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Issue
A&A
Volume 698, May 2025
Article Number A15
Number of page(s) 7
Section Extragalactic astronomy
DOI https://doi.org/10.1051/0004-6361/202553738
Published online 28 May 2025

© The Authors 2025

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

1. Introduction

Understanding the physical properties of the solar coronal plasma is critical for uncovering the mechanisms underlying a vast number of transient phenomena in space plasma physics and for forecasting space weather with a greater accuracy. Determining the properties of violent, large expulsions of plasma and magnetic flux known as coronal mass ejections (CMEs) before they impact Earth is crucial. This is because interplanetary CMEs, which propagate from the Sun to Earth in only a few days, typically cause strong space weather disturbances in the near-Earth space environment. When an impulsively accelerated magnetized plasma expands through the corona at a velocity exceeding the local characteristic speed, the outer boundary of its expanding ejecta forms a large-amplitude fast-mode magnetohydrodynamic (MHD) wave. Nonlinear effects can cause this wave to steepen into a shock. Both the shock and the ejecta contribute to geomagnetic disturbances and are significant accelerators of solar energetic particles (SEPs). Thus, understanding their independent evolution is crucial for making accurate space weather predictions.

Although global MHD models of the solar corona and solar wind have become increasingly sophisticated, our ability to predict the space-weather impact of CMEs and their associated shocks remains limited. The primary sources of uncertainty in predicting a CME's arrival time at Earth include the initial properties of the ejecta – its speed, mass, and propagation direction – as well as the characteristics of the ambient solar wind through which it propagates (Riley & Ben-Nun 2021; Verbeke et al. 2023). These MHD models are particularly sensitive to the pre-CME distribution and magnitude of coronal magnetic fields and plasma electron densities. Despite recent progress, significant discrepancies remain between the predicted and observed magnetic field strengths (Riley et al. 2021). In fact, MHD models of the corona often underestimate the magnitude of the open magnetic flux when compared to in situ measurements (Linker et al. 2017) and overestimate the solar electron density (Rouillard et al. 2016; Kouloumvakos et al. 2019). In the absence of in situ measurements – such as those recently provided by NASA's Parker Solar Probe (PSP), launched in 2018 – obtaining this information proves challenging.

Faraday rotation observations of linearly polarized extragalactic radio sources occulted by the solar corona and wind are a powerful method for constraining coronal density and magnetic field strengths. This technique measures the integrated product of the line-of-sight (LOS) component of the magnetic field B∥ and the electron density ne, which yield critical information about the plasma environment through which CMEs and shocks propagate. Faraday rotation relies on the physical effect in which a linearly polarized radio wave propagating through a plasma undergoes a rotation of its plane of polarization. The induced rotation of the radio wave's polarization plane is defined by Δχ=λ2RM, where λ is the wavelength of the radio signal and RM is the rotation measure, given by

RM = e 3 2 π m e 2 c 4 ∫ LOS n e B · ds . $$ {\textrm {RM}}={e^3\over {2\pi {m_e}^2c^4}}\int _{\mathrm {LOS}}{n_e}{{\textbf{B}}}\cdot {{\textbf{ds}}}. $$(1)

In this equation, ne is the electron density; B is the vector magnetic field; ds is the vector differential path length along the LOS; c is the speed of light; and e and me are the charge and mass of the electron, respectively. The contribution to the RM is positive for a magnetic field pointed toward the observer. In past decades, when using observations from the Very Large Array (VLA) radio interferometer of the National Radio Astronomy Observatory (NRAO) at GHz frequencies, this technique has provided valuable information on coronal plasma and solar wind structures (Sakurai & Spangler 1994; Mancuso & Spangler 1999, 2000; Spangler & Mancuso 2000; Spangler 2005; Ingleby et al. 2007; Mancuso & Garzelli 2013a, b; Kooi et al. 2014, 2017; Le Chat et al. 2014).

