Electron–Ion Temperature Ratio in Astrophysical Shocks

Collisional shock waves are characterized by a thickness on the order of the mean free path. Because the shock transitions are mediated by particle–particle collisions, they produce Maxwellian distributions and thermal equilibrium among the particle species. On the other hand, the thickness of a collisionless shock in a plasma is much smaller, on the order of the proton gyroradius or ion skin depth. Because they are mediated by electromagnetic fields and plasma turbulence instead of particle collisions, collisionless shocks can produce non-Maxwellian velocity distributions and very different temperatures among different particle species. Except for shocks inside stars or in molecular clouds, most astrophysical shocks are collisionless.

The electron and proton temperatures T_e and T_p predicted by the Rankine–Hugoniot jump conditions can be very different, because a shock thermalizes most of the kinetic energy of the particles flowing through it, and the electron and ion kinetic energies differ by their mass ratio. Collisions can eventually bring T_e and T_p into equilibrium, but over a timescale that can exceed the dynamical age of the astrophysical object (e.g., supernova remnants, SNR, galaxy cluster shocks, and structure formation shocks). However, energy dissipation in the shock front and precursor occurs by way of strong plasma turbulence, and this plasma turbulence is capable of transferring energy between ions and electrons more rapidly.

Reliable determination of T_e /T_p is important for understanding the physics of collisionless shocks. For instance, electron–ion thermal equilibration can affect the shock reformation process (Shimada & Hoshino 2005). T_e /T_p is also important as a diagnostic tool because T_e is measured from X-ray spectra. In partially neutral preshock conditions, shock speeds can also be determined from the Hα line profile, which effectively measures T_p (Chevalier & Raymond 1978; Bychkov & Lebedev 1979). In cases where protons keep all the thermal energy, T_p is twice as large as it will be in thermal equilibrium, when half the thermal energy is given to the electrons. Finally, electron heating may play a significant role in the injection of particles into the diffusive shock acceleration process (DSA), which produces radio and X-ray synchrotron emission as well as gamma-ray emission from inverse Compton scattering in SNRs.

Different estimates for the degree of electron–ion temperature equilibration in collisionless shocks give contradictory results. The in situ measurements of bow shocks in the solar wind at Earth and Saturn show T_e /T_p dropping from 1 at very low sonic Mach numbers or Alfvén Mach numbers (M or M_A) to about 0.1 at Mach numbers near 10. Although observations of SNRs show a parallel decline, the trend is systematically shifted to much higher Mach numbers. In SNRs, T_e /T_p is near 1 at M ∼ 30, and it drops to 0.05 at M ∼ 300, as shown in Figure 1 (adapted from Ghavamian et al. 2013).

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A second inconsistency arises at higher Mach numbers. Both particle-in-cell (PIC) simulations and solar wind shocks show a drop in T_e /T_p with Mach number between M = 1 and M = 15, but then T_e /T_p rises to about 0.2–0.3 (Bohdan et al. 2020; Hanusch et al. 2020; Wilson et al. 2020; Tsiolis et al. 2021). On the other hand, values of T_e /T_p from the Hα profiles observed in SNR shocks continue to decline with M to about 0.05 at M ≃ 300 (Ghavamian et al. 2013).

Given the importance of understanding the electron–ion temperature ratio for interpreting observations of SNRs and other shocks, as well as for understanding collisionless shock physics and the acceleration of nonthermal electrons in shock waves, we try to resolve these discrepancies. In principle, the values of T_e /T_p derived from the SNR observations, PIC simulations, and in situ measurements could all be correct, but the former pertain to a structure many orders of magnitude thicker than do the latter. The SNR measurements could be appropriate for tasks such as interpreting X-ray spectra, while the in situ and PIC results could be appropriate for studies of plasma processes on small scales. For instance, the PIC simulations and solar wind measurements could pertain to a subshock, while the SNR observations would include any precursor or postcursor effects.

On the other hand, the discrepancies could result from errors in either the theory or observations of the SNRs. In particular, the comparison in Ghavamian et al. (2013) relied on the Hα line profiles of shocks in partially neutral gas. The profiles consist of a narrow component from H atoms that are collisionally excited after they pass through the shock, and a broad component from atoms that undergo charge transfer with postshock protons, acquiring a velocity distribution like that of the protons before they are excited (Chevalier et al. 1980). The ratio of the broad and narrow-component intensities, I_B /I_N , depends on the electron temperature, and it is often used to determine T_e /T_p . However, this carries some uncertainty.

The aim of this paper is to compare T_e /T_p determinations based on the intensity ratios of the broad and narrow components of the Hα profiles of SNR shocks (Ghavamian et al. 2013) with determinations that use electron temperatures measured from X-ray spectra, T_e,x , proton temperatures based on Hα broad-component widths, T_p,w , and shock speeds based upon proper motions for SNRs at known distances, V_pm. We make four sets of determinations based on (1) I_B /I_N versus broad-component Hα width, (2) T_p,w versus T_e,x , (3) V_pm versus T_p,w , and 4) V_pm versus T_e,x , all based on published measurements of the relevant quantities. We will look for common trends of T_e /T_p with shock speed M or M_A among these different estimates.

We will not attempt a statistical comparison because the uncertainties are dominated by systematic errors in the measurements or the T_e /T_p determinations that we cannot quantify. For example, we try to choose a portion of an SNR shock where V_pm and T_e are both measured, but the values will not always pertain to exactly the same plasma. All the values of the basic parameters are taken from the literature, and when a range of values is given, it indicates the range that entered an average rather than a 1σ or 3σ uncertainty. The comparison is complicated by the fact that we often take, for instance, proper motions from one paper and electron temperatures based on X-ray spectra from another. Detailed considerations for each SNR are given in the appendix.

