The Role of Diffusive Shock Acceleration on Non-Equilibrium Ionization in Supernova Remnants
aa r X i v : . [ a s t r o - ph . H E ] F e b Draft version November 13, 2018
Preprint typeset using L A TEX style emulateapj v. 05/04/06
THE ROLE OF DIFFUSIVE SHOCK ACCELERATION ON NON-EQUILIBRIUM IONIZATION INSUPERNOVA REMNANTS
Daniel J. Patnaude , Donald C. Ellison , & Patrick Slane Draft version November 13, 2018
ABSTRACTWe present results of semi-analytic calculations which show clear evidence for changes in the non-equilibrium ionization behind a supernova remnant forward shock undergoing efficient diffusive shockacceleration (DSA). The efficient acceleration of particles (i.e., cosmic rays) lowers the shock temper-ature and raises the density of the shocked gas, thus altering the ionization state of the plasma incomparison to the test particle approximation where cosmic rays gain an insignificant fraction of theshock energy. The differences between the test particle and efficient acceleration cases are substan-tial and occur for both slow and fast temperature equilibration rates: in cases of higher accelerationefficiency, particular ion states are more populated at lower electron temperatures. We also presentresults which show that, in the efficient shock acceleration case, higher ionization fractions are reachednoticeably closer to the shock front than in the test-particle case, clearly indicating that DSA mayenhance thermal X-ray production. We attribute this to the higher postshock densities which lead tofaster electron temperature equilibration and higher ionization rates. These spatial differences shouldbe resolvable with current and future X-ray missions, and can be used as diagnostics in estimatingthe acceleration efficiency in cosmic–ray modified shocks.
Subject headings: cosmic rays – thermal emission: ISM – shock waves – supernova remnants – X-rays:ISM INTRODUCTION
In young supernova remnant (SNR) shocks, the ac-celeration of cosmic rays leads to a softening of theequation of state in the shocked plasma. This comesabout because the diffusive shock acceleration (DSA)process turns some non-relativistic particles into rela-tivistic ones and because some of the highest energy rel-ativistic particles escape from the shock. Both of theseeffects lead to lower post-shock plasma temperatures aswell as higher post-shock densities (e.g., Jones & Ellison1991; Berezhko & Ellison 1999). The ionization state ofshocked gas at a particular time is dependent upon boththe gas density and the electron temperature. In lightof this, DSA ought to leave its imprint on the ionizationstructure of the shocked gas. Toward this end, we presentwhat we believe to be the first self-consistent model forSNR evolution which includes the hydrodynamics, the ef-fects of efficient shock acceleration, and a full treatmentof the non-equilibrium ionization balance at the forwardshock.A number of young SNRs show both nonthermal andthermal emission in the region behind the forward shock,including SN 1006 (Vink et al. 2003; Bamba et al. 2008),Tycho (Hwang et al. 2002; Cassam-Chena¨ı et al. 2007),and Kepler (Reynolds et al. 2007). The thermal emissionarises when the forward shock sweeps up the circumstel-lar medium (CSM) and heats it to X-ray emitting tem-peratures. As pointed out in Ellison et al. (2007) (here-after DCE07), the thermal emission is often considerablyfainter than the nonthermal emission, but there are cer-tainly examples where the thermal emission is as bright Smithsonian Astrophysical Observatory, Cambridge, MA02138 Physics Department, NC State University, Box 8202, Raleigh,NC 27695; don [email protected] or brighter than any nonthermal emission (Vink et al.2006). In SNR RX J1713.7-3946, the lack of thermal X-ray emission is an important constraint on the ambientdensity and significantly impacts models for TeV emis-sion (e.g., Slane et al. 1999; Ellison, Slane & Gaensler2001; Aharonian et al. 2007; Katz & Waxman 2008).If the diffusive shock acceleration process in youngSNRs is as efficient as generally believed, with & T e ) andionization age (defined as n e t , where n e is the electrondensity and t is the time since the material was shocked)as a function of time in hydrodynamic simulations ofSNRs where the forward shock was