Sun-as-a-star Spectroscopic Observations of the Line-of-sight Velocity of a Solar Eruption on 2021 October 28

We derived the POS (from the STEREO-A viewpoint) velocity of the ejecta using a series of 304 Å images from STEREO-A observations. The EUVI images were rectified using secchi_prep.pro in the SolarSoftWare ⁴ package with the keyword rotated_on applied to rotate the solar north up in the images. We chose eight cuts (gray dashed lines in Figure 2(c)) along the directions of plasma motion. Then we derived the speeds of the ejecta along the eight directions from time-distance diagrams. The measurements are under the helioprojective-Cartesian (HPC) coordinate system in EUVI images. The HPC system owns a z-axis along the Sun-observer line and y-axis perpendicular to the z-axis in the plane containing both the solar north pole and the z-axis. The x-axis is perpendicular to both y-axis and z-axis, forming a right-handed coordinate system (see Thompson 2006 for more details).

No obvious acceleration was seen in the time-distance diagrams (Figure 2(d)), which can be attributed to the poor time cadence of EUVI as the initial acceleration process of the mass eruption was completed within 2 minutes. We then reasonably assumed that the ejecta moved along each cut at a constant speed. At each time step between 15:20 UT and 15:50 UT along each cut, the location of the moving plasma front was identified as the position where the intensity reaches its maximum value. For each cut, locations of the fronts form a track of the ejecta and we applied a linear fitting to the track to estimate the speed. The average velocity of the ejecta v_POS and its angle with respect to the +x-axis direction of the HPC system (i.e., Solar X in Figures 2(b)–(c)) were then computed using the following equations,

$$\begin{eqnarray}&&{v}_{\mathrm{POS}}=\displaystyle \frac{{\sum }_{i=1}^{8}{v}_{i}{P}_{i}}{{\sum }_{i=1}^{8}{P}_{i}}\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&\theta =\frac{{\sum }_{i=1}^{8}{\theta }_{i}{P}_{i}}{{\sum }_{i=1}^{8}{P}_{i}},\end{eqnarray} \tag{ 2 }$$

where v_i and θ_i are the speed along Cut i (i = 1, 2,..,8) and the angle between the +x-axis direction of the HPC system and the direction of Cut i, respectively. All angles are measured counterclockwise with respect to the +x-axis direction of the HPC system. P_i is the Pearson correlation coefficient that indicates the linear correlation between the front locations and the corresponding times. The identification of the ejecta fronts by maximum intensity values in the time-distance diagrams occasionally results in nonlinearity in the track of the ejecta, so we used P_i as weighting factors in Equations (1)–(2). The average POS velocity turned out to be around 587 km s⁻¹ with an angle of around −178 to the +x-axis direction of the HPC system.

We chose the spectra observed by SDO/EVE MEGS-B from 11:00 UT to 21:00 UT for analysis. First, we selected six spectral lines (listed in Table 1) for detailed analysis and checked line blends for each of them using the CHIANTI 10.0.1 (Dere et al. 1997; Del Zanna et al. 2021) database. Lines listed in Table 1 do not have obvious blends except for a red wing blend in Ne vii 46.52 nm with Ca ix 46.63 nm. Second, we calculated the intensity variation with time for each line. We applied a single Gaussian fitting (linear background for the continuum) to all the lines in Table 1 at each time step and obtained their line intensities. We defined a quiescent time, which was chosen to start from 11:00 UT to 13:00 UT when no obvious flare is observed (except for a small C2.2 flare at around 12:26 UT). The quiescent intensity level for each line was obtained by averaging the line intensities over the quiescent time period. The intensity variations for each line relative to their quiescent levels are displayed in Figure 3. The pre-flare relative intensities of all lines maintain a level around one, which validates the choice of the quiescent time to be reasonable. The intensities of all lines begin to rise right after the flare's onset. The lines formed below 10^5.8 K reach their peaks before the flare's peak, and the Ne viii 77.03 nm line with a formation temperature above 10^5.8 K reaches its peak around the time of the GOES soft X-ray peak. We noticed that almost all the spectral lines show dimming signatures (i.e., intensity dropping below the quiescent level) during and after the flare's gradual phase except for O iii 52.57 nm. We roughly quantified the dimming depth by averaging the decrease of the relative intensity from 19:00 UT to 21:00 UT when the flare almost ends. The dimming depths are marked in Figure 3, with positive values indicating dimming while negative enhancement. We suggest that the dimming is caused by the loss of plasma due to the mass ejection, in which the decrease of plasma density in the source region leads to the decrease of EUV emission. Signatures of CME-induced dimmings have been identified from EVE observations (e.g., Mason et al. 2014, 2016; Harra et al. 2016). In these previous observations, the dimmed lines usually have a minimum formation temperature of ∼10^5.8 K. Here we show that the dimming also exists in relatively cooler lines such as He i 58.43 nm, O v 62.96 nm, and O vi 103.1 nm, which suggests that some plasma in the lower atmosphere rises and erupts, resulting in the mass-loss dimming.

