# Jitter radiation as an alternative mechanism for the nonthermal X-ray
emission of Cassiopeia A

Emanuele Greco Anton Pannekoek Institute for Astronomy & GRAPPA, University of
Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands  Jacco Vink
Anton Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands  Amael Ellien Anton
Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands  Carlo Ferrigno Department of
Astronomy, University of Geneva, Chemin d’Ecogia 16, CH-1290 Versoix,
Switzerland  INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46,
I-23807, Merate, Italy

###### Abstract

Synchrotron radiation from relativistic electrons is usually invoked as the
responsible for the nonthermal emission observed in Supernova Remnants (SNRs).
Diffusive shock acceleration (DSA) is the most popular mechanism to explain
the process of particles acceleration and within its framework a crucial role
is played by the turbulent magnetic-field. However, the standard models
commonly used to fit X-ray synchrotron emission do not take into account the
effects of turbulence in the shape of the resulting photon spectra. An
alternative mechanism that properly includes such effects is the jitter
radiation, that provides for an additional power-law beyond the classical
synchrotron cutoff. We fitted a jitter spectral model to Chandra, NuSTAR,
SWIFT/BAT and INTEGRAL/ISGRI spectra of Cassiopeia A and found that it
describes the X-ray soft-to-hard range better than any of the standard cutoff
models. The jitter radiation allows us to measure the index of the magnetic
turbulence spectrum $\nu_{B}$ and the minimum scale of the turbulence
$\lambda_{\rm{min}}$ across several regions of Cas A, with best-fit values
$\nu_{B}\sim 2-2.4$ and $\lambda_{\rm{min}}\lesssim 100$ km.

##  1 Introduction

It is generally accepted that particle acceleration at the shock fronts of
supernova remnants (SNRs) is the result of diffusive shock acceleration (DSA)
(Bell, 1978; Malkov & O’C Drury, 2001). According to DSA, charged particles
repeatedly cross the shock front back and forth due to the magnetic-field
irregularities, gaining a few percent of energy with each crossing. Among the
accelerated particles are highly relativistic electrons, and those with
energies $\gtrsim 10$ TeV will produce X-ray synchrotron radiation commonly
observed in young SNRs (see Vink 2020 for a review). To reach those very high
energies, the electrons need to reside close to the shock front, enabling fast
repeated shock crossings. For this confinement close to the shock front a
highly turbulent magnetic field near the shock front is necessary.

Magnetic turbulence greatly affects the polarization of the light emitted.
Synchrotron radiation from a nonthermal population of electrons with a power-
law index of energy of 2–3 is intrinsically polarized at the 70–75% level (see
Eq. 3.29 in Ginzburg & Syrovatskii 1965), but Cassiopeia A (Cas A)
observations performed by the Imaging X-ray Polarimetry Explorer (IXPE)
(Weisskopf et al. 2022), showed that the degree polarization of the radiation
in the 4-6 keV energy band is roughly $3\%$ (Vink et al. 2022b), suggesting a
highly turbulent magnetic field.

Despite the crucial need for high turbulence to achieve nonthermal X-ray
emitting electrons and the recent X-ray polarization results, the most common
models used to fit X-ray synchrotron spectra do not take into account
turbulence effect in shaping the resulting photon spectra. The synchrotron
spectral shape emitted by a population of electrons having a power-law
distribution with high-energy cutoff with particle index $\xi$ is, in the
assumption of uniform magnetic field:

$n(\hbar\omega)\propto(\hbar\omega)^{-\Gamma}\rm{exp}\bigg{[}-\Big{(}\frac{\hbar\omega}{\hbar\omega_{c}}\Big{)}^{\beta}\bigg{]},$
(1)

where $\Gamma=(\xi+1)/2$. In the loss-limited scenario adopted in Zirakashvili
& Aharonian (2007), $\beta=0.5$ while $\beta\approx 1$ in the srcut model
available in XSPEC Reynolds & Keohane (1999). The latter has been the most
commonly used to fit X-ray synchrotron-dominated spectra of SNR— e.g. Bamba et
al. 2005; Reynolds et al. 2008; Miceli et al. 2013—but recently X-ray SNR
spectra were also analyzed using the relations from Zirakashvili & Aharonian
(2007), see e.g. Tsuji et al. (2021), Giuffrida et al. (2022) and Sapienza et
al. (2022). In particular, Zirakashvili & Aharonian (2007) derived the
analytical expression for the spectra of shock-accelerated electrons in the
scenario in which the electron energy is loss-limited. They also recovered the
non-thermal synchrotron spectra emitted by a population of such electrons. It
is also common to describe the synchrotron emission with a simple power-law
spectrum with the corresponding photon index $\Gamma\gtrsim 2.5$ (Bamba et al.
2005; Cassam-Chenaï et al. 2007; Helder et al. 2012). This is much steeper
than the typical radio spectral index of SNRs of $\alpha\approx 0.5$,
corresponding to the photon index $\Gamma=\alpha+1\approx 1.5$ or, for
energies beyond the cooling break, $\Gamma_{\rm cb}=\Gamma+0.5\approx 2$. This
discrepancy has been usually explained by taking into account the curvature of
the spectrum due to the cut-off expected in both age-limited (Reynolds &
Keohane 1999) and loss-limited scenarios (Zirakashvili & Aharonian 2007).
However, there are several detections of power-law spectra extending well
beyond the soft X-ray band (i.e. $\gtrsim 10$ keV), at odds with the
continuously steepening to be expected from a spectrum beyond the spectral
cutoff.

The best example is provided by the SNR Cas A, a young ($\sim$ 350 years old)
and bright core-collapse SNR located at a distance of 3.4 kpc (Reed et al.
1995). Cas A shows a power-law extending at least up to 50 keV in NuSTAR data
(Grefenstette et al. 2015) and up to 100 keV according to RXTE data (Allen et
al. 1997), BeppoSAX (Vink & Laming, 2003) and more recent publications based
on INTEGRAL and SWIFT (Renaud et al. 2006; Wang & Li 2016).

It is therefore natural to ask if a different mechanism, potentially linked to
the often overlooked small-scale magnetic-field turbulence, might be
responsible for this extended power-law component. The spectrum resulting from
a motion of a relativistic electron in a highly non-uniform magnetic field was
first investigated by Toptygin & Fleishman (1987) (hereafter TF87) and, more
recently, by Kelner et al. (2013) (hereafter K13). The analytical results have
also been numerically verified by Reville & Kirk (2010), but for the
emissivity of a single particle. These authors refer to this radiation as
_jitter radiation_. They show that an ensemble of relativistic electrons
embedded in a highly turbulent magnetic field leads to a more sophisticated
spectrum than the standard synchrotron one (see Fig. 1, adapted from TF87).

Figure 1: Radiation spectrum $P(\omega)$ produced by an ensemble of electron
in a turbulent magnetic field (adapted from TF87). See also Reville & Kirk
(2010) (their Fig. 5 and 6) for numerical results on the single particle
emissivity. See text for details.

Jitter radiation is sensitive to turbulence on length scales of

$\lambda\ll 170(B/100~{}{\rm\mu G})^{-1}~{}{\rm km},$ (2)

with $B=100~{}{\rm\mu G}$–$500~{}{\rm\mu G}$ being typical for the strengths
reported for Cas A (Helder et al., 2012). These are scales much smaller than
usually considered for magnetic-field turbulence responsible for cosmic-ray
acceleration ($\sim 10^{14}-10^{15}$ cm). The main factor limiting the
magnetic turbulence at short scale is Landau damping which is relevant at a
scale $\lambda_{Ld}=\sqrt{\frac{T_{e}}{4\pi e^{2}n_{e}}}$ with $T_{e}$ the
electron temperature and $n_{e}$ the electron density. In order to make jitter
radiation possible, it is then needed that photon formation length
$\lambda_{pf}\gg\lambda_{Ld}$. Considering that a typical temperature of Cas A
is 3 keV and an upstream density is $\approx 2$ cm-3 (Hwang & Laming 2012) we
obtain: $\lambda\gg\lambda_{Ld}\approx 0.15$ km.