efficiently produc-ing cosmic rays (CRs) and, as a result, was substan-tially modified from test-particle results. They foundthat, while both T e and n e t did differ between the test-particle and CR-modified cases, in the cases where DSAis highly efficient, the synchrotron emission in the X-rayrange is considerably stronger than the thermal X-rayspectrum, and any differences in the thermal X-rays asa result of CR-modification are likely to be missed. Inthis paper, we extend the work of DCE07 by explicitlytracking the non-equilibrium ionization state in a CR-modified shock. The lower shock temperature and higherdensity that result from efficient DSA combine to shortenboth the temperature equilibration and ionization equi-librium time-scale, and we show that this can have adramatic effect on the ionization structure between theforward shock (FS) and the contact discontinuity (CD).Although we don’t calculate the thermal X-ray emission Patnaude, Ellison, & Slanehere, the cases we study show that efficient DSA can in-crease the ionization fraction of important elements andpossibly enhance thermal X-ray emission.In §
2, we outline the changes to our model first pre-sented in DCE07 and discuss several caveats to our ap-proach. In §
3, we present our examples and discuss thequantitative and qualitative effects of efficient DSA onthe ionization state and SNR structure. We also showhow these effects might manifest themselves in currentand future X-ray observations. In §
4, we summarizeour results and outline our future enhancements to thismodel. CR-HYDRO + NEI MODEL
Our spherically symmetric model uses the semi-analytic DSA calculation developed by Amato & Blasi(2005) and Blasi et al. (2005) and is similar to thatused in DCE07, except that we now calculate the non-equilibrium ionization explicitly at every time step us-ing plasma parameters that are continually updated asthe SNR evolves. In DCE07, the NEI was calculated atthe end of the simulation using average plasma param-eters. We refer the reader to DCE07 for all details ofthe CR-hydro simulation apart for those discussed belowdetailing our dynamic NEI generalization.The DSA model used here differs from that describedin Ellison et al. (2007), Ellison & Cassam-Chena¨ı (2005),and previous papers, in two important ways. First, wereplace the “effective gamma,” γ eff , approximation witha more realistic model of the effect escaping particleshave on the shock dynamics. We now explicitly removefrom the shocked plasma the energy that escaping par-ticles carry away from the forward shock. The ratio ofspecific heats of the shocked gas used in the simulations, γ sk , is determined directly from the particle distributionfunction including the correct mix of relativistic and non-relativistic particles. While the old effective gamma hadthe range 1 < γ eff ≤ /
3, the ratio of specific heats γ sk isconstrained to lie between 4/3 and 5/3. These changes inthe way escaping particles are treated, and γ eff is calcu-lated, become important for later stages of the SNR evo-lution, but do not produce significant changes in timesas short as 1000 yr. The results reported in Ellison et al.(2007) are not modified significantly by these changes.The second difference is that instead of specifiying afixed injection parameter, χ inj (this is ξ in equation (25)in Blasi et al. 2005), which then determines the acceler-ation efficiency, we now specify a fixed diffusive shockacceleration efficiency, ǫ DSA , and then determine χ inj ac-cordingly. This change makes the parameterization ofthe acceleration efficiency more transparent but does notchange the basic approximation that is made.The semi-analytic DSA model we use does not cal-culate the acceleration efficiency self-consistently basedupon the Mach number, the available accelerationtime, and other relevant shock parameters; rather weparametrize the efficiency by χ inj , and the model thendetermines the shock structure self consistently. Fur-thermore, the DSA model assumes that the thermal par-ticles have a Maxwell-Boltzmann distribution with a su-perthermal tail. The actual shape of the quasi-thermaldistribution, and the shape at the point where the su-perthermal tail joins it, are approximated since the semi-analytic calculation only