Connecting the Properties of Coronal Shock Waves with Those of Solar Energetic Particles

For our analysis we use the SEP event list given in Table 3 of Paassilta et al. (2018), which consists of 46 SEP events with energies greater than 50 MeV that occurred from 2009 to 2016. This catalog is based on measurements of energetic protons by the Energetic and Relativistic Nuclei and Electron experiment (ERNE; Torsti et al. 1995) on board the Solar and Heliospheric Observatory (SOHO) and the High Energy Telescope (HET; von Rosenvinge et al. 2008) on board STEREO. An SEP event was selected when particle fluxes increased by a factor of ≥3.0 over the quiet-time background fluxes of at least two of the three spacecraft (i.e., SOHO, STEREO-A, or STEREO-B) over a short time interval. We added to this SEP list the 2017 September 10 event detected by STA, SOHO/ERNE, and the Geostationary Operational Environmental Satellites (GOES).

From the initial catalog of the 47 SEP events we rejected 14 events⁶ because of insufficient or bad-quality data (in situ or remote sensing). In 8 of the 14 events the energetic protons were detected in only two of the three spacecraft. Insufficient or missing data were retrieved at one or both instruments near the onset and/or the rise phase of the SEP events. This made it impossible to accurately determine the SEP characteristics; therefore, we do not perform any further analysis for those events. In two of the remaining six events there was insufficient coronal imaging to perform a reliable 3D fitting of the coronal pressure waves (see Section 2.3.1). Additionally, in three events multiple CMEs occurred in quick succession, making it difficult to determine the evolution of the shock wave and consequently to constrain accurately the shock fittings. The final list with the remaining 33 SEP events is given in Table 1.

The final catalog includes 12 SEP events with clear detection of energetic protons at the three different locations (e.g., ERNE in L1 or GOES, STA, and STB) and 21 events for which the energetic protons were detected clearly in at least two locations. It is worth noting that for those 21 events we have 12 events for which high-energy protons were clearly not detected at the third available location, despite the availability and good quality of the particle data, and 6 events with ambiguous detection, i.e., low intensities and delayed onsets at the third location. For the remaining 3 cases there was a data gap in the particle observations during SEP events at the third location. In events with data gaps on SOHO/ERNE we use the GOES data to complement the observations; this procedure is explained in Section 2.2. For the above separation we have already considered the GOES data. In summary, we have 84 S/C-specific SEP events (including the six ambiguous spacecraft (S/C)-specific cases).

A simple consideration of the source locations of the 84 S/C SEP events in our list by accounting for the Parker spiral for a wind speed measured near the onset of the SEPs reveals that 30 events measured at specific spacecraft were connecting magnetically to within ΔΦ_c = 50° of the flaring region, e.g., cases with presumably close magnetic connections and 54 cases with more distant magnetic connections (ΔΦ_c > 50°). Past observations have revealed that the onset of SEP events is prompt when magnetic connectivity is established with the nose of interplanetary shocks (i.e., along the central axis of the CME driver), but the rise in SEP fluxes is more gradual when magnetic field lines connect to regions far from the shock nose (Cane et al. 1988; Reames 1999; Cane & Lario 2006; Reames 2013). This trend has been confirmed from detailed reconstructions of shock waves, at least 30 minutes into the event, when the shock occupies already a significant portion of the solar disk (see Rouillard et al. 2012, 2016; Lario et al. 2014, 2016, 2017; Kouloumvakos et al. 2016; Patsourakos et al. 2016). It remains to be determined whether the flanks of the shock could be significant particle accelerators during the very early expansion phase of CMEs within 10–20 minutes of the CME onset.

In 20 SEP events the source region was located on the visible disk as viewed from Earth and a flare detection was possible by the GOES spacecraft. For those 20 events, 11 are associated with an X-class, 7 with an M-class, and 2 with a C-class flare measured in soft X-rays by GOES. In 13 SEP events the eruption was back-sided. Additionally, from an inspection of the available CME properties from the CDAW SOHO/LASCO CME Catalog⁷ and the Dual-Viewpoint CME Catalog from the STEREO coronagraphs⁸ (Vourlidas et al. 2017) we found that our SEP events are associated with wide halo CMEs, with projected speeds that range from 580 to 3163 km s⁻¹, with a mean value of ∼1519 km s⁻¹.

Additionally, from an inspection of the available radio spectral quick-look data from the SECCHI-radio-survey⁹ and the data from the RSTN radiospectrographs, we found that type II radio bursts (in metric, decametric, or kilometric wavelengths) occurred in all SEP events of our list. Furthermore, in 29 events the type II bursts have an interplanetary counterpart (i.e., observed by WIND/WAVES or SWAVES). For those 29 cases, in ∼65% (17/29 events) the IP-type II emission persists for longer than 3–4 hr. In some extreme events the type II emission can be clearly tracked over half a day; a clear example is the "Carrington-like" event of 2012 July 23. The presence of a type II radio burst in all SEP events in our list implies that shocks formed during all the events.

In our analysis we focus on the relationship between the shock properties and the SEP flux peak intensities. We use the 1-minute-averaged data of energetic proton fluxes from SOHO/ERNE and STEREO/HET instruments. SOHO/ERNE performs measurements of energetic protons in 20 individual energy channels spanning from 1.58 to 131 MeV. Its two separate detectors ERNE/LED and ERNE/HED provide 10 individual channels each. In our study we use data only from the HED detector in energies ranging from 13.8 to 131 MeV (i.e., channels 11–20). STEREO/HET performs measurements in 11 individual energy channels ranging from 13.6 to 100 MeV. Additionally, we use 5-minute-averaged particle data from GOES¹⁰ to perform a correction of the SOHO/ERNE observations from saturation due to its large geometric factor. In our analysis we focus on three energy ranges, 20–26 MeV, 40–60 MeV, and 60–100 MeV. Those were chosen to roughly match the individual energy channels of the different instruments we use. Also those ranges are frequently employed in previous statistical studies of SEP characteristics (Kahler 2001; Richardson et al. 2014; Papaioannou et al. 2016).