From Fig. 1 we can distinguish three main regimes: when the frequency $\omega$
is much lower than $\omega$break the radiation spectrum P($\omega$) is a
power-law with an exponential $\alpha=-\frac{\xi-1}{2}$, consistent with the
prediction for the synchrotron radiation in a homogeneous magnetic field (e.g.
Zirakashvili & Aharonian 2007); for $\omega$$~{}\approx$ $\omega$break
P($\omega$) faces a decline, whose characteristics depend on the cutoff
parameter $\beta$; for $\omega$$\gg$ $\omega$break, i.e. outside of this
intermediate regime, P($\omega$) is, again, a straight power-law but here the
exponential $\nu_{B}$ is the exponential of the magnetic-field turbulence
spectrum. We here do not focus on the mathematical aspects of jitter radiation
(the whole derivation can be found in TF87 and K13), but we just want to
highlight the following: $\omega$break is the break frequency of the jitter
radiation, related to the cutoff frequency $\omega_{\rm cut}$ of standard
synchrotron, i.e. with the electrons embedded in an homogeneous magnetic
field, as $\omega_{\rm break}=\omega_{\rm
cut}\times\frac{R_{\rm{L,j}}}{\lambda}$ with $R_{\rm{L,j}}$ the
nonrelativistic Larmor radius; $\omega$max = $\omega$${}_{\rm
cut}\times(R_{\rm L,j}/\lambda)^{3}$ is the maximum frequency to which the
additional power-law extends (K13).

In this paper, we analyzed Chandra/ACIS-S, NuSTAR/FPMA,B, INTEGRAL/ISGRI and
SWIFT/BAT data of Cas A, aiming at investigating the relative merits of the
standard model, consisting of a synchrotron spectrum with exponential cutoff,
versus its extension in the case of a highly turbulent magnetic-field, the
jitter radiation model. Interestingly, the jitter model can be used to
estimate both the magnetic-field turbulence spectral index and the minimum
length scale of magnetic-field turbulence.

The paper is divided as follows: in Sect. 2 we present all the X-ray data
considered, we describe their reduction and present results of image analysis;
in Sect. 3 we show the results of the spectral analysis; in Sect. 4 we discuss
the results and comment on their implications; in Sect. 5 we wrap up the main
findings and draw our conclusion.

##  2 X-ray data reduction and regions selection

We considered X-ray observations of Cas A in order to cover a wide X-ray
energy range, from 0.5 keV up to roughly 100 keV. We analyzed data collected
by Chandra/ACIS-S (operating in the 0.1-10 keV, see Sect. 2.1, Garmire et al.
2003), NuSTAR (3-79 keV, Sect. 2.2, Harrison et al. 2013), INTEGRAL/ISGRI (15
keV - 1 MeV, Sect. 2.3, Lebrun et al. 2003), SWIFT/BAT (20-200 keV, Sect. 2.4,
Barthelmy et al. 2005). All the observations considered are listed in Table 1.
Since INTEGRAL/ISGRI and SWIFT/BAT are coded masks and are not able to resolve
Cas A, it was possible to perform spatially resolved spectral analysis only
with Chandra and NuSTAR telescopes.

Table 1: Observation log table Telescope | Obs ID | PI | Exposure (Ms)
---|---|---|---
Chandra | 4638 | Hwang | 0.164
| 40001019002 | Harrison | 0.294
| 40021002002 | Harrison | 0.288
| 40021011002 | Harrison | 0.246
| 40021012002 | Harrison | 0.239
| 40021003003 | Harrison | 0.233
| 40021001005 | Harrison | 0.228
NuSTAR | 40021002008 | Harrison | 0.226
| 40021001002 | Harrison | 0.190
| 40021015003 | Harrison | 0.160
| 40021002006 | Harrison | 0.159
| 40021015002 | Harrison | 0.86
| Total | 2.3
INTEGRAL | See Sect. 2.3 | 10
SWIFT | See Sect. 2.4 | 180

###  2.1 Chandra

We considered the Chandra/ACIS-S single observation of Cas A with highest
exposure time (Obs ID 4638, PI Hwang). We reduced Chandra data with the task
chandra_repro available within CIAO v4.14.0, with CALDB v4.9.7. We produced
exposure-corrected images in the 0.5-8 keV and 4-6 keV energy bands with the
task fluximage, displayed in the left panel of Fig. 2 and, in red, in Fig. 3,
respectively.

Figure 2: Chandra and NuSTAR exposure/vignetting-corrected images of Cas A.
_Leftmost panel._ Chandra/ACIS-S count-rate image in the 0.5-8 keV band with a
pixel size of 1” and a square root scale. _Other panels._ Exposure/vignetting-
corrected and mosaicked NuSTAR images of Cas A in various energy bands,
reported on the images in units of keV, with a square root scale. Figure 3:
RGB Chandra and NuSTAR image of Cas A with a square root scale. In red,
Chandra/ACIS-S count-rate image in the 4-6 keV band with a pixel size of
$1^{\prime\prime}$, in green the 4-6 keV NuSTAR deconvolved count-rate image,
in blue the 10-20 keV NuSTAR deconvolved image. The region used for the
analysis of the whole remnant is shown in white and the regions selected for
the spatially resolved spectral analysis are shown in yellow.

We used the task specextract to extract the spectra from the Chandra/ACIS-S
event list with the proper RMF and ARF files. We selected different background
regions in order to cross-check the consistency of our analysis and we
verified that the choice of the background did not influence our results.

###  2.2 NuSTAR

We analyzed 11 NuSTAR observations of Cas A, listed in Table 1, for a total
exposure time of more than 2 Ms. The data have been reduced by using the
nupipeline task available within NuSTARDAS, version 2.1.2. We set
SAAMODE=strict and TENTACLE=yes, in order to be as conservative as possible
with the background contamination. We produced exposure and vignetting
corrected count-rate images of Cas A in several energy bands, using the same
ranges as in Grefenstette et al. (2015). We then mosaicked with XIMAGE the
images in the same energy band obtained from separate observations and from
the two different Focal Plane Modules (FPM) detectors. The resulting
mosaicked, exposure and vignetting corrected images are shown, with the
corresponding energy band, in Fig. 2.

The images shown in the right panel of Fig. 2 were deconvolved through the
task arestore, based on the Lucy algorithm (Lucy 1974, Richardson 1972), in
order to show that the morphology above 10 keV is consistent with the
filamentary structure detected by Chandra. The comparison between the
deconvolved NuSTAR (in green) and the Chandra (in red) images in the 4-6 keV
band is shown in Fig. 3. Indeed, the deconvolved NuSTAR count-rate image of
Cas A in the 4-6 keV closely matched the same image observed by Chandra,
proving the reliability of our region selection procedure. The main criterion
for selecting the extraction regions was the high surface brightness both in
the 4-6 keV and 10-20 keV bands, highlighted (in blue) in Fig. 3. This is
crucial, since high statistics over a wide energy range is required in order
to robustly discriminate between spectra with and without cutoff, especially
at energies higher than 10 keV. We highlight that, though the deconvolved
NuSTAR image show filamentary structures smaller than the region selected, we
could not safely reduce the size of the extraction regions because of the
NuSTAR Point Spread Function (PSF). Therefore, we extracted the NuSTAR spectra
from the regions shown in Fig. 3 through the task nuproducts following the
recipe by Grefenstette et al. (2017).

###  2.3 INTEGRAL

Unlike Chandra and NuSTAR, INTEGRAL/ISGRI does not spatially resolve Cas A and
therefore no image analysis is possible. INTEGRAL’s pointing strategy involves
a dithering on a time scale of a few kiloseconds to reduce coded mask
artifacts: each individual pointing is called _science window_. We selected
all the 5686 science windows in which the spacecraft was pointing at less than
10 degrees from Cas A. We built a catalog of the 7 detected sources detected
at more than 7$\sigma$ in an 20-80 keV image obtained from 10% of the full
sample, randomly selected. Using this catalog, we extracted the average
spectrum of the source as seen by the IBIS/ISGRI detector (Lebrun et al.,
2003) using version 11.2 of the Offline Science Analysis (Courvoisier et al.,
2003) served through the Multi-Messenger Online Data Analysis
(MMODA,111https://www.astro.unige.ch/mmoda/ Neronov et al., 2021). The
resulting equivalent on-axis exposure is about 10.8 Ms over 17 Ms of observing
time. We averaged spectra and responses through the mission: owing to the
detector evolution and the relative source faintness, we could reliable use
the data from 40 to 100 keV.