The paper is organized as follows. Section 2 describes predictions from numerical simulations, and Section 3 describes results from in situ measurements of shocks in the solar wind and in the laboratory. Section 4 describes the diagnostic measurements of SNR shocks, and Section 5 discusses the four methods of estimating T_e /T_p in SNR shocks, along with the uncertainties involved. Section 6 discusses the comparisons among SNRs, solar wind and laboratory shocks, and theory. It emphasizes the questions of whether the inclusion of shock precursors or postshock processes, cosmic-ray acceleration, or the presence of neutrals in the regions observed in the SNRs can account for the differences. Section 7 summarizes our results, and the appendix gives details of the T_e /T_p estimates for each SNR.

Early theoretical calculations for solar wind shocks based on energy transfer by various wave modes or on shock electric fields predicted T_e /T_p ∼ 0.2 (Cargill & Papadopoulos 1988; Hull & Scudder 2000). On the other hand, Ghavamian et al. (2007b) attempted to explain the observed decline on T_e /T_p in SNRs with shock speed based on lower hybrid waves in a cosmic-ray-generated precursor.

PIC simulations of shocks have explored part of the parameter space. For perpendicular shocks in plasma with β near 1, T_e /T_p drops steeply from 1 at very small M_A to 0.1–0.2 at M_A = 10 (Tran & Sironi 2020), then increases slowly from ∼0.1 at M_A = 20 to 0.2 at M_A = 100 (Bohdan et al. 2020), and perhaps tends asymptotically to 0.3 at higher M_A (Tsiolis et al. 2021). It should be noted that the prediction for Mach numbers above 100 depends on the scaling with M_A, and the saturation level may drop from 0.3 to 0.1. The results are summarized in Figure 2.

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Oblique shocks seem to behave like perpendicular shocks, even in the presence of the electron foreshock generated by the shock-reflected electrons (Morris et al. 2023). Lezhnin et al. (2021) report T_e /T_p ∼ 0.5 in a Mach 15 shock at Θ_BN = 60°, where Θ_BN is the angle between shock normal and the magnetic field. On the other hand, parallel shocks are analytically predicted to show a decline in T_e /T_p with Mach number similar to that seen in SNRs, as shown in Figure 12 of Arbutina & Zeković (2021), or to show a decline from near equilibration to around 0.3 at M_A around 20 (Hanusch et al. 2020). Tran & Sironi (2020) investigated low M shocks in the context of the solar wind. That work and subsequent simulations show that at low Mach numbers, the electron heating is not much above that predicted for adiabatic compression, but that the nonadiabatic heating decreases with increasing magnetic obliquity. It increases with Mach number, and neither plasma β nor the assumed electron–ion mass ratio is very important.

PIC simulations have also been performed for weak shocks (M ∼ 2–5) in high-β plasmas (β ≫ 1), as appropriate to shocks that form in the hot intracluster medium (ICM) during mergers of galaxy clusters. Electron-heating mechanisms and their efficiency were investigated for strictly perpendicular 2D shocks by Guo et al. (2017, 2018), who noted that T_e /T_p is independent of plasma β in the range β = 4–32, but decreases strongly with Mach number from T_e /T_p ∼ 0.8 at M = 2 to T_e /T_p ∼ 0.25 at M = 5. This result should hold at oblique superluminal shocks with ${{\rm{\Theta }}}_{\mathrm{BN}}\gt {{\rm{\Theta }}}_{\mathrm{BN},{cr}}={\cos }^{-1}({v}_{\mathrm{sh}}^{\mathrm{up}}/c)$. A recent calculation by Kobzar et al. (2021) for an M = 3 (M_A = 6.1) and β = 5 shock with a subluminal obliquity Θ_BN = 75°, close to Θ_BN,cr ≈ 81.4° for their parameters, gives T_e /T_p ∼ 0.4–0.45, in agreement with results by Guo et al. (2018). This is also compatible with 2D PIC simulations presented in Kang et al. (2019). They found (private communication) that T_e /T_p increases with plasma β from T_e /T_p ∼ 0.55 at β = 20 to T_e /T_p ≳ 0.7 at β = 100 at a subluminal angle Θ_BN = 63° for an M = 3 shock. T_e /T_p decreases in their simulations with the Mach number, but can reach T_e /T_p ≳ 1 in very weak shocks with M ∼ 2. It also depends quite strongly on Θ_BN, and it approaches T_e /T_p around 1 at high-obliquity angles in very high-β plasmas.

A limitation of some of the PIC simulations is the unrealistic electron–ion mass ratio. For instance, Shalaby et al. (2022) simulated very fast shocks (40,000 km s⁻¹) and found that a mass ratio of 100 suppresses an intermediate-scale instability in low M_A shocks. On the other hand, simulations by Tran & Sironi (2020 ; m_i /m_e = 20–625) and Bohdan et al. (2020 ; m_i /m_e = 50–400) demonstrate that the ion-to-electron mass ratio has a minor influence on T_e /T_p if the mass ratio is high enough to separate electron and ion scales (e.g., m_i /m_e > 100). They indicate that a changing shock velocity ratio with respect to c, with Mach and β fixed, does not affect this conclusion (Lezhnin et al. 2021)

They indicate that the shock velocity does not affect this conclusion as long as all plasma components remain nonrelativistic.

Extensive studies of T_e /T_p in the Earth's bow shock (Schwartz et al. 1988) and Saturn's bow shock (Masters et al. 2011) have been conducted, and the results are presented in Ghavamian et al. (2013) and in Figure 1. The recent work of Wilson et al. (2020) is based on a careful analysis of 15,000 velocity distribution functions in 52 interplanetary shocks seen by the Wind spacecraft. While there is a very large scatter, the average shows that the electron heating increases gradually with M_A, such that for solar wind shocks, T_e /T_p would be expected to drop from 1 at small Mach numbers to ∼0.05 at Mach numbers near 10, and then rise to around 0.4 at Mach numbers of order 40. They found little correlation with parameters such as preshock β or shock obliquity.

