THREE-DIMENSIONAL GEOMETRY OF A CURRENT SHEET IN THE HIGH SOLAR CORONA: EVIDENCE FOR RECONNECTION IN THE LATE STAGE OF THE CORONAL MASS EJECTIONS

We used coronagraph data taken from STEREO SECCHI-COR2 and SOHO LASCO-C3. The COR2 and C3 images provide observations in the extended solar corona with fields of view (FOVs) of 2.5–15 R_⊙ and 3.7–30 R_⊙, respectively. The twin STEREO spacecraft, named Ahead and Behind (hereafter STEREO-A and -B) orbit faster and slower than the Earth around the Sun. While SOHO provides the viewing perspective from the Earth, STEREO provides additional viewing perspectives separated from the Sun–Earth line. We studied an event observed on 2013 September 21, when the separation angles of STEREO-A and -B with the Sun–Earth line were 147° and 139°, respectively.

We used three coordinate systems: the reference coordinate system (x_ref, y_ref, z_ref), local coordinate system (x, y, z), and observational coordinate system (X, Y, Z). Figure 1(a) shows the relation between the reference and local coordinate systems. In the reference coordinate system, the origin O_ref is at Sun center, and the x_ref-axis lies on the plane defined by the central meridian of the Sun seen from the Earth. The z_ref-axis is the rotational axis of the Sun, and the y_ref-axis is determined by z_ref- and x_ref-axes satisfying the right-hand rule. The local coordinate system is used to construct the geometric models. The z-axis is a radial direction (from Sun center) passing through the origin of this coordinate system O. The x-axis is defined with a vector tangential to a circle connecting the z_ref- and z-axes, toward the south of the Sun. Figures 1(b) and (c) show the observational coordinate system. The image plane is defined as the Y–Z plane with the origin O_obs at disk center (Figure 1(b)). The X-axis points to the observer (Figure 1(c)). A more detailed description of the coordinate systems is given in Kwon et al. (2014).

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The post-CME ray is assumed to be a 3D slab characterized by the direction of its central axis and cross-section. Figures 1(b) and (c) demonstrates the geometry of a slab in the observational coordinate system for a given observer i. The direction is determined from the position angle Φ_i on the plane of the sky (POS; Figure 1(b)) and the angular distance Ψ_i from the POS (Figure 1(c)). It is assumed that the post-CME ray is radial, so that its 3D geometry is represented with a single pair of Φ_i and Ψ_i.

The cross-section is represented as width (w; or thickness) and depth (d). The width on an image plane (w_POS) is measured in the direction normal to the axis. If it is the exact edge-on view, the width projected on the image plane is the actual width (w) of the slab. As shown in Figure 1(c), the actual depth of the slab is given by,

$$\begin{eqnarray}&&d={d}_{\mathrm{LOS} \mbox{-} i}\cos {{\rm{\Psi }}}_{i},\end{eqnarray} \tag{ 1 }$$

where d_LOS-i and Ψ_i are the depth and angular distance of the slab for observer i.

Figure 1(d) shows the cross-section of the slab with the two LOS-i shown by the plus symbols in Figure 1(b). The two thin lines refer to the two LOSs of observer j (LOS-j). The two observers are separated by an angle Θ. The d_LOS-i is deduced by the projected width of the other observer j (${w}_{\mathrm{POS} \mbox{-} j}^{\prime }$) with separation angle Θ, namely,

$$\begin{eqnarray}&&{d}_{\mathrm{LOS} \mbox{-} i}=\displaystyle \frac{{w}_{\mathrm{POS} \mbox{-} j}^{\prime }}{\sin {\rm{\Theta }}}.\end{eqnarray} \tag{ 2 }$$

Note that ${w}_{\mathrm{POS} \mbox{-} j}^{\prime }$ is determined along the LOS-i on the image plane of observer j (POS-j). The width w_POS on an image plane is related to ${w}_{\mathrm{POS}}^{\prime }$ as,

$$\begin{eqnarray}&&{w}_{\mathrm{POS}}={w}_{\mathrm{POS}}^{\prime }\cos \zeta ,\end{eqnarray} \tag{ 3 }$$

where ζ is the angle between w_POS and ${w}_{\mathrm{POS}}^{\prime }$.