self-consistently describes par- ticles with speeds greater than the shock speed, i.e., v p ≫ v sk . The differences at low energies between what isassumed in the DSA model and the actual quasi-thermaldistribution are expected to be small, but these differ-ences may become more important if the contributuionto ionization from superthermal particles is considered.Despite the approximations of the semi-analytic calcula-tion at quasi-thermal energies, it is the state-of-the-artsince the actual quasi-thermal distribution can only bedetermined with plasma simulations and these are notyet available for SNR parameters.The ionization structure of shock heated gas at a par-ticular distance behind the shock in a SNR is determinedby the electron density n e , the electron temperature T e ,and the ionization and recombination rates for each ionof interest. The structure is determined by solving thecollisional ionization equations in a Lagrangian gas ele-ment behind the shock:1 n e D f ( X i )D t = C ( X i − , T e ) f ( X i − ) + α ( X i , T e ) f ( X i +1 ) − [ C ( X i , T e ) + α ( X i − , T e )] f ( X i ) . (1)Here, f ( X i ) is the fraction of element X in ion stage X i and C ( X i , T e ) and α ( X i , T e ) are the ionization andrecombination rates out of and into ion X i , respectively.We calculate the electron temperature by assumingthat the electrons are heated by Coulomb collisions withprotons and helium (Spitzer 1965). We adopt this simpleprescription, which gives a lower limit to the equilibra-tion time, knowing that the heating of electrons may, infact, be far more complicated. For instance, there is rea-son to believe that collisionless wave-particle interactionswith the magnetic turbulence will be important (e.g.,Laming 2001), and recent work interpreting hydrogenline widths suggests that the electron-to-proton tempera-ture ratio behind some SNR blast waves depends mainlyon the shock speed, a result implying a heating pro-cess substantially different from Coulomb collisions (e.g.,Ghavamian et al. 2007; Rakowski et al. 2008). However,there remain large uncertainties in connecting the mea-sured line widths to the electron-to-proton temperatureratio (see Heng & Sunyaev 2008), and until particle-in-cell (PIC) simulations are able to model non-relativistic,electron-proton shocks with parameters typical of SNRs,the plasma physics of electron heating will remain uncer-tain (see Vladimirov, Bykov & Ellison 2008, for a discus-sion of the limitations of PIC simulations in this regard).In order to model some of the complexity of electronheating, we scale the Coulomb equilibration time with aparameter, f eq , defined in Eq. (3) below.At the start of the simulation, we assume that the un-shocked electrons and ions are in equilibrium at a tem-perature T = 10 K. We also assume that unshockedH and He are both 10% singly ionized and all heavierelements are initially neutral. While we note that thisis not the precise equilibrium ionization state for 10 K,we emphasize that none of our results depend in anysignificant way on the ionization state of the unshockedmaterial as long as it is not fully neutral. In all of theresults shown here we fix the helium number density at10% of the proton number density, n p, .At each time-step, we track the ionic state X i withineach spherically symmetric fluid element by solving theon-Equilibrium Ionization in Cosmic Ray Modified Shocks 3time-dependent ionization equations for each abundantelement (H, He, C, N, O, Ne, Mg, Si, S, Ar, Ca, Fe, andNi). We solve the coupled set of equations with atomicdata extracted from Raymond & Smith (1977), as firstpresented in Gaetz et al. (1988) and updated by Edgar(2008).In Figure 1 we show an example of the time evolu-tion of the ionization fraction, f ( X i ), of high ionizationstates of oxygen (O , O and O ) in a mass shell thatis crossed by the forward shock 100 yr after the explo-sion. For this example, as in all we show in this paper,we have fixed parameters typical of Type Ia supernovae,i.e., the kinetic energy in ejecta from the supernova ex-plosion E SN = 10 erg, the mass of the ejecta M ej =1.4M ⊙ , the density of the ejecta follows an exponentialdensity profile as is generally assumed for Type Ia super-novae (Dwarkadas 2000), and we assume the