A Sun-to-Earth Analysis of Magnetic Helicity of the 2013 March 17–18 Interplanetary Coronal Mass Ejection

In this section, we compute MC unsigned magnetic flux and magnetic helicity (H_mc) of the MC using Lundquist's constant-α force-free field solution in cylindrical coordinates (Lepping et al. 1990). The solution provides the axial magnetic field component, ${B}_{z}={B}_{0}{J}_{0}(\alpha r)$, the poloidal (azimuthal) magnetic field components, ${B}_{\theta }={{DB}}_{0}{J}_{1}(\alpha r)$, and the radial field component, B_r = 0. Where B₀ is the axial magnetic field strength, $D=\pm 1$ is the flux rope handedness (plus for right-handed and minus for left-handed), J_n is the nth order Bessel function, and $\alpha ={x}_{01}/{R}_{0\mathrm{mc}}$ is the twist per unit length, where ${x}_{01}=2.4048$ is the location of the first zero of J₀.

The magnetic flux is defined as

$$\begin{eqnarray}&&\phi =\int {\boldsymbol{B}}\ {dA}\end{eqnarray} \tag{ 1 }$$

using cylindrical symmetry, ${dA}=2\pi {rdr}$.

The axial and poloidal components of magnetic flux (${\phi }_{z}$ and ${\phi }_{p}$) in a cylindrical flux rope is given by (e.g., Leamon et al. 2004; Qiu et al. 2007)

$$\begin{eqnarray}&&{\phi }_{z}=2\pi {\int }_{0}^{{R}_{0}}{B}_{z}\ {rdr}=\displaystyle \frac{2\pi {J}_{1}({x}_{01})}{{x}_{01}}{B}_{0}{R}_{0}^{2}\end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray}&&{\phi }_{p}=2\pi {\int }_{0}^{{R}_{0}}{B}_{\theta }\ {rdr}=\displaystyle \frac{L}{{x}_{01}}{B}_{0}{R}_{0},\end{eqnarray} \tag{ 3 }$$

where ${R}_{0},{B}_{0}$, and L are, respectively, the radius, axial magnetic field strength, and length of the cylindrical flux rope.

Within a volume V the magnetic helicity H of a field ${\boldsymbol{B}}$ is defined by

$$\begin{eqnarray}&&H={\int }_{V}{\boldsymbol{A}}\cdot {\boldsymbol{B}}\ {dV},\end{eqnarray} \tag{ 4 }$$

where ${\boldsymbol{A}}$ is the vector potential. This definition of helicity is meaningful only if the normal component of ${\boldsymbol{B}}$ (B_n) at any surface S surrounding the volume V is zero, i.e., B_n = 0. In the case of ${B}_{n}\ne 0$, relative magnetic helicity (H_r) is derived by subtracting the reference magnetic field (${{\boldsymbol{B}}}_{\mathrm{ref}}$) helicity from H (Berger & Field 1984). Thus, H_r is defined by

$$\begin{eqnarray}&&{H}_{r}={\int }_{V}{\boldsymbol{A}}\cdot {\boldsymbol{B}}\ {dV}-{\int }_{V}{{\boldsymbol{A}}}_{\mathrm{ref}}\,\cdot \,{{\boldsymbol{B}}}_{\mathrm{ref}}\ {dV}.\end{eqnarray} \tag{ 5 }$$

Here H_r is gauge-invariant if ${\boldsymbol{A}}\times \hat{n}={{\boldsymbol{A}}}_{\mathrm{ref}}\times \hat{n}$ on S of V. For a cylindrical flux tube, ${{\boldsymbol{B}}}_{\mathrm{ref}}$ can be chosen as ${{\boldsymbol{B}}}_{\mathrm{ref}}={B}_{z}\hat{{u}_{z}}$ and ${\boldsymbol{B}}={B}_{\theta }\hat{{u}_{\phi }}+{B}_{z}\hat{{u}_{z}}$. Considering ${\boldsymbol{A}}={\boldsymbol{B}}/\alpha$, we compute the magnetic helicity of cylindrical flux rope (DeVore 2000; Démoulin et al. 2002; Berger 2003; Dasso et al. 2003) as

$$\begin{eqnarray}H & = & 4\pi L{\displaystyle \int }_{0}^{{R}_{0}}{A}_{\theta }{B}_{\theta }\,\mathrm{rdr}\\ & = & \displaystyle \frac{4\pi {B}_{0}^{2}L}{\alpha }{\displaystyle \int }_{0}^{{R}_{0}}{J}_{1}^{2}(\alpha r)\ {rdr}\approx 0.7\ {B}_{0}^{2}{R}_{0}^{3}L.\end{eqnarray} \tag{ 6 }$$

