Electron Energy Partition across Interplanetary Shocks. III. Analysis

In this section, SEAs of s_ec/$\langle {s}_{{ec}}{\rangle }_{\mathrm{up}}$, κ_eh/$\langle {\kappa }_{{eh}}{\rangle }_{\mathrm{up}}$, and κ_eb/$\langle {\kappa }_{{eb}}{\rangle }_{\mathrm{up}}$ are introduced and discussed in Figure 1 (see Appendix A for definitions). First, 50% of the halo and beam/strahl exponents (i.e., between the lower and upper quartiles) satisfy 3.57 ≲ κ_eh ≲ 5.31 and 3.41 ≲ κ_eb ≲ 5.11, which are consistent with previous solar wind observations near 1 au (e.g., Maksimovic et al. 1997, 2005; Štverák et al. 2009; Pierrard et al. 2016; Tao et al. 2016a, 2016b; Lazar et al. 2017; Horaites et al. 2018). (Note there are no studies with which to compare the s_ec values).

The s_ec exponents are relatively constant across most IP shocks, with a few exceptions for Criteria LM and Criteria PA shocks showing values up near ∼1.5. The Criteria HM are especially sparsely populated in the downstream, as many of these VDFs were fit to the asymmetric self-similar model distribution (i.e., p_ec and q_ec exponents, not shown). However, the range of data between X_5% and X_95% is larger in the downstream, consistent with the interpretation of a wave-induced inelastic interaction discussed in Paper I.

The κ_eh panels (middle column) are more interesting and varied. For instance, the running median clearly increases across the shock. The change is the largest for Criteria HM and Criteria PE shocks. Theory and simulation suggest that stronger and more oblique shocks are better at energizing suprathermal electrons (e.g., Treumann 2009; Caprioli & Spitkovsky 2014; Park et al. 2015; Trotta & Burgess 2019). However, if the normalization used in Figure 1 is removed (not shown), the upstream only values of κ_eh between X_min and X_max cover a similar range for Criteria LM and Criteria HM shocks, with similar running medians as well. The downstream only κ_eh values are larger at Criteria HM than Criteria LM shocks, which explains the larger one-variable statistics values of κ_eh for Criteria HM than Criteria LM shocks shown in Paper I.

Note that the one-variable statistics of κ_eh reported in Paper I showed a slightly larger value for Criteria PE than Criteria PA shocks in agreement with the larger change across Criteria PE than Criteria PA shocks observed in Figure 1, consistent with theory and simulations (e.g., Wu 1984; Park et al. 2013; Trotta & Burgess 2019). While the change is not in the direction expected, the magnitude of the change is certainly larger for Criteria HM than Criteria LM shocks, suggesting stronger shocks have more of an impact on the halo than weaker. While the upstream kappa values have recently been predicted to increase with increasing Mach number for quasi-perpendicular shocks (Trotta & Burgess 2019), the normalization by the upstream median values for each shock removed the relative offsets to avoid this potentially misleading difference.

The increase in κ_eh across the shocks could result from several effects. For instance, it could result from core electrons being energized to suprathermal energies, initially starting with a larger equivalent kappa value, thus increasing the overall suprathermal exponent value. The increase in κ_eh could also result from acceleration through a quasi-static electric field (e.g., Scudder et al. 1986; Schwartz et al. 1988, 2011; Mitchell & Schwartz 2014; Schwartz 2014) for a one-dimensional VDF. The change in κ_eh for a two-dimensional VDF in velocity space is complicated if the quasi-static electric field is not uniformly aligned with both component directions in velocity space, as a kappa VDF is not a simple power law (e.g., Livadiotis 2015). Even so, the effect of a quasi-static electric field could result in the fit software finding a larger kappa value as the higher energy particles change velocity by a smaller fraction than the lower. However, the nonlinear relationship between the values of κ_s, n_s, and ${V}_{{Ts},j}$ in affecting the shape and peak phase space density of a bi-kappa VDF (e.g., see Figure 2 of Paper I) makes it difficult to interpret the change in κ_eh.

The κ_eb panels (right-hand column) are less dynamic than the κ_eh panels, but there is an important similarity. The change in the running median is largest for Criteria HM shocks, with all other selection criteria showing little-to-no change or a weak gradual change across the entire time range. The biggest difference between κ_eh and κ_eb is that the latter does not show a clear increase at the shock ramp for Criteria PE shocks. Similar to the discussion above for κ_eh, if the normalization is removed from Figure 1 for κ_eb, the difference between Criteria LM and Criteria HM shocks is mostly in the variation in the running median. For Criteria LM shocks, the running median of κ_eb is very smooth and does not change much at the shock ramp (i.e., changes by less than ∼0.5), while the running median for Criteria HM shocks fluctuates with a normalized amplitude at or above unity. Again, the change in the running median does not show a strong change at the shock ramp in the unnormalized SEA (not shown).

In this section, SEAs of ${n}_{s}/{n}_{\mathrm{eff}}$ are introduced and discussed, for the core (s = ec), halo (s = eh), beam/strahl (s = eb), and effective (s = eff). The halo-to-effective density ratios are presented in the first column of Figure 2, the halo-to-effective ratios are presented in the second column, and the beam-to-effective ratios are presented in the third column.

Figure 2 shows the ratio of each electron component number density to the effective number density, n_eff (see Appendix A for definitions), to illustrate the relative change in the fractional composition of the electron distribution. That is, the SEA plots show whether the core, halo, and/or beam/strahl number density fraction increase or decrease across the shock for each selection criteria. It is quite obvious that the median core fraction always increases across the shock, regardless of selection criteria. The difference between selection criteria is that magnitude of the change (e.g., Criteria PA) shows very little positive change in the running median and a skewness toward smaller values in the running 5th percentile.

Unlike the core, both the halo and beam/strahl fractional densities decrease across the shock for all selection criteria. Again, the weakest change is found in the Criteria PA shocks, which is not surprising as quasi-parallel shocks show smaller magnetic field compression ratios. Thus, the change in the core density is smaller which impacts the halo and beam/strahl fits. The largest change across the shock ramp occurred for Criteria LM and Criteria PE. It should also be noted that the gradient in the running median is sharpest for the beam/strahl component. The halo shows a short enhancement upstream of the shock ramp and then a decrease that continues shortly into the downstream past the ramp. The width of the upstream enhancement is larger for the Criteria HM than Criteria LM shocks, suggesting it results from some form of shock acceleration forming an electron foreshock. This is further evidenced by the stark difference in the running median profile between the Criteria PA and Criteria PE shocks, where the latter are expected to be better at energizing electrons (e.g., Wu 1984; Park et al. 2013; Trotta & Burgess 2019).