supernovaexplodes in a circumstellar medium (CSM) which is uni-form with proton number density n p, and magnetic fieldstrength B . In all of the models shown here, we take B = 15 µ G. The figure shows that the density and pro-ton temperature in the shell are dropping with time asthe electron temperature increases due to Coulomb col-lisions. After 1000 yr, the material is close to ionizationequilibrium for these ions.Figure 1 also compares results for test-particle (TP)and efficient DSA. In all of the examples in this paper,we define TP acceleration as being 1% efficient, i.e., 1%of the ram kinetic energy of the forward shock is placedinto superthermal particles. For all of our efficient ac-celeration cases, we assume 75% of the shock ram ki-netic energy is placed into superthermal particles, i.e., ǫ DSA = 75%. Figure 1 shows that efficient DSA producesa higher postshock density and lower postshock temper-ature, as expected. What is also clear is that the highionization states of oxygen become populated sooner inthe ǫ DSA = 75% case. This implies that, instead of sup-pressing thermal X-ray emission as has been suggested(e.g., Drury et al. 2008; Morlino, Amato & Blasi 2008),efficient DSA can possibly enhance it.We make the following approximations in the NEI cal-culation, noting that these are in addition to approx-imations made in the underlying CR-hydro model (asdescribed in Ellison et al. 2007, and references therein): • We assume that only electrons from the thermalpopulation contribute to the non-equilibrium ion-ization. In nonlinear DSA, the energetic popu-lation emerges smoothly from the thermal pop-ulation (a nice example from a relativistic PICsimulation is given in Spitkovsky 2008) and su-perthermal particles may contribute to ionization(see Porquet et al. 2001, for a test-particle calcu-lation involving a Maxwell-Boltzmann distributionwith nonthermal tail). As we discussed above, su-perthermal particles are expected to contribute to We refer the reader to Ellison et al. (2007) for a full discussionof the additional parameters required for the CR-hydro model. This value for B is somewhat higher than the typically as-sumed 3 µ G and reflects the possibility that magnetic field ampli-fication (MFA) may be taking place. We emphasize, however, thatwe do not include MFA in the DSA calculation performed here. Alarge upstream magnetic field, B , will reduce the effects of efficientDSA, as described in Berezhko & Ellison (1999). the ionization at some level. However, the sig-nificance of this nonthermal ionization, in shocksundergoing efficient particle acceleration, has notyet been determined and remains an area of activework. For the purposes of this paper, we assumeany nonthermal contribution is small. • We only model the interaction region betweenthe forward shock and the contact discontinuitywhere we assume cosmic elemental abundances.One reason for emphasizing the forward shock isthat it is not certain that significant CR produc-tion occurs at the reverse shock in SNRs (e.g.,Ellison, Decourchelle & Ballet 2005). • We only consider young SNRs and do not includethe effects of radiative cooling. In the high-densitylimit, radiative losses could be significant and thecooling time-scale could be comparable to other dy-namical time-scales. We will investigate this effectin a subsequent paper. RESULTS
In the following examples we investigate the effect theacceleration efficiency, ǫ DSA , and the CSM proton den-sity, n p, , has on the non-equilibrium ionization state ofsome selected elements. Ionization vs. Position
In Figure 2, we plot the ionization fractions of O and O and Si and Si in the top two panels as afunction of position behind the forward shock (FS). In allpanels, test-particle results ( ǫ DSA = 1%) are shown withdashed curves and efficient DSA results ( ǫ DSA = 75%)are shown with solid curves. The electron density andelectron and ion temperatures are shown in the bottomtwo panels. As the top two panels clearly show, higher ionizationfractions are attained closer to the shock front in the ef-ficient DSA cases, as compared to the TP cases. For in-stance, in the efficient case, the fraction of O peaks at adistance R/R FS ≃ .
98 behind the shock, while in the TPcase, this fraction peaks at
R/R FS ≃ .
