2. Method outline

1
Determination of Coronal Magnetic Helicity of
Solar Active Regions Using the Linear Force-Free
Field Model
Eun-Kyung Lim, Hyewon Jeong, Jongchul Chae Seoul
National University, Korea

• Abstract Magnetic helicity is a useful quantity
  in characterizing the magnetic system of solar
  active regions. We aim to measure the helicity of
  the coronal magnetic field of an active region
  based on the linear force-free field assumption.
  With a value of force-free a, the coronal field
  is constructed from the extrapolation of SOHO/MDI
  magnetograms, and the constructed field lines are
  compared with the coronal loops in the EUV images
  taken by SOHO/EIT. The force-free field that best
  fits the loops is used to calculate the helicity
  of the active region. We have applied this method
  to AR10696 during its first rotation. We have
  compared our results with the accumulated amount
  of the helicity transferred to the corona via the
  photsphere which is determined independently. Our
  results are summarized 1) we found that the value
  of force-free a and the coronal helicity was
  given as a range in stead of single value because
  of the uncertainty of the linear force-free field
  assumption. The uncertainty was about 3040. 2)
  The measured value of coronal helicity was close
  to the accumulated amount of injected helicity
  with discrepancy of 1030. That result shows
  that linear force-free field extrapolation method
  is more or less reasonable. 3) This research also
  showed that the coronal magnetic helicity
  decreased after the CMEs. The amount of decrease
  was close to twice of the magnetic helicity of a
  typical CME, 2x1042 Mx2. Our results support the
  conservation
• of magnetic helicity in the coronal in a general
  sense.

• 2. Method outline
• Equation of relative helicity (Berger 1985)
• - the fourier transform of
  normal component of magnetic field on
  the photospheric level
• -
• Finding a
• Construct the coronal magnetic field models by
  linear force-free extrapolation with various a.
• LFFF extrapolation Fourier transform method
  (Alissandrakis 1981)
• Fit individual coronal loops by a model field
  line to determine the best matches field lines
  and a.
• Fitting condition (Chae et al. 2005)
• minimizing the distance d between the field line
  and a single coronal loop
• - position of the ith selected
  point on the loop
• - the nearest position on the
  field line from .

• 1. Introduction
• Magnetic helicity is a well-conserved quantity
  in a general sense. .
• We compare the coronal magnetic helicity, Hcorona
  of AR10696, with the helicity injected through
  the photosphere, Hinjection, and the helicity
  carried away by CMEs, Hejection. We examine short
  time variations for a period of about one day and
  use Chae (2001)s method to calculate Hinjection.
  During the day of our interest, the active was
  rapidly growing and a couple of CMEs occurred, so
  that this study gives us a nice chance to compare
  Hcorona, Hinjection and Hejection.

• 3. Data
• The target active region AR10696
• MDI/SOHO boundary condition in LFFF
  extrapolation.
• - Solid line LFFF extrapolation
  computational box
• - Dashed line The region where coronal
  magnetic helicity will be computed.
• EIT/SOHO compare with the model field lines.
• - Figure b 5 Nov. 2004 (1911UT)
• - Figure c 6 Nov. 2004 (1911UT)

2

• CMEs
• A couple of CMEs occurred during the approaching
  time period.
• 6 Nov., 013151 UT
• 6 Nov., 020605 UT

• 5. Result
• The shape of coronal loops changed a lot during
  one day
• amax is mostly in the center of the active
  region.
• A couple of CMEs occurred between Figure b and c.
• Helicity increased before CMEs, decreased after
  CMEs.

• 4.1 Model fitting
• Select a number of coronal loops to be fitted by
  model field lines and express with signs.
• Fit the single coronal loop by the constructed
  field lines with arbitrary force-free a.
• Find the best fits field line that minimize the
  distance d between the field line and the coronal
  loop.

• Comparision with injected helicity calculated
  using Chaes method.
• Coronal helicity is given as a probable range.
• The uncertainty of LFFF method is about 2745.
• Coronal helicity decreases with CMEs in amount
  of 4x1042 Mx2.
• The value of helicity from LFFF method is
  comparable to the injected helicity with an
  discrepancy of less than 30.

• 4.2 Fitting example
• Blue thick/thin lines amax/amin.
• a -0.0116 -0.0236 Mm-1.
• Each coronal loop has its own a.
• Helicity of non-linear force-free active
  region may be bounded by
• that of linear force-free field with amax and
  amin.

