2. Method outline Equation of relative helicity (Berger 1985):

1. a b c Determination of Coronal Magnetic Helicity of Solar Active Regions Using the Linear Force-Free Field Model Eun-Kyung Lim, Hyewon Jeong, Jongchul ChaeSeoul National University, Korea • 2. Method outline Equation of relative helicity (Berger 1985): - : the fourier transform of normal component of magnetic field on the photospheric level - Finding α : • Construct the coronal magnetic field models by linear force-free extrapolation with various α. • LFFF extrapolation : Fourier transform method (Alissandrakis 1981) • Fit individual coronal loops by a model field line to determine the best matches field lines and α. 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 . Helicity injected through the photosphere (Chae 2001): - : determined by LCT method. - : calculated using the Fourier transform method. • 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. AbstractMagnetic 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 α, 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 αand 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 30~40%. 2) The measured value of coronal helicity was close to the accumulated amount of injected helicity with discrepancy of 10~30%. 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. • 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 (19:11UT) • - Figure c : 6 Nov. 2004 (19:11UT)

2. 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 α. Find the best fits field line that minimize the distance d between the field line and the coronal loop. a b c d a b c • CMEs • A couple of CMEs occurred during the approaching time period. • 6 Nov., 01:31:51 UT • 6 Nov., 02:06:05 UT • 5. Result • The shape of coronal loops changed a lot during one day • αmax 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. • Comparision with injected helicity calculated using Chae’s method. • Coronal helicity is given as a probable range. • The uncertainty of LFFF method is about 27~45%. • 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 : αmax/αmin. • α : -0.0116 ~ -0.0236 Mm-1. • Each coronal loop has its own α. • Helicity of non-linear force-free active region may be bounded by • that of linear force-free field with αmaxandαmin. • 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 27~45%. 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 Chae’s 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 α, 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 αand 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 30~40%. 2) The measured value of coronal helicity was close to the accumulated amount of injected helicity with discrepancy of 10~30%. 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, Hcorona=Hinjection-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 Hejection=NHCME. 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 α. 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 αthat makes the constructed linear-force free field best fit the transverse field measured in the photosphere. • However, the force-free αonly is not enough for quantifying the magnetic helicity, even though it gives the correct sign. • The linear force-free fit can give not only α, 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, A&A. 100, 197 • Berger, M.A. 1985, ApJ. 573:L133 • Chae, J., et al. 2001, ApJ. 560, 476 • Chae, J. 2001, ApJ, 560:L95 • Chae, J., et al., 2004, Sol. Phys. 223, 39 Chae, J., Moon, Y.-J. 2005, ApJ. 629, 1110 Demoulin, P., et al. 2002, A&A. 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 27~45%. 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 Chae’s 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 27~45%. 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 Chae’s 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