An Observationally Constrained Analytical Model for Predicting the Magnetic Field Vectors of Interplanetary Coronal Mass Ejections at 1 au

We determine the three-dimensional morphology and the propagation direction of CMEs by using the graduated cylindrical shell (GCS; Thernisien 2011) model. This model fits the geometric structure of CMEs as observed by white-light coronagraphs such as the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) on board the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995) mission and the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) on board the Solar Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008) mission. Using the GCS model, we obtain the propagation longitude (ϕ) and latitude (θしーた), half-angular width (βべーた), aspect ratio (κかっぱ), tilt angle (γがんま) with respect to the solar equator, and the leading-edge height (h) of the CME FR.

The parameter κかっぱ constrains the rate of expansion of the CME FR under the assumption of self-similar expansion. Therefore, the cross-sectional radius (r) of the self-similarly expanding FR at any heliocentric distance R (=h − r) can be obtained using the relation r = κかっぱh/(1+κかっぱ). On the other hand, the length (L) of the FR can be estimated from the relation L = 2βべーたR, where 2βべーた is the separation angle between the two legs of the CME in radians.

Observational approaches to determine the three magnetic parameters that completely define any force-free FR are discussed as follows.

2.2.1. Axial Field Strength (B₀)

Several studies have shown that the azimuthal (poloidal) flux of magnetic FRs formed due to reconnection is approximately equal to the low-coronal reconnection flux, which can be obtained from either the photospheric magnetic flux underlying the area swept out by the flare ribbons (Longcope et al. 2007; Qiu et al. 2007) or the magnetic flux underlying the PEAs (Gopalswamy et al. 2017). Combining the geometrical parameters of the FR obtained from the GCS fitting as discussed in Section 2.1 with the estimation of the reconnected magnetic flux, Gopalswamy et al. (2018b) introduced the FRED model which shows that the axial magnetic field strength of the FR can be determined using a constant alpha force-free FR model (Lundquist 1950). We thereby obtain the magnetic field strength (B₀) along the FR axis using the relation (Gopalswamy et al. 2018a, 2018b),

$$\begin{eqnarray}&&{B}_{0}=\displaystyle \frac{{\phi }_{p}{x}_{01}}{{Lr}},\end{eqnarray} \tag{ 1 }$$

where ϕ_p is the azimuthal magnetic flux taken as the reconnection flux, x₀₁ (=2.4048) is the first zero of the Bessel function J₀, L is the length, and r is the cross-sectional radius of the FR.

2.2.2. Direction of the Axial Magnetic Field and the Sign of Helicity

In order to estimate which part of the ICME will be intersected by the observing spacecraft at 1 au, it is important to first determine the impact distance (d) that is the closest distance between the MC axis and the location of the spacecraft. According to the geometry illustrated in Figure 1, we can write

$$\begin{eqnarray}&&\mathrm{BC}={R}_{\mathrm{SE}}\times \tan \phi ,\end{eqnarray} \tag{ 2 }$$

where R_SE is the distance between the Sun and Earth, and ϕ is the longitudinal direction of the line DA. As the plane perpendicular to the MC axis is tilted by an angle δでるた, we can further write

$$\begin{eqnarray}&&\mathrm{AC}=\displaystyle \frac{\mathrm{BC}}{\cos \delta }={R}_{\mathrm{SE}}\times \displaystyle \frac{\tan \phi }{\cos \delta }.\end{eqnarray} \tag{ 3 }$$

Using the value of AC from Equation (3), we can obtain the minimum separation angle ψぷさい between the axis of the MC and the Sun–Earth line from the following relation:

$$\begin{eqnarray}&&\tan \psi =\displaystyle \frac{\mathrm{AC}}{{R}_{\mathrm{SE}}}=\displaystyle \frac{\tan \phi }{\cos \delta }.\end{eqnarray} \tag{ 4 }$$

After determining the value of ψぷさい, the impact distance (d) of the MC at any heliocentric distance (R) along the Sun–Earth line can be obtained from the following equation:

$$\begin{eqnarray}&&d=R\times \sin \psi =R\times \sin ({\tan }^{-1}\displaystyle \frac{\tan \phi }{\cos \delta }).\end{eqnarray} \tag{ 5 }$$

