Particle Spectra from Acceleration at Forward and Reverse Shocks of Young Type Ia Supernova Remnants
aa r X i v : . [ a s t r o - ph . H E ] O c t Particle Spectra from Acceleration at Forward and Reverse Shocksof Young Type Ia Supernova Remnants
I.Telezhinsky a , V.Dwarkadas b , M.Pohl a,c a DESY, Platanenallee 6, 15738 Zeuthen, Germany b University of Chicago, Department of Astronomy & Astrophysics, 5640 S Ellis Ave, AAC 010c, Chicago,IL 60637, U.S.A. c Universit¨at Potsdam, Institut f¨ur Physik & Astronomie, Karl-Liebknecht-Strasse 24/25, 14476 Potsdam,Germany
Abstract
We study cosmic-ray acceleration in young Type Ia Supernova Remnants (SNRs) by meansof test-particle diffusive shock acceleration theory and 1-D hydrodynamical simulations oftheir evolution. In addition to acceleration at the forward shock, we explore the particleacceleration at the reverse shock in the presence of a possible substantial magnetic field,and consequently the impact of this acceleration on the particle spectra in the remnant.We investigate the time evolution of the spectra for various time-dependent profiles of themagnetic field in the shocked region of the remnant. We test a possible influence on particlespectra of the Alfv´enic drift of scattering centers in the precursor regions of the shocks.In addition, we study the radiation spectra and morphology in a broad band from radioto gamma-rays. It is demonstrated that the reverse shock contribution to the cosmic-rayparticle population of young Type Ia SNRs may be significant, modifying the spatial distri-bution of particles and noticeably affecting the volume-integrated particle spectra in youngSNRs. In particular spectral structures may arise in test-particle calculations that are oftendiscussed as signatures of non-linear cosmic-ray modification of shocks. Therefore, the spec-trum and morphology of emission, and their time evolution, differ from pure forward-shocksolutions.
Keywords:
Supernova Remnants; cosmic rays; cosmic-ray acceleration; hydrodynamics;forward and reverse shocks
1. Introduction
The energy density in local cosmic rays(CRs), when extrapolated to the wholeGalaxy, implies the existence of powerfulaccelerators inside the Galaxy. SupernovaRemnants (SNRs) have long been thoughtto be the main candidates on account ofenergetics [1]. Later, diffusive shock ac- celeration (DSA) of charged particles atthe SNR shocks was considered [2, 3, 4, 5]and is now thought to be the most likelymechanism that can accelerate CRs up to10 eV, where the cosmic-ray spectrumshows a break known as the ’knee’ [6]. Ifthey are very efficiently accelerated, CRscan produce a feedback on the plasma flow Preprint submitted to Astroparticle Physics October 23, 2018 n SNRs through their pressure, and canmodify the shock structure by deceleratingthe incoming plasma flow when streamingthrough it in the precursor region (see [7]for a review and references therein). Ob-servationally, it is unclear whether or notthe CR-acceleration efficiency is sufficientlyhigh for a significant feedback [8, 9], but ifso, SNR shocks may become modified, andthe particle spectra may deviate from theclassical power-law slope of s = 2 foundin test-particle calculations. The spectraproduced by modified shocks show concavestructure with softer ( s >
2) low-energy andharder ( s <
2) high-energy parts.The CR spectrum observed at Earth sug-gests that sources produce softer spectrathan predicted by non-linear DSA (NDSA)theory [10]. If the CR sources produce veryhard spectra, the energy dependence of thediffusion coefficient in the Galaxy needs tobe much stronger than is allowed by the ob-served CR anisotropy [11]. It should benoted, though, that the CR source spec-trum is that of the CR escaping from theirsources, the time integral of which couldbe considerably softer than the spectrumof particles contained in the sources at agiven time [12]. On the other hand, re-cent γ -ray observations of some shell-typeSNRs tend to show soft spectra (Γ > .
10% of the SNRenergy to CRs [34]. This number is roughlythat needed to explain the origin of CRs byacceleration in Supernova Remnants [37].Given the complexities faced by NDSAtheory, in this paper we propose to inves-tigate the diversity of particle spectra thatcan be produced by young SNR shocks ina test-particle mode, concentrating for sim-plicity on Type Ia SNRs in this paper.We consider acceleration by both forwardand reverse shocks in evolving spherically-symmetric systems. Many pieces of obser-vational evidence suggest that the reverseshock (RS) may also accelerate particles tohigh energies. This includes the detection ofnon-thermal X-rays from the RS of Cas A[38, 39], 1E 0102.2-7219 [40], and RCW 86[41], as well as radio emission from the RSof Kepler’s SNR [42]. CR acceleration atthe RS is also supported by theory [43, 44].In any case, high-energy particles producedby the FS may be re-accelerated at theRS. [45]. The preponderance of observa-tional and experimental evidence, combinedwith earlier theoretical implications, sug-gests that it is essential to at least consider2he effects of both shocks in acceleratingparticles. The total volume-integrated spec-tra in such a case will deviate from expec-tations based on plane-parallel shocks andsteady-state systems employing the FS only.The MF in the shocked interaction re-gion is essential to determining the parti-cle spectra, but is extremely hard to deduceobservationally. This has led to the param-eterization of the field in different models,which generally involves tying the field todynamically or kinematically determinableparameters. Our intent is not to study theamplification of the field itself, which is be-yond the scope of this paper, but to investi-gate the particle spectra that are producedunder widely used assumptions of the fieldin the literature. We also explore how theinclusion of Alfv´enic drift affects the totalparticle spectra in SNR. Given the obser-vational data on MF amplification [46, 47]as well as theoretical [4, 25] and numerical[48, 26] results, we consider two cases: amoderate (75 µ G) and a strongly amplified(300 µ G) MF. Besides calculating the to-tal particle spectra, we produce wide-bandspectral-energy distributions (SEDs), andprovide surface brightness profiles in the ra-dio, X-ray and γ -ray bands.This work intends to improve upon thepresent state of the field in many ways(1) We take into account the free expan-sion phase of the expanding SNR, when theshock velocities are the highest, whereasmost (though certainly not all) analyseshave totally neglected the initial phase,and started directly with the Sedov phase.(2) Our technique to solve the transportequation, and the corresponding high res-olution spherically-symmetric simulationsused, guarantee a very high level of ac-curacy and therefore an accurate render-ing of the particle spectra for the chosen MF and diffusion coefficient. (3) We havetaken into account the acceleration of CRsat the reverse shock, as suggested by re-cent observations. We have shown what ob-servable features in SNRs we may expectif a non-negligible MF permits CR accel-eration at the RS to high energies. Thishas not been explored in such detail be-fore, including spherical symmetry and fullytime-dependent cosmic-ray transport. Ear-lier studies [43] investigated CR accelerationat the RS using an approximate, analyti-cal model [49] and assumed there is no MFamplification at the RS. In particular, theyfocused on non-linear effects arising from ahigh injection efficiency in low MF. Otherauthors [44] concentrated on the study ofa specific object, which most probably isa core-collapse SNR, and have not madeany general conclusions regarding type-IaSNRs. (4) Our results therefore present amore complete study of the particle spec-tra and high-energy emission that is to beexpected from type-Ia SNe with moderateacceleration efficiency.