Since the division of the proton energy channels is not identical between ERNE and HET instruments, we combine adjacent channels to obtain matching energy ranges. For the energy range of 20–26 MeV, we use the channels ch-4 and ch-5 of STEREO/HET with energies spanning from 20.8 to 23.8 MeV (geometric mean: 22.2 MeV) and from 23.8 to 26.4 MeV (geometric mean: 25.1 MeV), respectively. The resulting mean energy¹¹ for the HET composite channel is 23.4 MeV, which closely matches the mean energy of SOHO/ERNE ch-13 (23.3 MeV). At the energy ranges 40–60 MeV and 60–100 MeV, we have considered ch-10 and ch-11, respectively, of STEREO/HET (mean energy: 49.0 and 77.5 MeV, respectively). Those channels have exactly the same energy range as the two pre-selected intervals of 40–60 MeV and 60–100 MeV. We combine the ERNE ch-16 with ch-17 and ch-18 with ch-19 to produce two individual channels in the energy ranges of 40.5–67.3 MeV and 63.8–101 MeV. The resulting mean energies for the two composite ERNE channels are 52.0 and 80.0 MeV, respectively, and reasonably match the energies of the corresponding HET channels we use.

We perform an intercalibration of our particle data to ensure that the measured proton intensities are comparable among the various instruments in the selected energy intervals. We follow the method in Richardson et al. (2014), and we compare the proton fluxes in the three energy ranges of 20–26 MeV, 40–60 MeV, and 60–100 MeV during 2006 December. At that time the two STEREO spacecraft were close to Earth; therefore, the in situ measurements between STEREO and the instruments in L1 should be comparable. To perform the calibration, we start the calibration with the SEP event that occurs at 2006 December 13, and we continue until the end of the month, where the intensities were close to the background values. In this way we have a large dynamic range of energetic particle intensities to perform the correlations. In Figure 1 we show a part of the results from the intercalibration. In panel (a) we show the energetic proton measurements from STEREO/HET and SOHO/ERNE in 40–60 MeV during the calibration interval (e.g., from 2006 December 13 to 31).

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In Figure 1(b) we compare the energetic proton fluxes between STEREO/HET-A and HET-B. From the comparison of the particle data between HET-A and HET-B we find that the particle intensities at both STEREOs are highly correlated and comparable in every energy range considered here, during the selected time interval. This result is expected and is in accordance with the results presented by Richardson et al. (2014). The residual differences in the particle fluxes may be due to anisotropies in the particle distribution and the different pointing directions of the individual particle telescopes (see von Rosenvinge et al. 2009). In 2006 December STEREO-B was inverted so that HET-B was observing perpendicular to the nominal Parker spiral direction whereas HET-A was observing along the spiral direction. Therefore, in our analysis we use the energetic proton data from STEREO-A and STEREO-B without applying any correction. We note that these particle intensity relations have a mean absolute percentage error of ∼12%–27%, depending on the energy channel. This sets a lower limit to the uncertainty in the determined peak proton intensities.

From the comparison of the particle data between STEREO/HET and SOHO/ERNE we find a rather complicated relation. In every energy range we observe (1) a break in the linear relationship between HET-A and ERNE at medium to high intensities (see Figure 3 of Richardson et al. 2014) and (2) a turnover at high intensities where SOHO/ERNE saturates. Before we proceed with the cross-calibration, we perform a correction of the ERNE saturation effect using GOES-SEPEM data from 2010 to 2014. The comparison of the particle data between GOES and ERNE is shown in Figure 1(c), before (red points) and after (black points) the saturation correction. The intensities we show in this panel are integral intensities over all ERNE and GOES energies (∼13–130 MeV). The ERNE response saturates like a paralyzable instrument: once the true flux exceeds a certain value, the measured one starts to decrease. The appropriate methodology is used to correct ERNE's intensities from saturation (see Wurz et al. 2007).

Before the final calibration between the HET and ERNE, we resample the data to 5-minute-averaged values. This averaging smooths out any time differences of the measurements at different positions. In Figure 1(c) we show the comparison of the particle data between the STA-HET and SOHO/ERNE data after the saturation correction and the calibration. After the calibration, the proton intensity relations result in a mean absolute percentage error of ∼30%–80%, depending on the energy channel. The final scaling functions for the intercalibrated ERNE intensities with STEREO/HET have the form $I^{\prime} ={aI}$. For the 20–26 MeV energy channel we have a = 2.15, for 40–60 MeV, a = 0.73, and for 60–100 MeV, a = 2.55. In our statistical analysis we use the corrected/calibrated ERNE observations, and only in cases where there are gaps in the observations do we use the GOES-SEPEM data instead.

From the intercalibrated particle data we determine the peak intensities (Ip) in the three selected energy ranges. For the purpose of this study we are limited to measuring the Ip when the associated shocks are still close to the Sun and exclude the peaks associated with the arrival of interplanetary shocks at the instruments, when detected in situ. To identify the possible passages of the interplanetary shock waves, we use solar wind measurements at Wind, ACE, and STEREO spacecraft from the OMNI database.¹² Before the time of the shock passage in situ we seek and register the maximum intensity for each spacecraft.

2.3.1. 3D Reconstruction Techniques

Thanks to the nearly simultaneous white-light and EUV observations of pressure/shock waves, we can carry out the forward modeling of coronal pressure waves during all the SEP events on our list. In the low corona we use full-disk image triplets from the STEREO Extreme UltraViolet Imager (Wuelser et al. 2004) at 195 Å and images from the Atmospheric Imaging Assembly (Lemen et al. 2012) on SDO at 193 Å. The field of view of the instruments extends to ∼1.7 and 1.3 R_☉, respectively. Additionally, we use total brightness coronagraph image triplets from the Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI; Howard et al. 2008) coronagraphs COR1 (FOV: ∼1.5–4 R_☉) and COR2 (∼3–15 R_☉) on board STEREO and the Large Angle and Spectrometric Coronagraph (LASCO; Brueckner et al. 1995) coronagraphs C2 (2.2–6 R_☉) and C3 (3.7–30 R_☉) on board SOHO. To enhance the visibility of the waves in the EUV and coronagraphic data, we use running difference images when we perform the fittings.

To perform the 3D reconstruction of the observed pressure waves, we deploy a forward modeling technique that follows a similar approach to Thernisien et al. (2009), Kwon et al. (2014), and Rouillard et al. (2016). These methods are widely used to perform the geometrical fitting of shock waves or CMEs using two or three simultaneous viewpoints from SECCHI and LASCO observations and employ different geometrical models (i.e., the spheroid or the graduated cylindrical model). In our analysis we are interested in 3D reconstructions of the pressure waves' frontal structure; we use an ellipsoid model and fit it at their outermost extent viewed from multiple vantage points. The geometrical shape and the position of the ellipsoid surface are defined, respectively, by a set of three parameters that control the length of the three axes of symmetry, which are pairwise perpendicular, and the center of the ellipsoid, which is defined in heliocentric coordinates. In the top panels of Figure 2 we show a set of coronagraphic observations for the 2012 July 23 CME and its associated density wave. In the bottom panels we show the corresponding running difference images onto which we overlay the ellipsoid fitting to the pressure wave front, which is represented by the red lines. For this event, the 3D reconstructions of the pressure wave provided a maximum deprojected speed of ∼2700 km s⁻¹ at the shock apex and ∼1850 km s⁻¹ at the flanks.