In our study, we used rare Faraday rotation measurements of a group of extragalactic radio sources that were occulted by the solar corona a few hours before the August 3, 2012, CME-driven shock event. These measurements allowed us to obtain the integrated product of the LOS component of the magnetic field and the electron density of the pre-event corona. By integrating these data into the Magnetohydrodynamics Around a Sphere Thermodynamic (MAST) model, and modeling the shock standoff distance to estimate the fast-mode shock speed, we lastly demonstrate how correction factors may be determined and applied to accurately scale the simulated magnetic field and electron density for that period. The paper is organized as follows: Section 2 presents an overview of the observations and reconstruction methods. Section 3 describes the MHD model and Faraday rotation observations. Section 4 provides our results. Section 5 outlines a summary and conclusion.

2. The August 3, 2012, CME-driven shock

2.1. Observations

The August 3, 2012, CME and associated shock event occurred during the Carrington rotation (CR) number 2126. It coincided with a weak, impulsive B-class flare that peaked at approximately 2:33 UT above the National Oceanic and Atmospheric Administration Active Region (NOAA AR) 11537. At the time of the eruption, this region was located at N13 E57. Around 2:30 UT, a group of coronal type III radio bursts at meter wavelengths marked the impulsive phase of the flare. These bursts, which are radio signatures of suprathermal electron beams propagating along open magnetic field lines, are typically associated with magnetic reconnection events at the flare site (Reid & Ratcliffe 2014).

As observed by the Atmospheric Imaging Assembly (AIA; Lemen et al. 2012) instrument on board the Solar Dynamics Observatory (SDO; Pesnell et al. 2012) in the 304 Å passband (Fig. 1), the initial appearance of the CME was jetlike. Farther up in the corona, a narrow CME surrounded by a diffuse halo was observed by the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) using the C2 and C3 coronagraphs aboard the Solar and Heliospheric Observatory (SOHO; Domingo et al. 1995). The C2 coronagraph covers the region from 2 to 6 R⊙, while C3 extends from 3.7 to 30 R⊙. Together, they provide white-light images of Thomson-scattered radiation from the extended corona. The CME event was listed as a partial halo in the SOHO LASCO CME catalog (CDAW; Yashiro et al. 2004; Gopalswamy et al. 2009) and was first observed at a height of 2.71 R⊙ at 3:12 UT within the LASCO C2 field of view. A linear fit to the CME's radial expansion indicated a projected speed of 520 km s−1, while a quadratic fit suggested an initial speed of about 650 km s−1, with a deceleration of –8.5 m s−2.

thumbnail Fig. 1.

Composite image of the solar corona on August 3, 2012, at 04:17 UT spanning from the solar disk center to the extended corona up to approximately 6 R⊙ (created using JHelioviewer; Müller et al. 2017). The running difference image from the SOHO/LASCO C2 white-light coronagraph shows a narrow CME (green arrow) propagating northeast, accompanied by the boundary of the spheroidal shock wave driven by the CME (red arrows). The inner image captured in the 304 Å passband by the SDO/AIA instrument depicts the earlier evolution of the CME. The blue circle marks the origin of the jetlike eruption on the Sun's surface.

The CME was also observed from two different viewpoints by the inner coronagraph COR1 (1.5–4 R⊙) and the outer coronagraph COR2 (2.5–15 R⊙), both part of the Sun-Earth Connections Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) suite on board the twin Solar Terrestrial Relations Observatory (STEREO; Kaiser et al. 2008) spacecraft. These spacecraft, STEREO-A and STEREO-B, are positioned on heliocentric orbits ahead of and behind Earth, respectively. At the time of the eruption, STEREO-A and STEREO-B were located at approximately W122 and E115, respectively, relative to Earth (see Fig. 2), a configuration that was particularly favorable for a 3D reconstruction of the eruption. Despite its initial jetlike appearance, the white-light images provided by the two STEREO coronagraphs in the extended corona show that the eruption exhibited a definite, albeit narrow, flux rope structure (see Fig. 3). Furthermore, the difference images from the three coronagraphs clearly show a faint, diffuse spheroidal region in front of the flux rope, interpreted as the shock surface marking the boundary of the sheath region.

thumbnail Fig. 2.

Positions of the twin STEREO-A and STEREO-B spacecraft relative to the Earth on August 3, 2012, at 03:00 UT. The dotted lines show the angular displacement from the Sun. The heliocentric Earth ecliptic (HEE) distance is expressed in AU. Courtesy of http://stereo-ssc.nascom.nasa.gov/where.shtml.

thumbnail Fig. 3.

