ENERGY DEPENDENCE OF SYNCHROTRON X-RAY RIMS IN TYCHO'S SUPERNOVA REMNANT

The energy and space distribution of electrons at a SNR's forward shock controls the synchrotron rims we see in X-ray and radio. We assume that DSA generates a power law distribution of electrons with an exponential cut-off at the forward shock and model 1D steady-state plane advection and diffusion of the electron distribution $f(E,x),$ where E is electron energy and x is distance downstream of the forward shock:

$$\begin{eqnarray}&&{v}_{{\rm{d}}}\displaystyle \frac{\partial f}{\partial x}-\displaystyle \frac{\partial }{\partial x}\left(D\displaystyle \frac{\partial f}{\partial x}\right)-\displaystyle \frac{\partial }{\partial E}\left({{bB}}^{2}{E}^{2}f\right)={K}_{0}{E}^{-s}{e}^{-E/{E}_{\mathrm{cut}}}\delta (x),\end{eqnarray} \tag{ 1 }$$

following Berezhko & Völk (2004), Cassam-Chenaï et al. (2007), Morlino et al. (2010), and Rettig & Pohl (2012). The forward shock is located at x = 0 with x > 0 increasing downstream of the shock; D is the diffusion coefficient, v_d is fluid velocity downstream of the shock, and the constant $b\equiv 4{e}^{4}/9{m}_{e}^{4}{c}^{7}=1.57\times {10}^{-3}$ in appropriate CGS units arises from synchrotron power loss (i.e., $\partial E/\partial t=-{{bB}}^{2}{E}^{2},$ averaged over pitch angles). The initial electron distribution is specified by an arbitrary normalization K₀, DSA cut-off energy ${E}_{\mathrm{cut}}$ (given in Section 2.3), and spectral index s = 2α + 1. Zirakashvili & Aharonian (2007) derive an electron energy spectrum with super-exponential cut-off ${e}^{-{(p/{p}_{\mathrm{cut}})}^{2}},$ but we use a simple exponential cut-off for simplicity and consistency with R14. We have not yet specified the cause of the injected spectrum cut-off (see Section 2.4), but the functional form of a power law with exponential cut-off is a good approximation to predictions for DSA spectra limited by synchrotron losses, remnant age, or particle escape (Webb et al. 1984; Reynolds 1998, 2008). All constants and equations are given in CGS (Gaussian) units. Our presentation is somewhat abbreviated, but a fuller exposition and literature review are given by R14.

For Tycho, we adopted radio spectral index α = 0.58 (Sun et al. 2011) and hence electron spectral index s = 2α + 1 = 2.16. We assumed a remnant distance 3 kpc (but see Hayato et al. 2010), which gives a shock radius of 1.08 × 10¹⁹ cm from the observed angular radius 240'' (Green 2014) and further sets shock velocities for each rim profile we consider. Tycho's forward shock velocity varies with azimuth by up to a factor of 2 (Katsuda et al. 2010); we linearly interpolated velocities reported by Williams et al. (2013; rescaled to 3 kpc) to estimate individual shock velocities for each region. We assume a compression ratio of 4 as for a strong shock (unmodified by cosmic-ray pressure), and take downstream velocities v_d to be one-fourth the interpolated shock velocities.

We assume isotropic diffusion and only consider particle transport downstream of the forward shock. The velocity is assumed constant, as is the magnetic field for loss-limited model rims, in contrast to the expected Sedov–Taylor similarity solution for velocity and density in an adiabatic blastwave. Our assumptions of constant velocity, magnetic field, and plane flow should be reasonable as we generally consider synchrotron emission within 10% of the shock radius, r_s, from the forward shock, though a few models are followed far enough inward that this approximation begins to break down. We consider profile emission strictly upstream of both the contact discontinuity and reverse shock (see Warren et al. 2005), and avoid regions where the contact discontinuity overruns the forward shock. Our modeling neglects flow and electron spectrum modification due to the nearby contact discontinuity and particle acceleration at the reverse shock. These effects could matter (and we do not quantify their importance), but we expect that at Tycho's age, X-ray emission is dominated by forward shock transport. Given the uncertainty in shock morphology, our model seeks only to capture the most relevant physics. More sophisticated work may treat, e.g., sphericity, shock precursors, anisotropic diffusion, and injection/acceleration efficiency (e.g., Reville & Bell 2013; Bykov et al. 2014; Ferrand et al. 2014 and references therein).

To determine rim profiles and widths, we compute the electron distribution using Green's function solutions by Lerche & Schlickeiser (1980) and Rettig & Pohl (2012), with the caveat that D(x) B²(x) is assumed constant; we discuss this assumption further below. The solutions are fully described in R14 using notation similar to ours. The electron distribution may be integrated over the one-particle synchrotron emissivity G(y) to obtain the "total" emissivity:

$$\begin{eqnarray}&&{j}_{\nu }(x)\propto B(x){\displaystyle \int }_{0}^{\infty }G(y)f(E,x){dE}\end{eqnarray} \tag{ 2 }$$

where y ≡ ν/(c₁E²B) is a scaled synchrotron emission ν frequency and $G(y)=y{\displaystyle \int }_{y}^{\infty }{K}_{5/3}(z){dz}$ with ${K}_{5/3}(z)$ a modified Bessel function of the second kind (Pacholczyk 1970); the constant c₁ = 6.27 × 10¹⁸ in CGS units. Integrating emissivity over lines of sight for a spherical remnant yields intensity as a function of radial coordinate r:

$$\begin{eqnarray}&&{I}_{\nu }(r)=2{\displaystyle \int }_{0}^{\sqrt{{r}_{{\rm{s}}}^{2}-{r}^{2}}}{j}_{\nu }\left({r}_{{\rm{s}}}-\sqrt{{s}^{2}+{r}^{2}}\right){ds}\end{eqnarray} \tag{ 3 }$$

where s is the line of sight coordinate and r_s is shock radius. We take the full width at half maximum (FWHM) of the resulting intensity profile as our metric for modeled rim widths. Using FWHM as opposed to, e.g., full width at three-quarters maximum, excludes measured rims and model parameters where X-ray intensity does not drop to half maximum immediately behind the rim. Our results thus focus on the most well-defined rims in Tycho rather than the global shock structure.

We consider two scenarios for post-shock magnetic field: (1) a constant field B(x) = B₀ corresponding to loss-limited rims, and (2) an exponentially damped field of form:

$$\begin{eqnarray}&&B(x)=\left({B}_{0}-{B}_{\mathrm{min}}\right)\mathrm{exp}\left(-x/{a}_{b}\right)+{B}_{\mathrm{min}}\end{eqnarray} \tag{ 4 }$$

following Pohl et al. (2005). Here B₀ is the magnetic field immediately downstream of the shock, i.e., B₀ = B(x = 0), and a_b is an e-folding damping lengthscale; our use of B₀ for downstream magnetic field departs from typical notation. A typical lengthscale for a_b is 10¹⁶–10¹⁷ cm (Pohl et al. 2005), corresponding to ∼0.1%–1% of Tycho's radius. Hereafter, we report a_b in units of shock radius r_s unless otherwise stated.

The distinction between damped and loss-limited rims is somewhat arbitrary; as ${a}_{b}\to \infty ,$ model results converge to loss-limited rims. Moreover, rims much thinner than the damping length are effectively loss-limited as electrons radiate in a nearly constant magnetic field. Furthermore, the dependence on observing frequency of electron losses and diffusion means that a model may be damping limited at one frequency and loss-limited at another. In the following analysis, we deem all fits with finite a_b to be damped, but we compare rim widths and damping lengths from such fits further below to better distinguish damped and loss-limited rim behavior, set by a combination of a_b, B₀, and other model parameters.