• 6. Conclusion
• We computed coronal magnetic helicity of
  AR10696 and compared it with the accumulated
  amount of helicity injected through the
  photosphere. The amount of coronal helicity in
  AR10696 was about -1043 Mx2. We found that
  magnetic field of active region is not exactly
  linear force-free so we computed possible range
  of coronal helicity rather than a single value.
  From the range we estimated the uncertainty of
  LFFF method as 2745. Although the magnetic
  field of active region is not LFFF, this method
  is not only simple but also reliable. The value
  of computed coronal helicity was similar to
  injected helicity calculated using Chaes method
  with an discrepancy less than 29. We detected
  the decrease of coronal helicity with CMEs from
  this approach. The amount of decrease by a
• couple of CMEs was very close to twice the
  helicity of a typical CME, 2x1042
  Mx2.

3
abstract

• Abstract
• Magnetic helicity is a useful quantity in
  characterizing the magnetic system of solar
  active regions. We aim to measure the helicity of
  the coronal magnetic field of an active region
  based on the linear force-free field assumption.
  With a value of force-free a, the coronal field
  is constructed from the extrapolation of SOHO/MDI
  magnetograms, and the constructed field lines are
  compared with the coronal loops in the EUV images
  taken by SOHO/EIT. The force-free field that best
  fits the loops is used to calculate the helicity
  of the active region. We have applied this method
  to AR10696 during its first rotation. We have
  compared our results with the accumulated amount
  of the helicity transferred to the corona via the
  photsphere which is determined independently. In
  brief, 1) we found that the value of force-free a
  and the coronal helicity was given as a range in
  stead of single value because of the uncertainty
  of the linear force-free field assumption. The
  uncertainty was about 3040. 2) The measured
  value of coronal helicity was close to the
  accumulated amount of injected helicity with
  discrepancy of 1030. That result shows that
  linear force-free field extrapolation method is
  more or less reasonable. 3) This research also
  showed that the coronal magnetic helicity
  decreased after the CMEs. The amount of decrease
  was close to twice of the magnetic helicity of a
  typical CME, 2x1042 Mx2. Our results support the
  conservation of magnetic helicity in the coronal
  in a general sense.
• Introduction
• Magnetic helicity is a well-conserved quantity
  in a general sense. The amount of helicity in the
  corona of an active region, Hcorona, should be
  directly related to the amount of helicity
  injected through the photospheric boundary,
  Hinjection , and the amount ejected out of the
  corona, Hejection, mostly with CMEs. That is,
  HcoronaHinjection-Hejection .
• Hinjection can be estimated if the velocity
  field and magnetic field on the photosphere are
  known as functions of time (Berger Field 1984).
  Hejection may be thought to have a simple
  relation as HejectionNHCME. HCME is assumed to
  be the same as that of the corresponding magnetic
  cloud (MC) observed near the Earth of which
  typical value often adopted by researches is
  2x1042 Mx2 (Devore 2001). Hcorona can be
  calculated using the equation of Berger (1985).
  Applying the linear force-free extrapolation and
  model fitting will give force-free a.
• In this paper, we compare the coronal magnetic
  helicity, Hcorona of AR10696, with the helicity
  injected through the photosphere, Hinjection, and
  the helicity carried away by CMEs, Hejection, as
  Demoulin et al. (2002) did. Our approach differs
  from Demoulin at al. (2002) in that we examine
  short time variations for a period of about one
  day and we use Chae (2001)s method to calculate
  Hinjection. During the day of our interest, the
  active was rapidly growing and a couple of CMEs
  occurred, so that this study gives us a nice
  chance to compare Hcorona, Hinjection and
  Hejection.

