11institutetext: Department of Astronomy, University of Maryland, College
Park, MD 20742 22institutetext: X-ray Astrophysics Laboratory NASA/GSFC,
Greenbelt, MD 20771 33institutetext: Center for Research and Exploration in
Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA
44institutetext: Center for Astrophysics — Harvard & Smithsonian, 60 Garden
St, Cambridge MA 02138, USA

# Comparing the three-dimensional morphological asymmetries in the ejecta of
Kepler and Tycho in X-rays

A. Picquenot1,2,3 T. Holland-Ashford4 B. J. Williams 22

###### Abstract

Aims. Recent simulations have shown that asymmetries in the ejecta
distribution of supernova remnants (SNRs) may be a reflection of asymmetries
left over from the initial supernova explosion. Thus, SNR studies provide a
vital means for testing and constraining model predictions in relation to the
distribution of heavy elements, which are key to improving our understanding
of the explosion mechanisms in Type Ia supernovae.

Methods. The use of a novel blind source separation method applied to the
megasecond X-ray observations of the historic Kepler and Tycho supernova
remnants has revealed maps of the ejecta distribution. These maps are endowed
with an unprecedented level of detail and clear separations from the continuum
emission. Our method also provides a three-dimensional (3D) view of the ejecta
by individually disentangling red- and blueshifted spectral components
associated with images of the Si, S, Ar, Ca, and Fe emission. This approach
provides insights into the morphology of the ejecta distribution in those two
remnants.

Results. Those mappings have allowed us to thoroughly investigate the
asymmetries in the intermediate-mass elements and Fe distribution in two Type
Ia supernova remnants. We also compared the results with the core-collapse
Cassiopeia A remnant, which we had studied previously. The images obtained
confirm, as expected for Type Ia SNRs, that the Fe distribution is mostly
closer to the core than that of intermediate-mass elements. They also
highlight peculiar features in the ejecta distribution, such as the Fe-rich
southeastern knot in Tycho.

###### Key Words.:

ISM: supernova remnants – ISM: individual objects: Kepler – ISM: individual
objects: Tycho – ISM: lines and bands – ISM: kinematics and dynamics – ISM:
structure

##  1 Introduction

The supernovae (SNe) of 1572 (hereafter, Tycho) and 1604 (hereafter, Kepler)
are undoubtedly two of the most famous historical supernova remnants (SNRs),
associated with ”stella novae” events, as documented in the literature (Brahe,
1573; Kepler, 1606). Both have left a lasting astronomical and philosophical
impact. However, despite their historical importance, the ignition mechanisms
and the deflagrating or detonating nature of the nuclear burning front in Type
Ia supernovae (SNe), including both Tycho and Kepler, are still poorly
understood. Discriminating among the single-degenerate (a white dwarf accretes
matter from a companion star) and the double-generate (explosion caused by the
merging of two white dwarfs) scenarios is still a challenge and we do not have
a precise idea of the relative quantity of single- and double-degenerate
explosions.

In this context, three-dimensional (3D) simulations have shown that
asymmetries are a key part of the explosion mechanisms (e.g., Seitenzahl et
al., 2013a; Stone et al., 2021). It has been argued that the diversity seen in
the spectral evolution of Type Ia supernovae may be a consequence of the
viewing angle on an asymmetric explosion (Maeda et al., 2010). Recently,
simulations presenting the evolution of a Type Ia SNR over a period spanning
one year after the explosion to several centuries afterward have shown that
asymmetries present in the original SN could still be observed for an extended
amount of time (Ferrand et al., 2019, 2022).

As a consequence, the study of the ejecta asymmetries in remnants from Type Ia
SNe is a logical way to probe simulations predictions and constrain explosion
mechanisms. In particular, models predict that Fe-group elements (e.g., Cr,
Mn, Ti, Fe, Ni) are located in the inner parts of the ejecta, surrounded by an
outer shell of intermediate-mass elements (e.g. Si, S, Ca and Ar); the same
models predict the velocity of intermediate-mass elements to be higher on
average than that of the Fe-group (Seitenzahl et al., 2013a; Stone et al.,
2021). While Miceli et al. (2015) confirmed this expected spatial distribution
in Tycho using XMM-Newton observations, Stone et al. (2021) discussed the case
of G1.9+0.3, which presents surprisingly high-velocity Fe emission.

As young, bright objects benefiting from deep Chandra observations, Tycho and
Kepler, both are prime targets for studying ejecta asymmetries in type Ia
SNRs. Tycho has been classified as a type Ia based on its observed light curve
and color evolution (Ruiz-Lapuente et al., 2004), and it is estimated to be at
a distance of $\sim 2.8$ kpc (Kozlova & Blinnikov, 2018). It is thought to
originate from a single-degenerate scenario, which is supported by the
discovery of a star near the center of the remnant (Ruiz-Lapuente et al.,
2004). Its environment is known to be inhomogeneous and Tycho shows traces of
interactions with the circumstellar medium (CSM; Lu et al., 2011; Williams et
al., 2013). Zhou et al. (2016) reports the observation of a clumpy molecular
bubble surrounding Tycho, which also supports a single-degenerate scenario to
explain its origin.

