A SPATIALLY RESOLVED STUDY OF THE SYNCHROTRON EMISSION AND TITANIUM IN TYCHO'S SUPERNOVA REMNANT USING NuSTAR

Figures 1 and 2 show the deconvolved, background-subtracted NuSTAR images of Tycho from the combined ∼750 ks observation in several energy bands. NuSTAR detects the "fluffy" Fe-rich ejecta found predominantly in the northwest of Tycho (Decourchelle et al. 2001; Yamaguchi et al. 2014) as well as non-thermal X-rays with up to energies of ∼40 keV. The hard (>8 keV) X-rays are predominantly distributed around the rim of the SNR, with the peak at the location of the "stripes" identified in Chandra observations (see Figure 3 for a comparison).

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In the right panel of Figure 3, we compare the 20-cm radio morphology seen by the VLA to the 4–6 keV Chandra and 10–20 keV NuSTAR images. The radio emission is distributed in a ring interior to the narrow, non-thermal filaments at the SNR periphery found by Chandra. Generally, the bright, hard X-ray knots detected by NuSTAR do not coincide with the radio emission. Overall, we find that the Chandra 4–6 keV band has a similar morphology as revealed in the NuSTAR images up to ∼20 keV. The hardest X-rays (20–40 keV) are concentrated around the stripes, as well as near a "hook" in the southwest of the SNR (denoted by the small red rectangle in Figure 3).

We also convolved the Chandra 4–6 keV image with a Gaussian to match the resolution of the NuSTAR data, and the comparison of the result to the 10–20 keV NuSTAR data is shown in Figure 4. As the 10–20 keV emission is almost exclusively non-thermal, the locations where the 4–6 keV emission is coincident reveals that the latter is substantially non-thermal in nature. The converse scenario (where 4–6 keV emission is found without 10–20 keV emission) indicates that the softer band may be dominated by thermal emission. We find that the 4–6 keV emission is largely coincident with the harder band, except in the northwest, where previous X-ray studies have found significant emission from the ejecta (Decourchelle et al. 2001; Yamaguchi et al. 2014). Thus, we conclude that the 4–6 keV Chandra band is likely dominated by non-thermal emission across Tycho, except in the northwest.

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In Figure 5, we compare the NuSTAR 20–40 keV emission to the GeV and TeV centroids reported by Giordano et al. (2012) and Acciari et al. (2011), respectively. The GeV localization (from Fermi observations with a 95% confidence level) coincides with the hard X-rays from the "hook" feature in the southeast of the SNR, although we note that the "stripe" region is just outside the 95% error circle. The TeV detection is consistent with a point source at the 011 ≈ 66 PSF of VERITAS, which is of the same order as the angular extent (∼8') of the SNR. Thus, it is not possible with the currently reported VERITAS observations to determine the origin of the TeV emission.

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We do not find any point-like or extended emission in the remnant using the 65–70 keV band containing the 68 keV ⁴⁴Ti emission line. We searched using the method from Grefenstette et al. (2014), where we compared the observed image to a simulated background image. We searched on multiple spatial scales by convolving both the NuSTAR image and the simulated image with a top hat function, varying the radius of the function from 3 to 15 NuSTAR pixels (roughly 7.5–30 arcsec). We computed the Poisson probability that the observed counts in each pixel could have been produced by the predicted background flux, where a low probability indicates that the pixel likely contains a source count. We do not find any evidence for excess emission on any spatial scales.

We place upper limits on the diffuse extended emission in Tycho arising from the 68 keV line. We extracted spectra and produced an ARF and RMF for source regions with radii from 1 to 5 arcmin centered on the remnant. For example, in Figure 6, we give the spectra and background model of the FPMA and FPMB data for a 4' radius extraction region. For each radius, we modeled the source using a combination of a power-law continuum with two Gaussian lines of fixed relative intensity centered at 68 and 78 keV and fit the data over the 10–79 keV bandpass. We do not find any evidence for line emission in any of the source regions.

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Our upper limits are dependent on the intrinsic width of the ⁴⁴Ti lines. We computed the 3σ confidence limits on the line normalizations for fixed line widths of 0.1, 1, or 3 keV, which correspond to velocities of ≈733, 7330, and 22,000 km s⁻¹, respectively. The resulting 3σ upper limits as a function of radius are shown in Figure 7. In all cases, the lower limits are zero.