Running difference images from STEREO-B/COR2 (left), SOHO/LASCO/C3 (middle), and STEREO-A/COR2 (right) taken around the same time on August 3, 2012, showing the CME apex reaching a height of approximately 10 R⊙. Top panels: GCS reconstruction of the flux rope shown in green. Bottom panels: Reconstruction of the shock surface depicted as a spheroidal surface superposed on the observed images. For further details, see the main text.

2.2. CME and shock 3D reconstruction

The eruption of CMEs is commonly attributed to the destabilization of coronal magnetic arcades, which contain semitoroidal magnetic flux ropes anchored in opposite photospheric magnetic polarities (e.g., Green et al. 2018). Once the pre-eruptive structure loses equilibrium, it accelerates due to the unbalanced Lorentz force and expands in a self-similar manner (e.g., Low 1984; Thernisien et al. 2009). The bright flux rope-like structure of the August 3, 2012, CME was fitted using the graduated cylindrical shell (GCS; Thernisien et al. 2009; Thernisien 2011) model. This 3D empirical model of a magnetic flux rope represents the structure as a hollow, croissant-like shell. The GCS model is widely used to reconstruct the large-scale geometry of flux rope-like CMEs in the solar corona. The 3D reconstruction was performed using the Python-based software package PyThea (Kouloumvakos et al. 2022). This involved overlaying a parameterized GCS model grid onto each image and adjusting its parameters until the fit closely matched the observations from the three spacecraft positioned along different LOSs. The flux rope's origin at a latitude of approximately 13° above the solar equator is indicated by the GCS reconstruction in the top panels of Fig. 3. Initially, the flux rope followed a nonradial trajectory, rapidly increasing its latitude by about 40° within an hour and reaching 5 R⊙ by 4:00 UT. After 4:00 UT, its trajectory became nearly radial. This pattern agrees with previous results, which find most CME deflections and rotations to occur below approximately 5 R⊙ (e.g., Kay & Opher 2015). Due to uncertainties when modeling the initial phase of abrupt nonradial expansion–and because the Faraday RMs in this study refer to heliocentric distances greater than 8 R⊙, we limited our analysis to heights greater than 6 R⊙ (i.e., where the trajectory became nearly radial). The faint and diffuse spheroidal shock wave in front of the expanding flux rope was modeled as a prolate spheroid bubble (Kwon et al. 2014), implying a fast-mode shock wave that expands self-similarly. The bottom panel of Fig. 3 shows a snapshot of the fits applied simultaneously to coronagraph observations from STEREO-A, STEREO-B, and SOHO.

Figure 4 shows the heliocentric distance-time profiles corresponding to the leading edge of the CME and the nose of the shock. As indicated by the inset of Fig. 4, the leading edge of the flux rope and the shock decelerate as they move farther from the Sun due to drag forces acting on the CME. Magnetic tension and pressure gradients also contribute to the deceleration of erupting flux ropes (Lin & Chen 2022). The standoff distance ΔR increases as the CME moves farther from the Sun because the shock speed exceeds the CME speed. Although driven bow shocks typically propagate at the same speed as their driver in homogeneous ambient media, this is not always the case. The anisotropic nature of the solar wind can lead to variations in behavior (Vršnak & Cliver 2008). This observation is consistent with 2.5D numerical MHD simulations of fast CMEs and their associated shock fronts (Savani et al. 2012).

thumbnail Fig. 4.

Kinematic evolution of the leading edge of the CME flux rope and the nose of the associated shock, as derived from observations by SOHO, STEREO-A, and STEREO-B. The solid and dot-dashed lines represent second-order polynomial fits to the data for each front. The inset shows the inferred speed as a function of time for both fronts.

3. The background corona

To characterize the coronal environment in which the CME and the shock propagated, we employed an MHD modeling approach, enhanced by Faraday rotation observations to calibrate the model.