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In order to infer the axial field strength of the MC from the conservation of magnetic flux, we need to estimate its cross-sectional area during its passage over the spacecraft. Figure 2 depicts a schematic picture of an MC cross section when the spacecraft just encounters its arrival. According to the geometry as illustrated in Figure 2, we can write

$$\begin{eqnarray}&&{R}_{c}\times \cos \psi +\sqrt{{R}_{i}^{2}-{{R}_{c}}^{2}\times {\sin }^{2}\psi }={R}_{\mathrm{SE}},\end{eqnarray} \tag{ 6 }$$

where R_C is the radial distance of the MC axis from the Sun center, ψぷさい is the separation angle between the MC axis and the Sun–Earth line, and R_i is the radius of the cross section of the MC. Assuming that the CME has evolved self-similarly between Sun and Earth, we can replace R_c in Equation (6) using the relation R_i = κかっぱR_c, where κかっぱ is the aspect ratio of the CME FR obtained from the observations as discussed in Section 2.1. Thereby, we can estimate the initial radius of the FR cross section upon its arrival at Earth using the following equation:

$$\begin{eqnarray}&&{R}_{i}=\displaystyle \frac{\kappa \times {R}_{\mathrm{SE}}}{\cos \psi +\sqrt{{\kappa }^{2}-{\sin }^{2}\psi }}.\end{eqnarray} \tag{ 7 }$$

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For ψぷさい = 0, Equation (7) reduces to Equation (8), which is the scenario when the spacecraft passes through the center of the FR cross section,

$$\begin{eqnarray}&&{R}_{i}=\displaystyle \frac{\kappa \times {R}_{\mathrm{SE}}}{1+\kappa }.\end{eqnarray} \tag{ 8 }$$

Figure 3 depicts the spacecraft trajectory inside the MC assumed to expand isotropically with expansion speed V_exp. The MC axis propagates with a speed V_pro along the direction depicted by the black arrows in Figure 3. Therefore, in the FR frame of reference, the spacecraft traverses from point A (lying on the front boundary of the MC) to point B (lying on the rear boundary of the MC) with speed V_pro. If t_p is the travel time for the spacecraft to complete the path AB, we can write

$$\begin{eqnarray}&&\sqrt{{R}_{i}^{2}-{d}^{2}}+\sqrt{{R}_{f}^{2}-{d}^{2}}={v}_{\mathrm{pro}}\times {t}_{p},\end{eqnarray} \tag{ 9 }$$

where R_i and R_f are the cross-sectional radii of the front and rear boundaries of the MC, respectively, and "d" is the impact distance of the spacecraft from the MC axis. By the time (t_p) the spacecraft traversed the path AB, the cross-sectional radius of the MC has increased from R_i to R_f with expansion speed V_pro. Therefore, we can write

$$\begin{eqnarray}&&{R}_{f}-{R}_{i}={v}_{\exp }\times {t}_{p}.\end{eqnarray} \tag{ 10 }$$

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Considering a general case, where the MC axis takes t_travel time to traverse a distance R_tip with a speed v_pro, we can write

$$\begin{eqnarray}&&{R}_{\mathrm{tip}}={v}_{\mathrm{pro}}\times {t}_{\mathrm{travel}}.\end{eqnarray} \tag{ 11 }$$

During the time t_travel, as the cross-sectional area of the MC also expands with speed v_exp, the final radius of the MC cross section after t_travel can be written as

$$\begin{eqnarray}&&{R}_{\mathrm{cross}}={v}_{\exp }\times {t}_{\mathrm{travel}}.\end{eqnarray} \tag{ 12 }$$

Using the properties of self-similar expansion, R_cross and R_tip can be related as R_cross = κかっぱR_tip. Therefore, using Equations (11) and (12), we can relate v_pro and v_exp through the following relation:

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{\mathrm{cross}}}{{R}_{\mathrm{tip}}}=\displaystyle \frac{{v}_{\exp }}{{v}_{\mathrm{pro}}}=\kappa .\end{eqnarray} \tag{ 13 }$$

Using Equations (9), (10), and (13), we can write

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{f}-{R}_{i}}{\sqrt{{R}_{i}^{2}-{d}^{2}}+\sqrt{{R}_{f}^{2}-{d}^{2}}}=\displaystyle \frac{{v}_{\exp }}{{v}_{\mathrm{pro}}}=\kappa .\end{eqnarray} \tag{ 14 }$$