Kepler’s type Ia identification is mainly based on X-ray observations showing
shocked ejecta with strong Si, S, and Fe emission and a near absence of O
emission (Kinugasa & Tsunemi, 1999; Cassam-Chenaï et al., 2004; Reynolds et
al., 2007). It is estimated to be at a distance of $\sim 5$ kpc (Sankrit et
al., 2016) and its relatively young age indicates that most of the emission is
due to heated-up ejecta material rather than swept-up interstellar medium (ISM
; Kinugasa & Tsunemi, 1999; Cassam-Chenaï et al., 2004). However, it seems to
be interacting with the dense CSM (Blair et al., 2007; Williams et al., 2012),
which would favour a single-degenerate scenario (Patnaude et al., 2012).
Nevertheless, a companion star is yet to be found (Kerzendorf et al., 2014;
Sato & Hughes, 2017; Ruiz-Lapuente et al., 2018)

The aim of this paper is to present a thorough study of the morphological
asymmetries in the Si, S, Ca, Ar, and Fe in Kepler and Tycho’s ejecta. Using a
blind source separation method (BSS), we were able to extract detailed and
unpolluted maps associated with a mean spectrum for each individual elements,
which we used to quantify and compare the morphological and velocity
asymmetries. In Section 2, we describe our method. In Sections 3.1 and 3.2, we
present and discuss our results for Kepler and Tycho, respectively. Finally,
since we are applying the same methodology as we did in Picquenot et al.
(2021) for the Cassiopeia A core-collapse supernova, we compare the
asymmetries we observed in the Tycho, Kepler, and Cassiopeia A SNRs in Section
4.

Object | ObsID | Date | ACIS-S | ACIS-I
---|---|---|---|---
Kepler | 116 | 2000 Jun 30 | 48.8 | -
| 4650 | 2004 Oct 26 | 46.2 | -
| 6714 | 2006 Apr 27 | 157.8 | -
| 6715 | 2006 Aug 3 | 159.1 | -
| 6716 | 2006 May 5 | 158.0 | -
| 6717 | 2006 Jul 13 | 106.8 | -
| 6718 | 2006 Jul 21 | 107.8 | -
| 7366 | 2006 Jul 16 | 51.5 | -
| 16004 | 2014 May 13 | 102.7 | -
| 16614 | 2014 May 16 | 36.4 | -
|  | Total | 975.1 | -
Tycho | 115 | 2000 Sep 20 | 48.9 | -
| 3837 | 2003 Apr 29 | - | 145.6
| 7639 | 2007 Apr 23 | - | 108.9
| 8551 | 2007 Apr 26 | - | 33.3
| 10093 | 2009 Apr 13 | - | 118.3
| 10094 | 2009 Apr 18 | - | 90.0
| 10095 | 2009 Apr 23 | - | 173.4
| 10096 | 2009 Apr 27 | - | 105.7
| 10097 | 2009 Apr 11 | - | 107.4
| 10902 | 2009 Apr 15 | - | 39.5
| 10903 | 2009 Apr 17 | - | 23.9
| 10904 | 2009 Apr 13 | - | 34.7
| 10906 | 2009 May 3 | - | 41.1
| 15998 | 2015 Apr 22 | - | 147.0
| 23538 | 2021 Sep 27 | - | 28.7
| 23541 | 2021 Oct 16 | - | 28.1
| 24391 | 2021 May 29 | - | 28.4
|  | Total | 48.9 | 1,254.0
Table 1: Data from Chandra used in our study.

##  2 Method

To retrieve accurate maps of the main thermal line components in Kepler and
Tycho, we used a BSS method based on the General Morphological Components
Analysis (GMCA, see Bobin et al., 2015), first introduced for X-ray
observations by Picquenot et al. (2019). This algorithm can disentangle
spectrally and spatially mixed components from an X-ray data cube of the form
$(x,y,E)$. In particular, the algorithm has proven capable of extracting
extremely faint components from X-ray data cubes. An updated version of this
algorithm, the pGMCA (see Bobin et al., 2020), has been developed to take into
account the Poissonian nature of X-ray data. It was first used on Cassiopeia A
data from Chandra,  demonstrating its suitability for producing clear,
detailed, and unpolluted images of both the ejecta (Picquenot et al., 2021)
and the synchrotron emission at different energies (Picquenot et al., 2023).
Given the pGMCA algorithm is primarily designed to focus on spatial
morphology, the extracted spectra are sometimes imperfectly reconstructed;
however, they can usually be used to assess the retrieved components’ nature
and physical properties.

###  2.1 Image analysis

In this paper, we used pGMCA on stacked Chandra observations of Kepler and
Tycho (see Table 1) with specific binnings to increase the photon counts in
each bin. For Kepler, we used a $16$” spatial binning and a $43.8$ eV energy
binning. For Tycho, we used a $30$” spatial binning and a $43.8$ eV energy
binning.

We applied the pGMCA algorithm on the resulting $(x,y,E)$ data cubes of Kepler
and Tycho around five energy bands surrounding main emission lines: Si
($1.5-2.2$ keV), S ($2.0-2.7$ keV), Ar ($2.7-3.7$ keV), Ca ($3.4-4.4$ keV),
and Fe ($5.0-8.0$ keV). We were able to retrieve maps of their spatial
distribution associated with spectra, successfully disentangling them from the
synchrotron emission or other unwanted components. The elements emission
distribution were retrieved as different images associated with spectra that
we interpret as being the same emission lines slightly red or blueshifted. We
labelled the components with the most extreme shifts as “redshifted” and
“blueshifted.” We conveniently labeled the other components as “other.” We
note that with pGMCA, the number of components to retrieve is user-defined.
After several tests, we chose the number that gave the most morphologically
different and physically meaningful components for each energy line.