The shaded region in Figure 1(d) illustrates the possible irregular-shaped cross-section of post-CME ray enclosed by the four LOSs. Since the true shape and orientation of the cross-section are unknown, only the projected width (w_POS) and depth (d_POS) are given by observations. For the sake of simplicity, we assume that the observer with the shortest projected width is seeing the exact edge-on view as shown in Figure 1(d).

In order to model the 3D geometry of the CME as a flux rope, we employ the Gradual Cylindrical Shell (GCS) model, developed in Thernisien et al. (2006). The GCS model is characterized by 7 parameters: the height of the local coordinate system, longitude and latitude of the center of the flux rope, the half angle, aspect ratio, height and rotation of the flux rope (Thernisien et al. 2006). The technique has been used extensively to derive 3D CME parameters in numerous publications. Because the CME fitting plays only an auxiliary role in this papers, we do not discuss it in detail. A more detailed description can be found, for example, in Kwon et al. (2014).

Since we have three different observers, STEREO-A, -B, and SOHO, we use "A," "B," and "E," respectively, instead of i and j. For instance, the LOSs of the three observers are LOS-A, -B, and -E, respectively.

The observed emission in white light observations results from the LOS integration of several components. In the case of the polarized brightness pB, the observed signal can be decomposed as (e.g., Hayes et al. 2001),

$$\begin{eqnarray}&&{pB}={{pB}}_{K}+{{pB}}_{F}+{{pB}}_{S},\end{eqnarray} \tag{ 4 }$$

where K, F, and S refer to the K coronal component, F coronal component, and stray light component, respectively. A similar relation holds for the total brightness, B. The K coronal component is due to Thomson scattering from free electrons in the solar corona and can be written as the integration of the volume electron density N_e [cm⁻³] along the LOS (Billings 1966),

$$\begin{eqnarray}&&{{pB}}_{K}(\rho ,s)={\int }_{\mathrm{LOS}}{\rm{\Gamma }}(\rho ,s){N}_{e}(s){ds}.\end{eqnarray} \tag{ 5 }$$

The function Γ(ρ, s) describes the Thomson-scattering geometry. It is determined by the geometric relation among the electrons, the photosphere, and the observer. We use the same expression for Γ(ρ, s) as shown in Wang & Davila (2014, and references therein).

Figure 2(a) illustrates the geometric relation. The LOS (dashed line) is assumed to be parallel to the X-axis. Since we deal with a narrow structure, we can assume that electrons are concentrated in a small volume approximated as a slab, and embedded in the background solar medium. The rectangle on the LOS in this figure refers to the small volume and the geometry of this volume is specified by ρ and s; the projected distance of the volume on the POS is ρ, and the distance from the POS is s. r is the radial distance from Sun center. In this manner, ${r}^{2}={\rho }^{2}+{s}^{2}$ and ${\rm{\Omega }}={\sin }^{-1}({R}_{\odot }/r)$.

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Since an observed intensity is the sum of various components as shown in Equation (4), the K coronal component should be extracted from the observed intensity. As seen in Figure 2(b), we can consider two pB intensities obtained along two adjacent LOSs, as follows,

$$\begin{eqnarray}&&{{pB}}_{1}={{pB}}_{K,1}+{{pB}}_{F,1}+{{pB}}_{S,1},\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{{pB}}_{0}={{pB}}_{K,0}+{{pB}}_{F,0}+{{pB}}_{S,0},\end{eqnarray} \tag{ 7 }$$

where pB₁ and pB₀ are pB intensities obtained along LOSs passing through the slab and passing right next to it, respectively.

We make the straightforward assumption that the F coronal component is identical for both LOSs, namely,

$$\begin{eqnarray}&&{{pB}}_{F,1}={{pB}}_{F,0}.\end{eqnarray} \tag{ 8 }$$

Also, since the two LOSs are close to each other, the stray light components will be very similar. Thus,