3.1. MHD model

The global coronal structure at the time of the eruption was modeled using the thermodynamic mode of the Magnetohydrodynamic Algorithm outside a Sphere (MAS) code (Lionello et al. 2009) developed by Predictive Science Inc1. The MAS model is a 3D, time-dependent, and finite-difference model that solves the Maxwell equations and the continuity, momentum, and energy equations. This MHD model requires photospheric radial magnetic field data and a heating mechanism as input conditions. To set the inner boundary conditions, the model uses magnetic field data from photospheric magnetograms obtained by the Heliospheric and Magnetic Imager (HMI; Scherrer et al. 2012) on board the SDO (Pesnell et al. 2012). The model incorporates detailed thermodynamics, including energy equations that account for thermal conduction along the magnetic field, radiative losses, and parameterized coronal heating (Lionello et al. 2001, 2009). The model's outer boundary is set at 30 R⊙. The MAS model is able to reproduce large-scale coronal features observed in white-light, EUV, and X-ray wavelengths and, as a result, effectively simulates the global coronal structure, including the location of streamers.

For this study, we used MAS model outputs that include global electron densities, magnetic field strengths, sound speeds, and solar wind speeds. These quantities are essential to compute the fast-mode Mach number for the eruption. However, it is important to note that the electron density (ne) and the magnetic field strength (B) in the model rarely match the empirical values. Consequently, derived quantities, such as the Alfvén speed and the fast-mode Mach number, may not be entirely reliable. In fact, both B and ne must be scaled using correction factors (fb and fn, respectively) to account for the changing conditions in the corona over time and during different phases of the solar cycle (e.g., Rouillard et al. 2016; Kouloumvakos et al. 2019).

3.2. Faraday rotation observations

As discussed above, Faraday rotation observations of extragalactic radio sources occulted by the solar corona offer a unique advantage in enabling a form of coronal tomography. These observations allow us to simultaneously calculate the RM along various LOSs and provide an empirical estimate of the product of the longitudinal component of the magnetic field (B∥) and the electron density (ne) along each one. Kooi et al. (2017) recently reported Faraday rotation observations performed on August 2, 2012, during the annual solar occultation of extragalactic radio sources 0842+1835, 0843+1547, 0846+1459, and 0900+1832. These observations were conducted at GHz frequencies using the upgraded Karl G. Jansky VLA of the NRAO and corrected for the negligible ionospheric Faraday rotation contribution. The data, with a resolution of approximately half an hour between each scan, are listed in Table 1 along with the heliocentric distances of the closest approach for each radio ray to the Sun. While three of the four radio sources were affected by a CME passage (as detailed in Kooi et al. 2017), all RM values reported are averages obtained prior to this event in the undisturbed corona. The left panel of Fig. 5 shows the positions of the Sun and the four radio sources at 16:00 UT on August 2, 2012. The middle panel of the same figure shows the synoptic map of the electron density (ne) at 10.8 R⊙ for CR 2126, as calculated by the MAS model. The curved lines represent the ray paths of the RMs from the four radio sources, projected to their point of closest approach to the Sun.

thumbnail Fig. 5.

Left panel: Location of the Sun on August 2, 2012, at 16:00 UT, as indicated by the bull's-eye. The colored points mark the positions of the radio sources. The dotted line marks the ecliptic. Middle panel: The synoptic map of the electron density is shown at 10.8 R⊙ for CR 2126, as calculated by the MAS model. The curved lines represent the ray paths of the Faraday RM from four radio sources at 16:00 UT, projected onto their points of closest approach to the Sun. The bull's-eyes mark the closest approach points; the crossed circle indicates the position of Earth; and the square represents the projected location of the radio source. Right panel: Comparison of model-predicted and observed RMs. The dashed line represents perfect agreement. The product of the MHD MAS model's electron densities and magnetic field intensities was scaled by a factor of 1.15 to match the Faraday rotation observations performed on August 2, 2012.

Table 1.

Faraday rotation observations obtained on August 2, 2012, from Kooi et al. (2017).

To calibrate the MHD model, we varied the correction factors fb and fn to achieve the best match between the modeled and observed RMs. The fb correction factor was applied to all components of the magnetic field to preserve the global topology. The right panel of Fig. 5 shows the comparison between the modeled and observed RMs. The dashed line represents perfect agreement. By fitting the RM data using a weighted least-squares method (with a slope of b = 1 and intercept a = 0), we obtained a value for the product of the correction factors: fn×fb = 1.15. In the next section, we discuss how these correction factors were determined based on the observed standoff distance reconstructed in Sect. 2.