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We examined the line profile variations for each line and identified emission enhancement in the blue wing during the impulsive phase and after the flare's peak. For each spectral line, we obtained a reference line profile by averaging the line profiles during the quiescent time, and then subtracted the reference line profile from the observed line profiles to obtain the pre-flare subtracted line profiles. Temporal evolutions of the pre-flare subtracted line profiles of O iii 52.57 nm, O v 62.96 nm, and O vi 103.1 nm are displayed in Figure 4. In order to diminish the effect of the sympathetic filament eruption at around 15:35 UT (as described in Section 2), we only display the pre-flare subtraction analysis results from 15:23:30 UT to 15:34:30 UT. The wavelength λ was converted into the offset velocity v_c through the following equation,

$$\begin{eqnarray}&&{v}_{c}=\frac{\lambda -{\lambda }_{0}}{{\lambda }_{0}}c,\end{eqnarray} \tag{ 3 }$$

where c = 2.9979 × 10⁵ km s⁻¹ is the light speed in the vacuum; λ₀ is the rest wavelength of the line, which was computed by averaging the central wavelengths, obtained from the single Gaussian fitting during the quiescent time period. Figure 4 shows the enhancement peaks near the rest wavelength and in the blue wing at around 500 km s⁻¹ in each of the spectral lines, which we suggested was caused by the outward movement of the heated plasma. The pre-flare subtraction was also conducted for the other three lines and the variations in He i 58.43 nm and Ne vii 46.52 nm are similar to those in the oxygen lines. The emission enhancements at the blue wings of those lines suggest that the heated filament contains plasma with a temperature range of about 0.02 MK–0.5 MK (i.e., the formation temperature of He i 58.43 nm; the formation temperature of Ne vii 46.52 nm). However, the variations in the Ne viii 77.03 nm line are much more fluctuated than the others and more than one peak was detected. The existence of fluctuation in the Ne viii 77.03 nm line still needs further investigation.

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In order to further investigate the LOS velocity of the moving plasma, we analyzed the observed spectral profiles of the six emission lines during the flare. We found obvious secondary components in the blue wings of O iii 52.57 nm, O v 62.96 nm, and Ne vii 46.52 nm during the impulsive phase and shortly after the flare's peak. The spectral profiles of these lines at a specific time (15:29:30 UT) are shown in Figure 5. The small bulges in the blue wings suggest the presence of secondary components. To the best of our knowledge, this is the first time that a distinct secondary emission component caused by a mass ejection is detected in the Sun-as-a-star spectra. Through a single Gaussian fitting, previous investigations of several flares (see, e.g., Hudson et al. 2011; Brown et al. 2016; Chamberlin 2016; Cheng et al. 2019) all found shifts of line centroids that are mostly smaller than 100 km s⁻¹. In these observations, no obvious secondary components were seen. The obvious bulges in the blue wings of the spectral lines (as shown in Figure 5) facilitate the conduction of double Gaussian fitting to derive the LOS speed of the ejecta.