In Equation (14), R_i, d, and κかっぱ are the known parameters. R_i is obtained from Equation (7), the impact distance "d" is obtained from Equation (5), and the value of κかっぱ is obtained from the observations as discussed in Section 2.1. Rewriting Equation (14), we get the following quadratic equation of R_f:

$$\begin{eqnarray}&&{R}_{f}^{2}+b\times {R}_{f}+c=0,\end{eqnarray} \tag{ 15 }$$

where

$$\begin{eqnarray*}\begin{array}{rcl}b & = & \displaystyle \frac{2\times ({R}_{i}+\kappa \times \sqrt{{R}_{i}^{2}-{d}^{2}})}{{\kappa }^{2}-1}\\ c & = & \displaystyle \frac{{\left({R}_{i}+\kappa \times \sqrt{{R}_{i}^{2}-{d}^{2}}\right)}^{2}-{d}^{2}\times {\kappa }^{2}}{1-{\kappa }^{2}}.\end{array}\end{eqnarray*}$$

Therefore, solving Equation (15), we can estimate the final radius (R_f) of the expanding FR when the spacecraft encounters the rear boundary of the MC. After estimating R_i (initial radius of the MC front boundary), R_f (final radius of the MC rear boundary), and "d" (impact distance), we can estimate the path AB as depicted in Figure 3. In order to capture the full expansion profile of the MC, next we need to determine the cross-sectional radius of the expanding FR at any distance x traversed by the spacecraft throughout the path AB (Figure 3). Let us consider that at any time t (0 ≤ t ≤ t_p), the spacecraft traverses a distance x with a speed v_pro along AB in the frame of reference attached to the MC axis. Therefore, we can write

$$\begin{eqnarray}&&x={v}_{\mathrm{pro}}\times t.\end{eqnarray} \tag{ 16 }$$

During the time t, the cross-sectional radius of the FR increases from R_i to R_t with a speed v_exp. Therefore, we can write

$$\begin{eqnarray}&&{R}_{t}-{R}_{i}={v}_{\exp }\times t.\end{eqnarray} \tag{ 17 }$$

Using Equations (14), (16), and (17), we can further write

$$\begin{eqnarray}&&\displaystyle \frac{{R}_{t}-{R}_{i}}{x}=\displaystyle \frac{{v}_{\exp }}{{v}_{\mathrm{pro}}}=\kappa .\end{eqnarray} \tag{ 18 }$$

Rewriting Equation (18), we get

$$\begin{eqnarray}&&{R}_{t}={R}_{i}+\kappa \times x.\end{eqnarray} \tag{ 19 }$$

Therefore, at any distance x along the path AB (Figure 3), we can estimate the cross-sectional radius (R_i ≤ R_t  ≤ R_f) of the expanding FR using Equation (19). It is noteworthy that we have started our formulation with the unknown parameters V_exp, V_pro, and t_p (see Equations (9) and 10) and finally arrived at Equations (15) and (18), which are independent of the aforementioned variables. This is the major advantage of this formulation as we have incorporated the FR expansion in such a way so as to get rid of the free or unknown parameters like the expansion speed (V_exp), propagation speed (V_pro), and time of passage (t_p) of the ICMEs at 1 au.

It is expected that the FR axial field strength (B₀) will decrease as the length ($L=\tfrac{2\beta }{\kappa }r$) and cross-sectional radius (r) of the FR will increase during its expansion and propagation throughout interplanetary space (see the expression of B₀ in Equation (1)). Assuming that the angular width (2βべーた) of the CME remains constant throughout its propagation and the nature of expansion is self-similar, we can consider that L ∝ r. Therefore, considering the conservation of magnetic flux (ϕ_p = constant), the axial magnetic field strength (B₀) of any FR having a cross-sectional radius r will follow the relation

$$\begin{eqnarray}&&{B}_{0}\propto \displaystyle \frac{1}{{r}^{2}}.\end{eqnarray} \tag{ 20 }$$

Therefore, knowing the cross-sectional radius R_t (R_i ≤ R_t ≤ R_f) of the FR during its passage through the spacecraft using the Equation (19), we can estimate its axial field strength (B_t) at any time t (0 ≤ t ≤ t_p) using the following relation:

$$\begin{eqnarray}&&{B}_{t}={B}_{{0}_{\mathrm{CME}}}\times \displaystyle \frac{{{r}_{\mathrm{CME}}}^{2}}{{{R}_{t}}^{2}},\end{eqnarray} \tag{ 21 }$$

where r_CME is the cross-sectional radius, and ${B}_{{0}_{\mathrm{CME}}}$ is the axial magnetic field strength of the near-Sun FR obtained from the observations as discussed in Section 2.