Most previous work has assumed Bohm-like diffusion in plasma downstream of SNR shocks. Bohm diffusion equates the particle mean free path λ is equal to the gyroradius ${r}_{{\rm{g}}}=E/({eB}),$ yielding diffusion coefficient ${D}_{{\rm{B}}}=\lambda c/3={cE}/(3{eB});$ here c is the speed of light, e is the elementary charge, and B is magnetic field. Bohm-like diffusion encapsulates diffusion scalings of $D\propto E,$ introducing a free prefactor η such that λ = ηr_g allows for varying diffusion strength. However, Bohm diffusion at η = 1 is commonly considered a lower limit on the diffusion coefficient at all energies.

We consider a generalized diffusion coefficient with arbitrary power law dependence upon energy following, e.g., Parizot et al. (2006):

$$\begin{eqnarray}&&D(E)=\displaystyle \frac{\eta {{cE}}^{\mu }}{3{eB}}={\eta }_{h}{D}_{{\rm{B}}}\left({E}_{h}\right){\left(\displaystyle \frac{E}{{E}_{h}}\right)}^{\mu }\end{eqnarray} \tag{ 5 }$$

where μ parameterizes diffusion-energy scaling and η now has units of ${\mathrm{erg}}^{1-\mu }.$ The right-hand side of Equation (5) introduces η_h, a dimensionless diffusion coefficient scaled to the Bohm value at a fiducial particle energy E_h. Note that η_h and η are related as $\eta ={\eta }_{h}{({E}_{h})}^{1-\mu },$ and η = η_h for Bohm-like diffusion (μ = 1). For subsequent analysis, we take fiducial electron energy E₂ = E_h corresponding to a 2 keV synchrotron photon and report results in terms of η₂ = η_h. Although E₂ varies with magnetic field as ${E}_{2}\propto {B}^{-1/2}$ and thus η₂ may vary around Tycho's shock for $\mu \ne 1,$ tying η_h to a fixed observation energy gives a convenient sense of diffusion strength regardless of the underlying electron energies.

The solutions to Equation (1) given by Lerche & Schlickeiser (1980) assume $D(x){B}^{2}(x)$ constant to render Equation (1) semi-analytically tractable. Although this assumption has no obvious physical basis, it contains the qualitatively correct behavior D constant if B is constant and D smaller for larger B. We enforce it by modifying the diffusion coefficient in the damping model as, following Rettig & Pohl (2012):

$$\begin{eqnarray}&&D(E,x)=\displaystyle \frac{\eta {{cE}}^{\mu }}{3{{eB}}_{0}}{\left[\displaystyle \frac{{B}_{\mathrm{min}}}{{B}_{0}}+\displaystyle \frac{{B}_{0}-{B}_{\mathrm{min}}}{{B}_{0}}{e}^{-x/{a}_{b}}\right]}^{-2}.\end{eqnarray} \tag{ 6 }$$

This strengthens the spatial-dependence of the diffusion coefficient as compared to the expected $D(x)\propto 1/B(x).$

We assume that the DSA process is limited by synchrotron losses at high energies and hence determine the ${E}_{\mathrm{cut}}$ by equating synchrotron loss and diffusive acceleration timescales. Here we ignore the possibility of age-limited or escape-limited acceleration (e.g., Reynolds 1998). In the former case, low magnetic-field strengths could mean that Tycho's age is less than a synchrotron loss time; the maximum energy is obtained by equating the remnant age and acceleration timescale. In the latter case, the diffusion coefficient upstream may increase substantially above some electron energy due to an absence of appropriate MHD waves. However, the magnetic field strengths we find below justify the assumption of loss-limited acceleration. For low energies and small synchrotron losses (cooling time longer than acceleration time), electrons are efficiently accelerated; near or above the cut-off energy, electrons will radiate or escape too rapidly to be accelerated to higher energies and the energy spectrum drops off steeply. The cut-off energy is given as:

$$\begin{eqnarray}{E}_{\mathrm{cut}} & = & {\left(8.3\;\mathrm{TeV}\right)}^{2/(1+\mu )}{\left(\displaystyle \frac{{B}_{0}}{100\;\mu {\rm{G}}}\right)}^{-1/(1+\mu )}\\ & & \times {\left(\displaystyle \frac{{v}_{{\rm{s}}}}{{10}^{8}\;\mathrm{cm}\;{{\rm{s}}}^{-1}}\right)}^{2/(1+\mu )}{\eta }^{-1/(1+\mu )}.\end{eqnarray} \tag{ 7 }$$

This result is derived by Parizot et al. (2006) for μ = 1 assuming a strong shock with compression ratio 4 and isotropic magnetic turbulence both upstream and downstream of the shock.

The cut-off energy and accelerated electron spectrum depend on the magnetic field, which varies in a damped model. Marcowith & Casse (2010) show that various turbulent damping mechanisms can modify the accelerated spectrum, as particles traveling downstream may not be effectively reflected back across the shock and further accelerated. Nevertheless, we make the simplifying assumption that particle acceleration is controlled by diffusion at the shock and neglect spatially varying diffusion and magnetic fields in the acceleration process; Equation (7) stands as evaluated with shock magnetic field strength B₀. Cut-off energies of ∼1–10 TeV can be plausibly achieved in the presence of damping due to Alfvén and magneto-sonic cascades (Marcowith & Casse 2010).

As the DSA imposed electron cut-off results in a cut-off of SNR synchrotron flux, the synchrotron cut-off frequency ${\nu }_{\mathrm{cut}}={c}_{m}{E}_{\mathrm{cut}}^{2}B$ with c_m = 1.82 × 10¹⁸ in cgs units (Pacholczyk 1970), which is the peak frequency emitted by electrons of energy ${E}_{\mathrm{cut}},$ provides an independent observable to estimate shock diffusion and is given by:

$$\begin{eqnarray}{\nu }_{\mathrm{cut}} & = & {c}_{m}{\left(13.3\;\mathrm{erg}\right)}^{\displaystyle \frac{4}{1+\mu }}(100\;\mu {\rm{G}}){\left(2657\;{\mathrm{erg}}^{2}\right)}^{-\displaystyle \frac{1-\mu }{1+\mu }}\\ & & \times {\left(\displaystyle \frac{{v}_{{\rm{s}}}}{{10}^{8}\;\mathrm{cm}\;{{\rm{s}}}^{-1}}\right)}^{\displaystyle \frac{4}{1+\mu }}{\left({\eta }_{2}\right)}^{-\displaystyle \frac{2}{1+\mu }}.\end{eqnarray} \tag{ 8 }$$

The cut-off frequency is independent of magnetic field B for all values of μ (but recall that the electron energies associated with η₂ will depend on B for $\mu \ne 1$). Parizot et al. (2006) previously used measurements of synchrotron cut-offs to estimate diffusion coefficients in Tycho and other historical SNRs.

We point out that the electron spectrum softens downstream of the shock due to synchrotron losses, but the local spectrum at any radial position will be a steeply cut-off power law (Webb et al. 1984; Reynolds 1998). No steepening from ${E}^{-s}$ to ${E}^{-(s+1)}$ is observed because the cut-off limits the electron spectrum at high energies. A homogeneous source of age t in which electrons are continuously accelerated throughout, with an initial straight power law distribution to infinite energy, will produce a steepened power law distribution above the energy at which the synchrotron loss time equals the acceleration time (Kardashev 1962). If we model emission without an initial exponential cut-off (i.e., inject a straight power law) in a constant magnetic field, then integrated spectra in our model would steepen by about one power. But, those assumptions do not apply to the current situation of continuous advection of electrons, with an energy distribution which is already a cut-off power law, through a region of non-constant magnetic field. See Reynolds (2009) for a fuller discussion of synchrotron losses in non-homogeneous sources.

X-ray rim widths are controlled by synchrotron losses, particle transport, and magnetic fields immediately downstream of the shock, each of which influences rim width-energy scaling differently. If diffusion is negligible and the downstream magnetic field is constant, loss-limited rims narrow with increasing energy as more energetic electrons radiate and cool more quickly. But diffusion will dilute this effect: more energetic electrons may diffuse further upstream or downstream than would be expected from pure advection, smearing out rims and weakening energy dependence at higher energies. Magnetic fields damped on a length scale comparable to filament widths should also weaken rim width-energy dependence—if the magnetic field turns off, synchrotron radiation turns off regardless of electron energy. Additionally, once B₀ varies, one observes electrons of different energy at different distances behind the shock.