###  2.4 SWIFT

SWIFT/BAT is a coded mask detector that cannot spatially resolve Cas A. We
used the spectra available online at SWIFT/BAT_CasA and the response matrix
was downloaded from SWIFT/BAT_RMF. The total exposure time of the spectra is
180 Ms.

##  3 Spectral analysis

We performed the spectral analysis with XSPEC (v12.12.1, Arnaud 1996) in
several energy ranges and setups. We analyzed Chandra/ACIS-S data in the 4-6
keV band; NuSTAR/FPMA,B spectra in the 4-5.5 keV, 9-15 keV and 15-40 keV;
INTEGRAL/ISGRI data in the 40-100 keV; SWIFT/BAT observations in the 15-200
keV. All the Chandra and NuSTAR spectra were optimally binned following the
procedure described in Kaastra & Bleeker (2016) through the FTool task
ftgrouppha. The corresponding background spectra were subtracted from
Chandra/ACIS-S and NuSTAR/FPMA,B source spectra, whereas the INTEGRAL/ISGRI
and SWIFT/BAT spectra were obtained already with proper background
subtraction. Our approach in analyzing the spectra is based on testing four
different nonthermal scenarios: i) straight power-law without any cutoff;
ii)srcut model (Reynolds & Keohane 1999); iii) a zira model (based on
Zirakashvili & Aharonian, 2007, see details in Appendix A); iv) a jitter
model, that we incorporated in XSPEC (see details in Appendix A), which
represents the jitter scenario. In order to select the model providing the
best description of the data we used three metrics: the standard $\chi^{2}$
statistic and its reduced version $\chi^{2}_{r}=\chi^{2}/n$ with $n$ the
degrees of freedom; the Bayesian Information Criterion (BIC, Schwarz 1978),
defined as BIC$=\chi^{2}+m\ln(n)$, $m$ being the number of free parameters in
the model; the Akaike Information Criterion (Akaike, 1974), defined as
AIC$=\chi^{2}+2m$. The model best describing the data is the one with the
corresponding lower value of either $\chi^{2}_{r}$, AIC or BIC metrics.

As discussed in Sect. 1, the overall spectrum of jitter radiation can be
divided in two components: the low-energy part, i.e. for
$\omega<\omega_{break}$, is similar in shape to the classical synchrotron
component; the high energy part, i.e. for $\omega>\omega_{break}$, which can
extend up to $\gamma$-ray energies (K13). In the following, we will refer to
the low-energy component as _synchrotron component/regime_ , to the high-
energy component as _jitter component/regime_ , while we keep the terms
_synchrotron radiation_ and _jitter radiation_ when referring to the global
radiation in the case of uniform and turbulent magnetic field, respectively.
It is evident that in order to firmly detect jitter radiation we need to look
for the absence of a cutoff in the spectra and a steepening of the photon
index between two energy bands and this is achievable only by considering wide
spectral ranges.

This section is divided in various subsections, corresponding to the different
regions of the SNR shown in Fig. 3.

###  3.1 Whole remnant

In this subsection we analyze the X-ray spectrum emitted by whole Cas A and
extracted from the white region shown in Fig. 3. In this scenario, we can
exploit the information collected by all the telescopes considered in this
project.

####  3.1.1 Hard band

We first focused on the hard energy band, $>15$ keV, where the X-ray thermal
contribution is negligible and the total measured flux is ascribable to purely
nonthermal emission processes. We only considered NuSTAR spectral bins where
the background emission is a small fraction ($\lesssim 10\%)$ of the Cas A
emission, till an energy of 40 keV.

We performed the analysis either by fitting a single combined NuSTAR spectrum,
obtained by processing the original NuSTAR spectra through the addascaspec
task, and by simultaneously fitting the NuSTAR spectra extracted from the 11
observations. These two slightly different approaches have pro and cons which
balance each other out. Summing the spectra lead to a single global spectrum
with much higher statistics, smaller error bars and, therefore, to higher
sensitivity to the spectral model adopted. However, since there were 11
separate NuSTAR observations performed in a time-lapse of 2 years, the
response matrix and the characteristics of the nonthermal emission might
slightly change between the first and the last observations. On the other
hand, the simultaneous analysis of the single spectra extracted from each
observation, provided very reliable results, at the expense of the sensitivity
to the spectral model.

We applied the spectral models introduced above to fit the X-ray synchrotron
spectra of Cas A, each coupled to a component modeling the Galactic absorption
(TBabs model in XSPEC) and to two gaussians, accounting for the radioactive
44Ti lines observed in Cas A at 68 keV and 78 keV(e.g., Grefenstette et al.,
2014). We also included a constant factor to take into account small variation
on the extracted spectra due to PSF effects, cross-calibration (see Madsen et
al. 2015) and to the temporal offset.

Figure 4: NuSTAR (black), INTEGRAL (green) and SWIFT (red) spectra of Cas A
fitted with different models. The normalization of each spectrum is readjusted
to the NuSTAR ones through the setplot area command available within XSPEC.
Solid line represents the model. _From top to bottom_ : spectra are fitted
with the zira, srcut and pow/jitter model, respectively (jitter model provides
same description as the pow). _Left column._ NuSTAR spectra from all the
observations are simultaneously fitted. All the spectra are rebinned for
graphical purposes _Right column._ NuSTAR spectra from all the observations
are combined. Table 2: Best-fit parameters for the 15-200 keV band

| Parameter | Sim | Comb
---|---|---|---
| norma | 0.192$\pm 0.004$ | 0.191 $\pm 0.04$
zira | Ecut (keV) | 1.064${}^{+0.018}_{-0.017}$ | 1.08$\pm 0.02$
|  $\chi^{2}$/d.o.f. | 2479/1869 | 317/120
| AIC | 2483 | 321
| BIC | 2494 | 327
| normb | 2180${}_{-100}^{+90}$ | 2140$\pm 0.80$
| Radio index | 0.77 (fixed)
srcut | Ebreak (keV) | 1.07 $\pm 0.03$ | 1.09$\pm 0.03$
|  $\chi^{2}$/d.o.f. | 2349/1869 | 193/120
| AIC | 2353 | 197
| BIC | 2364 | 203
| norma | 1.97$\pm 0.08$ | 1.94$\pm 0.08$
| $\Gamma$ | 3.336${}^{+0.016}_{-0.015}$ | 3.327$\pm 0.014$
pow |  $\chi^{2}$/d.o.f. | 2270/1869 | 129/120
| AIC | 2274 | 133
| BIC | 2285 | 139

Notes. aUnits of photons/keV/cm2 at 1 keV. b Flux at 1 GHz in units of Jy.
Simult and Comb mark the results obtained with the simultaneous analysis and
with the combination of the NuSTAR spectra, respectively (see text for
details). The jitter model is not reported since it is indistinguishable from
a simple power-law.