The beam/strahl gradients occur almost entirely in the thin region focused on the ramp. On the timescale shown in these figures, one minor tick mark is ∼180 s, so the halo gradient from upstream to downstream lasts upward of ∼15 minutes. The beam/strahl gradient is much shorter at ∼6 minutes (except for Criteria PA and Criteria HM shocks, which have a much gentler gradient). Recall the time windows for calculating the running median are 120 s and the windows are shifted by 22 s each time; thus the sharpest gradient one might expect would be slightly surpassing 2 minutes.

These plots also show that the fraction of suprathermal particles decreases across most IP shocks, which may explain why the suprathermal exponents tend to increase across the shocks. Although the suprathermal temperatures tend toward larger values in the downstream regions (see additional SEA plots of ${T}_{s,j}$ found in Wilson et al. 2020), the change is very weak except for Criteria HM shocks. However, the core temperature changes across the shocks are much more dramatic than both the halo and beam/strahl. Further, the beam/strahl shows the weakest changes in ${T}_{s,j}$ for all selection criteria (i.e., the beam/strahl component's mean kinetic energy is not strongly affected by the shock).

This section provides superposed epoch analyses (SEAs) of the electron plasma betas, ${\beta }_{s,j}$, for the core (s = ec), halo (s = eh), and beam/strahl (s = eb) components. The core betas are presented in Figure 3, the halo temperatures are presented in Figure 4, and the beam/strahl temperatures are presented in Figure 5.

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The next thing to examine is the plasma betas, ${\beta }_{s,j}$, for the core (s = ec), halo (s = eh), and beam/strahl (s = eb) components. The SEA plots in Figures 3–5 of ${\beta }_{s,j}$ show the largest change across the shock compared to all other electron fit parameters examined. Of the three components, the core shows the smallest change, but all three electron components decrease across the shock. The weak decrease in ${\beta }_{{ec},j}$, compared to ${\beta }_{{eh},j}$ and ${\beta }_{{eb},j}$, is dominated by the increase in the magnitude of ${{\boldsymbol{B}}}_{o}$ across the shock, since both ${T}_{{ec},j}$ and ${n}_{{ec}}$ increase. Although both ${T}_{{eh},j}$ and ${T}_{{eb},j}$ tend to increase across the shock, the increase is weak. This weak increase coupled with the stronger decrease in both the fractional n_eh and n_eb and increase in the magnitude of B_o makes for the corresponding decrease in suprathermal betas. Note, however, that the changes in n_eff and B_o are largely governed by the Rankine–Hugoniot conservation relations, while the change in any given ${T}_{s,j}$ is not. The change in ${T}_{{ec},j}$ is due to the partition of free energy available due to ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}\ne 0$, which will be discussed in Section 5.

Similar to the fractional n_eh, there is evidence of a foreshock in ${\beta }_{{eh},j}$ shown as a two-step decrease—the beta begins to drop ahead of the shock ramp, reaches a plateau or slight jump near the ramp, then continues to drop to the downstream values for all selection criteria except Criteria HM and Criteria PA. The interesting thing is that the upstream running median is roughly near unity for all ${\beta }_{s,j}$, by design of course, but the downstream running median drops to as low as ∼0.20 for ${\beta }_{{eh},j}$ and ${\beta }_{{eb},j}$. That is, the running median decreases by upward of ∼80% across the shock for the suprathermal electrons. Therefore, the ratio of the thermal-to-magnetic field energy density of the suprathermal electrons is upward of ∼80% smaller in the downstream than upstream. Another interesting feature is that ${\beta }_{{eh},j}$ begins to decrease upstream of the ramp center near the times when the fractional n_eh increases, which appears to be due to a local decrease in ${T}_{{eh},j}$.

There is some bias in the steepness of the gradient due to the cluster of data near the ramp center of the IP shocks. This is due to the burst mode trigger of the 3DP instrument, which was designed to capture shock ramps in high time resolution. Thus, the large spread of values near the ramp center is dominating many of the running median calculations through that time period. In effect, this is actually reducing the variance in the running median seen in the more sparsely sampled far upstream and downstream regions. Therefore, it is worth noting to avoid overinterpreting these gradients.

In this section, the instability thresholds for the WHFI and WTAI are examined based on the observed properties of the electron VDFs. Specifically, this section will focus on the numerical results of Gary et al. (1994) and Gary et al. (1999). To this end, the electron heat flux is calculated, which requires stable fit solutions for all three electron components (i.e., 10,983 of the total 15,210 VDFs examined satisfy this requirement; see Paper II for further details on the integration of the total electron model functions).

Figure 6 is an adaptation of Figures 7 and 8 from Gary et al. (1994), where the red (purple) line in the left (right) panel corresponds to the lower line in Figure 7(8) of the original work (i.e., the threshold for the WHFI). The threshold values shown in Figure 6 are defined as instabilities having a maximum positive growth rate, γ_max, satisfying γ_max > 10⁻¹ Ω_cp. Over-plotted are the electron VDF fit results from the present work. In the range of ${\beta }_{{ec},\parallel }$ shown, there are 9362 valid VDF fit results. For reference, 90% of the data for Criteria AT satisfy the following:

• 1.  
  0.28 rad s⁻¹ ≲ Ω_cp ≲ 2.03 rad s⁻¹;
• 2.  
  0.49 s ≲ ${\rm{\Omega }}{{}_{\mathrm{cp}}}^{-1}$ ≲ 3.52 s;
• 3.  
  0.76 km ≲ ρ_ceff ≲ 4.04 km;
• 4.  
  23.6 km ≲ ρ_p ≲ 167 km;
• 5.  
  0.93 km ≲ λ_ceff ≲ 3.65 km; and
• 6.  
  36.7 km ≲ λ_p ≲ 154 km.

Thus, the growth times corresponding to the threshold lines in Figure 6 satisfy 4.92 s ≲ $\gamma {{}_{\max }}^{-1}$ ≲ 35.2 s or ∼0.03%–0.24% of the total time examined around each IP shock.