Figure 2 plots model X-ray and radio profiles for a range of B₀ and a_b values to illustrate how damping and magnetic field strength impact width-energy dependence, which we will now explore. We discuss and incorporate model radio profiles into our analysis in Section 6.

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2.5.1. Undamped Models

Following R14, we parameterize rim width-energy dependence in terms of a scaling exponent ${m}_{{\rm{E}}}$ defined as:

$$\begin{eqnarray}&&w(\nu )\propto {\nu }^{{m}_{{\rm{E}}}}\end{eqnarray} \tag{ 9 }$$

where w(ν) is filament FWHM as a function of observed photon frequency ν, and the exponent ${m}_{{\rm{E}}}={m}_{{\rm{E}}}(\nu )$ is frequency dependent. We may refer to observed photons interchangeably by energy or frequency ν, but E is reserved for electron energy.

To better intuit the effects of advection and diffusion on rim widths, we introduce advective and diffusive lengthscales for bulk electron transport. These depend on electron energy; we write them in terms of the peak frequency radiated by electrons of energy E, $\nu ={c}_{m}{E}^{2}B$:

$$\begin{eqnarray}&&{l}_{\mathrm{ad}}={v}_{{\rm{d}}}{\tau }_{\mathrm{synch}}\propto {v}_{{\rm{d}}}{B}_{0}^{-3/2}{\nu }^{-1/2}\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{l}_{\mathrm{diff}}=\sqrt{D{\tau }_{\mathrm{synch}}}\propto {\eta }^{1/2}{B}_{0}^{-(\mu +5)/4}{\nu }^{(\mu -1)/4}.\end{eqnarray} \tag{ 11 }$$

The characteristic time is the synchrotron cooling time ${\tau }_{\mathrm{synch}}=1/({{bB}}^{2}E)$ with b = 1.57 × 10⁻³. For μ = 1, ${l}_{\mathrm{diff}}$ is independent of ν and both ${l}_{\mathrm{diff}}$ and ${l}_{\mathrm{ad}}$ scale as ${B}_{0}^{-3/2}.$ If both diffusion and magnetic field damping are negligible and electrons are only loss-limited as they advect downstream, ${m}_{{\rm{E}}}$ attains a minimum value ${m}_{{\rm{E}}}=-1/2$ as rim widths are set by ${l}_{\mathrm{ad}}\gt {l}_{\mathrm{diff}}.$ At higher energies where ${l}_{\mathrm{diff}}\gt {l}_{\mathrm{ad}},$ diffusion increases ${m}_{{\rm{E}}}$ from $-1/2$ to a value between $-1/4$ and 1/4 for μ = 0 and 2, respectively (R14, Figure 3). The presence of an electron energy cut-off decreases ${m}_{{\rm{E}}}$ slightly in all cases due to the decreased number of electrons and hence thinner rims at higher energies (R14, Figure 5), but the qualitative behavior is the same.

2.5.2. Field Damping Effects

We measured synchrotron rim FWHMs from an archival Chandra ACIS-I observation of Tycho (R.A.: 00^h25^m190, decl.: +64°08'100; J2000) between 2009 April 11 and May 3 (PI: J. Hughes; ObsIDs: 10093–10097, 10902–10904, 10906); Eriksen et al. (2011) present additional observation information. The total good exposure time was $734\;\mathrm{ks}.$ Level 1 Chandra data were reprocessed with CIAO 4.6 and CALDB 4.6.1.1 and kept unbinned with ACIS spatial resolution 0492. Merged and corrected events were divided into five energy bands: 0.7–1, 1–1.7, 2–3, 3–4.5, and 4.5–7 keV. We excluded the 1.7–2 keV energy range to avoid Si xiii (Heα) emission prevalent in the remnant's thermal ejecta which might contaminate our nonthermal profile measurements.

We selected 20 regions for profile extraction around Tycho's shock (Figure 1) based on the following criteria: (1) filaments should be clear of spatial plumes of thermal ejecta in Chandra images, which rules out, e.g., areas of strong thermal emission on Tycho's eastern limb; (2) filaments should be singular and localized, so multiple filaments should either not overlap or completely overlap; (3) filament peaks should be evident above the background signal or downstream thermal emission (rules out faint southern filaments). We accepted several regions with poor quality peaks in the lowest energy band (0.7–1 keV) so long as peaks in all higher energy bands were clear and well-fit. We grouped regions into five filaments by visual inspection of the remnant. Within each filament, we chose region widths to obtain comparable counts at the thin rim peak. All measured rim widths are at least 1''. The narrowest rim widths may be slightly over-estimated due to the Chandra point-spread function (PSF) at ∼4' off the optical axis, which has FWHM $\leqslant \;1\buildrel{\prime\prime}\over{.} 4$ at 6.4 keV and $\leqslant 1^{\prime\prime}$ at 1.5 keV. For simplicity, we neglect PSF effects in our analysis. Our observations of rim width-energy dependence are thus somewhat conservative at the highest energy.

We extracted spectra at and immediately behind thin rims in each region ("rim," "downstream" spectra, respectively) to confirm that rim width measurements are not contaminated by thermal line emission. The two extraction regions are determined by our empirical fits of rim profile shape (Section 3.3). The rim section is the smallest sub-region containing the measured FWHM bounds from all energy bands. The downstream section extends from the interior thin rim FWHM to the intensity minimum behind the rim (specifically, the downstream profile fit domain bound described in Section 3.3). To illustrate our selections, Figure 3 plots example rim profiles (4.5–7 keV) with the downstream and rim sections highlighted.

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Spectra were binned to a minimum of 15 cts/bin. We extracted background spectra from circular regions (radius ∼30'') around the remnant's exterior; for each region's rim and downstream spectra, we subtracted the closest background region's spectrum.

We fit each region's rim and downstream spectra to an absorbed power law model (XSPEC 12.8.1, phabs∗powerlaw) between 0.5 and 7 keV with photon index Γ, hydrogen column density ${N}_{{\rm{H}}},$ and a normalization as free parameters. Table 1 lists best fit parameters and reduced χ² values for all regions. Rim spectra are well-fit by the power law model alone; the best fit photon indices (2.4–3) and column densities (0.6–0.8 × ${10}^{22}\;{\mathrm{cm}}^{-2}$) are consistent with previous spectral fits to Tycho's nonthermal rims (Hwang et al. 2002; Cassam-Chenaï et al. 2007).