4. Results and discussion

When the 3D morphology of a CME and its associated shock is accurately known, white-light observations of the CME-driven shock can be used to probe the physical properties of the coronal plasma through which the eruption propagates (e.g., Mancuso et al. 2019, and references therein). The shape, size, and standoff distance of a CME-driven shock are influenced by several factors, including the CME's own shape and size, its velocity relative to the local plasma and fast-mode magnetosonic speed, and key properties of the medium, such as the ratio of specific heats (γ), magnetic field strength, and orientation.

Coronal-mass-ejection-driven shocks can generally be classified into two types: bow shocks and expansion (piston-driven) shocks (Vršnak & Cliver 2008; Warmuth 2015). Bow shocks form when solar wind flows around relatively narrow CMEs, while expansion shocks arise ahead of a piston driver that expands outward through and accumulates solar wind (Žic et al. 2008; Lulić et al. 2013). The plasma is pushed outward by the expanding piston with no material flowing behind. Once the piston decelerates, the shock detaches and continues to propagate. Although the distinction between piston-driven and bow shocks can be unclear, observational data suggest that fast CMEs with widths ≲ 60° are typically bow shocks, while those with widths > 60° are primarily piston-driven (Kahler et al. 2019; Fainshtein & Egorov 2019). Generally, both types of shocks are likely to occur simultaneously. In fact, rapid expansion in the early phase can generate a piston-driven shock, while continuous radial motion of the CME ejecta could produce a bow shock (Kwon et al. 2014).

Shocks propagate at speeds vsh that exceed the characteristic speed of the upstream medium. In a magnetized plasma, there are typically three modes: fast magnetosonic waves, slow magnetosonic waves, and intermediate Alfvén waves. However, CME-associated shocks are usually interpreted as collisionless fast-mode shock waves. The fast-mode speed, vfm is given by

v fm = 1 2 [ v A 2 + c S 2 + ( v A 2 + c S 2 ) 2 − 4 v A 2 c S 2 cos 2 θ Bn ] , $$ v_{\mathrm {fm}} = \sqrt {{1\over 2} \left [v_{\mathrm {A}}^2 + c_{\mathrm {S}}^2 + \sqrt {(v_{\mathrm {A}}^2 + c_{\mathrm {S}}^2)^2 - 4v_{\mathrm {A}}^2c_{\mathrm {S}}^2\cos ^2{\theta _{\mathrm {Bn}}}}\right ]}, $$(2)

where vA is the Alfvén speed, cS is the sound speed, and θBn is the angle between the wave vector and the magnetic field vector. This fast-mode speed is the characteristic speed we use to compare with the shock front speed. The fast-mode Mach number is defined as Mfm=(vsh−vsw)/vfm, where vsh is the shock speed in the observer frame. For parallel propagation (cosθBn = 1), the fast-mode speed simplifies to

v fm , ∥ = 1 2 ( v A 2 + c S 2 + | v A 2 − c S 2 | ) . $$ v_{\mathrm {fm,\parallel }} = \sqrt {{1\over 2} (v_{\mathrm {A}}^2 + c_{\mathrm {S}}^2 + |v_{\mathrm {A}}^2 - c_{\mathrm {S}}^2|)}. $$(3)

For propagation across magnetic field directions,

v fm , ⊥ = v A 2 + c S 2 , $$ v_{\mathrm {fm,\perp }} = \sqrt {v_{\mathrm {A}}^2 + c_{\mathrm {S}}^2}, $$(4)

and at intermediate angles, the fast-mode wave speed lies between these two extremes.

The standoff distance between the shock and the CME's leading edge, ΔR=rsh−rcme, depends on the CME's shape and the relative Mach number of the flow. Early gasdynamic studies with spherical obstacles found that ΔR is linearly related to the inverse of the compression ratio X=ρd/ρu, where ρu and ρd represent the densities upstream and downstream, respectively (Seiff 1962). However, this relationship is more complex for nonspherical structures such as those CMEs modeled as flux ropes. In a hydrodynamic regime based on mass conservation, the compression ratio X can be expressed in terms of the upstream Mach number Mu and the ratio of specific heats γ (Priest 2012):

ρ u ρ d = ( γ − 1 ) M u 2 + 2 ( γ + 1 ) M u 2 · $$ {\rho _u\over \rho _d} = {(\gamma - 1)M_u^2 + 2\over (\gamma + 1)M_u^2}\cdot $$(5)