We then applied the power-ratio method (PRM) to quantitatively analyze and
compare the asymmetries in the images extracted by the pGMCA algorithm. The
PRM was developed by Buote & Tsai (1995) and previously employed for use on
SNRs (Lopez et al., 2009a, b, 2011; Picquenot et al., 2021). It consists of
calculating multipole moments in a circular aperture positioned on the
centroid of the image, with a radius that encloses the whole SNR. Powers of
the multipole expansion $P_{m}$ are then obtained by integrating the $m$th
term over the circle. To normalize the powers with respect to flux, they are
divided by $P_{0}$, thus forming the power ratios $P_{m}/P_{0}$. For a more
detailed description of the method, see Lopez et al. (2009b).

The $P_{2}/P_{0}$ and $P_{3}/P_{0}$ terms convey complementary information
about the asymmetries in an image. The first term is the quadrupole power-
ratio and quantifies the ellipticity and elongation of an extended source,
while the second term is the octupole power-ratio and is a measure of mirror
asymmetry. Both are to be compared simultaneously to ascertain the asymmetries
in different images.

To compare asymmetries in the blue- and redshifted parts of the elements’
distribution, we proceeded as in Picquenot et al. (2021). In a first step, we
calculated the $P_{2}/P_{0}$ and $P_{3}/P_{0}$ ratios of each element’s total
distribution by using the sum of the shifted components maps as an image,
whose centroid is used as an approximation of the center-of-emission of the
considered element. Then, we calculated the power ratios of the blue- and
redshifted images separately using the same center-of-emission. Ultimately, we
normalized the power ratios thus obtained by the power ratios of the total
element’s distribution:

$P_{i}/P_{0\textit{ (shifted / total)}}=\frac{P_{i}/P_{0\textit{ (red or blue
image)}}}{P_{i}/P_{0\textit{ (total image)}}},$ (1)

where $i=2$ or $3$ and $P_{i}/P_{0\mbox{ (red or blue image)}}$ is calculated
using the centroid of the total image. That way, we can compare the relative
asymmetries of the shifted parts of different elements, without the comparison
becoming biased by the original asymmetries of the whole distribution.

Finally, to obtain an estimation of the errors in our analysis, we used the
constrained bootstrap introduced in Picquenot et al. (2021). This method does
not produce physically meaningful errors, but produces resamplings of the
original data to assess the robustness of the pGMCA algorithm. Unfortunately,
the algorithm is so sensitive to initial conditions that on some resamplings,
it often fails to capture the same components found in the original data. For
that reason, we defined our error bars as the standard deviation of the $50$
best results out of $110$ resamplings for each component. The resulting errors
constitute a tentative attempt at estimating the uncertainties when using an
advanced analysis tool, such as pGMCA, as a general and non-trivial
problematic. We note that the error values tend to increase with the total
number of counts, as more counts means more potential components to
disentangle, which reduces the algorithm’s robustness.

###  2.2 Spectral analysis

Although we classified the extracted pGMCA components by their 2D morphologies
and spectra, the differences in line centroid for each component do not
necessarily correspond to the plasma’s Doppler shift. The ionization state of
the plasma also affects spectral features: lower ionization plasmas typically
have lower-energy emission lines. Thus, in order to robustly classify certain
components as “redshifted” or “blueshifted,” we needed to disentangle the
effects of ionization and Doppler shifts. To do so, we fit the plasmas using
Xspec (version 12.13.0e). For each component, we simultaneously fit the
background-subtracted emission from all elements using a complex plasma model
TBabs*(vpshockSi-Ca+vvpshockFe+powerlaw), which includes a vpshock component
with variable abundances for the Si-Ca emission, a vvpshock component to
capture the Fe K-$\alpha$ and surrounding element emission, along with a power
law to capture any remaining nonthermal emission in these components. The
uncertainties for best-fit parameters reported reflect the 90% confidence
interval as determined by the Xspec error command. We used the redshift
parameter as a proxy for Doppler shift. Essentially, we wanted to ensure that
each of our identified components have distinct velocities, and that the
shifts we observe cannot be explained as similar plasmas in different
ionization states. This process also serves as a double check that the pGMCA
method is extracting physically plausible emission and can be understood as
shock-heated ejecta, ISM, or synchrotron emission. The spectra extracted by
pGMCA are sometimes imperfectly reconstructed and the resulting fits can be
burdened by large errors.

Figure 1: Total images of the different line emission spatial structure as
retrieved by pGMCA. Left: For Kepler. Right: For Tycho. The black star
represents the image centroid adopted in the PRM analysis. The color scale is
presented as a square root.