As with the values of T_e /T_p inferred from SNR observations and the values expected from PIC simulations, there are caveats. In particular, the shocks may not be steady, and the electrons may be mobile enough to reflect average conditions rather than the local shock parameters, so that measurements from a single spacecraft passing through a shock will not always reflect the expected local electron–ion equilibration.

Recent developments in high-power lasers are now creating important opportunities to probe the plasma microphysics of collisionless shocks in controlled laboratory experiments. The interest in using laser-produced plasmas to study the physics of collisionless shocks is not new. Early experiments in the 1970s and 1980s have used 100 J-class lasers to ablate solid targets and produce interpenetrating plasma flows to study their collisionless coupling (Dean et al. 1971; Cheung et al. 1973; Bell et al. 1988). However, for these laser energies, these earlier studies were limited to relatively low Mach numbers and electrostatic coupling. The development of high-energy laser facilities capable of delivering 10 kJ to MJ laser energy on target, such as OMEGA and the National Ignition Facility (NIF), are transforming the ability to study for the first time high-Mach number collisionless regimes dominated by electromagnetic processes as pertinent to most space and astrophysical shocks.

In the last few years, several experiments produced supersonic super-Alfvénic plasma flows in the laboratory to study the underlying shock formation processes (Ross et al. 2012; Fox et al. 2013; Niemann et al. 2014; Huntington et al. 2015; Schaeffer et al. 2017; Rigby et al. 2018; Fiuza et al. 2020; Swadling et al. 2020). The plasma flow velocities produced are typically in the range of 500–2000 km s⁻¹, and the corresponding Mach number is M = 2–400. Developments in plasma diagnostics are starting to allow resolved measurements of the evolution of the plasma temperature in these systems. In particular, recent experiments at NIF characterized the formation of a collisionless shock with M ∼ 12 and M_A ∼ 400, measuring a downstream T_e = 3 keV using Thomson scattering, and demonstrating the acceleration of nonthermal electrons to an energy >100 T_e (Fiuza et al. 2020). While these experiments did not measure T_i , considering that during shock formation, the plasma flow velocity varies between 1800 and 1000 km s⁻¹ (the velocity of the laser-produced flows varies in time) and using the standard conservation equations at the shock, one obtains ZT_e /T_i = 0.15–0.75 (note that the experiments use carbon plasmas with Z = 6). This is consistent with the PIC results summarized in Figure 2, which suggest T_e /T_p = 0.1–0.4 for M_A = 400. Planned follow-up experiments will simultaneously measure the electron plasma wave and ion acoustic wave features of the Thomson scattering spectrum (Ross et al. 2012), enabling a detailed characterization of T_e /T_i . Furthermore, by varying the field strength and orientation of an external magnetic field produced by coils, it will be possible to study T_e /T_i as a function of M, M_A, and θ_BN.

Here we list the three diagnostic measurements we will be using. Each measurement has systematic uncertainties. We discuss some of them here, and we will discuss others in more detail in the next section in the context of the specific estimates of T_e /T_p .

Some diagnostics for determining T_e /T_p in SNR shocks rely on the Hα profiles. A collisionless shock in partially neutral gas produces a thin filament of emission in Hα and other hydrogen lines, because some of the neutral H atoms are excited before they are ionized (Chevalier & Raymond 1978; Chevalier et al. 1980). The collisionless shock thickness is set by the proton gyroradius or the skin depth, and it is very thin: ∼10⁸⁻¹⁰ cm for typical ISM magnetic fields and densities. For comparison, the length scale for the excitation and ionization of H is about 10¹⁵ cm. The neutrals pass through the collisionless shock without feeling the electromagnetic fields or plasma turbulence, so many of them still have their preshock velocities when they are excited and emit Hα. On the other hand, some of the neutrals undergo charge-transfer reactions with postshock protons. These neutrals acquire a velocity distribution similar to the proton distribution, but weighted by the charge-transfer cross section and the relative velocity (Chevalier et al. 1980; Ghavamian et al. 2001; Heng & McCray 2007; van Adelsberg et al. 2008; Morlino et al. 2013). Thus the Hα profile consists of a broad component, whose FWHM is comparable to the shock speed, and a narrow component, whose FWHM is typically 20–50 km s⁻¹ (Sollerman et al. 2003).

The width of the Hα broad component is a direct measurement of T_p in the postshock region, and the narrow component width indicates the temperature in the preshock region. Both widths include any turbulent motion. The turbulence is likely to be significant in a shock precursor (e.g., Bell 2004). There may be significant turbulence in the postshock region as well. Small-scale (kinetic-scale) turbulence will decay over a distance much smaller than the size of the Hα emitting region, but we discuss turbulence in more detail below. The intensity ratio I_B /I_N depends on the ratio of ionization time to charge-transfer timescale, and it therefore depends on the electron temperature (Raymond 1991; Smith et al. 1991; Laming et al. 1996; Ghavamian et al. 2001).

A second diagnostic is the measurement of T_e,x from an X-ray spectrum. This is accomplished by folding a theoretical emission spectrum through the response function of an X-ray telescope and minimizing chi-squared between the model and the observation. To first order, T_e comes from the continuum shape, e⁽−hν/kT), where hν is the photon energy and kT is the thermal energy. Several factors can complicate the T_e,x measurement. It is necessary to separate the shocked ISM or CSM gas from shocked SN ejecta, which are heated by a separate reverse shock. The separation requires good spatial resolution, and that is particularly challenging for SNRs in the Magellanic Clouds. We use temperatures from analyses that were specifically meant to pertain to shocked ISM gas. The X-ray emitting plasma is not in ionization equilibrium. Models of shock-heated gas are available, but the ionization timescale is an additional free parameter in the fit. Elemental abundances introduce still more free parameters, because sputtering gradually liberates refractory elements from dust grains in the hot postshock gas. In addition, there are many cases where a single-temperature model does not provide an adequate fit, so the T_e,x determination is ambiguous. Finally, in some cases, X-ray synchrotron emission contaminates the spectrum of the thermal emission. That being said, observed temperatures of about 1 keV or lower are easily distinguished from the higher temperatures that the faster shocks would produce if they were in thermal equilibrium.