As the spacecraft intersects the MC along the path AB (see Figure 3), at any location (x) along AB, the magnetic field vectors of the FR can be obtained using a cylindrical FR solution (Lundquist 1950) in a local cylindrical coordinate ($r,\phi ,z$) attached to the MC axis. The magnetic vectors in the aforementioned (r, ϕ, z) coordinate system will be

$$\begin{eqnarray}&&{B}_{r}=0,\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{B}_{\phi }=H\times {B}_{t}\times {J}_{1}(\alpha r),\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{B}_{z}={B}_{t}\times {J}_{0}(\alpha r),\end{eqnarray} \tag{ 24 }$$

where H = ±1 is the handedness or sign of the helicity, which is the same as that of the near-Sun FRs according to the conservation of helicity rule, αあるふぁ is the constant force-free factor, and J₀ and J₁ are the Bessel functions of order 0 and 1, respectively. The boundary of the FR is located at the first zero of J₀, which leads to $\alpha =2.41/{R}_{t}$ and R_t is therefore the radius of the FR. B_t and R_t evolve according to the relation described in Equations (21) and (19), respectively.

As we have assumed that after 10R_S the CME does not suffer any significant rotation and deflection, therefore the final elevation angle (θしーた) of the MC axis at 1 au should follow the tilt angle (δでるた) of the CME and the azimuthal angle of the MC should follow the propagation longitude (ϕ) of the CME obtained from the GCS fitting as discussed in Section 2.1. In order to get the final magnetic field vectors in the Geocentric Solar Ecliptic (GSE) coordinate system (Hapgood 1992), we first transform the B_r, B_θしーた, and B_ϕ from the local cylindrical coordinates (r, θしーた, z) to the local Cartesian coordinates (${x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} }$) attached to the MC axis. Thereafter, knowing the azimuthal (ϕ) and elevation (θしーた) angles of the MC axis, we transform the magnetic vectors ${B}_{{x}^{{\prime} }},{B}_{{y}^{{\prime} }}$, and ${B}_{{z}^{{\prime} }}$ from the local Cartesian coordinates (${x}^{{\prime} },{y}^{{\prime} },{z}^{{\prime} }$) to the GSE coordinate system (x, y, z). Thus, we get the predicted magnetic vectors B_x, B_y, and B_z of the ICME as detected by the spacecraft at 1 au.

4.1.1. Poloidal Flux Content of the FR

4.1.2. Direction of the Axial Magnetic Field and the Chirality of the FR

The source location of the M6.6 flare that occurred in AR 11719 was associated with a pre-eruptive sigmoidal structure (Vemareddy & Mishra 2015; Joshi et al. 2017). Panel (a) of Figure 5 shows the highly skewed preflare sigmoid observed in EUV images of the AIA passbands (94, 335 and 193 Å). The observed inverse S-shaped morphology of the sigmoidal structure (indicated by the red dashed line) has been overlaid on the HMI line-of-sight magnetogram (panel (b) of Figure 5), which reveals the left-handed chirality of the associated FR. This follows the hemispheric helicity rule (Bothmer & Schwenn 1998) as the source region of the CME was located in the northern solar hemisphere.

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We identify the two boundaries as shown by the blue and green dashed lines in panel (a) of Figure 5, where the two ends of the bundle of sigmoidal field lines are rooted during the pre-eruptive phase. The two aforementioned boundaries are overlaid on the HMI line-of-sight magnetic field and the regions are marked by the yellow ellipses (see panel (b)). The simple connectivity (without considering any twist) between the two opposite magnetic polarities underlying the regions marked by the yellow ellipses suggests the northwest direction (as shown by the yellow arrow) as the axial orientation of the FR at higher heights in the corona (above ≈5 R_S). This is expected as the apex orientation of the left-handed FR should rotate in the counterclockwise direction to release the axial twist or writhe during its evolution in the lower corona below 5 R_S (Lynch et al. 2009).