97. We attributethe increased ionization fractions closer to the shock as adirect result of higher postshock densities in the efficientDSA case. Note that the curves extend from the forwardshock back to the contact discontinuity, indicating thatthe region between the forward shock and contact discon-tinuity is considerably narrower in the efficient accelera-tion case. This effect produces important morphologicalconsequences (e.g., Decourchelle, Ellison & Ballet 2000;Warren et al. 2005; Cassam-Chena¨ı et al. 2008).In Figure 3, we show the same quanitities as in Fig-ure 2, except that n p, = 0 . − . The lower CSMdensity results in lower shocked densities and in lessrapid collisional ionization behind the FS. For the ionswe show, higher ionization states (i.e., O and Si )are considerably less populated downstream from the FSwhen n p, is small. The differences resulting from DSAare less prominent but still evident; e.g., with n p, = 0 . − , O peaks behind the shock at R/R FS ≃ .
98 forthe efficient case, and at ≃ .
95 in the test particle case. In all results shown, we assume that shocked protons and otherions have the same temperature.
Patnaude, Ellison, & SlaneTo emphasize the importance of the different spatialstructures of ionization with ǫ DSA and n p, , we show, inFigure 4, a closeup view of the shock fronts in Figures 2and 3. Here, we have plotted the ionization fractionsas functions of angular distance behind the shock, as-suming a distance of 1 kpc. In the high density case( n p, = 1 cm − ; top panel), the fraction of O peaksright behind the shock at ∼ ′′ downstream, while itpeaks ∼ ′′ behind the shock in the test particle case.In the lower density case ( n p, = 0 . − ; lower panel),O peaks ∼ ′′ behind the shock in the efficient case,but peaks well beyond 50 ′′ behind the shock in the testparticle case. Similar results are found for silicon. Whilethese models are not scaled to match any particularGalactic SNR, we believe the angular separations shownhere would be easily resolvable in current and futurespace-based X-ray observatories even when line-of-sighteffects are taken into account. Thus, measuring the rela-tive fraction of H-like, He-like, and even Li-like chargestates would provide a useful diagnostic in studies ofGalactic SNRs undergoing efficient shock acceleration.Another interesting feature seen in Figures 2 and 3,is that the electron temperature is almost independentof ǫ DSA and only varies by a factor of ∼ n p, = 1 cm − and n p, = 0 . − cases. This is incontrast to the ion temperatures, where generally lowerion temperatures occur in the higher density models,due to the lower shock Mach number, and where thelarge ǫ DSA cases have considerably lower ion tempera-tures than the test-particle cases. The fact that lowerpostshock temperatures occur in efficient DSA is wellknown (e.g., Ellison 2000). The electron temperature isinfluenced by this and by the higher densities that occurwith efficient DSA. The higher postshock densities im-ply more collisions between electrons and ions, and thusmore rapid temperature equilibration. The higher elec-tron temperature combined with the higher postshockdensity leads to more rapid ionization, and thus highercharge states closer to the forward shock.
Ionization vs Equilibration Timescale
As is clear from Figures 1, 2, and 3, the ionization frac-tion for high charge state ions can increase with acceler-ation efficiency. Since the electron temperature is almostindependent of n p, in these cases, we attribute this effectmainly to the higher postshock densities. However, wehave assumed a particular model for temperature equili-bration between protons and electrons, namely that elec-trons start off cold and equilbration with the hot protonsoccurs only through Coulomb collisions where the equi-libration timescale is given by (Spitzer 1965, Eq. 5-31): t eq = 3 m p m e k / B π ) / n p Z Z e e ln Λ (cid:18) T p m p + T e m e (cid:19) / . (2)Here, m p is the proton mass and T p is the shocked pro-ton temperature and definitions of the other terms aregiven in Spitzer (1965). It’s important to note thatEq. (2) places strict limits on how low the electron toproton temperature ratio can be behind the shock (seeHughes, Rakowski & Decourchelle 2000); if other equili-bration mechanisms are important, such as plasma waveinteractions, equilibration will occur more rapidly. Toinvestigate the effects of more rapid temperature equili-bration, we define a parameter, 0 ≤ f eq ≤
1, and use the equilibration time t ′ eq in our calculations where, t ′ eq = f eq t eq . (3)In the results shown in Figures 1, 2, 3, and 4, we haveassumed f eq = 1.In Figure 5, we compare the ionization fraction ofO for ǫ DSA = 1% and ǫ DSA = 75% calculated with f eq = 1 (black curves in all panels) and f eq = 0 . ǫ DSA , f ( O )is larger immediately behind the shock for rapid equili-bration ( f eq = 0 .