The third diagnostic is the measurement of the shock speed based on the proper motion of shocks in an SNR whose distance is known, V_pm. Proper motions are mostly measured with Hα filaments because they are sharp and because high spatial resolution can be achieved, for instance, by the Hubble Space Telescope (HST). SNR filaments are tangencies of a line of sight to the thin sheet of shock-heated gas (Hester 1987), so their motion is perpendicular to the line of sight, and the proper motion should correspond to the actual shock speed. However, Shimoda et al. (2015) simulated an SNR shock in an inhomogeneous medium, and they found that the rippled shock front could add some energy to turbulence rather than heat. In their models, comparison of V_pm with T_p could overestimate the energy in cosmic rays or in electron thermal energy by up to 40%. Proper motions can also be measured from the expansion of a remnant in Chandra X-ray images, provided that a long temporal baseline is available and that there are enough point sources in the field for image registration, or from expansion seen in IR images. In many cases, uncertainty in the distance to the SNR is the main limitation. We will consider a few Galactic SNRs whose distances are well established, along with SNRs in the Magellanic Clouds.

Different methods can be used to to determine T_e /T_p , depending on the data available. Here we describe four methods, their uncertainties, and the trends of T_e /T_p with shock speed.

5.1. Hα Line Profiles: Line Width and I_B /I_N

5.2. Hα Line Width and T_e,x from X-Ray Spectra

5.3.  V_pm and T_e,x from X-Ray Spectra

5.4.  V_pm versus T_p from the Hα Broad Component

5.5. Evidence from Other Observations

Ultraviolet spectra are only available for a few SNRs, but they support the estimates of T_e /T_p given above. First, Laming et al. (1996) used the intensity ratios in SN1006 to show that T_e /T_p < 0.05. Second, kinetic temperatures of H, He, C, N, and O have been measured in a handful of SNRs that are observable in the UV and in Hα (Raymond et al. 1995; Korreck et al. 2004; Ghavamian et al. 2007a; Raymond et al. 2015, 2017), as well as in X-rays (Miceli et al. 2019). They show fairly complete equilibration in the Cygnus Loop, where the shocks are around 350 km s⁻¹, and mass proportional temperatures (no equilibration) in SN1006, where the shocks are faster than 2500 km s⁻¹. It is possible that the wave modes responsible for sharing energy between electrons and ions could be different from those that redistribute energy among the different ion species. However, the drop in ion–ion thermal equilibration seems to mirror that in electron–ion equilibration.

Another estimate of T_e /T_p is given by Matsuda et al. (2022), who find T_e /T_p < 0.15 in a knot on the NW limb of Tycho based on the X-ray temperature increases over 15 yr and a shock speed of >1500 km s⁻¹. There is also an estimate of T_e /T_p for the reverse shock in Tycho by Yamaguchi et al. (2014), who used the centroid of the Fe Kα line to determine the ionization state of iron. They found 0.003 ≤ T_e /T_p ≤ 0.02 for the reverse shock. Most recently, Ellien et al. (2023) were able to separate thermal from nonthermal X-ray emission at five positions in the south, west, and northeast regions of Tycho, and the 2.0–2.6 keV electron temperatures they derive are much lower than the 10–15 keV predicted from the shock speeds of Williams et al. (2016) and the assumption of equal electron and ion temperatures. This indicates that shocks that produce strong synchrotron X-rays also show T_e /T_p < 0.1 of the energy going into cosmic rays, and the magnetic field amplification is not too large (see Section 6.2).

We also note that, while Table 1 presents numbers for a 600 km s⁻¹/ shock in RCW86 from Ghavamian et al. (2001), which was analyzed in the greatest detail, several other positions in RCW86 were presented in Ghavamian (2000). Excluding three positions contaminated by radiative shocks, there were six other positions whose line widths and I_B /I_N ratios were similar to those in Table 1, so they would have similar shock speeds and values of T_e /T_p around 0.5. There were also two positions showing line widths of 325 km s⁻¹ and I_B /I_N = 1.0 ± 0.2, indicating shock speeds of around 350–400 km s⁻¹ and T_e /T_p > 0.6. These faster shock positions are in good agreement with the RCW86 value shown in Table 1, while the slower shocks agree with the Cygnus Loop value. Overall, these additional points support the trends in Table 1 and Figure 3.

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This paper has concentrated on shocks in SNRs, but pure Balmer-line spectra are also observed in the bow shocks driven through the ISM by some pulsars. Romani et al. (2022) measured an Hα profile in the bow shock of PSR J1959+2048. The shock speed is probably about 150 km s⁻¹, but it is not well determined. The I_B /I_N ratio is about 4, which would indicate a low value of T_e /T_p , probably lower than 0.2. This is much different than the Cygnus Loop shocks, where I_B /I_N is about 1. One possible cause of this difference is emission from a precursor in the Cygnus Loop. Another possible cause is that the T_e /T_p is lower in the bow shock because it is slower. Other possible causes include that the lower temperature in the bow shock favors excitation to the 3 s and 3d levels of hydrogen, as opposed to 3p, and that the smaller width of the broad component permits some conversion of Lyβ into Hα in the broad component. The latter might make T_e /T_p as high as 0.5, consistent with I_B /I_N = 4. If the shock speed is significantly lower than 150 km s⁻¹, the lower postshock temperature would reduce the ionization rate, which could strongly increase I_B /I_N . Since the shock is oblique, the effective shock speed could be lower than 150 km s⁻¹.

