The density compression ratio of shock fronts associated with coronal mass ejections
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THE DENSITY COMPRESSION RATIO OF SHOCK FRONTS ASSOCIATED WITH CORONAL MASSEJECTIONS
Ryun-Young Kwon and Angelos Vourlidas College of Science, George Mason University, 4400 University Drive, Fairfax, VA 22030, USA The Johns Hopkins University Applied Physics Laboratory, Laurel, MD 20723, USA Also at IAASARS, National Observatory of Athens, GR-15236, Penteli, Greece (Accepted for publication in the
Journal of Space Weather and Space Climate ) ABSTRACTWe present a new method to extract the three-dimensional electron density profile and density compression ratioof shock fronts associated with Coronal Mass Ejections (CMEs) observed in white light coronagraph images. Wedemonstrate the method with two examples of fast halo CMEs ( ∼ − ) observed on 2011 March 7 and 2014February 25. Our method uses the ellipsoid model to derive the three-dimensional (3D) geometry and kinematics ofthe fronts. The density profiles of the sheaths are modeled with double-Gaussian functions with four free parametersand the electrons are distributed within thin shells behind the front. The modeled densities are integrated along thelines of sight to be compared with the observed brightness in COR2-A, and a χ approach is used to obtain the optimalparameters for the Gaussian profiles. The upstream densities are obtained from both the inversion of the brightnessin a pre-event image and an empirical model. Then the density ratio and Alfv´enic Mach number are derived. We findthat the density compression peaks around the CME nose, and decreases at larger position angles. The behavior isconsistent with a driven shock at the nose and a freely-propagating shock wave at the CME flanks. Interestingly, wefind that the supercritical region extends over a large area of the shock and last longer (several tens of minutes) thanpast reports. It follows that CME shocks are capable of accelerating energetic particles in the corona over extendedspatial and temporal scales and are likely responsible for the wide longitudinal distribution of these particles in theinner heliosphere. Our results also demonstrate the power of multi-viewpoint coronagraphic observations and forwardmodeling in remotely deriving key shock properties in an otherwise inaccessible regime. Keywords: shocks – coronal mass ejections – 3D reconstruction
Corresponding author: Ryun-Young [email protected]
Kwon & Vourlidas INTRODUCTIONShocks associated with Coronal Mass Ejections (CMEs) are one of the sources responsible for highly energeticparticles, called Solar Energetic Particles (SEPs; e.g., see reviews by Reames 1999; Desai and Giacalone 2016). Whileparticle acceleration by flares (see Reames 1999) is expected to occur in a limited volume (i.e., magnetic reconnectionsite), fast-mode coronal shocks are able to directly inject SEPs over a broad range in heliolongitudes (e.g., Cliver et al.1995, 2005). SEPs are an important component of space weather as the cause of radiation hazards to astronautsand satellites, orbital degradation of satellites, communication disruptions, and electrical blackouts (e.g., Balan et al.2014, and references therein). To better understand how coronal shocks accelerate particles over a wide range ofheliolongitudes and hence improve our ability to predict their intensity, duration and energy spectrum, it is importantto understand the properties of coronal shocks associated with CMEs and their temporal and spatial relationship withthe SEPs measured in interplanetary space (IP).In IP, the association between CME-driven shocks and particles is known for a long time thanks to direct in-situmeasurements of shocks and SEPs (e.g., Cane et al. 1988). This has been difficult in the corona because direct measure-ments are currently unavailable, although the situation will soon be remedied (McComas et al. 2016). Understandingthe formation and evolution of shocks in the corona is crucial for discriminating between flare and CME origins,particularly for the highest energy (GeV) SEPs ejected when CMEs are at heliocentric distances of around 2-10 solarradii ( R ⊙ ; Tylka et al. 2005).Coronal shocks are observed via remote-sensing observations in Extreme Ultraviolet (EUV), white light, and radioimaging and via spectroscopy (Vourlidas and Bemporad 2012). They form large-scale spherical fronts seen as halos bywhite light observations, or as EUV waves propagating against the solar coronal base (e.g., Kwon and Vourlidas 2017,see also recent reviews by Patsourakos and Vourlidas (2012),Warmuth (2015), and Long et al. (2017) for debates onthe nature of EUV waves). Occasionally, type II radio bursts over a large spectral range accompany these shocks,particularly in IP space (e.g., Gopalswamy et al. 2008), while metric Type II emission seems to originate from theflanks of shock waves close to the Sun (e.g., D´emoulin et al. 2012).There is a growing amount of evidence for the association of CME-driven shocks with SEPs in the corona (seerecent papers, e.g., Rouillard et al. 2012, 2016; Carley et al. 2013; Lario et al. 2014, 2016; Salas-Matamoros et al.2016, and references therein) but it remains unclear whether such shock waves are capable of accelerating particles atthe observed energies (Bemporad and Mancuso 2010, 2011). An important shock parameter that could be accessiblefrom coronagraphic observations is the ratio between the downstream and upstream electron densities or densitycompression ratio, X . According to the diffusive shock acceleration theory, the slope of the SEP intensity spectrumdepends only on the density compression ratio (e.g. Eq. (11) in Desai and Giacalone 2016). Therefore, measurementsof the compression ratio and its temporal evolution in the corona can go a long way in understanding SEP in situobservations in the inner heliosphere.Because the coronagraph images provide only the projected density, some model or knowledge of the density distri-bution along the line of sight (LOS) is necessary to obtain an estimate of the true volumetric electron density acrossthe shock. Ontiveros and Vourlidas (2009) were the first to extract estimates of the compression ratio for a number ofshock fronts in the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) field of view usingforward modeling techniques. However, those were single viewpoint observations and hence the shock reconstructionsmight be subject to considerable uncertainties.Here, we return to this issue employing multi-viewpoint observations from the Solar Terrestrial Relations Observatory( STEREO ; Kaiser et al. 2008) Sun-Earth Connections Coronal and Heliospheric Investigation (SECCHI; Howard et al.2008) COR2 coronagraph and more sophisticated forward models and analysis methods. We reconstruct the three-dimensional (3D) electron density at the shock sheath using an ellipsoidal forward model and populating it with athin shell of Gaussian distributed electron density. The upstream density is derived with two methods: using theLeblanc et al. (1998) model and via the inversion of a pre-event polarized brightness image. The method is applied totwo CMEs, on 2011 March 7 and 2014 February 25, associated with intense SEP events (Park et al. 2013; Lario et al.2016).The paper is organized as follows. In Section 2, we present the model and method used to determine the electrondensity distributions at the shock sheaths. In Section 3, we present the resulting sheath (downstream) and upstreamelectron densities and density ratios and infer Alfv´enic Mach numbers. Our summary and conclusion are given inSection 4. ensity structure in sheath Figure 1. (a) An excess brightness COR2-A image taken at 2011 March 7 20:24 (UT). The projection of the 3D shock front(green lines in panel (b)) on the image plane is shown with the dotted line. The diamond refers to the geometric center of theellipsoid model projected onto the same plane. Two examples of radial cuts (relative to the geometric center), used to take thebrightness profile, are shown by the two rectangles marked as Cut A and Cut B. Thick lines in the rectangles show the lengthof L used for the calculation of χ (see Figure 4). (b) The excess brightness in panel (a) with the 3D shock front (green lines).The 3D geometry was modeled with the ellipsoid model in Kwon and Vourlidas (2017). (c) The geometric relation among theSun, shell-like sheath, and LOS. A partial circle around the origin O is the solar disk. A shell-like sheath is represented in graycolor. Arrows in black, blue and red are the LOS, the projected shock normal on the image plane, and the actual shock normalin 3D, respectively. 2. METHOD2.1.