Table 1.  Spectrum Fit Parameters

	Downstream, Power-Law			Rim, Power-Law			Rim, srcut
Region	${N}_{{\rm{H}}}$	Γ	${\chi }_{\mathrm{red}}^{2}$ (dof)	${N}_{{\rm{H}}}$	Γ	${\chi }_{\mathrm{red}}^{2}$ (dof)	${N}_{{\rm{H}}}$	${\nu }_{\mathrm{cut}}$	${\chi }_{\mathrm{red}}^{2}$ (dof)
	(${10}^{22}\;{\mathrm{cm}}^{-2}$)	(−)		(${10}^{22}\;{\mathrm{cm}}^{-2}$)	(−)		(${10}^{22}\;{\mathrm{cm}}^{-2}$)	(keV h−1)
1	0.53	2.72	2.25 (186)	0.72	2.84	1.20 (284)	0.62	0.30	1.17 (284)
2	0.68	2.99	5.07 (178)	0.69	2.83	1.10 (202)	0.60	0.30	1.08 (202)
3	0.67	2.96	1.97 (186)	0.77	2.80	1.15 (167)	0.68	0.33	1.14 (167)
4	0.59	2.97	1.43 (163)	0.70	2.84	1.21 (278)	0.61	0.29	1.15 (278)
5	0.62	2.93	4.76 (265)	0.73	2.88	1.18 (255)	0.64	0.27	1.11 (255)
6	0.68	3.00	2.12 (200)	0.74	2.85	0.96 (231)	0.64	0.29	0.93 (231)
7	0.65	3.02	1.00 (142)	0.82	2.97	1.14 (224)	0.71	0.23	1.14 (224)
8	0.74	2.93	1.38 (170)	0.75	2.71	0.98 (198)	0.66	0.41	0.96 (198)
9	0.78	3.03	1.12 (157)	0.82	2.83	0.90 (175)	0.73	0.30	0.88 (175)
10	0.62	2.86	1.40 (220)	0.77	2.76	0.98 (164)	0.68	0.36	0.96 (164)
11	0.67	2.94	2.56 (137)	0.69	2.60	1.10 (153)	0.61	0.55	1.07 (153)
12	0.61	2.79	2.65 (137)	0.64	2.44	0.90 (172)	0.57	0.88	0.90 (172)
13	0.61	2.98	3.12 (198)	0.67	2.73	1.12 (235)	0.59	0.38	1.09 (235)
14	0.46	2.93	1.37 (148)	0.63	2.93	0.96 (167)	0.54	0.23	0.95 (167)
15	0.43	2.92	1.33 (150)	0.65	2.84	1.05 (183)	0.57	0.29	1.03 (183)
16	0.48	2.94	2.04 (189)	0.67	2.80	1.13 (182)	0.58	0.32	1.12 (182)
17	0.48	2.86	1.70 (188)	0.68	2.83	0.97 (187)	0.59	0.30	0.94 (187)
18	0.44	2.87	1.91 (200)	0.64	3.02	1.20 (220)	0.55	0.19	1.13 (220)
19	0.40	2.84	1.31 (133)	0.66	2.78	1.01 (157)	0.57	0.34	0.99 (157)
20	0.40	2.75	3.01 (140)	0.63	2.81	1.11 (192)	0.55	0.31	1.07 (192)

Note. Absorbed power law fit parameters are photon index Γ and hydrogen column density ${N}_{{\rm{H}}}.$ srcut fits performed in log-frequency space; h is Planck's constant and ${\nu }_{\mathrm{cut}}$ a cut-off frequency. Horizontal rules group individual regions into filaments.

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Downstream spectra are poorly fit by the absorbed power law model due to thermal contamination from Si xiii and S xv Heα line emission at 1.85 and 2.45 keV. To confirm that thermal emission is dominated by these two lines near the shock, we also performed fits with (1) both lines excised (1.7–2.0 keV, 2.3–2.6 keV counts removed) and (2) with both lines fitted to Gaussian profiles. Fits with lines excised yield ${\chi }_{\mathrm{red}}^{2}$ values between 1–5. Fits with lines fitted to Gaussian profiles yield ${\chi }_{\mathrm{red}}^{2}$ values 0.83–1.6. In both fits (lines excised or modeled), we find somewhat smaller best fit column densities (0.3–0.8 × ${10}^{22}\;{\mathrm{cm}}^{-1}$) but similar best fit photon indices (2.6–3.1), compared to those of the rim spectra. The consistent photon indices indicate that the same synchrotron continuum is present beneath thermal line emission.

We also fitted rim spectra to the absorbed XSPEC model srcut, modified to fit in log-frequency space. srcut models a power law X-ray synchrotron spectrum set by a radio spectral index α with an exponential cut-off parameterized by cut-off frequency ${\nu }_{\mathrm{cut}}$ (Reynolds 1998; Reynolds & Keohane 1999). The fit values of ${\nu }_{\mathrm{cut}},$ in particular, permit an independent estimate of η₂ from Equation (8). The radio spectral index is fixed to α = 0.58 (Sun et al. 2011) as done in our transport modeling, and we fit for absorption column density ${N}_{{\rm{H}}}$ and cut-off frequency ${\nu }_{\mathrm{cut}}$ in each region. Table 1 lists best spectrum fit parameters for each region. The fitted cut-off frequency is typically $0.3\;\mathrm{keV}\;{h}^{-1},$ consistent with fits by Hwang et al. (2002), but Regions 11, 12 have unusually high cut-off frequencies 0.55, $0.88\;\mathrm{keV}\;{h}^{-1}$ consistent with harder rim spectra.

Our spectral fitting confirms that all selected region are practically free of thermal line emission, as already suggested by visual inspection (Figure 1). Excluding 1.7–2 keV photons in rim width measurements further limits thermal contamination as 1.85 keV Si line emission is over a third of Tycho's thermal flux as detected by Chandra (Hwang et al. 2002).

We obtained radial intensity profiles in five energy bands from ∼10''–20'' behind the shock to ∼5''–10'' in front for each region. To increase signal-to-noise, we integrate along the shock (5''–23'') in each region. Plotted and fitted profiles are reported in vignetting- and exposure-corrected intensity units; error bars were computed from raw counts assuming Poisson statistics. Intensity profiles peak sharply within ∼2''–3'' behind the shock, demarcating the thin rims, then fall off gradually until thermal emission picks up further behind the shock.

We fitted rim profiles to a piecewise two-exponential model:

$$\begin{eqnarray}h(r)=\left\{\begin{array}{ll}{A}_{u}\mathrm{exp}\left(\displaystyle \frac{{r}_{0}-r}{{w}_{u}}\right)+{C}_{u}, & r\geqslant {r}_{0}\\ {A}_{d}\mathrm{exp}\left(\displaystyle \frac{r-{r}_{0}}{{w}_{d}}\right)+{C}_{d}, & r\lt {r}_{0}\end{array}\right.\end{eqnarray} \tag{ 12 }$$

where h(r) is profile height and r is radial distance from remnant center. The rim model has six free parameters: ${A}_{u},{r}_{0},{w}_{u},{w}_{d},{C}_{u},$ and C_d; ${A}_{d}={A}_{u}+({C}_{u}-{C}_{d})$ enforces continuity at r = r₀. Our model is similar to that of Bamba et al. (2003, 2005) and differs slightly from that of R14. To fit only the nonthermal rim in each intensity profile, we selected the fit domain for each profile as follows. The downstream bound was set at the first local data minimum downstream of the rim peak, identified by smoothing the profiles with a 21-point (∼10'') Hanning window. The upstream bound was set at the profile's outer edge. Figure 4 illustrates the fit domain selections in each band for two example regions.

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From the fitted profiles we extracted a FWHM for each region and each energy band after subtracting a constant background term $\mathrm{min}({C}_{u},{C}_{d}).$ We could not resolve a FWHM in 8 of 20 regions at 0.7–1 keV (Table 2); in these regions, either the downstream FWHM bound would extend outside the fit domain or we could not find an acceptable fit to Equation (12). We were able to resolve FWHMs for all regions at higher energy bands (1–7 keV).

