THE DISTRIBUTION OF RADIOACTIVE ⁴⁴Ti IN CASSIOPEIA A

NuSTAR is the first focusing, hard X-ray observatory. It is composed of two co-aligned X-ray telescopes (FPMA and FPMB) observing the sky in the energy range 3–79 keV (Harrison et al. 2013). The field of view of each NuSTAR telescope is roughly 12' × 12' and has a point-spread function (PSF) with a FWHM of 18'' and a half-power diameter of 58''.

NuSTAR observed Cas A during the first 18 months of the NuSTAR mission (Table 1) with a total of 2.4 Ms of exposure time. These observations include the original ∼1 Ms of data that we have presented previously (Grefenstette et al. 2014). We reduced the NuSTAR data with the NuSTAR Data Analysis Software (NuSTARDAS) version 1.4.1 and NuSTAR calibration database (CALDB) version 20150316 to produce images, exposure maps, and response files for each telescope.

We examined the background reports from the NuSTAR Science Operations Centerand identified solar flares during three sequence IDs (40021003003, 40021011002, and 40021015002). As the flux from the solar flares only affects the spectrum below 10 keV (Wik et al. 2014) and the amount of time that is affected by solar flares is small compared to the duration of the observations, we do not apply any filtering to the data. We do not use the data for sequence IDs 40021001004, 40021002010, and 40021003002 because these short observations were significantly offset from the target pointing position.

We leverage the increased exposure time relative to our previous work to perform an analysis on smaller spatial scales, though at lower signal-to-noise ratio than when integrating over the remnant as a whole.

Figure 2 shows the 65–70 keV NuSTAR image of Cas A. To produce this image we combine all of the data (for all epochs and both telescopes) using ximage and then subtract the similarly combined background images. We smooth the result with an ≈18'' top-hat kernel to generate the underlying grayscale image in the left and center panels. This is comparable to the image that we used for the analysis in Grefenstette et al. (2014). While the band image is useful, it can contain some contamination from the strong non-thermal emission that is present in the remnant (Grefenstette et al. 2015) and is also not optimized to search for emission that is red- or blueshifted where some of the line flux may fall outside the 65–70 keV bandpass.

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Instead of using the band image, we instead perform a systematic, spatially resolved spectroscopic analysis by dividing the remnant into an 8 × 8 grid of regions (Figure 2). Each grid box is a square with 45'' sides. The grid is centered by eye on the spatial distribution of the ⁴⁴Ti ejecta. The right panel in Figure 2 shows the exposure maps computed at 68 keV (this accounts for the energy-dependent vignetting) and combined in the same way as the 65–70 keV band image. It demonstrates that the exposure is relatively uniform across the grid, only dropping by ≈30% near the corners of the grid.

We use the nuproducts FTOOL to extract source spectra and generate ancillary response files (ARFs), which describe the effective area of the optics, as well as response matrix files (RMFs), which describe the response of the detectors. We generate simulated background spectra using nuskybgd (Wik et al. 2014) following the procedure described in Grefenstette et al. (2014). This results in 22 sets of data files (11 epochs × 2 telescopes) for each region. The ARF and RMF files computed by nuproducts account for the variations in the effective exposure due to vignetting described above.

We integrate over all 11 epochs by using the addspec FTOOL, setting bexpscale = 1 when calling addspec to prevent overflowing the exposure keyword. This results in two sets of source, background, ARF, and RMF files (one for FPMA and one for FPMB) for each region.

Since the ⁴⁴Ti emission is extended (with a spatial distribution that we do not know a priori), we have to make a decision on how to normalize the ARF.

For the spectral analysis of point sources, nuproducts adjusts the normalization of the ARF (and thus the measured flux) to account for the fraction of the PSF that falls outside the source region. This "PSF correction" is not performed when observing extended sources because the correction assumes that the extraction region is precisely centered on the point source. Here the source flux is smeared out over the source extraction region, making an accurate PSF correction impossible. Instead we opt to simply apply no PSF correction to the ARF as the most conservative approach. We address the impact of this on the interpretation of the measured flux below.