##  3 Results

Figure 1 shows the total images for all five line emission in both remnants,
obtained by summing the shifted parts. It also indicates the centroid of each
image that is adopted in the PRM. In Kepler, Fig. 2 shows the differently
shifted parts together with their associated spectra, Fig. 3 shows the PRM
plot, and Fig. 4 gives the results of the spectral analysis. In Tycho, Fig. 5
shows the differently shifted parts, Fig. 6 shows the PRM plot, and Fig. 7
givds the results of the spectral analysis. From the images of Fig. 2 and Fig.
5, we derived the ratios presented in Table 2 (for Kepler) and Table 3 (for
Tycho), which highlight the relative importance of each component in the total
line emission. Table 4 presents the best fit parameters from the spectral
analysis from both remnants.

| Redshifted | Other 1 | Blueshifted
---|---|---|---
Si | 0.20 | 0.14 | 0.66
S | 0.19 | 0.08 | 0.73
Ar | 0.14 | 0.17 | 0.69
Ca | 0.30 | 0.23 | 0.47
Fe | 0.26 | - | 0.74
Table 2: Fractions of the counts in the total image that belong to the
redshifted, the blueshifted or the “other” parts, for each line in Kepler.
Figure 2: Red, blue-shifted, and “other” parts in the Si, S, Ar, Ca, and Fe
spatial distribution and their associated spectrum as found by pGMCA in
Kepler. The spectra in red correspond to the application of the algorithm on
real data, while the dotted gray spectra correspond to 50 selected
applications on constrained bootstrap resamplings, illustrating statistical
uncertainties. The x-axis is in keV and the y-axis in counts.

###  3.1 Kepler

In Kepler, the line emission total images from Fig. 1 appear very similar to
each other, with the exception of Fe that shows a more compact distribution,
generally closer to the geometrical center. The shifted images from Fig. 2
also show a great similitude between the differently shifted components for
each line emission. The most blue-shifted component always appears as a broad
rim following the forward shock in the north, and the “other” component
follows a similar but thinner distribution, closer to the edge near the shock
and with stronger emission in the west. The most extreme red-shifted component
is more diffuse, with peaks present in the north-east, the west and slightly
to the east of the geometrical center. The red component from the Fe line is
significantly different, with a clear peak near the center and comparatively
less prominent features other than the northern rim. The Si “other” component
is more complex and shows more features than the northern rim dominating
“other” components in S, Ar, and Ca emission. This might be due to the higher
number of counts at lower energies, which would allow pGMCA to reconstruct an
image endowed with more details.

The PRM plot shown in Fig. 3 highlights a clear trend, where the “red” and
“other” components are more asymmetric than the blueshifted parts, with the
exception of the “other” Si component. W note that the centroid used in the
PRM calculation is the centroid from the elements’ total emission, which in
the case of Kepler offsets it significantly to the north of the geometrical
center. The “blue” and “other” emission are dominated by the northern rim,
thus more offset than the “red” and total line emission for Si, S, Ar, and Ca.
The Fe line, which shows emission closer to the core, does not follow the
exact same trend.

Figure 3: Quadrupole power-ratios $P_{2}/P_{0}$ versus the octupole power-
ratios $P_{3}/P_{0}$ of the red, blueshifted, and “other” images of the
different line emission in Kepler shown in Fig. 2, normalized with the
quadrupole and octupole power-ratios of the total images. The dots represent
the values measured for the pGMCA images obtained from the real data, and the
error bars the standard deviation of fifty selected outputs from pGMCA on
constrained bootstrap resamplings. Figure 4: Xspec fits of the spectra
retrieved by pGMCA in Kepler. The parameter values and a description of the
fittings can be found in Table 4.

Katsuda et al. (2015) estimated that $\sim 87\%$ of the outer ejecta is
shocked, while an estimate based purely on the shocked and unshocked volumes
gives a value of $\sim 60\%$ (Holland-Ashford et al., 2023). In both cases,
the ratios from Table 2 show that most of the ejecta emission (used as a proxy
for ejecta mass) is slightly blueshifted (velocity estimated: -540 $\pm$ 120
km s-1), while a small part is significantly redshifted (velocity estimated:
$\sim$6900 $\pm$ 330 km s-1 for Si-Ca and $\sim$3000 $\pm$ 600 km s-1 for Fe).
This result is consistent with conservation of momentum; while most of the
ejecta mass is moving north, a smaller portion is ejected directly opposite
us, along the line of sight. Vink (2008) also found the proper motion of the
northern rim to be low, which is indicative of a low tangential velocity.

The low velocity of the northern rim might be a sign of a relatively recent
interaction with CSM, which is consistent with previous studies reporting a
north-south density gradient of an order of magnitude in Kepler’s surrounding
(Blair et al., 2007; Williams et al., 2012). Although an interaction with the
CSM would favour a single-degenerate scenario (Patnaude et al., 2012), a
companion star is yet to be found (Kerzendorf et al., 2014; Sato & Hughes,
2017; Ruiz-Lapuente et al., 2018).

Figure 5: Red-, blueshifted, and “other” parts in the Si, S, Ar, Ca, and Fe
spatial distribution and their associated spectrum as found by pGMCA in Tycho.
The spectra in red correspond to the application of the algorithm on real
data, while the dotted gray spectra correspond to fifty selected applications
on constrained bootstrap resamplings, illustrating statistical uncertainties.
The x-axis is in keV and the y-axis in counts.