Data
The 3D reconstruction of these CMEs and shocks has been reported in Kwon and Vourlidas (2017) using the Grad-uated Cylindrical Shell (GCS; Thernisien et al. 2006) model and the ellipsoid model model (Kwon et al. 2014), re-spectively. The fits were based on three viewpoint observations from STEREO-A, -B and SDO (Solar DynamicsObservatory; Pesnell et al. 2012)/SOHO (SOlar and Heliospheric Observatory; Domingo et al. 1995). For the densityratio analysis, we use the COR2-A images because of their higher signal-to-noise ratio. The COR2 white light imagescapture Thomson-scattered photospheric light by the electrons in the corona in a field of view (FOV) of 2.5-15 R ⊙ .We do not analyze any COR1 images for these events because CME-associated shock waves are generally too faint inthese heights to allow extraction of profiles.2.2.
3D Model of Density Structure in Sheaths
Our density analysis method uses the 3D geometry of shocks. Figure 1 shows the geometric relation among the2D image plane, the spherical shock front in 3D and the LOS. Panel (a) shows an excess brightness due to the CMEand shock, for example, taken at 2011 March 7 20:24 (UT) by COR2-A. The halo front on the image plane (dottedline in panel (a)) is the projection of a bubble-shaped shock front (green lines in panel (b)). The 3D geometry ofthe bubble-shaped shock fronts is modeled with the ellipsoid model (Kwon et al. 2014). The sheath structure is givenas a shell behind the ellipsoid model. Panel (c) shows a cross-section of the shell-like sheath. The image plane ofobservations (panel (a)) is defined as the Y - Z plane, and the X -axis is towards the observer. The origin O is at Suncenter. L with an arrow in blue color is the distance normal to the projected shock front on the image plane. L ismeasured from the edge of the shock front on the image plane (dashed LOS arrow in this panel). Note that the shocknormal on the image plane differs from the actual shock normal in 3D. The shock normal in 3D is shown by arrows inred, and τ denotes the distance normal to the shock front in 3D from the edge of the shock where the electron densitydue to the shock begins increasing. The excess brightness on the image plane results from the integration over theLOS passing through the shell-like sheath.We would like to emphasize that the electron density jump along the 3D shock normal τ cannot be determineddirectly from the brightness profile along the projected shock normal L . As shown in Figure 1(c), a LOS passesthrough various τ , and the observed excess brightness is the integration over the LOS. A way to overcome this is tomodel the excess brightness I ′ ( L ) integrating the sheath electron density ( N e ( τ )) along the LOS ( s ) and compare it Kwon & Vourlidas -2 -1 0 1 2 3 4 τ / R O • ×10 N e ( τ ) [ c m - ] Upstream Downstream S hea t h F oo t + R a m p -4 -3 -2 -1 0 1 2 L/R O • I ′ ( L ) [ × - B O • ] -2 -1 0 1 2 3 4 τ / R O • ×10 N e ( τ ) [ c m - ] -4 -3 -2 -1 0 1 2 L/R O • I ′ ( L ) [ × - B O • ] -2 -1 0 1 2 3 4 τ / R O • ×10 N e ( τ ) [ c m - ] -4 -3 -2 -1 0 1 2 L/R O • I ′ ( L ) [ × - B O • ] (a) (b) (c) Figure 2.