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We conducted the double Gaussian fitting for line profiles of O iii 52.57 nm, O v 62.96 nm, and Ne vii 46.52 nm observed during the impulsive phase and several minutes after the flare's peak. We adopted the following expression for the double Gaussian fitting:

$$\begin{eqnarray}&&{I}_{\mathrm{fit}}(\lambda )={b}_{0}+{i}_{1}{e}^{-\tfrac{{\left(\lambda -{\lambda }_{1}\right)}^{2}}{{w}_{1}^{2}}}+{i}_{2}{e}^{-\tfrac{{\left(\lambda -{\lambda }_{2}\right)}^{2}}{{w}_{2}^{2}}},\end{eqnarray} \tag{ 4 }$$

where b₀ was a constant background representing the continuum; i_n and λ_n (n = 1, 2) were the line peak intensity and central wavelength of the nth component, respectively. We followed the definition of exponential width in Tian et al. (2012) and used w_n (n = 1, 2) to represent the exponential width (width hereafter) of the nth component. We regarded the primary component as the first component, which is located near the rest wavelength and represents emission from the background solar atmosphere. The secondary (or second) component is located in the blue wing of the line, which results from the outward moving ejecta. We adopted the double Gaussian fitting procedure used by Tian et al. (2011). The procedure requires initial guess values of centroids, widths, and peak intensities of the two Gaussian components and their variation ranges as inputs and iterates until a global minimization of the difference between the observed spectrum and the fitted one is reached. The initial values of the parameters were set as follows. We took the minimum spectral irradiance in an observed line profile as the initial constant background b₀ and allowed it to vary from 75% to 125% of the initial value during the iterations. The initial central wavelengths of the primary and secondary components were set to be the rest wavelength and the wavelength corresponding to an offset velocity of −400 km s⁻¹, respectively. The central wavelength of the primary component was allowed to vary from −20 km s⁻¹ to 20 km s⁻¹, a rough range of the shifts of line centroids derived from single Gaussian fitting found in previous EVE observations (Hudson et al. 2011; Brown et al. 2016). The centroid of the secondary component was allowed to vary within the blue wing. The initial peak intensity values of the primary and secondary components were set to be the spectral irradiances at the offset velocities of zero and −400 km s⁻¹, respectively. The peak intensity of the primary component was allowed to vary within [75%, 125%] of the initial value, and that of the secondary component was allowed to vary from zero to the maximum spectral irradiance of the line profile. The initial widths of both components were set to equal the average width during the quiescent time, and their variation ranges during the iterations were [50%, 150%] and [50%, 125%] of the initial value, respectively. All the initial value settings were the results of trial and error. The uncertainty of the double Gaussian fitting coming from a large number of free parameters causes the failure of fitting, sometimes. We selected the valid fitting results based on the principle that the fitted profiles are located within the measurement error ranges of the observed profiles.

The double Gaussian fitting results at 15:29:30 UT for the three lines with obvious secondary components (i.e., O iii 52.57 nm, O v 62.96 nm, and Ne vii 46.52 nm) are shown in Figure 5, where we interpolated the wavelength points to make the fitted curves look much smoother. All three lines reveal a high-speed component with a velocity of 400–500 km s⁻¹. The fitting results for O v 62.96 nm at different time steps are shown in Figure 6. The double Gaussian fitting failed in the first 2 minutes after the flare's onset, so we only show the results from 15:26:30 UT to 15:34:30 UT. Our analysis indicates that the plasma at around 0.2 MK (the formation temperature of the O v 62.96 nm line) moves at a speed around 400 km s⁻¹ during the flare's impulsive phase and shortly after the flare's peak. The width of the primary component remains relatively constant, whereas that of the secondary component generally increases with time. This could be attributed to the expansion of the ejecta, which causes a wide velocity distribution and then results in a broadening of the line profiles. The variation of the peak intensity ratio of the two components shows that the secondary component reaches its peak at around 15:29:30 UT, which is ∼3 minutes before the flare's peak. The intensity ratio then gradually decreases. The velocities of the secondary components in all the three lines are plotted in Figure 8(a). We can see a slight increasing trend of the speeds from 15:26 UT to 15:30 UT in all the three lines, likely indicating the continuous acceleration process of the ejecta.