2. Cosmic-Ray Acceleration
We consider a test-particle approach todiffusive shock acceleration of CRs in SNRs[2, 3, 4, 5]. It has been recently [34] shownthat a test-particle description is applica-ble if the CR pressure at the shock is lessthan 10% of the shock ram pressure. There-fore, if one limits the amount of energy con-tained in CRs and the CR pressure at theshock, the acceleration can be treated byindependently solving the cosmic-ray trans-port equation and the hydrodynamic equa-tions of SNR evolution. The cosmic-raytransport equation is a diffusion-advection3quation in both space and momentum [31]: ∂N∂t = ∇ ( D r ∇ N − ~v N ) − ∂∂p (cid:18) ( N ˙ p ) − ∇ ~v N p (cid:19) + Q (1)where N is the differential number densityof cosmic rays, D r is the spatial diffusion co-efficient, ~v is the advective velocity given bya 1-D hydrodynamical simulation, ˙ p are theenergy losses, and Q is the source term rep-resenting the injection of the thermal parti-cles into the acceleration process given as Q = η i n u | V sh − v u | δ ( r − R sh ) δ ( p − p inj ) , (2)where η i is the injection efficiency parame-ter, n u is the number density of plasma inthe shock upstream region, V sh is the shockspeed, v u is the plasma velocity in the shockupstream region, r is the distance from theSNR center, R sh is the radius of the shock, p is the particle momentum, and p inj is themomentum of the injected particles.We assume the thermal leakage injectionmodel [50], where only the particles withmomentum p inj > ψp th can be accelerated,with ψ being a multiple of the particle ther-mal momentum, p th . The efficiency of in-jection is determined as η i = 43 √ π ( σ − ψ e − ψ (3)where σ is the shock compression ratio. Al-though σ and p th are obtained from hydrosimulations, ψ remains a free parameter inthe model. In our calculations we assume ψ ≃ .
45, keeping η i ≃ × − sufficientlylow to stay within the framework of the test-particle approximation. We note that sincewe are not interested here in the absolute ra-tio of electron to proton spectra, and lackingsecure knowledge of the details of electron injection, as well as the time-scale of ther-malization between electrons and protons,we assume that electrons and protons areinjected equally and are at the same tem-perature, i.e. the injected electron to pro-ton ratio, K e/p,i = 1. Thus, K e/p obtainedin our simulations is purely a result of theelectron-to-proton mass ratio and correctedfor standard ISM abundances.We impose a free-escape spatial bound-ary upstream of the FS that should accountfor escape of the highest-energy particlesfrom the system. We assume that all par-ticles crossing the boundary leave the sys-tem, therefore we set the CR number den-sity, N , to zero at the free-escape boundary,at R esc = 2 R F S , where R F S is the FS radius.We also assume that the highest en-ergy particles generate MHD waves in theshock upstream region, and thus may am-plify the MF [25, 26, 27, 35]. Being scat-tered by MHD waves, the CR particles un-dergo Bohm diffusion everywhere inside theescape boundary [51]. Also, the advec-tive velocity in Eq. 1, given the presenceof amplified MF, may include the Alfv´enicdrift of scattering centers and thus maynot coincide with the plasma flow velocity[32, 35, 34, 33].In the current paper we assume spher-ical symmetry and do not consider themomentum-diffusion term representingstochastic (second order Fermi) accelera-tion.The main difficulty for the numerical so-lution of Eq. 1 is that the diffusion coeffi-cient D r is strongly dependent on the parti-cle momentum ( D r = D r ( p ) ∼ p , in caseof Bohm diffusion) and therefore spans awide range of magnitudes, ln( p max /p inj ) ≈
25. Formally, the spatial grid must re-solve the smallest diffusion length definedas D r ( p inj ) /V sh , otherwise the modeled ac-4eleration is artificially slow. The lowest-energy particles have a very small diffusionlength, and thus a uniform grid must con-tain millions (or even billions) of cells whichis computationally impossible.Several approaches are known to date toovercome this numerical difficulty. The so-called normalized grid is used in [52], wherea substitution of variables is performed andthe spatial coordinates are normalized tothe diffusion scale. Another approach pre-sented by [53] consists of introducing a co-moving frame, in which the shock is station-ary, coupled with a mesh refinement - a finediscretization is used only near the shockregion, where the low-energy particles areinjected.Here we use the second approach. Wenormalize the spatial coordinate in Eq. 1to the shock radius, introducing coordinate x = r/R sh , and then substitute the coordi-nate x with a new coordinate, x ∗ , for whichwe use a uniform grid when solving Eq. 1:( x −
1) = ( x ∗ − (4)with Jacobian dxdx ∗ = 3( x ∗ − (5)These simple transformations allow us i)to place the shock at the center of a knowncell at any given time and ii) to resolve theshock vicinity for low-energy particles withonly a few hundred bins in the spatial coor-dinate, x ∗ . We use parametrizations for theradial dependence of the MF as describedin Section 4, and we obtain the hydrody-namical parameters (the plasma flow veloc-ity, density and the shock speed) needed tosolve Eq. 1 from 1-D hydrodynamic simu-lations of the SNR evolution described inSection 3. Then, Eq. 1 is solved in spherically-symmetric geometry using implicit finite-difference methods implemented in the FiPy [54] library modules. We make separateruns for the FS and the RS, using the ap-propriate coordinate transformations to ob-tain the highest resolution where injectionoccurs. Then we sum the results becausethey are linear and independent. Someof the highest-energy particles may reachthe other shock and be re-accelerated there.However, the energy of these particles is al-ready so high, and their number so low, thatthe poor resolution at the other shock doesnot visibly affect the results. Finally, wecheck the CR-to-ram pressure ratio as wellas energy containment in CRs a posteriori .In our calculations these values are neverallowed to exceed 10%.