To derive the axial and poloidal components of MC flux (${\phi }_{z\mathrm{mc}}$ and ${\phi }_{\mathrm{pmc}}$) and the magnetic helicity (H_mc) of the MC, we apply ${B}_{0}={B}_{0\mathrm{mc}},{R}_{0}={R}_{0\mathrm{mc}}$, and $L={L}_{\mathrm{mc}}$ in Equations (2), (3), and (6). Here L_mc is the estimated length of the MC.

The largest uncertainty in calculating the flux and helicity of an MC arises from the MC flux rope length approximation. Larson et al. (1997) estimated the length of MC as 2.5 au by measuring the travel time of suprathermal electrons moving along with the twisted magnetic field lines. The presence of bidirectional electrons in MCs observed at 1 au suggests the possibility of MCs rooted on the Sun when they reach at 1 au (Shodhan et al. 2000). We consider the MC a half torus that has a circular cross section as an approximation to the expanded flux rope, extending from Sun to Earth. Then the length of MC at 1 au is π au if the major axis length of the torus is 1 au. We note that the length is only 21% higher than the statistical value (2.6 ± 0.3 au) reported in Démoulin et al. (2016). In our study, we measure the cloud's flux and helicity using each of the approximated MC flux rope lengths. In column 8–11 of Table 1, we show the twist density (${\alpha }_{\mathrm{mc}}$) of the magnetic field in MC flux rope, ${\phi }_{z\mathrm{mc}},{\phi }_{\mathrm{pmc}}$, and H_mc corresponding to different L_mcs.

The CME is a halo CME observed from the LASCO/C2 coronagraph at 05:12 UT on 2013 March 15. It appears as a flux-rope-like structure at 08:08 UT in the field of view of SECCHI/COR2 A and B. Gopalswamy et al. (2017b) showed that the magnetic properties of the coronal flux rope can be obtained by combining the reconnection flux and the geometric properties obtained from flux rope fits to white-light data. Gopalswamy et al. (2017b) used the Krall & Cyr (2006) flux rope model to obtain the geometrical properties of coronal flux ropes. Here we fit the CME with the forward-modeling technique, the Graduated Cylindrical Shell (GCS) model, developed by Thernisien et al. (2006) because three views are available from STEREO and SOHO. Thus we get the half angular width (γ), aspect ratio (κ), height of the leading edge (h), tilt angle (λ) with respect to the equator and the propagation longitude (ψ), and latitude (θ) of the CME. From these results, we calculate the radius (${R}_{0\mathrm{cme}}$) of the CME's circular annulus at its leading edge point using the relation, ${R}_{0\mathrm{cme}}/h=1/(1+(1/\kappa ))$ derived using Equation (1) in Thernisien et al. (2006) and 3D speed from the time evolution of CME leading edge. In Figure 5, we show the GCS model in the green wire frame overplotted on the white-light CME, observed by LASCO/C3, and SECCHI/COR2 A and B around the same time.

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We estimate the axial magnetic field strength (${B}_{0\mathrm{cme}}$) of the CME from its poloidal flux component (${\phi }_{\mathrm{pcme}}$) by taking ${\phi }_{p}={\phi }_{\mathrm{pcme}},L={L}_{\mathrm{cme}}$ (length of CME flux rope from the photosphere) and ${R}_{0}={R}_{0\mathrm{cme}}$ in Equation (3). Thus, ${B}_{0\mathrm{cme}}$ can be defined as

$$\begin{eqnarray}&&{B}_{0\mathrm{cme}}=\displaystyle \frac{{\phi }_{\mathrm{pcme}}{x}_{01}}{{L}_{\mathrm{cme}}{R}_{0\mathrm{cme}}}.\end{eqnarray} \tag{ 7 }$$