We performed spectral fitting using XSPEC (Arnaud 1996) using the cstat statistic for the model fitting. In general, the observed (source+background) spectra satisfy the requirement that each bin contains at least one count, so we do not arbitrarily rebin the spectra before fitting. We simultaneously fit the spectra for each telescope, allowing a standard cross-normalization constant to account for variations in the overall effective area between the two telescopes.

The broadband hard X-ray spectrum of Cas A is dominated by thermal emission in the interior of the remnant along with a non-thermal tail throughout the remnant (e.g., Grefenstette et al. 2015). The non-thermal tail varies spatially across the remnant in both flux and spectral shape, so we fit each box with the srcut spectral model. We keep the radio spectral index fixed to 0.77 (which is the radio spectral index integrated over the remnant, Baars et al. 1977) and then fit for the break frequency and the normalization of the continuum. In the interior of Cas A, the data also require a thermal component, which we model using a simple bremss component. This thermal continuum can contribute significantly up to ∼15 keV. We mask the spectrum over the Fe K line features in the 5–7 keV band in the NuSTAR spectrum. This results in a fit range of 3–4.5 keV and 8.5–79 keV.

To model the ⁴⁴Ti decay lines we include two Gauss components with the line widths and fluxes tied together and require that both lines have the same observed Doppler shift. The NuSTAR optics have an absorption edge at ∼78.4 keV (Madsen et al. 2015), so the 78.32 keV decay line is visible only when the material is stationary or redshifted. Where the 67.87 keV line is blueshifted we fit with only a single Gaussian component rather than the two components tied together. To determine whether the line is detected we compare the cash fit statistic with and without the ⁴⁴Ti lines and require that the change in fit statistic is >9.0 to declare the ⁴⁴Ti emission to be consistent with the data. We then use the error command to generate 90% and 1σ confidence regions for the line centroid, Gaussian width, and line flux. Unless otherwise stated, all uncertainties quoted in the text are 90% error estimates.

Overall, 10 of the 64 regions satisfied our detection conditions (the fit parameters are given in Table 2).

Table 2.  Results of NuSTAR Fits

			bremss		srcut, α = 0.77		Gaussian
ID	R.A.	Decl.	kT (keV)	Norm. (10−3)	Break (1017 Hz)	Norm.a	Centroid (keV)	1σ Width (keV)	Fluxb
19	350.8401	58.8355	2.34 ± 0.06	7.3 ± 0.3	2.07 ± 0.09	90.9 ± 6.5	67.35 ± 0.29	<0.3	8.8 ± 2.8
20a	350.8643	58.8355	${2.14}_{-0.06}^{+0.15}$	8.3 ± 0.3	${2.1}_{-0.1}^{+5.7}$	${65}_{-20}^{+5}$	67.15 ± 0.2	<0.3	12.3 ± 4.9
20bc	...	...	...	...	...	...	69.5 ± 0.6	${0.9}_{-0.4}^{+0.6}$	${17.2}_{-7.4}^{+8.6}$
27	350.8401	58.8230	2.26 ± 0.09	8.9 ± 0.3	2.8 ± 0.3	60 ± 10	66.6 ± 0.3	${0.61}_{-0.25}^{+0.4}$	${11.3}_{-3.8}^{+4.6}$
28	350.8643	58.8230	2.1 ± 0.1	8.2 ± 0.4	2.7 ± 0.2	56.5 ± 8	67.6 ± 0.5	<0.92	7.8 ± 5.0
29	350.8884	58.8230	2.1 ± 0.2	6 ± 0.3	${2.37}_{-0.48}^{+0.25}$	${37}_{-5}^{+14}$	${67.8}_{-0.9}^{+0.4}$	0.7 ± 0.3	12.5 ± 5.5
30c	350.9125	58.8230	2.3 ± 0.1	6.8 ± 0.3	${3.3}_{-0.8}^{+0.5}$	${16}_{-3}^{+9}$	67.8 ± 0.6	${0.73}_{-0.6}^{+0.4}$	10.6 ± 5.5
34	350.8160	58.8105	2.5 ± 0.1	7.1 ± 0.3	3.5 ± 0.2	82 ± 5	66.8 ± 0.2	<0.6	${6.4}_{-2.5}^{+4.1}$
35	350.8401	58.8105	2.4 ± 0.1	7 ± 0.3	3.5 ± 0.4	${46}_{-4}^{+9}$	67.4 ± 0.3	0.64 ± 0.3	14.4 ± 4.5
36	350.8643	58.8105	2.2 ± 0.1	7.4 ± 0.3	3.6 ± 0.2	48 ± 4	67.6 ± 0.3	0.43 ± 0.3	9.6 ± 4
43	350.8401	58.7980	2.13 ± 0.1	5.9 ± 0.1	2.72d	51d	67.1 ± 0.2	<0.34	${7.3}_{-3.2}^{+1.9}$