As described in Section 2.2, we performed spectral fitting to ensure that the
labels we attributed to the components retrieved by pGMCA in Kepler’s SNR were
meaningful. For our “red” pGMCA components in Kepler’s SNR, our spectral fits
returned ionization timescales of a few times $10^{10}$ s cm-3 for the Si, S,
Ar, & Ca emission, and $\sim$1 $\times 10^{10}$ for Fe ejecta. Our best-fit
redshifts corresponded to velocities of 6900 $\pm$ 300 km s-1 for the Si–Ca
emission and 3000 $\pm$600 km s-1 for the Fe-component (90% confidence
intervals; see Fig. 4 and Table 4). We conclude that these pGMCA components
are strongly redshifted, matching our labels. These high velocities match with
our 2D maps, as a large portion of the emission is from the knot of emission
in the center of the SNR: a region that should have a low tangential velocity
but a high line-of-sight velocity. Our estimated Fe velocity matches well with
the 2531${}^{+289}_{-290}$ km s-1 ejecta velocity of the “CS-E Center” region
reported by Kasuga et al. (2021) and the $\sim$500-3200 km s-1 for an ejecta
knot near this central SE knot reported by Millard et al. (2020), confirming
its redshift.

For our “other” pGMCA components in Kepler’s SNR, our spectral fits returned
an ionization timescale of 1.2 $\times 10^{11}$ s cm-3 and a redshift
corresponding to a line-of-sight velocity of 420${}^{+150}_{-20}$ km s-1:
slightly redshifted (see Fig. 4 and Table 4). A low velocity for this region
makes sense as emission in these components is mostly coming from the northern
rim of Kepler’s SNR, and thus should have a high tangential but low line-of-
sight velocity. Importantly, our fits prove that this pGMCA component has a
different Doppler-shifted velocity than our labeled “red” components. Millard
et al. (2020) found small velocities for ejecta in the northern rim,
consistent with zero or slight (hundreds of km s-1) doppler shifts.

For our “blue” pGMCA components in Kepler’s SNR, our spectral fits returned
ionization timescales of $\sim 1\times 10^{11}$ s cm-3 for the Si, S, Ar, & Ca
emission, and $\sim$3 $\times 10^{10}$ for Fe ejecta. Our fits produced
redshifts corresponding to velocities in the range of -720 to 100 km s-1 for
the Si–Ca emission and -480 to 660 km s-1 for the Fe-component (90% confidence
intervals; see Fig. 4 and Table 4). These values are significantly lower than
the Doppler shifts in our “red”-shifted material and their centers are
slightly lower than the velocities of our pGMCA components labeled “other 1.”
Visually, there is a lot of overlap in our 2D ejecta maps for our “other 1”
and “blue” pGMCA components, but the latter have more emission in the western
and central regions of Kepler’s SNR. Again, it makes sense that our measured
Doppler-shifted velocities are small because the emission in these components
is mainly in the outer rim of Kepler’s SNR. Kasuga et al. (2021) reported
edge-ejecta velocities of -300 to 700 km s-1, matching our findings. Our
findings of high redshifted velocities and only moderate blueshifted
velocities match the findings of Millard et al. (2020), supporting the
existence of bulk asymmetry in the ejecta of Kepler’s SNR .

###  3.2 Tycho

In Tycho as well, the total line emission images from Fig. 1 appear very
similar to each other, with the exception of Fe that shows a more compact
distribution. The shifted images from Fig. 5 show also a great similitude
between the differently shifted components from Si and S. For Ar and Ca, the
algorithm could not retrieve as many components, so the results appear to be a
mix of the different emission extracted from Si and S. In particular, the
“other 1” and “other 2” components from Ar are endowed with poorly
reconstructed spectra that we could not use in our spectral analysis, so they
were labeled uniquely on the basis of their morphological distribution.

| Redshifted | Other 1 | Other 2 | Blueshifted
---|---|---|---|---
Si | 0.23 | 0.40 | 0.25 | 0.12
S | 0.11 | 0.29 | 0.50 | 0.10
Ar | - | 0.11 | 0.10 | 0.79
Ca | 0.22 | - | - | 0.78
Fe | - | 0.21 | 0.57 | 0.22
Table 3: Fractions of the counts in the total image that belong to the
redshifted, blueshifted, or the “other” parts, for each line in Tycho.

The most blueshifted component appears as a collection of clumps mainly
present in the north, while the most extreme redshifted component appears a
collection of evenly distributed clumps throughout the SNR. A detailed study
of the proper motion of these ejecta clumps was carried out with GMCA in
Godinaud et al. (2023). In Si and S, the “other 1” components look like
parenthesis-shaped rims surrounding the remnant east and west, while the
“other 2” components mainly consist of a bright clump in the northwest. We
only found two components in the Ca emission and three in the Ar, and their
blueshifted emission appears like a mix from the “blue,” “other 1,” and “other
2” components from Si and S. The “other 1” and “other 2” components from Ar
appear similar to those from Si and S. As the spectra appear especially poorly
reconstructed for Ar, we can infer that the pGMCA struggled to properly
disentangle them. The Fe components appear significantly different. Most of
the blueshifted emission appears to be in a small clump in the east previously
studied by Yamaguchi et al. (2017), while the “other 1” and “other 2”
components appear as broad rims in the southwest and northwest of the remnant,
respectively. There is no proper redshifted emission.