3. Hydrodynamics
To study the evolution of the SNRshock waves over time, we have performedspherically-symmetric hydrodynamic sim-ulations using the VH-1 code [55], a 3-dimensional hydrodynamic code based onthe Piecewise Parabolic Method [56]. Thesimulations are based on an expanding grid[57, 58], which tracks the motion of theouter shock and expands along with it.Thus, a high resolution is maintained rightfrom the start of the simulations, which isessential considering that the remnant sizeincreases by about 6 orders of magnitudeduring the simulation. The expanding gridalso has another advantage - the position ofthe forward shock is almost stationary onthe grid over most of the evolution, whichis useful in computing the particle spec-trum. Furthermore, the grid expansion isadjusted such that, as far as possible, theshocked interaction region between the in-5er and outer shocks occupies most of thegrid. Although cooling of the material is in-corporated into the simulations via a cool-ing function, the velocities and densities aresuch that cooling is not effective and doesnot play a role.In order to study the evolution of youngSNe in the ambient medium, we need atminimum two parameters - a description ofthe density profile of the material ejectedin the explosion, and that of the mediuminto which the SN is expanding. In thisfirst paper, we consider the evolution of aType Ia SN, which are thought to arise fromthe thermonuclear deflagration of a whitedwarf. Since the progenitor star is a low-mass star and (presumably) does not un-dergo significant mass-loss, the medium sur-rounding it is assumed to be a constant-density interstellar medium.The ejecta density structure is moreuncertain. However, by comparingspherically-symmetric models of Type IaSN explosions, it was found [59] that theejecta structure of a Type Ia SN afterexplosion can best be represented by anexponential ejecta density profile. This isthe profile that we have used in our simu-lations. Since an exponential introduces anadditional dimensional parameter to satisfya non-dimensional exponent, the resultingSN evolution is no longer self-similar. TheSN is homologously expanding, thereforethe ejecta velocity simply increases linearlywith radius.The initial conditions then depend onthree parameters: (i) The energy in theejected material, which we take as thecanonical energy of a SN explosion, 10 erg,(ii) the mass of the ejecta, which we take tobe the Chandrashekhar mass, 1.4 M ⊙ , and(iii) the density of the surrounding medium,which we take to be 9.36 × − g cm − , or l og V [ c m / s ] r/R FS Velocity 8.5 9 9.5 l og T [ K ] Temperature-8-7.5-7-6.5 l og P [ d y n / c m ] Pressure-25-24-23 l og ρ [ g / c m ] Density t= 100 yrt= 400 yrt=1000 yr
Figure 1: Hydrodynamical parameters of the SNRat different ages. The x-axis is scaled to the radiusof the forward shock. × − pc. By the end of the sim-ulation, which is carried on for about 1000years, it has grown to more than 10 pc. Thehydrodynamic parameters at the age of 100,400, and 1000 years are shown in Fig 1. Theouter shock, inner shock and contact discon-tinuity are all clearly delineated as sharpdiscontinuities in the density panel (top).After a few hundred years the shocked re-gion occupies more than 90% of the grid.This resolution is essential for computingthe velocity differential across the shock,required in the solution of the cosmic-raytransport equation. Note that, while thevelocity varies smoothly across the contactdiscontinuity (CD), and the pressure showsa slight change in slope, there is a dra-matic change in density across the CD. Theshocked ejecta reach a maximum density onthe inner side of the contact discontinuity,while the shocked ambient medium reachesa minimum on the outer side. The temper-ature therefore reaches a maximum at theCD for the shocked ambient medium. This distinguishes it from the structure obtainedfor a power-law profile expanding into theambient medium, as would be more appro-priate for a Type II or core-collapse SN [59].Finally, the numerical solution of the hy-drodynamic equations is mapped onto thespatial coordinate x ∗ (cf. Eq 4) in which wehave written the particle transport equation(Eq. 1). The shocks, which are invariablysmeared out in the diffusive hydrodynamiccalculation, are re-sharpened by interpola-tion toward a point-like jump in density,flow velocity, and pressure. This is neces-sary to maintain a realistically fast acceler-ation at GeV-to-TeV energies.