Table 2.  Measured Full Widths at Half Maximum (FWHMs)

	FWHM (arcsec)					${m}_{{\rm{E}}}$ (−)
Region	Band 1	Band 2	Band 3	Band 4	Band 5	Bands 1–2	Bands 2–3	Bands 3–4	Bands 4–5
	(0.7–1 keV)	(1–1.7 keV)	(2–3 keV)	(3–4.5 keV)	(4.5–7 keV)	(1 keV)	(2 keV)	(3 keV)	(4.5 keV)
1	⋯	${8.80}_{-0.15}^{+0.18}$	${6.34}_{-0.21}^{+0.26}$	${7.40}_{-0.23}^{+0.30}$	${5.57}_{-0.42}^{+0.47}$	⋯	−0.47 ± 0.06	0.38 ± 0.13	−0.70 ± 0.22
2	⋯	${4.22}_{-0.09}^{+0.12}$	${2.36}_{-0.09}^{+0.12}$	${3.00}_{-0.12}^{+0.16}$	${4.11}_{-0.30}^{+0.34}$	⋯	−0.84 ± 0.08	0.59 ± 0.16	0.77 ± 0.23
3	⋯	${2.47}_{-0.07}^{+0.08}$	${1.78}_{-0.07}^{+0.09}$	${2.10}_{-0.11}^{+0.11}$	${1.32}_{-0.09}^{+0.10}$	⋯	−0.47 ± 0.08	0.41 ± 0.17	−1.15 ± 0.22
4	${5.85}_{-0.33}^{+0.37}$	${4.35}_{-0.08}^{+0.09}$	${3.26}_{-0.09}^{+0.11}$	${3.69}_{-0.11}^{+0.12}$	${3.20}_{-0.18}^{+0.21}$	−0.83 ± 0.18	−0.41 ± 0.05	0.31 ± 0.11	−0.35 ± 0.17
5	⋯	${4.52}_{-0.12}^{+0.11}$	${3.06}_{-0.11}^{+0.11}$	${3.25}_{-0.13}^{+0.15}$	${3.04}_{-0.18}^{+0.21}$	⋯	−0.56 ± 0.06	0.15 ± 0.14	−0.17 ± 0.19
6	${2.48}_{-0.18}^{+0.18}$	${2.32}_{-0.06}^{+0.05}$	${2.98}_{-0.09}^{+0.11}$	${2.05}_{-0.09}^{+0.08}$	${2.21}_{-0.14}^{+0.15}$	−0.19 ± 0.21	0.36 ± 0.06	−0.92 ± 0.13	0.18 ± 0.19
7	${2.69}_{-0.17}^{+0.20}$	${2.33}_{-0.05}^{+0.05}$	${2.31}_{-0.08}^{+0.08}$	${1.81}_{-0.07}^{+0.09}$	${1.83}_{-0.08}^{+0.11}$	−0.39 ± 0.20	−0.01 ± 0.06	−0.60 ± 0.14	0.02 ± 0.17
8	${2.33}_{-0.20}^{+0.21}$	${2.72}_{-0.08}^{+0.08}$	${2.38}_{-0.09}^{+0.10}$	${2.10}_{-0.09}^{+0.10}$	${2.37}_{-0.17}^{+0.20}$	0.43 ± 0.26	−0.19 ± 0.07	−0.30 ± 0.15	0.29 ± 0.22
9	${2.16}_{-0.23}^{+0.24}$	${2.35}_{-0.06}^{+0.07}$	${2.47}_{-0.11}^{+0.11}$	${1.91}_{-0.09}^{+0.09}$	${2.20}_{-0.16}^{+0.17}$	0.24 ± 0.31	0.07 ± 0.07	−0.63 ± 0.16	0.34 ± 0.22
10	${2.38}_{-0.23}^{+0.24}$	${1.99}_{-0.06}^{+0.07}$	${1.76}_{-0.08}^{+0.09}$	${1.59}_{-0.08}^{+0.09}$	${1.58}_{-0.12}^{+0.13}$	−0.50 ± 0.29	−0.18 ± 0.08	−0.24 ± 0.18	−0.02 ± 0.23
11	⋯	${3.23}_{-0.13}^{+0.15}$	${2.52}_{-0.13}^{+0.16}$	${1.90}_{-0.13}^{+0.14}$	${3.09}_{-0.38}^{+0.45}$	⋯	−0.36 ± 0.10	−0.70 ± 0.22	1.21 ± 0.37
12	⋯	${3.86}_{-0.16}^{+0.17}$	${2.61}_{-0.13}^{+0.15}$	${3.02}_{-0.21}^{+0.22}$	${2.23}_{-0.17}^{+0.21}$	⋯	−0.56 ± 0.10	0.36 ± 0.22	−0.74 ± 0.27
13	${2.85}_{-0.17}^{+0.22}$	${2.43}_{-0.05}^{+0.05}$	${2.36}_{-0.05}^{+0.08}$	${1.95}_{-0.10}^{+0.09}$	${1.84}_{-0.14}^{+0.11}$	−0.45 ± 0.20	−0.04 ± 0.05	−0.47 ± 0.13	−0.15 ± 0.20
14	${2.86}_{-0.16}^{+0.17}$	${2.42}_{-0.04}^{+0.06}$	${2.23}_{-0.07}^{+0.08}$	${2.38}_{-0.08}^{+0.10}$	${2.19}_{-0.10}^{+0.12}$	−0.47 ± 0.17	−0.12 ± 0.06	0.17 ± 0.12	−0.20 ± 0.15
15	${2.71}_{-0.16}^{+0.17}$	${1.99}_{-0.04}^{+0.05}$	${1.80}_{-0.05}^{+0.06}$	${1.87}_{-0.05}^{+0.07}$	${1.52}_{-0.08}^{+0.09}$	−0.85 ± 0.18	−0.15 ± 0.05	0.09 ± 0.11	−0.51 ± 0.16
16	${1.87}_{-0.13}^{+0.14}$	${1.73}_{-0.03}^{+0.04}$	${1.52}_{-0.05}^{+0.06}$	${1.25}_{-0.04}^{+0.06}$	${1.23}_{-0.06}^{+0.08}$	−0.22 ± 0.21	−0.18 ± 0.06	−0.49 ± 0.13	−0.04 ± 0.17
17	${1.65}_{-0.12}^{+0.13}$	${1.92}_{-0.05}^{+0.05}$	${1.54}_{-0.07}^{+0.06}$	${1.45}_{-0.06}^{+0.07}$	${2.05}_{-0.14}^{+0.16}$	0.43 ± 0.22	−0.31 ± 0.07	−0.16 ± 0.15	0.86 ± 0.21
18	⋯	${4.45}_{-0.12}^{+0.13}$	${3.18}_{-0.16}^{+0.17}$	${2.96}_{-0.19}^{+0.20}$	${1.65}_{-0.16}^{+0.21}$	⋯	−0.49 ± 0.09	−0.17 ± 0.21	−1.45 ± 0.32
19	⋯	${2.30}_{-0.06}^{+0.08}$	${2.28}_{-0.08}^{+0.11}$	${2.16}_{-0.11}^{+0.12}$	${1.60}_{-0.14}^{+0.17}$	⋯	−0.02 ± 0.08	−0.13 ± 0.17	−0.74 ± 0.27
20	${4.81}_{-0.31}^{+0.31}$	${1.84}_{-0.03}^{+0.06}$	${1.87}_{-0.06}^{+0.08}$	${1.56}_{-0.06}^{+0.07}$	${2.14}_{-0.23}^{+0.23}$	−2.68 ± 0.19	0.02 ± 0.07	−0.44 ± 0.14	0.77 ± 0.28
Mean	2.89 ± 0.35	3.11 ± 0.37	2.53 ± 0.23	2.47 ± 0.30	2.35 ± 0.23	−0.46 ± 0.24	−0.25 ± 0.06	−0.14 ± 0.10	−0.09 ± 0.15

Note. Mean values computed for all regions; mean ${m}_{{\rm{E}}}$ values are averages for region ${m}_{{\rm{E}}}$ values (i.e., not computed from mean FWHMs). Errors on mean values are standard errors of the mean. Horizontal rules group individual regions into filaments.

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To estimate FWHM uncertainties, we horizontally stretched each best fit profile by mapping radial coordinate r to $r^{\prime} (r)=r(1+\xi (r-{r}_{0})/(50^{\prime\prime} -{r}_{0}))$ with ξ an arbitrary stretch parameter and r₀ the best fit rim center from Equation (12), yielding a new profile $h^{\prime} (r)=h(r^{\prime} (r)).$ We varied ξ (and hence rim FWHM) to vary each profile fit χ² by 2.7 and took stretched FWHMs as upper/lower bounds on reported FWHMs. This procedure again follows R14.

