EVIDENCE FOR PARTICLE ACCELERATION TO THE KNEE OF THE COSMIC RAY SPECTRUM IN TYCHO'S SUPERNOVA REMNANT

Before investigating the nature of the stripes, we first must locate them within the three-dimensional volume of the remnant. The canonical picture of a young SNR consists of three distinct fluid discontinuities: the blast wave, which marks the shock propagating into the ambient medium, a Rayleigh–Taylor (R–T) unstable contact discontinuity (CD) at the ejecta-interstellar material boundary, and a reverse shock that propagates into the stellar remains. Warren et al. (2005) set an upper limit for the azimuthally dependent projected radius of the reverse shock in Tycho using the location of the Fe Kα emission. Adopting their center of expansion, the western stripes peak at a radius of 220'', well outside the 190'' position of the reverse shock at that azimuth. While the position of the stripes does coincide with the Warren et al. estimate of the CD, the regularly spaced, linear morphology of the non-thermal stripes does not correspond to any features in the R–T plumes of thermal emission tracing the ejecta boundary, nor is the CD a prominent feature elsewhere in the 4–6 keV band. Conversely, the blast wave is a bright source of 4–6 keV emission, and we identify the stripes as projected features of this forward shock.

Tycho's blast wave is traced by a very thin shell of X-ray emission, with a typical thickness only 1%–2% of its radius (Warren et al. 2005). Since the stripes are seen in projection away from the rim, their line-of-sight path length through the shell is small. Thus, their intrinsic emissivity must be high relative to the limb regions. We examined two regions of different brightness on the western rim with minimal ejecta contamination, and assumed a spherical shell model with a radius of 214'', chosen to match the curvature of the western limb. The radial brightness profile of the brighter rim region is too narrow and peaked to be consistent with a simple projected shell geometry, meaning that its high surface brightness likely arises from a local enhancement in emissivity, perhaps similar to the stripes. In contrast, the profile of the fainter region is well fit by a shell with thickness 2% of its radius and is likely more typical of the blast wave as a whole. This simple model predicts that features at 80% of the shock radius (where the stripes are located) would have a surface brightness 20% that of the limb. The observed peak surface brightness of the stripes is, however, five times that of the faint limb region. Thus, the peak intrinsic emissivity of the synchrotron-emitting plasma in the stripes must be a factor of 25 higher than is typical for the blast wave. In principle, this brightening could be due to a local enhancement in the ambient density and/or magnetic field. While we cannot entirely exclude this possibility, we consider the existence of a pre-existing structure around Tycho with the right combination of increased density, magnetic field, and, as we show below, turbulence necessary to produce the observed correlations between the ordered structure and spectral variations improbable. In contrast, models of CR-driven magnetic field amplification produce structure in the precursor density and field, and we find it more likely that the stripes mark a region extensively modified by the acceleration process. This assumption is implicit in the analysis in this Letter.

The spatial position of the stripes, seen in projection against the bright thermal X-rays of the shocked ejecta, greatly complicates spectroscopy. This can be mitigated by restricting analysis to the 4.2–10 keV range, where the emission is dominated by the synchrotron continuum and the Fe Kα line complex at ∼6.45 keV. Throughout this work we set the absorbing column to N_H = 7 × 10²¹ cm^-2, which has negligible effect at these energies. The spectrum of the brightest stripe is well fit with a power law of photon index of Γ = 2.11^+0.08_−0.10, significantly harder than the spectrum of the entire SNR at these energies (Γ = 2.72 ± 0.02; this fit includes faint lines identified by Tamagawa et al. 2009), and a co-addition of several of the lower surface brightness stripes (Γ = 2.52 ± 0.05; this fit includes an Fe K line).

While photon indices are useful for characterization of the non-thermal spectrum, the broadband spectrum gradually departs from a simple power law over several decades in energy. Detailed DSA models of synchrotron spectra show a small degree of curvature through the radio to UV, with a cutoff in the X-rays, set (in young SNRs like Tycho) by the short cooling time of TeV-scale electrons (Ellison et al. 2000). Absent detailed spectral calculations for Tycho, we adopt a common approximation and assume a power-law electron momentum distribution with an exponential cutoff, which produces a photon spectrum that is nearly a power law from the radio through the UV and is curved in X-rays (Reynolds & Keohane 1999). For this model, the low energy index (α_radio) and the flux normalization are set by radio observations, while the location of the high energy cutoff is constrained with the X-ray spectrum. For the spectrum of the entire remnant, assuming a radio spectral index of α_radio = 0.65, a flux density at 1 GHz of ∼50 Jy (Kothes et al. 2006), and again including the faint X-ray emission lines from Tamagawa et al. (2009), the fit to the Chandra data gives ν_cut = 1.9 × 10¹⁷ Hz(hν_cut ∼ 0.79 keV). This provides a remarkably good fit (χ²_ν = 1.05), and is consistent with measurements from Suzaku. (The statistical error on this number is so small as to be irrelevant; the precision to which we measure it is far greater than the efficacy of our emission model or the accuracy of the radio flux and spectral index measurements.) If we instead assume α_radio = 0.61 (Klein et al. 1979), the fit is slightly worse (χ²_ν = 1.10), and the cutoff moves to lower energies (ν_cut = 0.9 × 10¹⁷ Hz ; hν_cut  ∼ 0.37 keV). For the stripes, we estimated the radio flux in these regions from a Very Large Array (VLA) map. The fits favor slightly shallower indices (α_radio ∼ 0.60) and higher cutoff energies, ν_cut = 19⁺¹³₋₉ × 10¹⁷ Hz(7.9^+5.4_−3.7 keV) for the bright stripe and ν_cut = (2.8 ± 0.2) × 10¹⁷ Hz(1.16 ± 0.04 keV) for the ensemble of fainter stripes. We plot the broadband synchrotron spectra of these regions in Figure 3.