4. Magnetic Field and Models Setup
The strength of the magnetic field in theSNR is one of the crucial parameters forCR acceleration, particularly so in the shockregions. Since we assume Bohm diffusionhere, the MF determines the mean free pathof the particle before it is scattered. Forhigher MF a particle will be more quicklyaccelerated and it can attain a higher energybefore it may escape to the far-upstreamregion of the forward shock. Therefore,the MF determines the maximum energyof both protons and electrons, the latter inaddition suffering energy losses determinedmainly by the same MF.The actual mechanism of MF amplifica-tion (MFA) is a subject of ongoing research.Despite recent progress [60, 26, 35], it is notyet clear what mechanism (or most prob-ably a combination of mechanisms) is re-sponsible for induction of MHD turbulenceon scales needed for the particle accelera-tion (for a recent review see [61] and refer-ences therein). Due to the uncertainty in7 B [ µ G ] r/R FS P075I/A 0.1 1 10 100 D075I/A 100 yr400 yr1000 yr
Figure 2: Magnetic field profiles in the SNR for Dand P type models at different ages. the dominant mechanism of MFA, and thedifficulty of taking into account the com-plexiety of the MFA process, the magneticfield in SNRs is often phenomenologicallyparametrized. If the MF is assumed to befrozen in, the energy density of the field maycorrelate with the thermal and relativisticparticle energy densities. Here we investi-gate the impact of widely used MF profileson the spectra of accelerated particles. Weintroduce a set of simple scalings which de-scribe the temporal and spatial evolution ofthe magnetic field in a SNR. The models arelisted in Table 1 and are explained below insubsection 4.2.Prompted by recent high resolution X-ray observations, which claim to detect non-thermal emission from the reverse-shock re-
Table 1: Nomenclature of MF models used in thisstudy.
PPPPPPPPP
MF Profile Density Pressure75 µ G D075I P075I+ Alfv´enic drift D075A P075A300 µ G D300I P300I+ Alfv´enic drift D300A P300Agion of some SNRs [38, 39, 40, 41], we havechosen to consider particle acceleration atthe reverse shock in addition to that at theforward shock. The major argument againstthe ability of the RS to accelerate parti-cles is that an exceptionally low MF existsin the ejecta, a theoretical argument ob-tained under the assumption that the mag-netic flux at the surface of progenitor star isconserved during the expansion of the SNR.However, the detection of non-thermal X-rays from the RS implies that there may besome processes acting against rapid dilutionof the MF, and that a large MF at the re-verse shock can exist. These processes maybe the fluid-dynamical MHD turbulence inthe ejecta and CD region - the initial ac-celeration of CRs, while the MF was highenough, with subsequent streaming througha confined ejecta region (accelerated CRs donot leave the upstream region of the RS aseasy as that of the FS). It is also not clearwhether the MF in the ejecta, to be scaledaccording to flux conservation, must be thefield at the surface of SN progenitor. Fi-nally, the fact that shocks in SNRs are col-lisionless implies the existence of a finite MFwhich mediates the collisionless shock. Ad-ditional justification for the presence of MFat the RS can be found in [44].8 .2. Parametrizations of Magnetic FieldProfiles
The actual amplification of the mag-netic field is a complicated, turbulent pro-cess that is not well understood. Manypublished studies of particle accelerationin SNRs assume that the MF is propor-tional to the density of the plasma (e.g.,[33, 44, 62, 52]). An alternative approachis generally adopted in models of SNe radioemission, where the MF energy density inthe remnant is proportional to the thermalenergy density, and thereby the gas pres-sure [63] (for review see [64]). Followingthese scalings, we set up 8 MF models, 4 ’D’models following the density profiles and 4’P’ models scaling with the square root ofpressure profiles. All our models are timedependent, but the matching at the FS isdifferent for the two classes of MF models.D models assume a constant value of theMF at the FS, B = B F S , and only thescaling inside the SNR proportionally fol-lows the time-dependent density distribu-tion, B ( r, t ) = B ρ ( r, t ) /ρ ( R F S , t ). In con-trast, P models assume that the MF at theFS and inside the SNR both evolve withtime according to the square root of pres-sure distribution, B ( r, t ) = p C πP ( r, t ).Here, ρ ( r, t ) and P ( r, t ) are the density andthe pressure profiles of the plasma insidethe SNR, and C is a proportionality con-stant chosen so that B ( R F S , t m ) = B F S ,where t m = 500 years is the mid-time ofsimulations. We assume two possible val-ues of the MF at the SNR forward shock, B F S = 75 µ G and B F S = 300 µ G. Thesevalues are taken in accordance with aver-age limits given by the observational dataon the MF for young SNRs in X-rays [47]and gamma-rays [46]. The proportionalityconstant, C , is rather small in both cases, C = 0 .
19% for B F S = 75 µ G and C = 3% for B F S = 300 µ G. The amplified valueof the MF at the RS varies between ≃ ≃ µ G depending on the age andmodel. Even if we start from a rather di-luted MF, this requires amplification factorsof ≃ − B F S in D modelsand B ( R F S , t ) in P models). We thereforeassume that the MF falls off to the inter-stellar field (5 µ G) at the distance of 5% ofthe FS radius ahead of the FS, and downto a very small ejecta field (0.01-0.1 µ G)at the distance of 5% of the RS radius to-ward the interior [44]. This transition maybe explained by an exponential decrease inthe number density of high energy particles,which may cause MF amplification. In gen-eral, one can also expect that the diffusioncoefficient becomes larger than Bohm diffu-sion, albeit still much lower than the Galac-tic one due to the enhanced flux of CRs.The transition to the Galactic diffusion oc-curs at the escape boundary, where the con-centration of CRs tends to zero. Here we as-sume that Bohm diffusion operates up to es-cape boundary, R esc . In our simulations, weassume that in young SNRs the MF is fullyisotropic on account of efficient turbulentfield amplification, therefore the MF com-pression ratio is σ MF = p (1 + 2 σ ) / √
11, where σ is the gas compression ratio.The distributions of the MF for different9odel types and different times are givenat Fig 2. As noted earlier, we also investigate theinfluence on the particle spectra of a possi-ble drift of the scattering centers upstreamof the shocks. Resonant streaming insta-bilities in the precursor region of the shockgenerate outward-moving Alfv´en waves [26]that may have a significant phase velocityin the presence of amplified MF. As par-ticle are scattered on excited MHD waves,the Alfv´en velocity of the waves may effec-tively change the compression ratio felt bythe particles, and the spectrum becomes no-ticeably softer [65, 32, 35, 33]. In cases in-volving an Alfv´enic Mach number M A . s <
2, thanthe classical solution [34]. It should benoted that non-resonant instabilities ofteninvolve turbulence with very small phase ve-locity [25] for which Alfv´enic drift is ignor-able. Which type of instability dominatesand what the turbulence properties are inthe nonlinear regime is the subject of ongo-ing research [66, 67, 68]. Here we paramet-rically study the impact of Alfv´enic drift.In half of our models we introduce Alfv´enicdrift in the upstream regions of both shocksassuming that the effective velocity of thescattering centers equals the Alfv´en veloc-ity, v A = B F S / √ πρ ISM , plus the upstreamgas velocity. In the downstream region,we assume that the scattering centers areisotropized and there is no Alfv´enic drift.We mark the models including Alfv´enic driftwith “A ” while the others are marked with“I”.