The upper three panels show the dependency of the sheath electron density profiles ( N e ( τ )) on the three parameters, ρ e (panel (a)), d sheath (b), and d front (c), while the other two parameters are fixed. The lower panels are the modeled Thomson-scattering brightness profiles I ′ ( L ) in mean solar brightness ( B ⊙ ), resulting from the given N e ( τ ) in the same line-styles in theupper panels. Lines in solid, dotted, dashed, dash-dotted, dash-dot-dotted refers to the cases when ρ e = (1.6, 3.1, 4.7, 6.2, 7.8) × , d sheath = 1 R ⊙ , and d front = 0.25 R ⊙ in panel (a); ρ e = 6.2 × , d sheath = (0.6, 1.0, 1.4, 1.8, 8.0) R ⊙ , and d front =0.25 R ⊙ in panel (b); and ρ e = 6.2 × , d sheath = 1 R ⊙ , and (0.0, 0.1, 0.3, 0.4, 0.5) R ⊙ in panel (c). Because the Gaussian tailis not zero even at large distances, N e ( τ <
0) is not exactly zero, if d front = 0. By definition, we simply set N e ( τ <
0) = 0. The3D geometry of the shock used to calculate I ′ is shown in Figure 1(b). with the observed excess brightness I ( L ). Once we obtain the best I ′ ( L ) for I ( L ), N e ( τ ) along the shock normal isdetermined. To model N e ( τ ), we use Gaussian function that is generally used for the density structure of a wave, butallow asymmetry by using two Gaussian functions, namely, N e ( τ ) = ρ e exp − ( τ − τ ′ ) d , (1) d = d front , if τ < τ ′ ,d sheath , if τ ≥ τ ′ , with τ ′ = 2 d front √ , where ρ e and d are constants. τ ′ is the full width at half maximum of the Gaussian function with d front . The outermostedge of the density structure is defined at L = 0, so that N e ( τ <
0) = 0. ρ e is the peak electron density excess (densityjump).The upper panels in Figure 2 show the resulting sheath electron density distributions N e ( τ ) from various sets ρ e , d front , and d sheath . For instance, panel (a) shows N e ( τ ) obtained when varying ρ e while the other two parameters arefixed. In general, N e ( τ ) starts increasing at τ = 0, reaches its maximum at τ = τ ′ , and decreases gradually where τ> τ ′ . The key features of the function and the physical interpretations can be summarized as follows.1. If d front = d sheath , it becomes a linear wave.2. If d sheath is very large (dash-dot-dotted line in panel (b)) and d front = 0 (solid line in panel (c)), N e ( τ ) becomes astep function near the shock front, followed by a gradually decreasing tail. This profile can represent steepeningwaves and shock waves. ensity structure in sheath
53. if d front = 0 and d front < d sheath (panel (a)), it resembles the in-situ measurements and the theoretical expectationsof collisionless shocks (see a review by Treumann 2009). In this case, d front serves as the density in the foot andramp, with d sheath being the density in the sheath including overshoot and turbulence (if any).The real density distribution may deviate from our double-Gaussian model. Note, however, that the total numberof electrons will largely depend on the density excess ρ e , rather than the detailed shape of the function or fine scalestructures (e.g., foot and overshoot). Since we fit the density model to observed brightnesses that result from the LOSintegrations, we expect that the details of the profile is negligible when determining the density jump at shocks (seeFigure 5). Note that Equation (2) is similar to the function in Thernisien et al. (2006). Thernisien et al. (2006) usedthe function to reproduce the enhanced brightness due to the pileup plasma of an expanding flux rope, whereas thefunction is used for the density structure of waves and shocks in this paper.Since observations do not resolve density distributions along the shock normal as discussed above, we calculate theThomson-scattering brightness corresponding to the given density distribution N e ( τ ) to be compared with the actualobservation. We use the ellipsoid fits in Kwon and Vourlidas (2017) as the 3D geometry of the shock fronts. Giventhe 3D geometry as illustrated in Figure 1(c), the 3D coordinates of all points along the LOS are known, and thus τ along the LOS is also known.Once the sheath electron distribution N e ( τ ) and the coordinates of the LOS are given, Thomson-scattering bright-nesses resulting from the integration can be calculated using the standard Thomson-scattering equations (e.g. Billings1966). We assume that the sheath electron density structure varies slowly along the shock surface so that a singledensity distribution model can represent the density structure for the region where the LOS are passing through (seeFigure 1(c)). This assumption is valid because the fits of the model to observations are done only close to the shockfront (see Section 3.1).The lower panels of Figure 2 show the calculated Thomson-scattering brightnesses I ′ ( L ) from electron densitydistributions N e ( τ ) given in the upper panels, together with the 3D geometry of the shock shown in Figure 1(b). Notethat the shapes of I ′ ( L ) differ from the sheath density distributions N e ( τ ). For instance, I ′ ( L = 0) at the shock frontis zero, although the electron density has the maximum at the shock front (see the solid line in Figure 2(c)). Thesedifferences are due to the integration along the LOS.As shown in Figure 2, the shape of I ′ varies with the density model N e ( τ ) for the given geometries. In this sense,we find the optimal parameters of N e ( τ ) by minimizing χ defined as χ = 1 m (cid:18) I ( L ′ − ∆ L ) /I ′ ( L ) − w (cid:19) , (2)where I and I ′ are the observed and modeled excess brightnesses, respectively. m is the total number of the pointscompared, and w is the weight function. ∆ L is the displacement on the sky plane of the true shock front from theinitially estimated one, i.e., L = L ′ - ∆ L . Because of this, our fit has the four parameters, ρ e , d front , d sheath , and ∆ L .Note that the difference between I and I ′ is normalized by I ′ . Instead of the error in I , we use a weight function w for the denominator, and the weight function is defined as, w = " (cid:18) | L | R ⊙ (cid:19) / − . (3)2.3. Upstream Density Profile
We estimate the upstream electron density ρ u via inversion of a pre-event polarized brightness image. The method as-sumes an axisymmetric density distribution that can be described with an n th-order polynomial function (van de Hulst1950; Hayes et al. 2001). We use the polynomial form in Leblanc et al. (1998), ρ u ( r ) = A (cid:18) rR ⊙ (cid:19) − + B (cid:18) rR ⊙ (cid:19) − + C (cid:18) rR ⊙ (cid:19) − , (4)where A , B , and C are constants. The constants are found by the fit to the observed profiles at each position angle.Equation (4) is essentially equivalent to the standard polynomial expansion used traditionally in the field. Note thatit reduces the number of constants while it includes the higher-order terms. Kwon & Vourlidas
Cut ACut B20:24 (UT) 20:39 (UT) 20:54 (UT) - - - - Figure 3.
Excess brightness images used to determine the density excess ρ e . The upper and lower panels show two differentevents on 2011 March 7 and 2014 February 25, respectively. Dots in red are the points where the ρ e measurements have beenmade. Solid lines refer to the projected shock front determined with the ellipsoid model. Dashed lines represent the ±
4% errorin the geometry of the shock fronts.