We left free to vary the following parameters for each nonthermal model:
photon index $\Gamma$ and normalization for the power-law; energy break Ebreak
and normalization for the srcut model, with the radio spectral index $\alpha$
fixed to 0.77 (Green 2019); cutoff energy Ecut and normalization for the zira
model. For the jitter model, we left free to vary the photon indices
$\Gamma_{1}$ and $\Gamma_{2}$, the normalization, the cutoff shape parameter
$\beta$ and the energy break Ebreak whereas the ratio between the
normalization of the two components was kept fixed to 1. Values of reduced
$\chi^{2}$ and best-fit parameters for each adopted model are shown in Table
2. Fig. 4 show the NuSTAR, INTEGRAL and SWIFT spectra fitted with the
different models and the corresponding residuals. The power-law model provided
the best description of the spectra at a very significant ( $>5\sigma$)
confidence level, as can be clearly seen by the residuals in Fig. 4 relative
to the zira and cutoff models fitted on the combined spectra. The simultaneous
analysis provided best-fit parameters perfectly consistent with those obtained
with the combined spectra, guaranteeing the reliability of our results. The
absence of any cutoff in such a wide (15-100 keV) energy range is the first
hint of jitter radiation being a potentially important mechanism for the
nonthermal X-ray emission from Cas A . If the turbulence length scale
$\lambda$ is smaller than the gyroradius but larger than the photon formation
length $\lambda_{pf}$, the slope of the spectrum is directly linked to the
exponential of the magnetic-field distribution $\sigma_{B}$ (see K13) rather
than to the turbulence spectrum. Given the purpose of this work, we will focus
on the jitter scenario, i.e. assuming that $\lambda<\lambda_{pf}$. To
investigate the jitter scenario, we fitted the same spectra with the jitter
model, finding that the synchrotron component is unconstrained and that the
jitter component has same characteristics as the simple power law scenario,
i.e. it is not possible to distinguish between a single power-law and a jitter
model. By interpreting the 15-100 keV emission as the jitter component, the
synchrotron component must then be detectable at lower energies, outside this
range222The 100-200 keV band is covered only by SWIFT and because of the wide
error bars only upper limits on the flux above 100 keV could be estimated (see
Appendix C).. Naturally, this is true not only for this hard energy range but
for a generic X-ray spectrum. Given that the jitter component necessarily
comes with a low-energy counterpart (the synchrotron component), the absence
of a cutoff in the spectra simply reflects that the synchrotron regime is
dominant at energies/frequencies lower than the ones considered. In this
framework, the best-fit photon index of $\Gamma\sim 3.3$ implies a turbulence
spectrum with an index $\nu_{B}=2.3$, higher than typical values for
Kolmogorov $\nu_{\rm{Kol}}=5/3$ and Kraichnan $\nu_{\rm{Kra}}=3/2$ turbulence.
It is worth noticing that if $\lambda_{pf}<\lambda<R_{L,j}$, then $\Gamma\sim
3.3$ would imply $\sigma_{B}\sim 4.3$, from Eq. A5 in K13. Going into details
of implications for this scenario is out of the goals of the paper and we
leave it for a future study. We will discuss possible explanations for this
discrepancy in the jitter scenario in Sect. 4.2.

####  3.1.2 Soft band

In Sect. 3.1.1 we focused on the hard ($\gtrsim 15$ keV) band of the spectra,
finding hints for only one of the two components expected in the jitter
radiation. If this component is actually the jitter component, we expect the
synchrotron one to contribute at lower energies. The typical energy range in
which soft X-ray nonthermal emission is studied in SNRs is between 4 and 6
keV, where there are no bright emission lines333Due to the poor spectral
resolution of NuSTAR, some tail of the Fe K emission line contaminated the
region between 5.5 and 6 keV. Therefore, we performed the NuSTAR spectral
analysis between 4 and 5.5 keV..

We adopted the same methodology used for the hard energy band, i.e. we
analyzed the Chandra and NuSTAR spectra in the 4-6 keV range with each of the
different nonthermal emission models. We included an absorbing component
(TBabs model in XSPEC), though its effect is minor at 4 keV. We kept the NH
value fixed to $0.9\times 10^{22}$, a value found from performing a fit of the
Chandra 0.5-8 keV spectra. Given the narrow energy band, we did not consider
the jitter model for the 4-6 keV range. The best-fit values with the
corresponding errors bars (at 90% confidence level) of $E_{\rm{break}}$,
$\Gamma$ and $E_{\rm{cut}}$ are displayed in the rightmost set of points in
Fig. 5.

The best-fit values of $E_{\rm{break}}$, $\Gamma$ and $E_{\rm{cut}}$ show that
the 4-6 keV NuSTAR and Chandra spectra are flatter than above 15 keV, where
the nonthermal models adopted require higher values of $\Gamma$,
$E_{\rm{break}}$ and $E_{\rm{cut}}$. While the steepening $\Delta\Gamma\approx
0.4$ is compatible with what we expect in the jitter framework, it is
surprising to observe an increase of the break/cutoff energy with increasing
energy range, challenging the picture predicting a unique cutoff for the
spectrum of the Whole Cas A, and in agreement with the results obtained in the
15-100 keV band.

###  3.2 Spatially resolved spectral analysis

Figure 5: Best-fit values of the nonthermal parameters in the zira, srcut, and
power-law scenarios, as a function of the region and for different analysis
setups. Error bars are estimated at 90% confidence level. Each color represent
the instrument used and the energy range considered: black for Chandra in the
4-6 keV band; red for NuSTAR data in the 4-5.5 keV band; purple for NuSTAR in
the 9-15 keV band; cyan for NuSTAR in the 15-40 keV band.  Figure 6: Values of
$\chi^{2}_{r}$ for fits performed with different nonthermal models in the 4-6
keV, 9-15 keV and 15-40 energy bands of the NuSTAR spectra. Black, green and
blue represent models with a power-law, srcut and zira component,
respectively.

In Sect. 3.1 we showed the results of the analysis of the global X-ray
spectrum of Cas A, providing us with information about the overall remnant. In
order to investigate the spatial variation of the parameters characterizing
the nonthermal emission, we extracted Chandra and NuSTAR spectra444SWIFT and
INTEGRAL are not able to spatially resolve Cas A and cannot be used for this
analysis from the other seven regions shown in yellow in Fig. 3. The main
criterion for the region selection was the high surface brightness both in the
4-6 keV and 10-20 keV that lead to us the identification of five regions close
to the shock front: NorthWest (NW), North (N), NorthEast (NE), SouthEast (SE),
SouthWest (SW). Moreover, we also considered two additional regions, Inner (I)
and West (W): the former being characterized by particularly bright clumps at
energies higher than 15 keV, the latter corresponding to the location where
the emission of nonthermal X-ray radiation indicates fast electron
acceleration at the reverse shock (see e.g. Orlando et al. 2022; Vink et al.
2022a).

We adopted the same procedure used for the whole remnant, by analyzing the
Chandra and NuSTAR spectra in the 4-6 keV band and the NuSTAR spectra in the
9-15 and 15-40 keV bands. We here focus only on the spectra combined, in order
to increase statistics particularly at energies above 20 keV. The best-fit
parameters are shown in Fig. 5 and the corresponding reduced $\chi^{2}$ values
are shown in Fig. 6.

The best-fit values of the nonthermal parameters (with the corresponding 90%
confidence error bars) provide the following relevant information: i) the cut-
off energy measured with the zira model systematically increases with
increasing energy; ii) the break energy measured with the srcut model shows
the same trend but less significantly than the zira scenario, with the NW
region marginally consistent at a 90$\%$ confidence level with a fixed break
energy; iii) most of the regions are characterized by a steepening of the
photon index; iv) independently of the spectral model adopted, the Chandra
data were systematically requiring flatter spectra than the NuSTAR spectra in
the 4-6 keV energy band.

The average best-fit value of $E_{\rm{break}}$ changed from 0.5 keV in the 4-6
keV energy band up to 1.5 keV at energies higher than 9 keV. A similar
argument is valid also for $E_{\rm{cut}}$, with an even higher discrepancy
since the spectrum in the zira model falls off quicker than in the srcut
model. The fit performed by adopting the simple power-law model provided more
accurate information on the steepening of the spectra across the SNR. Half of
the regions (SE, SW, NE, I) showed spectra with a constant photon index from 4
up to 40 keV, whereas the other half (NW, N, W and the already discussed Whole
remnant) presented a clear steepening of $\Delta\Gamma\sim 0.4$.

We also measured a significant discrepancy between the best-fit values
obtained from the Chandra and the NuSTAR spectra in the 4-6 keV energy band,
with NuSTAR predicting steeper slope. We checked that this issue was not due
to errors in the spectral extraction procedure (see Appendix B) and we
concluded that this is probably an effect of cross-calibration between the two
telescopes. Given this inconsistency, we did not consider the Chandra spectra
in the broadband 4-40 keV analysis presented in the following and we opted for
a more conservative approach, based on the analysis of only the NuSTAR
spectra.