1) but drops below the f eq = 1 valuefurther downstream as O becomes populated. Thetemperature plots in the bottom two panels show thatthe electrons and protons have come into equilibriumfor a range of radii (i.e., 0 . . R/R FS . .
98) when ǫ DSA = 75% and f eq = 0 .
1, but remain far from equilib-rium for f eq = 1 regardless of ǫ DSA . The equilibrationrate changes the ionization structure for this particularion, producing changes comparible in scale to those pro-duced by efficient DSA.To quantify these effects further, we look at a pointmidway between the contact discontinuity and FS, i.e., at
R/R FS ≃ .
89 for ǫ DSA = 75% and at
R/R FS ≃ .
83 for ǫ DSA = 1% in Figure 5. At these locations, the electronto proton temperature ratios are: ( T e /T p ) TP ≃ .
11 and( T e /T p ) NL ≃ .
36, for f eq = 1, and ( T e /T p ) TP ≃ . T e /T p ) NL = 1 for f eq = 0 .
1, i.e., the ratios areabout 3 times larger with rapid equilibration. At thesemidpoint locations, the ionization fractions of O rangefrom f (O ) ≃ .
05 for f eq = 1 and ǫ DSA = 75%, to f (O ) ≃ .
23 for f eq = 0 . ǫ DSA = 1%, i.e., abouta factor of five span.The electron temperature ratio for f eq = 1 is( T e, NL /T e, TP ) f eq =1 = 1 . × K/ . × K ≃ . f eq = 0 . T e, NL /T e, TP ) f eq =0 . =3 × K/ × K ≃ .
5. For the particular parame-ters used in this example, the electron temperature stayswithin a factor of ∼ ǫ DSA and equi-libration time, while f (O ) varies by a factor of ∼ Emission Measure vs. Acceleration Efficiency
As seen in Figures 2 or 3, the plasma density is greatestimmediately behind the shock where the electron tem-perature is lowest. Since the rate for electron temper-ature equilibration depends on the proton temperatureand density and both the temperature and density de-pend on ǫ DSA , the NEI calculation will depend in a com-plicated fashion on the forward shock dynamics and theevolution of the interaction region between the CD andFS. Of course, the important property is the emissionthe plasma produces and this can be characterized bythe emission measure (EM) and the differential emissionmeasure (DEM).In Figure 6 we plot the emission measure for individ-ual ions, EM = N X f ( X i , R ) n e n p d V , and in Figure 7we plot ionic differential emission measures, DEM = P N X f ( X i , R ) n e n p d V / d(log T e ), where N X is the abun-dance of element X relative to hydrogen, f ( X i , R ) is theionization fraction for the ion X i at a distance R behindthe shock, and dV is the volume of the shell where EMor DEM is determined. The EM plotted in Figure 6 is aline-of-sight projection normalized to 1 cm surface area,on-Equilibrium Ionization in Cosmic Ray Modified Shocks 5and the DEM is obtained by summing over the regionbetween the CD and FS.Figure 6 clearly shows that the emission for these ionspeaks much closer to the FS and is considerably strongerwith efficient DSA than in the TP case. Figure 7 showsthat the peak emission for these two ions shifts down intemperature by about a factor of ∼ ∼ DISCUSSION AND CONCLUSIONS
We have presented a calculation of non-equilibriumionization in a hydrodynamic simulation of SNRs un-dergoing efficient DSA. While we have only explored alimited range of parameters in this paper, it’s clear thatthe production of CRs by the outer blast wave modifiesthe SNR evolution and structure enough to produce sig-nificant changes in the ionization of the shocked materialbetween the forward shock and contact discontinuity. Inparticular, higher ionization states are reached at lowerelectron temperatures (compared to the test particle case)because of the increase in post shock density due to theincreased shock compression. The calculation of thermalX-ray line emission requires the additional step of cou-pling the resultant ionization state vectors to a plasmaemissivity code, work which is in progress. Nevertheless,our results clearly show that taking DSA into accountand dynamically calculating the NEI produces changesin the ionization fractions of important elements thatshould translate into noticeable changes in the interpre-tation