Probing Shock Properties with Non-thermal X-ray Filaments in Cas A
Miguel Araya, David Lomiashvili, Chulhoon Chang, Maxim Lyutikov, Wei Cui
aa r X i v : . [ a s t r o - ph . H E ] A p r Probing Shock Properties with Non-thermal X-ray Filaments in Cas A
Miguel Araya † , , David Lomiashvili † , § , , Chulhoon Chang † , , Maxim Lyutikov † , and Wei Cui † , † Department of Physics, Purdue University, West Lafayette, USA § E. Kharadze Georgian National Astrophysical Observatory, Ilia Chavchavadze State University, Tbilisi, Georgia
ABSTRACT
Thin non-thermal X-ray filaments are often seen in young supernova remnants. Weused data from the 1 Ms
Chandra observation of Cassiopeia A to study spectral prop-erties of some of the filaments in this remnant. For all the cases that we examined, theX-ray spectrum across the filaments hardens, at about 10% level, going outward, whileobserved filament widths depend only weakly on the photon energy. Using a model thatincludes radiative cooling, advection and diffusion of accelerated particles behind theshock, we estimated the magnetic field, turbulence level, and shock obliquity.
Subject headings: shock: diffusion — advection, supernova remnant
1. Introduction
Young supernova remnants (SNRs) have long been thought to be the main source of galacticcosmic rays (Shklovskii 1953). Evidence for the existence of high-energy electrons in SNRs firstcame with the detection of non-thermal emission in the radio and later in X-rays (e.g., Koyama et al.1995; Bamba et al. 2000; Slane et al. 2001). With the use of the
Chandra X-Ray Observatory , de-tailed images of SNRs have revealed very thin structures (Long et al. 2003) near the forward shock.The spectral and spatial properties of such structures, often referred to as filaments, are consis-tent with synchrotron emission from highly relativistic electrons. High-energy protons and nucleiare also believed to be produced within SNRs, although no direct evidence has been conclusivelyfound. However, a recent analysis of the
Fermi -LAT spectrum of the SNR W51C suggests that themain component of emission in the GeV band from this object is produced through interactions ofhigh-energy hadrons (Abdo et al. 2009).In SNRs, charged particles may gain energy by repeatedly crossing the shock (Bell 1978b;Blandford & Ostriker 1978; Drury 1983). The process is thought to be facilitated by scattering [email protected] [email protected] Now at the Pennsylvania State University, PA, USA; [email protected] [email protected] [email protected] Chandra (Hwang et al. 2004)to carry out a detailed spectral analysis of non-thermal filaments in the outermost region of theremnant. Specifically, we are interested in the energy dependence of the width of the filaments andspectral variation across them. In the context of advection and synchrotron radiative cooling, thewidths are expected to decrease with increasing energy and the spectrum softens going downstream.However, diffusion may significantly modify the behaviors.
2. The Observation
We extracted spectra from nine non-thermal filaments of Cas A from the archival
Chandra
Chandra softwarepackage CIAO v3.4 and
Chandra calibration database (CALDB) version 3.5.2. Data were accumu-lated in GRADED mode to avoid telemetry loss, therefore the effects of charge transfer inefficiencyin the spectra cannot be corrected. This loss of charge affects the measured pulse-height distri-bution and the energy resolution, although the effect is small for the backside-illuminated CCDs(Townsley et al. 2000).The image of Cas A, shown in Figure 1, was obtained by combining events in the energy rangefrom 0 . −
10 keV and then correcting the count map by effective exposures. Since the effective areais energy dependent, weighted exposure maps were calculated at different energies and combined.We focused on regions that had previously been identified as being non-thermal (Stage et al. 2006)and been thought to be associated with the forward shock (Gotthelf et al. 2001). However, dueto low statistics (even with a 1 Ms exposure), faint filaments mainly located in the western andeastern sides of the remnant were not included in the analysis. The image also shows the ninefilaments chosen for this work along with off-source regions for background estimates.
