure 1). The black
color corresponds to the north polarity of the magnetic field and the white
color corresponds to its south polarity. The thick black line is the magnetic
neutral line.
To apply this model to the flare, we begin by approximating its photospheric flux. Let us reproduce the observed line-of-sight magnetic field
component on the model magnetogram (Figure 2). The red color corresponds to the southern polarity, the blue one - to the nothern polarity.
The green line is the magnetic neutral line. In the case of our flare, we
have restricted the number of charges to 5. This small number allows us to
reproduce the most important features on the SOHO magnetogram and,
at the same time, makes the model simple and clear. The used values and
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TOPOLOGICAL MODEL
3
AR 0488
N 09 W 77
4
2
2
4
6
8
10
12
14
16
18
Figure 1. TOP: Observed magnetograms from MDI/SOHO; BOTTOM: Real proportion
of the AR in accordance with its position on the Sun.
the radius-vectors of the charges are following:
e1
e2
e3
e4
e5
= −20;
= −15;
= 35;
= 30;
= −07;
r1
r2
r3
r4
r5
= ( 9.5, 1.0, −1.5);
= ( 11.5, 1.5, −1.5);
= ( 16.6, 1.2, −1.5);
= ( 11.0, 4.0, −1.5);
= ( 1.0, 1.0, −1.5).
(2)
The comparison of the positive magnetic flux through the photosphere
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I. V. ORESHINA, B. V. SOMOV, O.G. DEN
Figure 2.
Model magnetogram
for the model and for the observed magnetgram gives us the correspondence
between model length and magnetic units and their real values:
1 length unite = 1.45 · 109 cm;
1 magnetic unite = 68 G
(3)
The next step of the modelling is to approximate the coronal magnetic
field. On the base of the formulas (1) and (2), we are able to to develop
the topological model of the AR 0488, i.e. to separate the magnetic field
into interacting fluxes and to find separators. The Figure 3 presents the
vector field with the separatrixes (thick lines) in the plane of charges. The
reconstruction of the coronal magnetic field has been performed on the base
of the Euler method. In particular, the field line connecting null points np1
and np2 is a separator. Its image after the necessary rotation is over-plotted
on the observed magnetogram in the Figure 4, left. The right figure shows
the X-ray sources from RHESSI (Liu,Jiang,Liu and Petrosian, 2004). It
allows us to conclude that the observed X-ray sources are the top and foot
points of the separator. Assuming that the flare is caused by the magnetic
reconnection in a current layer (Oreshina and Somov, 2000; Litvinenko and
Somov, 1993; Somov, 2000) at the separator top where the particles are
accelerated and then descend to the dense chromosphere following the field
lines, we can explain the appearance of all the X-ray sources. Thus, the
topological model obviously can describe the photospheric/chromospheric
observations as well as the coronal ones. Furthermore, the formulas (1) with
the estimations (3) allows us to conclude that the separator top is situated
at the height of H ∼ 2 · 109 cm where magnetic field value is B ∼ 150 G.
Let us now consider the question on the quantitative ratio of the potential and non-potential parts of the magnetic energy in the AR 0488. As is
well known, the energy release during an ordinary X-ray solar flare is about
1032 erg. It is believed to be accumulated by current layers on separators
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TOPOLOGICAL MODEL
5
5
+
+
0
+
–5
–10
0
10
20
Figure 3. Magnetic vector field with the separatrixes (thick lines) and the null points
(crosses) in the plane of charges.
Figure 4. RIGHT: The reconstructed separator; LEFT: The loop-top X-ray source (LT)
with the northern (N-FP) and the southern (S-FP) foot-point X-ray sources observed by
RHESSI.
(Somov, 1992; Somov, 2000), i.e. is concerned with the non-potential part
of the magnetic field energy. Let us calculate its potential part.
The energy E of the magnetic field B in a volume V can be estimated
with the help of the integral
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I. V. ORESHINA, B. V. SOMOV, O.G. DEN
Z
E=
V
B2
dv.
8π
(4)
In our case, B is assumed to be described by the formulas (1) and (2).
As an volume V we have chosen a parallelepiped with a height equal to
the height of the separator and including all the separatrix lines on the
photospheric level (Figure 3), i.e. the volume
x = (0; 20);
y = (−10; 10);
z = (0; 1.3).
(5)
As the integrand is a smooth function, there is no problem to intergate,
and we obtain the value of the energy in the dimensionless variables:
Edl = 80 energy unites.
In order to find the correspond dimensional value, we have used the
next obvious equality:
E = Edl · B02 · L30 ≈ 1 · 1033 erg.
This is an order of magnitude greater than the non-potential part of
energy.
3. Summary
On the base of the flare on 03.11.2003, we have presented two new arguments in favour of the potential approximation for investigations of the
coronal field topology in ARs on the Sun:
− The direct comparison of the topological model with hard X-ray images
from RHESSI shows a very well coincidence between the loop top and
foot points of the separator, on the one hand, and the photospheric and
coronal hard X-ray sources, on the other hand. This fact is in a good
agreement with the theory of magnetic reconnection on the separator
top.
− It is shown that the potential part of magnetic energy in the AR 0488
is an order of magnitude greater than its non-potential part. So, we
conclude that the potential part plays the key role in the topology of
the solar ARs.
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TOPOLOGICAL MODEL
7
Furthermore, the topological model allows us to deduce a simple analytical formula (1) for calculate the magnetic field value in any point of the
corona. Note that, in contrast to other models, the advantages of our way
are its simplicity and clarity in describing topology of the magnetic field in
ARs.
Acknowledgements
Inna V. Oreshina would like to express her sincere thanks to the Organizing Committee of the JENAM-2004 The many scales in the Universe for
the financial support and hospitality in Granada. The research has been
supported by RFBR grant 04-0216125.
References
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