ELECTRON HEATING, MAGNETIC FIELD AMPLIFICATION, AND COSMIC-RAY PRECURSOR LENGTH AT SUPERNOVA REMNANT SHOCKS

As mentioned above, preshock magnetic field may be amplified by either resonant or nonresonant instabilities. Resonant amplification occurs when an individual cosmic-ray gyrofrequency is Doppler shifted into resonance with a specific wave mode (Alfvén or fast mode, both commonly referred to as Alfvén waves when parallel propagating) and energy is transferred from particle to wave or vice versa. This typically saturates when δB ∼ B or less, because at this stage the wave-particle resonance is lost and amplification ceases. Nonresonant magnetic field amplification is essentially an MHD process. The current associated with the drifting cosmic rays (or more properly the return current so induced in the background plasma) amplifies Alfvén waves, typically with k_||r_g ≫ 1 (in the resonant case k_||r_g ∼ 1, where k_|| is the parallel wavevector and r_g is the cosmic-ray gyroradius). There is no individual wave-particle resonance. Bell (2004, 2005) has discussed nonlinear nonresonant saturation of this process, whereby unmagnetized cosmic rays and their associated magnetic field expel the ambient plasma from cylindrical filaments, shutting down the return current as the plasma becomes magnetized. Other authors (e.g., Luo & Melrose 2009; Gargaté et al. 2010) have discussed a resonant means of saturating the nonresonant magnetic field growth, as the cosmic-ray current is limited by pitch-angle scattering in turbulence. In the following subsections we estimate the magnetic field amplification level and precursor distance over which this amplification occurs in both regimes of saturation.

Various kinetic simulations of nonresonantly amplified magnetic field (Riquelme & Spitkovsky 2009; Stroman et al. 2009; Gargaté et al. 2010) suggest that cosmic-ray-induced magnetic field saturates not according to the Bell (2004, 2005) scenario outlined above but by the generation of turbulence that scatters the cosmic rays and reduces their drift relative to the upstream medium. We discuss here the level of magnetic field amplification and the length of the associated precursor to be expected in such a scenario.

We follow in part the analytical description of this saturation process given by Luo & Melrose (2009). The rate of change of the cosmic-ray streaming speed ahead of the shock, v_CR = ∫fv_zd³p, where f is the cosmic-ray distribution function normalized to unity and v_z is the z-component of an individual cosmic-ray velocity vector, due to scattering in turbulence is usually given as

$$\begin{equation} {dv_{{\rm CR}}\over dt}=\int f{dv_z\over dt}d^3p=-\int f\left<D\right>v_zd^3p, \end{equation} \tag{ 1 }$$

where $\left<D\right>=\int _{-1}^1 (1-\mu ^2)Dd\mu /2$ with D = (π/4)(Ω_i0/ < γ > )(δB²/B²), the cosmic-ray pitch-angle scattering diffusion coefficient. Here Ω_i0 is the proton cyclotron frequency calculated with the initial magnetic field B (Ω_i will be the same quantity calculated with the amplified field δB), <γ > is the average cosmic-ray Lorentz factor, and δB²/B² ≪ 1 is assumed. We extrapolate this to the case of magnetic field amplification by the nonresonant instability, where δB²/B² ≪ 1 no longer holds, as follows. Since the nonresonant instability preferentially grows magnetic field with kr_g ≫ 1, only cosmic rays with pitch-angle cosines μ ≃ ±1/kr_g interact with the turbulence. The limits on the integral over μ become ±1/kr_g and <D > =(π/4)(Ω_i0/ < γ > )(δB²/B²)/kr_g. The particle scattering rate remains less than the gyrofrequency, even though δB²/B² ≫ 1, due to the presence of the factor kr_g in the denominator. Assuming δB∝exp (Γt), we can integrate Equation (1). Luo & Melrose (2009) insert a growth rate $\Gamma =\sqrt{2}k_{\Vert }v_A\sqrt{4\pi J_{{\rm CR}}/k_{\Vert }cB -1}\simeq (6\eta kr_{g0})^{1/2}\left(v_s/c\right)^{3/2} \Omega _{{\rm CR}}$ and derive δB²/B² in the range 10–100. These values are lower than the nonresonant saturation, and so saturation by resonant diffusion, suppressing the cosmic-ray flow speed with respect to the upstream medium, will occur first if allowed.

The approximate expression for the growth rate above neglects the last term in the square root and so overestimates the growth rate. This is not a very serious problem, except where the cosmic-ray drift velocity is approaching the saturation value where Γ → 0. We prefer to take $\Gamma =\sqrt{2}k_{\Vert {\rm max}}v_A$, the maximum value of the growth rate at parallel wavevector $k_{\Vert {\rm max}}=J_{{\rm CR}}B/2\rho cv_A^2=2\pi J_{{\rm CR}}B/c\delta B^2$, which may both be rewritten with ω_pi and Ω_i as the upstream ion plasma and cyclotron frequencies, respectively, as

$$\begin{eqnarray} \Gamma &=&{3\eta \over \sqrt{2}}{\omega _{pi}\over \gamma _1\gamma _{{\rm max}}}{\gamma _{{\rm max}}- 3\gamma _1/4 \over \ln \gamma _{{\rm max}}-1}{v_s^2v_{{\rm CR}}\over c^3}{B\over \delta B}\nonumber \\ & =& 0.013{\eta \sqrt{n_i}\over \gamma _1\gamma _{{\rm max}}}{\gamma _{{\rm max}}-3\gamma _1/4\over \ln \gamma _{{\rm max}}-1}{B\over \delta B} \nonumber\\ && \times \left(v_s\over {\rm 5000\, km\, s}^{-1}\right)^2\left(v_{{\rm CR}}\over {\rm 5000\, km\, s}^{-1}\right) {\rm s}^{-1} \end{eqnarray} \tag{ 2 }$$

and

$$\begin{eqnarray} k_{\Vert {\rm max}}r_g &=& {3\eta \over 2}{\omega _{pi}^2\over \Omega _i^2}{v_s^2v_{{\rm CR}}\over c^3}{\gamma _{{\rm max}}-3\gamma _1/4\over \ln \gamma _{{\rm max}}-1}{\gamma \over \gamma _{{\rm max}}\gamma _1}\nonumber \\ & =& 15340\eta n_i\left(v_s\over {\rm 5000\, km\, s}^{-1}\right)^2\left(v_{{\rm CR}}\over {\rm 5000\, km\, s}^{-1}\right) \nonumber\\ && \times \left(3\,\mu {\rm G}\over B\right)^2\left(B\over \delta B\right)^2 {\gamma _{{\rm max}}-3\gamma _1/4\over \ln \gamma _{{\rm max}}-1}{\gamma \over \gamma _{{\rm max}}\gamma _1}. \end{eqnarray} \tag{ 3 }$$