Each of the non-thermal filaments was divided into an ‘inner’ and an ‘outer’ region, with the‘inner’ region being closer to the interior of the remnant. The division between the inner and outer 4 –regions for each filament was set at the peak of its linear intensity profile. This division made itpossible for us to quantify the difference in the spectral properties of the radiation emitted by theelectrons at different locations. One may naively attribute the difference to the fact that electronsin the inner region have had more time to evolve after interacting with the forward shock than theelectrons in the outer region. However, as we will show, there seems to be a fair amount of mixing,implied by the inferred diffusion coefficients.As an example, Figure 2 shows the linear profile of Filament 5 (for the 0 . −
10 keV band).This profile was obtained from a 1 ′′ . × ′′ . ′′ .
5. The top panel shows the division between the inner and outer emitting zones.
The widths of each filament were estimated in three energy bands: 0 . − − −
10 keV. In order to quantify the width of a linear profile, we fitted the profile around the peakwith a Gaussian function, as shown in Figure 3. The results of the fits are summarized in Table 1.No strong dependence of the widths on energy is apparent.We should note that the overall linear profiles of the filaments are highly non-Gaussian (see,e.g., Figure 2). Nevertheless, we think that the derived Gaussian widths reflect fairly accuratelythe widths of the filaments.
To carry out the spectral analysis, we reprojected the event 2 files to a common tangent pointand used the CIAO tool acisspec to extract events between 0.3 keV and 10 keV from each regionshown in Figure 1, calculate weighted Auxiliary Response Files, and combine the spectra from theindividual segments of the observation. Consistent calibration was used separately to produce theexposure-weighted responses by applying time-dependent gain corrections appropriate for − ◦ CGRADED mode data on the back-illuminated S3 chip. After the individual files were combined, webinned each spectrum such that each bin contained at least 100 counts and proceeded to individuallymodel them with XSPEC version 11.3.2 (Arnaud 1996).All spectra show, with varying degrees of prominence, the presence of emission lines (seeFigure 4), indicating the existence of thermal photons in the regions. For most cases, we addedtwo Gaussian components to model the lines at around 1.85 and 2.38 keV, which we attribute to SiK XIII and S K XV, respectively. Other weaker lines also appear to be present in some filaments.Filaments 5 and 6 show two additional lines at 1.3 keV and 1.0 keV, most likely associated withMg XI and Fe XXI, respectively. Filament 9 also shows the line at 1.0 keV.The inner and outer extraction regions have a typical extension of about 7 ′′ each, which cor- 5 –responds to a physical size of roughly 0.1 pc (assuming a distance of 3.4 kpc to the remnant; (seeReed et al. 1995)). The regions for Filaments 2 and 5 are smaller (4 ′′ and 3 ′′ , respectively) sincethere seems to be a considerable amount of thermal emission in these areas. To assess possible“contamination” from thermal emission, we also experimented with thinner extraction regions foreach filament as well as on-source background regions. In the first case, we failed to remove thelines seen, while in the second one it becomes difficult to determine the appropriate locations ofbackground regions necessary to avoid subtraction of non-thermal photons. The resulting lack ofstatistics after the subtraction generally does not allow to carry out a satisfactory analysis of thenon-thermal X-rays. It is possible that the thermal and non-thermal emissions are cospatial, butwe do not rule out that the detection of thermal photons might be due (at least partly) to scatteredX-rays.The photon spectra of all filaments were satisfactorily fitted with an absorbed power-law, withindices ranging from 2.2 to 3. Figure 4 shows the spectral fits, as well as residuals, for both theinner and outer regions of each filament. The results are summarized in Table 2.Although in most cases the error intervals of the photon indices for the inner and outer regionsoverlap, we note that the spectrum of the inner region is in general softer than that of the outerone. The difference in photon indices between the inner and outer regions is on the order of 10%.The hydrogen absorption column values obtained from the fits are typically 0 . − . × cm − in all regions except for a larger value of 1 . × cm − found for Filament 8, at the western edgeof the remnant, where it is believed that it is interacting with a molecular cloud (Keohane et al.1996).