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To effectively compare these NuSTAR upper limits with results from other missions, we convert all previous measurements and limits to epoch 2014. In Table 2, we list these values for observations with the Compton Gamma-ray Observatory/COMPTEL, INTEGRAL/IBIS, and Swift/BAT. To obtain epoch 2014 fluxes (which are also given in Table 2), we adopt a half-life of 58.9 years (Ahmad et al. 2006). The NuSTAR upper limits may be used in conjunction with the previous results to assess the likely spatial distribution of the ⁴⁴Ti. In particular, our upper limits for radii ≲2' are less than the reported Swift/BAT detection (at epoch 2014) of (1.2 ± 0.6) × 10⁻⁵ ph cm⁻² s⁻¹, suggesting that the ⁴⁴Ti is distributed over a larger radius of Tycho.

Recently, Miceli et al. (2015) reported detection of the Ti–K line complex at ∼4.9 keV using stacked XMM-Newton spectra from Tycho. In particular, these authors identified the Ti feature in two regions of enhanced emission from Fe-group elements, in the northwest and in the southeast of the SNR (regions #1 and #3 in Figure 2 of Miceli et al. 2015). Motivated by this result, we searched for the 68 keV line at these locations in the NuSTAR data (the region files were provided graciously by M. Miceli). We did not detect any signal above the background from either region, with lower limits of zero in all cases. The 3σ confidence flux upper limits are 3.9 × 10⁻⁶ ph cm⁻² s⁻¹ and 7.3 × 10⁻⁷ ph cm⁻² s⁻¹ for regions #1 and #3, respectively, assuming a line width of 1 keV.

We performed spatially resolved spectroscopic analyses by extracting and modeling spectra from sixty-six 1' by 1' boxes across the SNR (see Figure 8). To evaluate the non-thermal emission, we fit the spectra in the 10–50 keV range with an absorbed srcut model. In our fits, we have fixed the absorbing column to N_H = 7 × 10²¹ cm⁻² (Eriksen et al. 2011), and we have grouped the spectra to a minimum of 30 counts bin⁻¹.

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The srcut model describes the spectrum as that radiated by a power-law energy distribution of electrons with an exponential cutoff at energy E_max (Reynolds & Keohane 1999). That spectrum cuts off roughly as $\mathrm{exp}(-\sqrt{\nu /{\nu }_{{\rm{rolloff}}}})$ (though the fitting uses the complete expression). The rolloff photon energy E_rolloff = hν_rolloff is related to the maximum energy of the accelerated electrons E_max by

$$\begin{eqnarray}&&{E}_{{\rm{max}}}=120{\left(\displaystyle \frac{h{\nu }_{{\rm{rolloff}}}}{\text{1 keV}}\right)}^{1/2}{\left(\displaystyle \frac{B}{\mu {\rm{G}}}\right)}^{-1/2}\;{\rm{TeV}},\end{eqnarray} \tag{ 1 }$$

where B is the magnetic field strength and we have corrected for a small numerical error in Reynolds & Keohane (1999).

In XSPEC, the srcut model is characterized by three parameters: the rolloff frequency ν_rolloff, the mean radio-to-X-ray spectral index α, and the 1 GHz radio flux density F_1GHz. To limit the free parameters in the fit, we estimated F_1GHz in each of the sixty-six regions by measuring the 1.505 GHz flux density in the NRAO/VLA data assuming a radio spectral index of α = −0.65 (Kothes et al. 2006). The derived flux densities at 1 GHz using this procedure are listed in Table 3. We note that without additional single-dish observations, the interferometric VLA data are missing flux from the largest scales. Based on previous radio studies of Tycho, we estimate the missing flux is of the order of ∼10% of the values listed in Table 3.