However, this equation is only applicable when the shock velocity and magnetic field are parallel. In an MHD regime, the assumption of quasi-parallelity is reasonable for shock propagation at higher altitudes in the corona, where the magnetic field is predominantly radial. Thus, for CME-driven shocks, the hydrodynamic case provides a plausible approach. For low Mach numbers (Mu≲3), Farris & Russell (1994) derived the following expression for ΔR:

Δ R R c = 0.81 ( γ − 1 ) M u 2 + 2 ( γ + 1 ) ( M u 2 − 1 ) , $$ {\Delta R\over R_{\mathrm {c}}} = 0.81 {(\gamma - 1)M_u^2 + 2\over (\gamma + 1)(M_u^2 - 1)}, $$(6)

where Rc is the CME's radius of curvature. In the astrophysical context, hydrodynamic relations offer a good approximation for low Mfm (Russell & Mulligan 2002a).

For a CME modeled as a flux rope with a croissant-like shape, the standoff distance is governed by the geometric mean of the two radii of curvature perpendicular to the direction of propagation (Stahara et al. 1989; Russell & Mulligan 2002a; Kay et al. 2020). The minor and major radii of curvature are provided by the azimuthal extent of the rope (perpendicular to the solar wind flow and the rope axis), and the major axis is provided by the curvature of the axial field of the rope (Russell & Mulligan 2002b). The minor and major radii can be inferred from the coronagraph images. For this event, the difference images from COR2-A provide a clear face-on view of the flux rope, so that we were able to retrieve the minor and major radii of curvature as they evolved self-similarly with time. Given the quantities ΔR and Rc deduced from the coronagraphic images, and assuming a reasonable value for γ, the fast-mode Mach number Mfm can be easily obtained by inverting Eq. (6):

M fm = 2 × 0.81 + Δ R R c ( γ + 1 ) Δ R R c ( γ + 1 ) − 0.81 ( γ − 1 ) · $$ M_{\mathrm {fm}} = \sqrt {{2 \times 0.81 + {\Delta R\over R_{\mathrm {c}}}(\gamma + 1)}\over {\Delta R\over R_{\mathrm {c}}}(\gamma + 1) -0.81(\gamma - 1)}\cdot $$(7)

Figure 6 shows the standoff distance ΔR of the shock derived from the 3D reconstruction as a function of time. The figure also shows the temporal dependence of the effective radius of curvature of the flux rope, Rc, and of the ratio ΔR/Rc. These parameters clearly increase quasi-linearly over time. The same figure further illustrates the time evolution of the fast-mode Mach number, calculated using Eq. (7) for the two common cases of adiabatic indices γ = 4/3 and 5/3. In the above analysis, we assumed quasi-parallel propagation above 6 R⊙, where the angle θBn between the shock wave vector and the magnetic field vector was varied between 0° and 25°. To derive our result, we also assumed that the shock formed as the perturbation propagated through the densest region of the corona. This region approximately corresponds within a few degrees to the nose of the shock front. We also averaged the data over CR longitudes between 78° and 90° with a mean value of 84°, inferred from the shock reconstruction. Adjusting the correction factors, fn and fb–constrained by the previously determined product, fn×fb = 1.15, derived from the Faraday RMs–and comparing the result with that obtained from Eq. (7), we find that Mfm from the MAS model is best described with fn≃2.4 and fb≃0.5, with γ = 5/3 (see Fig. 6). Our results, deduced from Fig. 6, indicate that the fast-mode Mach number decreased from about 2 to nearly 1 over a period of 4 hours. This suggests that the shock gradually dissipated and could not propagate above approximately 20 R⊙. Moreover, the low values we obtained for the fast-mode Mach number appear to explain the nondetection of type II radio emission both in the extended corona and in interplanetary space. Type II radio bursts are characterized as emission bands that slowly drift from high to low frequencies in radio dynamic spectra, with a duration of several minutes, and are usually interpreted as signatures of CME-driven shocks (e.g., Mancuso et al. 2002; Mancuso & Avetta 2008; Bemporad & Mancuso 2010). The Mach number was presumably subcritical (e.g., Edmiston & Kennel 1984; Bemporad & Mancuso 2011), and therefore implies that the particle acceleration efficiency was likely very low (Mann et al. 2003). Consequently, the weak shock might not accelerate a sufficient number of electrons required to produce a type II burst (see Gopalswamy et al. 2008, 2010 for a discussion about radio-quiet CME-driven shocks).

thumbnail Fig. 6.