In order to confirm the axial orientation of the FR, we further investigate the morphology of the associated PEA formed during the flare. Panel (c) of Figure 5 shows that the eastern part of the PEA channel is tilted toward the southwest direction and further bends toward the northwest direction at the location indicated by the yellow arrow, forming a nearly U-shaped morphology. This is certainly a complex morphology, which makes the event more complicated. Considering the apex orientation of the FR inferred only from the eastern part of the PEA channel, Palmerio et al. (2018) found a contradiction between the solar and 1 au Bz direction. However, we focused on the full U-shaped morphology of the PEA channel in this study. Considering the full extent of the PEAs allows us to analyze the FR structure beyond the sigmoidal pre-eruptive configuration and, therefore, to capture the complete evolution of the FR in the lower corona during the phase of sigmoid to arcade formation. According to the standard flare model in three dimensions (Shibata et al. 1995; Moore et al. 2001; Priest & Forbes 2002), the footpoints of the eruptive FRs are believed to be located on either side of the two ends of the PEA channel. Therefore, considering the left-handed chirality, we mark the expected locations of the two footpoints of the FR as shown by the yellow circles at the two ends of the U-shaped PEA channel. The red dashed curve connecting the two yellow circles indicates the possible writhe presented in the FR during the formation phase. This is in agreement with the observed writhing motion of that FR during the eruptive phase as reported by Joshi et al. (2017). Therefore, due to the writhing motion, the FR would have relaxed the axial twist during its evolution in the lower corona, resulting in an orientation following the straight connectivity (shown by the blue dashed line in panel (d) of Figure 5) between the two footpoint locations. In such a scenario, the magnetic polarities underlying the two yellow circles clearly indicate that the axial orientation of the FR is directed toward the northwest.

From the GCS fitting (Figure 6) of the observed white-light morphology of the CME at ≈10R_S, we estimate the tilt angle of the CME axis as 73° ± 10° with respect to the ecliptic plane. Minimizing the difference in angle between the GCS tilt and the axial direction (northwest) of the FR inferred from the on-disk observations, we obtain the axial magnetic field direction of the CME FR at ≈10R_S along 73° ± 10°, measured in the counterclockwise direction with respect to the solar equator. Assuming that no major rotation occurred after 10R_S, we consider this axis orientation as the final orientation of the associated MC axis at 1 au.

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4.1.3. Axial Field Strength of the FR

4.1.4. Propagation Direction of the CME

We notice that the sign of the B_x component for the estimated magnetic vectors of the ICME as detected by any spacecraft aligned along the Sun–Earth line is very sensitive to the propagation direction of the CME. The three panels in Figure 7 depict the location of Earth (denoted by blue dots) with respect to the magnetic axis (denoted by black solid lines) of the CME propagating along three different directions which are within the error limits as estimated in Section 4.1.4. In each of the three panels, the Sun grids are shown within ±30° longitude and latitude where the projected location of Earth on the solar disk resides at 0° longitude and 0° latitude. Keeping the tilt angle as 73°, we project the magnetic axis of the CME on the solar disk as shown by the black solid lines in each panel. The green dots and arrows on the magnetic axis denote the propagation direction of the CME and the direction of the axial magnetic field of the associated FR, respectively. The arrows along the black dashed lines surrounding the CME magnetic axis depict the direction of the poloidal magnetic field according to the left-handed chirality of the associated FR.

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Notably, at any projected location on the solar disk which lies on the left/right side of the CME axis, the direction of the poloidal magnetic field will be toward/outwards from the Sun. Accordingly, the sign of B_x will change at any location on either side of the magnetic axis, which we have shown by the pink and blue regions, where B_x possesses positive and negative values, respectively. Panel (a) in Figure 7 shows that the projected location of Earth lies on the region of negative B_x for the estimated direction (10° E, 5° S) and tilt (73° with respect to the ecliptic plane) of the MC axis as obtained from the GCS fitting. However, a small shift in the propagation direction from 10° E, 5° S to 3° E, 12° S results in a zero impact distance between the MC axis and the Sun–Earth line (see panel (b) in Figure 7) for which the estimated B_x component turns out to be zero. If we further shift the propagation direction of the MC axis from 3° E, 12° S to 0° E, 15° S within the error limits, the sign of B_x becomes positive as the location of Earth or any spacecraft aligned along the Sun–Earth line lies on the left side of the MC axis where the direction of the poloidal magnetic field is toward the Sun (see panel (b) in Figure 7). Therefore, our analysis shows that within the error limits of the propagation direction of the CME, B_x can have both positive and negative components in the estimated magnetic vectors of the ICME at 1 au.

