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Register a new userCosmic Ray Spectrum Steepening in Supernova Remnants -- I. Loss-Free Self-Similar Solution
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The direct measurements of cosmic rays (CRs), after correction for the propagation effects in the interstellar medium, indicate that their source spectra are likely to be significantly steeper than the canonical $E^{-2}$ spectrum predicted by the standard Diffusive Shock Acceleration (DSA) mechanism. The DSA has long been held responsible for the production of galactic CRs in supernova remnant (SNR) shocks. The $gamma$-ray probes of the acceleration spectra of CRs on-the-spot, inside of the SNRs, lead to the same conclusion. We show that the steep acceleration spectrum can be attributed to the $combination$ of (i) spherical expansion, (ii) tilting of the magnetic field along the shock surface and (iii) shock deceleration. Because of (i) and (ii), the DSA is efficient only on two ``polar caps of a spherical shock where the local magnetic field is within $simeq45^{circ}$ to its normal. The shock-produced spectrum observed edge-on steepens with the particle energy because the number of freshly accelerated particles with lower energies continually adds up to a growing acceleration region. We demonstrate the steepening effect by obtaining an exact self-similar solution for the particle acceleration at expanding shock surface with an arbitrary energy dependence of particle diffusivity $kappa$. We show that its increase toward higher energy steepens the spectrum, which deeply contrasts with the standard DSA spectrum where $kappa$ cancels out.
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In this note, we present a detailed self-similar solution to the interaction of a uniformly expanding gas and a stationary ambient medium, with an application to supernovae interacting with preexisting circumstellar media (Type IIn SNe). We implement the generalized solution into the Modular Open Source Fitter for Transients (MOSFiT), an open-source Python package for fitting extragalactic transient light curves.
Supernova remnants (SNRs) are believed to accelerate particles up to high energies through the mechanism of diffusive shock acceleration (DSA). Except for direct plasma simulations, all modeling efforts must rely on a given form of the diffusion coefficient, a key parameter that embodies the interactions of energetic charged particles with the magnetic turbulence. The so-called Bohm limit is commonly employed. In this paper we revisit the question of acceleration at perpendicular shocks, by employing a realistic model of perpendicular diffusion. Our coefficient reduces to a power-law in momentum for low momenta (of index $alpha$), but becomes independent of the particle momentum at high momenta (reaching a constant value $kappa_{infty}$ above some characteristic momentum $p_{rm c}$). We first provide simple analytical expressions of the maximum momentum that can be reached at a given time with this coefficient. Then we perform time-dependent numerical simulations to investigate the shape of the particle distribution that can be obtained when the particle pressure back-reacts on the flow. We observe that, for a given index $alpha$ and injection level, the shock modifications are similar for different possible values of $p_{rm c}$, whereas the particle spectra differ markedly. Of particular interest, low values of $p_{rm c}$ tend to remove the concavity once thought to be typical of non-linear DSA, and result in steep spectra, as required by recent high-energy observations of Galactic SNRs.
The spectrum of cosmic ray protons and electrons released by supernova remnants throughout their evolution is poorly known, because of the difficulty in accounting for particle escape and confinement in the downstream of a shock front, where both adiabatic and radiative losses are present. Here we calculate the spectrum of cosmic ray protons released during the evolution of supernovae of different types, accounting for the escape from upstream and for adiabatic losses of particles advected downstream of the shock and liberated at later times. The same calculation is carried out for electrons. The magnetic field in the post-shock region is calculated by using an analytic treatment of the magnetic field amplification due to non--resonant and resonant streaming instability and their saturation. We find that when the field is the result of the growth of the cosmic-ray--driven non--resonant instability alone, the spectrum of electrons and protons released by a supernova remnant are indeed different, but such a difference becomes appreciable only at energies $gtrsim 100-1000$ GeV, while observations of the electron spectrum require such a difference to be present at energies as low as $sim 10$ GeV. An effect at such low energies requires substantial magnetic field amplification in the late stages of the supernova remnant evolution (shock velocity $ll 1000$ km/s), perhaps not due to streaming instability but hydrodynamical processes. We comment on the feasibility of such conditions and speculate on the possibility that the difference in spectral shape between electrons and protons may reflect either some unknown acceleration effect, or additional energy losses in cocoons around the sources.
We discuss processes in galactic cosmic ray (GCR) acceleration sites - supernova remnants, compact associations of young massive stars, and superbubbles. Mechanisms of efficient conversion of the mechanical power of the outflows driven by supernova shocks and fast stellar winds of young stars into magnetic fields and relativistic particles are discussed. The high efficiency of particle acceleration in the sources implies the importance of nonlinear feedback effects in a symbiotic relationship where the magnetic turbulence required to accelerate the CRs is created by the accelerated CRs themselves. Non-thermal emission produced by relativistic particles (both those confined in and those that escape from the cosmic accelerators) can be used to constrain the basic physical models of the GCR sources. High resolution X-ray synchrotron imaging, combined with GeV-TeV gamma ray spectra, is a powerful tool to probe the maximum energies of accelerated particles. Future MeV regime spectroscopy will provide unique information on the composition of accelerated particles.
Supernova remnants are known to accelerate cosmic rays on account of their non-thermal emission of radio waves, X-rays, and gamma rays. Although there are many models for the acceleration of cosmic rays in Supernova remnants, the escape of cosmic rays from these sources is yet understudied. We use our time-dependent acceleration code RATPaC to study the acceleration of cosmic rays and their escape in post-adiabatic Supernova remnants and calculate the subsequent gamma-ray emission from inverse-Compton scattering and Pion decay. We performed spherically symmetric 1-D simulations in which we simultaneously solve the transport equations for CRs, magnetic turbulence, and the hydrodynamical flow of the thermal plasma in a volume large enough to keep all CRs in the simulation. The transport equations for cosmic-rays and magnetic turbulence are coupled via the cosmic-ray gradient and the spatial diffusion coefficient of the cosmic rays, while the cosmic-ray feedback onto the shock structure can be ignored. Our simulations span 100kyrs, thus covering the remnants evolution until the beginning of the post-adiabatic phase. At later stages of the evolution cosmic rays over a wide range of energy can reside outside of the remnant, creating spectra that are softer than predicted by standard DSA and feature breaks in the 10-100 GeV-range. The total spectrum of cosmic rays released into the interstellar medium has a spectral index of s~2.4 above roughly 10 GeV which is close to that required by Galactic propagation models. We further find the gamma-ray luminosity to peak around an age of 4,000 years for inverse-Compton-dominated high-energy emission. Remnants expanding in low-density media emit generally more inverse-Compton radiation matching the fact that the brightest known supernova remnants - RCW86, Vela Jr, HESSJ1731-347 and RXJ1713.7-3946 - are all expanding in low density environments.
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