of X-ray line emission observed from young SNRs.Our main results are the following: • Compared to the test-particle case, the increasein ionization that accompanied DSA in our ex-amples suggests that efficient DSA will result inan increase in the overall thermal X-ray emission(see Figure 6). We note that an increase in ther-mal emission with increasing acceleration efficiencyis evident in our eariler results which explored aslightly different parameter space (i.e., Figures 7and 8, Ellison et al. 2007). The actual increasemay depend importantly on other model param-eters, such as the CSM density, and it is importantto explore a more expanded parameter space to de-termine how broadly valid our results are. Thiswork is in progress. However, regardless of whetheror not efficient DSA increases the integrated ther-mal emission over the test-particle case, some ther-mal emission is expected because ionization is notsuppressed when efficient DSA occurs. As Fig-ure 1 shows, electrons reach X-ray emitting tem-peratures well before they come into equilibrationwith protons and nearly as rapidly with or withoutefficient DSA. This occurs even if only Coulombequlibration is assumed. This is in contrast torecent claims (e.g., Morlino, Amato & Blasi 2008;Drury et al. 2008) that very weak thermal X-rayemission might result from efficient shock acceler-ation. • Compared to the test-particle case, ionization oc-curs more rapidly and, therefore, closer to the FS,with efficient acceleration (see Figures 4 and 6).The differences in spatial structure should be largeenough to observe and may be used as a discrimi-nant for the level of CR-modification, if a particularion state is coupled to other known properties, suchas the dynamics and ambient conditions. • Efficient DSA leads to more efficient Coulomb heat-ing of electrons and faster equilibration with ions,relative to the test particle case. This results be-cause the shocked plasma temperature is lower andthe shocked density is higher when efficient DSAoccurs. We showed, with a simple parameteriza-tion of the thermal equilibration time, that the sig-nature of efficient DSA on the ionization state re-mains apparent for equilibration more rapid thanoccurs with just Coulomb collisions. • Using the differential emission measure, we showedthat the maximum emission from a particular ionstate occurs at a significantly lower electron tem-perature with efficient DSA. For the ions shown inFigure 7, the difference in T e for peak emission ison the order of 1 keV while the maximum DEM re-mains almost constant. A difference this large willhave an important impact on the interpretation ofthermal X-ray emission in young SNRs.Currently, we do not treat radiative or slow shocks,but these regimes are easily explored. For instance in aradiative shock, the cooling time might be comparableto the energy loss time in a cosmic ray modified shock.Increases in the density will enhance the cooling to thepoint where radiative losses might rival losses from effi-cient DSA (Wagner et al. 2006). We intend to explorethis regime in a forthcoming paper.While we only considered shocked CSM here, we willconsider shocked ejecta in future work. In the ejecta,the electron density can be higher and the tempera-ture may be lower but, more importantly, the abun-dance structure is far more complicated than for CSMand calculations of X-ray emission are intrinsically moredifficult. Furthermore, simple arguments based on theexpansion of the ejecta material suggest that the mag-netic field may be too low to support DSA by the reverseshock. Nevertheless, there has been speculation thatparticles are accelerated there (e.g., Gotthelf et al. 2001;Uchiyama & Aharonian 2008; Helder & Vink 2008) andif DSA is efficient at the reverse shock, it will likely alterthe ionization balance of the shocked ejecta as much asshown here for the shocked CSM.Finally, while we have limited our examples here toSNRs expanding into a uniform medium typical of TypeIa supernovae, we emphasize that a wider parameterspace should be explored, in terms of both the struc-ture of the ambient medium (i.e. pre-SN winds) and theparameters which determine the cosmic ray accelerationefficiency. These cases will be addressed in a follow-uppaper.We would like to thank Dick Edgar, John Raymond,and Cara Rakowski for several useful discussions on how Patnaude, Ellison, & Slaneto thoughtfully display and interpret ionization fractions.This work was partially supported through a Smith-sonian Endowment Grant. D. J. P. and P. O. 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Fig. 1.—