$$\begin{eqnarray}&&{{pB}}_{S,1}={{pB}}_{S,0}.\end{eqnarray} \tag{ 9 }$$

As shown in Figure 2(b), the intensity of ${{pB}}_{K,1}$ can be thought as the sum of two components, one emitting from the slab (${{pB}}_{K,\mathrm{slab}};$  solid line), and the other from the background of corona (${{pB}}_{K,{bg}};$  dashed lines), i.e., ${{pB}}_{K,1}$ = ${{pB}}_{K,\mathrm{slab}}$  + ${{pB}}_{K,{bg}}$. Since the depth is small, ${{pB}}_{K,{bg}}\approx {{pB}}_{K,0}$. In this respect, ${{pB}}_{K,1}$ and pB₁ could be written as,

$$\begin{eqnarray}&&{{pB}}_{K,1}={{pB}}_{K,\mathrm{slab}}+{{pB}}_{K,0},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{{pB}}_{1}={{pB}}_{K,\mathrm{slab}}+{{pB}}_{0}.\end{eqnarray} \tag{ 11 }$$

Therefore, the pB_K component due to the electrons in a slab can be determined by subtracting pB₀ from pB₁,

$$\begin{eqnarray}&&{{pB}}_{K,\mathrm{slab}}={{pB}}_{1}-{{pB}}_{0}.\end{eqnarray} \tag{ 12 }$$

Similarly to the pB, the K coronal component of the total brightness in a slab can be determined. The total brightness B along the two LOSs can be defined as,

$$\begin{eqnarray}&&{B}_{1}={B}_{K,\mathrm{slab}}+{B}_{K,0}+{B}_{F,1}+{B}_{S,1}+{B}_{u,1},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{B}_{0}={B}_{K,0}+{B}_{F,0}+{B}_{S,0}+{B}_{u,0}.\end{eqnarray} \tag{ 14 }$$

By using the same assumptions, the ${B}_{K,\mathrm{slab}}$ component can be obtained by subtracting the intensity B₀ from B₁,

$$\begin{eqnarray}&&{B}_{K,\mathrm{slab}}={B}_{1}-{B}_{0}.\end{eqnarray} \tag{ 15 }$$

The intensities of ${{pB}}_{K,\mathrm{slab}}$ and ${B}_{K,\mathrm{slab}}$ result from the electrons distributed along a LOS within the depth of the slab, [s₁, s₂]. Note that if the depth is known as shown in Section 2.2, then the electron density can be obtained by taking the integral in Equation (5) within the depth, d_LOS = [s₁, s₂]. However, we do not know the density distribution N_e(s) along the LOS. We can replace it by an averaged density $\bar{{N}_{e}}$ because the d_LOS is small. When ${{pB}}_{K,\mathrm{slab}}$ is given together with the 3D geometry, $\bar{{N}_{e}}$ can be determined as,

$$\begin{eqnarray}&&\bar{{N}_{e}}=\displaystyle \frac{{{pB}}_{K}}{{\int }_{{s}_{1}}^{{s}_{2}}{\rm{\Gamma }}(\rho ,s){ds}}.\end{eqnarray} \tag{ 16 }$$

In the case that ${B}_{K,\mathrm{slab}}$ is given, the function Γ should be replaced with a proper function for total brightness. The function can be found in the literature, for instance, in Billings (1966).

In addition to the volume electron density [cm⁻³] given by Equation (16), we determine the surface (or column [cm⁻²]) and linear [cm⁻¹] electron densities. The surface density is obtained by multiplying $\bar{{N}_{e}}$ by d_LOS. Once the surface densities are determined across the post-CME ray on the POS, the linear density can be determined by integrating the surface densities over the projected width.

In practice, the differences between the two intensities, passing outside and through a slab, in Equations (12) and (15) could be obtained in various ways. If the signal-to-background ratio is high enough to distinguish the signal within the slab, we use a least-squares fit with a Gaussian function as follows,

$$\begin{eqnarray}&&I(\tau )={A}_{0}{e}^{\tfrac{-{z}^{2}}{2}}+{A}_{3}+{A}_{4}\tau ,\end{eqnarray} \tag{ 17 }$$

$$\begin{eqnarray}&&z=\displaystyle \frac{\tau -{A}_{1}}{{A}_{2}},\end{eqnarray} \tag{ 18 }$$

where τ is the distance from a reference point, along an arc across the axis of a post-CME ray as shown in Figure 1(b). The linear function, ${A}_{3}+{A}_{4}\tau$, is regarded as the intensity integrated along the LOS passing by a ray, such as pB₀ and B₀. On the other hand, the pure Gaussian function, ${A}_{0}{e}^{\tfrac{-{z}^{2}}{2}}$ is used for the intensity obtained along the LOS passing through the ray, ${{pB}}_{K,\mathrm{slab}}$ and ${B}_{K,\mathrm{slab}}$.