We fit model FWHM predictions given by Equation (1) to measured rim widths as a function of energy by varying several physical parameters: magnetic field strength B₀, normalized diffusion coefficient η₂, diffusion-energy scaling exponent μ, and minimum field strength ${B}_{\mathrm{min}}$ and lengthscale a_b for a damped magnetic field. We mapped each width measurement to the lower energy limit of its energy band; e.g., 0.7–1 keV is assigned to 0.7 keV and fitted to model profile widths at 0.7 keV. Width errors in our least squares fits average the positive and negative errors on each FWHM measurement. For a given set of model parameters, we numerically computed intensity profiles and hence model FWHMs as detailed in Section 2.1; we then used a Levenberg–Marquardt fitter to seek model parameters yielding best fit FWHMs. To assist the nonlinear fitting, we tabulated model FWHM values on a large grid of model parameters and used best fit grid parameters as initial guesses for fitting. We required η₂ to be positive and deemed best fit values with η₂ ≥ 10⁵ and ${B}_{0}\geqslant 10\;\mathrm{mG}$ to be effectively unconstrained. In subsequent analysis we focus on fits with μ = 1 and η₂ = 1 fixed, though we discuss the effects of varying both μ and η₂ (and fitting for η₂) as well (recall that for μ = 1, η = η₂ is energy-independent).

The purely loss-limited model (constant downstream magnetic field B₀) has three parameters μ, η₂, and B₀. To make nonlinear fitting tractable, we fixed μ in all fits and considered μ = 0, 1/3, 1/2, 1, 1.5, and 2. In particular, nonlinear diffusion-energy scalings with μ = 1/3 and 1/2 may arise from Kolmogorov and Kraichnan turbulent energy spectra respectively (Reynolds 2004).

For damped magnetic field fits, we held the remnant interior field strength ${B}_{\mathrm{min}}$ constant at $5\;\mu {\rm{G}},$ slightly higher than typical intergalactic values of ∼2–3 μG (Lyne & Smith 1989; Han et al. 2006). We stepped a_b through 14 different values between 0.5 and 0.002 (sampling most finely between 0.01 and 0.002). To ensure that damped fits generate rims influenced by magnetic damping, we arbitrarily require that the rim FWHM at 2 keV be strictly greater than the fit value of a_b; we revisit this requirement below. For best fits, we report the value of a_b yielding the smallest χ² value, with the caveat that that our a_b sampling is relatively coarse.

Predicted rim widths are subject to resolution error in the numerical integrals (discretization over radial coordinate, line of sight coordinate, electron distribution, Green's function integrals). We chose integration resolutions such that the fractional error in model FWHMs associated with halving or doubling each integration resolution is less than 1% for the parameter space relevant to our filaments. The maximum resolution errors in a sample of parameter space are typically 0.1%–1%, but mean and median errors are typically an order of magnitude smaller than maximum errors.

Measured rim widths decrease with energy in most regions and energy bands. Table 2 reports FWHM measurements for all of our regions. We also report ${m}_{{\rm{E}}}$ values for all but the lowest energy band, computed point-to-point between discrete energy bands as:

$$\begin{eqnarray}&&{m}_{{\rm{E}}}({E}_{2})=\displaystyle \frac{\mathrm{ln}({w}_{2}/{w}_{1})}{\mathrm{ln}({E}_{2}/{E}_{1})}\end{eqnarray} \tag{ 13 }$$

where w₁, w₂ and E₁, E₂ are FWHMs and lower energy values for each energy band—e.g., ${m}_{{\rm{E}}}$ at 1 keV is computed using FWHMs from 0.7–1 keV and 1–1.7 keV, with E₂ = 1 keV and E₁ = 0.7 keV. Errors on ${m}_{{\rm{E}}}$ are propagated in quadrature from adjacent FWHM measurements.

Although the measurement scatter is quite large, shown dramatically in the point-wise computed ${m}_{{\rm{E}}}$ values, the mean rim width decreases consistently with increasing energy. Furthermore, mean ${m}_{{\rm{E}}}$ values are consistently negative and tend smoothly toward 0 (weaker energy-dependence) with increasing energy. Errors on FWHM measurements are typically ≲10%, reflecting the high quality of the underlying Chandra data. Scatter in FWHM measurements may be attributed in part to (1) our measurement procedure, which depends on an empirical choice of profile fit function, and (2) variation in Tycho's rim morphology (e.g., Figure 4).

Best fit results for loss-limited and damped cases are summarized in Figure 5 and Table 3. In all tables, we report values of χ² rather than ${\chi }_{\mathrm{red}}^{2}$ to ease comparison of different fits, as we often manually stepped parameters that were held constant in fitting.

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Table 4 reports model fits with η₂ fixed to values derived from srcut spectrum fits for ${\nu }_{\mathrm{cut}}$ and hence η₂ from Equation (8), with B₀ varying freely; cutoff-derived η₂ values of ∼10 require increased magnetic fields compared to the η₂ = 1 case if rims are purely loss-limited. We derive different η₂ values for varying μ and find that fitted B₀ varies only weakly with μ; for example, Regions 1 and 16 have best fit values of B₀ increasing over 252–266 μG and 660–700 μG respectively as μ ranges between 0–2, for loss-limited fits.

Magnetically damped rims are able to fit width-energy dependence in our measurements at least as well as purely loss-limited rims for an acceptable range of damping lengths including a_b values small enough to exert influence on rim widths. In both loss-limited and damped fits, we do not observe systematic variation of fit parameters with azimuth around the remnant.

We briefly explore the effects of varying η₂ and other fit parameters, before focusing on results with μ = η₂ = 1 for simplicity. Fixing η₂ from measured ${\nu }_{\mathrm{cut}}$ values does not yield any obvious insight. But, diffusion coefficients computed in this manner may give more credible derived estimates of B₀ if the DSA assumptions invoked are accurate.

Loss-limited model values of B₀ and η₂ may covary without strongly altering fit quality in many regions. If diffusion is the primary control on rim width—i.e., ${l}_{\mathrm{diff}}\gt {l}_{\mathrm{ad}}$—then η₂ and B₀ become degenerate as the product ${\eta }^{1/2}{B}_{0}^{-3/2}$ exerts most control on rim width $w\sim {l}_{\mathrm{diff}}$ (Equation (11)). If advection is the primary control on rim width (widths narrow rapidly with energy; i.e., ${m}_{{\rm{E}}}\sim -0.5$), then ${\eta }_{2}\ll 1$ becomes unimportant and fits are well-behaved with effectively one free parameter.

We also performed loss-limited model fits with μ fixed between 0 and 2 and both η₂ and B₀ free. Fits with μ ≲ 1 generally yield larger parameter values and errors for both η₂ and B₀, and they are more likely to be ill-constrained (e.g., η₂ > 10³ or B₀ > 10³ μG). Fits to the same data with varying μ can yield η₂ varying by 1–2 orders of magnitude. The χ² values for individual fits are variable and large ($\gg 1$), so we cannot favor or disfavor particular values of μ and η₂. But, neglecting the magnitude of our χ² values, values of $\mu \geqslant 1$ are qualitatively favored by χ² in most regions. This trend may be partially an artifact of the correlation between B₀ and η₂ as they are not entirely independent parameters, but they are less correlated for larger μ; fits at smaller μ may have fewer (non-integer) degrees of freedom.

Damping modifies the degeneracy in B₀ and η₂. For small values of a_b (strong damping), increased diffusion (η₂) can cause rim widths at all energies to narrow counterintuitively. Speculatively, spatial variation of the diffusion coefficient may oppose downstream advection through a decreased effective velocity ${v}_{{\rm{d}}}-(\partial D/\partial x)$ in our transport equation:

$$\begin{eqnarray}&&\left[{v}_{{\rm{d}}}-\displaystyle \frac{\partial D(x)}{\partial x}\right]\displaystyle \frac{\partial f}{\partial x}-D(x)\displaystyle \frac{{\partial }^{2}f}{\partial {x}^{2}}-\displaystyle \frac{\partial }{\partial E}\left({{bB}}^{2}{E}^{2}f\right)\\ &&\quad ={K}_{0}{E}^{-s}{e}^{-E/{E}_{\mathrm{cut}}}\delta (x),\end{eqnarray} \tag{ 14 }$$

requiring thinner rims as η₂ and hence $\partial D(x)/\partial x$ increase. Moreover, the effective velocity may impact rim widths even if advection dominated. In practice, if η₂ varies freely in damped fits, we observe that smaller values of a_b permit and favor smaller best fit values of both B₀ and η₂.