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Another method for estimating the velocity of the secondary component is to use the red-blue (RB) asymmetry of the line profile (see, e.g., De Pontieu et al. 2009; Tian et al. 2011). The RB asymmetry profile (hereafter, RB profile) was calculated from the following equation

$$\begin{eqnarray}&&{RB}({v}_{c})={\int }_{-{v}_{c}-\delta v/2}^{-{v}_{c}+\delta v/2}I(v){dv}-{\int }_{{v}_{c}-\delta v/2}^{{v}_{c}+\delta v/2}I(v){dv},\end{eqnarray} \tag{ 5 }$$

where v_c is the offset velocity computed using Equation (3) (here positive only) and δv is the velocity step in the RB profile calculation; I(v) is the observational or interpolated spectral irradiance around v_c . The RB profile can be normalized to the peak intensity of the interpolated spectrum. From the peak of the RB(v_c ) profile, we can infer parameters of the secondary component. We interpolated an observational line profile and adopted δv = 5 km s⁻¹ to obtain the RB profile at each time step. We regarded the velocity where the RB profile peaks as the LOS velocity of the ejecta. Figure 7 shows the RB profile of the O v 62.96 nm line at 15:29:30 UT. The RB profile peaks at around 380 km s⁻¹, which indicates a mass ejection moving toward Earth at the speed around 380 km s⁻¹. Due to the blend at the red wing of Ne vii 46.52 nm, we only conducted an RB asymmetry analysis for the O iii 52.57 nm and O v 62.96 nm lines. The velocities of the secondary component derived from the RB analysis are shown in Figure 8(a).

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We then calculated the average LOS velocity of the ejecta by averaging all the velocities obtained from different lines using both methods at each time step during the period of 15:26:30–15:34:30 UT. The average LOS velocity is 423 km s⁻¹ with a standard deviation of 40 km s⁻¹, which we take as the uncertainty of the LOS velocity.

We have obtained the POS velocity v_POS from the viewpoint of STEREO-A using time-distance diagrams in Section 3.1. We have also estimated the LOS velocity v_LOS using SDO/EVE observations around the Earth in Section 3.3. The full velocity of the bulk motion of the ejecta was then computed by combining these two velocities.

As the POS velocity is measured in the HPC system, we first converted it to the HEE system. We projected the POS velocity in the ecliptic plane (i.e., the x–y plane of the HEE system) and perpendicular to the ecliptic plane (i.e., the z-axis direction of the HEE system). The deviation of STEREO-A from the ecliptic plane, which is 0075 at 15: 30 UT on 2021 October 28, can often be ignored as it is less than around 013 throughout the STEREO era. Hence the z-axis of the HPC system is in the ecliptic plane. Because the z-axis is perpendicular to the x–y plane, the ecliptic plane containing the z-axis is perpendicular to the x–y plane of the HPC system. Therefore, the projections of the POS velocity v_POS in and perpendicular to the ecliptic plane are still in the x–y plane of the HPC system. The angle between the +x-axis direction of the HPC system and the ecliptic plane (denoted as α) and the angle between the POS velocity and +x-axis direction (i.e., θ in Equation (2)) can be directly added together to obtain the angle between the POS velocity and the ecliptic plane (denoted as θ_POSP). From the angle between the +z-axis of the HEE system and +x-axis of the HPC system, α was derived to be around 16, indicating that the +x-axis of the HPC system is north to the ecliptic plane. Therefore, θ_POSP turned out to be around −162, which is negative meaning that the POS velocity is south to the ecliptic plane. The projected speeds of v_POS in and perpendicular to the ecliptic plane can be calculated by,

$$\begin{eqnarray}&&{v}_{\mathrm{POSE}}={v}_{\mathrm{POS}}\cos | {\theta }_{\mathrm{POSP}}| ,\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{v}_{\mathrm{POSP}}={v}_{\mathrm{POS}}\sin | {\theta }_{\mathrm{POSP}}| ,\end{eqnarray} \tag{ 7 }$$

where v_POSE, around 564 km s⁻¹, is in the ecliptic plane and v_POSP, around 163 km s⁻¹, is perpendicular to the ecliptic plane. It is also worth noting that the direction of v_POSE is perpendicular to the Sun-STEREO-A line (i.e., the z-axis of the HPC system).