Figure 6 shows the reduced $\chi^{2}$ for each of the model adopted as a
function of the regions. Since all the models adopted have the same number of
free parameters, a direct comparison of the $chi^{2}$ values allowed us to
identify the more appropriate model for each of the spectra. We found no
significant preference for spectral models with or without cutoff in the 4-6
keV energy band. In regions N, NE, NW, SW and W different nonthermal scenarios
provided the same description already at a 1$\sigma$ confidence level, with a
$\Delta\chi^{2}\leq 1$ among different models. Region SE was marginally better
described by a straight power law, at a confidence level lower than $90\%$
($\Delta\chi^{2}<2.71$) and regions I and Whole were described better by a
cutoff at the $90\%$ confidence level but not at 3$\sigma$. Overall, these
results indicated that it was not possible to robustly state whether the 4-6
keV emission was characterized by a cutoff or a straight power-law. An
additional source of uncertainty comes from the contribution of thermal
continuum emission, which is neglected in our analysis and might contribute to
the total flux, though this relative contribution could be as low as 5% as
reported by Vink et al. (2022b) and it is in any case at least a factor 3
lower than the nonthermal one (see Appendix D).

Similar points were also valid for the 9-15 keV and 15-40 keV bands. Regions
N, NE, NW, SE, SW were equally described by any of the spectral models adopted
in both the energy bands. A fit with a power-law provided a slightly better
$\chi^{2}$ value in region W though not significant at more than 90$\%$
confidence. Only region I and Whole showed a significant preference for the
power-law model both in the 9-15 and 15-40 keV energy bands, at odds with the
4-6 keV range. Overall, the results obtained from 15-40 keV band suggested an
absence of cutoff in the spectra for region I, N, W, Whole, whereas the other
regions did not allow to discriminate between the three different adopted
models.

The results of the spectral analysis presented so far could be summarized in
two main points: i) The $\chi^{2}$ values did not indicate a significant
preference for either cutoff or non-cutoff models. This is most likely due to
the narrow energy bands considered: the spectral differences between a cutoff
and a non-cutoff models are enhanced when looking at the evolution of the
shape in wide energy bands. ii) The best-fit values of the break/cutoff
energies increased with energy, pointing against the presence of cutoff in the
spectra. iii) Best-fit values of photon indices indicated a steepening in at
least half of the regions considered. In order to further investigate this
point, we analyzed the curvature of the continuum between the soft (4-6 keV)
and hard (15-40 keV) energy ranges (see Fig. 3.3).

###  3.3 Curvature of the continuum

We investigated the change of the curvature of the NuSTAR spectra by
extrapolating the best-fit nonthermal component measured from the 4-6 keV
range in the 9-40 keV band: if the broadband nonthermal emission was
characterized by a cutoff, then the best-fit model in the 4-6 keV band should
well describe the data points in the 9-40 keV band as well. In Fig. 7 we show
the results of this comparison for regions NW and SW, that were characterized
by the two most extreme behaviors, with the best-fit values of the nonthermal
parameters obtained from the 4-6 keV band analysis. The same plot for all the
regions is available inAppendix E.

Figure 7: NuSTAR spectra in the 4-5.5 keV and 9-40 keV compared with the best-
fit model (solid line) obtained by fitting only the 4-5.5 keV band for regions
NW (on the left) and SW (on the right). The yellow area marks the spectral
region in which the fit is performed. Each panel shows the 4-6 keV best-fit
value of the nonthermal parameter considered: $\Gamma$, $E_{\rm{break}}$ (in
keV units) and $E_{\rm{cut}}$ (in keV units) for pow, srcut and zira models,
respectively.

The adoption of the zira model for the 4-6 keV energy range leads to a heavy
underprediction of the higher energy data points, indicating that this kind of
cutoff was not suited to describe the broad nonthermal spectrum. The fact that
the power-law in the 4-6 keV band was systematically flatter than in the 9-40
keV was already clear in Fig. 5. The only regions that did not show a clear
steepening from the soft to hard X-rays were NW and SW (see Fig. 7). Region SW
was characterized by the highest $\Gamma\sim 3.2$ and showed the highest
underprediction of the data 9-40 keV data points when considering cutoff
models, suggesting that in this region the jitter component might be
detectable already in the 4-6 keV band, while in other regions it might become
so only at higher energies. The spectrum extracted from NW presented an
opposite behavior, being well described by the srcut model in the whole 4-40
keV range (see Fig. 7), indicating that in this region jitter radiation is not
at work or that its synchrotron component is dominating the spectrum.

We also repeated this analysis by fitting the 15-40 keV data and extrapolating
the obtained best-fit model down to 3 keV. As done in Fig. 7 we here show the
plots for SW and NW regions (Fig. 8) and we report the results for all the
regions in Appendix E.

Figure 8: Same as Fig. 7 but here the fit is performed in the 15-40 keV band
and extrapolated down to 3 keV.

The cutoff models, i.e. zira and srcut, systematically underpredicted the
observed data points, with the best-fit value of $E_{\rm{break}}$ and
$E_{\rm{cut}}$ higher than the corresponding values obtained from the 4-6 keV
spectra only (as already observed in Fig. 5). The region with the smallest
variation in any of these parameter is NW, confirming that its emission is
most likely dominated by synchrotron radiation. The extrapolation of the
power-law model in region SW lead to a significant overprediction of the data
in the 4-6 keV band. It is worth mentioning that the results from the 15-40
keV are only indicative and cannot be taken as firm proof of the presence or
absence of steepening in the spectra, given that the wide error bars in this
range lead to broad uncertainty on the slope of the high-energy tail, as in
the case of region SW. In fact, while extrapolating the power-law with
$\Gamma=3.4$ resulted in an overprediction of the 4-6 keV band, by taking
$\Gamma=3.2$, a value within the 90% confidence level, the resulting
extrapolated power-law very well described the 4-6 keV data points.

Overall, we found that the spectra face a steepening between the soft and the
hard band and that the cutoff models describe the data systematically and
significantly worse than the standard power-law. However, the results
presented so far did not take into account the shape of the spectrum in the
two bands simultaneously. Therefore, we also fitted the the 4-6 keV and 9-40
keV energy bands simultaneously (Sect. 3.4), excluding the 6-9 keV range that
is dominated by the Fe K emission line.

###  3.4 Jitter model

The results presented in Sect. 3.3 provided a qualitative estimate of how
coherent is the continuum shape between the soft, 4-6 keV, and hard, 9-40 keV,
energy bands. We fitted the 4-6 plus 9-40 keV energy band for all the regions
with our jitter model (see Appendix A) and the standard srcut, zira and pow
models. We performed the analysis either by leaving the $\beta$ free to vary
and by keeping it fixed to 0.5 (as in the zira model), 1 (as in the srcut
model) and 2. We found that leaving the $\beta$ free to vary significantly
improved the quality of the fit for all the regions but NW, with an average
value of roughly 0.5 and systematically lower than 1. The best-fit parameters
for all the models with corresponding $\chi^{2}_{r}$, BIC and AIC are listed
in Table 3.

Figure 9: NuSTAR spectra from regions NW (top line), N (central line), SW
(bottom line) fitted with srcut (left column), jitter (central column) and pow
(right column) models in the 4-5.5 keV and 9-40 keV and corresponding
residuals.  Table 3: Best-fit values on the 4-5.5 plus 9-40 keV bands