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Figure 6 shows that >80% of all 10,983 VDFs are at or above the threshold for either the WHFI or WTAI. Limiting to the 9362 VDFs shown in Figure 6, ∼54% are unstable to the WHFI and ∼43% are unstable to the WTAI. That is, only ∼3% of the VDFs are stable for these criteria. Further, Gary et al. (1999) noted that in the presence of a finite electron heat flux and ${{ \mathcal A }}_{{eh}}\gt 1.01$, the heat flux carrying electrons are always unstable to the WHFI.¹⁷ Of the 10,983 VDFs with a calculated heat flux, ∼62% satisfied ${{ \mathcal A }}_{{eh}}\gt 1.01$ (i.e., ∼62% are at least linearly unstable to the WHFI). These rates are significantly larger than recent work using data from the ARTEMIS mission (e.g., Tong et al. 2019), which found occurrence rates of whistler waves to be ≲2%. However, the ARTEMIS work limited the observations to the pristine solar wind and used ${{ \mathcal A }}_{\mathrm{eff}}$ instead of ${{ \mathcal A }}_{{eh}}$ for anisotropy threshold calculations. When they limit their occurrence rate estimates to intervals satisfying ${{ \mathcal A }}_{\mathrm{eff}}\gt 1$, the rates jump to ∼15%.

Note that Gary et al. (1994) only used two bi-Maxwellian electron components while this work uses three non-Maxwellian electron components, which makes the comparison subject to scrutiny. For instance, the use of bi-Maxwellian instead of bi-kappa is known to cause differences in the WHFI (e.g., Shaaban et al. 2018; Lee et al. 2019) and other instabilities (e.g., Lazar et al. 2012, 2013, 2014, 2017, 2018, 2019; Shaaban et al. 2019a). In addition, the definition of ${{{ \mathcal T }}_{{ec}}^{{eh}}}_{\parallel }$ used to create the left-hand panel in Figure 6 relied upon the halo and core components only, not a single suprathermal population like that used by Gary et al. (1994).

More than half of the VDFs are unstable to the WHFI, which is consistent with the observation that both ${{ \mathcal A }}_{{eb}}$ and κ_eb are generally smaller than ${{ \mathcal A }}_{{eh}}$ and κ_eh (see Paper II for values and statistics), respectively. That is, the radiation of the wave reduces ${{ \mathcal A }}_{{eb}}$ and κ_eb and subsequent scattering can increase ${{ \mathcal A }}_{{eh}}$, though it should be noted that κ_eh ≳ κ_eb has been found to be true in previous solar wind studies. Statistically, the halo exponent tends to be larger than the beam/strahl at 1 au in some studies (e.g., Štverák et al. 2009), though others show the converse and a solar cycle dependence (e.g., Tao et al. 2016a). Therefore, the larger κ_eh may be a remnant of the typical conditions in the solar wind and not directly related to local instabilities. Alternately, the differences between the two exponents may be observed with this relationship precisely because the two populations are intrinsically coupled through whistler-like instabilities. It may also be that quasi-static fields are affecting the core more than the halo by energizing them to suprathermal energies, thus effectively increasing the suprathermal electron exponents (e.g., Scudder et al. 1986; Schwartz et al. 1988, 2011; Mitchell & Schwartz 2014; Schwartz 2014). Kinetic simulations are required to resolve this discrepancy and are beyond the scope of this study.

In this section, the instability thresholds for short wavelength,¹⁸ electrostatic IAWs will be calculated and discussed. The purpose is to calculate the probability of occurrence and determine whether such modes could be playing a role in the evolution of the electron VDFs across the examined IP shocks.

A long-standing problem in solar wind physics is the occurrence of short wavelength IAWs (e.g., Fuselier & Gurnett 1984; Gurnett et al. 1979a, 1979b; Wilson et al. 2007) despite the commonly observed electron-to-ion temperature ratios satisfying ${{{ \mathcal T }}_{i}^{\mathrm{eff}}}_{\mathrm{tot}}\lt 3$. The temperature ratio threshold derives from the assumption of single, isotropic Maxwellian VDFs for both the electrons and ions, which shows that current-driven IAWs are heavily Landau damped if ${{{ \mathcal T }}_{i}^{\mathrm{eff}}}_{\mathrm{tot}}\lt 3$ (e.g., Fried & Gould 1961; Fried & Wong 1966; Gould 1964; Gurnett et al. 1979a). However, temperature gradients (e.g., Priest & Sanderson 1972; Allan & Sanderson 1974; Dum 1978a, 1978b), shear flow (e.g., Agrimson et al. 2001; Gavrishchaka et al. 1999), nonlinear whistler wave decay (e.g., Saito et al. 2017), finite electron heat flux (e.g., Dum et al. 1980), nonlinear Langmuir wave decay (e.g., Zakharov 1972; Zakharov & Rubenchik 1972; Kellogg et al. 2013; Cairns & Layden 2018), and ion-ion instabilities (e.g., Ashour-Abdalla & Okuda 1986; Winske et al. 1987; Goodrich et al. 2018, 2019) have all been shown to reduce or eliminate this temperature ratio threshold.

Regardless, the occurrence rates of ${{{ \mathcal T }}_{p}^{{ec}}}_{j}\geqslant 3$ and ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$ were calculated for reference. For Criteria AT, ${{{ \mathcal T }}_{p}^{{ec}}}_{j}\geqslant 3$ is satisfied for ∼29.5%, ∼22.6%, and ∼24.4% of the VDFs for j = ∥, ⊥, and tot, respectively. These occurrence rates jump to ∼34.8%, ∼27.2%, and ∼28.6% for ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$. If the data are limited to Criteria UP, the occurrence rates for ${{{ \mathcal T }}_{p}^{{ec}}}_{j}\geqslant 3$ increase to ∼38.2%, ∼38.2%, and ∼37.7%, and similarly the rates for ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$ also increase to ∼47.4%, ∼46.1%, and ∼42.8%. If the time range is limited to −120 s ≤ Δt ≤ +3 s (where Δt is the time from ramp center), the rates for ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$ are ∼41.2%, ∼32.3%, and ∼26.4%. Thus, even if the analysis is limited to times near the shock ramps, the occurrence rates of temperature ratios meeting or exceeding three are smaller than all upstream observations. The occurrence rate of IAWs has been shown to peak within collisionless shock ramps (e.g., Fuselier & Gurnett 1984; Wilson et al. 2007, 2014a, 2014b; Goodrich et al. 2018; Cohen et al. 2019; Davis et al. 2020), and this rate was found to be independent of ${{{ \mathcal T }}_{p}^{e}}_{\mathrm{tot}}$ (e.g., Wilson et al. 2007). That the rate of ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$ does not peak near the ramp regions of the 52 IP shocks examined herein is consistent with the apparent lack of dependence on ${{{ \mathcal T }}_{i}^{e}}_{\mathrm{tot}}$ found in previous work (e.g., Rodriguez & Gurnett 1975; Wilson et al. 2007). Note that these rates are significantly higher than those estimated for the ambient solar wind in a recent study (e.g., Wilson et al. 2018).