The PRM plot shown in Fig. 6 does not highlight any trend. Table 3 shows that
most of the emission is in the intermediate components “other 1” and “other 2”
for Si, S, and Fe. For Ar and Ca, most of the emission is in the blueshifted
component, that appears like a mix of the “blue”, “other 1,” and “other 2”
emission from Si and S. The “other 2” component is largely dominated by a
small clump in the northwest, but other components show a roughly spherical
distribution.

Williams et al. (2017) and Sato & Hughes (2017) performed ejecta velocity
measurements of Tycho’s SNR using Chandra ACIS observations. Our distribution
maps seem to roughly match their findings: the blueshifted ejecta are mainly
present in the center-east and north, while the redshifted ejecta are present
in the center-west and south; we both find ionization timescales to be in the
range of $\sim 10^{9}-10^{11}$ s cm-3 and $\sim$1.0–few keV electron
temperatures. Sato & Hughes (2017) found velocities of around -3000 to -4000
km s-1 for the blueshifted Si ejecta, and velocities of -1000 to -2500 km s-1
km s-1 for the eastern knot. Our spectral fits report similarly strong
blueshfited velocities: -1600 $\pm$ 100 km s-1 for the Si & S emission and
-2400 $\pm$ 1100 km s-1 for the Ar, Ca, and Fe emission: emission dominated by
the eastern knot (90% CI; see Fig. 7 and Table 4). As our maps have most of
their emission on the edges of Tycho’s SNs – namely, regions with lower line-
of-sight velocities, but higher tangential ones — it makes sense that our
measured Doppler shifts are smaller in magnitude. Thus, it is reasonable to
conclude that our “blue” labeled pGMCA components are accurately capturing
ejecta moving towards us. These results are also consistent with the line-of-
sight velocities found by Godinaud et al. (2023), which were retrieved in a
similar way using GMCA components.

Figure 6: Quadrupole power-ratios $P_{2}/P_{0}$ versus the octupole power-
ratios $P_{3}/P_{0}$ of the redshifted, blueshifted, and “other” images of the
different line emission in Tycho shown in Fig. 5, normalized with the
quadrupole and octupole power-ratios of the total images. The dots represent
the values measured for the pGMCA images obtained from the real data, and the
error bars the standard deviation of fifty selected outputs from pGMCA on
constrained bootstrap resamplings. Figure 7: Xspec fits of the spectra
retrieved by pGMCA in Tycho. The parameter values and a description of the
fittings can be found in Table 4.

The “low-velocity ejecta blobs” from Sato & Hughes (2017) are around the edges
of Tycho’s SNR and should thus be captured by our “other1” components. Our
fits to Si, S, and Ar emission produce best-fit Doppler velocities of 780–1530
km s-1 and our fits to Fe produce Doppler velocities of -1530 to 210 km s-1
(90% CI; see Fig. 7 and Table 4). These velocities are moderate to lower in
magnitude than those in our “blueshifted” ejecta maps. Interestingly, even
though our “Other 1” Fe pGMCA component displays emission clearly to the left
of the Fe K-$\alpha$ complex, we find that it is slightly blueshifted. We note
that the pGMCA FeK-$\alpha$ spectral line complex is visially cutoff when
plotted, showing a sharp drop-off and small line width that does not match up
with any plasma model we tried. It is possible that this emission is part of a
larger plasma region (perhaps, in combination with the similarly cutoff “Fe-
Blue” component) that is all blueshifted.

Our “other 2” components are dominated by a northwestern knot and more diffuse
emission. Our Si & S pGMCA components are best fit by a plasma with a moderate
($\sim$5.5e10 cm s-3) ionization timescale and our Fe pGMCA component is best
fit by a plasma with a low (few times 109 s-3 cm s-3) ionization timescale.
However, the best-fit Doppler shifts were very uncertain, settling on a spread
of values from different fits. The Si & S emission had Doppler shifts
corresponding to -3600 to 2100 km s-1, while the Fe emission had a Doppler
shift corresponding to -600 to 2400 km s-1 (90% CI; see Fig. 7 and Table 4).
These are consistent with zero or moderate blue- and redshifts, possibly
greater in magnitude than our “other 1” emission. The large spread and
consistency with low-to-moderate Doppler shifts is consistent with our 2D
maps; there seems to be more emission from both the center and rim of Tycho’s
SNR in these components, representing the NW $\sim$half of Tycho’s SNR in Si &
S emission as opposed to only the rim.

For Si–Ca, “red” labeled components (excluding Ar as its spectra are too poor
in quality), we find extreme velocities of 4500–9500 km s-1 (90% CI; see Fig.
7 and Table 4). These velocities closely match the 4000–7500 km s-1 velocities
found by Sato & Hughes (2017), providing support that our “red” labeled pGMCA
components are indeed capturing ejecta moving away from us.

Additionally, our pGMCA components for Tycho’s SNR generally match up with a
previous GMCA analysis on Si emission in Tycho’s SNR (Godinaud et al., 2023).
They found that the redshifted emission is largely present in the southern
hemisphere of the SNR, with central velocities for both red- and blueshifted
ejecta of a few thousand km s-1, as we do. Visually, our combined ”ed” and
“other 1” pGMCA Si components look like their identified redshifted map, and
our combined “other 2” and “blue” pGMCA Si components look like their
identified blueshifted map.