If the signal-to-background ratio is so low that the Gaussian fit is not appropriate, we can create a minimum image by taking the minimum value of each pixel, over all images within two days before and after the time of the image we analyze. This minimum image is then subtracted from the original image. After this, the intensity of a slab could be obtained with the Gaussian fit. In this paper, we create and use the minimum images for total brightness (B) images because of the high background emission. Since the minimum image may contain remnant scattered light, the resulting densities may be underestimated.

A post-CME ray was observed with two CMEs on 2013 September 21 by three different spacecraft, SOHO and STEREO-A and -B. Figure 3 shows the snapshot images of observations as the running difference composite images constructed with the COR1-COR2 (left and right panels) and the C2–C3 (middle) white light images, observed around 10:08 UT. The C3 image clearly shows that two CMEs overlap while it is not so clear on the COR2 images. The CMEs were first seen at 02:50 UT in COR1-B. We determined the 3D properties of the CMEs with the GCS model (Thernisien et al. 2006), using a forward modeling method developed in Kwon et al. (2014). The two GCS models are overlaid in the bottom panels in this figure. The CME shown by yellow color was seen prior to the CME in orange color. Note that the dashed GCS fits represent the structures on the far side of the POS. It was found that the first CME is faster, and their average speeds are 322 ± 124 km s⁻¹ and 281 ± 111 km s⁻¹, respectively. The SOHO LASCO CME catalog (Yashiro et al. 2004) has reported these CMEs as a single event of which the speed determined with a linear fit is 387 km s⁻¹.

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Figure 4 shows the time-series COR2-B images of a post-CME ray taken from 02:03 UT on September 21 to 22:08 UT on the next day. Each panel was taken parallel to the axis of the ray with a FOV of 2 R_⊙  in width and 5–15 R_⊙  in height. The central axis of the ray observed at 15:08 UT on September 21 was determined as a radial line, and the axis was traced backward and forward in time with Carrington longitude, and used to construct this figure (see Section 3.2). In this way, we found what was a pre-existing streamer-like structure around the axis before the eruption from 02:08 UT to 06:08 UT, indicating that this may have been a streamer blowout event (e.g., Howard et al. 1985). However, it was not immediately obvious whether either of our CMEs were responsible for it. We will not discuss this issue further, since it is not a focus of this paper.

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In the aftermath of the CMEs (10:08 UT), the streamer-like structure seems to disappear, and a ray appeared in the lower part of the FOV at 12:08 UT in Figure 4. The ray increased in height, and exits the FOV at 16:08 UT. At 15:08 UT, the ray was observed to be straight and thick up to 12 R_⊙. After that, the ray became faint and turbulent. The observed ray seemed to move from the initial central axis in the azimuthal direction from 01:28 UT on September 22, and it looks coaxial again from 19:08 UT.

One of the notable features shown in Figure 4 is the formation of successive blobs moving outward along the ray axis. The first blob was observed at the bottom of the FOV at 18:08 UT on September 21, and two more blobs were found, as shown by the black arrows. The three blobs were seen in the FOV in the time ranges of 18:08–21:08 UT, 20:08–22:08 UT, and 21:08–00:08 UT, respectively. The speed of each blob was determined by a least-squares linear fit, as 374 ± 22 km s⁻¹, 336 ± 35 km s⁻¹, and 337 ± 22 km s⁻¹, respectively. The average speed and its standard deviation of the three blobs is 349 ± 22 km s⁻¹.

We selected the ray as observed at 15:08 UT (15:06 UT for C3) on 2013 September 21 to analyze its detailed properties, such as geometry and electron density. Figure 5 shows white light observations of the post-CME ray. The two CMEs leading the ray had left these FOVs. The COR2-B and C3 images show thin and bright rays but the image of COR2-A does not show a clear ray-like structure. We used the COR2-B and C3 images in order to determine the 3D properties of the ray.