Fast-mode Mach number Mfm as a function of time calculated using Eq. (7) for γ = 4/3 (dashed blue curve) and γ = 5/3 (dashed red curve). The shaded green bands represent Mfm values computed using MAS with the given correction factors, as the angle θBn between the wave vector and the magnetic field vector varies from 0° to 25°. The plot also shows the temporal evolution of the shock standoff distance ΔR, the effective radius of the flux rope's curvature Rc, and the ratio ΔR/Rc.

This outcome is consistent with Kouloumvakos et al. (2019), who also applied estimated correction factors to scale up the coronal magnetic field and scale down the density values provided by the MHD models. These correction factors were obtained by extrapolating the averaged values of the unsigned radial component of the magnetic field and solar wind density at an outer boundary within the MHD model together with an inverse square dependence. These correction factors were then compared with in situ measurements at 1 AU. According to Kouloumvakos et al. (2019), the magnetic field scaling factor, fb, ranges between 1.6 and 2.4, averaging at 1.8, while the density correction factor, fn, ranges between 0.3 and 0.6, averaging at 0.45. Our results show that the MAS model similarly overestimates densities and underestimates magnetic fields near the Sun, particularly at heliocentric distances greater than 8 R⊙ (the shortest proximity along the LOS; see Table 1).

5. Summary and conclusion

In this work, we introduced a novel approach that combined advanced MHD modeling with Faraday rotation observations to improve our understanding of CMEs, their associated shocks, and their impact on space weather. Faraday rotation measurements of extragalactic radio sources occulted by the solar corona provide a powerful complementary tool for probing the pre-eruption electron density and magnetic field structure of the solar corona and thus improve predictions from global MHD models. In fact, these observations offer independent information about the integrated product of the LOS magnetic field component and the electron density. Using the August 3, 2012, CME as a case study, we investigated the 3D geometry and kinematics of the shock driven by a narrow flux rope. Observations were obtained from three spacecraft: SOHO, STEREO-A, and STEREO-B. Strategically positioned at the time of the event, these instruments allowed us to capture the shock's propagation in space.

The fortunate availability of Faraday RMs, obtained a few hours before the transient, allowed us to properly calibrate the MHD MAS model in its thermodynamic mode (Lionello et al. 2009). To perform this calibration, we adjusted for the plasma conditions using correction factors for the magnetic field (fb) and electron density (fn). These were derived from Faraday rotation data. The application of these factors (fb≃2.4 and fn≃0.5) produced a strong match between the fast-mode Mach number predicted by MAS and that obtained from the analysis of the 3D reconstruction of coronagraphic data. The reconstruction utilized the model from Farris & Russell (1994) and Russell & Mulligan (2002a). This result is consistent with Kouloumvakos et al. (2019), who also applied estimated correction factors to scale up the coronal magnetic field and to scale down the density values provided by the MHD models. In fact, the Mach number was probably subcritical, and we expected the particle acceleration efficiency to be particularly low. Our analysis shows that the shock's fast-mode Mach number decreased over time, approaching unity as the shock dissipated. This is consistent with the expected physical behavior of shocks as they lose energy during propagation.

In conclusion, this study underscores the critical role of integrating observational techniques, such as Faraday rotation and MHD modeling, to improve the understanding of CMEs and their impact on space weather. The results of our analysis contribute to a more reliable framework for predicting space weather events, particularly the timing and intensity of geomagnetic storms caused by solar eruptions. Further refinements in these models, alongside continued observational efforts, are vital to enhance space weather forecasting capabilities and mitigate the risks to space-based technologies and the Earth's magnetosphere caused by solar events.