Time evolution of a spherically symmetric Lagrangian mass shell which is crossed by the forward shock at 100 yr. The toppanel shows the evolution of high ionization states of oxygen, the middle panel shows the electron number density, and the bottom panelshows the electron and proton temperatures, assuming Coulomb equilibration. In all panels, the solid curves correspond to a model with75% DSA efficiency, while the dashed curves are for a TP model with ǫ DSA = 1%. The CSM proton number density for this exampleis n p, = 1 cm − . Here, and in all other examples, the unshocked CSM temperature is T = 10 K, and the unshocked magnetic field is B = 15 µ G. Patnaude, Ellison, & Slane
Fig. 2.—
Spatial profiles of H- and He-like oxygen and silicon, electron density, and temperature as a function of distance behind theforward shock. In the bottom panel, the curves labeled T i are ion (or proton) temperatures and those T e are electron temperatures. Here,and in figures 3-5 that follow, we show values from spherically symmetric shells as a function of R or ∆ R , not line-of-sight projections. Inall panels, solid curves correspond to models with 75% efficiency, while the dashed lines correspond to TP models. These models are for aCSM proton density of n p, = 1 cm − and are calculated at t SNR = 1000 yr. In the model with 75% efficiency, the forward shock velocityis ≈ − , while in the test particle model, it is ≈ − at t SNR = 1000 yr. on-Equilibrium Ionization in Cosmic Ray Modified Shocks 9
Fig. 3.—
Spatial profiles of oxygen and silicon ions, electron density, and temperature as a function of distance behind the forward shock.In all panels, solid curves correspond to models with 75% efficiency, while the dashed lines correspond to TP models. These models are fora CSM proton density n p, = 0 . − and are calculated at t SNR = 1000 yr. In the model with 75% efficiency, the forward shock velocityis ≈ − , while in the test particle model, it is ≈ − at t SNR = 1000 yr.
Fig. 4.—
Top:
Ionization fraction as a function of distance behind the forward shock for O and O with n p, = 1 . − . Bottom:
Ionization fractions of O and O with n p, = 0 . − . In both panels, the solid curves are for ǫ DSA = 75% and the dashed curves arefor ǫ DSA = 1%. The angular scale is determined assuming the SNR is at a distance of 1 kpc and the results are calculated at t SNR = 1000 yr. on-Equilibrium Ionization in Cosmic Ray Modified Shocks 11
Fig. 5.—
Ionization fraction and temperature calculated between the contact discontinuity and FS. All calculations are at t SNR = 1000 yrand assume n p, = 1 cm − . In all panels, black curves assume f eq = 1 and red curves assume f eq = 0 .
1. In the bottom two panels, thesolid curves are the shocked electron temperature, T e , and the dashed cuvres are the shocked proton temperature, T p . As in Figures 2 and3, the left end of each curve is at the position of the CD. Fig. 6.—
Line-of-sight projection of the emission measure (EM) for O and Si as labeled. The solid curves are for ǫ DSA = 75% andthe dashed curves are for ǫ DSA = 1%. The angular distance, ∆ R , from the FS is determined assuming the SNR is at 1 kpc and the resultsare calculated at t SNR = 1000 yr with n p, = 1 cm − and t eq = 1. on-Equilibrium Ionization in Cosmic Ray Modified Shocks 13 Fig. 7.—
Differential emission measure (DEM) vs. electron temperature for O and Si as labeled. The solid curves are for ǫ DSA = 75% and the dashed curves are for ǫ DSA = 1%. The results are calculated at t SNR = 1000 yr with n p, = 1 cm − and t eqeq