5. Results and Discussion
In this section we present our main re-sults. We discuss spectra of particles ac- celerated at the reverse and the forwardshocks, analyze their properties and evolu-tionary differences, and explore the intrin-sic differences in particle spectra imposedby different parametrizations of the MF andhow the Alfv´enic drift of scattering centersaffects particle spectra. Finally, we calcu-late the emission from the particles in SNRand discuss the resulting observable proper-ties which could be compared to the exper-imental data.
The non-thermal spectra of acceleratedparticles are characterized by a power-lawindex and a cut-off energy. The power-law slope according to DSA theory de-pends solely on the compression ratio ofthe plasma flow (velocity jump across theshock). At the location of the shock, wherethe plasma is compressed, particles adiabat-ically gain energy and then lose it every-where in the expanding flow. In the evolv-ing system, the average compression ratioshould be considered. Our hydrodynamicalsimulations were performed with sufficientlyhigh resolution that the deviations of theshock compression ratio do not affect thespectral slope through the entire evolution.The compression ratios in our simulationare σ F S = 4 . ± .
09 and σ RS = 3 . ± . s ≃ E max,p ∼ V sh B t . Therefore, the increase of E max,p due to system expansion (higher energies10 l og N E [ G e V / c m ] log E [GeV]t= 100 yrt= 400 yrt=1000 yr -12-11-10 0 1 2 3 4 5 l og N E [ G e V / c m ] log E [GeV]t= 100 yrt= 400 yrt=1000 yr Figure 3: The FS (thick lines) vs. the RS (thin lines) contributions to the spectra of protons (left) andelectrons (right) for different SNR ages. are needed to leave the system) is eventu-ally compensated by the decrease of V sh , so E max,p decreases after reaching a maximum.This is illustrated by the time evolution ofFS proton spectra in Fig. 3 (left), whichshows the D300I model, in which B is keptconstant, thus isolating E max,p ∝ V sh t . Forelectrons, E max,e evolves in the same man-ner up to the time when the energy gainper cycle no longer exceeds the energy lossper cycle due to synchrotron radiation, i.e, t acc = t rad . A second, typically lower, char-acteristic energy, E rad , is approximately setby equality of the radiative-loss timescalewith the age of the remnant. Above E rad ,the electron spectra assume a quasi steady-state, changing slowly on account of theevolution of the gas-flow profiles. Besides,above E rad the electron spectra steepen byone power approximately (Fig. 3, right).The ability of the RS to accelerate par-ticles depends on the MF configuration atthe location of the RS. With the MF profilesassumed here and discussed above, it is ob-served that the RS is capable of acceleratingparticles to sufficiently high energies and in-tensities to contribute to the total volume- integrated particle spectra. These resultsare at least consistent with many of the ob-servations noted above. In order to obtainthe volume-integrated spectra, N ( E ), oneperforms an integral of the spatially differ-ential particle distribution, N ( x, E ), overthe entire volume of the simulation grid upto R esc = 2 R F S : N ( E ) = 4 πR F S Z N ( x, E ) x dx (6)where x = r/R F S . In all plots featur-ing particle spectra we show the quantity N = N ( E ) / πR F S . In the absence ofAlfv´enic drift, the RS spectra show slopessimilar to those at the FS, with the ex-ception of some hardening at higher ener-gies. The maximum energies for protons arelower than at the FS on account of a lowerMF strength, which allows particles to es-cape the acceleration region at lower ener-gies. In contrast, the maximum energies forelectrons are nearly as high as at the FSmainly because a weaker MF also reducesthe rate of radiative losses of electrons. Theintensity of the volume-integrated RS spec-tra falls with time relative to the FS spectra11s the number of injected particles becomessmaller. Also the ratio of the RS surfaceto FS surface decreases with time, whichmeans fewer particles that could be poten-tially injected cross the RS.The stronger spectral hardening andmore significant bump at high energiesmake the RS volume-integrated spectra dif-ferent from those of the FS. The spectralhardening is most visible at early evolution-ary epochs and gradually disappears withtime for both shocks, but at a somewhatfaster rate for the FS. The bump in the for-ward and the reverse shock spectra is lo-cated close to E max , which is beyond E rad for electrons. The presence of these spec-tral features and their evolution can be ex-plained by the geometry of our spherically-symmetric system.In order to get volume-integrated spec-tra we integrate over the entire simulationdomain, so both upstream and downstreamregions contribute to the volume-integratedspectra. The volume-integrated upstreamspectra usually