Here the current J_CR is assumed to be carried by unmagnetized cosmic rays with Lorentz factors γ₁ < γ < γ_max, as given in the Appendix, with an additional flux coming from cosmic rays escaping the shock upstream at γ ⩾ γ_max, given by Bell et al. (2013) approximately as n_CRv_s/4γ_max. The ratio of cosmic-ray pressure to the shock ram pressure η and γ₁ are assumed to be independent of δB. We substitute Equation (3) into Equation (1) and integrate the right-hand side over p, assuming f∝1/p⁴. In Equation (1), we rewrite dv_CR/dt → v_CRdv_CR/dz. With δB/B ∼ exp (Γz/v_CR), we put dv_CR/dz ≃ −v_CR/z ≃ Γ/ln (δB/B) in Equation (1) to find

$$\begin{eqnarray} \ln \left(\delta B\over B\right){\delta B^5\over B^5} &\simeq & {9\sqrt{2}\over \pi }{\omega _{pi}^3\over \Omega _{i0}^3}{v_s^6\over c^6}\eta ^2 \nonumber \\ && \times \left(\gamma _{{\rm max}}-3\gamma _1/4\over \ln \gamma _{{\rm max}}-1\right)^2 \left(1+\gamma _{{\rm max}}\gamma _1\over \gamma _{{\rm max}}^2\gamma _1^2\right) \nonumber\\ &=& 8349\left(v_s\over {\rm 5000\, km\, s}^{-1}\right)^6\left(3\,\mu {\rm G}\over B\right)^3n_i^{3/2}\eta ^2 \nonumber\\ && \times \left(\gamma _{{\rm max}}-3\gamma _1/4\over \ln \gamma _{{\rm max}}-1\right)^2\left(1+\gamma _{{\rm max}}\gamma _1\over \gamma _{{\rm max}}^2\gamma _1^2\right). \end{eqnarray} \tag{ 4 }$$

This evaluates to δB/B ∼ 14(v_s/5000 km s⁻¹)^6/5(γ_max/10⁶)^1/5 for η ∼ 0.1, n_i ∼ 1, and γ₁ = 1, similarly to Luo & Melrose (2009), but with different dependencies of δB/B on the shock velocity and other parameters. Comparing with Gargaté et al. (2010), for example, where γ_max ∼ 10³, plugging in numbers for their model B1, B2, or B3 yields δB/B ∼ 14, comparable to their simulation results.

This degree of magnetic field amplification is smaller than that inferred in the observations cited above, though within uncertainties, once the magnetic field compression by the shock is taken into account, the lower end of the postshock magnetic fields given above approaching ∼100 μG might be accessible.

The length scale over which this precursor develops is given by

$$\begin{eqnarray} L &=& {v_{{\rm CR}}\over \Gamma }\ln \left(\delta B\over B\right)={\sqrt{2}\over 3}{c^3\over v_s^2}{\delta B\over B}\ln \left(\delta B\over B\right) \nonumber\\ && \times {\gamma _1\gamma _{{\rm max}}\over \gamma _{{\rm max}}-3\gamma _1/4}{\left(\ln \gamma _{{\rm max}}-1\right)\over \eta \omega _{pi}}, \end{eqnarray} \tag{ 5 }$$

which evaluates to approximately $4\times 10^{10} (5000\, {\rm km\, s}^{-1}/v_s)^2 (\delta B/B)\ln (\delta B/B)\gamma _1\ln \gamma _{{\rm max}}/\eta \sqrt{n_i}$ cm for γ_max ≫ γ₁. With unmagnetized cosmic rays, γ₁ ≃ 1, and L ∼ 10¹²(5000 km s⁻¹/v_s)²ln γ_max/η cm. The precursor to the shock in model B2 of Gargaté et al. (2010) is thus predicted to be 8 × 10¹² cm deep, in good agreement with the simulated value (v_s/Γ_max)ln (δB/B) ≃ 4 × 10¹² cm. Where δB/B is approximately proportional to $v_s^{6/5}$, $L\propto v_s^{-4/5}\ln v_s$, which is close to the L∝1/v_s discussed above. Calculating r_g with δB rather than B in Equation (3) would give δB/B∝v_s and $L\propto v_s^{-1}\ln v_s$, even closer to the behavior seen in the data. We emphasize that L represents the cosmic-ray precursor associated with magnetic field amplification and is of course much smaller than the cosmic-ray precursor associated with resonant wave generation and scattering, given approximately by <D > /v_s.

If the unmagnetized cosmic-ray spectrum extends down to nonrelativistic energies so that γ₁ ∼ 1, then this precursor is predicted to be too small to be spatially resolvable, ∼10¹⁴ cm for η ∼ 0.1 and n_i ∼ 1. Significant magnetization of cosmic rays, increasing γ₁, lengthens the precursor but also reduces the eventual magnetic field so long as resonant saturation remains effective. Such magnetization, though, is more likely in the case of nonresonant saturation discussed below, where higher magnetic fields can result.

Bell (2005) argues that cosmic-ray-induced magnetic field amplification should saturate as the unmagnetized cosmic rays and their associated magnetic field expel the ambient plasma from cylindrical filaments.

We can examine the expected magnitude of the amplified field at saturation by considering the nonresonant instability (Bell 2004, 2005). At saturation, $1/r_g < k_{\Vert } < J_{{\rm CR}}B/n_im_i cv_A^2=4\pi (n_{{\rm CR}}^{\prime }+n_{{\rm CR}}/4\gamma _{{\rm max}})qv_sB/c\delta B^2$, where $n_{{\rm CR}}^{\prime }$ is the number density of unmagnetized cosmic rays with gyroradius >r_g. Consequently,

$$\begin{eqnarray} && { \delta B < \displaystyle\sqrt{4\pi n_{{\rm CR}}^{\prime }v_sm_ic\left<\gamma ^{\prime }\right>} < \displaystyle\sqrt{12\pi \eta n_im_i v_s^3/c} \displaystyle\sqrt{\displaystyle\ln \gamma _{{\rm max}}-\ln \gamma _1+1/4\over\displaystyle \ln \gamma _{{\rm max}}- 1} }\nonumber \\ &&\;\; {< 5\times 10^{-4} \displaystyle\sqrt{\eta n_i}\left(v_s/{\rm 5000\, km\, s}^{-1}\right)^{3/2} \displaystyle\sqrt{\displaystyle\ln \gamma _{{\rm max}}-\ln \gamma _1+1/4\over \displaystyle\ln \gamma _{{\rm max}}- 1} {\rm G.} } \end{eqnarray} \tag{ 6 }$$

This represents an amplification over the initial field, assumed to be 3μG, of a factor of 10–50 (taking η ∼ 0.1, n_i ∼ 1 cm⁻³), depending on the value assumed for the maximum Lorentz factor of magnetized cosmic rays, γ₁. This is about an order of magnitude higher than magnetic field amplification limited by resonant scattering.

We also calculate the magnetic precursor depth in the case that the growth is nonresonantly saturated, according to Bell (2004, 2005). We assume k||B, as seen, for example, even at oblique shocks, in simulations (Gargaté & Spitkovsky 2012), and in in situ observations (Bamert et al. 2004). Starting from Bell's expression for the growth rate, we write the time evolution of the amplified magnetic field δB as

$$\begin{equation} {d\delta B\over dt} = \sqrt{{J_{{\rm CR}}Bk\over \rho c}-k^2v_A^2}\delta B = \sqrt{{J_{{\rm CR}}Bk\over \rho c}-{k^2\delta B^2\over 4\pi \rho }}\delta B. \end{equation} \tag{ 7 }$$

We integrate to find the time t over which this magnetic field develops,

$$\begin{equation} t=\int _B^{\delta B}\sqrt{\rho c\over J_{{\rm CR}}Bk}{1\over \sqrt{1-{kc\over 4\pi J_{{\rm CR}}B}\delta B^2}}{d\delta B\over \delta B}. \end{equation} \tag{ 8 }$$

