Microphysics of Shock Acceleration from Observations of X-ray Synchrotron Emission from Supernova Remnants

Recent observations of non-thermal X-rays from supernova remnants have been attributed to synchrotron radiation from the loss-steepened tail of a non-thermal distribution of electrons accelerated at the remnant blast wave. In the test-particle limit of diffusive shock acceleration, in which the energy in shock-accelerated particles is unimportant, the slope of a shock-accelerated power-law is independent of the diffusion coefficient κ, and on how κ depends on particle energy. However, the maximum energy to which particles can be accelerated depends on the rate of acceleration, and that does depend on the energy-dependence of the diffusion coefficient. If the time to accelerate an electron from thermal energies to energy E ≫ mec is τ(E), and if κ ∝ E, then τ(E) ∝ E in parallel shocks, and τ ∝ E2−β in perpendicular shocks. Most work on shock acceleration has made the plausible assumption that κ ∝ rg (where rg is the particle gyroradius), so that β = 1 at relativistic energies, implying a particular (wavelength-independent) spectrum of MHD turbulence, where Kolmogorov or Kraichnan spectra might be more physically plausible. I derive the βdependence of the maximum electron energy resulting from limitations due to radiative (synchrotron and inverseCompton) losses and to finite remnant age (or size). I then exhibit calculations of synchrotron X-ray spectra, and model images, for supernova remnants as a function of β and compare to earlier β = 1 results. Spectra can be considerably altered for β < 1, and images are dramatically different for values of β corresponding to Kolmogorov or Kraichnan spectra of turbulence. The predicted images are quite unlike observed remnants, suggesting that the turbulence near SNRs is generated by the high-energy particles themselves. MICROPHYSICAL PARAMETERS OF STANDARD SHOCK-ACCELERATION THEORY Diffusive (first-order Fermi) shock acceleration is presumed to provide the relativistic-electron distributions demanded by radio and, in a few cases, X-ray observations of supernova remnants (SNRs). In the test-particle limit in which the fast particles exert no influence on the shock, the spectrum is independent of the value and energydependence of the diffusion coefficient κ. When the energy in accelerated particles becomes non-negligible, however, various nonlinear effects appear. Most directly, the preshock gas is pre-accelerated by a shock precursor whose extent depends on the diffusion distance rD = κ/ushock which does depend on particle energies. One observable effect is concave spectral curvature if κ(E) increases with E (Ellison and Reynolds 1991; Reynolds and Ellison 1992). For both test-particle and nonlinear acceleration, the acceleration rate for relativistic electrons depends on the diffusion coefficient. Macrophysical parameters, those required for hydrodynamic and thermal-shock modeling of SNRs, include the remnant age t, shock velocity ush (ideally ush(t)), preshock density n0, and compression ratio r. (A piece of important thermal-shock microphysics concerns non-Coulomb electron heating at shock, in the absence of which Te ≪ Ti initially). For standard shock-acceleration theory, we need in addition the upstream magnetic field B1, the obliquity angle θBn between the shock normal and B1, and the diffusion coefficient upstream and downstream, κ1 and κ2 (in general κ(E, r, t)). In the standard picture (e.g., Blandford and Eichler 1987), in strong shocks (MA, Ms >∼ 10), once particles have energies well above thermal, the primary scattering is from resonant MHD waves (wavenumber k ∝ particle gyrofrequency Ω: kR = Ω/c). Quasi-linear theory allows the inference of a diffusion coefficient. Assume a wave spectrum I(k) = Ak−n erg cm−3 (cm−1)−1 (so the energy density at k ∼ kI(k)). Then κ‖ = 1 3 λmfpv = c 3 rg (