| Region
---|---
Model | Parameter | Inner | North | NorthEast | NorthWest | SouthEast | SouthWest | West | Whole
TBabs | NHa | 1.7 | 1.1 | 1.0 | 1.13 | 1.5 | 1.4 | 2.0 | 0.9
|  $E_{\rm{cut}}$(keV) | 0.447${}_{-0.002}^{+0.002}$ | 0.421${}_{-0.004}^{+0.004}$ | 0.68${}_{-0.01}^{+0.01}$ | 0.51${}_{-0.006}^{+0.006}$ | 0.547${}_{-0.007}^{+0.007}$ | 0.488${}_{-0.009}^{+0.009}$ | 0.497${}_{-0.002}^{+0.002}$ | 0.494${}_{-0.001}^{+0.001}$
| normb | 73.2${}_{-0.4}^{+0.4}$ | 8.5${}_{-0.1}^{+0.1}$ | 3.38${}_{-0.05}^{+0.05}$ | 6.03${}_{-0.08}^{+0.08}$ | 5.46${}_{-0.07}^{+0.07}$ | 2.9${}_{-0.06}^{+0.06}$ | 57.0${}_{-0.2}^{+0.3}$ | 599.0${}_{-2.0}^{+2.0}$
zira |  $\chi^{2}$r | 46.5 | 6.5 | 5.1 | 5.25 | 8.25 | 4.42 | 62.32 | 178.75
| AIC | 5723.37 | 731.79 | 560.24 | 581.76 | 960.48 | 472.71 | 7918.88 | 25268.8
| BIC | 5728.99 | 737.23 | 565.62 | 587.16 | 965.99 | 478.04 | 7924.57 | 25274.99
|  $E_{\rm{break}}$(keV) | 0.462${}_{-0.004}^{+0.004}$ | 0.424${}_{-0.007}^{+0.007}$ | 0.9${}_{-0.02}^{+0.02}$ | 0.57${}_{-0.01}^{+0.01}$ | 0.65${}_{-0.01}^{+0.01}$ | 0.53${}_{-0.02}^{+0.02}$ | 0.545${}_{-0.003}^{+0.003}$ | 0.486${}_{-0.002}^{+0.002}$
| normc | 930${}_{-10}^{+10}$ | 114${}_{-3}^{+3}$ | 28.0${}_{-0.8}^{+0.8}$ | 66${}_{-2}^{+2}$ | 55${}_{-1}^{+1}$ | 33${}_{-1}^{+1}$ | 650${}_{-5}^{+5}$ | 8050${}_{-60}^{+60}$
srcut |  $\chi^{2}$r | 17.68 | 2.48 | 2.25 | 1.78 | 3.79 | 2.39 | 19.08 | 62.26
| AIC | 2178.41 | 281.75 | 248.94 | 199.77 | 443.95 | 257.58 | 2427.71 | 8710.71
| BIC | 2184.03 | 287.19 | 254.32 | 205.17 | 449.46 | 262.91 | 2433.4 | 8716.9
| $\Gamma$ | 3.295${}_{-0.004}^{+0.004}$ | 3.333${}_{-0.009}^{+0.009}$ | 2.99${}_{-0.01}^{+0.01}$ | 3.186${}_{-0.009}^{+0.009}$ | 3.124${}_{-0.009}^{+0.009}$ | 3.22${}_{-0.01}^{+0.01}$ | 3.219${}_{-0.003}^{+0.003}$ | 3.347${}_{-0.002}^{+0.002}$
| normb | 198${}_{-2}^{+2}$ | 22.8${}_{-0.4}^{+0.4}$ | 7.8${}_{-0.2}^{+0.2}$ | 15.2${}_{-0.3}^{+0.3}$ | 13.3${}_{-0.2}^{+0.2}$ | 7.5${}_{-0.2}^{+0.2}$ | 150.0${}_{-0.8}^{+0.8}$ | 2070${}_{-10}^{+10}$
pow |  $\chi^{2}$r | 1.5 | 2.28 | 1.05 | 1.83 | 1.11 | 1.38 | 2.75 | 2.14
| AIC | 188.13 | 259.9 | 118.1 | 205.77 | 132.39 | 150.58 | 353.74 | 391.93
| BIC | 193.75 | 265.34 | 123.48 | 211.17 | 137.9 | 155.91 | 359.43 | 398.12
| $\Gamma_{1}$ | 2.87${}_{-0.3}^{+0.02}$ | 2.87${}_{-0.04}^{+0.02}$ | 2.45${}_{-0.05}^{+0.05}$ | 3.0${}_{-0.08}^{+0.08}$ | 2.5${}_{-0.3}^{+0.3}$ | 2.9${}_{-0.07}^{+0.2}$ | 2.71${}_{-0.01}^{+0.01}$ | 2.724${}_{-0.01}^{+0.006}$
|  $E_{\rm{break}}$(keV) | 11.8${}_{-0.3}^{+0.3}$ | 5.2${}_{-0.1}^{+0.1}$ | 5.2${}_{-0.2}^{+0.3}$ | 40.4${}_{-0.8}^{+0.5}$ | 6${}_{-1}^{+4}$ | 4.4${}_{-0.4}^{+0.2}$ | 5.36${}_{-0.03}^{+0.08}$ | 4.59${}_{-0.1}^{+0.04}$
| $\beta$ | 0.57${}_{-0.04}^{+0.04}$ | 0.38${}_{-0.02}^{+0.03}$ | 0.6${}_{-0.2}^{+0.2}$ | 1.2${}_{-0.4}^{+0.7}$ | 0.65${}_{-0.06}^{+0.06}$ | 0.3${}_{-0.3}^{+0.1}$ | 0.439${}_{-0.003}^{+0.02}$ | 0.485${}_{-0.009}^{+0.007}$
| $\Gamma_{2}$ | 3.27${}_{-0.02}^{+0.02}$ | 3.429${}_{-0.009}^{+0.02}$ | 3.03${}_{-0.02}^{+0.03}$ | 2.6${}_{-0.2}^{+0.5}$ | 3.05${}_{-0.03}^{+0.04}$ | 3.5${}_{-0.3}^{+0.1}$ | 3.277${}_{-0.008}^{+0.004}$ | 3.359${}_{-0.003}^{+0.004}$
jitter | normb | 186${}_{-10}^{+10}$ | 29${}_{-1}^{+2}$ | 8.6${}_{-0.4}^{+0.6}$ | 12.2${}_{-0.3}^{+0.8}$ | 10.8${}_{-0.9}^{+1.0}$ | 12${}_{-4}^{+4}$ | 173${}_{-2}^{+1}$ | 2130${}_{-20}^{+20}$
|  $\chi^{2}$r | 1.19 | 1.06 | 0.95 | 1.03 | 0.93 | 1.38 | 1.14 | 1.04
| AIC | 152.99 | 125.79 | 110.94 | 119.85 | 115.13 | 151.74 | 150.94 | 181.61
| BIC | 166.93 | 139.25 | 124.26 | 133.21 | 128.77 | 164.91 | 165.04 | 196.99

Notes. a Units of 1022 cm-2, fixed in the fitting procedure. bUnits of 10-3
photons/keV/cm2/s at 1 keV. c Flux at 1 GHz in units of Jy.

The recurrent striking feature evident in Table 3 is that the jitter model
systematically provides a better description of the data than any cutoff
model, as witnessed not only by the $\chi^{2}_{r}$ but also by the BIC and AIC
metrics. While in the 4-6 keV and in the 15-40 keV bands analyzed separately
we were not able to pinpoint a favored model for the nonthermal emission (see
Fig. 6), by simultaneously analysing such a wide spectral ranges we are more
sensitive to the emission pattern. Hints for this result were already
available in Fig. 5 where we noticed a (monotonic) increase in the photon
index values and break energies with increasing energy ranges. We found values
of $\Gamma_{1}$ slightly, but systematically, lower than those obtained with
the analysis of the 4-6 keV band alone (red points in Fig. 5). This is not
surprising given that the jitter model provides the slope of the power-law
corrected by the effect of the cutoff, while a simple power-law applied to a
cutoff spectrum leads to an overall softer spectrum. On the other hand, best-
fit values for $\Gamma_{2}$ were perfectly compatible with the $\Gamma$ best-
fit values obtained from the analysis of the 15-40 keV spectra with the
standard power-law model, since in this regime no curvature is present. Thanks
to the simultaneous 4-6 keV and 9-40 keV analysis we were now able to quantify
and directly compare different models. We highlight this in Fig. 9 where we
show a comparison between the srcut, zira and jitter model for the regions NW,
N and SW, representing three different regimes.