It is noteworthy that the above-mentioned scenario is true for any tilt angle of the FR orientation where the propagation direction is very close to the Sun–Earth line. Interestingly, the sign or the direction of the variation (positive to negative or vice versa) of the estimated B_y and B_z components is not sensitive to small variations (±10°) in the propagation direction and tilt angle of the CME. Therefore, we expect less uncertainty in the prediction of the B_y and B_z components of the MC.

Using the near-Sun FR properties of the associated CME as described in Section 4.1, we estimate the magnetic vectors of the ICME as intersected by the spacecraft at 1 au. The curves shown by the black solid lines in Figure 8 depict the observed magnetic vectors of the ICME as detected by the Wind spacecraft (Ogilvie & Desch 1997). The red vertical lines denote the front and rear boundaries of the MC which we have estimated from the observed magnetic field and plasma parameters of the ICME.

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Incorporating the uncertainties in the GCS parameters involved in the modeling, we generate all of the possible input data sets from the range of values of the input parameters, i.e., the propagation direction (0°–10°E, 5°–15° S), tilt angle (63°–83°), and aspect ratio (0.20–0.24) of the CME. Further, considering an error of 2 × 10²⁰ Mx (standard deviation of the two values of reconnection flux obtained from the two different methods as discussed in Section 4.1.1) in determining the poloidal flux and ±0.02 in determining the CME aspect ratio, we get a 20% error in estimating the axial field strength (${B}_{{0}_{\mathrm{CME}}}$) of the CME. This yields the estimated range of ${B}_{{0}_{\mathrm{CME}}}$ at 10R_s as 42–62 mG, with a mean value of 52 mG. This is consistent with the average value of the distribution of axial fields at 10R_S (Gopalswamy et al. 2018b). Using these sets of input data, we run our model and generate synthetic magnetic profiles of the MC. Among these sets of predicted magnetic vectors, we find that the magnetic profiles (shown by the blue dashed lines in each panel of Figure 8) which best match the observed magnetic vectors of the MC can be obtained by using the propagation direction along 0° E, 15° S, the tilt angle as 73°, the aspect ratio as 0.22, and the axial field strength at 10R_s as 52 mG. For this set of input parameters, we show the spacecraft trajectory through the MC and the magnetic field profiles of the FR cross section when the spacecraft intersects the front and rear boundaries of the MC (Figure 9). The uncertainty in predicting the magnetic vectors as shown by the gray shaded region in each panel of Figure 8 is obtained by overplotting all sets of output magnetic profiles.

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In order to overplot the modeled magnetic vectors within the temporal window of the observed MC, we identify the front and rear boundaries of the modeled MC from the hodogram analysis. Figure 10 shows the scattered plots among the magnetic field vectors within the MC for both observed and modeled data values. The yellow dots drawn over the plots for the modeled data values denote the data points that approximately match the front and rear boundaries of the observed MC. Therefore, we take the observed MC boundary as a reference boundary and overplot the data points of the modeled magnetic vectors that lie in between the two yellow dots.

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Figure 8 shows that the predicted magnetic field profiles of the MC obtained from our model are in good agreement with those of the observed profiles as detected by the Wind spacecraft. In comparison to the B_y and B_z components, the larger uncertainty that arises in predicting the B_x component is due to its sensitivity toward the propagation direction of the CME, which we have discussed in Section 4.2. Nevertheless, the predicted profiles for the B_y and B_z components show good agreement with the observed profiles. The predicted strength of the B_z component has been found to be 10.5 ± 2.5 nT when the MC axis makes its closest approach to the spacecraft. This is in agreement with the maximum observed strength (11 nT) of the B_z component obtained from the in situ data. Therefore, our model successfully predicts both the strength and the general profile of the B_z component of the MC with good accuracy.