Notes.

^aFlux at 1 GHz in Jy. ^b10⁻⁷ photons cm⁻² s⁻¹. ^cOnly fit with a single Gaussian line. ^dNot well constrained.

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In most of these cases the measured Gaussian 1σ widths of the lines are consistent with the energy resolution of the detectors (≈0.6 keV FWHM at 68 keV, Harrison et al. 2013). This implies that the ejecta within each 45'' region samples ⁴⁴Ti ejecta traveling in roughly the same direction or having a spread in velocities below the ability of NuSTAR to detect (i.e., the top panel in Figure 3).

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The one exception is region 20 (Figure 3, bottom panel), which contains line emission clearly broadened beyond the instrument response. In this case we added a second line and achieved an improved fit to the data, resulting in one red-redshifted component (20a) and a broad (∼1 keV Gaussian width) blueshifted component (20b).

Systematic errors in the calibration of detector gain could influence the measured Doppler shift of the lines. The systematic uncertainties for NuSTAR are roughly 2 × 10⁻⁴ in gain and 40 eV in offset (Madsen et al. 2015). At 67.86 keV, the gain uncertainty yields a systematic error of 13 eV, or 60 km s⁻¹, while the offset uncertainty of 40 eV offset results in a systematic uncertainty in velocity of 180 km s⁻¹. Both of these effects are significantly smaller than the statistical errors, so we neglect them below.

"Look-back" effects can change the apparent bulk location of the ejecta. This is entirely an effect of the difference in light travel time between blueshifted and redshifted ejecta, where the redshifted ejecta are "younger" than the blueshifted ejecta. For unresolved sources, this can cause spherically symmetric sources to appear redshifted, especially for the gamma-ray lines from rapidly expanding supernovae (e.g., Chan & Lingenfelter 1988). For Cas A, each region represents the integration over the line-of-sight distribution of the ejecta in the remnant. We can estimate the difference in flux due to look-back effects for ejecta. The maximum observed redshifted line had a centroid of 67.13 keV (or 1% c). This produces a 1 lt-yr line-of-sight offset per 100 yr. For a symmetric distribution of ejecta 340 yr after the explosion, the difference in apparent age between the front and rear extreme ejecta is 6.84 yr. For an e-folding time of 86.54 yr, this results in a difference in observed flux due to look-back effects of only ∼8%. We conclude that look-back effects do not significantly affect our results.

We can combine the distance from the center of expansion of the remnant and the observed Doppler shift of the ejecta to determine the 3D velocity of each ejecta knot. We measure the projected offsets in the plane of the sky from the center of expansion of the remnant (α(J2000) = 23^h23^m2777 ± 005, δ(J2000) = 58°48'494 ± 04, Thorstensen et al. 2001). If we assume the material is freely expanding, then the observed velocity is proportional to the distance from the center of expansion of the remnant. DeLaney et al. (2010) found a proportionality constant for undecelerated ejecta of 0022 per km s⁻¹. However, if a knot of ⁴⁴Ti ejecta has encountered the reverse shock, then the knot will be decelerated by some amount that will depend on the density of the knot and the local speed of the reverse shock at the time of the encounter. For simplicity, we will adopt the undecelerated proportionality constant for undecelerated ejecta for all of the ⁴⁴Ti knots because this is the most conservative projection into 3D.