All the regions but NW and SW showed indication for the presence of both
synchrotron and jitter components of jitter radiation as all the parameters
were satisfyingly constrained and the energy breaks lied within the spectral
range investigated. Region N is shown in Fig. 9 as clearly depicts this most
common regime, with $4.5<E_{\rm{break}}<6$ keV, and an average value of
roughly 5 keV. Regions I was characterized by slightly higher, but quite
unconstrained $E_{\rm{break}}$value. In particular, it was compatible, at a
$3\sigma$ confidence level, with 5 keV and could then safely safely included
in this class of regions. Region NW showed a $E_{\rm{break}}\sim 40$ keV, at
the upper edge of the spectral energy considered indicating that only the
synchrotron component was detectable in the spectrum. This result nicely fits
the picture discussed in Sect. 3.3 where NW was the region better described by
srcut model. Moreover, this is also the only region better reproduced by such
a model rather than by a simple power-law. Region SW showed a $\beta$ poorly
constrained with a $E_{\rm{break}}$ of $\sim$ 4 keV, in the bottom edge of the
spectral range considered, indicating that in this region the synchrotron
component of jitter radiation might be non detectable. Again, this feature was
already suggested by the plot in Fig. 5 where no clear steepening was observed
between the 4-6 keV and 9-40 keV band for the SW. We also noticed that this
was the only region that shows no preference for the jitter model with respect
to a classical power-law in terms of $\chi^{2}$, though the best-fit values of
the two jitter photon indices differed significantly from the best-fit found
with the single pow-law. We investigated this discrepancy by imposing an
energy break lower than 4 keV, representing a pure jitter component scenario.
The resulting best-fit model showed a $\Delta\chi^{2}=2$, well within the 90%
confidence range and a best-fit value for $\Gamma_{2}=3.2$, consistent with
the results of the single power-law scenario. In conclusion, regions NW and SW
represented two extreme possible regimes for jitter radiation, showing absence
of hard (jitter) and soft (synchrotron) components, respectively.

Since in the radio band a spectral index of $\sim 0.8$ is observed, we
expected to measure $\Gamma_{1}\sim 1.8$, or, by taking into account cooling
break effects and the subsequent steepening $\Delta\Gamma=0.5$ (K13),
$\Gamma_{1}\sim 2.3$. Therefore, we repeated the analysis described above by
setting priors on the upper limit of $\Gamma_{1}$, i.e. leaving $\Gamma_{1}$
free to vary but forced to be lower than 2.5, leaving some space for spatial
variation of the radio index across Cas A. We found energy break values lower
than 4 keV, i.e. outside of the X-ray domain considered, and we obtained an
overall worse description of the spectra for all the regions but NE and SE.
However, a significantly ($>3\sigma$) worse description was detected only for
regions N ($\Delta\chi^{2}\sim 30$), W ($\Delta\chi^{2}\sim 70$) and Whole
($\Delta\chi^{2}\sim 200$). Overall, these results indicated that: i) in
regions N, W and Whole, some additional mechanism must be responsible for the
radio-to-X-ray steepening of the synchrotron spectrum; ii) in the other
regions, the X-ray synchrotron spectrum is compatible (within 3$\sigma$) with
being the natural extrapolation of the radio one; iii) given the low value of
$E_{\rm{break}}$ outside of the X-ray domain considered, the broadband 4-40
keV spectrum is dominated by the jitter component in all regions but NE and
SE. We will discuss in more detail this point in Sect. 4.

##  4 Discussion

We here reported on a spectral analysis of the Chandra/ACIS-S, NuSTAR/FPMA,B,
SWIFT/BAT and INTEGRAL/ISGRI data of the SNR Cas A, demonstrating that the
nonthermal emission is better described by the jitter model than by a model of
a power-law with an exponential cutoff. In fact, in all regions but NW the
9-40 keV spectra are not well fitted at all by a simple cutoff. The latter
result was particularly evident for the whole remnant for which X-ray emission
extends up to at least 100 keV, in disagreement with what we expect from a
synchrotron regime beyond the cutoff. Additionally, there is supporting
evidence at the $4\sigma$ level for nonthermal emission above 100 keV from
CGRO/OSSE observations (The et al., 1996).

The absence of a cutoff in the hard X-ray spectra is indicative of the
presence of a radiation mechanism different from the standard synchrotron
radiation. The jitter radiation model provides a well-argued and physically
motivated alternative radiation mechanism for the origin of the nonthermal
hard X-ray radiation. The photon spectrum of jitter radiation self-
consistently takes into account the effect of the magnetic-field turbulence
and directly connects it to the shape of the hard X-ray emission. In contrast,
the standard model of a power-law spectrum with some form of exponential
cutoff assumes an homogeneous magnetic-field, whereas, paradoxically, the high
cutoff energy itself requires a highly turbulent magnetic field. In the
following sections, we discuss in detail the implication of these results on
the turbulent magnetic field in which the electrons are embedded.

###  4.1 Synchrotron versus jitter regimes

The jitter radiation model (TF87, K13) is an extension of synchrotron
radiation in presence of an highly turbulent magnetic field. For synchrotron
radiation the acceleration is perpendicular to the magnetic-field causing the
electrons to follow a helical path, with the magnetic field assumed to be
constant at length scales comparable to the gyroradius of the electrons.
Jitter radiation is a consequence of magnetic-field fluctuations on scales
smaller than the photon formation length, causing deviations from the regular,
helical path.

At frequencies below $\omega_{\rm break}$ jitter radiation simply reduces to
classical synchrotron radiation which dominates the radiative spectrum from
the populations of relativistic electrons. If the electron energy distribution
is a power law in energy—$n(E)\propto E^{-\xi}$, then the synchrotron radio
spectral index is $\alpha=(\xi-1)/2$, translating into a photon index
$\Gamma=\alpha+1$. For Cas A $\alpha=0.77$ and $\Gamma=1.77$, although a
flattening of the average radio-to-infrared nonthermal spectrum is likely
happening above 10 GHz (see the discussion in Domček et al., 2021).

At the shock front the electron distribution is expected to have an
exponential cutoff, caused by the fact that energy gains due to DSA are at
high energies limited by radiative losses —in the socalled loss-limited
scenario—or alternatively by a limited time available for acceleration, the
time-limited scenario (Reynolds, 1996; Helder et al., 2012). Further
downstream of the shock, the maximum energy of electrons is lower as after
having participated in the DSA process electrons keep losing energy. Given the
limited life time of a SNR, there is a maximum energy for the electrons that
were accelerated at the beginning. This gives rise to a break—the cooling, or
age break—in the synchrotron spectrum with steepening of
$\Delta\alpha=\Delta\Gamma=0.5$ (e.g. Longair, 2011). For Cas A the spectrum
beyond the cooling break is expected to have an index $\alpha\approx 1.1$—1.3,
or $\Gamma=2.1$–2.3. This is clearly flatter than the power-law of the hard
X-ray radiation, which is closer to $\Gamma\approx 3$ (see Table 3). Moreover,
the nonthermal X-ray emission is clearly associated with narrow filaments at
the shock front (Vink & Laming, 2003) for which we expect the synchrotron
spectrum to have an exponential cutoff, rather than a broken powerlaw. The
cutoff photon energy and the cutoff electron energy are related by the
following:

$\hbar\omega_{\rm cut}\approx 0.19(B/100~{}{\rm\mu G})(E_{\rm e,cut}/10~{}{\rm
TeV})^{2}~{}{\rm keV},$ (3)

supporting the idea of a cutoff photon energy in the range of 0.2–10 keV,
given the uncertainty in $B$ and $E_{\rm e,cut}$.

Based on the X-ray synchrotron model of the filaments, the measured photon
cutoff energy corresponds to an electron cutoff energy of around 10 TeV (Vink
& Laming, 2003). This is also in agreement with the cutoff energy inferred
from the very-high energy gamma-ray emission measured by MAGIC (Ahnen et al.,
2017). Although, surprisingly, the cutoff inferred from the gamma-ray spectrum
pertains to the hadronic (proton) cosmic-ray cutoff, rather than the electron
cutoff energy. Clearly the lack of a clear cutoff energy in the hard X-ray
spectra, the steepness of the spectra and the generally good fits provided by
the jitter model lends support to the hypothesis that both the synchrotron and
jitter components are contributing to the nonthermal X-ray emission from Cas
A. An exceptional case is provided by the spectrum from the NW, where the
jitter model suggests an unusually high break energy of $E_{\rm break}\sim 40$
keV, compared to 4–5 keV elsewhere.