Because of the presence of F-corona polarized component, especially above 5 Rs (Hayes et al. 2001), the determined ρ u in our height range would be overestimated. In contrast, the measured ρ u in coronal holes could be suppressed because of the calibration (e.g., Hayes et al. 2001; Thompson et al. 2010). We have checked other empirical coronalelectron density models that have been used widely, i.e., Newkirk (1961), Saito et al. (1977), and Leblanc et al. (1998),to infer ρ u . Since the Leblanc et al. (1998) model provides the lowest electron density among them, we use as alower limit. In this way, the upstream electron density ρ u is obtained as a range between our measurement and theLeblanc et al. (1998) model. 2.4. Error in Electron Density Excess
The main sources of error in ρ e are the uncertainties of brightness given by white light observations, the LOSintegration relying on the ellipsoid model, the Gaussian description of the density distribution, and the relationbetween the white light structure and the true shock.First of all, we use the excess brightness to determine ρ e . An excess brightness image is obtained by subtracting aprevious image, which is temporally closest but at least 30 minutes apart from the image. In this manner, it is expectedthat the noise level, including the calibration error, the F-corona polarized component, the stray light component, andthe brightness due to the background electron density, is similar in the two images, since they are temporally close.In this sense, the error in the excess brightness would be less than the error (20%; Frazin et al. 2012) of the directbrightness images. In addition, the CME and shock structure in the previous image cannot affect the excess brightnessclose to the shock front, because the two images are at least 30 minutes apart. Note that the shocks/waves in whitelight observations are generally faint and diffuse (e.g., Sheeley et al. 2000), so that the signal-to-nose level is still aissue. We use the COR2-A images because of their higher signal-to-noise ratio than the COR2-B images. ensity structure in sheath ± δ ) in the 3D geometry, wehave the three ellipsoid models (see dashed lines in Figure 3), and the lower and upper estimates of the 3D geometryare used to estimate the error in ρ e .The sheath electron density distribution model in Equation (2) is also highly idealized, and the real density distri-bution may deviate from our model. However, the brightness is obtained by taking the integration of the density overthe LOS. It is obvious that the small scale structure will not affect much the total number of electrons and, therefore,the observed brightness. As we will see in Section in 3.1, the most significant source causing the brightness increase isthe peak electron density excess ρ e (density jump).The imperfection of the background subtractions will result in error in excess brightness. If we assume that the true ρ e varies slowly along the shock surface, the error can be estimated by comparing ρ e with those in the neighboringposition angles. We repeat the same analysis for every position angles and take the average and standard deviationover 7 ◦ . The standard deviation serves as the error. Note that we discard the cases where the brightness profiles arecontaminated by the following pileup plasma or deflected streamers.As discussed above, the two errors in ρ e are given from the uncertainties of the ellipsoid model and the excessbrightness. We use the larger one. RESULTS AND DISCUSSIONWe apply our method to two fast CMEs on 2011 March 7 and 2014 February 25. The details of the 3D reconstructionswith the ellipsoid model are given in Kwon and Vourlidas (2017) but we summarize the modeling results here forcompleteness. The shock speeds in 3D are ∼ − and ∼ − for the March and February events,respectively. The minimum angular widths of the shock envelopes are 192 ◦ and 252 ◦ , whereas the CME angular widthsare 58 ◦ and 90 ◦ , respectively. The shock speeds vary with position angle. The maximum speeds are seen near theCME noses, in which the speeds are well correlated with the CME nose speeds. The speeds in the far-flanks tendtowards the local fast magnetosonic speed.This 3D modeling considered only the outline of the CME and its outer shock envelope. Here, we take the nextstep and attempt to fit the observed brightness of the shock by modeling the shock envelope as a thin shell withsome electron density distribution. Thus we can determine the density excess ρ e at the sheaths and the downstream–upstream density ratios X , using the 3D geometry of the shocks. The 3D geometries and kinematics and the estimatedelectron densities enable us to also estimate the Mach numbers M A and upstream Alfv´en speeds v A .3.1. Density Excess ρ e at Shock Front Figure 3 shows the excess brightness COR2-A images for two events on 2011 March 7 (upper panels) and 2014February 25 (lower panels). The previous images (at least 30 minutes apart) are subtracted from these images, sothe brightness is largely due to the electrons in the CME and sheath. To derive the density structure of the sheathwe take radial cuts (relative to the geometric center of the shock front; see the diamond symbols in the first panel ofFigure 3) along all position angles at 1 ◦ intervals. The position angle, ζ , is measured counterclockwise from the shockleading edge (directional axis of the ellipsoid model projected onto the image plane). The two rectangles in the firstpanel of Figure 3 (see also Figure 1(a) zoomed in on the shock front) are two example radial cuts, far from and closeto the CME nose (Cut A and Cut B, respectively). To increase the signal-to-nose ratio, we average the brightnessprofiles across the width of the rectangle. The width ω is chosen as, ω = 2 e r sin θ , where e r is the average distanceof the projected shock front from its geometric center, and θ = cos − (1 − λ/e r ). λ is the distance corresponding to0.5 pixel. In this way, the difference in distance L of the curved shock front across the width is at most 0.5 pixel, andthus the curvature is negligible when taking the average over the width.The upper and lower panels of Figure 4 show the excess brightness profiles of Cut A and Cut B, respectively. Sincethe background brightness has been subtracted, the profiles are flattened, and the brightness in the region where L (= L ′ − ∆ L ) > Kwon & Vourlidas -4 -3 -2 -1 0 1 2( L ′ - ∆ L )/ R O • B r i gh t ne ss [ × - B O • ] -4 -3 -2 -1 0 1 2( L ′ - ∆ L )/ R O • B r i gh t ne ss [ × - B O • ] Cut ACut B
Figure 4.
Excess brightness profiles I ( L = L ′ − ∆ L ) taken from Cut A (upper panel) and Cut B (lower) shown in Figure 3.The dotted line in each panel indicates the zero brightness. The red line in each panel is the modeled excess brightness profile I ′ ( L ) obtained from χ fits shown in Figure 5. The blue lines indicate the slope of the observed brightness profiles. The twodashed lines in each panel demarcate the region where the χ values are calculated. ρ e [ × cm -3 ]0.51.01.52.0 d s hea t h / R O • ρ e [ × cm -3 ]0.000.050.100.15 d f r on t / R O • ρ e [ × cm -3 ]-8-6-4-20246 ∆ L [ p i x e l ] d sheath / R O • d f r on t / R ρ e [ × cm -3 ]0.51.01.52.0 d s hea t h / R O • ρ e [ × cm -3 ]0.000.050.100.15 d f r on t / R O • ρ e [ × cm -3 ]-6-4-20246 ∆ L [ p i x e l ] d sheath / R O • d f r on t / R (a) (b) (c) (d)(e) (f) (g) (h) Figure 5.