All four estimates of T_e /T_p in SNRs agree that the electron–ion temperature ratio declines sharply with shock speed. The discrepancies between SNR observations and solar wind measurements appeared when T_e /T_p was plotted against V_S , M_A or M_ms in Figure 1, which is adapted from Figure 7 of Ghavamian et al. (2013). Measurements at the bow shocks of Earth and Saturn appeared to disagree when plotted against V_S , but agreed nicely with each other when plotted against M_A or M_ms, while the discrepancy with SNR shocks was magnified by plotting against M_A or M_ms.

The plots just described assumed generic sound speeds and Alfvén speeds for the SNRs of 11 km s⁻¹ and 9 km s⁻¹, respectively. Here, we consider the possibility that electron heating occurs only in the narrow collisionless shock, while protons are also heated in the precursor. In this case, the relevant sound speed is the sound speed in the precursor, which is given by the narrow-line width, rather than the sound speed in the ambient ISM. Therefore, we use the widths of the Hα narrow components to determine temperatures and sound speeds. Because nearly all of the observed shocks are Balmer-line filaments, we assume that the gas is 50% ionized, except for SN1006 and 1E102.2-7219, which have low neutral fractions. As shown in Table 5, the narrow-component widths vary, and the corresponding temperatures range from T₄ = 1.0 to 4.6, where T₄ = T/10,000 K. We note that the Tycho SNR shows both a narrow and an intermediate component that account for 35% and 65% of the narrow component. We have taken the weighted average of the corresponding temperatures in quadrature.

The Alfvén speed is even more problematic, given that neither the ambient magnetic field nor the compression in the precursor is known. For Milky Way SNRs, we assume a field strength of 10 μG (Sofue et al. 2019). The Kepler SNR and RCW 86 are expanding into their own circumstellar shells, which could have very low magnetic fields or very high fields due to compression, and the preshock densities vary enormously around their peripheries. For lack of reliable numbers, we do not compute the Alfvén or magnetosonic speed for these remnants. For the Large and Small Magellanic Clouds, we also assume 10 μG based upon Mao et al. (2012), Seta & Federrath (2021), Seta et al. (2023), and Livingston et al. (2022). These are, of course, average magnetic fields, so the local field will be different. To obtain Alfvén speeds, we use preshock densities generally derived from X-ray observations and an assumed compression ratio of 4 at the shock. As seen in Table 5, the preshock densities vary by over an order of magnitude. The Alfvén speed may change in the precursor, but as the plasma and the perpendicular component of the field are compressed together, the change in V_A should not be drastic.

Based on the estimates of sound speed and Alfvén speed in Table 5 and the values of T_e /T_p in Tables 1–4, we plot in Figure 3 the equilibration as a function of shock speed, sonic Mach number, Alfvén Mach number, and magnetosonic Mach number, using different colors for the different estimation methods. To reduce the clutter, we plot only a single point for each SNR for each estimation method, generally an average if a range of values is present, but in some cases, such as RCW 86, the best determination for that SNR. We also introduce slight shifts along the X-axis in cases where a symbol in one color would cover that in another color.

Figure 3 shows that the trends of T_e /T_p with shock speed shown in Figure 7 of Ghavamian et al. (2013; our Figure 1) hold, but that the Mach numbers are considerably smaller. This is due to the generally higher temperatures assumed here based on the narrow-component line widths, to the generally higher Alfvén speeds due to smaller preshock density estimates, and to an assumed 10 μG magnetic field. However, the changes in Mach number, Alfvén Mach number, and magnetosonic Mach number are not nearly enough to bring the SNR results into agreement with the solar wind results (T_e /T_p ∼ 0.15 at V ∼ 400 km s⁻¹, M_A ∼ 10, M_MS ∼ 7), or with the PIC prediction that T_e /T_p rises to about 0.1 to 0.3 when M_A exceeds about 20 to 30.

There are, of course, still ways in which our Mach numbers could be overestimates. For very short-period Alfvén waves, the density of the ionized gas rather than the total density should be used. The ionization fraction is about 90% in SN1006 (Ghavamian et al. 2002), and it cannot be much lower than about 0.5 because the shock itself produces ionizing photons (Medina et al. 2014). We also use mean magnetic fields to estimate the Alfvén speed, and the local fields can be higher or lower. We also use the preshock temperature estimated from the narrow-component line width, and it is possible that protons are further heated in the precursor after the last charge transfer (on average) communicates the proton temperature to the neutrals. It is unlikely that these effects would reduce the Mach numbers by more than a factor of 2.

We therefore consider some ways in which the comparison among SNR, solar wind, and PIC simulated shocks might be comparing different things. Then, we consider possible differences in the physics.

6.1. Different Length Scales

6.2. Jump Condition Energetics, Cosmic Rays, and Magnetic Fields

6.3. Other Parameters

We have used used four measured parameters of SNR shocks to determine T_e /T_p : the Hα broad-component widths, the Hα broad- to narrow-intensity ratios, the electron temperatures derived from X-ray spectra, and the shock speeds derived from proper motions. They provide four distinct estimates of T_e /T_p whose systematic uncertainties are different, and we have discussed the uncertainties for each method. We have used published values for the measured parameters, although in some cases, we reinterpreted the data using newer models (Morlino et al. 2012, 2013) or newer distances to individual SNRs.

The result is that the trend of T_e /T_p obtained from the Hα profiles by Ghavamian et al. (2013) is robust, and the disagreements with measurements of solar wind shocks remain, along with disagreement with PIC simulations. First, the in situ measurements and PIC simulations indicate that T_e /T_p is close to 1 at very low M numbers, but drops to ∼0.2 around M_A near 15, while the SNR shocks are still at T_e /T_p near 1 at M = 15–25. Second, both PIC simulations and in situ measurements indicate that as the shock speed increases for M above about 20, T_e /T_p increases to about 0.1 to 0.3, while in the SNR shocks, T_e /T_p continues to decrease to 0.05 or less.