Using this proportionality we can convert between observed offsets (in arcsec) and velocity (in km s⁻¹). Figure 4 shows the projected distance in the plane of the sky for each region and the measured velocity along the line of sight, together with a fiducial reverse shock radius of 95'', while Figure 5 and its associated animation show a proper 3D representation of the data.

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Nearly all the ⁴⁴Ti ejecta are seen traveling away from the observer. However, unlike in SN1987A, where the ⁴⁴Ti ejecta appear to be traveling in the same direction with the same velocity (Boggs et al. 2015), here we see that the ejecta are expelled into a large solid angle. We also see ejecta that are traveling at different speeds in roughly the same direction. There is also evidence for high-velocity ⁴⁴Ti ejecta that have passed beyond the reverse shock (Figure 6). However, one of these knots is region 20b, which we cannot indisputably identify as a single coherent feature because it is broadened along the line of sight. Even so, there is clearly significantly blueshifted emission in this region, so there must be some ⁴⁴Ti ejecta that have been ejected beyond the reverse shock radius. The fact that these ejecta are beyond the reverse shock implies that our assumption that these ejecta are freely expanding is probably incorrect since they should have been decelerated as they traversed the reverse shock. However, the amount of deceleration the ejecta experience depends on the density of the ejecta, and we have no way of measuring the density of the ⁴⁴Ti ejecta knots. We therefore consider the positions along the line of sight to be lower limits for these data.

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Simply combining the fluxes from the detected regions in Table 2 does not result in a good estimate of the total flux measured by NuSTAR.

As described above, for point sources a PSF correction is applied to account for the fraction of the NuSTAR PSF that falls outside the extraction radius. For extended sources it's not possible to apply an accurate PSF correction (and thus correctly normalize the flux) when combining neighboring regions because a region may contribute flux to its neighbors and correcting for this "loss" will overcorrect the flux.

This can be avoided by simply integrating over a larger region that covers all of the ⁴⁴Ti emission. Here we compute the total flux by integrating over a circular extraction region with a radius of 120'' centered on the remnant (Figure 2, center panel). The standard behavior of nuproducts when producing extended ARFs is to assume that the spatial distribution of the counts for an extended source is described by the observed distribution of counts over a given energy range (i.e., that the spatial distribution of low-energy counts is the same as the distribution of high-energy counts). Here, since the source is background-dominated and since the ⁴⁴Ti emission does not follow any other energy band where the images are source-dominated, we instead use the flatflagarf keyword to produce the ARFs. This explicitly assumes the prior of a "spatially flat" distribution of source flux across the 120'' source region. We note that this option was not available for the previous analysis.

We fit the data with a power-law continuum and a single Gaussian line (fitting over the 10–72 keV band), and a power-law continuum with two Gaussian lines with the Doppler shift of the lines, the line width, and line flux tied together (fitting over the 10–80 keV band), though the choice of model does significantly affect our results (Table 4).

We find a 68 keV line flux that is slightly higher than we previously reported ((1.84 ± 0.25) × 10⁻⁵ photons cm⁻² s⁻¹ compared with (1.53 ± 0.31) × 10⁻⁵ photons cm⁻² s⁻¹), though the line centroid(s) and line width(s) are both consistent with our previous results. The change in flux likely arises from the improvements in the generation of the ARF (we consider the new method to be superior). Taking a distance of 3.4 kpc, an explosion date of 1671, and an average epoch of the observations of 2013 gives a total initial mass of ⁴⁴Ti of (1.54 ± 0.21) × 10⁻⁴ M_⊙ (compared with our previous value of (1.25 ± 0.30) × 10⁻⁴ M_⊙).