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We estimated the 3D central axis of the ray via forward modeling. It has been done with visual inspection as all other forward modeling fits in SECCHI-LASCO. We determined the position angle Φ_B of the axis on the POS of COR2-B (POS-B) first (arrows in Figure 5(a)), and then determined the angular distance Ψ_B, comparing with the position and length of rays likely to be the counterpart on the C3 image. The Φ_B is fixed while the Ψ_B varies because the axis moves only along the LOS of COR2-B (LOS-B). However, the position angles of the axis on the POS-E and -A vary as Ψ_B is adjusted. At the same time, the projected lengths vary on all image planes due to the changing angular distances from the image planes. In Figure 5(b), there are two rays which can be considered as the counterpart of the COR2-B ray, and the arrows point at their axes. These arrows denote the position angles and the projected lengths of the axis. The ray pointed by the black arrows is discarded because of the discrepancy between the lengths of the axis and observed ray, and the Ψ_B determined with the other one was selected.

Since we assumed that the ray is radial relative to Sun center, its central axis was determined with a single pair of the position angle and angular distance (Φ, Ψ), and it can be converted to a set of co-latitude and (Carrington) longitude (θ, ϕ) with simple geometric treatments. It was found that the co-latitude and Carrington longitude are 59° and 18°, respectively. In addition, the position angles and angular distances in the observational coordinate systems of COR2-B, C3, and COR2-A are (−62°, 14°), (56°, 19°), and (−47°, −50°), respectively. Note that the COR2-A image does not show the ray. This is consistent with the large and negative value of Ψ_A indicating that the ray is far from the POS-A, and hence its emission will be too weak.

Once the 3D geometry of a ray is specified, the electron density can be estimated as shown in Section 2.3. We assumed that the COR2-B image shows the exact edge-on view of the slab because the width on the POS-B is shorter than that on the POS-E. In this respect, the width w can be determined from the intensity profiles across the axis of the ray on the POS-B. Figure 6(a) shows an example of the pB profile across the axis of the ray. The pB values were taken at a fixed projected distance (ρ_B = 7.06 R_⊙) from disk center on the image plane as illustrated in Figure 1(b). Note, in this way, that all geometric parameters in Equation (16), s and ρ, are fixed for a single profile. A least-squares Gaussian fit in Equation (17) was used to extract the properties, such as ${{pB}}_{K,\mathrm{slab}}$ and w_POS-B. Circles refer to the intensities obtained from the pB image of COR2-B, and solid and dashed lines show the best result of the fit. The full width was defined as two times the full-width at half-maximum (FWHM). By this definition, the inner part of the ray is selected as shown by the closed circles and solid line in this panel. It was repeated in the de-projected height range of 5–9 R_⊙. The ${{pB}}_{K,\mathrm{slab}}$ and the background pB₀ at the axis along the ray are shown as plus symbols and the solid line, respectively, in Figure 6(b). Figure 6(c) shows the determined FWHM (w/2) and error along the ray. It is interesting to note that the FWHMs seem to be constant with height. The dashed line represents the result of the least-squares fit to the linear model and the inclination of the linear function is 6.0 × 10⁻³. The average and standard deviation of FWHMs over the height range are 0.21 R_⊙  and 0.04 R_⊙, respectively. In other words, the full widths are 0.42 R_⊙ ± 0.08 R_⊙.

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In addition to the width, we determined the depth (d) in three different ways based on the geometric relation shown in Section 2.2. Columns 2–4 in Table 1 list the values for w_POS, d_POS, w, and d obtained from the different ways. As shown in Figure 1(d), a d_LOS of the edge-on view can be determined by the corresponding ${w}_{\mathrm{POS}}^{\prime }$ of the other observer. In our case, the projected depth of COR2-B (d_LOS-B) was determined from ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$ (Table 1, column 2). We determined the FWHMs on the C3 total brightness image with the Gaussian fit and found the average and standard deviation of the full width (w_POS-E) as 0.54 ± 0.12 R_⊙  over the height range. Equation (3) gives, ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$ = 0.82 ± 0.18 R_⊙, where ζ_E = 49°. Using Equations (2) and (1), d_LOS-B = 1.25 ± 0.28 R_⊙  with Θ = 139°  and d = 1.22 ± 0.27 R_⊙  with Ψ_B = 14°.