Acknowledgments

The author thanks the anonymous referee for a thorough review, providing constructive comments and suggestions. The SECCHI data are produced by an international consortium of the NRL, LMSAL, and NASA GSFC (USA), RAL and U. Bham (UK), MPS (Germany), CSL (Belgium), and IOTA and IAS (France). The SOHO/LASCO data used here are produced by a consortium of the Naval Research Laboratory (USA), Max-Planck-Institut für Aeronomie (Germany), Laboratoire d’Astronomie (France), and the University of Birmingham (UK). SOHO is a project of international cooperation between ESA and NASA. STEREO is a mission in NASA's Solar Terrestrial Probes program. The LASCO CME catalog is generated and maintained at the CDAW Data Center by NASA and The Catholic University of America in cooperation with the Naval Research Laboratory. We also acknowledge Predictive Science Inc. for publicly providing the data used in this study. This study has used the JHelioviewer software provided by ESA. This research has used PyThea v0.12.0, an open source and free Python package to reconstruct the 3D structure of CMEs and shock waves (Zenodo: https://doi.org/10.5281/zenodo.5713659). Finally, the author wishes to express his sincere gratitude to Prof. Steven R. Spangler for being a source of great inspiration and wisdom.

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All Tables

Table 1.

Faraday rotation observations obtained on August 2, 2012, from Kooi et al. (2017).

In the text

All Figures

thumbnail Fig. 1.

Composite image of the solar corona on August 3, 2012, at 04:17 UT spanning from the solar disk center to the extended corona up to approximately 6 R⊙ (created using JHelioviewer; Müller et al. 2017). The running difference image from the SOHO/LASCO C2 white-light coronagraph shows a narrow CME (green arrow) propagating northeast, accompanied by the boundary of the spheroidal shock wave driven by the CME (red arrows). The inner image captured in the 304 Å passband by the SDO/AIA instrument depicts the earlier evolution of the CME. The blue circle marks the origin of the jetlike eruption on the Sun's surface.
In the text
thumbnail Fig. 2.

Positions of the twin STEREO-A and STEREO-B spacecraft relative to the Earth on August 3, 2012, at 03:00 UT. The dotted lines show the angular displacement from the Sun. The heliocentric Earth ecliptic (HEE) distance is expressed in AU. Courtesy of http://stereo-ssc.nascom.nasa.gov/where.shtml.
In the text
thumbnail Fig. 3.

Running difference images from STEREO-B/COR2 (left), SOHO/LASCO/C3 (middle), and STEREO-A/COR2 (right) taken around the same time on August 3, 2012, showing the CME apex reaching a height of approximately 10 R⊙. Top panels: GCS reconstruction of the flux rope shown in green. Bottom panels: Reconstruction of the shock surface depicted as a spheroidal surface superposed on the observed images. For further details, see the main text.
In the text
thumbnail Fig. 4.

Kinematic evolution of the leading edge of the CME flux rope and the nose of the associated shock, as derived from observations by SOHO, STEREO-A, and STEREO-B. The solid and dot-dashed lines represent second-order polynomial fits to the data for each front. The inset shows the inferred speed as a function of time for both fronts.
In the text
thumbnail Fig. 5.

Left panel: Location of the Sun on August 2, 2012, at 16:00 UT, as indicated by the bull's-eye. The colored points mark the positions of the radio sources. The dotted line marks the ecliptic. Middle panel: The synoptic map of the electron density is shown at 10.8 R⊙ for CR 2126, as calculated by the MAS model. The curved lines represent the ray paths of the Faraday RM from four radio sources at 16:00 UT, projected onto their points of closest approach to the Sun. The bull's-eyes mark the closest approach points; the crossed circle indicates the position of Earth; and the square represents the projected location of the radio source. Right panel: Comparison of model-predicted and observed RMs. The dashed line represents perfect agreement. The product of the MHD MAS model's electron densities and magnetic field intensities was scaled by a factor of 1.15 to match the Faraday rotation observations performed on August 2, 2012.
In the text
thumbnail Fig. 6.

Fast-mode Mach number Mfm as a function of time calculated using Eq. (7) for γ = 4/3 (dashed blue curve) and γ = 5/3 (dashed red curve). The shaded green bands represent Mfm values computed using MAS with the given correction factors, as the angle θBn between the wave vector and the magnetic field vector varies from 0° to 25°. The plot also shows the temporal evolution of the shock standoff distance ΔR, the effective radius of the flux rope's curvature Rc, and the ratio ΔR/Rc.
In the text

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