3. Theoretical modeling
We developed a model to explain the observational results. The model takes into accountsynchrotron radiative losses and diffusion of particles in the forward shock region. We assumedthat the injected particles follow a power-law spectral distribution with index Γ and the particlespectrum subsequently evolves.We approximated Cas A as a sphere with radius R = 10 cm (Reed et al. 1995). The non-thermal emission is assumed to come from a thin shell near the edge of the sphere and integratedalong the line of sight. The evolution of the non-thermal electron distribution is given by thediffusion-loss equation. We used the solution derived by Syrovatskii (1959), but also included theadvection process. For Cas A, the advection speed of the plasma downstream of the shock is V adv = 1300 km s −
1, equal to the shock speed V sh divided by a shock compression ratio of 4,( V sh ∼ −
1; Vink et al. 1998). This value agrees with X-ray Doppler shift measurements,which imply a velocity relative to the shock of about 1400 km s − P ν ( γ ) = ( σ T cB γ / π ) δ ( ν − ν c ) , (1)where ν c = 3 qBγ / πmc = 3 ν L γ /
2, and ν L is the Larmor frequency; here, γ is the Lorentz factorof the particle, m is its mass, q is its charge, and B is the magnitude of the magnetic field. We willdiscuss the effects of this approximation in Section 5.We assumed that all of the emission originates behind the shock, where the magnetic field isbelieved to be amplified with respect to the ambient field. The diffusion coefficient was taken as D ( γ ) = κ mc γ qB , (2)where κ is a proportionality constant to be determined. The case when κ = 1 is referred to as Bohmdiffusion. Other types of diffusion are also being studied, including Kolmogorov and Kraichnanturbulences (Kolmogorov 1941; Kraichnan 1965), but will be discussed in detail elsewhere (D.Lomiashvili et al. 2010, in preparation).There are four main parameters in the model: magnetic field, spectral index of electrons (Γ),diffusion length ( l dif) and advection length ( l adv). The diffusion and advection lengths are definedas l dif = (cid:18) κmc ψqB (cid:19) / , (3) l adv( γ ) = V adv ψB γ , (4)respectively, where ψ = σ T / πmc . For convenience, we combined these quantities to define twonew parameters, Λdif = l dif /R and ζ = l adv(1 keV ) /l adv, which can be determined from the data.
4. Results
We implemented the model in XSPEC as a table model and applied it to the spectra of thefilaments. For each spectral fit, the line features and the hydrogen column density were fixed tovalues found in the corresponding power-law fits. The quality of the model fits is the same as thatof the fits by this phenomenological (power-law) model. 7 –
From the best-fit Λdif and ζ , we derived the magnetic field and diffusion coefficient for eachfilament (Equations (3) and (4)): B ≈ µ G (cid:18) Λdif0 . (cid:19) − / (cid:18) ζ . (cid:19) − / (cid:18) V adv1 . × cm s − (cid:19) / (cid:18) R cm (cid:19) − / , (5) κ ≈ . (cid:18) ζ . (cid:19) − (cid:18) V adv1 . × cm s − (cid:19) . (6)The results are summarized in Table 3. The indices of the injected electron spectrum found varyfrom 2.6 to 4 and the magnetic field ranges from ∼ µ G to 70 µ G, while κ stays around 0 . . − . ζ . The accelerated electrons will lose their energy due to synchrotron radiation. The evolution ofthe particle’s Lorentz factor, γ , is given by (cid:18) γ dγdt (cid:19) loss = − ψB γ . (7)A maximum energy will be reached by the particle when this loss becomes equal to the accel-eration rate. If we assume that the mean magnetic field is perpendicular to the shock normal, thenfor a compression ratio of 4 we can write the acceleration rate as (Jokipii 1987) (cid:18) γ dγdt (cid:19) acc = V sh32 κD B , (8)where D B = r g c/ r g = ( mc /qB ) γ is the particle gyroradius.Our assumption about the direction of the magnetic field is justified by our estimates of thediffusion coefficient, which constrain the obliquity angle to be nearly 90 ◦ (see below).The maximum energy for an electron then is given by E max ≈ (660 T eV ) (cid:16) κ . (cid:17) − / (cid:18) B µ G (cid:19) − / (cid:18) V sh5 . × cm s − (cid:19) . (9) 8 – We considered diffusion in the radial