Table 1.  Geometric Parameters for the Cross-section of the Slab Determined with Three Different Methods

Geometric	With Geometric Relation	With Linear Density	With Geometric Relation
Parameter	on POS-B	on POS-B	on POS-E
wPOS-B	0.42 ± 0.08	0.42 ± 0.08	0.42 ± 0.08
${w}_{\mathrm{POS} \mbox{-} B}^{\prime }$	N/A	N/A	0.66 ± 0.13
dPOS-B	1.25 ± 0.28	1.28 ± 0.36	N/A
wPOS-E	0.54 ± 0.12	N/A	0.54 ± 0.12
${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$	0.82 ± 0.18	N/A	N/A
dPOS-E	N/A	N/A	1.01 ± 0.19
w	0.42 ± 0.08	0.42 ± 0.08	0.54 ± 0.12
d	1.22 ± 0.27	1.24 ± 0.35	0.95 ± 0.18

Note. The actual cross-section (w, d) of a slab is determined from the projected cross-section (w_POS, d_POS). Columns 2–3 list the geometric parameters determined with three different methods; the geometric relation shown in Figure 1(d), assuming that the edge-on view of the slab is on the POS-B; the comparison of the linear densities [cm⁻¹] between COR2-B and C3; and the geometric relation assuming that the edge-on view is on the POS-E (see Section 3.3). The listed values are divided by the solar radius (R_⊙). "N/A" denotes undetermined parameters.

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However, it has been often argued that an observed ray is the projection of multiple ray-like structures or the side view of a post-CME ray may be invisible (Lin et al. 2015). In this regard, the ray on the C3 image may not be the counterpart of the ray on the COR2-B image, and the actual w_POS-E cannot be determined directly from the C3 image with the Gaussian fit. In order to test this, we determined the linear electron densities [cm⁻¹], from the COR2-B (${N}_{e}^{{\rm{COR2}} \mbox{-} B}$) and C3 (${N}_{e}^{{\rm{C}}3}$) images. If these two rays in the two images are truly the observations of a single post-CME ray and the width and depth are identified correctly, then the linear electron densities determined from both images should be equal.

We determined the average ratio of the linear electron densities, ${N}_{e}^{{\rm{C}}3}$/${N}_{e}^{{\rm{COR2}} \mbox{-} B}$, over the height range (5–9 R_⊙). To do this, we perturbed d_LOS-B in the intervals of 0.01 R_⊙, determined the average ratio for each d_LOS-B, and selected the d_LOS-B when the average ratio is closest to 1. If, for example, the ray on the COR2-B images is the projection of multiple ray-like structure, the determined d_LOS-B will pass through several rays on the C3 image (cf. Figure 7). Note that it was assumed that the d_LOS-B is a constant with height, since it is assumed to be a slab.

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In the case of the linear density determined from COR2-B, the measurement can be done straightforwardly by using the pB and B images of COR2-B. The ${{pB}}_{K,\mathrm{slab}}$ or ${B}_{K,\mathrm{slab}}$ was taken from each pixel (LOS) along w_POS-B, and it was used for the measure of surface electron density [cm⁻²]. By doing this, we came to have the surface densities for all LOSs across the axis on the POS-B. The ${N}_{e}^{{\rm{COR2}} \mbox{-} B}$ was obtained by taking the integral of the surface densities over the full width w_POS-B. It was repeated over the height range.

As for ${N}_{e}^{{\rm{C}}3}$, d_LOS-B defines the integration section (${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$) on the POS-E. In Figure 7, the arrow shows the segment of a LOS-B projected on the POS-E. If the segment is associated with the depth of the slab observed from COR2-B, it is the d_LOS-B and its projected length on the POS-E is the ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$. We took the ${B}_{K,\mathrm{slab}}$ at pixels along the ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$, and the surface densities were derived from the ${B}_{K,\mathrm{slab}}$. Finally, ${N}_{e}^{{\rm{C}}3}$ was determined by integrating the surface densities over the ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$. We used only the B image for C3 because there were no pB observations at this time. Each linear density ${N}_{e}^{{\rm{COR2}} \mbox{-} B}$ determined from the pB or B image of COR2-B was compared with the density determined from the B image of C3 (Figure 8).