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The density/shock wave fitting procedure can be summarized as follows: (1) We start the fitting in the high corona (WL images) at times when coronagraphic images are temporally synchronized at the different spacecraft. (2) When the 3D fittings at these points have been performed, we then proceed at times with data from only one or two viewpoints. We make sure that the parameters that define the ellipsoids vary smoothly between optimal three-viewpoint fittings and those obtained from one/two viewpoints. (3) Finally, we address fittings in the low corona (EUV images), when the pressure wave forms, during which we often see near the base the formation of an EUV wave.

Once the ellipsoid fittings are performed, we apply a spline fitting to the ellipsoid parameters over time. With this method we reduce the variability of the parameters produced by random errors in the triangulation. This typically leads to a smooth displacement and expansion of the ellipsoid positions from one frame to another. From the spline fittings we produce a new set of ellipsoid parameters at steps of 1 minute to generate a final sequence of regularly time-spaced ellipsoids.

Application of the forward modeling technique with uninterrupted multiviewpoint observations results in very precise localization of the shocks at times when the STEREO and near-Earth spacecraft were equally separated around the Sun (such as from 2010 to 2012). Latitudinal and longitudinal shifts of as little as a degree in shock position are typically detected during the fitting. Different issues often affect the overall accuracy of the fittings from one event to another, such as the quality of the remote-sensing observations and the associated ability to identify and model accurately the same features in the different viewpoints. For the first type of uncertainties the accuracy of the fittings depends on (1) the general availability of simultaneous observations from at least three viewpoints, (2) the overall cadence of the imaging, (3) the existence of previous events and in general the overall contamination of the images (e.g., by high-energy particles), and (4) the separation angle of the viewpoints with respect to the source region. For items 1 and 2 the resulting uncertainty can be compensated by the smoothing of the ellipsoid parameters over time; for item 3 it is difficult to make robust fittings at high solar elongations when the features' contrast drops dramatically and the edges are not easily resolved especially in a contaminated background. However, this usually occurs above ∼20 R_☉, and beyond these heights the shock geometry departs from that of an ellipsoid and it is difficult to achieve a robust fitting. We therefore stop the shock modeling near ∼20 R_☉.

For item 4 the separation angle of the viewpoints may affect the accuracy of the fittings depending also on the source position. An example is given in Figure 2, where the green/red/blue circles mark the regions on the ellipsoid that are tightly constrained from each viewpoint. They essentially map on the surface of the ellipsoid the locations along the line of sight where the line of sight is tangent to that surface. For this event the multiviewpoint imaging led to a robust fitting of both the radial and lateral expansion of the pressure wave. However, there are cases where it is nearly impossible to perform forward modeling with such accuracy because the viewing angles may not provide tight constraints in radial and lateral extension of the wave at the same time. The worst-case scenario is when the separation of all viewing pairs is significantly narrower than ∼20°. An example of this unfortunate case is the SEP event of 2014 September 25. In this event the CME has been observed in every viewpoint head-on (i.e., as a HALO-CME). Therefore, in this event only the lateral expansion of the shock wave can be modeled accurately since the edges of the shock apex are not captured in any of the viewpoints. We have discarded this event from our analysis. In no other events have we faced this problem.

Additionally, the expansion characteristics of the shock wave itself can affect the accuracy of the fittings. With the current geometrical fitting methods we can model robustly the large-scale envelope of the wave's frontal structure and also include in the models asymmetries like rotations or departures from the self-similar expansion. However, it is hard to model accurately small- or large-scale distortions to the modeled pressure waves. From the fittings of the pressure wave that we present in Figure 2 it is obvious that the fitted ellipsoid can capture accurately a large part of the observed WL pressure/shock wave except from a region that is located at the south flank. This region is highly asymmetrical compared to the wave's overall frontal structure, and no treatment was here applied to improve the fitting in such regions. Despite these few difficulties faced with the simplicity of the 3D reconstruction, a rough estimate of the possible uncertainties in speed determination, for instance, gives errors of about 10%–20% in the toughest cases implied by poorer tie pointing and asymmetries.

To derive the 3D properties of the modeled coronal pressure/shock waves, we use the methods presented by Rouillard et al. (2016) and Plotnikov et al. (2017). We start with the sequence of the regularly time-spaced ellipsoids, and we consider a set of 71 × 71 grid points distributed over their surface. We compute the pressure waves' 3D expansion speed from the minimum distance between the grid points of two consecutive ellipsoids divided by their time interval, in our case $\delta t=1$ minute. This approach slightly underestimates the wave speed during the acceleration phase. For a mean acceleration of 500 ${\rm{m}}\,{{\rm{s}}}^{-2}$ the difference would be 30 km s⁻¹ in the time interval $\delta t$ of our consecutive computations, which yield a maximum error of ∼6% even for the slowest parts of the wave (i.e., for speed ∼500 km s⁻¹). This error is negligible above ∼2–3 R_☉ when the acceleration phase has ended and the shock has reached a maximum or constant speed.

We compute the Mach numbers (${M}_{{\rm{A}}},{M}_{\mathrm{fm}}$) and the magnetic field obliquity with respect to the shock normal (${\theta }_{\mathrm{Bn}}$), using the plasma and magnetic field properties provided from the Magneto-Hydrodynamic Around a Sphere Thermodynamic (MAST) model (Lionello et al. 2009). Those parameters are inferred along the surface of the pressure waves. The MAST model is an MHD model developed by Predictive Sciences Inc.¹³ that makes use of the photospheric magnetograms from SDO/HMI as the inner boundary condition of the magnetic field and includes detailed thermodynamics with realistic energy equations accounting for thermal conduction parallel to the magnetic field, radiative losses, and parameterized coronal heating. This thermodynamic MHD model produces more accurate estimates of plasma density and temperature in the corona (Lionello et al. 2009; Riley et al. 2011). The outer boundary of the model is at 30 R_☉. From the MAST 3D cubes we include at the front location the magnetic field and solar wind speed vector and the plasma temperature and density. The resolution of the MAST data cubes is 150 × 100 × 181 in r × θ × ϕ components. We calculate the Mach Alfvén (M_A), the fast magnetosonic Mach number (M_fm), and ${\theta }_{\mathrm{Bn}}$ from