The critical drift speed between electrons and ions for a current-driven IAW can be analytically derived for two isotropic Maxwellians that generate a current. The critical drift is known to depend upon ${{{ \mathcal T }}_{i}^{e}}_{\mathrm{tot}}$ (e.g., Gurnett & Bhattacharjee 2005). This critical drift threshold for current-driven IAWs, ignoring the details of the ion and electron populations, is satisfied for ∼4.78% of the 12,081 VDFs for Criteria AT that had solutions for both electron and proton data. However, as discussed in Priest & Sanderson (1972), the presence of gradients in the temperature and density reduce this idealized critical drift by factors of ∼2–8. This increases the number of VDFs satisfying the critical drift threshold to ∼5.2%–10.4%. These rates require context to appreciate their magnitudes.

To provide context, some statistical calculations will be performed based upon the observations and relying upon the near ubiquity of IAWs in and around collisionless shock waves (e.g., Gurnett et al. 1979b; Fuselier & Gurnett 1984; Wilson et al. 2007; Breneman et al. 2013; Chen et al. 2018; Goodrich et al. 2018). The typical collisionless shock ramp thickness is anywhere from $\gtrsim 1\langle {\lambda }_{e}{\rangle }_{\mathrm{up}}$ to $\lesssim 43\langle {\lambda }_{e}{\rangle }_{\mathrm{up}}$ ∼ $1\langle {\lambda }_{i}{\rangle }_{\mathrm{up}}$ (e.g., Hobara et al. 2010; Mazelle et al. 2010). For the 52 IP shocks examined herein, the following are satisfied for 90% of the events:

• 1.  
  1.2 km ≲ $\langle {\lambda }_{e}{\rangle }_{\mathrm{up}}$ ≲ 4.2 km;
• 2.  
  155 km s⁻¹ ≲ $\langle | {V}_{\mathrm{shn}}| {\rangle }_{\mathrm{up}}$ ≲ 700 km s⁻¹; and
• 3.  
  1 km ≲ L_sh ≲ 180 km.

These shock-ramp thicknesses correspond to timescales of ∼0.002–1.2 s in the spacecraft frame, or ∼10⁻⁵–0.008% of the total time window examined for each IP shock. The duration of each electron VDF is ∼3 s,¹⁹ which means the shock ramps are at most ∼30% of the minimum cadence of the Wind 3DP instrument. Thus any given VDF cannot parameterize only the shock ramp, and the instruments cannot directly measure the shock-ramp currents.

Despite the limitation of the particle data time resolution and unrealistic VDF profile assumptions in the theory, there were still nearly 600 VDFs that satisfied the critical drift threshold for current-driven IAWs. This corresponds to roughly 11 VDFs for each of the 52 IP shocks examined herein or a total duration more than 27 times that of the longest shock ramp in this study. Further, there were nearly 3500 VDFs (or ∼67 per IP shock) that satisfy ${{{ \mathcal T }}_{p}^{\mathrm{eff}}}_{j}\geqslant 3$ (i.e., the often-quoted temperature ratio threshold for IAWs). The IAWs of interest here should affect the electron VDFs on timescales much shorter than the integration time; thus the Wind 3DP instrument should only observe the post-instability form.

The Criteria DN VDFs have larger core self-similar exponents (s_ec, p_ec, and q_ec) than Criteria UP VDFs, with the strongest shocks consistently showing flattops (i.e., p_ec ≥ 4) for minutes to hours in the downstream, similar to terrestrial bow shock observations. The larger core exponents in the downstream regions combined with IAW amplitudes positively correlated with Mach number (e.g., Wilson et al. 2007) is consistent with IAWs stochastically accelerating the core electrons (e.g., Dum et al. 1974; Dum 1975). This type of stochastic acceleration is qualitatively referred to as inelastic scattering throughout this three-part study. However, such a change in VDF profile has also been interpreted as due to the acceleration by quasi-static cross-shock electric fields (e.g., Scudder et al. 1986; Schwartz et al. 1988, 2011; Hull et al. 2001; Mitchell & Schwartz 2014; Schwartz 2014). Note that for both s_ec and p_ec, the differences between Criteria LM and Criteria HM shocks are not statistically significant. That is, theory and observation suggest that the magnitude of both the quasi-static cross-shock electric fields and IAW amplitudes should increase with increasing Mach number. Therefore, if IAWs are affecting the core electrons, are they increasing the exponent or the temperature or both? If so, why and how? Similar questions arise regarding the quasi-static cross-shock electric field explanation. Kinetic simulations are required to resolve this discrepancy and are beyond the scope of this study.

Finally, two more instabilities will be discussed that also affect the ions since most previous instability work has focused on the ions (e.g., Maruca et al. 2012; Kasper et al. 2013; Klein et al. 2017, 2018). The purpose is to examine the differences in the particle populations near IP shocks versus what is typically considered ambient solar wind. The threshold for both the mirror and firehose instabilities can also be calculated following the approach²⁰ in Chen et al. (2016). Using only VDF solutions when all five particle populations have finite velocity moments, the plasma is unstable to the firehose instability ∼1.3% of the time and mirror ∼13.5%. These rates are ∼10 and ∼20 times larger, respectively, than the rates found by Chen et al. (2016). It may not be surprising to find that VDFs are statistically more unstable near IP shocks than the ambient solar wind, since the shock itself is a constant source of multiple types of free energy. However, these rates do not significantly change if the time range of analysis is limited to ±20 minutes of the shock ramp center, suggesting the enhanced rates are either due to the separation of electron components or the lack of inclusion of a secondary proton beam. Despite this, the Criteria UP VDFs should be treated as generally being more unstable than time periods that intentionally avoid IP shocks.