In regions where we do not detect ⁴⁴Ti emission we instead define the upper limit to be "the flux at which 50% of the time we would have detected the ⁴⁴Ti." We determine this by repeatedly simulating the source and background spectra using a power-law continuum (fit to the observed data between 20 and 60 keV) and inserting a single narrow Gaussian line at 67.87 keV at a given flux level. The synthetic spectrum is then fit over the 20–75 keV bandpass to see if the Gaussian component is detected as described above.

We produce 1000 simulations for each flux level using fakeit in XSPEC and determine how many of the simulations resulted in the detection of the line. We declare the flux level at which 50% of the simulations produce detections to be the upper limit. The upper limits vary spatially over the remnant (Figure 7) due to the vignetting of the NuSTAR optics (i.e., the varying exposure as seen in Figure 2, right panel) and the fact that the pointing strategy was optimized to cover the interior of the remnant with a uniform response.

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Comparing the distributions of ⁴⁴Ti and ⁵⁶Ni is vital to understanding the physical conditions and kinematics of the innermost region of the supernova explosion. Most of the iron should result from the decay of ⁵⁶Ni, so comparing the distributions of ⁴⁴Ti and iron yields information on the initial distributions of nickel-rich ejecta and titanium-rich ejecta from the supernova explosion.

⁴⁴Ti is produced in a variety of nuclear processes, though the dominant process depends on the nature of the explosion (thermonuclear versus core collapse) and the structure of the star.

For thermonuclear supernovae, much of the ⁴⁴Ti is formed in the burning of the He shell or the C/O core. If the density and temperature are sufficiently low (temperatures <2–3 × 10⁹ K and densities <10⁶ g cm⁻³), the material does not burn all the way to ⁵⁶Ni. Instead, the explosion produces ⁴⁰Ca, ⁴⁴Ti, and ⁴⁸Cr (Holcomb et al. 2013). In these conditions, ⁴⁴Ti can be produced in regions where very little or no ⁵⁶Ni is synthesized.

In contrast, for core-collapse supernovae like Cas A, the dominant ⁴⁴Ti production occurs when the shock passes through the innermost silicon layer. The densities and temperatures are typically higher than those found in the ⁴⁴Ti sites for thermonuclear supernovae, with peak temperatures >4 × 10⁹ K and densities >10⁶ g cm⁻³ (Magkotsios et al. 2010).

The burning of silicon proceeds through photodisintegration when the rearrangement of the nuclei produces clusters of nuclei. This drives the material into nuclear statistical equilibrium, when the composition is determined by a balance between forward and reverse nuclear reaction rates. Depending upon the exact peak temperatures (and densities at these peak temperatures), Magkotsios et al. (2010) identified six regions where different processes and reactions (e.g., different quasi-equilibrium clusters and different nuclear statistical equilibrium freezeout conditions) dictate the final nucleosynthetic yields. These conditions of higher density/temperature mean that, in nearly all cases for core-collapse supernovae, at least some ⁵⁶Ni is produced when ⁴⁴Ti is produced. There are, however, scenarios in which the ⁵⁶Ni/⁴⁴Ti ratio can fall to ∼100 and others where it is very high (>10⁸). Measuring this ratio (or the Fe/Ti ratio as a proxy) and its spatial variations can provide detailed clues to the nature of the explosion.

4.1.1. Comparison of ⁴⁴Ti and ⁵⁶Ni Ejecta in 3D

Figure 8 shows the comparison of the NuSTAR⁴⁴Ti data and the Fe K emission observed by Chandra. The latter data are taken from DeLaney et al. (2010), and because the iron is visible in the X-rays only after it has encountered the reverse shock we adopt the value derived by those authors to convert the line-of-sight velocity to distance of 0032 per km s⁻¹, which is appropriate for the decelerated, reverse-shocked material.

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In nearly all cases where we see ⁴⁴Ti ejecta at or beyond the reverse shock (the green and blue data points in Figure 8) we also see emission from shocked iron (e.g., the regions labeled northwest and southwest Iron). The one exception to this is the blueshifted region 20b, which does have any obvious analog in the 3D distribution of shocked iron.