have hard slopes, s ≃ E max is comparable with that ofthe downstream spectra. Thus, in spectraintegrated over both upstream and down-stream regions one can see the contributionfrom the hard upstream spectra in high-energies close to E max , whereas the con-tribution of low-energy particles is negligi-ble. Besides, the E max -region of the up-stream spectra exhibits a bump which ap-pears on account of the exponential de-crease of the MF in the upstream direc-tion. As a result, the diffusion time of par-ticles located close to the shock is longerthan that of particles further away fromthe shock. Therefore, the recently accel-erated particles around E max are accumu-lated near the shock in the upstream re- gion. The spectral hardening and the bumpis strongly visible in the RS spectra (Fig. 3,right) because the volume of the upstreamregions of both shocks are significantly dif-ferent in the spherically-symmetric geome-try assumed here, contrary to the case ofplane-parallel shocks. This also explains thedifferent time evolution of the spectral fea-tures. The upstream region of the FS ex-pands significantly faster than the upstreamregion of the RS, thus the CR number den-sity in the upstream region of the FS de-creases faster and contributes less to thetotal volume-integrated spectra. We notethe presence of a peculiar wiggle in the elec-tron spectra. The MF in the upstream re-gion is lower than downstream, resulting inweaker synchrotron losses for the electronsand therefore E rad ≃ E max,e . Consequently,the upstream electron spectra have a sig-nificantly higher intensity near E max,e thanthe downstream electron spectra. There-fore, in the total volume-integrated spectrathe bump stands out beyond E rad in the E max,e region. For protons, E max,p in theupstream and downstream spectra nearlycoincide, and a small bump appears near E max,p in the total volume-integrated pro-ton spectra. Again, due to the faster de-crease in the CR number density ahead ofthe FS with time, the feature is more promi-nent in the RS spectra. Most important for DSA is the configura-tion, strength, and evolution of the MF atthe location of the shocks and their precur-sor regions. Since here we adopted similarprofiles of MF in the precursor regions ofthe shocks, only a different time evolutionof the MF strength at the location of theshocks may change the shape of the volume-12 l og N E [ G e V / c m ] D075IP075ID300IP300I-20-19-18-17-16-15-14 t= 100 yr 0.4 0.5 0.6 0.7 0.8 0.9 1r/R FS t= 400 yr 0.4 0.5 0.6 0.7 0.8 0.9 1 t=1000 yr Figure 4: The time evolution of the spatial distribution of protons (top row) and electrons (bottom row) atenergy of 20 TeV. integrated spectra. It was noted, however,that the differences between models in thespectra of particles created by the RS andthe FS are marginal, in fact only E rad and E max truly depend on the MF strengths.While for protons a higher MF would meana higher E max,p , for electrons this wouldmean a lower E rad and E max,e due to ra-diative losses.The MF profiles in the SNR interior affectthe radial distribution of CR particles andtheir subsequent emission, especially so forleptons. In Fig.4 we plot the evolution ofthe radial distribution of protons and elec-trons at the energy of 20 TeV, which couldbe a characteristic energy of the particlesproducing high-energy emission, discussedin more detail in subsection 5.2.2. We usemodels without Alfv´enic drift as it doesnot change the distribution significantly -it would only slightly reduce the intensity at the FS. One can clearly see that elec-trons suffer significant losses in the shockedregion where the MF is high. The num-ber density of protons falls nearly exponen-tially from the FS towards the CD. There isa noticeable peak in the CR distributionsnear the RS. Whereas electrons peak ex-actly at the RS, protons are accumulatedbetween the RS and the CD. At 100 years,the CR number density close to the RS iseven higher than at the FS. At 400 yearsthe peak is still clearly visible, however al-ready at 1000 years only a minor bump isobserved. In the ejecta region, where theMF is low, both particle species show nearlyuniform distributions. Although the D075Imodel follows the trends described above, itis a clear outlier. The cut-off energy of par-ticles accelerated by the RS in this modelis around 4 TeV, so one finds at 20 TeValmost no particles from the RS. This ex-13 l og N E [ G e V / c m ] t= 100 yrt= 400 yrt=1000 yr Figure 5: The spectra of the FS (thick lines) andthe RS (thin lines) for different SNR ages in P300A(top) and D300A (bottom) models. plains the absence of a peak near the RS andthe low CR number density in the interiorof the SNR. Besides, the electrons have suf-fered little energy losses in the initial phasesof SNR evolution ( t < t rad ) compared withother models (remember that P075I modelhave initially a high MF).