$$\begin{eqnarray}&&\cos \ {\theta }_{\mathrm{BN}}=\displaystyle \frac{{\boldsymbol{B}}\cdot \hat{{\boldsymbol{n}}}}{| {\boldsymbol{B}}| },\end{eqnarray} \tag{ 1 }$$

$$\begin{eqnarray}&&{M}_{{\rm{A}}}=\displaystyle \frac{{V}_{\mathrm{sh}}-{{\boldsymbol{V}}}_{\mathrm{sw}}\cdot \hat{{\boldsymbol{n}}}}{{V}_{{\rm{A}}}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{M}_{\mathrm{fm}}=\displaystyle \frac{{V}_{\mathrm{sh}}-{{\boldsymbol{V}}}_{\mathrm{sw}}\cdot \hat{{\boldsymbol{n}}}}{{V}_{\mathrm{fm}}},\end{eqnarray} \tag{ 3 }$$

where $\hat{{\boldsymbol{n}}}$ is the shock-normal vector, V_sh is the shock speed, V_A is the Alfvén speed, V_fm is the fast magnetosonic speed, and V_sw is the background solar wind speed vector. Our calculations are in the local solar wind frame (see Section 4 in Plotnikov et al. 2017, for more details). Before we proceed with the calculation of the shock properties from the 3D shock fittings and the MAST model, we apply a calibration of the 3D MHD magnetic field and density data using in situ measurements. For the 3D MHD magnetic field data, Rouillard et al. (2016) showed that the average radial field measured at the outer boundary of the MAST model (and also in the potential field source surface (PFSS) model) is lower than the measured radial field values near 1 au; therefore, a calibration (correction factor) is necessary. The correction factor for the magnetic field can be obtained by comparing the radial magnetic field components of the MHD cubes at an outer boundary with the values measured near 1 au. From the MHD models we calculate the average of the unsigned radial field component over an entire source surface and apply an inverse square dependence to compare with in situ measurements near 1 au. The height of the source surface is defined as the distance where the magnetic field lines are mostly open to the interplanetary medium and magnetic fields have become very radial. In our analysis we estimate the correction factor using different heights for the surface situated between 1 and 15 R_☉. The distance of the source surface where the correction factor starts to have a constant value is on average located near ∼2.5–3.5 R_☉, hence very close to the source surface of the PFSS model.

To derive the magnetic correction factors, we process the in situ measurements in the same manner as in Rouillard et al. (2016) to compare them with the extrapolated average values from the MHD data (also see Rouillard et al. 2007). We determine the absolute magnetic field values considering a full solar rotation period and using magnetic field measurements from the OMNI database and the STEREO spacecraft. To preserve the global topology of the field, we apply the correction factor to both open- and closed-field regions of the corona. However, the closed-field regions cannot be related to the in situ measurements made near 1 au. Since we are interested in the SEP propagation along open magnetic field lines to 1 au, we adopt this technique. From the overall analysis of the magnetic correction factors of the MHD cubes we have that their values vary from ∼1.6 to ∼2.4 with an average of ∼1.8. Since the Alfvén speed is proportional to the ambient magnetic field strength, the correction of magnetic field will have for effect to decrease the computed Mach numbers of the shock, thereby providing low estimates.

The plasma density structures simulated by the MAST model are capable of reproducing global coronal features observed in WL, EUV, and X-ray emission (see Lionello et al. 2009; Rušin et al. 2010). Systematic comparisons over several solar rotations between the simulated coronal density and the density from the inversion of polarized brightness WL observations have shown that the density distributions are consistent overall with each other within a factor of 1/2 to 2 (e.g., de Patoul et al. 2015; Wang et al. 2017). To derive the density correction factors, we follow the same procedure as the one described above to derive the magnetic correction factors. We average the density values of the MAST data over the an entire source surface at different distances, and we extrapolate those values at 1 au. This region was situated beyond the region where the solar wind forms, i.e., at the very least >20 R_☉. Then, we compare the derived densities with the average densities calculated from the in situ observations over a complete solar rotation. To make our calculations more robust, we used in situ solar wind measurements from STA, STB, and L1. We found that the density correction factors vary from ∼0.30 to ∼0.63, with an average value of ∼0.45.

From the determined values of the density correction factors it is evident that the MAST model may systematically overestimate the densities when we extrapolate the simulated values to 1 au and compare them with the in situ measurements. However, it is worth noting that the intercomparisons between the observed and the simulated densities might be in reality more complex. Wang et al. (2017) presented in their Figure 7 a comparison between the 3D coronal electron densities derived by the spherically symmetric polynomial approximation (SSPA) inversion method using remote-sensing observations from STEREO coronagraphs and the MHD simulated densities. The density ratio between the MAST model and SSPA inversion method ranges from 1/8 to 5 at heights between 2.0 and 3.0 R_☉. We note that the coronal brightness below 3.0 R_☉ is mainly controlled by closed-field regions; however, in our analysis what matters is the open field, and therefore the coronal brightness beyond the region where the solar wind forms.

Therefore, if we neglect for the moment where those differences arise, it is not certain whether the simulated densities overestimate or underestimate the "true densities" (if we could measure them in situ). Those differences could be attributed either to the WL image inversion methods or to the MAST simulation. This is an interesting question that could be addressed when PSP returns its first in situ measurements below 30 R_☉.

Considering these uncertainties, we assume in our analysis that the simulated densities are accurate within 1/2 to 2 times the "true coronal densities," and we will apply the density correction factors as a separate step to our statistical analysis. We note that since the Alfvén speed is inversely dependent on the square root of the plasma density, the correction of density will have the effect of decreasing the computed Mach numbers of the shock, thereby providing conservative estimates.

Using the method presented in A. Kouloumvakos et. al. (2019, in preparation), we calculate the density compression ratio (X) from the explicit solutions of Rankine–Hugoniot (R-H) jump conditions for MHD and the plasma and magnetic field properties derived at the front location. From the R-H jump conditions we derive a third-degree polynomial of the compression ratio as a function of the M_A, plasma beta, and ${\theta }_{\mathrm{Bn}}$. We assume that the polytropic indices upstream and downstream from the shock are the same (see Section 3 in Livadiotis 2015). The solutions of the cubic trinomial can be obtained analytically and are given by Equation (25)α in Livadiotis (2015). We note that these three solutions essentially describe the compressional fast- and slow-mode MHD waves and/or intermediate waves. Additionally, from the derived density compression ratios we also calculate the magnetic compression ratio (MR) again from the explicit solutions of R-H conditions and the scattering center compression ratio (rc) from the relation ${r}_{c}=X(1-\cos {\theta }_{\mathrm{Bn}}{M}_{{\rm{A}}}^{-1})$ (see Vainio et al. 2014; Afanasiev et al. 2018).