We consider several possible avenues to resolve this discrepancy: the scale lengths of the regions in which the diagnostic emission is formed, the presence of energetic particles and magnetic turbulence, the roles of shock precursors and postcursors, the Alfvén speeds used to derive Alfvén Mach numbers, and the presence of neutrals. While some of these factors decrease the discrepancies, they do not provide definitive explanations.

Thus, we have utterly failed to resolve the discrepancies among SNR shocks, solar wind shocks, and PIC simulations in electron–ion thermal equilibration. It is not clear whether the answer lies in more theory or more observations. Systematic observational studies are needed in which Hα profiles, proper motion measurements and high-resolution UV and X-ray spectra of individual shocks are obtained. On the theoretical side, simulations that include neutrals and the important plasma processes are needed, both upstream and downstream of the shock jump.

This research was supported by the International Space Science Institute (ISSI) in Bern, through ISSI International Team project #520. The effort was supported by HST Guest Observer grant GO-15285.0001_A to the Smithsonian Astrophysical Observatory. The work of J. N. has been supported by Narodowe Centrum Nauki through research project No. 2019/33/B/ST9/02569. The work of D.R. was supported by the National Research Foundation of Korea through 2020R1A2C2102800. A.T. was supported by NASA grant FINESST 80NSSC21K1383. A.B. was supported by the German Research Foundation (DFG) as part of the Excellence Strategy of the federal and state governments—EXC 2094-390783311. P.G. would like to acknowledge HST Grant GO-15989.005A to Towson University. F.F. was supported by the U.S. DOE Early Career Research Program under FWP 100331.

SN1006 has been extensively studied. Hα is present around the entire circumference, but it is too faint for good line profiles to be obtained, except in the NW, where Nikolić et al. (2013) obtained excellent IFU data. Proper motions have been measured both in Hα and X-rays (Winkler et al. 2003; Miceli et al. 2012; Katsuda et al. 2013; Raymond et al. 2017). The distance to SN1006 has been estimated in a number of ways to be 1.6–2.2 kpc. High-velocity absorption from the unshocked ejecta of SN 1006 has been seen in the spectrum of the Schweizer-Middleditch star, whose Gaia parallax indicates 1.43–2.0 kpc. A distance shorter than 1.6 kpc would imply shock speeds too small to account for the observed Hα line widths. Therefore, we adopt D = 1.85 ± 0.15 kpc to convert proper motions into shock speeds. X-ray temperatures are taken from Winkler et al. (2013) in the NW and Miceli et al. (2012) in the SE. We note that the regions observed by Miceli et al. (2012) do not show the relatively bright Hα seen in the NW, but very faint Hα is present. This might indicate a neutral fraction even lower than the value of 0.1 in the NW found by Ghavamian et al. (2002). Miceli et al. (2012) found values of n_e t of only 2–7×10⁸ cm⁻³ s, so T_Coul is only about 1/3 the observed X-ray temperatures. In the NW, Winkler et al. (2013) find a higher value of n_e t, so Coulomb collisions could provide a substantial fraction of the temperature obtained from the X-ray spectra. We also note that other measurements of T_e,x and V_pm yield similar results for T_e /T_p (Long et al. 2003; Winkler et al. 2003).

The Tycho SNR has also been studied extensively. For the T_e /T_p estimate from I_B /I_N , we use Hα widths and the broad- to narrow-intensity ratios of Ghavamian et al. (2001) and Kirshner et al. (1987), and we interpret them with the models of Morlino et al. (2012) and Morlino et al. (2013). X-ray temperatures of the shocked ISM are available in several regions from Hwang et al. (2002), and while Hα profiles are not available at these positions, we can compare them with proper motions from Williams et al. (2016). Distance estimates range from 2.3 to 3.7 kpc (Chevalier et al. 1980; Black & Raymond 1984; Albinson et al. 1986). A distance of 2.3 kpc gives a lower limit to the shock speed of 3200 km s⁻¹, which leads to an upper limit to T_e /T_p of 0.15. The strongest limit comes from the combination of the X-ray temperatures from Hwang et al. (2002) with the proper motions on the western rim from Williams et al. (2016) and a distance of 3 kpc, which gives T_e /T_p = 0.027. We note that this region does not show Balmer-line filaments, so the preshock neutral fraction is small. The proper motion of knot g of Kamper & van den Bergh (1978) is 020 ± 001 per year. With the Hα width of Ghavamian et al. (2001) and the distance of 2.3–3.7 kpc, this implies T_e /T_p < 0.4. Another spectrum in the knot g area is given by Raymond et al. (2010). Within the uncertainties, the width of the broad line agrees with the other values, but a Gaussian broad component did not provide an adequate fit to the data. Several interpretations are possible, including multiple shocks within the aperture, a non-Maxwellian velocity distribution, a shock precursor, or pickup ions.

Another estimate of T_e /T_p is given by Matsuda et al. (2022). They see a substantial increase in T_e from 0.30 to 0.69 keV in a knot on the NW limb of Tycho between 2000 and 2015, which they attribute to Coulomb heating. They find T_e /T_p < 0.15 at the shock.

Individual values of I_B /I_N from Fesen et al. (1989) and Blair et al. (1991) vary considerably, and some are inconsistent with the models of Morlino et al. (2012). This is likely to be the result of contamination by radiative shocks. We exclude position P2D2 of Blair et al. (1991) because its very large FWHM would imply a shock speed above 5000 km s⁻¹, and we exclude positions in which I_B /I_N cannot be matched by the models. The average FWHM and I_B /I_N give T_e /T_p = 0.8 if T_e = T_p upstream. Overall, the profiles are suspect, given the possible contamination by radiative shocks. For the comparison of T_e,x with the Hα broad-component line width, we use Blair et al. (1991) and Katsuda et al. (2015). Coffin et al. (2022) have measured proper motions, but the distance uncertainty is large enough that we do not use proper motion velocities.