Table 3 lists the spectroscopic results, giving the best-fit rolloff frequency ν_rolloff and the χ² and degrees of freedom (dof) for all sixty-six regions. We find that all of the regions are fit well by the srcut models, with χ²/dof = χ_r² ≈ 0.82–1.26 with ∼100–200 dof per region. Figure 9 shows example spectra from six of the regions, and Figure 10 maps the best-fit rolloff frequency ν_rolloff values across Tycho. We find substantial variation of a factor of five in ν_rolloff. The largest values of ν_rolloff ≳ 3 × 10¹⁷ Hz (corresponding to a rolloff energy of E_rolloff = 1.24 keV) are concentrated in the west of the SNR where Eriksen et al. (2011) identified the "stripes" in Tycho. Furthermore, we find elevated rolloff frequencies in the northeast of the SNR, with ν_rolloff ≈ 2 × 10¹⁷ Hz (corresponding to E_rolloff ≈ 0.83 keV), where the SNR is thought to be interacting with molecular material (Reynoso et al. 1999; Lee et al. 2004). The lowest values of ν_rolloff ∼ 10¹⁷ Hz (corresponding to E_rolloff ≈ 0.42 keV) are found in the southeast and north of the SNR. For the range of E_rolloff ≈ 0.4–2.0 keV, we find in our spatially resolved spectroscopic analyses, Equation (1) gives a maximum energy E_max ≈ (75–170) (B/μG)^−1/2 TeV. Assuming a magnetic field B = 200 μG across Tycho (Parizot et al. 2006), this relation gives E_max = 5–12 TeV.

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Previous X-ray studies of Tycho have also measured ν_rolloff over the entire SNR or along prominent synchrotron filaments. Using ASCA data from Tycho, Reynolds & Keohane (1999) found ν_rolloff ≈ 8.8 × 10¹⁶ Hz, and using the first Chandra observation of Tycho, Hwang et al. (2002) derived even lower ν_rolloff toward the northeast, northwest, and southwest rim, with ν_rolloff ∼3–7 ×10¹⁶ Hz. More recently, in their analysis of the deep Chandra observations of Tycho, Eriksen et al. (2011) fit absorbed srcut models to the hard "stripes" and a series of filaments projected in the south of Tycho (corresponding to our regions #51 and 52) which they called the "faint ensemble." They found that the stripes have ν_rolloff ∼ 2 × 10¹⁸ Hz while the faint ensemble had softer spectra, with ν_rolloff ≈ 3 × 10¹⁷ Hz.

Our NuSTAR results confirm that the stripes have harder emission than other regions in Tycho, but our 90% confidence limits exclude ν_rolloff ≳ 5 × 10¹⁷ Hz there. The softest regions (with ν_rolloff ≈ 10¹⁷ Hz) are comparable to the values found by Reynolds & Keohane (1999) integrated over Tycho. The discrepancy between our 90% confidence limits and the hard stripes analyzed by Eriksen et al. (2011) may be due to two issues. First, the angular resolution of NuSTAR is not sufficient to resolve the individual stripes, and thus our spectra appear softer as they are averaging over larger areas of the SNR. Second, Eriksen et al. (2011) fit the Chandra data from 4.2 to 10 keV, and these spectra may not have had reliable leverage to model the high-energy rolloff. Additionally, we note that the effective area of Chandra drops precipitously above 8 keV.

We note that we have assumed a constant radio spectral index of α = −0.65, the value derived by Kothes et al. (2006) based on the integrated flux densities at 408 and 1420 MHz. However, the radio spectrum is actually concave, and thus α may flatten at higher frequencies (Reynolds & Ellison 1992). If we instead adopted α = −0.5 in our analysis, the best-fit values of ν_rolloff change substantially, decreasing by about a factor of five. However, the assumption of α = −0.5 is not favored statistically based on the χ² computed from those fits. For example, region #21 has χ² = 235 (147) with 160 dof in the α = −0.5 (−0.65) case. Additionally, the α = −0.5 fits dramatically over-predict the continuum flux below 10 keV, whereas the α = −0.65 fits adequately describe the continuum down to 3 keV (see Figure 9). Generally, regardless of the choice of α, the qualitative trend in the spatial distribution of ν_rolloff (e.g., with greater ν_rolloff in the west of the SNR) remains the same.