Here L_cme is estimated from the GCS model and can be computed as ${L}_{\mathrm{cme}}=2{h}_{\mathrm{leg}}+y(h-{h}_{\mathrm{leg}}/\cos \gamma )/2-2\,{R}_{\odot }$, where h_leg is the height of the legs of the CME flux rope computed using Equation (3) in Thernisien et al. (2006), ($(h-{h}_{\mathrm{leg}}/\cos \gamma )/2$) is the radius of the arc of the flux rope, $y=2(\pi /2+\gamma )$ is the arc angle in radians, and R_⊙ represents the solar radius. Longcope et al. (2007) demonstrated that ${\phi }_{\mathrm{pcme}}$ is approximately equal to the magnetic reconnection flux (${\phi }_{\mathrm{RC}}$) in the low corona. Gopalswamy et al. (2017a) proposed a process to estimate the reconnection flux considering the half of the unsigned photospheric flux underlying the area occupied by the PEA (as discussed in the next section).

In Equation (6), we use ${B}_{0}={B}_{0\mathrm{cme}},{R}_{0}={R}_{0\mathrm{cme}}$ and $L={L}_{\mathrm{cme}}$ to obtain the magnetic helicity (H_cme) of the CME flux rope. In Table 2, we present the GCS fitting results along with ${R}_{0\mathrm{cme}}$ at 10 ${R}_{s}$ (${R}_{0\mathrm{cme}}^{10{R}_{s}}$), L_cme, and 3D speed (${V}_{\mathrm{cme}}^{3{\rm{D}}}$) of the CME estimated from the fitting.

Table 2.  CME Parameters Determined by GCS Model Fitting

CME Properties	Values
(1)	(2)
${\psi }_{\mathrm{cme}}$ (Carrington coordinate)	$68\buildrel{\circ}\over{.} 82$
${\theta }_{\mathrm{cme}}$ (Carrington coordinate)	$-6\buildrel{\circ}\over{.} 15$
γ	$25\buildrel{\circ}\over{.} 16$
λ	$-74\buildrel{\circ}\over{.} 35$
κ	0.27
${R}_{0\mathrm{cme}}^{10{R}_{s}}$	2.13 ${R}_{s}$
Lcme	17.07 ${R}_{s}$
${V}_{\mathrm{cme}}^{3{\rm{D}}}$	1321 km s−1

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To analyze the source active region (AR 11692) of the ICME, we consider the SDO/HMI LOS photospheric magnetogram together with the SDO/AIA 193 Å image and the solar X-ray imager on board the Geostationary Operational Environmental Satellite (GOES) system (Hill et al. 2005), which spatially maps soft X-ray emission of solar corona. We calculate the reconnection flux at the decay phase of the flare (when the post-eruption arcade is almost matured) by using SDO/HMI LOS, AIA 193 Å data and GOES SXI data. We identify the post-eruption arcade (PEA) region in both AIA 193 Å and GOES SXI images, find the pixels associated with the arcade area, and overlay it on HMI LOS data. Considering B_LOS and the area of each of those pixels from HMI LOS data, we derive the total reconnection flux (${\phi }_{\mathrm{RC}}$) using the Equation (Gopalswamy et al. 2017a),

$$\begin{eqnarray}&&{\phi }_{\mathrm{RC}}=\displaystyle \frac{1}{2}{\int }_{\mathrm{PEA}}| {B}_{\mathrm{LOS}}| {dA}.\end{eqnarray} \tag{ 8 }$$

To estimate the uncertainty in the arcade area measurement, we consider a range of arcade area (A_PEA). We derive the upper and lower limits of the range by selecting the arcade foot points in GOES SXI and AIA 193 Å images. The X-ray imager has a broader temperature response than the EUV image at a particular wavelength. In Figures 6(a) and (b), we show the estimated arcade foot points in red and blue lines superposed on AIA 193 Å and GOES SXI images, respectively, around 15:00 UT. In Figure 6(d), we overlay the foot points derived from AIA 193 Å and GOES SXI images on the differential-rotation corrected HMI LOS magnetogram in their respective colors at 6:11 UT (time of available HMI data just at the begining of the flare). In Figure 6(c), we show the ribbon structures by arrows in the AIA 1600 Å image around 7:35 UT. Figure 6(e) shows the temporal evolution of GOES X-ray flare intensity. The solid vertical lines a, b, and d on the plot indicate the epochs of measuring arcade in AIA 193 Å, SXI, and HMI LOS magnetogram. The vertical line c indicates the time when a faded ribbon structure is shown in the AIA 1600 Å image. The shaded interval between the dotted vertical line and line c shows an SDO data gap. Due to the absence of SDO/AIA observations during the impulsive phase of the co-produced flare, it is not possible to analyze ${\phi }_{\mathrm{RC}}$ using the flare ribbon method as in Kazachenko et al. (2017). In principle, one expects an overestimate of the area of the PEA (and hence ${\phi }_{\mathrm{RC}}$) if the ribbons start at a finite distance from the polarity inversion line (PIL). However, Gopalswamy et al. (2017a) considered this issue using events that had both ribbon and PEA data and found no evidence of an overestimate. Another possible uncertainty in ${\phi }_{\mathrm{RC}}$ is in identifying the boundaries of PEAs: the PEA is identified in the corona, but superposed on the photosphere. This was also considered by Gopalswamy et al. (2017a) and found that the difference is not significant because ribbon and arcade methods give the same value. However, there may be other uncertainties when the arcade appears differently in EUV and X-ray images. This is explicitly shown in Figure 6: the AIA arcade has a smaller area than the SXI area, giving a 37% lower ${\phi }_{\mathrm{RC}}$.