In Figure 3 we show a 3D view of the reconstructed pressure wave for the event of 2014 February 25. Panels (a)–(d) present the 3D distributions of different parameters, i.e., the speed, ${\theta }_{\mathrm{Bn}},{M}_{{\rm{A}}}$, and X that are depicted with overplotted color maps along the ellipsoid surface. The pressure wave speed varies from ∼2100 km s⁻¹ at the apex to ∼1500 km s⁻¹ on the flank (see panel (a)). The geometry also changes from quasi-parallel to quasi-perpendicular, as expected, in the same regions. The M_A values range from ∼0.5 to ∼8, with the highest values measured in the vicinity of the neutral line, located for this particular event near the western flank and mostly over a broad region south of the apex. There is also a region in the west flank near the neutral line with M_A values lower than ∼2. In this region it is unlikely that a strong shock wave formed. This is more apparent in panel (d) of Figure 3, where we show the 3D density compression ratio. The low values of M_A and possibly the geometry do not give density compression ratios higher than 1 for the solutions of the R-H conditions; therefore, these regions are unlikely to produce a shock. The compression ratio ranges between ∼3 and 4 in the region from the shock flanks to the apex.

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To compare the properties of energetic particles measured near 1 au with the 3D properties of shocks inferred in the corona, we must consider the magnetic field lines that connect our reconstructed shock front surface with the points of in situ measurements. Using the magnetic field vector data from the MAST 3D cubes, we perform a field line tracing starting at the wave front and antisunward until the outer boundary of MAST simulation domain (30 R_☉). In this paper we consider only the open-field lines that intersect the reconstructed front surface and connect in situ spacecraft. The field line tracing is performed at every time step between launch until ∼90 minutes into the event.

Some weak points of using the MAST data to trace magnetic field lines are low grid resolutions above 10 R_☉ and the numerical diffusion. The resulting uncertainties in field line tracing can be estimated by considering the intersection point between the Parker spiral and the traced field lines at heights below 10 R_☉. The top panels of Figure 4 show a 3D view of the resulting magnetic field line tracing for the 2014 February 25 event, around 01:05 UT. Here we present 20% of the 5041 open and closed magnetic field lines that we traced. In panels (a)–(c), the different colors for the traced field lines are defined by the shock parameters at the point of intersection of the field lines with the wave surface.

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A way to sample the regional and temporal variability of shock parameters near the shock locations that are magnetically connected to in situ detectors is to map this variability on latitude–longitude maps or "Carrington maps." For a particular height in the solar atmosphere (that defines a sphere) we compute the latitude and longitude of each field line connected to the shock. We then display on that map the parameters of the shock wave at the intersection point of the field line and the shock. Essentially, the maps we use here are similar to those introduced by Plotnikov et al. (2017) to study solar gamma-ray events. With this representation it is more straightforward to evaluate how uncertainties in magnetic connectivity may affect our comparison of shock properties with SEPs measured at specific points in the inner heliosphere. Examples of the produced maps are given in the middle panels of Figure 4 at a distance of 5 R_☉ and a bin size of 20° × 15° in longitude and latitude, respectively. From left to right we show in the middle panels of Figure 4 the average values of the shock speed, M_A, and density compression ratio, respectively.

In this study we use a bin size of 20° × 15° when we perform the statistical analysis, and we examine the correlations in Section 3. Considering the uncertainty faced to establish magnetic connections with the instruments, the bin size we will use provides a good upper limit. In terms of the uncertainty in connectivity induced by the variability of the solar wind speed, the selected bin size corresponds to an uncertainty of ∼150–200 km s⁻¹. Additionally, a bin size of 20° × 15° provides a good statistical sample in every region in the map. We performed some further tests to validate these analysis uncertainties. Lower bin sizes up to ∼15° × 10° gave us results that were qualitatively and quantitatively similar.

As a next step we use these maps to derive the temporal evolution of the shock parameters only at the magnetically well-connected regions with the instruments at 1 au. First, we assume that the interplanetary field lines can be roughly modeled as an Archimedian spiral, and we calculate the location of the magnetically well-connected footpoints for each spacecraft. The locus of the spirals and therefore the location of the footpoints of the field lines of interest are controlled by the speed of the solar wind carrying them and the solar rotation period. We use the average solar wind speed measured in situ near the onset time of the SEP event. The assumption of an Archimedian spiral to connect near-1 au observations with the shock requires that the SEP event occurred during quiet solar wind conditions in the region situated between the Sun and the instrument making the in situ measurements. This might not be the case for SEP events that occur at periods of extreme solar activity. We overplot on the maps the location of the spiral footpoint for each spacecraft (see Figure 4, middle row); to select the bins, we perform the statistics of the shock parameter. An example of the temporal evolution of the shock parameters, at the well-connected regions for each spacecraft, is given in the bottom panels of Figure 4.

The uncertainty of the derived shock parameters at the magnetically well-connected field lines is evaluated from the parameters' statistics. The upper and lower limits are derived from the distributions' values at the 10th and 90th percentiles. We use those estimates of the shock parameter uncertainties in the next sections, where we compare the modeled shock parameters with SEP measurements.

We start this analysis with the correlation of the SEP peak intensities at 20–26 MeV (Ip20), 40–60 MeV (Ip40), and 60–100 MeV (Ip60) with the shock speed (see Sections 2.2 and 2.4). We use throughout our analysis logarithms when we perform regressions and correlations because of the dynamic range of the peak intensities and the other parameters. The log−log regression fittings yield the following relation between our measurements: Ip ∼ ${k}_{2}^{{\prime} }{V}^{{k}_{1}^{{\prime} }}$. The coefficients can be directly comparable to those derived in previous studies (e.g., Kahler & Vourlidas 2013). To determine the correlation between the parameters, we use the Pearson's correlation coefficient (CC), which measures the amount of linear dependence between two variables, and it is defined as the covariance of the two variables divided by the product of their standard deviations.