Fits of the sheath density distribution model to the observed brightness profiles of Cut A (upper) and Cut B (lower).From the leftmost to the rightmost, the panels show the χ values varying with the parameters, d sheath – ρ e , d front – ρ e , ∆ L – ρ e and d front – d sheath , respectively, and the χ values are indicated by the color bars. Dashed lines in each panel indicate the parametersat the minimum χ . is only for the excess brightness due to the shock, the brightness profiles also contain the brightnesses owing to theplasma pileup ahead of the following CME and deflected streamers.In order to discriminate the shock part from the following pileup plasma in the excess brightness profiles, we use theslope of the profiles. The brightness at the outermost part ( L ∼
0) is expected to be mostly due to the density jump, ensity structure in sheath ζ = -80 ° r/R O • N e [ × c m - ] ζ = -80 ° r/R O • N e [ × c m - ] ζ = -80 ° r/R O • N e [ × c m - ] Figure 6.
Evolution of the sheath taken at Cut A ( ζ = 80 ◦ ) of the 2011 March 7 event (rectangles in the top panels of Figure3). From the top to bottom, each panel shows the modeled sheath electron density distribution plus the background electrondensity at 20:24 UT, 20:39 UT, and 20:54 UT, respectively. Note that the ranges of the abscissas are different. but the following pileup plasma enhances the brightness. The top panel in Figure 4 shows the case that the brightnessprofile is taken at a position angle far away from the CME. While the brightness due to the pure sheath (red line; seeFigure 5 and the description) is expected to decrease at large distances ( | L | ), the observed brightness increases withdistance. Since the CME is far away from the shock front at this position angle, the effect of the pileup plasma on thebrightness profile would be gentle. As indicated by the three blue lines in this panel, it results in the slower increasein brightness in the intermediate part (second blue line; -2 ≤ L ≤ -0.6) than the outermost part (third blue line; 0 ≤ L ≤ -0.6). Since the innermost part (first blue line) is mostly due to the pileup plasma, the gentle slope drops off intoa steep slant as the distance increases. When the CME is close to the shock front, as seen in the lower panel of Figure4, the slope keeps increasing with distance. The three-part slope enables us to discriminate the part due to the shockand minimizes the effect of the following pileup plasma on the fits. We only use the first increase part as demarcatedby the two dashed lines in each panel (see also the thick lines in the rectangles in Figure 3). The red dots in Figure3 refer to the position angles where we have been able to determine ρ e using χ minimizations. In the position angleswhere the following CME is too close, we simply discard them.Figure 5 describes our χ approach to obtain the optimal parameters of the double-Gaussian density distributionmodel from the observed brightness profiles I in Figure 4. The best model brightness profiles I ′ are shown as thered lines in Figure 4. We perturb the parameters ( ρ e , d front , d sheath , ∆ L ), obtain the brightness I ′ ( L ), and thencalculate χ comparing I ′ ( L ) with the observed brightness I ( L ), as discussed in Sec. 2.2. The various density modelsare obtained by the various sets of parameters which are given with the intervals of ∆ ρ e = 0.5 N Leblanc ( r ), ∆ d front =0.1 R ⊙ , ∆ d sheath = 0.5 R ⊙ and ∆∆ L = 0.02 R ⊙ . The full ranges of the input parameters are shown in Figure 5.Note that ∆ ρ e shown is normalized to the Leblanc model at the corresponding height r because the electron density0 Kwon & Vourlidas -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0.51.01.52.02.53.0 ρ e [ × c m - ] r/R O • ρ e [ × c m - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0.51.01.52.02.53.0 ρ e [ × c m - ] r/R O • ρ e [ × c m - ] Figure 7.
Determined electron density excess ρ e for the 2011 March 7 and 2014 February 25 events. The results were obtainedat three different times for each event, and they are shown in different colors, black, red, and cyan in chronological order (20:24UT, 20:39 UT, and 20:54 UT for the 2011 March 7 event and 01:24 UT, 01:39 UT, and 01:54 UT for the 2014 February 25event). The upper row shows the ρ e distributions over position angle ζ . The position angle that ζ = 0 ◦ is at the projecteddirectional axis of the ellipsoid model. The lower row shows the ρ e distributions over height. Lines in solid and dashed representtwo empirical electron density models in Leblanc et al. (1998) and Saito et al. (1977), respectively. in the corona varies in several orders of magnitude. Once the minimum χ is obtained, we calculate χ again with thedifferent intervals of the parameters. The full ranges of the parameters used for the second calculation are given as ± ′ , where ∆ ′ is the initial intervals, centered on the parameters at the minimum χ . The intervals of the parametersare ∆ ρ e = 0.1 N Leblanc ( r ), ∆ d front = 0.02 R ⊙ , ∆ d sheath = 0.1 R ⊙ , and ∆∆ L = 0.02 R ⊙ . Not only the minimum χ values, we check the resultant profiles visually as shown in Figure 4.The dependency of χ on the four parameters ρ e , d sheath , d front , and ∆ L is shown in Figure 5. The top and bottompanels are χ values for Cut A and Cut B, respectively. Since χ is normalized by the model brightness I ′ , the minimum χ value is less than unity (see color bars). As shown in Figure 2(a), the value and the shape of I ′ are sensitive to ρ e .Because of this, as shown in Figures 5(a)-(c) and 5(e)-(g), χ converges at a ρ e . It is also shown that d front = 0 (panels(b), (d), (f) and (h)), being consistent with the density profile of shock waves. It may imply that the shock thicknessis very small and thus cannot be resolved in our observation and method. It is evident that d sheath (panels (a) and(e)) and ∆ L (panels (c) and (g)) do not affect the measure of ρ e . The low χ values (black colors) are aligned alongthe best ρ e value (see the vertical dashed lines in these panels). It demonstrates that our χ approach can provide thereasonable measure of ρ e .We repeat the χ fitting at all position angles. Since it is applied also in the time series, the temporal evolution ofthe sheath structures can be investigated. Figure 6 shows the time evolution of the sheath propagating at the positionangle ζ = 80 ◦ (Cut A). The position angle in three different images is shown in the upper panels of Figure 3. The ensity structure in sheath r/R O • ρ u [ × c m - ] r/R O • ρ u [ × c m - ] Figure 8.