We then put $\delta B = \sqrt{4\pi J_{{\rm CR}}B/kc}\cos \theta$ to write

$$\begin{eqnarray} t &=& -\sqrt{\rho c\over J_{{\rm CR}}Bk}\int _{\arccos \sqrt{kcB/4\pi J_{{\rm CR}}}} ^{\arccos \sqrt{kc\delta B^2/ 4\pi J_{{\rm CR}}B}}\sec \theta d\theta \nonumber\\ &=& \sqrt{\rho c\over J_{{\rm CR}}Bk}\left[\ln \left|\tan \theta +\sec \theta \right|\right]^{\arccos \sqrt{kcB/4\pi J_{{\rm CR}}}} _{\arccos \sqrt{kc\delta B^2/ 4\pi J_{{\rm CR}}B}}. \end{eqnarray} \tag{ 9 }$$

Evaluating,

$$\begin{eqnarray} t &=&\sqrt{\rho c\over J_{{\rm CR}}Bk}\left[\ln \left|{\sqrt{1-kcB/4\pi J_{{\rm CR}}}\over \sqrt{kcB/4\pi J_{{\rm CR}}}} +\sqrt{4\pi J_{{\rm CR}}/kcB}\right|\right. \nonumber\\ &&\left. -\ln \sqrt{4\pi J_{{\rm CR}}B/kc\delta B^2}\right]\nonumber\\ &=&\sqrt{\rho c\over J_{{\rm CR}}Bk}\ln \left|{\delta B\over B}\sqrt{1-kcB/4\pi J_{{\rm CR}}} +{\delta B\over B}\right|. \end{eqnarray} \tag{ 10 }$$

At saturation, where k = 4πJ_CRB/cδB², with δB ≫ B,

$$\begin{eqnarray} t &=& \sqrt{n_im_i\over 4\pi }{\delta B\over B}{c\over n_{{\rm CR}}^{\prime }qv_{{\rm CR}}+n_{{\rm CR}}qv_s/4\gamma _{{\rm max}}}\ln \left(2{\delta B\over B}\right) \nonumber\\ &\simeq & {c^3\gamma _1\gamma _{{\rm max}}\over 3\eta v_s^2v_{{\rm CR}}\omega _{pi}}{\ln \gamma _{{\rm max}}-1\over \gamma _{{\rm max}}-3\gamma _1/4}{\delta B\over B}\ln \left(2{\delta B\over B}\right). \end{eqnarray} \tag{ 11 }$$

The precursor depth is then

$$\begin{equation} L={c^3\over 3\eta v_sv_{{\rm CR}}\omega _{pi}}{\gamma _1\gamma _{{\rm max}}\over \gamma _{{\rm max}}-3\gamma _1/4}\left(\ln \gamma _{{\rm max}}-1\right) {\delta B\over B}\ln \left(2{\delta B\over B}\right). \end{equation} \tag{ 12 }$$

This is similar in expression to the case of resonant saturation of the nonresonant instability. There is a missing factor of $\sqrt{2}$ in the numerical constant, and here $\delta B \propto v_s^{3/2}$. Also, for nonresonant saturation, γ₁ is likely to be significantly larger, yielding a longer cosmic-ray magnetic field amplification precursor, as well as a higher magnetic field than in resonant saturation. Bell et al. (2013) argue that γ₁ ∼ γ_max, in which case $L\sim 10^{13}\gamma _{{\rm max}}\ln \gamma _{{\rm max}}/\eta \sqrt{n_i}$ cm.

We can estimate the physical size of the precursor by taking γ_max ∼ 10⁵ (e.g., Bell et al. 2013; Vink & Laming 2003), to find L ∼ 10¹⁷ − 10¹⁸ cm, which is potentially resolvable by, for example, the 05 angular resolution of Chandra for relatively nearby galactic SNRs. Morlino et al. (2010) and Winkler et al. (2014) consider the case of SN 1006 specifically. The different dependence of δB on v_s in this saturation case leads to $L\propto v_s^{-1/2}\ln v_s$, assuming that γ₁ and γ_max are independent of v_s. This is farther from the L∝1/v_s that produces the constant electron heating with shock velocity and suggests that one should look at SNRs possibly subject to the Bell magnetic field amplification to see whether evidence can be found for enhanced postshock electron temperatures because the magnetic precursor length decreases less quickly with increasing shock speed.

Another means of nonresonant saturation is discussed by Niemiec et al. (2010), following Winske & Leroy (1984). The background plasma can be accelerated by the nonresonant mode, gradually shutting down the cosmic-ray current with respect to it. The principal requirement for this is a cold cosmic-ray "beam," where the perpendicular temperature is much lower than the parallel temperature, and when the beam is cold, filamentation does not occur. Niemiec et al. (2010) argue that such a case may occur ahead of relativistic shocks, where only cosmic rays focused along the shock velocity vector are able to outrun the shock and contribute to magnetic field amplification. The application to lower velocity SNR shocks appears less clear.

Clearly, resonant saturation, if allowed, will restrict magnetic field amplification to only a factor of a few over the preshock ambient field. However, observations of SNRs generally reveal much larger magnetic field amplification than this, suggesting that resonant saturation does not occur. Either nonresonant magnetic field amplification proceeds until nonresonant saturation sets in, or some other mechanism of magnetic field amplification is at work.

Assuming that nonresonant field amplification is at work, what should determine whether resonant or nonresonant saturation will occur? We argue that when the parallel wavevector of the amplification satisfies the inequality

$$\begin{equation} {1\over r_g} < k_{\Vert } < {J_{{\rm CR}}B\over \rho cv_A^2} = {\gamma n_{{\rm CR}}\over n_i}{v_s^2\over v_A^2}{\Omega _{{\rm CR}0}\over v_s} < {\Omega _{{\rm CR}0}\over v_s}{\delta B\over B}, \end{equation} \tag{ 13 }$$

resonant scattering should begin to be inhibited. This is because when k_|| < Ω_CR/v_s = (Ω_CR0/v_s)(δB/B) at a parallel shock, the minimum drift velocity cosmic rays need to outrun the shock and form the precursor also moves them out of resonance with parallel propagating waves.⁵ Substituting for δB/B from Equation (4) or Equation (6), we get v_s in the range

$$\begin{eqnarray} & & \left(v_s\over 5000\, {\rm km\, s}^{-1}\right)< 0.21\left(B\over 3\,\mu {\rm G}\right)^{-13/14}n_i^{-1/4} \eta ^{-3/14} \nonumber\\ && \quad\quad \times \left(\ln \gamma _{{\rm max}}-1\over \ln \gamma _{{\rm max}}-\ln \gamma _1 +1/4\right)^{5/14} \nonumber \\ & &\rightarrow 0.64\left(B\over 3\,\mu {\rm G}\right)^{-6/5}n_i^{-1/5} \eta ^{-1/5}\left(\ln \gamma _{{\rm max}}-1\over \ln \gamma _{{\rm max}}-\ln \gamma _1 +1/4\right)^{1/5}, \nonumber\\ \end{eqnarray} \tag{ 14 }$$

respectively. To satisfy the conditions discussed by Bell (2004, 2005), we take γ₁ → γ_max, η ∼ 0.1, and the upper limit on δB derived from Equation (6). Equation (14) shows that resonant saturation is inhibited when v_s < ∼ 10, 000 km s⁻¹ (dropping the explicit dependencies on other parameters).