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Using the reconnection flux limits, we derive a range of ${B}_{0\mathrm{cme}}$ at 10 R_S (${B}_{0\mathrm{cme}}^{10{R}_{s}}$) and H_cme. In Table 3, we show the upper and lower limits of A_PEA (A_PEA^u and A_PEA^l), ${\phi }_{\mathrm{RC}}$ (${\phi }_{\mathrm{RC}}^{u}$ and ${\phi }_{\mathrm{RC}}^{l}$), ${B}_{0\mathrm{cme}}^{10{R}_{s}}$ (${B}_{0\mathrm{cme}}^{10{R}_{s}u}$ and ${B}_{0\mathrm{cme}}^{10{R}_{s}l}$), and H_cme (H_cme^u and H_cme^l).

Table 3.  Magnetic Parameters of 2013 March 15 CME and Its Source AR Derived Using HMI LOS, AIA 193 Å, and GOES SXI Data

APEA		${\phi }_{\mathrm{RC}}$		${B}_{0\mathrm{cme}}^{10{R}_{s}}$		Hcme
$({10}^{19}$ cm2)		$({10}^{21}$ Mx)		(mG)		$({10}^{42}$ Mx2)
(1)		(2)		(3)		(4)
APEAl	APEAu	${\phi }_{\mathrm{RC}}^{l}$	${\phi }_{\mathrm{RC}}^{u}$	${B}_{0\mathrm{cme}}^{10{R}_{s}l}$	${B}_{0\mathrm{cme}}^{10{R}_{s}u}$	Hcmel	Hcmeu
9.63	20.0	4.1	6.55	55.3	88.4	8.5	21.6

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In Table 4, we show the helicity difference between the near-Sun and 1 au flux rope for L_mc = 2, 2.5, and π au. For each approximated MC length, H_mc is less than H_cme which suggests that the helicity of the flux rope at 1 au is primarily invoked by magnetic reconnection at low corona during the eruption. We find a minimum helicity difference in the case of ${L}_{\mathrm{mc}}=\pi$ au with ${H}_{\mathrm{cme}}={H}_{\mathrm{cme}}^{l}$. So, the lower boundary of source region helicity is consistent with H_mc calculated using the MC length $\approx 3.14\,\mathrm{au}$. We note that we have used ${L}_{\mathrm{mc}}=2.5\,\mathrm{au}$, which is very close to the average length reported in Démoulin et al. (2016). If we use ${L}_{\mathrm{mc}}=2.6\,\mathrm{au}$, the helicity value changes only by a small amount (4%). We notice a large helicity difference between H_cme^u and H_mc for each of the L_mcs.

Table 4.  Comparison of Magnetic Helicity of CME and Its Interplanetary Flux Rope Counterpart at 1 au

$({H}_{\mathrm{cme}}^{l}-{H}_{\mathrm{mc}})/{H}_{\mathrm{cme}}^{l}( \% )$			$({H}_{\mathrm{cme}}^{u}-{H}_{\mathrm{mc}})/{H}_{\mathrm{cme}}^{u}( \% )$
(1)			(2)
Lmc = 2 au	Lmc = 2.5 au	${L}_{\mathrm{mc}}=\pi$ au	Lmc = 2 au	Lmc = 2.5 au	${L}_{\mathrm{mc}}=\pi$ au
47.4	34.4	17.5	79.3	74.2	67.5

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