The yield, spatial distribution, and velocity distribution of ⁴⁴Ti in a young SNR are a direct probe of the underlying explosion mechanism of the originating supernova (see, e.g., Magkotsios et al. 2010). Among core-collapse SNRs, Cassiopeia A remains the only Galactic source with a confirmed detection of ⁴⁴Ti (Iyudin et al. 1994; Grefenstette et al. 2014), and SN 1987A in the Large Magellanic Cloud also has a robust ⁴⁴Ti detection (Grebenev et al. 2012; Boggs et al. 2015). To date, the only probable detection of ⁴⁴Ti in a Type Ia SNR was reported by Troja et al. (2014), who found a 4σ level excess above the continuum in the 60–85 keV band in Swift/BAT observations of Tycho. Wang & Li (2014) also found a marginal 2.6σ detection of excess emission in the 60–90 keV band using INTEGRAL/IBIS, although their 3σ upper limit is consistent with the previous INTEGRAL studies of Tycho (Renaud et al. 2006a). Other searches for ⁴⁴Ti in Type Ia SNRs have yielded upper limits, such as the recent work by Zoglauer et al. (2015) using NuSTAR observations of the young SNR G1.9+0.3.

Based on our NuSTAR analyses, we do not detect ⁴⁴Ti in Tycho. We looked both for small scale "clumpy" emission as found in Cassiopeia A (Grefenstette et al. 2014) as well as for diffuse emission throughout the SNR. We find upper limits which are more strict than the previous COMPTEL (Dupraz et al. 1997; Iyudin et al. 1999) and INTEGRAL (Renaud et al. 2006a; Wang & Li 2014) values, particularly if the ⁴⁴Ti is concentrated in the central 2' and/or has slow or moderate velocities (i.e., the 0.1 and 1.0 keV line width cases). For example, our 3σ limit in the 0.1 keV (1.0 keV) case at an enclosed radius of 2' is 5.0 × 10⁻⁶ (1.0 × 10⁻⁵) ph cm⁻² s⁻¹. To be consistent with the possible detection of ⁴⁴Ti with Swift/BAT (Troja et al. 2014), our results indicate that the ⁴⁴Ti is expanding at moderate-to-high velocities (≳7000 km s⁻¹) and/or is distributed across the majority (≳3') of the SNR.

We have also found no ⁴⁴Ti in the regions where Miceli et al. (2015) reported detection of the Ti–K line complex using XMM-Newton. This ∼4.9 keV feature arises from shock-heated, stable ⁴⁸Ti and ⁵⁰Ti. Thus, the lack of a co-located detection of ⁴⁴Ti may indicate that the radioactive isotope does not trace the Fe-rich ejecta. Alternatively, the divergent results may suggest that the 4.9 keV feature arises from transitions of Ca xix and Ca xx rather than from stable Ti. In the future, these interpretations can be tested via deeper NuSTAR observations to reveal the spatial distribution of ⁴⁴Ti, and through high-resolution X-ray spectroscopy (with e.g., Astro-H) which may distinguish between the Ca and Ti spectral features.

The upper limit flux that we have measured of the 67.9 keV line can be employed to set an upper limit on the ⁴⁴Ti yield in the SN explosion. The ⁴⁴Ti mass M₄₄ is related to the flux F₆₈ in the 67.9 keV ⁴⁴Ti line (e.g., Grebenev et al. 2012) by

$$\begin{eqnarray}&&{M}_{44}=4\pi {d}^{2}(44{m}_{{\rm{p}}})\tau {\rm{exp}}(t/\tau ){F}_{68}{W}^{-1},\end{eqnarray} \tag{ 2 }$$

where d is the distance to the source, m_p is the mass of a proton, τ is the mean lifetime of the radioactive titanium (which is related to the half-life t_1/2 by τ = t_1/2 / ln 2 ≈ 85 years for ⁴⁴Ti: Ahmad et al. 2006), t is the age of the source, and W is the emission efficiency (average numbers of photons per decay) for a given emission line (W = 0.93 for the 67.9 keV ⁴⁴Ti line). Assuming a distance of D = 2.3 kpc and an age of t = 442 years for Tycho, an upper limit of F₆₈ < 2 × 10⁻⁵ ph cm⁻² s⁻¹ corresponds to M₄₄ < 2.4 × 10⁻⁴ M_⊙. Here, we adopted the upper limit F₆₈ in the case where the ⁴⁴Ti is located within a radius of 3' and has a line width of 1 keV, as these most closely match the expected scale and velocity of the shell. For the range of distance estimates in the literature of D = 1.7–5.0 kpc to Tycho, we find upper limits of M₄₄ < 1.3 × 10⁻⁴ M_⊙ and M₄₄ < 10⁻³ M_⊙, respectively.