We combined v_LOS, v_POSE, and v_POSP to acquire the full velocity of the bulk motion of the ejecta. First, we calculated the projected speed of full velocity of the ejecta in the ecliptic plane. A sketch for the geometry used in the calculation is displayed in Figure 8(b). We denoted the projected speed in the ecliptic plane as v_E and the angle between the propagation direction of the ejecta and the Sun–Earth line in the ecliptic plane as θ_E. The angle between the Sun-STEREO-A line and the Sun–Earth line measured in the ecliptic plane (θ_STA) is around −375. So we have,

$$\begin{eqnarray}&&\left\{\begin{array}{c}{v}_{\mathrm{LOS}}={v}_{{\rm{E}}}\cos {\theta }_{{\rm{E}}}\\ {v}_{\mathrm{POSE}}={v}_{{\rm{E}}}\cos (90^\circ +{\theta }_{\mathrm{STA}}-{\theta }_{{\rm{E}}})\end{array}\right.,\end{eqnarray} \tag{ 8 }$$

from which we derived

$$\begin{eqnarray}&&\begin{array}{c}\frac{{v}_{\mathrm{LOS}}}{\cos {\theta }_{{\rm{E}}}}=\frac{{v}_{\mathrm{POSE}}}{\cos (90^\circ +{\theta }_{\mathrm{STA}}-{\theta }_{{\rm{E}}})}\\ \Longrightarrow {\theta }_{{\rm{E}}}=\arctan \frac{{v}_{\mathrm{POSE}}/{v}_{\mathrm{LOS}}+\sin {\theta }_{\mathrm{STA}}}{\cos {\theta }_{\mathrm{STA}}}.\end{array}\end{eqnarray} \tag{ 9 }$$

The projected speed in the ecliptic plane (v_E) is then obtained by Equation (8). Second, we acquired the propagation direction and the full speed of the bulk motion of the ejecta by combing the velocities in and perpendicular to the ecliptic plane. The projected speed perpendicular to the ecliptic plane (denoted as v_P) equals v_POSP; thus the full velocity of the bulk motion of the ejecta can be calculated as

$$\begin{eqnarray}&&v=\sqrt{{v}_{{\rm{E}}}^{2}+{v}_{{\rm{P}}}^{2}},{v}_{{\rm{P}}}={v}_{\mathrm{POSP}},\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{\theta }_{{\rm{P}}}=\arctan \frac{{v}_{\mathrm{POS}}\sin {\theta }_{\mathrm{POSP}}}{{v}_{{\rm{E}}}},\end{eqnarray} \tag{ 11 }$$

where v is the full speed of the ejecta and θ_P is the angle between the true velocity vector and the ecliptic plane. Negative θ_P indicates that the full velocity is south to the ecliptic plane. The direction of the bulk motion of the ejecta is described by θ_E and θ_P. A 3D interactive version of the full velocity calculation is available with a snapshot shown in Figure 8(c). All the velocity vectors are moved to the origin of the HEE system for plotting but it is worth noting that the ejecta originates from the AR, which is located near the central meridian in the southern hemisphere of the Sun. The full velocity of the bulk motion of the ejecta was found to be around 596 km s⁻¹ with a propagation direction of around 424 west to the Sun–Earth line in the ecliptic plane and 160 deviating to the south of the ecliptic plane.

The above derivation of the full velocity becomes unusable when ∣θ_STA∣ equals 90°. Under this condition, the POS velocity will lie in the x-z plane of the HEE system and the solution of the full velocity is not unique. Also, the uncertainty in the full velocity estimation becomes fairly large when ∣θ_STA∣ is close to 90°.