In considering the competition between resonant and nonresonant magnetic field amplification, Marcowith & Casse (2010) argue for a gradual transition between the two mechanisms over an order of magnitude in shock velocity, at similar values to those in Equation (14), with resonant amplification dominating at lower shock velocities, since at saturation δB∝v_s, whereas for nonresonant amplification $\delta B\propto v_s^{3/2}$. Conversely, we have just shown that resonant saturation is favored over nonresonant saturation at higher shock velocities, although this depends on assumptions about the dependence of η on v_s.

The arguments we have made above also tacitly assume a parallel shock geometry. Bell (2005) gives the angular dependence of the nonresonant growth rate, Γ. Keeping k||B,

$$\begin{eqnarray} && x^3 +x^2\left[2+\beta \right]+x\left[1+2\beta -{B^2j^2\over k^2v_A^4\rho ^2}\right]+\beta -{B^2j^2\over k^2v_A^4\rho ^2} \nonumber\\ && \quad\quad \times [(\beta -1)\cos ^2\theta +1]=0, \end{eqnarray} \tag{ 15 }$$

where $x=\Gamma ^2/k^2v_A^2$, $\beta = c_s^2/v_A^2$, and θ is the angle between k and j. At β = 1, the explicit angular dependence disappears and $\Gamma = kv_A\sqrt{Bj/k\rho v_A^2 -1}$. Putting k = k_||max from Equation (3), $Bj/k_{\Vert {\rm max}}\rho v_A^2 =2$ and Γ = k_||maxv_A. For 0 ⩽ β ⩽ 1, k_||maxv_A ⩽ Γ ⩽ 1.25k_||maxv_A. We emphasize "explicit" angular dependence above, because k_||max may depend on shock obliquity through its proportionality to $\eta = \left(\ln \gamma _{{\rm max}}-1\right)c^2n_{{\rm CR}}/3n_iv_s^2$. The variation of cosmic-ray pressure η with shock obliquity is very uncertain. The threshold energy for injection into the diffusive shock acceleration process increases with increasing obliquity (e.g., Zank et al. 2006), reducing the cosmic-ray number density, but the acceleration rate in such geometry increases. The generation of turbulence, however, is much more likely to decrease with increasing obliquity (e.g., Laming et al. 2013), as does the cosmic-ray current. These both make it likely that resonant scattering and saturation are even more inhibited in favor of nonresonant saturation in these cases of oblique shocks. Simply reinstating the obliquity in Equation (13) leads to v_s gaining an extra factor cos ^−5/14θ_bn → cos ^−2/5θ_bn in Equation (14).

In cases where cosmic-ray acceleration is efficient Reville & Bell (2013) show that amplified magnetic fields become highly disorganized and essentially isotropic. The initial shock obliquity has little effect on the final magnetic field. This might support some of our arguments about the transition to turbulence above, but by assuming that the cosmic-ray acceleration is "efficient," Reville & Bell (2013) avoid the shock injection issue. Caprioli & Spitkovsky (2013) show that nonresonant saturation is still obtained at shock obliquities of 20°, using large-scale hybrid simulations, but they comment that the effect is particularly evident at parallel shocks. This implication is contrary to our speculation above.

In summary, whether nonresonant magnetic field amplification saturates resonantly or nonresonantly depends on the value of η and its dependence on v_s and remains unclear. Much of this uncertainty can be traced to the problem of particle injection into diffusive shock acceleration, and so a simple solution appears unlikely. However, one clear difference between these two regimes appears to be in the extent of the cosmic-ray-induced magnetic field shock precursor to be expected. This is something that could possibly be exploited observationally to identify the saturation mechanism, and we return to a discussion of this further below.

Following the inference of Ghavamian et al. (2007b) that electrons at collisionless shocks may be heated in the cosmic-ray precursor, Rakowski et al. (2008) estimated the growth rate of lower-hybrid waves in the cosmic-ray-amplified magnetic field. Lower-hybrid waves are electrostatic ion oscillations directed almost perpendicularly to the magnetic field in conditions where the electrons are magnetized (electron gyroradius ≪ wavelength), inhibiting the electron screening of the oscillation that would otherwise occur. The wave phase velocity perpendicular to the magnetic field is much smaller than that along it, allowing the wave to simultaneously be in resonance with unmagnetized ions and magnetized electrons.

Rakowski et al. (2008) calculated a kinetic growth rate of lower-hybrid waves in a cosmic-ray precursor, showing that the reactive growth for plausible cosmic-ray distribution functions is zero. We correct here a small error in the final step of their calculation. We give a revised version of their Equation (5) for the growth rate of lower hybrid waves in a cosmic-ray precursor, where the cosmic rays are assumed to be in a "kappa"-distribution with index κ, where κ = 2 corresponds to the p⁻⁴ spectrum familiar in first-order Fermi acceleration,

$$\begin{eqnarray} \gamma &=& \left(\frac{\pi }{\kappa }\right)^{3/2}\frac{q^{2}}{p_{t}k} \frac{\omega ^{3} n_{{\rm CR}}}{\omega _{pi}^{2}+ \omega _{pe}^{2}\cos ^{2}\theta } \frac{(2\kappa -3)\Gamma (\kappa)}{\sqrt{2}\Gamma (\kappa -1/2)} \nonumber \\ & & \times \int \delta (\omega - {\bf k} \cdot {\bf v}) \left\lbrace -\left[1+\frac{(p_{x}-mv_{s})^{2}}{2\kappa p_{t}^{2}}\right]^{-\kappa } \frac{(p_{x}-mv_{s})}{p_{t}^{2}}\right. \nonumber\\ &&\left. \times {k\over \left|k\right|} + \frac{\kappa }{\kappa -1}\frac{xv_{s}}{\varkappa ^{2}}\frac{\partial \varkappa }{\partial p_{x}} \left[ 1 + \frac{(p_{x}-mv_{s})^{2}}{2\kappa p_{t}^{2}} \right]^{1-\kappa } \right\rbrace \nonumber \\ & & \times e^{-xv_{s}/D} dp_{x}. \end{eqnarray} \tag{ 16 }$$

This is modified from the original version by the factor k/|k| multiplying the first term in curly brackets. This arises from the scalar product k · ∂f/∂p and restricts the region of p_x where the growth rate may be positive to 0 < p_x < mv_s. When the momentum dependence of ϰ is neglected, maximum growth is found at p_x = 0.7mv_s for κ = 2 with rate

$$\begin{equation} \gamma _1= 0.04 {n_{{\rm CR}}\over n_i}\omega. \end{equation} \tag{ 17 }$$

It is not possible in this case for lower-hybrid waves to stay in contact with the shock, as in Laming (2001a) and Laming (2001b), but in the context of an extended cosmic-ray precursor (as opposed to a narrow precursor due to shock-reflected ions extending about one gyroradius upstream), this does not greatly affect their growth. If the second term involving ∂ϰ/∂p_x dominates the growth rate, maximum growth is found at $p_x=\pm \sqrt{13}mv_s/2$, also for κ = 2, with rate

$$\begin{equation} \gamma _2= 0.17r {n_{{\rm CR}}\over n_i}\omega, \end{equation} \tag{ 18 }$$

where the cosmic-ray diffusion coefficient ϰ∝p^r. At a parallel shock r = 1/3–1/2 for Kolmogorov or Kraichnan turbulence, respectively, whereas r = 1/9–1/6 at a perpendicular shock (see Appendix A in Rakowski et al. 2008). Thus, the two terms are likely to contribute a similar order of magnitude to the growth rate. This is a factor of about two smaller than that originally given (γ = 0.14(n_CR/n_i)ω) in the most appropriate case of a perpendicular shock. Rakowski et al. (2008) compared the growth rate for lower-hybrid waves with that for magnetic field amplification, assuming that the same cosmic rays are responsible for both. They thus determine a critical Alfvén Mach number (of order 10) where magnetic field amplification ceases and lower-hybrid wave generation takes over. This argument is most appropriate for resonant saturation, where the magnetization of cosmic rays is not effective in saturating the growth. To explain other cases, the growth rate of Rakowski et al. (2008) needs to be modified to include only the unmagnetized cosmic rays.