These upper limits can be compared to the predictions of ⁴⁴Ti synthesized in SN Ia models. Generally, the amount of ⁴⁴Ti produced in an SN Ia depends on both the progenitor systems as well as the ignition processes. Simulations predict ⁴⁴Ti masses ranging from ∼10 ${}^{-6}{M}_{\odot }$ in centrally ignited pure deflagration models and ≲6 × 10⁻⁵ M_⊙ for off-center delayed detonation models (e.g., Iwamoto et al. 1999; Maeda et al. 2010). In sub-Chandrasekhar models, greater ⁴⁴Ti yields are expected, with estimates of ≳10⁻³ M_⊙ (e.g., Fink et al. 2010; Woosley & Kasen 2011; Moll & Woosley 2013). The mass limits estimated above are most consistent with the models that produce small or moderate yields of ⁴⁴Ti and disfavor the sub-Chandrasekhar models.

To explore the relationship between the non-thermal X-ray emission detected by NuSTAR and shock properties, we compare the spatially resolved spectroscopic results of Section 3.3 to measurements of the shock expansion and the ambient density ${n}_{{\rm{o}}}$ in Tycho from previous studies.

Expansion measurements of Tycho's rim have been done at radio, optical, and X-ray wavelengths. Using 1375-MHz VLA data over a 10-year baseline, Reynoso et al. (1997) measured expansion rates of ∼01–05 yr⁻¹, depending on the azimuthal angle. These authors found that the slowest expansion occurs in the east/northeast, where the forward shock is interacting with a dense cloud (Reynoso et al. 1999; Lee et al. 2004), and the fastest expansion is along the western rim. Quantitatively similar results were obtained in the optical by Kamper & van den Bergh (1978) and in the X-ray by Katsuda et al. (2010). Using the measured expansion index and SNR evolutionary models, Katsuda et al. (2010) inferred a pre-shock ambient density of n_o ≲ 0.2 cm⁻³, with probable local variations of factors of a few based on the substantial differences in the proper motion around the rim.

The shock expansion is described by the radius of the shock with time, R ∝ t^m, where m is the expansion parameter. From this expression, the shock velocity is related to m by v_s = mR/t. Thus, the m is a useful indicator of shock velocity, yet it does not require assumptions about the distance to the source. Generally, m is close to unity when the expansion is undecelerated (as in e.g., the free expansion phase of a young SNR). As the shock undergoes deceleration, m ∼ 0.6–0.8 (Chevalier 1982a, 1982b), and m = 0.4 once the shock reaches the Sedov phase. Finally, as the shock becomes radiative, m drops below 0.4. While interior pressure is still important, m = 0.33 ("pressure-driven snowplow"; e.g., Blondin et al. 1998), and then falls to m = 0.25 when the shock transitions to a pure momentum-conserving stage (Cioffi et al. 1988). In the case of Tycho, Reynoso et al. (1997) demonstrated the expansion parameter m varies around the rim from m ∼ 0.25 in the east (where the shock is decelerating due to the collision with a dense clump known as knot g: Kamper & van den Bergh 1978; Ghavamian et al. 2000) up to m ≲ 0.8 in the southwest.

In Figure 11, we show the rolloff energy (found in Section 3.3) versus the expansion parameter m obtained by Reynoso et al. (1997) using VLA data. To produce this plot, we have considered only our regions around the rim of Tycho, and we have adopted the mean of the expansion parameters over the azimuthal angles which correspond to those regions using the data in Table 2 of Reynoso et al. (1997). At m > 0.4, the rolloff energy appears to increase with m, suggesting the least decelerated (fastest) shocks of Tycho are the most efficient at accelerating particles to high energies. The minimum rolloff energy is around m = 0.4 where the SNR has reached its Sedov phase. At m = 0.25, where the shock is encountering dense material, the SNR is slightly more efficient at accelerating particles, reaching half of the rolloff energy of the high-m regions.

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In Figure 12, we plot the rolloff energy E_rolloff versus the post-shock density n_o (left panel) and versus the shock velocity v_s (right panel). Both of these quantities were derived by Williams et al. (2013) using Spitzer Space Telescope 24 and 70 μm data as well as the radio and X-ray proper motions of Reynoso et al. (1997) and Katsuda et al. (2010). We find that the regions of large E_rolloff have the lowest post-shock densities (with n_o ≲ 0.25 cm⁻³) and the largest shock velocities (with v_s ≳ 3500 km s⁻¹). The two outliers to these trends (those with n₀ > 1 cm⁻³, v_s < 2200 km s⁻¹, and E_rolloff > 0.8 keV) correspond to regions #12 and 21, where the shock is interacting with knot g.