The top row of Figure 5 shows the correlation between the Ip and the maximum speed derived at the shock apex during its evolution, for the three energy bands we consider for this study. Here, we consider only S/C-specific SEP events with a magnetic connection longitude that is less than 50°. We also show the same correlations with the projected CME speed. As already discussed in Section 2.1, to account for (1) the random errors in the estimation of the peak intensities and (2) any errors produced from the calibration of the different instruments at the energy channels, we use for the uncertainty in Ip a standard error of 50% of their measured values. Additionally, for the uncertainty of the determined shock speed at the apex we use a standard error of 10%, which is a rough estimate based on the accuracy of 3D ellipsoid fittings (see Section 2.3.1). We use the same error for the CME speed; however, we recognize that this might be greater.

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From the Pearson's correlation between the Ip and the shock speeds at the apex we found that the CCs hardly exceed 55% in every case, despite the fact that the speeds do not suffer from projection effects since they are derived from the shock fittings. The highest CC is observed for Ip20, $\mathrm{CC}=54.4 \%$. As noted in Section 2.3.1, for our analysis we use the speeds that are directly derived from the 3D shock fittings and not from measurements in the plane of the sky. The lack of a good correlation is more or less expected since the speed at the shock apex is unlikely to be, for most events, a representative parameter of the region at which the particles are accelerated. A first expectation is that when the shock parameters are derived directly at the well-connected field lines, the correlations will improve significantly. Additionally, we found similar CC values for the CME speeds. The best CC is observed for Ip20, $\mathrm{CC}=58.8 \%$.

In the bottom row of Figure 5 we present the correlation of Ip with the maximum shock speed determined at the well-connected regions. We exclude from the correlations the events where the SEP onset times are registered >3–6 hr later (depending on the energy) than the time when shock waves pass the magnetically well-connected regions. For those SEPs, the the link between shock wave expansion and the timing of SEPs may be affected by transport processes. We examine those cases in more detail in Section 3.6. From here on, we will use gray circles to depict those cases that are rejected from our statistical analysis.

From the CCs of Figure 5 (bottom row) it is clear that we have a significant improvement of the CCs and their p-values in every energy range, when we compare Ip with the shock speed at the well-connected regions (i.e., when we included the magnetic connectivities to our analysis). The best correlation is observed for Ip20 with ${\mathrm{CC}}_{{\rm{V}}20}=64.9 \%$, while for Ip40 and Ip60 we have ${\mathrm{CC}}_{{\rm{V}}40}=65.6 \%$ and ${\mathrm{CC}}_{{\rm{V}}60}=61.6 \%$, respectively. The calculated p-values for our correlations are on the order of ${10}^{-8}\mbox{--}{10}^{-9}$, so the CCs are statistically significant. The aggregate results for the derived Pearson's CCs and their p-values are summarized in Table 2. From the fitted function, $\mathrm{log}({I}_{p})={k}_{1}\mathrm{log}(V)+{k}_{2}$, we have the following coefficients, k = [k₁, k₂]: k = [3.30, −23.19] for Ip20, k = [3.36, −26.06] for Ip40, and k = [3.14, −25.60] for Ip60.

Table 2.  Pearson's Correlation Coefficients between the SEP Peak Fluxes in Three Energy Channels and the Different Shock Parameters

CCs	Ip20	Ip40	Ip60
(1)	(2)	(3)	(4)
VCME	58.8% (10−4)	51.3% (0.002)	49.7% (0.003)
	[41%, 72%]	[33%, 68%]	[30%, 66%]
Vsha	54.4% (0.001)	53.0% (0.002)	53.4% (0.002)
	[36%, 69%]	[34%, 68%]	[33%, 68%]
Vsh	64.9% (10−9)	65.6% (10−9)	61.6% (10−8)
	[56%, 74%]	[56%, 74%]	[51%, 71%]
MA	71.4% (10−11)	75.6% (10−13)	68.9% (10−10)
	[63%, 79%]	[68%, 82%]	[59%, 77%]
MAb	71.4% (10−11)	75.0% (10−13)	68.6% (10−10)
	[63%, 79%]	[67%, 81%]	[59%, 76%]
Mfm	67.7% (10−10)	69.7% (10−11)	64.3% (10−9)
	[59%, 76%]	[61%, 77%]	[54%, 73%]
Mfmb	67.6% (10−10)	69.1% (10−10)	63.9% (10−9)
	[59%, 76%]	[60%, 77%]	[54%, 73%]
X	60.4% (10−8)	59.4% (10−7)	53.4% (10−6)
	[50%, 70%]	[48%, 69%]	[41%, 64%]
MR	50.3% (10−5)	55.0% (10−6)	47.4% (10−5)
	[38%, 62%]	[42%, 65%]	[33%, 58%]
rc	55.8% (10−6)	62.0% (10−7)	53.3% (10−6)
	[46%, 69%]	[53%, 72%]	[45%, 67%]
${\theta }_{\mathrm{Bn}}^{M}$	−15.7% (0.15)	−10.5% (0.3)	−12.3% (0.3)
	[−33%, 0%]	[−27%, −1%]	[−29%, 0%]
${\theta }_{\mathrm{Bn}}^{C}$	26.8% (0.14)	28.1% (0.1)	30.0% (0.1)
	[0%, −45%]	[1%, 47%]	[0%, 47%]

Notes. Next to the Pearson's correlation coefficients we include the p-value of the statistics, and below we give in parentheses the upper and lower bounds of the CCs from the Fisher transformation, calculated for a confidence interval of 90%.

^aMaximum shock speed measured at the apex. ^bWith the application of the density correction factors.

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Previous studies have shown that there is a slight tendency for the CCs of Ip with shock speed to decrease at higher energies (see Papaioannou et al. 2016). This trend appears in our analysis, but it is too small an effect to tell whether this dependence is statistically important. We can evaluate how significant it is by examining the confidence interval of the Pearson's coefficient using the so-called Fisher r-to-z transformation. This method has been applied in statistical analysis of SEP characteristics by Papaioannou et al. (2016) and cosmic rays (see also Rouillard & Lockwood 2004). Essentially, for a specified value of a confidence interval, it is possible to calculate the limits of the confidence interval around our CCs from the Fisher transformation. This transformation is defined as z_r = artanh(r), where r is the Pearson's CC. For a confidence interval of 90% (significance level: α = 0.1) we determine upper and lower limits of our CCs using the relation ${r}_{+/-}=\tanh ({z}_{u/l})=\tanh ({z}_{r}\pm {z}_{1-\alpha /2}{\sigma }_{z})$, where ${z}_{1-\alpha /2}=100(1-a/2),{\sigma }_{z}$ is the standard deviation ${\sigma }_{z}=1/\sqrt{N-3}$ and N is the sample size used for the CC determination. In Table 2 we include the calculated upper and lower limits for the CCs. From the small differences between the CCs at the different energies and the width of the calculated bounds from the Fisher transformation it is rather inconclusive whether the speed–Ip correlations have any energy dependence.