To examine the effect of electron heating, we compare the growth rates for magnetic field amplification and lower-hybrid waves. At full nonresonant saturation, the unmagnetized cosmic rays and ambient plasma do not overlap, and the growth of lower-hybrid waves should be suppressed. In such a case, the treatment of Rakowski et al. (2008) may become invalid. However, it is important to be aware that the inference $T_e/T_i\propto 1/v_s^2$ is determined from SNR shocks that have neutral material in their preshock media and generally do not show strong X-ray synchrotron emission. As such, these are unlikely to be the strongest cosmic-ray acceleration sites and presumably have not saturated their magnetic field amplification in this manner. It is even possible that the electron heating itself prevents the magnetic field amplification from saturating.

We compare the growth rate for waves that heat electrons, Γ ≃ 0.07ωn_CR/n_i, with the growth rate for magnetic field amplification, $\Gamma =\sqrt{2}k_{\Vert {\rm max}}v_A$. We find

$$\begin{eqnarray} {\delta B^2\over B^2} &=& {\omega _{pi}\over \Omega _{i0}}\sqrt{m_e\over 2m_i}{v_{{\rm CR}}\over c}{1\over 0.07} \nonumber\\ &\simeq & 180\left(v_s\over 5000\ {\rm km\ s}^{-1}\right)\sqrt{n_i}\left(3\,\mu G\over B\right) \left(\gamma _{{\rm max}}-3\gamma _1/4\over \gamma _{{\rm max}}-\gamma _1\right). \nonumber\\ \end{eqnarray} \tag{ 19 }$$

The predicted amplified magnetic field is higher than that coming from nonresonant amplification saturated by resonant scattering, and so that result is unlikely to change. But in this case the magnetic field never grows to the level where lower-hybrid wave growth competes with magnetic field amplification, so we do not expect significant electron heating when resonant saturation is important. However, the field in Equation (19) is lower than that expected from nonresonant saturation, and so the electron heating in a cosmic-ray precursor might prevent the full nonlinear stage of that instability from developing, and allow electron heating by lower-hybrid waves to take over from magnetic field amplification in dissipating the free energy of the cosmic-ray current. From Equation (12), we would also expect it to reduce the depth of the magnetic field shock precursor by a similar factor, i.e., about one order of magnitude.

The level of magnetic field saturation imposed by electron heating on the amplification by preexisting turbulence is harder to assess. In the case that the magnetic field is amplified by the shock itself (Giacalone & Jokipii 2007), the electron heating should have no effect. In the case discussed by Beresnyak et al. (2009) and Drury & Downes (2012), corresponding to the interaction of the cosmic-ray precursor with pre-shock turbulence, the growth rate is approximately given by the vorticity resulting in the preshock flow following deceleration by the cosmic-ray pressure gradient. This is Γ ∼ v_sηδρ/ρλ, where δρ/ρ expresses the amplitude of the preshock turbulence and λ is the length scale over which the density varies. Equating this to the growth rate for lower-hybrid waves yields δB ∼ 2  ×  10⁷(ln γ_max − 1)δρ/ρλ G. Specializing to the forward shock of Cas A, where λ ∼ 10¹⁸ cm and δρ/ρ ∼ 10³, δB ∼ 2 × 10⁻⁸(ln γ_max − 1) G. This is much lower than the observed field of order 10⁻⁴G, suggesting that such instabilities are not operating. Values of λ as low as 10¹⁵ cm would be required. As Beresnyak et al. (2009) comment, the reason such instabilities may compete with the nonresonant current-driven instability is that although the growth rate is intrinsically weaker, all the cosmic rays participate in the former. In the latter, only the very highest energy cosmic rays contribute. But because the growth rate is intrinsically weaker, the Beresnyak et al. (2009) instability is more easily saturated by electron heating.

We briefly recap. Ghavamian et al. (2007b) and van Adelsberg et al. (2008) report a dependence of electron temperature on ion temperature, T_e/T_i, immediately postshock approximately proportional to $1/v_s^2$ for a selection of SNRs exhibiting Hα emission. If $T_i\propto v_s^2$, then T_e is constant. Ghavamian et al. (2007b) put forward an explanation that electron heating in a cosmic-ray precursor of length L ∼ ϰ/v_s, where ϰ is the cosmic-ray diffusion coefficient, would lead to such a dependence. The time spent in the precursor by preshock gas is then $t\sim \varkappa /v_s^2$, leading to electron heating $E=0.5 mv_e^2 = 0.5 mD_{\Vert \Vert }t$ independent of v_s if the electron velocity diffusion coefficient, D_||||, in waves excited in the precursor is proportional to $v_s^2$. In the specific case of lower-hybrid waves considered by Ghavamian et al. (2007b), this is the case. In this discussion, no reference was made to the specific form of the cosmic-ray precursor, but the required dependence ϰ∝1/B suggests a Bohm-like diffusion coefficient and magnetic field amplification via the resonant instability.

Thus far, it has been the purpose of this paper to investigate how far such behavior should persist into the regime of nonresonant magnetic field amplification. Rakowski et al. (2008) invoked magnetic field amplification as a means of ensuring a locally quasi-perpendicular shock (necessary for the generation of lower-hybrid waves) and showed that at high Alfvén Mach number magnetic field amplification has the higher growth rate, while at lower M_A the cosmic-ray generation of lower-hybrid waves wins. They estimated a critical M_A ∼ 6v_inj/v_s, which would become M_A ∼ 3v_inj/v_s ∼ 30 with the correction to the lower-hybrid wave growth rate discussed above in Equation (18), where v_inj ∼ 10v_s at a perpendicular shock is the injection velocity for diffusive shock acceleration. This is very similar to the estimate above in Equation (19). In both of these (resonant and nonresonant) cases, the magnetic field amplification and lower-hybrid wave generation are dependent on the streaming velocity of unmagnetized cosmic rays through the upstream medium.

Figure 1 illustrates schematically how we envisage the electron heating to vary with shock velocity. At relatively low shock velocities, where the cosmic rays amplify magnetic field by resonant interactions, T_e is constant, according to the arguments above. At higher shock velocities, where nonresonant amplification can occur, T_e can increase with shock velocity, as the precursor lengths over which magnetic field amplification can occur become larger (Equations (5) and (12) for resonant and nonresonant saturation, respectively). The curve for resonant saturation of the nonresonant instability is shown as a dotted line, because in this case magnetic field amplification probably always dominates and quenches the electron heating.

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By way of contrast, Matsukiyo (2010) describes a calculation of instabilities excited by shock-reflected ions at perpendicular shocks, which at low M_A < 10 predicts T_e approximately independent of shock velocity and increasing as $v_s^2$ at higher M_A. This arises because at low M_A, the modified two-stream instability grows fastest, generating lower-hybrid waves, and is driven to saturation where the electron gyroradius equals the electron inertial length, $v_{T_e}/\Omega _e\sim c/\omega _{pe}$, giving n_ek_BT_e ∼ B²/8π. Thus, so long as B²/n_e is constant, shocks of different velocity heat electrons to the same T_e. At higher M_A, the Buneman instability with faster growth rate takes over, leading to the sequence of growing Langmuir waves, heating electrons, and then ion acoustic waves taking over, as modeled by Cargill & Papadopoulos (1988), leading to $T_e\propto v_s^2$.