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We note that the conversion of proper motions to shock velocities requires an assumption about the distance to Tycho, which is still uncertain, with estimates in the literature ranging from 1.7 to 5.0 kpc (e.g., Albinson et al. 1986; Schwarz et al. 1995; Völk et al. 2008; Hayato et al. 2010; Tian & Leahy 2011; Slane et al. 2014). For the data plotted in Figure 12, Williams et al. (2013) employed a distance of D = 2.3 kpc, as suggested by Chevalier et al. (1980). The shock velocities scale linearly with the assumed distance, so the choice of distance will shift the points in the right panel of Figure 12 accordingly.

The relationship between E_rolloff and v_s in Figure 12 can be used to examine the mechanism limiting the maximum energy of electrons E_max undergoing DSA. The timescale t_acc of the acceleration is set by the lesser of three relevant processes: the age of the SNR (i.e., the finite time the electrons have had to accelerate), the timescale of the radiative losses of the electrons, and the timescale for the particles to escape upstream of the shock. In each case, E_max (and consequently, E_rolloff) has different dependences on physical parameters (from Reynolds 2008, Equations (26)–(28)), specifically v_s:

$$\begin{eqnarray}&&{E}_{{\rm{max}}}({\rm{age}})\propto {v}_{{\rm{s}}}^{2}{tB}{(\eta {R}_{{\mathcal{J}}})}^{-1}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{E}_{{\rm{max}}}({\rm{loss}})\propto {v}_{{\rm{s}}}{(\eta {R}_{{\mathcal{J}}}B)}^{-1/2}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&{E}_{{\rm{max}}}({\rm{esc}})\propto B{\lambda }_{{\rm{max}}}.\end{eqnarray} \tag{ 5 }$$

In the above relations, t is the age of the SNR, η is the gyrofactor (where η = 1 is Bohm diffusion and η > 1 is "Bohm-like" diffusion), and λ_max is the maximum wavelength of MHD waves present to scatter electrons. ${R}_{{\mathcal{J}}}$ parametrizes the obliquity dependence of the acceleration: given that the shock obliquity angle θ_Bn is the angle between the mean upstream B-field direction and the shock velocity, then the quantity ${R}_{{\mathcal{J}}}$ is defined as ${R}_{{\mathcal{J}}}\equiv {t}_{{\rm{acc}}}({\theta }_{{\rm{Bn}}})/{t}_{{\rm{acc}}}({\theta }_{{\rm{Bn}}}=0).$ Since ${E}_{{\rm{rolloff}}}\propto {E}_{{\rm{max}}}^{2}B,$ the above relations can be rewritten in terms of E_rolloff:

$$\begin{eqnarray}&&{E}_{{\rm{rolloff}}}({\rm{age}})\propto {v}_{{\rm{s}}}^{4}{t}^{2}{B}^{3}{(\eta {R}_{{\mathcal{J}}})}^{-2}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{E}_{{\rm{rolloff}}}({\rm{loss}})\propto {v}_{{\rm{s}}}^{2}{(\eta {R}_{{\mathcal{J}}})}^{-1}\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{E}_{{\rm{rolloff}}}({\rm{esc}})\propto {B}^{3}{\lambda }_{{\rm{max}}}^{2}.\end{eqnarray} \tag{ 8 }$$