Best fit B₀ values are smallest for η₂ approaching 0 and μ = 2; intuitively, μ > 1 strengthens diffusion at energies >2 keV for fixed η₂, permitting a smaller best fit η₂ and hence smaller B₀. Fixing η₂ = 1 (Table 3), however, ties the range of B₀ values to the range of observed rim widths. Our minimum loss-limited values of B₀ are consistent with prior estimates of ∼200–300 μG for advection-dominated transport (Völk et al. 2005; Parizot et al. 2006; Morlino & Caprioli 2012). If loss-limited, Tycho's rims require strong magnetic field amplification to ∼100× typical galactic field values of ∼2–3 μG, versus the expected 4× amplification from a strong shock.

Magnetic damping fits permit much smaller values of B₀, as expected. The minimum value of B₀ is around 20 μG, which would require no field amplification beyond that from a strong shock. Fixing ${B}_{\mathrm{min}}=2\;\mu {\rm{G}}$ instead of $5\;\mu {\rm{G}}$ permits smaller fit B₀ values, though fit values for both η₂ and B₀ still display considerable scatter.

Width-energy dependence is not as sensitive a discriminant between damping and loss-limited models as may be intuitively expected. Both loss-limited and damped fits do not perfectly capture sharp drop-offs (especially between 0.7–1 and 1–2 keV), and the data show large scatter (Figure 5). In most regions, best fit width-energy curves have ${m}_{{\rm{E}}}\sim -0.2.$ Only sharp decreases in rim width (${m}_{{\rm{E}}}\sim -0.5$) can disfavor damping as a control on rim widths.

When width-energy dependence is weak, both loss-limited and damped models give best fits with similar profiles—amplified magnetic fields in both models cause rim intensities to drop sharply. When width-energy dependence is stronger, damping profiles yield energy dependent rims if magnetic fields are small (say, $\lesssim 50\;\mu {\rm{G}}$) and the damping lengthscale is much smaller than rim FWHMs. Figure 6 shows this contrasting behavior using best fit parameters for Regions 1 and 16.

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The best damped fit parameters for Regions 1–5, 10, 12, and 18 predict a 1.375 GHz radio profile that does not drop below 50% of peak intensity within $\sim 20^{\prime\prime}$ downstream of the shock. These regions are all best fit with field B₀ < 40 μG. We refer to this model behavior as "weak-field" damping, associated with weak magnetic fields and stronger emission intensity immediately downstream of the thin rim. As observation frequency decreases, rim and trough contrast decreases, causing rim FWHMs to increase and eventually become unmeasurable. Rim width-energy dependence strengthens dramatically (${m}_{{\rm{E}}}\to -\infty$), permitting model fits to replicate strong width-energy dependence in observed rim widths. Weak-field fit parameters require energy losses downstream of the shock to produce energy dependence in a damped field, as discussed in Section 2.5.2.

Damping model fits to all other regions predict consistently thin rims with measurable FWHMs at decreasing energy. The width-energy dependence parameterized by ${m}_{{\rm{E}}}$ trends toward zero at low energy, indicating that rim widths are comparatively energy independent. At higher energy, such rims narrow slightly with increasing energy (${m}_{{\rm{E}}}\sim -0.2$) to match the observed width-energy dependence in X-rays. The gradual increase in $| {m}_{{\rm{E}}}|$ with energy is expected in the damping model as advection and/or diffusion take control of rim widths at increasing energy, as discussed in Section 2.5. The best damped fits for these regions all have larger B₀ values than in the "weak-field" damping case. We refer to these fits as giving rise to "strong-field" damping, associated with stronger magnetic fields, weaker emission intensity behind the thin rim, and clear rims with measurable FWHMs at low photon energies (at and below soft X-rays).

We are unable to determine whether weak- or strong-field damping is descriptive of Tycho's shock magnetic field, given the large χ² values on our model fits. Several regions may be well-fit by "weak-field" and "strong-field" damping alike. But, the qualitative behavior of B₀ is suggestive; very strong fields at the shock (≳100 μG) with magnetic field damping on a scale comparable to rim FWHMs appear incompatible with significant energy dependence. The distinction between weak- and strong-field damping may be equally well described as strong versus weak damping. The dichotomy reflects additional degeneracy between a_b and B₀—for a given rim width, increased damping length a_b requires increased B₀ to reproduce the same width, and vice versa.

Although model fits are poorly constrained, we attempt to further explore how damping and synchrotron losses set rim widths for damped fits with finite a_b. Table 5 compares rim widths at 2 keV to advection lengthscale ${l}_{\mathrm{ad}}$ and damping lengthscale a_b derived from the loss-limited and damped fits of Table 3. As discussed in Section 2.5, loss-limited rim widths are qualitatively set by either damping or the dominant transport process as $w\sim \mathrm{min}\left({a}_{b},\mathrm{max}\left({l}_{\mathrm{ad}},{l}_{\mathrm{diff}}\right)\right).$ At 2 keV with η₂ = 1 and μ = 1 fixed, the ratio ${l}_{\mathrm{ad}}/{l}_{\mathrm{diff}}$ is nearly constant; variation arises solely from azimuthal variation in shock velocity. If rims are loss-limited and diffusion is negligible, we anticipate rim widths $w=4.6{l}_{\mathrm{ad}},$ where the factor 4.6 may be derived assuming spherical symmetry and an exponential synchrotron emissivity (Ballet 2006). At 2 keV, we find that loss-limited rim widths $w\sim 5{l}_{\mathrm{ad}}$ due to diffusion. Enforcing $w(2\;\mathrm{keV})/{a}_{b}\gt 1$ for damped fits still permits ${l}_{\mathrm{ad}}/{a}_{b}\lt 1$ as $w(2\;\mathrm{keV})\lesssim 5{l}_{\mathrm{ad}},$ with damping decreasing w(2 keV) below the expected loss-limited width at a given B₀. We may impose a tighter or looser bound on a_b in damped fits, but our results should not be greatly affected given large uncertainty in our fits and associated degeneracy between B₀ and a_b.

Table 5.  Lengthscale Analysis

	Measurements		Loss-lim. Fit		Damped Fit
Region	vd	w (2 keV)	${l}_{\mathrm{ad}}$	${l}_{\mathrm{ad}}/{l}_{\mathrm{diff}}$	${l}_{\mathrm{ad}}$	ab	${l}_{\mathrm{ad}}/{a}_{b}$	$w(2\;\mathrm{keV})/{a}_{b}$
	(${10}^{8}\;\mathrm{cm}\;{{\rm{s}}}^{-1}$)	(%rs)	(%rs)	(−)	(%rs)	(%rs)	(−)	(−)
1	1.30	2.64	0.61	1.40	10.26	0.80	12.83	3.30
2	1.29	0.99	0.27	1.39	11.54	0.30	38.48	3.28
3	1.29	0.74	0.17	1.38	10.80	0.20	54.00	3.72
4	1.28	1.36	0.31	1.38	10.08	0.40	25.20	3.40
5	1.28	1.27	0.30	1.37	12.07	0.30	40.24	4.25
6	1.28	1.24	0.18	1.37	2.61	0.40	6.53	3.10
7	1.27	0.96	0.17	1.36	0.29	0.60	0.49	1.61
8	1.27	0.99	0.19	1.36	0.90	0.50	1.80	1.98
9	1.26	1.03	0.17	1.36	2.09	0.40	5.23	2.57
10	1.26	0.73	0.14	1.35	9.21	0.20	46.05	3.66
11	1.25	1.05	0.21	1.34	0.25	1.00	0.25	1.05
12	1.24	1.09	0.25	1.33	10.80	0.30	35.98	3.62
13	1.23	0.98	0.18	1.32	0.34	0.60	0.57	1.64
14	1.14	0.93	0.17	1.23	2.84	0.40	7.10	2.32
15	1.14	0.75	0.15	1.22	3.95	0.30	13.16	2.50
16	1.13	0.63	0.12	1.21	0.14	0.60	0.24	1.06
17	1.12	0.64	0.13	1.20	0.36	0.40	0.91	1.61
18	1.15	1.32	0.28	1.23	11.29	0.30	37.64	4.42
19	1.18	0.95	0.17	1.27	1.99	0.40	4.97	2.37
20	1.17	0.78	0.14	1.26	3.08	0.30	10.28	2.59