4
introduction

• Magnetic helicity is an important quantity in
  characterizing the magnetic system of solar
  active region. And it is a well-conserved
  quantity in a general sense.
• The amount of helicity in the corona of an active
  region, Hcorona, should be directly related to
  the amount of helicity injected through the
  photospheric boundary, Hinjection , and the
  amount ejected out of the corona, Hejection,
  mostly with CMEs.
• The amount of the helicity injected through the
  photosphere, Hinjection, can be estimated if the
  velocity field and magnetic field on the
  photosphere are known as functions of time
  (Berger Field 1984).
• Differential rotation is one of the previously
  known shear motions. Devore (2000) investigated
  quantitatively the generation of magnetic
  helicity by solar differential rotation using
  analytical and numerical methods. However,
  differential rotation alone is not enough to
  explain the total magnetic helicity injected
  through the photosphere (Chae et al 2001
  Demoulin et al 2002).
• Chae (2001) proposed a practical method of
  observationally determining the rate of magnetic
  helicity transport through the photosphere in the
  presence of horizontal motions other than
  differential rotation. Later Chae et al. (2004)
  showed that this method in fact determines not
  only the helicity changes by the horizontal
  motion of field lines but also the contributions
  by the flux emergence.
• CMEs are believed to be the dominant process that
  takes away the magnetic helicity out of its
  source active region.
• N total number of CMEs that occurred in the
  active region and HCME is the average helicity of
  a single CME.
• The magnetic helicity of a CME is assumed to be
  the same as that of the corresponding magnetic
  cloud (MC) observed near the Earth, which ranges
  from 1041 to 1043 Mx2 (Lepping at al. 1990 Zhao
  et al. 2001). The typical value often adopted by
  researches is 2x1042 Mx2 (Devore 2001) which is
  based of the assumption that the effective lengh
  of a MC is 0.5 AU. The number of CMEs is usually
  determined from the CME catalogue of SOHO/LASCO.
  (Demoulin et al. 2002 Nindos et al. 2002)

5
introduction

• The existence of non-negligible magnetic helicity
  in the corona has been well-known from vector
  magnetograph observations of magnetic field in
  active regions.
• Pevtsov et al. (1995) quantified the magnetic
  helcity of an entire active region by a single
  value of force-free a that makes the constructed
  linear-force free field best fit the transverse
  field measured in the photosphere.
• However, the force-free a only is not enough for
  quantifying the magnetic helicity, even though it
  gives the correct sign.
• The linear force-free fit can give not only a,
  but also coronal magnetic helicity itself Hcorona
  with the formula presented by Berger (1985).
• Demoulin et al. (2002) carried out an interesting
  study of examining the variation of the coronal
  helicity for a very long time of about 6
  rotations. They also examined Hinjecction due to
  differential rotation during the same period.
• As a result they found that differential rotation
  is not enough for Hcorona and Hejection (Green et
  al. 2002 Mandrini et al. 2002).
• In this paper, we compare the coronal magnetic
  helicity, Hcorona of AR10696, with the helicity
  injected through the photosphere, Hinjection, and
  the helicity carried away by CMEs, Hejection, as
  Demoulin et al. (2002) did.
• Our approach differs from Demoulin at al. (2002)
  in that we examine short time variations for a
  period of about one day
• And we use Chae (2001)s method to calculate
  Hinjection.
• During the day of our interest, the active was
  rapidly growing and a couple of CMEs occurred, so
  that this study gives us a nice chance to compare
  Hcorona, Hinjection and Hejection.

6
method

• Eun-Kyung Lim, Haewon Jeong, Jongchul ChaeSeoul
  National University, School of Earth and
  Environmental Science

• reference
• Alissandrakis, C. E. 181, AA. 100, 197
• Berger, M.A. 1985, ApJ. 573L133
• Chae, J., et al. 2001, ApJ. 560, 476
• Chae, J. 2001, ApJ, 560L95
• Chae, J., et al., 2004, Sol. Phys. 223, 39

Chae, J., Moon, Y.-J. 2005, ApJ. 629,
1110 Demoulin, P., et al. 2002, AA. 382, 650
7
result

• We computed coronal magnetic helicity of AR10696
  and compared it with the accumulated amount of
  helicity injected through the photosphere. The
  amoun of coronal helicity in AR10696 is about
  -1043 Mx2. We found that magnetic field of active
  region is not exactly linear force-free field
  field so we computed possible range of coronal
  helicity rather than a single value. From the
  range we estimated the uncertainty of LFFF method
  as 2745. Although the magnetic field of active
  region is not LFFF, this method is not only
  simple but also reliable. The value of computed
  coronal helicity was similar to injected helicity
  calculated using Chaes method with an error of
  less than 29. We detected the decrease of
  coronal helicity with CMEs from this approach.
  The amount of decrease by a couple of CMEs was
  very close to twice the helicity of a typical
  CME, 2x1042 Mx2.
• The amount of helicity in AR10696 is about -1043
  Mx2. We computed possible range of coronal
  helicity with an uncertainty of 2745. That is,
  magnetic field of active region is not linear
  force-free field. Nevertheless, LFFF method is
  simple and gives us reliable result. The value of
  computed coronal helicity was similar to injected
  helicity calculated using Chaes method with an
  error of less than 29. We detected the decrease
  of coronal helicity with CMEs from this approach.
  The amount of decrease by a couple of CMEs was
  very close to twice the helicity of a typical
  CME, 2x1042 Mx2.

8
conclusion