In the right panel of Figure 12, we plot curves for ${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{2}$ (solid light blue line) and ${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{4}$ (dashed red line), for comparison. Additionally, since ${v}_{{\rm{s}}}={mR}/t\propto m$ (where R is the radius of the shock and t is the age of the SNR), we have plotted the same curves on Figure 11 as well. The two dependencies of E_rolloff on v_s reflect those of the loss-limited and age-limited relations above, assuming constant B and ${R}_{{\mathcal{J}}}.$ We find that the $\propto {v}_{{\rm{s}}}^{2}$ curve is too shallow to match the high-velocity (v_s ≳ 3500 km s⁻¹) points, and the $\propto {v}_{{\rm{s}}}^{4}$ curve is a better descriptor of the data. We note that if B is not constant and is instead amplified by a constant fraction of the shock kinetic energy, then B ∝ v_s, and Equation (6) gives ${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{7}.$ We have plotted this steeper curve in Figures 11 and 12 as well, and it adequately describes the data. Due to the relatively large error bars on the shock velocity v_s, we are not able to distinguish which curve (${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{4}$ or ${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{7}$) is better statistically, but both are much more successful than the loss-limited case (with ${E}_{{\rm{rolloff}}}\propto {v}_{{\rm{s}}}^{2},$ with no dependence on magnetic-field strength B) and the escape-limited case (with no E_rolloff dependence on v_s). Thus, our spatially resolved spectroscopic results are most consistent with the age-limited scenario, whereby the maximum electron energy E_max is determined by the finite time the electrons have had to accelerate. In this case, the accelerated electrons achieve the same maximum energy as the protons/ions because the electrons are not limited by their radiative losses.

However, previous studies of Tycho have argued that the SNR is likely loss-limited instead of age-limited (although see Reynolds & Keohane 1999). In particular, Parizot et al. (2006) and Helder et al. (2012) calculate the synchrotron loss timescale and found it to be a few percent of the age of Tycho. Furthermore, recent hydrodynamical simulations by Slane et al. (2014) show that the loss-limited scenario is consistent with the observed broadband spectrum of Tycho. We note that these models were matched to the global spectra across Tycho as the gamma-ray emission is not resolved. Thus, local variations in the gamma-ray emission is possible, with the implication that the electrons and protons may achieve the same maximum energies in certain locations.

For the acceleration to be age limited, the synchrotron loss time t_syn for X-ray emitting electrons must be larger than the remnant age, t = 442 years for Tycho. As a consistency check, we estimate the magnetic field strength B necessary for t_syn > t and compare it to that measured observationally. The synchrotron loss time t_syn is related to the E_max and E_rolloff by (adapted from Vink 2012)

$$\begin{eqnarray}&&{t}_{{\rm{syn}}}=12.5{\left(\displaystyle \frac{{E}_{{\rm{max}}}}{100\;{\rm{TeV}}}\right)}^{-1}{\left(\displaystyle \frac{B}{100\;\mu {\rm{G}}}\right)}^{-2}\;{\rm{year}}\end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray}&&{t}_{{\rm{syn}}}=104{\left(\displaystyle \frac{{E}_{{\rm{rolloff}}}}{1\;{\rm{keV}}}\right)}^{-1/2}{\left(\displaystyle \frac{B}{100\;\mu {\rm{G}}}\right)}^{-3/2}\;{\rm{year}}.\end{eqnarray} \tag{ 10 }$$

Adopting the highest rolloff frequency from our spectral analysis, ν_rolloff = 5.38 × 10¹⁷ Hz (from region #20), we derive B = 29 μG. This value is far below the measurements by Parizot et al. (2006) of B = 200 μG based on the assumption that the X-ray filament widths are limited by the synchrotron loss time. However, it is possible that filament widths are governed instead by magnetic field damping (e.g., Pohl et al. 2005; Rettig & Pohl 2012; Ressler et al. 2014), and this mechanism is the only possible one if the rims are equally thin at radio wavelengths, as is true in some regions of Tycho. Thus, the age-limited interpretation of our NuSTAR results would demand low magnetic field strengths and magnetic-field damping to be responsible for the X-ray rim morphology. However, a recent detailed analysis of Tycho (Tran et al. 2015) finds that even in damping models, magnetic field strengths ≫30 μG are required to fit the radio and X-ray profiles of the thin rims.

Alternatively, the electrons are loss limited, and the apparent trend of E_rolloff with v_s arises because of obliquity effects. In the scalings relating E_rolloff to v_s, we have assumed constant ${R}_{{\mathcal{J}}},$ yet the shock obliquity θ_Bn is expected to change around the periphery of the SNR as the shock encounters a roughly uniform magnetic field in the environs of an SN Ia (Reynolds 1998). For example, changes in obliquity are thought to be the origin of the X-ray bright, synchrotron rims aligned in the northeast and southwest directions of SN 1006 (Rothenflug et al. 2004; Reynoso et al. 2013). The magnetic field is not as ordered in Tycho as in SN 1006 (see Figure 8 of Reynoso et al. 1997), and further work is necessary to assess the obliquity dependence of the particle acceleration in Tycho.