Overall, the mostly symmetrical line-of-sight velocity distribution, the
similar ejecta morphologies, and the strong quadrupole power-ratios suggest a
bipolar explosion with a north and south elongation tilted towards the
observer and even amounts of ejecta accelerated in each direction.

Parameter | Redshifted | Other 1 | Other 2 | Blueshifted
---|---|---|---|---
|  | Kepler |  |
$\tau_{Si-Ca}$ (s cm-3) | 5.3e10${}^{+4.3e10}_{-1.3e10}$ | 1.2e11${}^{+7.4e10}_{-3.8e10}$ | - | 1.7e11${}^{8.5e9}_{-1.7e10}$
$\tau_{Fe}$ (s cm-3) | 1.1e10${}^{+1.9e10}_{-4.7e-9}$ | - | - | 2.3e10${}^{+3.4e9}_{-2.6e9}$
redshiftSi-Ca | 2.3e-2${}^{+1.4e-3}_{-6.3e-4}$ | 1.4e-3${}^{5e-4}_{-7e-5}$ | - | 2.2e-4${}^{+1e-4}_{-2.2e-3}$
redshiftFe | 9.9e-3${}^{+1.9e-3}_{-2.5e-3}$ | - | - | -1.6e-3 to 2.2e-3
VSi-Ca (km s-1) | 6900${}^{+420}_{-190}$ | 420${}^{+150}_{-20}$ | - | 66${}^{+30}_{-660}$
VFe (km s-1) | 3000${}^{+570}_{750}$ | - | - | -480 to 660
|  | Tycho |  |
$\tau_{Si-Ca}$ (s cm-3) | 8e10 $\pm$ 3e10 | 4.3e10${}^{+8.8e9}_{-5.5e9}$ | 5e10 to 7e10 | 1.07e11${}^{+1.2e10}_{-1.0e10}$
$\tau_{Fe}$ (s cm-3) | - | 6.3e9${}^{+6.3e9}_{-3e9}$ | 6e8 to 7e9 | 1.77e10${}^{+5.1e9}_{-5.3e9}$
redshiftSi-Ca | 2.9e-2 ${}^{+2.8e-3}_{-1.2e-2}$ | 2.8e-3${}^{+2.3e-3}_{-2.0e-4}$ | -1.4e-2 to 7e-3 | -5.2e-3${}^{+2.2e-4}_{-3.5e-4}$
redshiftFe | - | -2.4e-3${}^{+2.7e-3}_{-3.1e-3}$ | -2e-3 to 8e-3 | -8.2e-3${}^{+3.9e-3}_{-3.4e-3}$
VSi-Ca (km s-1) | 8700${}^{+840}_{-4200}$ | 840${}^{+690}_{-60}$ | -3600 to 2100 | -1560${}^{+66}_{-105}$
VFe (km s-1) | - | -720${}^{+930}_{-810}$ | -600 to 2400 | -2460${}^{+1170}_{-1020}$
Table 4: Results of fitting the spectral components retrieved by pGMCA. In
Kepler, we see that the measured Doppler shifts are all consistent with
positive (redshifted) velocities. Our “red” component is highly redshifted,
while the other components have small velocities. For the Fe velocity in the
“blue” component, multiple equally-valid fits returned different best-fit
Doppler shifts; we report the spread between these models.
In Tycho, we see that our measured Doppler shifts span large positive and
negative velocities. Our “red” component is highly redshifted, our “blue”
component is moderately blueshifted, and our “other 1” component shows a mix
of both slight redshifts and blueshifts. As noted in Section 3.2, the apprent
cut-off of the Fe K-$\alpha$ line in this component suggest that this “other
1” Fe emission might actually be part of the “blue” component. As for the
“other 2” Tycho component, multiple equally valid fits returned different
best-fit parameters; we report the spread between these models instead of the
90% CI from a single model.

##  4 Comparison with Cassiopeia A

The PRM plot in Fig. 8 shows a comparison of the asymmetries of the total line
emission in Kepler and Tycho, to which we added the PRM values we found for
Cassiopeia A in Picquenot et al. (2021). It appears that the total line
emission in Cassiopeia A are all significantly more elliptical asymmetric than
those in Kepler and Tycho and mostly more mirror-asymmetric. This is
consistent with the results of Lopez et al. (2009a), highlighting that core
collapse SNRs had an overall morphology that tended to be more asymmetric than
Type Ia SNRs.

In the case of Cassiopeia A, we had noted in our previous paper that heavier
elements were ejected more asymmetrically than lighter elements, which is
consistent with core-collapse simulations (Wongwathanarat et al., 2013; Janka,
2017; Gessner & Janka, 2018; Müller et al., 2019). Here, we can notice that
Kepler and Tycho do not follow the same trend. Furthermore, Si, S, and Ar even
follow a reverse trend in both Tycho and Kepler, were the lighter elements are
the most asymmetric. The Fe in Kepler appears to be the most symmetrically
distributed element and the Ca in Tycho is one of the two most mirror-
symmetric elements. Interestingly, the Ca emission is the most elliptically
asymmetric element in both remnants. With a sample size in this work of only
two objects, it is premature to draw specific conclusions regarding the link
between asymmetries and elemental atomic numbers in Type Ia SNRs. Subsequent
studies, with the application of the same method on a larger number of SNRs
would allow us to explore this question more thoroughly.