direction, with a corresponding diffusion coefficient givenby (Jokipii 1987; Blandford & Eichler 1987): D = D k cos θ + D ⊥ sin θ , (10)where θ is the angle between the mean magnetic field and the normal direction of the shock. Here,we assumed that the kinetic theory relations, D k = ηD B for the diffusion coefficient along themean direction of the field and D ⊥ = ηD B / (1 + η ) for the component of the diffusion coefficientperpendicular to the mean direction of the field, hold (e.g., Forman et al. 1974). In these equations, η ≡ λ mfp /r g is the particle’s gyrofactor, which is the ratio of the scattering mean free path, λ mfp,to the particle gyroradius, r g (Hayakawa 1969; Melrose 1980). Since isotropic Bohm-type diffusionis assumed here, we can rewrite Equation (2) in the form D = κD B . From Equation (10) we have κ = η (cid:18) cos θ + sin θ η (cid:19) . (11)When diffusion is taken as a perturbation in the particle orbits, the fraction η can be writtenin terms of the energy content in the resonant MHD waves (e.g., Blandford & Eichler 1987), η = ( δB/B ) − , of amplitude δB . We can then use Equation (11) to constrain θ and the levelof turbulence. For most cases, we found that κ ≈ .
02 which requires that 86 ◦ ≤ θ ≤ ◦ and6 ≤ η ≤
16. This implies a relatively high turbulence level,0 . ≤ δBB ≤ . . The results indicate that most of the radiation is originated from behind the forward shock(see Figure 2). However, the model could not explain the observed linear profile of the filaments(see Figure 5 for an example). The model predicts a sharp decline after the peak, which is notobserved.We speculated that some of the emission may come from a precursor (Ellison et al. 1994). Weestimated the contribution from the precursor by requiring that the distribution function should becontinuous across the shock. The precursor would consist of particles that have diffused across theshock but remain energetic. Specifically, it evolves in the presence of a magnetic field consistentwith that of the un-shocked medium surrounding the SNR, here assumed to be 4 times lower thanthe compressed field estimated downstream; however, we assumed that κ remains the same.Addition of this component substantially improves the predicted profile shape, as shown inFigure 5. On the other hand, we found that the inclusion of a precursor hardly affects the spectralparameters. More details will be discussed in a future publication (D. Lomiashvili et al. 2010, inpreparation). 9 –
5. Discussion
From the power-law fits to the spectra of the filaments in Cas A, it is seen that the emissionfrom the inner regions is consistently softer, by about 10%, than that from the outer regions. Thisseems to be consistent with the effect expected from radiative cooling, since the outer regions havehad less time since they interacted with the shock. When only synchrotron losses and advection aretaken into account, however, the predicted difference between the inner and outer photon indicesis the same in all filaments. The data show that this difference can change from one filament toanother.Also, from a consideration of synchrotron losses, one might expect that the widths of thefilaments get narrower at higher energies. In fact, if synchrotron cooling and advection were theonly processes controlling the plasma distribution, the width of a non-thermal filament can beestimated as w = V adv τ s , or the distance the particles are advected before radiating away theirenergy. This can also be written as w = V adv /ψB γ , where ψ = σ T / πmc , with σ T the Thompsoncross section for electrons, B the magnetic field, and γ the Lorentz factor of the accelerated particle,which when assuming emission peaked at the Larmor frequency ν L can be written as ( ν/ν L ) / .Therefore, an important dependence of the widths on the frequency of the radiation, of the form w α ν − / , would result. However, the data suggest that no dependence exists.These observations seem to point at the existence of additional mechanisms affecting theevolution of the plasma distribution and are found to be consistent with the model used. In thismodel, the difference between the photon indices of the inner and outer regions is