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Although it is clear that the bright stripe and faint ensemble spectra require different best-fit parameter values, in principle these differences might be due to either different cutoff frequencies or radio spectral indices. Our spectral analysis (detailed in Table 1) allows us to discriminate between these two options. When the spectra for the bright and faint regions are fit with radio flux densities fixed at their nominal values, both regions prefer an index very close to the average for the entire remnant. Conversely, if we instead require that both regions have the same cutoff, the fitted radio indices are α = 0.565 (bright) and α = 0.623 (faint). We cannot reject this level of variation outright since measurements of the spatial variation of the radio spectral index are not available for Tycho, though in Cas A (a plausible comparison object) the radio index does vary spatially by much as ±0.1 (Wright et al. 1999). On the other hand, the best fit for a single cutoff frequency is clearly worse (Δχ² = 13.8) than for independent cutoff values. An F-test shows that the fit including independent cutoff frequencies is an improvement at a significance level of more than 3σ. Thus, we conclude that the cutoff frequencies are significantly different in the bright and faint spectral regions analyzed.

DSA requires magnetic field inhomogeneities strong enough that particles can scatter multiple times across the shock front. This diffusion of CRs in the magnetized turbulence is generally assumed to be close to the Bohm regime, where the mean free path is the particle gyroradius. Departure from this limit can be parameterized as k₀ = D₀/D_0,Bohm, where D₀ is the diffusion coefficient at the electron cutoff energy. Parizot et al. (2006) show that for an electron distribution limited by synchrotron losses, the photon cutoff energy (E_γ, cut = hν_cut) is set by the diffusion coefficient, shock velocity (V₃, in units of 1000 km s⁻¹), and compression ratio, r:

$$\begin{equation} k_0 = 0.14\; E^{-1}_{\gamma,{\rm cut,keV}}\; V^{2}_{3,{\rm shock}}\; \frac{16\,(r-1)}{3\,r^{2}}. \end{equation} \tag{ 1 }$$

At a distance of 4.0 ± 1.0 kpc (Hayato et al. 2010), with a blast wave proper motion of 03 yr^-1 (Katsuda et al. 2010), v_s = 5700 km s^-1. For an unmodified shock (r = 4) and our fit cutoff energies, k₀= 5.8 (whole SNR), 3.9 ± 0.1 (faint stripes), and 0.6^+0.4_−0.3 (bright stripe). With a higher compression more applicable to Tycho (r = 7), these numbers become 3.8, 2.6 ± 0.1, and 0.4^+0.3_−0.2. (The error bars reflect only the statistical error in ν_cut, and not the ∼25% uncertainty in the distance to Tycho.) Our value for the whole SNR is near to those found by Parizot et al. (2006) for Tycho (k₀ = 4.9–10), while the very small numbers for the brightest stripe are similar to their results for G347.3-0.5 and SN 1006 (k₀ = 0.2–0.9). Since k₀>1 for isotropic turbulence, this may be due to our oversimplified emission model, and could indicate that the photon spectrum falls more gradually to high energies, as seen in some DSA calculations (Ellison et al. 2010), or may be indicative of anisotropic diffusion (Reville et al. 2008). Regardless, our measured photon cutoff energies require an order of magnitude decrease in the diffusion coefficient in the stripes relative to the average value in Tycho, and indicate that diffusion in the stripes is very near the Bohm limit. Furthermore, since Bohm diffusion requires δB/B ∼ 1, k₀ ∼ 1 implies fully developed turbulence, at least on the scale of the gyroradius of the highest energy (∼TeV) electrons in the brightest stripes.

Might the spatial structure and enhanced turbulence in the stripes be a manifestation of the plasma instability driving the CR acceleration? The largest possible characteristic scale of the acceleration process is set by the gyroradius of the highest energy particles present. If we identify the gaps between the stripes (l_gap ∼ 8'') with twice the proton gyroradius, an estimate of the magnetic field yields a measurement of the energy, given by Equation (2):

$$\begin{equation} E_{{\rm CR}} = 9\; \Bigl (\frac{l_{\rm gap}}{1 \mbox{$^{\prime \prime }$}} \Bigr) \Bigl (\frac{D}{4.0\; {\rm kpc}} \Bigr) \Bigl (\frac{B}{{\rm \mu G}} \Bigr) \times 10^{12}\; {\rm eV}. \end{equation} \tag{ 2 }$$

The choice of B is somewhat uncertain. If the stripes are a consequence of the highest energy CRs interacting with the non-amplified ISM field (B ∼ 3 μG), for a gap spacing of 8'', E_CR = 2 × 10¹⁴ eV. However, these features may arise from a region where the field has already been somewhat amplified. For a shock velocity of 5700 km s^-1, fits to models of Tycho's synchrotron emission (Cassam-Chenaï et al. 2007) predict an upstream field strength of B ∼ 30 μG, indicating E_CR = 2 × 10¹⁵ eV—just at the knee of the Galactic CR spectrum.