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In our original manuscript, the calculation of the electron spectrum near maximum momentum produced an overestimate of the electron pileup from radiation losses due to the use of insufficiently fine spatial and energy gridding in the region where the electron energy losses are rapid. This has no effect on the dynamical evolution of the system, nor on the proton spectrum. A corrected spectrum, shown in Figure 1 (upper left panel, to be compared with Figure 4 in the original paper) results in a reduction of the inverse-Compton (IC) emission in the GeV band (and also the nonthermal bremsstrahlung emission in the 100 GeV–1 TeV range). The modified broadband spectra for the models presented in the paper are shown in the remaining panels of Figure 1 (to be compared with lower panels in Figures 3, 5, and 11 in the original paper). The corrected calculation provides a modestly improved fit for our preferred Model A, but changes the conclusion that IC emission provides a similar contribution to the γ-ray emission in the GeV band as that from π0-decay; in fact, the π0-decay emission is dominant across the γ-ray band in our Model A. Figure 2 contains updated X-ray spectral fits contained in Figures 7–9 and 12 in our original paper. The predicted X-ray spectrum from the model is slightly flatter, and also slightly brighter, because the slope of the electron spectrum in the X-ray emitting region beyond the cutoff is flatter than in our original calculation. This results in slightly better fits to the X-ray data, with modified values for our best-fit column density and in the relative normalization of the nonthermal to thermal model components. These improved values are reported in Table 1 (to be compared with Table 2 of the original manuscript) and do not change any of our conclusions. The normalizations for the NW/NE/W regions differ from the fractional geometric sizes of the regions by factors of 1.4/0.6/1.6, representing better agreement than in our original fits, although the original fit for the NE region provided almost exactly the geometric ratio. The updates do not produce any significant change in the reported radio and X-ray brightness profiles.
Figure 1. Upper left: proton (red) and electron (black) spectra p4 f(p) for Model A. Remaining panels: broadband spectral energy distribution for indicated models, compared with observed emission; see the text of the original paper for a description of different curves.
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Figure 2. Chandra spectra from the NE, NW, and W rims of Tycho, compared with the predictions from Model A. The upper (black) spectrum corresponds to a region directly along the shock (region 1 in Figure 1 of the original manuscript) while the lower (red) spectrum is taken from a region immediately behind the shock (region 2). The lower right panel shows the spectrum from the NE region 3, residual emission from Ne, Si, and S is evident, indicating the presence of ejecta extended nearly all the way to the FS. The red histogram corresponds to a CR-hydro-NEI model with Gaussians added at the energies expected for He-like emission from Ne, Si, and S, although we note that significant Ne emission is not expected in spectra of Type Ia supernova remnants.
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| Parameter | NW a | NE a | W a | ||||||
|---|---|---|---|---|---|---|---|---|---|
| A | B | C | A | B | C | A | B | C | |
| NH b | 5.1 ± 0.1 | 4.6 ± 0.1 | 8.9 ± 0.1 | 5.3 ± 0.2 | 4.8 ± 0.3 | 8.0 ± 0.2 | 6.4 ± 0.1 | 5.7 ± 0.2 | 9.2 ± 0.1 |
| K1 c | 39.0 ± 0.8 | 26.1 ± 0.5 | 48.0 ± 1.0 | 12.8 ± 0.7 | 7.4 ± 0.1 | 16.7 ± 0.4 | 52.8 ± 0.7 | 34.9 ± 0.5 | 68.8 ± 1.0 |
| K2 c | 10.3 ± 0.4 | 5.0 ± 0.2 | 10.9 ± 0.3 | 7.9 ± 0.3 | 3.9 ± 0.2 | 7.8 ± 0.3 | 26.9 ± 0.6 | 13.2 ± 0.3 | 27.0 ± 0.6 |
| 1.1 | 1.2 | 1.3 | 1.0 | 1.0 | 1.3 | 1.1 | 1.2 | 1.4 |
| dof | 398 | 373 | 533 | ||||||
Notes. aRegion sizes as fraction of total azimuth are 0.027 (NW), 0.023 (NE), and 0.032 (W). bColumn density × 1021 cm−2. cModel normalization × 10−3.
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In Figures 2 and 6 of the original paper, the reverse shock (RS) expansion speed was incorrectly plotted in the ejecta frame rather than the explosion frame. In addition, measurements of the ejecta expansion by Katsuda et al. (2010) were incorrectly presented as bounds on the RS velocity. In fact, the ejecta knots are not necessarily located at the RS. In Figure 3, we present a modified version of Figure 6 from the original paper (which also updates information plotted in the blue panels in Figure 2 of the original manuscript). Here, the cyan region indicates the range of measured values for the forward shock (FS), and the blue region indicates that for shocked ejecta. These ejecta must reside somewhere between the RS and the contact discontinuity (CD). The model predictions for these quantities are shown in red and green. The conclusions of the paper are not affected by the modified calculations; the ejecta expansion measurements are marginally consistent with our Model A, with the ejecta knots falling close to the CD. The measurements are also consistent with our Model B, but rule out our Model C.
Figure 3. Time evolution of the FS (black), RS (red), and CD (green) angular speeds for models described in the text. The cyan (blue) box corresponds to the range of expansion speeds measured for the FS (ejecta). Solid lines correspond to Model A while short (long) dashed lines correspond to Model B (C).
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We thank Jean Ballet for pointing out these issues.