Derived upstream electron densities ρ u for the 2011 March 7 and 2014 February 25 events. The densities determinedat three different frons are shown in black, red, and cyan, in accordance with Figure 7. Lines in solid and dashed represent twoempirical electron density models in Leblanc et al. (1998) and Saito et al. (1977), respectively. background density in this figure has been determined independently in Section 3.2. As the shock propagates outward, ρ e falls off.Figure 7 shows the derived ρ e for the 2011 March 7 and 2014 February 25 CMEs. The upper panels show ρ e versus ζ for three consecutive times (marked by the different colors). Because of the shock–streamer interaction at 20:24 UT on2011 March 7 (see the arrow in Figure 3), we do not attempt to derive ρ e beyond ζ ∼ ◦ . As expected, ρ e decreaseswith time, and tends to maximum around the shock leading edge ( ζ ∼ ◦ ). In the lower panels, we plot ρ e versusheliocentric distance for all position angles. ρ e decreases with height but remains high at the leading edge (markedby the downward arrows). This is a clear indication, at least to us, that the super-magnetosonic CME keeps drivingshocks and it results in the higher density jump than the lateral parts that are likely decoupled from the driver andturning into a freely-propagating shock wave (Kwon and Vourlidas 2017). Kwon and Vourlidas (2017) showed thatthe speed of the shocks ahead of the CME noses is well correlated with the CME nose speeds while the shock speedin the far-flanks is not. The far-flank speeds tend rather towards the local fast magnetosonic speed. For comparison,the empirical density models of Saito et al. (1977) and Leblanc et al. (1998) are shown by the dashed and solid lines,respectively.The errors in ρ e are shown by error bars in Figure 7. To determine the errors, we take into account the two sourcesof the error in ρ e as described in Section 2.4. First, the error can arise from the uncertainty of the excess brightness.Assuming that ρ e varies slowly along the shock surface in the real corona, the fluctuations in ρ e determined within asmall range of ζ could be due to the uncertainty. We have repeated our fits for every 1 ◦ intervals of ζ independently,and we take the average and standard deviation over ζ spanning 7 ◦ . The other source of error is the 3D geometry ofthe shock modeled with the ellipsoid. The error of the ellipsoid model has been determined as shown in the Appendix2 Kwon & Vourlidas -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] r/R O • X S peed [ × k m s - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] -150 -100 -50 0 50 100 150Position Angle ζ [ ° ]0123456 X S peed [ × k m s - ] r/R O • X S peed [ × k m s - ] Figure 9.
Density ratios of downstream over upstream X = ρ d / ρ u . The left and right columns are the results for the 2011March 7 and 2014 February 25 events, respectively. The three rows from the top show X over position angle ζ . The bottompanels show the determined X over height. Solid lines in black, red and cyan are the speeds of the shock fronts derived fromthe 3D ellipsoid model, in accordance with the ordinate on the right-handed side. of Kwon et al. (2014). The error is ∼
4% for all images we have analyzed. The two additional ellipsoid modelsconsidering ±
4% error are obtained, and their projections on the images are shown as dashed lines in Figure 3. Werepeat the same χ calculation with these ellipsoid models, and the results are considered as the error in ρ e due to the3D geometry of the shock. Once the two errors are calculated, we use the larger one for the error in ρ e . ensity structure in sheath Density Compression Ratio
The density compression ratio, X = ρ d / ρ u , where ρ d = ρ e + ρ u , can only be determined if ρ u is known.We determine ρ u using the polarized brightness images prior to each event as described in Section 2.3. In practice,we have used the polarized brightness images observed from 00:08 UT to 14:08 UT on 2011 March 7 and from 19:08UT on 2014 February 24 to 00:08 UT on 25. The standard polarized brightness inversion method is applied to theaveraged polarized brightness images. Figure 8 shows the determined ρ u along the shock fronts where ρ e has beendetermined. Given the inhomogeneity of the background corona, the derived ρ u varies considerably along the shockfronts resulting in the ‘U-shaped’ curves. The errors in ρ u are determined from the errors in r due to the uncertainty ofthe 3D geometry of the shock. The errors in r and ρ u are shown by the horizontal and vertical error bars, respectively.Because of the uncertainty in the background electron density (e.g., Hayes et al. 2001), we also employ an empiricalcoronal electron density model of Leblanc et al. (1998) that provides the lowest values among the other empiricalmodels (solid lines in Figure 8). In general, measured ρ u is greater than the Leblanc model probably due to F-coronapolarized component, but a portion of the 2011 March 7 shock propagates in the northern polar coronal hole in thesky planes, and the measured ρ u in this region is lower than that of the Leblanc model due to the calibration (e.g.,Thompson et al. 2010). The measure of electron density in coronal holes is even more uncertain (e.g., Hayes et al.2001). We also consider the error in r when taking ρ u from the empirical model. We use the full ranges covering theobservations and empirical model as well as their errors, as the upstream electron densities ρ u .Figure 9 shows the ratios between the downstream and upstream electron densities, i.e., compression ratios X . Thespeeds are also shown in colored lines for the fronts where X is determined. The error bars are the propagation errorsgiven by the errors in ρ e and ρ u . The compression ratio peaks around the shock leading edge, as also seen in ρ e inFigure 7. Note that there is no dramatic increase in X around the shock leading edge of the 2014 February 25 shock asthe 2011 March 7 event. It may be because of the upstream electron density ρ u and the proximity of the shock frontto the CME. As seen in Figure 8, ρ u at the leading edge of the February event is lower than that of the March event.The 3D angular distances between the CME noses and the the projected shock leading edges are ∼ ◦ (2011 March7) and ∼ ◦ (2014 February 25). In addition, X seems to be correlated with speed. However, it is interesting to notethat X is nearly constant in time (cf. Bemporad and Mancuso 2011), while the speed decreases with time, as clearlyseen in the bottom panels. Figure 6 indicates that this is due to the decrease in ρ u with height (see also Figure 8). Asthe shocks propagate outward, ρ e is falling off together with ρ u . Table 1 shows the full ranges of the determined X with the averages.The ratios X in Figure 9 are found to be larger than 4 around the shock leading edge. The upper limit of X isgiven as ( γ + 1) / ( γ − γ is the adiabatic index, and the upper limit is 4 if γ = 5/3. It might imply thatthe adiabatic index γ could be close to 1 in the low corona (Van Doorsselaere et al. 2011). If γ = 4/3, for example,the upper limit is 7. Alternatively, the regions where X > ρ e and/or ρ u , and the actual X is less then 4. As discussed in Section 3.1, the brightness profiles taken closeto the shock leading edge could be slightly contaminated by the pileup plasma of the following CME.Several attempts have been made in the past literature to determine the density ratios X from white light(Ontiveros and Vourlidas 2009; Bemporad and Mancuso 2010, 2011) and EUV (Kozarev et al. 2011; Kouloumvakos et al.2014; Long et al. 2015), and radio observations (Ma et al. 2011). Ontiveros and Vourlidas (2009) determined X , rang-ing between 1.2 and 2.8 for 11 CMEs that were faster than 1500 km s − observed by LASCO C2 coronagraph.Bemporad and Mancuso (2010, 2011) also used the LASCO C2 images and found that X ≈ X values are consistent with past results, their temporal evolution differs in that the values aremore or less constant in contrast to the results in Bemporad and Mancuso (2010, 2011). Bemporad and Mancuso4 Kwon & Vourlidas -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n -150 -100 -50 0 50 100 150Position angle ζ [ ° ]1.01.52.02.53.0 M θ B n Figure 10.