However, simulation results predict that Fe-group elements are located in the
inner parts of the ejecta, surrounded by an outer shell of intermediate-mass
elements (e.g., Si, S, Ca, and Ar; Seitenzahl et al., 2013a; Stone et al.,
2021). Our results are consistent with this, since the Fe emission appears
more compact for both remnants and closer to the core than the other emission,
while also presenting a significantly different morphology. The line-of-sight
velocities are harder to compare, as our fits are endowed with large error
bars. However, the velocity we retrieved are mostly compatible with larger
velocities for the intermediate-mass elements than for Fe, as expected. The
main discrepancy is the blueshifted component in Tycho, where $V_{Fe}$=-2460
$\pm$ 1100 km s-1 and we $V_{Si-Ca}$= -1500 $\pm$ 100 km s-1. The blueshifted
Fe component mainly consists of a southeastern knot, that Millard et al.
(2020) found to move radially at about 5000 km $s^{-1}$. This knot was
thoroughly studied by Yamaguchi et al. (2017). They found its ionization
timescale to be highly different from that of its Si-rich surrounding,
favoring the hypothesis that during the explosion, this knot detached from the
rest of the Fe-rich material and formed a protrusion at the edge of the
ejecta.

Figure 8: Quadrupole power ratios, $P_{2}/P_{0}$, versus the octupole power-
ratios, $P_{3}/P_{0}$, of the total images of the different line emission in
Kepler and Tycho. The dots represent the values measured for the pGMCA images
obtained from the real data, and the error bars the standard deviation of
fifty selected outputs from pGMCA on constrained bootstrap resamplings.

##  5 Conclusions

By using a new methodology and applying it to the Kepler and Tycho Chandra
X-ray data, we were able to revisit the mapping of the intermediate-mass
elements and Fe and separate them into differently red- or blue-shifted parts,
allowing us to investigate the three-dimensional (3D) morphology of these
SNRs. These new maps and the associated spectra could then be used to quantify
the asymmetries of each component, their mean direction and their line-of-
sight velocity. The main findings of the paper are consistent with the general
results found in the previous studies cited throughout our study.

In Kepler, the most blueshifted component appears as a broad rim following the
forward shock in the north, generally considered in previous studies to be
trace of a CSM interaction. This component appears to have a low velocity,
that makes it compatible both with a slight blueshift or with a slight
redshift. A smaller portion of the the ejecta is significantly redshifted,
which is consistent with the conservation of momentum: most of the ejecta is
moving slowly to the north, while a small portion is ejected directly opposite
us at higher speeds. An extremely off-center explosion site (Maeda et al.,
2010) could result in this ejecta distribution.

In Tycho, the redshifted emission appears as a collection of clumps slightly
more present in the center-east and south, while the blueshifted emission
seems mainly present in the north and west of the remnant. The PRM plot does
not show any clear trend, and the mostly symmetrical line-of-sight velocity
distribution from past studies, consistent with our own findings, suggests a
bipolar explosion with a north-south elongation tilted towards the observer,
and even amounts of ejecta accelerated in each direction. Such a bipolar
explosion might be the result of a double-degenerate origin due where close
interactions between the white dwarfs and/or instabilities in material
accreted along the orbital plane led to an asymmetric explosion.

In both remnants, the Fe emission appears more compact, closer to the core
than the other emission and presents a significantly different morphology. The
line-of-sight velocities we derive are endowed with large error bars, but they
are mostly compatible with lower values for the Fe emission, with the notable
exception of a southeastern clump in Tycho.

A comparison with the results obtained in a previous paper on the core-
collapse SNR Casiopeia A also proved meaningful. The total line emission in
Cassiopeia A are all significantly more elliptical asymmetric than those in
Kepler and Tycho, and mostly more mirror asymmetric, which is consistent with
past studies stating that the global morphology of core collapse SNRs tended
to be more asymmetric than that of Type Ia.

In Cassiopeia A, heavier elements are ejected more asymmetrically than lighter
elements, which is consistent with core-collapse simulations. Kepler and
Tycho’s SNRs, however, seem to exhibit almost a reverse trend, where many
heavier elements present more symmetric morphologies than lighter elements in
these SNRs. This might be a result of the detonation processes and/or pre-
explosion dynamics in Type Ia SNe; for instance, an initial detonation of an
outer He-shell in the “double-detonation” mechanism or a differing number of
explosion kernels (potentially off-center) for the central explosion (e.g.,
Maeda et al. 2010; Seitenzahl et al. 2013b) could result in these observables.
We note that a sample of two SNRs is too small to conclude anything and
suggest that studies, with the application of the same method on a larger
number of targets, would enable exploration of this question more thoroughly.
More than ten SNRs already benefit from exposure above 250 ks from Chandra and
several millions of photons detected, making them great candidates for pGMCA
applications. Good targets with deep observations would include: the three
large Magellanic cloud young SNRs (0509-67.5, 0519-69.0, and N103B, each
$\sim$400 ks from 2017-2020 observations) and SN 1006 ($\sim$500 ks) for Type
Ia; G292.0+1.8 ($\sim$1 Ms), N132D ($\sim$1 Ms), and W49B ($\sim$270 ks total)
for core-collapse.

###### Acknowledgements.

The material is based upon work supported by NASA under award number
80GSFC21M0002.

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