Helder et al. (2011) studied the electron–ion equilibration in RCW 86 based on the broad-component widths and electron temperatures from XMM spectra. In their sample, the three positions with low uncertainties showed T_e /T_p < 0.1, two positions showed T_e /T_p ≃ 0.45, and two positions showed T_e /T_p > 0.58. One position showed an extremely narrow Hα width that corresponded to T_e /T_p >10, but that is probably a case of a slow shock dominating the Hα profile while a faster shock produces the X-rays. The shock speeds ranged from 580 to 1100 km s⁻¹, but there was no trend of T_e /T_p with shock speed. Table 1 gives the value of T_e /T_p from Ghavamian et al. (2000) based on the Hα profile. As described under the corroborating observations, Ghavamian (2000) presented several other Hα profiles, and they indicate T_e /T_p in accord with values in Table 1.

The distance to RCW 86 is 2.4 ± 0.2 kpc based on the kinematic distance to the associated atomic and molecular shell (Sano et al. 2017), so proper motion measurements by Helder et al. (2013) and Yamaguchi et al. (2016) can be translated into shock speeds of the thermal filaments of 700–2200 km s⁻¹, with speeds of 1200 ± 200 km s⁻¹ in the NE and SE that correspond very well to the Hα broad-component widths for T_e /T_p = 0.4.

Morlino et al. (2014) combined the proper motion, Hα line width, and X-ray temperature to obtain tight limits on the cosmic-ray acceleration efficiency and T_e /T_p at one position. We use their value for Tables 2, 3, and 4.

For estimates of T_e /T_p based on the Hα FWHM and I_B /I_N , we use the observations of Medina et al. (2014) with the Ghavamian et al. (2001) predictions for I_B /I_N as a function of T_e /T_p . We exclude a point with a large uncertainty (FWHM = 347 ± 89 km s⁻¹) and positions with line widths of 141 km s⁻¹and 178 km s⁻¹, which would correspond to shocks slow enough to be radiative. We average the remaining nine positions and adopt T_e /T_p = 0.8–1.0.

Salvesen et al. (2009) also measured electron temperatures from ROSAT spectra. Two-temperature fits were required, but we use the lower temperatures because that component dominated the total emission. They are in line with other X-ray temperatures in the northern Cygnus Loop (Raymond et al. 2003; Katsuda et al. 2008; Sankrit et al. 2010). To estimate the contribution of Coulomb collisions to T_e for the X-ray observations, we assume a preshock density of 0.2 and therefore a postshock density of 0.8 cm⁻³, along with a thickness of 5 × 10¹⁷ cm corresponding to the spatial resolution of the ROSAT spectra. The Hα filaments are formed much closer to the shock, around 10¹⁵ cm, so Coulomb collisions can only heat the electrons to about 3 × 10⁵ K in the Hα-emitting region. We exclude four outlying points whose apparent speeds were lower than 300 km s⁻¹, as they would be ineffective in producing X-rays and may be transitioning to become radiative shocks. The observed X-rays probably originate in faster nearby shocks. This leaves 14 positions whose speeds range from 300 to 470 km s⁻¹, and whose values of T_e /T_p range from 0.73 to 1.62 and average to 1.10.

To determine T_e /T_p from the Hα width and T_e,x , we combine the Salvesen et al. (2009) and Medina et al. (2014) results. Of the six positions where both are available, we exclude the five with line widths below 250 km s⁻¹, leaving only Salvesen's position 8. It yields a value of T_e /T_p of 0.8, but the uncertainty in the Hα line width is 20%, and the uncertainty in T_e /T_p is correspondingly large.

We also combine the data from Salvesen et al. (2009) and Medina et al. (2014) to determine T_e /T_p from V_pm and the Hα width. Here, we average three positions, excluding the slowest shocks and the position whose line width is most uncertain and obtain an average T_e /T_p of 1.0.

Although the relatively bright filament in the NE first discussed by Fesen et al. (1982) has been studied extensively (Long et al. 1992; Hester et al. 1994; Sankrit et al. 2000; Blair et al. 2005; Katsuda et al. 2016), we do not include it here because it is partially a radiative shock.

There are several sources for Hα profiles and proper motions, but we have been unable to find an X-ray temperature derived from fits to a region that definitely corresponds to the shocked ISM (Smith et al. 1994; Helder et al. 2010; Hovey et al. 2018; Knežević et al. 2021). The broad-component line width varies around the remnant, ranging from around 3000 km s⁻¹ in the E and to over 4000 km s⁻¹ in the NE and NW. The value of I_B /I_N is 0.06 ± 0.01 according to the MUSE observations of Knežević et al. (2021). Although some positions show large uncertainties and provide less stringent upper limits to T_e /T_p , some positions indicate T_e /T_p < 0.1 and probably lower; see Figure 3 of Knežević et al. (2021).

This SNR also has well-determined proper motions (Hovey et al. 2018), and the distance to the LMC is known, so reliable shock speeds can be derived. Very recent proper motion measurements were made from HST Hα images (Arunachalam et al. 2022) and from Chandra images (Guest et al. 2022b). They show average shock speeds of 6315 and 6120 km s⁻¹, respectively, with speeds as high as 8000 km s⁻¹ in some regions. The limit on T_e /T_p given by Hovey et al. (2018) combines the Hα profile information with the proper motions, and we use that value for both determinations.