However, as we noted above, the line associated with region 20b is Doppler-broadened beyond the nominal energy resolution of the detectors. This implies that the PSF of NuSTAR is blending together several knots (or a shell) of ⁴⁴Ti ejecta rather than a resolving a single knot. In this case our conversion from the Doppler velocity to a 3D position may not be correct.

There is also some evidence for a trace amount of iron in the northern shell that is stationary with respect to the center of expansion of the remnant or slightly blueshifted toward the observer. It may be the case that there is a tenuous amount of iron that would be associated with region 20b but is too faint to observed when seen in projection against the (brighter) redshifted iron (i.e., the region labeled northwest Iron in Figure 8).

We remind the reader that we need to be careful when we interpret the observed distribution of iron. First, we only observe the iron that has been shock-heated (i.e., it has passed through the reverse shock), so X-ray measurements provide a partial observation of the iron produced in the supernova. Second, estimates of the iron mass based on X-ray measurements depend upon the excitation states of the iron and so will be affected by deviations from coronal equilibrium. Since the observed X-ray flux is proportional to the product of the iron mass and electron density, highly clumped material can also produce higher X-ray flux for the same iron mass than smoothly distributed material. Finally, if iron is present in the star and the circumstellar medium, the ejecta will contain "swept-up" iron that cannot be distinguished from the iron synthesized in the explosion.

We can avoid the ambiguities in the "swept-up" iron versus iron synthesized in the explosion if we consider only iron that was producing in the explosion. A subset of the Chandra Fe K-emitting knots contain "pure" iron; that is, the knots are characterized by a lack of associated silicon emission (Hwang & Laming 2003, 2012). The lack of observed emission from lighter elements suggests that these regions are associated with α-rich freezeout during the explosive nucleosynthesis (Hwang & Laming 2012) rather than incomplete silicon burning. We can thus attempt to quantify the relative production rates of ⁴⁴Ti and iron by comparing the distribution of ⁴⁴Ti observed by NuSTAR with the distribution of pure iron observed by Chandra (Figure 9).

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4.1.2. Beyond the Reverse Shock

4.1.3. The Unshocked Interior

We can also compare the ⁴⁴Ti ejecta with the emission seen at optical and infrared wavelengths. This again broadly falls into two categories: ejecta that have encountered the reverse shock and ejecta that are interior to the reverse shock.

For the shocked ejecta, we can use the [Ar ii] 6.99 μm 3D maps from Spitzer (DeLaney et al. 2010; Isensee et al. 2010). Seen in the plane of the sky, the ejecta forms the feature known as the "bright ring," while in 3D there are circular structures in the plane of the sky (labeled as the "north ring" and the "NE jet" structures, Figure 10).

We find that the ⁴⁴Ti ejecta appear to correlate with both the north ring and the NE jet (we recommend the movie available via the online journal for a more complete picture of these complex data). In the north ring (where we also find Fe K emission), this could have been the result of a bubble being blown in the material by the radioactive decay of clumps of ⁵⁶Ni (e.g., Li et al. 1993; Blondin et al. 2001). The north ring happens to be coincident with the direction of one of the light echoes from Cas A, which showed that the photosphere of the supernova to the rear/northwest direction was moving faster than along the other lines of sight to the northeast and southeast (Rest et al. 2011).

There is also emission from unshocked ejecta seen in NIR [S iii] line emission (906.9 and 953.1 nm) in the interior of the remnant (Milisavljevic & Fesen 2015). These ejecta form bubble-like structures in the interior of the remnant (labeled the "north" and "south" cavities in Figure 11). We find that the ⁴⁴Ti ejecta may be associated with the northern cavity seen in the [S iii] data, though we do not see any evidence for ⁴⁴Ti being associated with the southern, blueshifted cavity. We expect these bubbles to be associated with the decay of ⁵⁶Ni and so we may be seeing variations in the resulting Fe/Ti ratio in the interior of the remnant similar to the variations that we observe beyond the reverse shock.