We have explored the influence ofAlfv´enic drift on the particle spectra of theSNR. Interesting is that it affects our mod-els in a different way during the SNR evo-lution. We observe that the FS and the RSspectra in P models are noticeably softerduring all evolutionary times. This is be-cause in P models the Alfv´enic Mach num-ber, M A , is almost constant for both shocksduring the evolution. The MF strength, B , evolves as the square root of the pres- sure, √ P , and hence roughly scales withthe shock velocity. Therefore, spectra shownearly the same softening at all times asshown at the top of Fig. 5. Contrary to Pmodels, in D models the effect of Alfv´enicdrift is different for the FS and the RS. Atthe FS B is kept constant during the evolu-tion, and M A is very large at early times be-cause the Alfv´en velocity, v A , is small com-pared to the shock speed. At later times v A remains constant, but the shock speeddrops, and so M A becomes small. Thisimplies that the softening of the FS spec-tra in D models was small at early timesand increased during the evolution of theSNR. In contrast to the FS, B is evolvingat the RS as density, ρ . The initially largeejecta density caused a high Alfv´en velocity( v A ∼ √ ρ in D models) and rather small M A , though larger than at the FS, whichcaused some softening of the spectra. Overtime ρ dropped significantly, and therefore M A became large. Therefore, the softeningof the RS spectra in D models is stronger atearlier times and almost not noticeable atlater times (Fig. 5, bottom). Obviously, allthese effects and characteristics are strongerin models with high MF strength. The shape of the total volume-integratedspectra depends on the SNR age on accountof the changing importance of the contribu-tions from the forward and reverse shocks.For very young Type Ia SNRs, and keep-ing in mind the MF profiles assumed herein,a significant contribution to the total spec-trum arises from the RS. However, alreadyafter a few hundred years the bulk of par-ticles is provided by the FS. Nevertheless,particles accelerated at the RS continue tocreate peculiar spectral features up to latetimes of the SNR free expansion phase (see14ig. 6). Added to this is the different im-pact of Alfv´enic drift on the spectra pro-duced with different MF models and thetime evolution of this impact, especially inD models. The total spectra are very dif-ferent from a single-shock test-particle solu-tions as seen at Fig. 6. We plotted 6 out of8 models because D075A and P075A mod-els show similar shapes as D075I and P075Imodels, apart from being slighter softer.At 100 years the contribution of the RS tothe total spectra is significant in all models.With age this contribution decreases. Af-ter 400 and 1000 years, the proton spectrain P models look like broken power laws orvary gradually, much smoother than expo-nentially cut-off. Since the RS proton spec-tra in D models have a somewhat smallercut-off energy and are harder than the FSspectra, if Alfv´enic drift is assumed, the RSmaintains a significant contribution to thetotal spectrum until late epochs. Because ofit and because of the bump contributed bythe upstream region of the FS, the protonspectra of D models show some small-scaleconcavity at high energy through all times,however at later times it is shallower andbroader (bumps in both RS and FS spectrabecome less significant).The total electron spectra (Fig. 6, bot-tom row) also have interesting features. Onaccount of energy losses, the contribution ofthe RS in the cut-off region is stronger. Thiscreates either small-scale concavity or a bro-ken power-low in the loss-affected region(between E rad and E max,e ). The strongerthe contribution from the RS is, the morepronounced a concavity is observed. Whenthe contribution of the RS is mild, then onesees a broken power-law. In some models,(i.e., D300I and D300A), the shallow con-cavity in the loss-affected region is visiblethroughout all late times. It is striking that the electron spectra in single-shock NDSAsimulations [70] have similar concavity inthe loss-affected spectral band. Addition-ally, one can note that near E rad the spectraof P models are smoother than the spectraof D models.We must make an important remark con-cerning particle spectra. The plotted spec-tra are integrated over the entire simulationdomain. However, one finds (5.2.1) that themain contribution to the emission is doneby the particles confined inside the remnant.These particle spectra are typically softer athigh energies because they do not includethe hard upstream particles. For illustra-tion see the thin lines of D075I models inFig. 6, which show separately SNR-confined(downstream) particles and peaked at highenergies upstream particles. Only inverseCompton emission because of low MF inthe upstream region can benefit from theupstream particle population. We consider three basic non-thermalemission processes, synchrotron and inverseCompton (IC) emission of electrons, andneutral pion decays originating in collisionsof CR protons with protons at rest. Thesynchrotron emission is calculated accord-ing to [71]. We consider only the CMB pho-ton field for the IC scattering. Therefore,we may use a simple rescaling of the syn-chrotron emission to IC scattering [72, 73].We calculate the gamma-ray emission pro-duced by pion decays according to the pro-cedure described in [74]. Thermal emissionis not considered. However, we made es-timates that even assuming instant equili-bration the maximum of the thermal emis-sion does not exceed 10% of the synchrotronemission at the respective energy for all con-sidered ages. In particular, at 3 keV the15 l og N E [ G e V / c m ] D075IP075ID300IP300ID300AP300A-8-7 1 2 3 4 5 6 t= 100 yr 0 1 2 3 4 5log E [GeV] 1 2 3 4 5 6 t= 400 yr 0 1 2 3 4 5 1 2 3 4 5 6t=1000 yr
Figure 6: The time evolution of volume-integrated proton (top row) and electron (bottom row) spectra fordifferent models and SNR ages. Thin lines for the D075I model in the top row display the downstream andupstream components of the proton spectra. thermal component may comprise ≃
3% ofthe X-ray emission of the models consideredhere. The reader should be aware thoughthat we used the same injection efficiencyfor electrons and protons which results inhigh amplitudes of leptonic emission. If theelectron injection efficiency is lowered signif-icantly, below 3 keV the brightness profilesmay be affected.
The radiation spectra show diversity dur-ing SNR evolution (see Fig. 7). In general,the radiation spectra are much smootherthan the parent particle distributions. Theleptonic spectra show broad features, whichare the result of energy losses experiencedby electrons during their acceleration his-tory. A noticeable feature of the leptonicspectra is concavity, which appears in dif- ferent models at different times. At 100years it is strongly visible in P300I andP300A models, but in D075I and D075A itis nearly invisible. Later, at 400 years, itappears in D300I and D300A models whileit disappears in D075I and D075A, and itstarts to vanish in P300I and P300A mod-els. At 1000 years it is visible only in D300Iand D300A models. The concavity reflectsthe change in electron slope on account ofthe contribution of the RS between E rad and E max,e . It interesting to see that thevolume-integrated synchrotron and IC spec-tra have different shapes. While the syn-chrotron emission comes mainly from theshocked region, a significant fraction of ICemission comes from the ejecta and up-stream region with low MF, where the elec-tron spectrum is different. The pion-decay16 l og F [ e r g c m - s - s r - ] -7-6-5-4 -2 -1 0 1 2 3 4 5t= 100 yr 7 8 9 10 11 12 13 14log E [eV] -2 -1 0 1 2 3 4 5t= 400 yr 7 8 9 10 11 12 13 14 -2 -1 0 1 2 3 4 5t=1000 yrD075IP075ID300IP300ID300AP300A Figure 7: The time evolution of volume-integrated emission from the SNR due to synchrotron (top row),inverse Compton (middle row) and pion-decay (bottom row) for different models and SNR ages. spectra are much smoother than their lep-tonic counterparts and can be characterizedby a power-law with a very gradual andbroad cut off at relatively low energy be-tween a few 100 GeV to a few TeV. Thismakes the spectra rather soft in the en-ergy band observed with modern Cherenkovtelescopes. The pion-decay emission comesmainly from inside the SNR, with a smallamount of upstream contribution.