For the value of T_e /T_p from I_B /I_N and the Hα line width, we use the value of Hovey et al. (2018), which folds in proper motion measurements as well. We quote 1σ upper limits rather than the 95% confidence values given by Hovey et al. (2018). We also use the 95% upper limit from Hovey et al. (2018) (converted to 1σ) for the constraint from the Hα line width and proper motion velocity.

To determine T_e /T_p from X-ray observations and proper motion velocities, we combine the X-ray temperatures of Schenck et al. (2016) with proper motions from Hovey et al. (2018) and Williams et al. (2022). The Hovey et al. (2018) positions correspond better to the Schenck et al. (2016) extraction regions, while the Williams et al. (2022) proper motions should be more accurate because of their longer baseline. The average electron temperature of 1.4 keV and n_e t = 5.0 × 10¹⁰ cm⁻³ s are taken from Schenck et al. (2016). The extraction regions for the X-ray spectrum correspond approximately to regions east, 1-S, 2-S, and 1-N of Hovey et al. (2018), so we use the average Hα broad-component line width of those regions for comparison. We use the average proper motion speeds of Hovey et al. (2018) to predict an average proton temperature of 11.6 keV if there is no equilibration. More recently, Williams et al. (2022) measured proper motions in 0519–69.0, and their regions 5, 6, 16, and 20 approximately correspond to the regions where Schenck et al. (2016) extracted the X-ray spectrum. Their velocities give an average proton temperature of 16.7 keV. We use both to estimate a range T_e /T_p = 0.04–0.07. This is a case in which Coulomb heating could account for all the electron thermal energy. Guest et al. (2023) have very recently remeasured the X-ray expansion rate, and they find an average value of 4760 km s⁻¹, which is at the upper end of the range reported by Williams et al. (2022).

The estimate of T_e /T_p from I_B /I_N and the Hα line width uses the measurements of Smith et al. (1991). The X-ray temperature of 0.42 keV from Schenck et al. (2016) and the shock speed of 760 ± 40 from the line width given by Smith et al. (1991) give T_e /T_p = 0.28–0.67. However, Schenck et al. (2016) also give n_e t = 2.6 × 10¹¹ cm⁻³ s, which is long enough for Coulomb collisions to heat the electrons to 0.5 keV. We consider the value of n_e t to be the most uncertain of the measurements, and indeed somewhat large for the likely age of the SNR. Simply taking T_p = 1.02 keV from the Hα line width and the X-ray temperature T_e = 0.42 keV gives T_e /T_p = 0.42, and since Coulomb equilibration is probably significant, we take that as an upper limit. Proper motions are not available for this SNR.

We use the Hα profile measurements of Smith et al. (1991) and Rakowski et al. (2009) to find the values of T_e /T_p from I_B /I_N . Alan & Bilir (2022) fit X-ray spectra of the shocked ISM in DEM L71, and we have interpolated among positions where Rakowski et al. (2009) measured the Hα line widths. For seven positions, the broad-component widths range from 450 to 950 km s⁻¹, with an average of 785 km s⁻¹. The corresponding proton temperatures range from 0.39 to 1.72 keV, while the electron temperatures from the X-ray fits range from 0.26 to 0.43 keV. The average value for T_e /T_p is 0.23. The values of n_e t obtained by Alan & Bilir (2022) are higher than 10¹¹ cm⁻³ s, so substantial Coulomb heating of the electrons has probably occurred. We note that Rakowski et al. (2003) also compared Hα line widths with T_e,x at five positions around DEM L71. Their values of n_e t at two positions preclude useful determination of T_e /T_p , but three other positions with shock speeds from 750 to 1000 km s⁻¹ show T_e /T_p averaging 0.7, where Coulomb collisions might account for perhaps half of T_e . No proper motions are available for this SNR.

Hα profiles at four positions are available from a MUSE data cube, and they indicate shock speeds between 1160 and 3560 km s⁻¹, depending on how much electron–ion equilibration occurs (Ghavamian et al. 2017). Unfortunately, the values of I_B /I_N range between 0.23 and 0.48, and they are all below the range predicted by Morlino et al. (2012). It is likely that the narrow components are contaminated by Hα from a stronger precursor than is predicted by Morlino et al. (2012) or by some other background. Guest et al. (2022a) fit the X-ray spectrum of the region selected as shocked circumstellar medium on the basis of X-ray line ratios, but this region may be dominated by slower shocks in regions far from the positions of the MUSE spectra. The proton temperatures from the Hα profiles and the values of n_e t from the X-ray fits would give T_e from Coulomb collisions alone of 1.6 keV, which is well above the value of 0.9 keV from the fit. The average proper motion expansion speed is 4170 km s⁻¹ (Williams et al. 2022), and this speed combined with T_e,x = 0.9 keV from Guest et al. (2022a) yields T_e /T_p = 0.03. Because T_Coul is almost twice the electron temperature measured from the X-ray spectrum, and because slower shocks probably contribute to the X-ray emission averaged over much of the SNR and reduce the temperature determined from the X-ray spectra, we do not include N103B in the tables or Figure 3, but it is consistent with the low values of T_e /T_p in other fast shocks.

There are no Hα profiles for this SNR, apparently because the shock is passing through ionized material. Xi et al. (2019) measured the expansion proper motion of 1E102.2–7219, and they fit the X-ray spectra of the shocked ISM in five sectors covering over half the circumference of the SNR. They found temperatures in a narrow range between 0.61 and 0.75 keV and values of n_e t of 1.1–2.3 × 10¹¹ cm⁻³ s. For a shock speed of 1610 km s⁻¹ and n_e t = 1.7 × 10¹¹ cm⁻³ s, Coulomb collisions alone could heat the electrons to 1.7 keV, which is above the observed electron temperature. The value of n_e t is not unreasonable given an estimated age of 1740 yr (Banovetz et al. 2021), but it is still the least reliably determined parameter in this comparison.