Note. All lengthscales computed at fiducial energy 2 keV from best fits with ${\eta }_{2}=1$ and μ = 1. Ratio of ${l}_{\mathrm{ad}}/{l}_{\mathrm{diff}}$ is same for loss-limited and damped fits and depends only on plasma velocity v_d and observation energy, as ${l}_{\mathrm{ad}}/{l}_{\mathrm{diff}}$ is independent of B₀ for μ = 1. Values of a_b are given here in units of %r_s instead of r_s.

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Our modeling can also reassess the significance of rim width-energy dependence in the remnant of SN 1006 presented by R14. We fit averaged filament widths in SN 1006 measured by R14 to our damping model using the same procedure as for Tycho. We fix ${B}_{\mathrm{min}}=5\;\mu {\rm{G}},$ although best fit magnetic fields for SN 1006 are smaller than those of Tycho due to the much wider filaments of SN 1006. A more extensive search of parameter space produces acceptable damping (hybrid) models with more rapid shrinkage than found in R14. Damped fits are comparable to or better than loss-limited fits in 3 of 5 filaments in SN 1006. Fits to two filaments with strong energy dependence (${m}_{{\rm{E}}}\sim -0.5$) favor a loss-limited model with sub-Bohm diffusion (${\eta }_{2}\ll 1$), though sub-Bohm diffusion may be an unphysical result pointing to oversimplifications in the model. The width-energy dependence in SN 1006 (${m}_{{\rm{E}}}\sim -0.3$ to $-0.5$) is overall slightly stronger than in Tycho, and the best damped fits for SN 1006 all fall into the "weak-field" case. The best fit B₀ values are less than $40\;\mu {\rm{G}}$ in the damped model, compared to ∼100–200 μG in the loss-limited model. If thin radio rims in SN 1006 (Reynolds & Gilmore 1986) indicate magnetic field damping on lengthscales comparable to X-ray filament widths, some magnetic damping is not incompatible with rim width narrowing. A future study of SN 1006 with recent multi-frequency Karl G. Jansky VLA data (PI: D. Green) may verify whether radio and X-ray rims can be jointly described by an amplified and subsequently damped magnetic field.

Our models adopted a distance d to Tycho of 3 kpc, but estimates for Tycho's distance range between 2–5 kpc (Hayato et al. 2010). As a larger remnant distance would increase both physical filament widths and shock velocity estimates from proper motion, we may derive expected scalings for modeled η₂ and B₀ as functions of assumed distance d. The advective lengthscale, Equation (10), may be rearranged to obtain:

$$\begin{eqnarray}{B}_{0} & = & (3.17\;\mu {\rm{G}}){\left(\displaystyle \frac{{v}_{{\rm{d}}}}{{10}^{8}\;\mathrm{cm}\;{{\rm{s}}}^{-1}}\right)}^{2/3}\\ & & \times {\left(\displaystyle \frac{{l}_{\mathrm{ad}}}{0.01\;\mathrm{kpc}}\right)}^{-2/3}{\left(\displaystyle \frac{h\nu }{1\;\mathrm{keV}}\right)}^{-1/3}\end{eqnarray} \tag{ 15 }$$

or, more simply,

$$\begin{eqnarray}&&{B}_{0}\propto {\left({v}_{{\rm{d}}}\right)}^{2/3}{\left({l}_{\mathrm{ad}}\right)}^{-2/3}{\nu }^{-1/3}.\end{eqnarray} \tag{ 16 }$$

Both ${l}_{\mathrm{ad}}$ and v_d scale linearly with remnant distance d and thus their effects cancel in determining the magnetic field. If diffusion is the primary control on filament lengthscales, Equation (11) yields:

$$\begin{eqnarray}&&{\eta }_{2}\propto {\left({l}_{\mathrm{diff}}\right)}^{2}\;{B}_{0}^{(\mu +5)/2}{\nu }^{-(\mu -1)/2}.\end{eqnarray} \tag{ 17 }$$

Model fits with varying distance obey both scalings, ${\eta }_{2}\propto {d}^{2}$ and B₀ constant. When comparing model fits with remnant distances of 3 and 4 kpc, the deviation from the idealized scaling is ≲1% for B₀ and ∼1%–5% for η₂. As varying remnant distance d leaves width-energy scaling ${m}_{{\rm{E}}}$ invariant, the relative contributions of ${l}_{\mathrm{ad}}$ and ${l}_{\mathrm{diff}}$ should also be invariant. Then both lengthscales should scale simultaneously with d, yielding the observed behavior. In the damped model, a larger distance d will require larger physical damping lengths from magnetic turbulence. But, fitted values of a_b should remain unchanged as we report a_b in units of shock radius r_s.

The exponential variation in $D(x)$ (Equation (6)) may not be physically reasonable; as noted in Section 4.3, the assumption $D(x)\propto 1/{B}^{2}(x)$ gives rise to sharp gradients in diffusion coefficient and may even invert the effect of η₂ on our model profiles (where larger η₂ can cause rim widths to narrow). Nevertheless, the modeled behavior appears physically reasonable. Only X-ray profiles are impacted by the assumption on $D(x)$ as radio profiles assume no diffusion. And, model behavior driven by advection and magnetic damping—namely, rim width-energy dependence in a damped magnetic field—occurs for D = 0, and cannot be an artifact of our assumption that $D(x){B}^{2}(x)$ is constant.

We emphasize that additional counts from averaging measurements or selecting larger regions will likely not improve our ability to constrain B₀ and η₂ from width-energy modeling. This is easily seen from Figure 5, and from the large χ² values in Table 3 that reflect tight errors on our FWHM measurements (Table 2).

Rim width-energy dependence is a morphological manifestation of spectral softening downstream of the forward shock; previously, Cassam-Chenaï et al. (2007) also sought to distinguish loss-limited and damped rims with a careful spectral study and 1D hydrodynamical model. Although knowledge of radial spectral variation, in principle, fully determines rim profiles, there is not a clear relationship between observed spectral variation and rim widths alone. Our model, for example, replicates observed rim width variation but underpredicts the observed spectral variation. The more sophisticated model of Cassam-Chenaï et al. (2007) similarly has trouble reproducing the observed radial gradient in spectral photon index. Photon indices predicted in our model, for integrated rim spectra similar to those of Figure 3 and Table 1, are also somewhat ill-constrained. Enforcing limits on acceptable model photon indices (e.g., Figure 16, Cassam-Chenaï et al. 2007) could help constrain shock parameters or identify discrepancies between model assumptions and measurements.

Estimates of magnetic field strength that depend upon global models for Tycho's evolution (e.g., Morlino & Caprioli 2012) may not help constrain our rim model results. In such models, assumptions on particle acceleration and magnetic field evolution downstream of the shock (e.g., further damping or amplification at the reverse shock and contact discontinuity) would affect the spatially integrated spectrum of Tycho. We have focused on effects immediately behind the forward shock, making as few assumptions as possible about particle acceleration and magnetic fields throughout the remnant. It may not make sense to extrapolate our results to estimate magnetic fields throughout the remnant or, conversely, to use global models to constrain our results given all assumptions involved.