The interior unshocked ejecta is also seen in the infrared via [Si ii] (34.8 μm) emission (DeLaney et al. 2010). These ejecta are apparently arranged into a "tilted thick disk" (identified in Figure 12) with a significant gap between the redshifted and blueshifted faces. We do not see any evidence for ⁴⁴Ti ejecta associated with the blueshifted half of the thick disk, though we do see that the redshifted ⁴⁴Ti ejecta are reasonably consistent with the redshifted half of the thick disk.

A future instrument with spatial resolution comparable to what is achieved by Chandra or Spitzer and spectral resolution better than NuSTAR will likely be required to study the relative associations of the optical/infrared and ⁴⁴Ti in the interior of the remnant in further detail.

The discovery of a point-like X-ray source (hereafter the central compact object, or CCO) in Cas A was one of the major results of the first images of Cas A taken at high spatial resolution with Chandra (Tananbaum 1999). The CCO is now thought to be consistent with a slowly rotating neutron star (e.g., Pavlov et al. 2000; Chakrabarty et al. 2001; Mereghetti et al. 2002) or, perhaps, a neutron star with a low surface magnetic field (e.g., Pavlov & Luna 2009).

If we accept that the CCO is a neutron star, then we can use it to study the natal kick of the neutron star (e.g., Burrows & Hayes 1996; Burrows et al. 2004; Wongwathanarat et al. 2013). The CCO is offset from the center of expansion of the remnant by 70 ± 08 with a position angle of 169° ± 8.4° in an observation made in 2004 (Fesen et al. 2006). For an explosion date of ∼1671 and a distance of 3.4 kpc, this corresponds to a velocity in the plane of the sky of ∼330 km s⁻¹. None of the major features in the remnant observed in the optical, infrared, or radio appears to match the CCO motion, though the bulk motion of the X-ray-emitting ejecta in the remnant (210 km s⁻¹ east and 680 km s⁻¹ north, Hwang & Laming 2012) is ∼150° from the direction of motion of the CCO.

The bulk of the ⁴⁴Ti, however, is traveling in the west/northwest direction away from the observer. Using the flux-weighted mean of the 3D velocities given in Table 3, we find that the average direction of motion of the ⁴⁴Ti ejecta has a position angle of ∼340° ± 15° (measured clockwise from celestial north in the plane of the sky) and is tilted by 58° ± 20° into the plane of the sky away from the observer (Figure 13). The angle in the plane of the sky is almost exactly opposite to the inferred direction of the CCO motion.

Table 4.  NuSTAR Cas A Integrated Fits

	Power-law Parameters		67.87 keV Gauss Parameters			Initial 44Ti
Line Model (Fit Range)	Γ	Norm.a	Centroid	Width	Flux	Mass
		(photons cm−2 s−1)	(keV)	(keV)	(10−5 photons cm−2 s−1)	(10−4 M⊙)
Single line (10–72 keV)	3.36 ± 0.05	1.229 ± 0.015	${67.44}_{-0.14}^{+0.11}$	0.68 ± 0.15	1.84 ± 0.25	1.52 ± 0.2
Two lines (10–80 keV)	3.36 ± 0.05	1.229 ± 0.015	${67.41}_{-0.12}^{+0.10}$	0.69 ± 0.15	1.87 ± 0.24	1.54 ± 0.2

Notes.

^aFlux at 1 keV.

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If we assume that the 3D distribution of the ⁴⁴Ti ejecta is a tracer of the 3D mass of the innermost ejecta at the time of the explosion and that the neutron star kick is related to the asymmetries in the ejecta mass distribution, then we can make a prediction about the velocity of the neutron star along the line of sight. We can then proceed assuming that there is a simple scaling relation between the total momentum of the ⁴⁴Ti and the momentum of the neutron star:

$$\begin{eqnarray}&&C\times {{\boldsymbol{\rho }}}_{\mathrm{Ti}}={{\boldsymbol{\rho }}}_{\mathrm{NS}}\end{eqnarray} \tag{ 1 }$$