As a final note, the instability analysis presented in this section should be interpreted with care. The rates calculated are based upon thresholds and do not directly imply anything about whether an observable wave amplitude would result. For instance, the threshold for the whistler instabilities are for growth rates at 10% of the proton cyclotron frequency (i.e., >18,000 times longer than a single electron cyclotron period). Therefore, that only ∼3% of the VDFs are stable does not imply large amplitude whistler waves should be observed for nearly all intervals examined herein.²¹ That is, in the cases where the instability thresholds are barely surpassed, the resulting fluctuation amplitudes may be so small that they are below the noise floor of many instruments. Further, the separation of the electron VDF into three populations, all non-Maxwellian, is different than the two drifting bi-Maxwellians assumed by Gary et al. (1994). Thus, the large fraction of VDFs satisfying instability thresholds presented herein should not be interpreted as most time intervals exhibiting large amplitude electromagnetic fluctuations.

In this section, the changes in the fit results across all shocks are examined. The data are presented as the median of all permutations of the difference, $\widetilde{{\rm{\Delta }}Q}$, of parameter Q between the upstream and downstream values for each IP shock. The uncertainties for the fit parameters are half the magnitude of the difference between X_5% and X_95%. The uncertainties for the macroscopic shock parameters result from the standard propagation of uncertainties given by the values in the Wind shock database.

These medians with uncertainties were then fit to a model power-law function, Y = $A\ {X}^{B}$ + C, assuming Poisson weights, where X is one of the macroscopic shock parameters and Y is one of the medians of all permutations of the difference (or ratio) across the shock. Given that the shock must transform the change in bulk kinetic energy, the obvious shock parameter to examine is the change in kinetic energy, ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ (see Appendix A), across the shock.²²

Figure 7 shows the relationship between the change in temperature versus the change in shock kinetic energy, and Table 1 shows the values of the fit parameters with goodness of fit estimates. Here, the ${\widetilde{{\rm{\Delta }}T}}_{s,j}$ values represent the median²³ of all the permuted temperature differences between downstream and upstream for each IP shock. The ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ values are just computed from the $\langle | {U}_{\mathrm{shn}}| {\rangle }_{j}$ values from the Wind shock database. The uncertainties are calculated using the standard propagation of uncertainties.

Table 1.  Fit Parameters for Figure 7

Y	A × 10−3a	Bb	Cc	${\tilde{\chi }}^{2}$ d	Σfite
${T}_{\mathrm{eff},\parallel }$	0.4 ± 0.3f	2.13 ± 0.11	0.89 ± 0.27	1.7	4.5
${T}_{\mathrm{eff},\perp }$	0.3 ± 0.4	1.67 ± 0.27	0.74 ± 0.39	1.1	2.7
${T}_{\mathrm{eff},\mathrm{tot}}$	5.0 ± 4.0	1.59 ± 0.11	0.41 ± 0.31	1.4	3.3
${T}_{{ec},\parallel }$	0.5 ± 0.5	2.11 ± 0.10	0.87 ± 0.28	1.8	4.7
${T}_{{ec},\perp }$	3.0 ± 3.0	1.71 ± 0.22	0.78 ± 0.37	1.3	3.0
${T}_{{ec},\mathrm{tot}}$	1.0 ± 1.0	1.96 ± 0.20	0.89 ± 0.34	1.3	3.1
${T}_{i,\parallel }$	2.0 ± 3.0	1.77 ± 0.23	1.44 ± 0.41	3.8	9.1
${T}_{i,\perp }$	8.0 ± 8.0	1.61 ± 0.16	2.22 ± 0.60	2.1	5.5
${T}_{i,\mathrm{tot}}$	3.0 ± 4.0	1.76 ± 0.18	2.10 ± 0.52	2.5	6.4

Notes. For symbol definitions, see Appendix A.

^aConstant multiplier in power law. ^bExponent in power law. ^cConstant offset in power law. ^dReduced chi-squared of fit. ^eStandard error between fit and data. ^fValues shown are larger by a factor of 10⁺³.

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The first thing to notice is that the relationship is not linear. In fact, several previous studies examined the change in temperature and found positive correlations between ${\rm{\Delta }}{\bar{T}}_{e,\mathrm{tot}}$ and variants of ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ (e.g., Thomsen et al. 1987, 1993; Schwartz et al. 1988; Hull et al. 2000). However, if one examines the plots of ${\widetilde{{\rm{\Delta }}T}}_{e,\mathrm{tot}}$ versus ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ in each of these studies, it is not clear whether the trend is linear or otherwise. For instance, examination of Figures 5 and 6 in Feldman et al. (1983c) do not appear to have a linear trend. The linear relationship between ${\rm{\Delta }}{\bar{T}}_{e,\mathrm{tot}}$ and ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ found by Hull et al. (2000) required that they ignore shocks with ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}\lt 100$ eV in the fit. The relationship between ${\rm{\Delta }}{\bar{T}}_{e,\mathrm{tot}}$ and ${\rm{\Delta }}{\bar{U}}_{\mathrm{shn}}^{2}$ found by Fitzenreiter et al. (2003), however, did appear to be linear. Note that many of the previous studies examined the higher Mach number terrestrial bow shock.

However, it is not unreasonable to assume that a linear relationship would exist if the energy conversion was linear. That is, if the irreversible transformation of excess kinetic energy across the shock²⁴ into particle heating occurred directly with no intermediary processes,²⁵ one may expect that a linear relationship would exist with the change in electron and ion temperature with some associated efficiency. A linear relationship also requires, in general, fewer assumptions to model. That is, the scatter plot of points in previous studies could have just as easily been fit to a nonlinear function like a power law, though it is likely these authors chose a linear relationship, as that is the simplest possibility.

One should also note that the exponents for ${\widetilde{{\rm{\Delta }}T}}_{{ec},\parallel }$ and ${\widetilde{{\rm{\Delta }}T}}_{\mathrm{eff},\parallel }$ are larger than all exponents of ${\widetilde{{\rm{\Delta }}T}}_{i,j}$. However, if the five magenta points at large ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ are included in the fits, the exponents for ${\widetilde{{\rm{\Delta }}T}}_{{ec},j}$ and ${\widetilde{{\rm{\Delta }}T}}_{\mathrm{eff},j}$ would decrease more than ${\widetilde{{\rm{\Delta }}T}}_{i,j}$. Note that previous work has either inferred (e.g., Ghavamian et al. 2007, 2014) or showed with in situ measurements that stronger shocks heat ions more than electrons (e.g., Schwartz et al. 1988; Thomsen et al. 1993; Masters et al. 2011). In fact, Schwartz et al. (1988) and others have even noticed that ${\rm{\Delta }}{\bar{T}}_{e,\mathrm{tot}}$/${\rm{\Delta }}{\bar{T}}_{t,\mathrm{tot}}$ ∝ $\langle {M}_{A}{\rangle }_{\mathrm{up}}^{-1}$. Therefore, the change in slope/trend at larger ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ may be indicative of differences in shock energy dissipation at stronger shocks, as suggested by trends in previous work.