We have argued that the shock velocity dependence of the length of shock precursors due to cosmic-ray magnetic field amplification means that higher electron heating should be expected at shocks where nonresonant magnetic field amplification is dominant over resonant processes. Following Riquelme & Spitkovsky (2010), another possibility for electron heating at shocks undergoing nonresonant magnetic field amplification may exist, involving a drift instability of magnetized cosmic rays.

As cosmic rays streaming ahead of a shock amplify magnetic field, more and more cosmic rays at the lower end of the energy spectrum become magnetized. Their gyroradii in the stronger field are too small to allow them to stream ahead and contribute to the current that amplifies the field. Thus, according to the inequality (15), a shock with strong cosmic-ray current that saturates resonantly may transition to nonresonant saturation (and higher magnetic field) if sufficient cosmic rays become magnetized to reduce γn_CR and hence satisfy the inequality. The resonant saturation of nonresonantly generated field occurs because streaming cosmic rays are scattered and isotropized by the turbulence. In this case, no electron heating by cosmic rays should occur, because the streaming motion is inhibited. Riquelme & Spitkovsky (2010) discuss an interesting exception to this rule. A drift current associated with magnetized cosmic rays may also amplify magnetic field. We argue below that it might also generate lower-hybrid waves and heat electrons.

Riquelme & Spitkovsky (2010) give a theory of the perpendicular current-driven instability. In the presence of cross-B density gradients, a cosmic-ray drift current may develop along the vector ∇f_CR × B. Riquelme & Spitkovsky (2010) estimate the cosmic-ray drift velocity to be ∼c/2, so other instabilities besides the magnetic field amplification that they discuss could operate, including the generation of waves (e.g., lower-hybrid waves) that could heat electrons.

Therefore, we briefly investigate a reactive instability driven by cosmic-ray drift. In perpendicular propagation, the longitudinal part of the cold plasma dielectric tensor is

$$\begin{equation} K^L=1+{\omega _{pe}^2\over \Omega _e^2}+{\omega _{pi}^2\over \Omega _i^2-\omega ^2}+{\omega _{p{\rm CR}}^2\over \Omega _{{\rm CR}}^2-\omega ^2}{\omega - {\bf k}\cdot {\bf v}_d\over \omega }=0, \end{equation} \tag{ 20 }$$

where ω_pCR and Ω_CR are the cosmic-ray plasma and cyclotron frequencies, respectively, and v_d is the cosmic-ray drift velocity. We have assumed ω ≪ Ω_e, the electron gyrofrequency. The dispersion relation is

$$\begin{eqnarray} \omega ^5 &-& \omega ^3\Omega _{{\rm LH}}^2+\omega ^2{\bf k}\cdot {\bf v}_d{n_{{\rm CR}}\over \left<\gamma \right>n_i}\Omega _{{\rm LH}}^2+\omega \Omega _{{\rm LH}}^2\Omega _{{\rm CR}}^2 \nonumber\\ &-& {\bf k}\cdot {\bf v}_d{n_{{\rm CR}}\over \left<\gamma \right>n_i}\Omega _{{\rm LH}}^2\Omega _i^2=0 \end{eqnarray} \tag{ 21 }$$

where $\Omega _{{\rm LH}}^2=\Omega _i\Omega _e$. As k · v_d → 0, the solutions are

$$\begin{equation} \omega =0, \quad \omega ^2={\Omega _{{\rm LH}}^2\over 2}\pm {1\over 2}\sqrt{\Omega _{{\rm LH}}^4-4\Omega _{{\rm LH}}^2\Omega _{{\rm CR}}^2}\simeq \Omega _{{\rm LH}}^2 {\rm or} \Omega _{{\rm CR}}^2. \end{equation} \tag{ 22 }$$

Reinstating the cosmic-ray drift current, we find growing solutions at ω = ±Ω_LH when k · v_dn_CR/ < γ > n_i > 0.39Ω_LH. With $n_{{\rm CR}}/\left<\gamma \right>n_i\simeq 3\eta v_s^2/\left<\gamma \right>^2c^2$, k ≃ Ω_LH/v_i, where v_i ∼ 10⁶ cm s⁻¹ is the ion thermal speed in the precursor, v_d ≃ c/2, we find $v_s^2 > 0.26cv_i\left<\gamma \right>^2/\eta$, or v_s > 2.8 × 10⁸ < γ > cm s⁻¹, assuming η ≃ 0.1. So depending on the value of <γ > assumed, one might expect to see a contribution to electron heating from magnetized cosmic rays start to become effective at shock velocities of order a few thousand km s⁻¹, which should exhibit itself as a break from the $T_e/T_i\propto 1/v_s^2$ law reported by Ghavamian et al. (2007b) and van Adelsberg et al. (2008). Such a contribution also has to compete with magnetic field amplification by magnetized cosmic rays, which has growth rate (Riquelme & Spitkovsky 2010)

$$\begin{equation} \Gamma _B= 2{J_{{\rm CR}}\over c}\sqrt{\pi \over \rho }{v_A/c_s\over 1+v_A/c_s} = {\omega _{pi}n_{{\rm CR}}v_s\over n_ic}{v_A/c_s\over 1+v_A/c_s}. \end{equation} \tag{ 23 }$$

The resulting electron temperature may be estimated as follows. The electron velocity will be

$$\begin{equation} v_e\simeq {\Omega _{{\rm LH}}\over k_{\Vert }}\simeq {\Omega _e\over k} \end{equation} \tag{ 24 }$$

since lower-hybrid waves propagate within a cosine $\sqrt{{m_e}/ {m_i}}$ of the perpendicular to the magnetic field. If we put k ≃ 0.39Ω_LH < γ > n_i/n_CRv_d, the electron kinetic energy is

$$\begin{equation} {1\over 2}m_ev_e^2 \simeq {m_iv_d^2\over 2\times 0.39^2}{n_{{\rm CR}}^2\over n_i^2\left<\gamma \right>^2} \simeq 8\times 10^8 {n_{{\rm CR}}^2\over n_i^2\left<\gamma \right>^2}, \end{equation} \tag{ 25 }$$

where we have put v_d ≃ c/2. For n_CR/n_i < γ > ∼10⁻³, this gives electron energies of order 100–1000 eV, constant with shock velocity if this ratio is also constant. Alternatively, requiring Ω_LH/v_e < k < Ω_LH/v_i for lower-hybrid waves yields the following inequality:

$$\begin{equation} {0.78 v_i\over c} < {n_{{\rm CR}}\over n_i\left<\gamma \right>} < {0.78 v_e\over c}, \end{equation} \tag{ 26 }$$

which at T = 10⁴ K gives 2 × 10⁻⁵ < n_CR/n_i < γ > <10⁻³, consistent with Equation (25).