Mach numbers M A (shaded regions) derived from the full ranges of X in Figure 9. Red lines indicate the criticalMach number for the determined θ Bn . θ Bn is also plotted by dashed lines (see the y -axes on the right-handed side). (2011) showed that the peak X value declines from ∼ ∼ ∼ R ⊙ to ∼ R ⊙ . Note that our CMEs are very fast, and the speeds at the times of the latest images are still over 1500 km s − .Kwon and Vourlidas (2017) showed that the speeds of these driver CMEs are much faster than the local fast-modewave speeds and are able to generate shocks during our measurements. The fast speeds may be responsible for thehigh X even in higher altitudes. 3.3. Alfv´enic Mach Number, M A To relate our results to modeling effort and past literature, we proceed to compute the Alfv´enic Mach number, M A ,under the assumption of γ = 5/3. M A can be estimated from the compression ratio, X (e.g., Bemporad and Mancuso2011, and references therein). M A is a function of plasma β and θ Bn , and we could simplify to the case of β → θ Bn is the angle between the magnetic fields and the shock. Weassume that the coronal magnetic fields are purely radial in our height range and determine θ Bn using the ray-liketrajectories of the shocks shown in Kwon and Vourlidas (2017). The Mach number of an oblique shock is given by(e.g., Bemporad and Mancuso 2011, and references therein) M A = ( M A ⊥ sin θ Bn ) + ( M A k cos θ Bn ) , (5)where M A ⊥ = [0 . X ( X + 5) / (4 − X )] / and M A k = X / , assuming the adiabatic γ = 5/3. ensity structure in sheath r/R O • v A ( r ) [ k m s - ] r/R O • B ( r ) [ G ] Figure 11.
Upper panel: Local Alfv´en speed, v A (plus symbols), inferred from the Mach numbers in Figure 10. Dashed anddash-dotted lines refer to v A that are modeled with the empirical electron density models of Leblanc et al. (1998) and Saito et al.(1977), respectively, together with the empirical magnetic field function of B = 2.2( r/R ⊙ ) − . Lower panel: Magnetic fieldstrength B inferred from v A in the upper panel together with the determined ρ u . Lines in dashed and dash-dotted are radialmagnetic field profiles given in Mann et al. (1999) above the quiet Sun and Dulk and McLean (1978) above an active region,respectively. Figure 10 shows the derived M A from the full ranges of X in Figure 9. Dashed lines are the estimated θ Bn . Becauseof the assumption of purely radial magnetic fields, θ Bn decreases monotonically with time. The full ranges of M A andthe averages are given in Table 1. Similarly to X , M A seems to be more or less constant in time, which is contrary tothe prevailing wisdom (as shown in Bemporad and Mancuso 2011). This does not seem unreasonable since the localAlfv´en speed should decrease with height. As a shock propagates outward the shock speed should also fall, and itresults in a (relatively) constant M A since M A = V sh / v A (Figure 11).Since M A = V sh / v A , we can use the Mach number to estimate the Alfv´enic speed across the shock front. To thatend, we need to move to the shock frame by estimating the ambient solar wind speed, V SW . Then, the true shockspeed V sh = V ′ sh - V SW · cos θ Bn , where V ′ sh is the measured one. Since V SW measurements does not exist in these coronalheights, we turn to a formula given in Sheeley et al. (1997), V SW = 2 A ( r − r ) , (6)where A = 3.6 m s − and r = 2.1 R ⊙ . In this manner, v A can be estimated and is shown in the upper panel of Figure11. The upstream v A decreases with height and time but our estimates are higher than those predicted by empiricalmodels of density and magnetic field (see the dashed and dash-dotted lines). This may not be unreasonable since ourtwo case studies are very fast CMEs with speeds higher than 2000 km s − . Note that the magnetic field strengthsinferred from the determined v A and ρ u in the lower panel of Figure 11 are consistent with the empirical models given6 Kwon & Vourlidas in Mann et al. (1999) and Dulk and McLean (1978). We need further investigations including slower CMEs to see thegeneral characteristics of shocks associated with CMEs.In addition, plasma β can also be determined from the estimated upstream v A by a relation that β = (2/ γ )( c S / v A ) ,if sound speed c S is known (e.g., Kwon et al. 2013b). We obtain that β = 0.06 ± γ = 5/3 and c S = 200 km s − . It is consistent with the low β assumption for the corona widely used in this field.3.4. Physical Implications for SEP Acceleration
One of the challenges in SEP research is to explain the wide longitudinal distribution of SEPs originating in a singleflare and CME event (Desai and Giacalone 2016, and references therein). The large widths of SEP-associated CMEsare the obvious candidate. While the spatial and temporal relationships between the CME-driven shock wave and SEPinjections seem to support this assertion (e.g., Rouillard et al. 2012, 2016; Lario et al. 2014, 2016), it remains unclearwhether the shocks are capable of accelerating particles at these locations since the properties of the local plasmaenvironment (i.e., seed particle populations, turbulence levels) in the coronal heights are largely unknown. However,we have presented a method that allows us to derive the electron density distribution (in 3D) in the shock and henceobtain the density compression ratio and consequently the Mach number, Alfv´en speed and other parameters (underfurther assumptions). So we can, at least, deduce whether, and most importantly where, our shocks have the potentialto accelerate particles by comparing our Mach numbers to the critical Mach number, M c , (e.g., Bemporad and Mancuso2011).Red lines in Figure 10 show the critical Mach number corresponding to the estimated θ Bn . The critical Mach number M c for a collisionless shock is taken from Figure 2 in Treumann (2009), for the case of β ≈ β< M A > M c ) over significant portion of their extent. We also seethat the condition extends for longer periods than those reported by Bemporad and Mancuso (2011). Note that ourcase events are associated with longitudinally-wide SEP events (Park et