Even when the five magenta points at large ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ (i.e., magenta points in upper right-hand corner of each panel of Figure 7) were included in the fit, the linear fit had much larger ${\tilde{\chi }}^{2}$ or Σ_fit values (not shown). Though it is also fair to argue that the fits shown in Figure 7 are not really good fits despite the low ${\tilde{\chi }}^{2}$ or Σ_fit values shown in Table 1. That is, the data have large uncertainties and large relative spread for each ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$, as evidenced by the green lines on either side of the red fit lines. Therefore, it is likely that if even stronger shocks were added to this data set, the power-law trend presented in Figure 7 would need to be modified by either an exponential roll-over or higher-order terms or the trend would entirely fall apart. Thus, these fits should be treated with caution and/or skepticism.

To verify that the use of the median on the permutations of all differences was not causing the nonlinearity, the same temperature differences were plotted versus ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ but now using ${\rm{\Delta }}{\bar{T}}_{s,j}$ instead of ${\widetilde{{\rm{\Delta }}T}}_{s,j}$ (not shown). That is, the average over each region was calculated as a single scalar prior to finding the difference between the regions. The nonlinear fit lines were still a better match to the results than a linear line. Thus, the nonlinear relationship between the temperature increase and the change in kinetic energy across the shock appears to be real at least for low ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$.

Most of the earlier work looking at the dependence of ${\rm{\Delta }}{\bar{T}}_{s,j}$ on ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ focused on the Earth's bow shock, which is typically higher Mach number (i.e., $\langle {M}_{f}{\rangle }_{\mathrm{up}}\gtrsim 3\mbox{--}10$) than those examined herein. The Mach numbers in this study satisfy 1.01 ≤ $\langle {M}_{f}{\rangle }_{\mathrm{up}}$ ≤ 6.4, with 90% satisfying ∼1.15–4.00, and a median of ∼1.86. Note that 45 of the 52 IP shocks examined herein satisfied $\langle {M}_{f}{\rangle }_{\mathrm{up}}\lt 3$. Further, the relationship between $\langle {M}_{f}{\rangle }_{\mathrm{up}}$ and ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ is not linear. This begs the question of what could cause the energy transformation from ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ into ${\rm{\Delta }}{\bar{T}}_{s,j}$ to be nonlinear, if the trend is real.

Regardless, the fraction of ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ distributed to each of the populations shown in Figure 7, for 90% of events (i.e., X_5% to X_95% range), satisfy the following:

• 1.  
  1.5% ≲ ${\widetilde{{\rm{\Delta }}T}}_{{ec},j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ ≲ 34%;
• 2.  
  0.8% ≲ ${\widetilde{{\rm{\Delta }}T}}_{\mathrm{eff},j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ ≲ 41%; and
• 3.  
  1.4% ≲ ${\widetilde{{\rm{\Delta }}T}}_{i,j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ ≲ 63%.

Although these ratios are mostly uniform for all j for both electron populations, the ${\widetilde{{\rm{\Delta }}T}}_{i,\parallel }/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ range is systematically smaller than the other two components. That is, if one separates the perpendicular and total from the parallel components then 90% of the events would satisfy

• 1.  
  5.8% ≲ ${\widetilde{{\rm{\Delta }}T}}_{i,\{\perp ,{tot}\}}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ ≲ 63%; and
• 2.  
  1.4% ≲ ${\widetilde{{\rm{\Delta }}T}}_{i,\parallel }/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ ≲ 40%.

Similarly the median values for ${\widetilde{{\rm{\Delta }}T}}_{{ec},j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ and ${\widetilde{{\rm{\Delta }}T}}_{\mathrm{eff},j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ are in the ∼7.6%–9.3% range while the ${\widetilde{{\rm{\Delta }}T}}_{i,j}/{\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$ median values satisfy ∼10.7%–17.9%, with the parallel component being the smallest. In summary, the electron core and effective populations only gain ∼40%–80% of what the ions do in thermal energy across the shock for these events.

In this section, the partition of energy among the five primary constituent particle populations will be discussed. These five are the electron core (s = ec), halo (s = eh), beam/strahl (s = eb), and ion proton (s = p) and alpha-particle (s = α) populations. Similar to Section 5.1, the median of all permutations of the differences will be discussed.

The normalized pressures, ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{s,j}$ and ${\widetilde{{\rm{\Delta }}\psi }}_{s,j}$, were plotted (not shown) versus θ_Bn, $\langle | {V}_{\mathrm{shn}}| {\rangle }_{\mathrm{up}}$, $\langle | {U}_{\mathrm{shn}}| {\rangle }_{\mathrm{up}}$, $\langle {M}_{f}{\rangle }_{\mathrm{up}}$, $\langle {M}_{A}{\rangle }_{\mathrm{up}}$, $\langle {M}_{{Te}}{\rangle }_{\mathrm{up}}$, ${\rm{\Delta }}{\bar{U}}_{\mathrm{shn}}$, ${\rm{\Delta }}{\overline{{KE}}}_{\mathrm{shn}}$, and ${\rm{\Delta }}{\bar{\xi }}_{\mathrm{shn}}$ (i.e., every macroscopic shock parameter predicted to be of importance here). In the following, weak correlations imply there is a trend, but the large scatter and multiple outliers make interpretation difficult. Moderate correlations are for clear trends but still with a significant spread in data (e.g., similar to ${\widetilde{{\rm{\Delta }}T}}_{s,j}$ plots in Figure 7). There were no good correlations observed for any pair of parameters examined—only a few moderate correlations and several weak correlations (not shown).

For reference, the following will show parameters as X_5% ≲ X ≲ X_95%, $\tilde{X}$, for the 52 IP shocks examined herein:

• 1.  
  4.07% ≲ $\widetilde{{\rm{\Delta }}{\rm{\Pi }}}$_{ec, j} ≲ 41.0%, ∼12.7%;
• 2.  
  1.15% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eh},j}$ ≲ 10.6%, ∼3.85%;
• 3.  
  0.30% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eb},j}$ ≲ 7.46%, ∼3.22%;
• 4.  
  4.36% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\mathrm{eff},j}$ ≲ 36.1%, ∼12.8%;
• 5.  
  4.05% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{p,j}$ ≲ 36.1%, ∼12.5%; and
• 6.  
  0.15% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\alpha ,j}$ ≲ 9.05%, ∼2.33%.