regulated bydiffusion, and it is determined mostly by the ratio of advection to diffusion lengths, ζ , whereasit is found that varying Λdif ( ≡ l dif /R ) produces changes mainly in the calculated width of thenon-thermal filaments without considerably affecting the model spectra. This was also seen whencarrying out the fits, since the values of χ did not change appreciably for a wide range of valuesof Λdif, and therefore additional constraints on this parameter were necessary. As a constrain, weused the values of Λdif that were calculated to match the filamentary widths at half intensity tothe actual data as initial values for the fits. This parameter ranges from 0.014 to 0.034, while ζ varies from 3.3 to 8.9.Due to the role that ζ plays in the model, it should be possible to correlate the spatial differencesbetween the photon indices with the value of the proportionality constant in the diffusion coefficient, κ . It is seen that Filaments 1, 6, and 7 show the highest values for κ (although considerableuncertainties were obtained) and that the relative change in photon spectral index from inner toouter regions is the lowest for these filaments (although Filament 5 shows a similar change). Theamount of diffusion can change 1 order of magnitude for the different filaments depending on thedegree of spatial spectral variation observed. It can be argued that the diffusion of particles tendsto homogenize the plasma distribution and lower the difference between the inner and outer photonindices. This was also seen in the simulations where higher values of κ were used.The particle spectral indices are found close to 3, although steeper values are also seen (up to 10 –4 for Filament 6). This index corresponds to the power-law distribution of electrons resulting fromshock acceleration processes. After the particles evolve in the magnetic field, one might expectto see steeper spectral indices, especially at higher X-ray energies. Such steep distributions mighthave been seen before. For instance, when comparing the 10-32 keV RXTE
Proportional CounterArray spectrum of Cas A with the predicted synchrotron emission spectrum dominating the bandfrom 0.3 keV to 7 keV, the observed excess can be accounted for by an additional contribution ofnon-thermal Bremsstrahlung radiation from a steep power-law (index ∼ ′′ . − ′′ ) is determined by synchrotron losses and advection only (Vink & Laming2003), B sync ∼ − µ G. The difference might be explained by the fact that our estimates takeinto account these two processes but additionally consider diffusion. The turbulent magnetic field isconstrained to be 0 . ≤ δBB ≤ .
4. Moreover, the ordered field is found to be nearly perpendicularto the shock front (86 ◦ ≤ θ ≤ ◦ ), consistent with an expansion inside a toroidal external fieldproduced originally by the progenitor star. The inferred magnetic field in the filaments is stillhigher than that expected from magnetic field amplification of the interstellar field ( ∼ µ G), aswas also pointed out by Vink & Laming. Perhaps the interstellar field surrounding Cas A is higher,or the field has been amplified by the high-energy particles near the shock front through nonlinearwave growth.In some cases, the magnetic field might be much higher in other filaments of Cas A (Atoyan et al.2000). Patnaude & Fesen (2007) and Uchiyama & Aharonian (2008) have observed X-ray vari-ability of some of the non-thermal filaments seen in projection near the reverse shock on a timescale of a few years. Assuming that this variability is related to synchrotron cooling and DSA, andthat the diffusion is close to the Bohm limit with the field parallel to the shock front, Uchiyama &Aharonian estimated that the field required would be ∼ ν c (see Equation(1)), which is clearly an oversimplification. When we incorporated the full synchrotron spectruminto the model, we saw changes in the model parameters. For instance, a fit to the spectrum ofFilament 5 with the revised model leads to B ≈ µ G and κ ≈ .
06, which are different to thevalues shown in Table 3. However, the changes do not qualitatively modify our conclusions. Theobliquity angle is still close to 90 ◦ and the turbulent field is δBB ≈ .