To permit morphological studies of SNRat different ages, we computed radial inten- sity profiles at characteristic wavebands forcurrent instrumentation (radio @1.4 GHz,X-rays @3 keV, gamma-rays @1 TeV). InFig. 8 we separately plot the emission of lep-tonic and hadronic origin. The profiles arenormalized to the emission maximum alongthe SNR radius. The absolute intensity maybe obtained from the spectra given at Fig. 7.We plot here “I” models only. Their “A”counterparts show similar profiles, but theintensity of the FS is a bit lower, makingthe profiles look somewhat flatter.It is astonishing to see how the brightnessprofiles undergo severe changes with time.17 F / F m a x D300I 0 0.2 0.4 0.6 0.8 1 P075I 0 0.2 0.4 0.6 0.8 1 D075I t= 100 yr 0.2 0.4 0.6 0.8 1r/R FS P300I D300I P075I D075I t= 400 yr 0.2 0.4 0.6 0.8 1P300I D300I P075I D075I t=1000 yr
Figure 8: The radial intensity profiles of the SNR in different models at different wavelength and ages. [email protected] GHz (dotted-pink), X-rays @3 keV (solid red), IC @1 TeV (long-dashed green), pion-decay @1 TeV(dashed blue).
At 100 years both shocks are bright andvisible in all bands except for the FS in ra-dio waves, which are emitted by low-energyelectrons. The contribution of the RS to thetotal emission is high at the initial stages.The low-energy electrons cannot propagatefar from their point of origin, and thereforethe relatively high injection rate at the RSin the early phase renders it much brighterthan the FS in radio band. Pion-decaygamma rays are copiously produced in theejecta region on account of the high targetdensity there. At 400 years the contributionof the RS region becomes less prominent,but is still visible. Only in synchrotron X- rays (and IC in D075I/A models) the RS be-comes insignificant. The 3 keV synchrotronX-rays are created by electrons of ratherhigh energy ( ∼ . B = 300 µ Gand ∼
27 TeV for B = 75 µ G). At thisage the RS is no longer able to boost elec-trons up to these energies. The electronsaccelerated at the FS suffer from losses ontheir way to the SNR interior, and there-fore the synchrotron emission becomes no-ticeably filamentary, especially in high-MFmodels. A characteristic feature developsin IC emission of the models with high MF.A plateau of high intensity appears in theejecta region and a low intensity between18he two shocks. In the shocked region, high-energy electrons are few on account of se-vere energy losses in the high MF, so theIC emission is suppressed. At the sametime, the electrons that propagated into theejecta region, where the MF is low, accumu-late there and account for bright IC emis-sion. This trend continues with age, and af-ter 1000 years we observe a bright plateau ofIC emission from the SNR center, while theFS appears very filamentary. There is al-most no trace of emission from the RS after1000 years in the other wavebands. X-raysynchrotron emission is seen as a thin fila-ment near the FS, especially in the modelswith high MF.
6. Conclusions
We have studied particle acceleration byboth forward and reverse shocks in youngtype-Ia Supernova Remnants. Rather thanstarting with Sedov models, as is most com-monly done, we used gas-flow profiles inthe ejecta-dominated phase, derived from1-D hydrodynamical simulations, to solvethe cosmic-ray transport equation in a test-particle approach. We analyzed how differ-ent phenomenological parametrizations forthe magnetic-field distribution and strengthaffect the maximum energy and spatial dis-tributions of accelerated particles. Addi-tionally, we have explored the influence ofpossible Alfv´enic drift of scattering centerson the total particle spectra in the upstreamregion of both SNR shocks and find our re-sults for the forward shock in agreementwith previous findings [32, 35, 34]. We in-vestigated the properties of the resultingnon-thermal radiation, i.e., its spectra andthe spatial distribution of the intensity atcharacteristic wavelengths. We have demonstrated that it is impor-tant to account for the contribution of theRS to cosmic-ray particle population in theinitial 400 - 600 years of SNR evolution,given the widely used parametrizations ofthe MF that are assumed here. At thattime the RS is able to accelerate particlesup to very high energies, and the number ofaccelerated particles is comparable to thatat the FS. This is well visible in volume-integrated particle spectra, the distributionof high-energy particles in the SNR, as wellas in the morphology of particle radiation.The significance of the RS contribution fallswith time. Although the RS contribution isstill visible in particle spectra up to 1000years or so, it is barely noticeable in theemission spectra and morphology, which isin agreement with observations.We found that the total volume-integrated particle spectra of the SNRcan be very different from single-shocktest-particle solutions, especially for veryyoung SNRs. The choice of MF profilesaffected the relative contribution to thetotal spectrum of the RS and the FS, andconsequently the emission spectra in thehigh-energy region varied. Additionally,Alfv´enic drift of scattering centers in theshock precursors gives rise to additionalspectral feature, whose appearance andtime evolution depends on the choice ofMF profile. Generally, P models are moresensitive to Alfv´enic drift than D models.Obviously, the models with high MF areaffected stronger than those with low MF,for which the effect is subtle.The particle distributions computed hereshow a variety of spectral shapes. Interest-ing is that some of the spectra (both elec-tron and protons) exhibit features whichmay be considered similiar to ones shownusing NDSA, such as spectral concavity and19igh-energy bumps. However, concavity inour spectra arises in a much smaller high-energy band compared to the broadbandconcavity of NDSA. Besides, it is not re-flected in pion-decay spectra as these areproduced predominantly by confined par-ticles in the SNR. Additionally, our test-particle spectra are softer at high energiesthan those of NDSA, even if both are mod-ified by Alfv´enic drift, which may providebetter agreement with gamma-ray spectraobserved from SNRs.
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