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Plus symbols in the top panel of Figure 8 show the extracted intensities B taken from the C3 image along the arrow (d_LOS-B) in Figure 7. Once a d_LOS-B is given, the B profiles obtained along the corresponding ${w}_{\mathrm{POS} \mbox{-} E}^{\prime }$ are considered as B₁ in Equation (15). The intensity B₀ passing right next to the ray was determined as a linear function B^model₀ = ${A}_{3}+{A}_{4}\tau$  in Equation (17). It was assumed that the inclination of the B₀^model (A₄) is the same as that of the observed intensities B₁. It is true in such circumstances that a slab structure is embedded in the background that is represented with a linear function. The dashed line in this panel is the result of the least-squares linear fit, ${B}_{1}^{\mathrm{model}}$ = ${A}_{3}^{\prime }+{A}_{4}^{\prime }\tau$, of the B₁. By the assumption, A₄ = ${A}_{4}^{\prime }$. The parameter A₃ can be found by the difference between ${A}_{3}^{\prime }$ and the minimum value of ${B}_{1}-{B}_{1}^{\mathrm{model}}$. The solid line shows the B^model₀ found with this way. The ${B}_{K,\mathrm{slab}}$ was determined as ${B}_{1}-{B}_{0}^{\mathrm{model}}$ and used for the measurement of the surface density of C3 as described above.

The bottom panel in Figure 8 shows the comparison of two linear densities, ${N}_{e}^{{\rm{C}}3}$ and ${N}_{e}^{{\rm{COR2}} \mbox{-} B}$, when d_LOS-B is 1.64 R_⊙, and the average of ${N}_{e}^{{\rm{C}}3}$/${N}_{e}^{{\rm{COR2}} \mbox{-} B}$  is 0.99. In this figure, the ${N}_{e}^{{\rm{COR2}} \mbox{-} B}$ was determined with the pB image. This figure indicates that the two values are close to each other but with a large scatter. The large scatter may be due to the small variation of the density with height (see the top panel of Figure 9). The average and standard deviation over the height range is (6.9 ± 1.8) × 10²⁵ cm⁻¹. The standard deviation is 26% of the average. It could be interpreted as that the intrinsic variation of the density is comparable to the uncertainty of the density measurements.

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In addition to the comparison between the C3 B and COR2-B pB images, we obtained that d_LOS-B = 0.92 R_⊙  when the average of ${N}_{e}^{{\rm{C}}3}$/${N}_{e}^{{\rm{COR2}} \mbox{-} B}$  is 1.01, from the comparison between the B images of C3 and COR2-B. The discrepancy in d_LOS-B may result from the uncertainty of the density measurements. The linear density determined from the B image of COR2-B could be underestimated because of the subtraction of the minimum image (see Section 2.3). Considering the discrepancy as the uncertainty, we obtained that d_LOS-B = 1.28 ± 0.36 R_⊙. The third column in Table 1 shows the results. From Equation (1), we obtained d = 1.24 ± 0.35 R_⊙.

Note that this result is consistent with the result determined with the geometric treatment with the Gaussian fit shown in the second column in Table 1. In Figure 7, it is evident that the d_LOS-B determined in this way is consistent with the morphology of the ray on the C3 image. Two dotted lines show the heads and tails of the best d_LOS-B (1.28 R_⊙) as shown by the arrow. It demonstrates that the ray observed with C3 is the actual counterpart of that on the COR2-B image.

The two widths, w_POS-B and w_POS-E, are not much different. In this regard, we determined w and d assuming that the ray on the C3 image is the exact edge-on view for a comparison, and the result is shown in the fourth column in Table 1. It was found that ζ_B = 51°, and the ${w}_{\mathrm{POS} \mbox{-} B}^{\prime }$ is derived as 0.66 ± 0.13 R_⊙  from the w_POS-B determined with the Gaussian fit. From the ${w}_{\mathrm{POS} \mbox{-} B}^{\prime }$, we obtained that d_LOS-E = 1.01 ± 0.19 R_⊙, using Ψ_E = 19°. Finally, we found that d = 0.95 ± 0.18 R_⊙.