where C is some proportionality constant that is roughly the ratio of the ⁴⁴Ti ejecta mass to the total inner ejecta mass. Since we can directly measure the velocity components of both the ⁴⁴Ti and the neutron star in the plane of the sky, we can fold the neutron star mass into the proportionality constant itself and make the comparison:

$$\begin{eqnarray}&&\langle {{\boldsymbol{v}}}_{\mathrm{Ti}}{\rangle }_{\mathrm{Sky}}={C}^{\prime }\times {{\boldsymbol{v}}}_{\mathrm{NS},\mathrm{Sky}}\end{eqnarray} \tag{ 2 }$$

where $\langle {{\boldsymbol{v}}}_{\mathrm{Ti}}{\rangle }_{\mathrm{Sky}}$ is the flux-weighted mean velocity of the ⁴⁴Ti ejecta in the plane of the sky. The ⁴⁴Ti has an observed velocity in the plane of the sky of 1450 ± 380 km s⁻¹, so we can compute the proportionality constant C^' that will produce a velocity of the neutron star in the plane of the sky of 330 km s⁻¹. Assuming the same proportionality constant applies to the line-of-sight direction we can convert the flux-weighted average of the ⁴⁴Ti velocity along the line of sight (920 ± 510 km s⁻¹) into an estimate for the line-of-sight velocity of the neutron star (205 ± 125 km s⁻¹).

Unfortunately, testing this hypothesis is non-trivial, the uncertainties we quote here are large, and the physical scaling that we have performed here to convert between the ⁴⁴Ti momentum and the neutron star kick is likely overly simplistic. However, the fact that the ⁴⁴Ti ejecta are apparently moving in the direction opposite to that of the neutron star strongly suggests that the two are related. When we also consider that the bulk ejecta is recoiling in roughly the same direction as the ⁴⁴Ti and opposite to the direction of the neutron star (Hwang & Laming 2012) and that the light echo in this region indicates that the exploding star was moving faster than along the other lines of sight to the northeast and southeast (Rest et al. 2011), then we start to construct a more complete picture of the explosion. If the ejecta expands rapidly (perhaps as the result of a more energetic explosion), then the density can quickly drop into a region where α-rich freezeout may occur, resulting in a high yield of ⁴⁴Ti.

One of the most important unresolved issues currently facing the supernova simulation community is whether supernova explosions can be adequately modeled in 2D (i.e., the explosion can be described by axisymmetric simulations) or whether they require 3D simulations to fully capture the relevant instabilities (see, e.g., recent reviews by Fryer et al. 2014; Janka et al. 2016). The favored interpretation for a core-collapse supernova is that neutrino heating drives shock instabilities in the collapsing star (e.g., Bethe & Wilson 1985). These instabilities give rise to large spatial structures (i.e., those that can be described by "low-mode" spherical harmonics), or bubbles, in the ejecta that carry enough momentum to revive the stalled shock and explode the star. The NuSTAR 2D ⁴⁴Ti map of Cas A strongly supports this low-mode convection model for the supernova engine (Grefenstette et al. 2014).

We have also previously argued that even the 2D images of the ⁴⁴Ti ejecta suggest that large-scale structures dominate the ejecta distribution rather than small turbulent eddies (i.e., features that can be described by "high-mode" spherical harmonics) or "jet"-like features that can result from the collapse of a rapidly rotating massive star like those that are present in Type Ib/Ic supernovae and/or gamma-ray bursts (e.g., Mösta et al. 2015). The generation of these large structures in 3D may be related to the standing-mode accretion shock instability (SASI), which redistributes power to lower spherical harmonics in 3D simulations while turbulence will drive power to higher order modes (Janka et al. 2016). This can also occur in Rayleigh–Taylor driven convection (e.g., Herant 1995).

The fact that we now see large, coherent structures in the 3D distribution of the ejecta is further evidence that the large spatial instabilities do not cascade down to small spatial scales in less than a dynamical timescale. This is especially true when we consider the spatial variations of the measured Fe/Ti abundance, which is nearly bipolar in structure and may be the best tracer for density asymmetries in the innermost ejecta during explosive nucleosynthesis.