An analytic expression for the lower-hybrid wave growth rate can be derived by dropping the last two terms in Equation (22), which are of order m_e/m_i relative to the others through their dependence on Ω_CR, and solving the resulting cubic equation

$$\begin{equation} \omega ^3-\omega \Omega _{{\rm LH}}^2+{\bf k}\cdot {\bf v}_d{n_{{\rm CR}}\over \left<\gamma \right>n_i}\Omega _{{\rm LH}}^2=0. \end{equation} \tag{ 27 }$$

Using standard procedures (Abramowitz & Stegun 1984), the condition for complex roots becomes ${\bf k}\cdot {\bf v}_dn_{{\rm CR}}/\left<\gamma \right>n_i > (2/3\sqrt{3})\Omega _{{\rm LH}}=0.385 \Omega _{{\rm LH}}$, in good agreement with the numerical solution above, and with predicted growth rate $\Gamma _{{\rm LH}}=2^{-1/3}3^{-1/2}A^{-2/3}\sqrt{A^2/4-1/27}\Omega _{{\rm LH}}$, where A = k · v_dn_CR/ < γ > n_iΩ_LH.

In Figure 2 we plot Γ_LH against log₁₀A as a solid line. It increases monotonically from 0 at A = 0.385 (log₁₀A = −0.415). We compare with Γ_B plotted as dashed lines for values of c_s/v_A = 1, 0.3, 0.1, and 0.03 (from Equation (22), in units of Ω_LH < γ >). As the magnetic field becomes stronger, the lower-hybrid wave growth rate becomes stronger relative to the growth rate for magnetic field amplification. At sufficiently large A, magnetic field amplification always wins, but there is always a range of lower A where lower-hybrid wave growth dominates. These dashed curves take a realistic electron–proton mass ratio. Riquelme & Spitkovsky (2010) present simulations with m_e/m_i = 0.1, and for this reason we give as dotted curves Γ_B for the reduced mass ratio and the same values of c_s/v_A. The magnetic field amplification is much stronger in such circumstances, and the "window" in A where lower-hybrid wave growth dominates is much reduced, consistent with Riquelme & Spitkovsky (2010), who do not report any evidence of electron heating in their simulations. Riquelme & Spitkovsky (2011) also report a mass ratio dependence in their treatment of electron injection by whistlers. A detailed assessment of the electron heating at saturation will require a dielectric tensor accounting for electron thermal motions in place of Equation (21), which will be deferred to a separate paper. However, our simple treatment illustrates that in realistic conditions, lower-hybrid wave growth may compete with magnetic field amplification and provide extra electron heating at fast efficient cosmic-ray accelerating shocks.

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We have argued that electron heating at SNR shocks should break from the behavior discovered in shocks with v_s < 3000 km s⁻¹ by Ghavamian et al. (2007b) and van Adelsberg et al. (2008) at sufficiently high shock velocity where nonresonant magnetic field amplification sets in. Higher electron temperatures should be expected in this regime, because the shock precursor over which electron heating develops becomes longer, allowing more time for such heating, than in the purely resonant case. As a first step, we therefore investigate the electron heating behind the forward shocks of SN 1006 and Cas A, which are faster than those studied in the samples above.

The likely remnant of a Type Ia supernova, SN 1006 is located 500 pc above the galactic plane, expanding into unusually low density interstellar medium. Having sustained high shock speeds over a long period of time, it is the most effective accelerator of cosmic rays of all known SNRs.

SN 1006 was the first SNR from which the X-ray emission was conclusively identified as synchrotron radiation (Koyama et al. 1995; Reynolds 1996). This emission is strong on the NE and SW limbs. It has been demonstrated that the NE and SW synchrotron limbs represent "polar caps" (Willingale et al. 1996; Rothenflug et al. 2004; Reynoso et al. 2013), suggesting that particle acceleration (at least of electrons) is favored in this particular direction, which coincides with the likely direction of the galactic magnetic field (Leckband et al. 1989). Thus, we should tentatively conclude that particle acceleration at quasi-parallel shocks (magnetic field aligned along the shock velocity vector) is more favored than at quasi-perpendicular. Cassam-Chenaï et al. (2008) find the contact discontinuity closer to the forward shock in these regions (NE and SW) than elsewhere in the remnant, though Miceli et al. (2009) reach different conclusions. Rakowski et al. (2011) study knots of emission apparently ahead of the blast wave in the SE region. While the regular spacing between the three knots is consistent with what one would expect from the nonlinear saturation of the Bell instability, the spectra of these knots indicate that they are most likely ejecta. However, in plasma with an adiabatic index of 5/3, such knots are not expected to move anywhere close to the blast wave, let alone overtake it. Rakowski et al. (2011) suggested, following Jun et al. (1996), that density perturbations ahead of the shock advected downstream induce extra vorticity that allows such knots to move ahead of the forward shock. In the case of SN 1006, situated high above the galactic plane, perturbations induced by a cosmic-ray precursor are a plausible origin of such density structures.

A HESS detection of SN 1006 has been reported (Acero et al. 2010), consistent with an ambient gas density of 0.05 cm⁻³ (Acero et al. 2007). Proper motions have been measured in the optical, 028 yr⁻¹ (in the NW, see Figure 1; Winkler et al. 2003), corresponding to a shock velocity of v_s = 2900 km s⁻¹ (assuming a distance of 2.2 kpc), and in X-rays at 048 yr⁻¹ in the NE (Katsuda et al. 2009), giving 5000 km s⁻¹, and at 03–049 in the NW (Katsuda et al. 2013), indicating higher density in the NW than elsewhere. Hamilton et al. (2007) derive a reverse shock velocity of 2700 km s⁻¹ and determine the expansion velocity of ejecta entering the reverse shock to be 7000 km s⁻¹ from Hubble Space Telescope observations. The NW limb is believed to have encountered higher densities (∼0.4 cm⁻³), including partially neutral material giving rise to optical and UV spectra.

We model SN 1006 as expanding into a uniform-density interstellar medium with density 0.05 amu cm⁻³, explosion energy 1 × 10⁵¹ erg, and ejecta mass 1.4 M_☉, yielding a blast-wave velocity of 4700 km s⁻¹ and radius 8 pc at a distance of 2.2 kpc (Ghavamian et al. 2002). Figure 3 shows similar loci of electron temperature and ionization age behind the forward shock for the cases of no preshock electron heating and heating to 1 × 10⁶, 3 × 10⁶, and 1 × 10⁷ K, respectively, moving upward. We show data points for the SE synchrotron dim region given by Miceli et al. (2012, including their 95% confidence limits). These indicate a temperature of 5 × 10⁶ to 1 × 10⁷ K in the precursor. This is significantly higher than the electron heating of 3 × 10⁶ K determined at the NW limb by Ghavamian et al. (2007b), supporting our hypothesis. Note that this is an electron temperature immediately postshock, including any heating by adiabatic compression occurring upon shock passage, and so implies a temperature of ∼1 × 10⁶ K in the precursor.

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Assuming otherwise similar cosmic-ray acceleration properties at the NW and SW limbs of SN 1006, the presence of neutrals in the NW, presumably indicative of a region of increased density in the interstellar medium that decelerates the shock, must be making the difference. In our view, the shock has been decelerated in the NW to velocities where resonant magnetic field amplification dominates, whereas in the SE, nonresonant amplification is still operating, giving rise to the increased electron heating. The direct damping of Alfvén waves by charge exchange reactions is probably insignificant compared to other mechanisms of damping.