Figure 9 (top) shows the linear electron density determined from the pB of COR2-B (diamond symbols) and B of C3 (plus symbols). There is weak height dependency of the electron density. The solid line shows a power law function fit, N_e = a(r/R_⊙)^b, with a = 1.4 × 10²⁶ and b = −0.37. The power law function found by Vršnak et al. (2009) is shown as dashed line. The difference in the power index could be interpreted as event-to-event variation. The top panel of Figure 5 in Vršnak et al. (2009) shows the linear density with height for five post-CME rays, and some cases show very weak height dependency.

Symbols in the bottom panel of Figure 9 represent the volume electron density [cm⁻³]. It was determined from the electron density shown in the top panel, divided by the cross-sectional area. We applied the power law function to a least-squares fit, and obtained a = 8.7 × 10⁴ and b = −0.75. The average density with the standard deviation was found to be (2.0 ± 0.5) × 10⁴ cm⁻³. Two dashed lines shows the one- and ten-fold empirical coronal density model of Leblanc et al. (1998). The determined electron density is generally higher than the empirical model. Note that the volume electron density could be underestimated. As indicated by Figure 1, the area determined from the LOSs (slab model) gives only the upper limit, and thus the volume density inferred from the area would be the lower limit. In addition, we assumed a constant depth with height. If the depth increases with height, the volume density in lower heights will be greater than what we obtained, while it will be less than the determined value in higher heights. If so, the volume density may decrease with height with a higher rate than that in the bottom panel of Figure 9.

As shown in Figure 6(a), the intensity profiles across the ray are reproduced well with a Gaussian function, implying that the density at the center would be higher than that at the edge. The shaded region between two thick lines in the bottom panel of Figure 9 shows the range of the electron density from the center (upper thick line) to the edge (lower thick line), determined from the COR2-B image (pB). The density at the center varies from (2.1–5.3) × 10⁴ cm⁻³. The density variation from the center to the edge implies that the ray is not a perfect slab.

Figure 10 shows the comparison of the geometries of the CMEs and the ray in three different viewing perspectives. Note that the geometries were determined at different times. We determined the 3D geometry of the central axis of the ray at 15:08 UT and it was compared with the geometries of the CMEs at 10:08 UT shown in Figure 3. The geometries of the CMEs were traced forward in time with Carrington longitude to the time of the ray.

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Figures 10(a)–(c) show the 3D geometries of the CMEs as seen from STEREO-B, Earth, and the top of the second CME, respectively. The directional axes of the CMEs and the central axis of the ray are represented with the colored arrows. The directional axis of the second CME is closer to that of the ray. The separation angles between the ray and the first and second CMEs are 21° and 9°, respectively. This geometric comparison suggests that the second CME is the most likely parent CME of the ray. Webb et al. (2003) investigated the alignments of 27 post-CME rays with the parent CMEs, and 78% of rays were found to be coaxial on 2D image planes. Our results imply that the rays in the aftermath of CMEs may be coaxial with the parent CMEs in 3D space as well (see also, Patsourakos & Vourlidas 2011).

Figures 10(d)–(f) compare the orientations among the CMEs, ray, and PIL. The images were taken in the 195 Å passband of the Extreme UltraViolet Imager (EUVI; Wülser et al. 2004) on board STEREO-B, 193 Å passband image of Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) of Solar Dynamics Observatory (Pesnell et al. 2012), and magnetogram synoptic map (Carrington Rotation 2141) of Global Oscillation Network Group (GONG), respectively. The lines in yellow and orange are the intersection of the axes of the GCS models (thick lines in the top panels) with the solar surface. The circles refer to the points where the directional axes of the CMEs and the central axis of the ray intersect with the solar surface. The black and white lines represent the d_LOS-B and d_LOS-E, respectively, projected on the solar surface.

The orientation of the second CME is consistent with that of the PIL pointed by the arrow in panel (f). A rising loop system along the PIL was observed from around 09:50 UT as pointed by the arrow in panel (d), and it lasted the rest of the day. This loop system seems to be associated with the successive blobs observed with the white light observation from 18:08 UT to 00:08 UT, as shown in Figure 4. Panel (f) also indicates that the central axis of the ray (black circle) is close to the PIL, whereas the orientation is not consistent with that of the PIL. Note that we were not able to find the orientation of the ray directly in our method, implying a large uncertainty in the orientation of the ray. Our results support that this PIL is the source of the second CME and possibly the ray.