Next, we examine the correlation between the SEP peak intensities and the maximum Alfvénic or fast magnetosonic Mach number, both determined at the well-connected field lines. A first expectation is that the correlations should be significant based on the results of Reames (2012) and Rouillard et al. (2016). Additionally, recent simulations of DSA (e.g., Afanasiev et al. 2018) show that the acceleration efficiency of the shock tends to be higher for larger values of the scattering center compression ratio that depends on M_A.

In Figure 6 we show the correlations between Ip20, Ip40, Ip60, and M_A. The uncertainty (i.e., the error bars in Figure 6) of the derived M_A values is evaluated from the statistics on the well-connected field lines, and for the Ip we use a standard error of 50% as explained earlier in the text. In the top panels of Figure 6 we show the correlations between the SEPs' peak intensity and M_A without considering the density correction factors (see Section 2.4 for details). In the bottom panels we show the same correlations with the application of the density correction. The CCs remain almost unchanged after the application of the density correction factors. The correction of the simulated densities acts as a scale factor for the M_A values and has no effect on the statistical output.

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From the correlations of Figure 6 we find that M_A correlate extremely well with Ip in all three energy ranges, with CCs greater than 70%. Essentially, the resulting relations and correlations show that the magnetically well-connected shock regions that have high Alfvénic Mach number relate well to SEP events observed in situ with high peak intensities and vice versa. This is an important result that shows a direct connection of the properties of the shock region that the energetic particles accelerated and released into the open magnetic field lines and the characteristics of the SEPs measured in situ in terms of their peak intensity. Another aspect that might be of importance arises from the rejected points (see Section 3.1), which we depict with the gray circles in Figure 6. In all cases except for some outliers (two to four cases depending on the energy) the points concentrate in a region of low peak intensities and M_A less than ∼2, while almost half of rejected cases (e.g., 7/15 for 40–60 MeV) have M_A less than one for which a shock was unlikely to form along the connected magnetic field lines (see also Section 3.6). From the fitted function, $\mathrm{log}({I}_{p})={k}_{1}\mathrm{log}({M}_{{\rm{A}}})+{k}_{2}$, we have the following coefficients: k = [1.94, −1.81] for Ip20, k = [2.07, −4.38] for Ip40, and k = [1.88, −5.29] for Ip60.

The best correlation is found for Ip40, ${\mathrm{CC}}_{\mathrm{MA}40}=75.6 \%$. We find smaller correlations for Ip20, ${\mathrm{CC}}_{\mathrm{MA}20}=71.4 \%$, and for Ip60, ${\mathrm{CC}}_{\mathrm{MA}60}=68.9 \%$. Additionally, the calculated p-values for the CCs are on the order of ${10}^{-10}\mbox{--}{10}^{-13}$ (see Table 2). We also calculate the CC bounds from the Fisher transformation in Table 2. Comparing the correlations between M_A and the shock speed with the Ip, it seems that the CCs are significantly improved for M_A. The relative improvement of the CCs ranges from ∼6% to 10%, depending on the energy. Considering the upper and lower bounds of the CCs from the Fisher transformation, we evaluate the significance of the difference between the CC_V and CC_MA. The derived difference probabilities range between ∼0.13 and ∼0.23 (i.e., confidence 87%–76%). For the CC_V40 and CC_MA40 we find that there is a significant difference with a confidence of ∼87%, while for the CCs at the other two energy ranges considered here, the null-hypothesis test gives lower confidence intervals. It is rather inconclusive whether the difference is significant or not in those cases. This is likely an effect of the low statistical sample that we have (65–70 points depending on the energy). Additionally, it is not clear in our statistics whether there is an energy dependence on the CCs; however, the weaker correlation is found for the high-energy channel, 60–100 MeV.

In Figure 7 we show the correlations between the SEPs' peak intensities (Ip20, Ip40, Ip60) and the fast magnetosonic Mach number, M_fm. Comparing the CCs of CC_Mfm with the shock speed correlations, we find a relative improvement of ∼4%. Again the resulting CC values are similar either with the application of the density correction factors or not (see Table 2); however, the CCs are lower than those found for M_A.

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In Figure 8 we present the correlations of Ip with the shocks' compression ratios. From top to bottom we show the density (X), the MR, and the scattering center (rc) compression ratio,¹⁴ respectively. For the density and the scattering center compression ratio the correlations are good, with CCs ranging from ∼53% to ∼62%. For the magnetic compression ratio we find lower CCs, from ∼47% to ∼55%. In all the cases the CCs for the compression ratios are lower than the CCs we find for the shock speed or the Mach numbers.

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From the statistics in Figure 8 it is important to note the following: the issue encountered with the correlations is that the density compression ratio saturates fast with increasing values of the peak intensities, and the points cluster in a narrow region between X ∼ 3.5 and 4. This clustering results from the solutions of the R-H jump conditions. When the Mach number tends to values beyond 10, the compression ratio tends to an asymptotic value of 4 when the polytropic index γ = 5/3.

The relatively good correlations in both the scattering center and the density compression ratio suggest that efficient acceleration is expected to occur in regions of increased Mach number. Both compression ratios, X and r_c, depend on the magnitude of M_A and the shock geometry. Therefore, our survey based on 33 SEPs detected supports the conclusions of the case studies in Rouillard et al. (2016) and Plotnikov et al. (2017). Along with the M_A number, r_c is a critical parameter in simulations of DSA (Vainio et al. 2014; Afanasiev et al. 2018). The best CCs are found for the scattering center compression ratio at 40–60 MeV (CC ∼ 64%), while in any energy range the CCs of the magnetic compression ratio are the lowest found here.

We follow a different approach to investigate the link between ${\theta }_{\mathrm{Bn}}$ and Ip. Instead of using the maximum value of θ_BN in the well-connected region, we use the value of θ_BN at the time when M_A is maximum or the shock becomes supercritical. The reason that drives us to switch our method is the time dependence of the θ_BN in the well-connected regions. In most of the cases, when the shock connects for the first time with the open magnetic field lines, the geometry is quasi-perpendicular in the well-connected regions. Progressively as the shock further evolves, its geometry at the connected region becomes more oblique, and finally it switches to quasi-parallel shock.