If the analysis is limited to Criteria LM shocks, then these relations go to

• 1.  
  2.36% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{ec},j}$ ≲ 38.3%, ∼12.8%;
• 2.  
  1.06% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eh},j}$ ≲ 9.49%, ∼3.57%;
• 3.  
  0.74% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eb},j}$ ≲ 6.87%, ∼2.43%;
• 4.  
  4.36% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\mathrm{eff},j}$ ≲ 35.3%, ∼12.4%;
• 5.  
  4.36% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{p,j}$ ≲ 35.3%, ∼12.4%; and
• 6.  
  0.15% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\alpha ,j}$ ≲ 9.05%, ∼2.35%.

Finally, if the analysis is limited to Criteria HM shocks, then these relations go to

• 1.  
  4.51% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{ec},j}$ ≲ 43.8%, ∼10.3%;
• 2.  
  1.83% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eh},j}$ ≲ 11.6%, ∼6.21%;
• 3.  
  0.97% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{{eb},j}$ ≲ 7.46%, ∼5.96%;
• 4.  
  4.40% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\mathrm{eff},j}$ ≲ 40.4%, ∼13.4%;
• 5.  
  3.73% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{p,j}$ ≲ 40.4%, ∼13.7%; and
• 6.  
  0.43% ≲ ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\alpha ,j}$ ≲ 3.48%, ∼1.68%.

Note that the Wind SWE Faraday cups have difficulty separating alpha particles from protons and finding good nonlinear fit solutions in the immediate downstream of the stronger shocks in this study, which is likely affecting the upper bounds of both ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{p,j}$ and ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{\alpha ,j}$. The reason for emphasizing this point is that previous work (e.g., Schwartz et al. 1988; Thomsen et al. 1993; Masters et al. 2011) found more energy going to the ions as the Mach number increases but the core electrons seem comparable to the protons here. A possible difference may be that previous work examined the total ion and electron VDFs, which may blur the differences in trends between the various components of each species. It is also worth noting that the ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{s,j}$ values result from the absolute value of the differences (i.e., the actual change may be negative for one population).

To determine which population gained more thermal energy density across the shocks, the ratio of the electron component to proton thermal energy densities are examined where the parameters are shown as X_5% ≲ X ≲ X_95%, $\tilde{X}$. The values the 52 IP shocks examined herein are as follows:

• 1.  
  16.9% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{ec},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 391%, ∼101%;
• 2.  
  1.11% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eh},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 41.3%, ∼5.41%;
• 3.  
  0.47% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eb},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 18.3%, ∼2.55%; and
• 4.  
  19.6% ≲ ${\widetilde{{\rm{\Delta }}P}}_{\mathrm{eff},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 453%, ∼107%.

If the analysis is limited to Criteria LM shocks, then these relations go to

• 1.  
  15.8% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{ec},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 391%, ∼89.7%;
• 2.  
  0.94% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eh},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 28.8%, ∼5.41%;
• 3.  
  0.47% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eb},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 22.2%, ∼2.37%; and
• 4.  
  17.3% ≲ ${\widetilde{{\rm{\Delta }}P}}_{\mathrm{eff},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 453%, ∼102%.

Finally, if the analysis is limited to Criteria HM shocks, then these relations go to

• 1.  
  36.4% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{ec},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 372%, ∼182%;
• 2.  
  2.82% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eh},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 49.3%, ∼8.76%;
• 3.  
  0.54% ≲ ${\widetilde{{\rm{\Delta }}P}}_{{eb},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 10.0%, ∼3.42%; and
• 4.  
  22.0% ≲ ${\widetilde{{\rm{\Delta }}P}}_{\mathrm{eff},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ ≲ 391%, ∼199%.

Therefore, the change in core electron thermal pressure is comparable to or larger than that for the protons, even for the Criteria HM shocks, unexpectedly. For the Criteria LM shocks, the core electrons receive roughly the same heating as the protons, consistent with previous results (e.g., Schwartz et al. 1988). Note that despite the use of pressure here as a measure of thermal energy density, none of these processes are occurring in a thermodynamic system. That is, there is no well-defined equation of state for each of the particle populations.

In summary, none of the energy density ratios showed a clear dependence upon any macroscopic shock parameter. None of the ${\widetilde{{\rm{\Delta }}{\rm{\Pi }}}}_{s,j}$ or ${\widetilde{{\rm{\Delta }}\psi }}_{s,j}$ cared about ${\theta }_{\mathrm{Bn}}$—that is, the change in the ratio of thermal energy density to both the total pressure and total energy density is independent of shock geometry. In other words, the shock geometry does not appear to affect the change in the partition of energy among the five major particle populations. The most correlations were found with ${\rm{\Delta }}{\bar{U}}_{\mathrm{shn}}$, but again none of them were good. The only good correlation was observed between ${\rm{\Delta }}{\bar{\xi }}_{\mathrm{shn}}$ and ${\widetilde{{\rm{\Delta }}\epsilon }}_{j}$ (not shown; i.e., the change in total energy density increases with increasing change in shock kinetic energy density). This merely shows that as the available free energy increases, the total internal energy increases, as expected.

Finally, the absolute changes in normalized partial pressures were dominated by the core electrons and protons with the suprathermal electrons and alpha-particles serving as minor constituents, which is again expected. However, the absolute differences discussed in this section do not inform us whether the change is positive or negative. Further, somewhat unexpectedly, the core electron pressure changes were often larger than those for the protons for Criteria HM shocks, while for Criteria LM shocks, the changes were closer to previous observations (e.g., Schwartz et al. 1988). Again, some of this is most likely due to the issues facing the Wind SWE Faraday cups downstream of the strongest shocks in this study. However, the larger-than-unity ratios of ${\widetilde{{\rm{\Delta }}P}}_{{ec},j}$/${\widetilde{{\rm{\Delta }}P}}_{p,j}$ for even Criteria LM shocks were not really expected. To help address this, SEA plots are shown to illustrate the trends in ${\rm{\Delta }}{{\rm{\Pi }}}_{s,j}$ versus time in the following section.

In this section, the partition of energy among the five primary constituent particle populations will be discussed. These five are the electron core (s = ec), halo (s = eh), beam/strahl (s = eb), and ion proton (s = p) and alpha-particle (s = α) populations. Similar to Section 3, SEA plots will be presented.