al. 2013; Richardson et al. 2014; Lario et al.2016). Our work here provides significant additional support to these previous works that the SEPs are produced atthe CME shocks and that the wide extent of these shocks is the reason for the distribution of SEPs over a very widerange of heliospheric longitudes. SUMMARY AND CONCLUSIONWe present a new method that, using multi-viewpoint observations and forward modeling techniques, enables usto model the observed brightness of shock fronts in white light coronagraphic observations and extract the three-dimensional electron density distribution across the fronts. We apply the method to two case studies; the CME eventson 2011 March 7 and 2014 February 25. Both CMEs were fast ( > − ) and wide ( ≥ ◦ ) halo events. The 3Dreconstructions of the CMEs and their shock envelopes were reported in a separate study (Kwon and Vourlidas 2017).The shock envelope is based on an ellipsoidal fitting and its white light emission is assumed to arise from electronsdistributed in a thin shell over the surface of the shock. The electron distribution is assumed to be a double-Gaussiandescribed by three free parameters. By varying the parameters and integrating the resulting density profiles along theknown (from the ellipsoid model) LOS, we can derive modeled brightness profiles to be compared against the observedones. We locate the best set of free parameters via a χ minimization approach. We use two methods to estimatethe background density; the empirical Leblanc et al. (1998) model and the standard polarized brightness inversiontechnique using the pre-event polarized brightness images. In this way, we obtain an upper and lower estimate for thebackground density and thus we can bound the resulting density compression ratio. The final results of this exerciseare the excess electron density ρ e , density ratio X and Alfv´en Mach number M A across the full shock front. This is thefirst time that the properties of white light shocks are quantified in this way. Our results corroborate past estimates,mostly based on single viewpoint observations (Ontiveros and Vourlidas 2009; Bemporad and Mancuso 2010, 2011)and are fully consistent with the presence of radio emissions and SEPs in the two cases.Our findings can be summarized as follows,1. The density excess, ρ e peaks at and around the CME leading edge and the peak is maintained during the timeinterval we analyzed (Figure 7). This finding indicates that the two CMEs drive bow-type shock waves aroundtheir noses while the shock waves in the lateral flank propagate nearly freely, (i.e. as blast waves). This is thesame conclusion we reached in the previous paper (Kwon and Vourlidas 2017) using the 3D speeds and a simplemodel. ensity structure in sheath Table 1.
The determined density compression ratios X and Mach numbers M A with the corresponding ranges of height r . Thistable provides the full ranges of these values with the averages and standard deviations.Date Time r / R ⊙ X M A ± ± ± ± ± ± ± ± ± ± ± ±
2. The density compression ratio, X (Figure 9), and Alfv´enic Mach number, M A (Figure 10), also vary with theposition angle ζ , i.e., their maximum occurs around the CME nose and correlates with the shock speed. Both X and M A remain relatively constant in time and distance despite the decrease of the shock speed (see also Table1). It is not unreasonable because the local Alfv´en speed v A also decreases as the shocks propagate further out(Figure 11).3. The shocks are supercritical over a wider spatial range, and they last longer than those of what has been reportedby Bemporad and Mancuso (2011).4. The averages of X and M A are 2.1–2.6 and 1.1–1.8, respectively (see Table 1).Once again, we show (this time via density analysis) that the diffuse white light emission ahead of fast CMEs outlinesthe shock sheath. Our 3D analysis suggests that the density compression is sufficiently high for the production of highenergy particles and that the conditions last for, at least, tens of minutes over a wide range of position angles aroundthe CME nose. It suggests that the CME-associated shocks can account both for the SEP production and theirlongitudinal distribution.Our method builds upon and greatly extends past work (Thernisien et al. 2006; Ontiveros and Vourlidas 2009) byusing the 3D reconstruction information of the shock envelope to deconvolve the density distribution from projectioneffects and to derive the 3D speed, upstream and downstream densities, Mach number and other parameters across thefull shock front. The method holds promise for improving the extraction of quantitative information from shocks in thecorona using remote sensing observations. It can readily use observations from different vantage points, including fromthe imagers (Howard et al. 2013; Vourlidas et al. 2016) aboard the upcoming Solar Orbiter (M¨ueller et al. 2013) andSolar Probe Plus (Fox et al. 2016) missions to be compared with direct in situ measurements from these missions. Thetechnique will also greatly improve the results from any future joint coronagraphic imaging and off-limb spectroscopy ofshocks (Bemporad and Mancuso 2011; Vourlidas and Bemporad 2012). We plan to test the double-Gaussian densityprofile along the shock normal, 3D geometry and the χ approach with numerical simulations. Further analyses,including slower CME–shock events, will lead to a better understanding of the shock properties and its relation toSEPs and other phenomena such as EUV waves and radio bursts.We would like to thank Rob Decker and David Lario for insightful discussions. This work of R.Y. K. and A.V. issupported by NASA Grant NNX16AG86G issued under the HSR Program. The SECCHI data are produced by aninternational consortium of the NRL, LMSAL and NASA GSFC (USA), RAL and Univ. Bham (UK), MPS (Germany),CSL (Belgium), IOTA and IAS (France). The editor thanks two anonymous referees for their assistance in evaluatingthis paper. REFERENCES Balan, N., R. Skoug, S. Tulasi Ram et al. CME front andsevere space weather.
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