In Figure 9 we show the correlations between Ip and the ${\theta }_{\mathrm{Bn}}$ value at the time of (1) maximum M_A (red points) and (2) ${M}_{{\rm{A}}}={M}_{{\rm{A}}}(C)$ (black points). For condition 1 we find a weak anticorrelation, while for condition 2 we find a weak correlation of ${\theta }_{\mathrm{Bn}}$ with the Ip. The CC values in any of those cases are the worst found so far (see Table 2). The absolute CC values are lower than 30%. From the results of Figure 9, it is inconclusive whether the shock geometry can be connected with the acceleration efficiency. A more detailed time-dependent analysis is needed to further elucidate this aspect. Some predictions of the DSA theory show that highly oblique shocks have a higher acceleration efficiency compared to shocks that are nearly aligned along the magnetic field (e.g., Jokipii 1987; Giacalone 2005), while other DSA theoretical models (Zank et al. 2000; Lee 2005) and simulations (Vainio et al. 2014; Afanasiev et al. 2018) predict that protons accelerated by quasi-parallel shocks stream upstream away from the shock and generate self-excited Alfvén waves that scatter particles back to the shock for further acceleration (Vainio & Laitinen 2007).

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We further examine the relative time difference between the estimated release time of the S/C-specific SEP events and the time when the shock waves reach the well-connected field lines for each S/C. We use Tables 4 and 5 of Paassilta et al. (2018), which include the SEP release times for the events in our list. Those times were inferred from the application of either the velocity dispersion analysis (VDA) or the time-shifting analysis (TSA) to the SEPs' onset times.

We perform this timing analysis by selecting a characteristic time when the shock waves are both connected to the S/C and sufficiently strong to accelerate particles to high energy. From the temporal evolution of the shock parameters at the well-connected field lines we register for each S/C-specific SEP event the time when the shock wave becomes supercritical (M > M_c) (and connected to the S/C). Since the threshold for criticality depends on local plasma conditions β and geometry ${\theta }_{\mathrm{BN}}$, and because for $\beta \lt 1$ the M_c varies from ∼1.53 for parallel shocks and ∼2.76 for perpendicular shocks, we choose an average value of ${M}_{c}\sim 2$ to infer the time when a well-connected region at the shock front is supercritical or not.

In the top panel of Figure 10 we show the distributions of the computed time difference, ${\rm{\Delta }}T$, between the SEP release time and the time when the shock becomes supercritical. We have rejected the cases in Paassilta et al. (2018) when the release times have an uncertainty greater than 60 minutes. From the timing distributions of the top panel, we find a maximum at –15 minutes for both VDA and TSA and a median around −11 minutes from VDA and −23 minutes from TSA, i.e., the SEP release occurs on average later than the time when the well-connected shock waves become supercritical along specific lines. This time difference must to a large extent be attributed to the time for the particles to be accelerated to high energy by the shock. The scatter of timings around the distribution maximum/mean values arises mainly from two likely sources: (1) the uncertainty in the methods used to estimate SEP release time (mean errors of ∼30 minutes; see Paassilta et al. 2018), and (2) the inherent uncertainty from the selection of the critical Mach number threshold. Additionally, there are cases for which release times cannot be explained from the evolution of the shock wave parameters only.

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In the bottom panel of Figure 10 we show the time difference as a function of the absolute longitudinal angular separation between the S/C magnetic connection point at the shock and the eruptive active region, ${\rm{\Delta }}{\rm{\Phi }}$. We have found for all the triangulated shock waves that the heliocentric position of the eruptive active region is generally close to the direction of propagation and therefore the apex of the CME. For ${\rm{\Delta }}{\rm{\Phi }}\lt 70^\circ$ the absolute time differences range from a few minutes up to ∼80 minutes, with median values of ∼10–15 minutes (excluding the three outliers). For ${\rm{\Delta }}{\rm{\Phi }}\gt 70^\circ$ the time differences become greater than 90 minutes and the particle release is difficult to explain from the evolution of the shock wave. Those extreme cases (three for VDA and eight for TSA) with high time differences must be a product of at least two effects. There are typically large uncertainties involved for the estimation of the SEP release time when the particle flux measured in situ rises very gradually in comparison with other events that exhibit prompt and sharp onsets in particle fluxes. The other is likely due to the actual acceleration/transport mechanisms at play. The pressure waves we model far from the apex are no longer shocks but rather fast-mode waves; consequently, spacecraft at these longitudes connect magnetically to likely inefficient accelerators. We discuss these points further in the next section.

In the left panel of Figure 11 we show the relation between the maximum Alfvén Mach number M_A (black points) and maximum shock speed (red points) with respect to the separation angle ${\rm{\Delta }}{\rm{\Phi }}$ defined in the previous paragraph. The two horizontal lines depict the thresholds for M_A > 1 (dot-dashed line) and M_A > M_c (dashed line).

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An anticorrelation between M_A/shock speed and ${\rm{\Delta }}{\rm{\Phi }}$ is observed in the left panel of Figure 11. In addition to the anticorrelation, the range of M_A/shock speed values decreases as ΔΦ increases. For SEPs with good magnetic connectivity, i.e., low ΔΦ values, the magnetic field lines are connected closer to the wave apex where the shock is strongly driven. The dynamic range of possible shock parameters is controlled by the driver (flux rope) characteristics. As ΔΦ increases, the magnetic connectivity is displaced toward the flanks of the CME event, where the pressure wave becomes detached from the driver and progressively evolves into a freely propagating wave. The observed reduction of the range of M_A/shock speed therefore corresponds to the transition from a fast-driven motion to a more freely propagating motion at the local characteristic speed (see Figure 11).

The results of Figure 11 (left) show that for ${\rm{\Delta }}{\rm{\Phi }}\lt 120^\circ$ supercritical shocks are measured in all but eight of the magnetic connections. For ΔΦ ∼ 90°–120°, there are cases where a connection to a supercritical shock region is still measured; however, for higher ΔΦ values (>120°) we find only four instances of supercritical shocks.

In the middle panel of Figure 11 we present, as a function of ΔΦ, the height where the shock becomes supercritical (red points) and where M_A maximizes (black points) along the field lines connected to spacecraft measuring SEPs. Supercriticality is reached between ∼1.2 and 6 R_☉, and the shock waves reach their maximum strength (in terms of maximum M_A) at heights well below ∼20 R_☉. We also examined whether there is any dependence of the observed particle intensity on the connection height where the shock becomes supercritical (Figure 11, right panel). We found a weak anticorrelation between the connection height and the Ip, which suggests that the most intense SEPs are produced in shock regions that become supercritical low in the corona.