2. The details of the full modelwill be presented in a future publication.
6. Summary
We summarize our main results as follows.1. Spectral evolution is seen across non-thermal filaments in Cas A, with the spectra of the outerregions being harder by about 10% on average.2. The widths of the filaments show no significant dependence on photon energy.3. To account for the observational results (1 and 2), we needed to include the effects of diffusion.If we restrict to Bohm-type diffusion, we could quantify the level of turbulence (0 . ≤ δBB ≤ .
4) as well as the diffusion itself ( κ ≈ . − . µ G, varying from filament to filament, and that the field isnearly perpendicular to the shock front.These results are in overall agreement with models of cosmic ray acceleration in the shocks ofSNRs. They imply that there is a high level of magnetic turbulence in the non-thermal filamentsassociated with the forward shock of Cas A, as well as magnetic field amplification. Both of theseconditions are necessary to efficiently accelerate cosmic rays.Regarding the shock orientation, our analysis of a sampling of non-thermal filaments, whichhave good azimuthal coverage of the remnant, implies that the obliquities are close to 90 ◦ , whichis consistent with the expansion of Cas A in the wind environment produced by the progenitor(Chevalier & Oishi 2003).Efficient cosmic ray acceleration in the shock of Cas A would have implications regardingthe acceleration of protons (and heavier ions), which may interact with cold ambient protons andproduce neutral pions that would decay into gamma rays, leaving a signature in the spectrum ofthe remnant in the GeV to TeV energy range. This signature could in principle be detected.We thank M. Laming, M. Pohl, and S. Reynolds for useful discussions. This research has madeuse of data obtained from the Chandra Data Archive and the
Chandra Source Catalog , and softwareprovided by the
Chandra X-ray Center (CXC) in the application package CIAO. This work has 12 –also made use of NASA’s Astrophysical Data System. We gratefully acknowledge financial supportfrom NASA and Purdue University. 13 –
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This preprint was prepared with the AAS L A TEX macros v5.2.
16 –Table 1: Widths of Linear Profiles of Non-Thermal Filaments.Filament 0.3 - 2.0 keV 3.0 - 6.0 keV 6.0 - 10.0 keV1 1.58 +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +3 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +6 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +3 . − . Note. — Error intervals at the 90% confidence level and widths are in arcseconds.
Table 2: Best-Fit Spectral Power Laws.Inside OutsideFilament γ a Norm b χ / dof c γ a Norm b χ / dof c +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . +0 . − . Note. — Error intervals at the 90% confidence level a Power-law index b Power-law normalization, in units of 10 − photons keV − cm − s − c Reduced chi-squared of the fit
17 –Table 3: Key Parameters of the Diffusion-Advection Model.Filament B ( µ G) κ Γ1 72 +24 − +0 . − . +0 . − . +10 − +0 . − . +0 . − . +10 − +0 . − . +0 . − . +15 − +0 . − . +0 . − . +26 − +0 . − . +0 . − . +20 − +0 . − . +0 . − . +40 − +0 . − . +0 . − . +16 − +0 . − . +1 . − . +10 − +0 . − . +0 . − . Note. — Error intervals at the 90% confidence level
Table 4: Equivalent Widths of Si K XIII and S K XV Emission LinesInside OutsideFilament
Si K XIII S K XV Si K XIII S K XV +90 − +120 − +140 − +190 − +168 − +225 − +98 − +157 − +64 − +109 − +100 − +170 − +144 − +228 − +100 − +160 − +94 − − +70 − +130 − +115 − +142 − +150 − +200 − +92 − +150 − +140 − +220 − . +150 − +200 − +290 − +370 − . +148 − +205 − +400 − − Note. — Error intervals at the 90% confidence level
18 –Fig. 1.— Exposure-corrected image of Cas A in the range from 0.3 keV to 10 keV. The selected non-thermal filaments are indicated, along with source and background regions for spectral extraction. 19 –Fig. 2.— Linear intensity profile of Filament 5 in the energy range 0.3 keV-10 keV. The top panelshows a
Chandra image of the filament with the inner and outer regions labeled and the positionof the shock indicated.Fig. 3.— Linear profiles of the filaments near the peak, for three energy bands (from top to bottom):0 . − − −−