Since none of the $\widetilde{{\rm{\Delta }}Q}$ seemed to show very good correlations with any macroscopic shock parameter predicted to be of importance, the statistical trend of the normalized energy densities directly using SEA were examined. The purpose is to see if any clear trend versus time(space) emerges that is not reflected in macroscopic differences or ratios. Note that the SEA plots involve the calculation of one-variable statistics for all points within a given time bin, while the $\widetilde{{\rm{\Delta }}Q}$ values are calculated on a shock-by-shock basis.

Figure 8 shows the running median only from SEA of the partial thermal pressures, ${{\rm{\Pi }}}_{s,j}$, of each of the major particle populations in the solar wind, including the core electrons (blue lines), halo electrons (red lines), beam/strahl electrons (orange lines), protons (magenta lines), and alpha-particles (purple/violet lines). First, ${{\rm{\Pi }}}_{{ec},j}$ and ${{\rm{\Pi }}}_{p,j}$ dominate at all times for all selection criteria, as expected. Second, the general trend of all electron populations is for their fractional thermal energy density to reduce across the shock ramp except for ${{\rm{\Pi }}}_{{ec},j}$ for Criteria HM shocks. Both the ${{\rm{\Pi }}}_{{eh},j}$ and ${{\rm{\Pi }}}_{{eb},j}$ electrons consistently decrease across the shock, with the weakest change across Criteria PA shocks. The ${{\rm{\Pi }}}_{{eb},j}$ show a continual decreasing trend across Criteria HM shocks with a distinct jump at the shock ramp in contrast to ${{\rm{\Pi }}}_{{eh},j}$, which basically levels off and recovers downstream. There are also several intervals where ${{\rm{\Pi }}}_{{ec},j}$ and ${{\rm{\Pi }}}_{p,j}$ seem to oscillate exactly out of phase from each other, likely owing to pressure balance features near these shocks. The interesting aspect is that they are common/strong enough to show up in a running median constructed from SEA on 52 different shocks.

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In general, the following are satisfied for ∼90% of all data for Criteria AT (i.e., between X_5% and X_95%), from largest to smallest (see Table B2):

• 1.  
  25% ≲ ${{\rm{\Pi }}}_{{ec},j}$ ≲ 92%;
• 2.  
  13% ≲ ${{\rm{\Pi }}}_{p,j}$ ≲ 72%;
• 3.  
  1% ≲ ${{\rm{\Pi }}}_{{eh},j}$ ≲ 18%;
• 4.  
  0.3% ≲ ${{\rm{\Pi }}}_{{eb},j}$ ≲ 11%; and
• 5.  
  0.2% ≲ ${{\rm{\Pi }}}_{\alpha ,j}$ ≲ 11%.

These are the partitions of thermal energy density for all time periods. When the data are separated into upstream and downstream, things change slightly but not tremendously. For Criteria UP, sorted from largest to smallest, the following are satisfied:

• 1.  
  27% ≲ ${{\rm{\Pi }}}_{{ec},j}$ ≲ 85%;
• 2.  
  11% ≲ ${{\rm{\Pi }}}_{p,j}$ ≲ 68%;
• 3.  
  1% ≲ ${{\rm{\Pi }}}_{{eh},j}$ ≲ 23%;
• 4.  
  0.7% ≲ ${{\rm{\Pi }}}_{{eb},j}$ ≲ 15%; and
• 5.  
  0.1% ≲ ${{\rm{\Pi }}}_{\alpha ,j}$ ≲ 9%.

For Criteria DN , sorted from largest to smallest, the following are satisfied:

• 1.  
  24% ≲ ${{\rm{\Pi }}}_{{ec},j}$ ≲ 95%;
• 2.  
  15% ≲ ${{\rm{\Pi }}}_{p,j}$ ≲ 74%;
• 3.  
  1% ≲ ${{\rm{\Pi }}}_{{eh},j}$ ≲ 15%;
• 4.  
  0.2% ≲ ${{\rm{\Pi }}}_{{eb},j}$ ≲ 8%; and
• 5.  
  0.2% ≲ ${{\rm{\Pi }}}_{\alpha ,j}$ ≲ 13%.

To illustrate that the energy partitions shown as running medians in Figure 8 can be characteristic of most of the data, Figure 9 shows the same parameters for the same populations but as shaded regions bounded by ${X}_{5 \% }$ and ${X}_{95 \% }$. What is immediately clear is that again ${{\rm{\Pi }}}_{{ec},j}$ and ${{\rm{\Pi }}}_{p,j}$ dominate at all times, except for some transient excursions to large values for ${{\rm{\Pi }}}_{{eh},j}$. Both ${{\rm{\Pi }}}_{{eb},j}$ and ${{\rm{\Pi }}}_{\alpha ,j}$ have relatively large spreads in the range between percentiles, but they still follow the same trends illustrated by the running median. That is, the suprathermal electron fractional thermal energy density decreases while both ion species increase. The abrupt end in ${{\rm{\Pi }}}_{\alpha ,j}$ data for Criteria PA shocks in both Figures 8 and 9 are due to low statistics owing to difficulties in finding high quality fits for the alpha-particle peak (e.g., see the SWE fit flag requirements in Wilson et al. 2018).

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It is also clear that Criteria HM shocks have the largest separation between ${{\rm{\Pi }}}_{{ec},j}$ and ${{\rm{\Pi }}}_{p,j}$ for all time periods compared to other selection criteria. The downstream running median values for Criteria LM and Criteria PE shocks oscillate out of phase, but this is not directly reflected in ${X}_{5 \% }$–${X}_{95 \% }$. Criteria PA shocks exhibit the most overlap in ${X}_{5 \% }$–${X}_{95 \% }$, and Criteria HM shocks have the least. In both the running $\tilde{X}$ and ${X}_{5 \% }$–${X}_{95 \% }$, ${{\rm{\Pi }}}_{{eh},j}$ shows a very clear decrease across the shock except for Criteria PA shocks where the change in ${X}_{5 \% }$ and ${X}_{95 \% }$ is more difficult to observe. The abrupt drop in ${{\rm{\Pi }}}_{{eh},j}$ and ${{\rm{\Pi }}}_{{eb},j}$ across shocks is likely related to the drop in ${n}_{{eh}}/{n}_{\mathrm{eff}}$ and ${n}_{{eb}}/{n}_{\mathrm{eff}}$ across the shocks because the associated temperatures show slight increases across the shock (see additional SEA plots of ${T}_{s,j}$ found in Wilson et al. 2020).