Cas A offers a complementary case to SN 1006. It is a core-collapse SNR, which is expanding into dense remnant stellar wind. Since this ambient medium was presumably photoionized by the radiation from the supernova itself, neutrals are absent from the preshock gas. The preshock magnetic field could well be considerably different from the "canonical" interstellar medium values of ∼3 μG. The expansion of the stellar wind from the stellar atmosphere to the position where it is now encountered by the forward shock of the SNR dilutes any preexisting magnetic field to negligible strength. Any preshock magnetic field must be generated by motions within the stellar wind itself. The equipartition field for a wind moving at 20 km s⁻¹ would be 10 μG, taking a density of 2 amu cm⁻³ from Hwang & Laming (2012). These parameters correspond to a mass-loss rate of 10⁻⁴ M_☉yr⁻¹. A lower mass-loss rate, or the absence of means for the magnetic field to reach equipartition, would imply a (much) lower preshock magnetic field.

To search for and obtain measured temperatures at the forward shock, we examined 13 regions at and interior to the outermost filaments of Cas A, as shown in Figure 4. This search is complicated by two factors: the filaments are mostly dominated by synchrotron emission (Gotthelf et al. 2001; Helder & Vink 2008), and the faint outer regions of the remnant are strongly contaminated by scattering of emission from the bright ejecta ring.

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As a dust scattering model is beyond the scope of our work, we chose a conservative approach in subtracting this scattered emission to identify secure examples of thermal X-ray emission associated with the forward shock. We select a local background region in the vicinity of each filament of interest and model the spectrum in that region including a component for the particle background. The astrophysical component of the background spectrum is generally well described by a thermal plane-parallel shock model (vpshock in XSPEC) with variable abundances characteristic of the ejecta and fitted temperatures and ionization ages similar to those obtained for ejecta regions by Hwang & Laming (2012). This background model is then frozen and included in the spectral model for the source region of interest, with only its overall normalization freely fitted. This is a simple and conservative way to remove the effect of the scattered emission from the bright ejecta ring, as this scattered emission will vary depending on the relative positions of the source and background region (it is also energy dependent). In most cases, the background region is outside the source region, and the fitted normalization is numerically larger, though of comparable scale, to the ratio of geometrical areas on the detector. This is at least consistent with the idea that scattered intensity is higher closer to the ejecta ring, but we found that the scale factor was always fitted higher than the geometrical scale factor, regardless of the relative position of source and background. Thus, the thermal background emission is more likely to be oversubtracted than undersubtracted in our approach, and our identification of thermal emission associated with the forward shock should be conservative.

The source spectrum itself is fitted with a variety of models, including a pure power law and vpshock models with either ejecta-type element abundances or abundances characteristic of Cas A's expected circumstellar environment. We also considered models combining a power-law with a forward-shock-type model. Regions were rejected for the purpose of this study if they were better fitted with ejecta-type abundances or a power law or if the best-fitting thermal model for the forward shock had ionization age consistent with zero within its 90% confidence errors. The actual fitted ionization ages cannot be taken too seriously, as the spectral models are not likely to be very accurate at these very low ionization ages. The most that can be gleaned is that the thermal emission is present and that the ionization age is low. Possibly some of the rejected regions could actually contain detectable emission from the forward shock, but this could not be confidently determined without a sophisticated and careful treatment of the scattered background. Having been chosen conservatively, the two regions that remain are likely to represent instances of thermal emission associated with the forward shock. The best fitting of the models that we examined for these two regions was a combined power-law and thermal forward shock model. Their fitted temperatures and ionization ages are given in Table 1, with 90% confidence limits.

We model Cas A very similarly to the preferred model for Cas A in Hwang & Laming (2012): 3 M_☉ ejecta, 2.4 × 10⁵¹ erg explosion energy, a product of blast-wave radius and circumstellar density $\rho r_f^2 = 16$ amu pc², and the radius of a circumstellar bubble R_bub = 0.3 pc. The blast wave has velocity 5000 km s⁻¹ (at the current epoch) and is at a radius of 2.6 pc. The preshock medium is taken to be 49% H, 49% He, and 2% N by mass, reflecting the enrichment of the outer stellar layers in N by the CNO process. Relative to H, He and N are enhanced over solar values by factors of approximately 2 and 20, respectively.

Figure 5 shows the locus of electron temperature and ionization age (the product of electron density and time, n_et) behind the forward shock for different values of electron heating in the precursor. From the lowest to the highest, they represent no heating and heating to temperatures of 1 × 10⁶, 3 × 10⁶, 1 × 10⁷, and 3 × 10⁷ K, respectively. Electrons are further heated by adiabatic compression on passage through the shock. The two points labeled "a" and "b" come from fits in Vink & Laming (2003), one from a synchrotron-bright portion of the blast wave with higher electron temperature, and one from a synchrotron-dim region with lower electron temperature. Also shown in Figure 5 are points from fits to regions labeled "s4" and "sw," with locations within Cas A shown in Figure 4. These both have lower n_et than regions "a" and "b" and so have undergone passage through the forward shock more recently. They also broadly support the inference from the other two, that of a precursor electron temperature in the range 1 × 10⁷ to 3 × 10⁷ K, again significantly higher than the corresponding value from an extrapolation of the survey of Ghavamian et al. (2007b). Considering that the preshock magnetic field may be lower, possibly considerably lower, than that in SN 1006, the inference of postshock magnetic fields of order 100 μG (Vink & Laming 2003) suggests that nonnresonant magnetic field amplification and saturation should be at work, if cosmic rays are responsible. Unfortunately, it does not appear possible to detect such a precursor observationally, because of scattered light from the bright remnant interior. We also note that the circumstellar medium of Cas A, unlike that presumed for SN 1006, is naturally clumpy, and that in such circumstances magnetic field amplification by the interaction of the shock with such preexisting turbulence (Giacalone & Jokipii 2007), or variations upon this mechanism (e.g., Beresnyak et al. 2009; Drury & Downes 2012), cannot so easily be ruled out.

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In our focus on higher shock velocities than those considered by Ghavamian et al. (2007b), results derived from SN 1006 and Cas A, while supportive of our hypothesis, are limited by the fact that the emission we observe comes from a region downstream of the shock itself and electron heating at the shock front is derived by model extrapolation. This is presumably because the Hα emission studied by Ghavamian et al. (2007b) is strongest in regions where the shock encounters denser regions of interstellar medium, sufficiently dense that neutrals can survive against ionization, and the shock naturally decelerates in such regions. One possible example of a faster shock displaying Hα emission is SNR 0509-67.5 (Helder et al. 2010). The remnant has forward shock velocities in the range 5200–6300 km s⁻¹ (Ghavamian et al. 2007a), derived from the width of the Lyβ line.

The Hα line profile consists of two components: a narrow feature arising as preshock neutrals diffuse through the shock and are excited before being ionized by shocked ions and electrons, and a broad component with an origin in charge exchange, as a preshock neutral has its electron captured by a shocked proton. The intensity ratio between these two can be a diagnostic of the electron temperature. If the neutrals are primarily destroyed by electron impacts, then the broad/narrow intensity ratio will be small (i.e., not enough slow neutrals survive long enough to become fast neutrals by charge exchange), but if the electrons are inefficient at ionizing neutrals, because of insignificant electron heating at the shock, the broad component becomes comparable in intensity to the narrow feature.

In SNR 0509-67.5, the broad-component intensity is significantly smaller than that of the narrow component, indicating the presence of electron heating. The analysis of Helder et al. (2010) also accounts for shock energy losses to cosmic rays, corroborating the shock speeds derived by Ghavamian et al. (2007a), and finding electron/proton temperature ratios in the range 0.2–1, significantly larger than those found by Ghavamian et al. (2007b). While more uncertain than the SN 1006 or Cas A results given above, the elevated electron temperature is determined at the forward shock itself and further supports the argument of this paper.