###  4.2 Magnetic-field turbulence spectrum and length scales

In this Section we take advantage of the results of the spectral analysis to
infer significant information on the turbulence energy distribution and
spatial scale. Magnetic-field turbulence is an integral part of both
collisionless shock heating and particle acceleration through the DSA
mechanism. For DSA magnetic-field turbulence is needed at the scales of the
Larmor radius. For the highest energy electrons this corresponds to $R_{\rm
L}=E_{\rm e}/eB\approx 3\times 10^{14}(E/10~{}{\rm TeV})(B/100~{}{\rm\mu
G})^{-1}$ cm. This is much larger than the turbulence length scales relevant
for jitter radiation, which are below the non- or mildly relativistic Larmor
radius, $R_{\rm L,j}\equiv m_{\rm e}c^{2}/eB=170(B/100~{}{\rm\mu G})$ km
(K13). This scale is therefore not so much relevant for the acceleration of
relativistic particles through DSA, but it is for the plasma heating of
collisionless shocks (and for the initial stages of particle acceleration),
taking place on the ion inertial length scale, $\lambda_{\rm ii}=c/\omega_{\rm
pi}\approx 230n_{\rm p}^{-1/2}$ km.

For synchrotron radiation the photon index $\Gamma_{1}=\frac{\xi+1}{2}$
reflects the spectral index $\xi$ of the electron spectrum. In contrast, in
the jitter regime $\Gamma_{2}$ reflects the spectral index $\nu_{\rm B}$ of
the magnetic-field fluctuation spectrum (defined as in Eq. 69 of K13, with the
relation $\Gamma_{2}=\nu_{\rm B}+1$. The spectral indices reported here (Table
3) range from $\Gamma_{2}=3$ (NE) to 3.4 (N), implying $\nu_{\rm B}\approx
2.0$– 2.5. Models of magnetic-field fluctuation spectra that are often
considered are $\nu_{\rm B}=5/3$ for Kolmogorov turbulence (Kolmogorov, 1941)
and $\nu_{\rm B}=3/2$ Kraichnan turbulence (Kraichnan, 1965). Both are flatter
than what is implied by the jitter model. However, Kolmorogov and Kraichnan
turbulence assume injection of large scale turbulence at a large length scale
cascading down to smaller length scales. At shock fronts the generation of
turbulence is different. For example, Takabe (2023) argues that the Weibel
instability results in a saturated spectrum with $\nu_{\rm B}=2$. This would
correspond to $\Gamma_{2}=3$, compatible with what we found here, and in line
with the hard X-ray spectra of other young SNRs, which are also generally also
around $\Gamma=3$ (e.g. for SN1006, Li et al., 2018).

For synchrotron radiation the cutoff photon energy relates to the highest
electron energy, see Eq. 3. For jitter radiation the maximum photon energy
$E_{\rm{max}}$ is (K13)

$E_{\rm{max}}\sim(R_{\rm{L,j}}/\lambda)^{3}\hbar\omega_{\rm{cut}}\Rightarrow\lambda/R_{\rm{L,j}}=\sqrt{E_{\rm{break}}/E_{\rm{max}}},$
(4)

for which we take advantage of the relation (K13)

$E_{\rm{break}}\sim\hbar\omega_{\rm{cut}}\frac{R_{\rm{L,j}}}{\lambda},$ (5)

with $\lambda$ the turbulence scale and

$\omega_{\rm{cut}}=\frac{3}{2}\frac{eB}{mc}\Big{(}\frac{E_{\rm{e,cut}}}{mc^{2}}\Big{)}^{2}$
(6)

being the cutoff frequency of the standard synchrotron regime (i.e. when
R${}_{\rm{L,j}}<<\lambda$) with B the average magnetic field.

From Eq. 6 we can resolve with respect to the cutoff energy of the population
of electrons $E_{\rm{e,cut}}$ emitting jitter radiation obtaining:

$E_{\rm{e,cut}}=\sqrt{\frac{2}{3}\frac{\omega_{\rm{cut}}m^{3}c^{5}}{eB}}=\sqrt{\frac{2}{3}\frac{m^{3}c^{5}}{eB}\frac{E_{\rm{break}}\lambda}{\hbar
R_{\rm{L,j}}}}$ (7)

All the quantities in Eq. 7 are known, though we only have an upper limit on
$\lambda$ and we then could put only upper limits on the particles’ cutoff
energy as well. The resulting values are shown in Table 4.

For Cas A the downstream magnetic-field strength has been estimated to be in
the range of 100–$500~{}{\rm\mu G}$ (Vink & Laming, 2003; Berezhko & Völk,
2004; Helder et al., 2012). For that reason we normalize to $B_{100}\equiv
B=100~{}{\mu G}$, so that we can translate the best fit values of $E_{\rm
break}$ and our lower limits for $E_{\rm max}$ into an upper limit on the
minimum scale for the magnetic-field fluctuation length scale, $\lambda_{\rm
min}$, as shown in Table 4. For calculating $E_{\rm e,cut}$ we included an
additional factor 2/3 in Eq. (3), as suggested by K13.

The table shows that the condition 2 required for jitter to be at work is
satisfied for all the regions but NW, indicating that magnetic-field
fluctuations must be present to scales of $\sim 100$ km and smaller. Such
values are remarkably small, and at a scale barely probed by in situ
measurements of interplanetary shocks, which measure magnetic-field
fluctuations with a resolution of seconds to minutes, which translates into
length scales of 100–6000 km, for plasma speeds of 100 km/s. Noticeably, the
cutoff energy of the particles in the jitter scenario does not differ much
from the standard synchrotron, which predicts $E_{\rm e,cut}\sim 10$ TeV.

Table 4: Upper limits on the minimum turbulence scale and particles’ cutoff
energy.

Region | $E_{\rm break}$ |  $E_{\rm max}$ (LL) |  $\lambda/B_{100}$ (UL) |  $E_{\rm ph,cut}$(UL) |  $E_{\rm e,cut}B_{100}^{-1/2}$ (UL)
---|---|---|---|---|---
| (keV) | (keV) | (km) | (keV) | (TeV)
Inner | 4.5 | 40 | 57 | 1.5 | 15
North | 5.2 | 40 | 61 | 1.9 | 17
Northeast | 5.2 | 40 | 61 | 1.9 | 17
Northwest | 40.4 | 40 | 171 | 41 | 78
SouthEast | 6.0 | 40 | 66 | 2.3 | 19
Southwest | 4.4 | 40 | 56 | 1.5 | 15
West | 4.7 | 40 | 59 | 1.6 | 16
Whole | 4.5 | 100 | 36 | 0.9 | 12

Notes. Scaled by $B_{100}=B/100~{}{\rm\mu G}$. LL and UL stand for lower and
upper limit, respectively.

###  4.3 Polarization and filaments’ width

If the nonthermal hard X-ray emission from Cas A and other young SNRs is
indeed due to jitter radiation then we need to rethink about some of the
phenomenology of nonthermal X-ray emission that is often taken for granted. In
particular, there are two aspects for which jitter radiation may have
important consequences: X-ray polarization, and the widths of X-ray
synchrotron filaments.

So far two X-ray polarization measurements by IXPE of young SNRs have been
reported, for Cas A (Vink et al., 2022b) and Tycho’s SNR (Ferrazzoli et al.,
2023). For Tycho’s SNR the X-ray rim was reported to have 9% polarized X-ray
emission, whereas for Cas A the polarization fraction was even lower: no
isolated regions with a large polarization fraction could be found, and the
overall polarization fraction on the nonthermal emission must be below 4%,
with a polarization angle topology consistent with radially aligned magnetic
fields. Again this argues for a highly turbulent magnetic fields at scales
$\lesssim 10^{17}$ cm in the context of synchrotron radiation. However, it is
surprising that there is not a large scale anisotropy in the magnetic field,
due to plasma shear—incurring radially oriented magnetic fields—or to shock
compression causing tangential magnetic fields. Since jitter radiation is
sensitive to very small-scale magnetic-field fluctuations, which are largely
isotropic, it should not be intrinsically polarized.555Private communication
Brian Reville. Note that Prosekin et al. (2016) calculated the expected
polarization of jitter radiation, but assuming only 2 dimensional
fluctuations, and showed that an anisotropic turbulence could still lead to a
considerable amount of polarized light. In the context of Cas A (and to a
lesser extent Tycho’s SNR), the low X-ray polarization fraction is simply a
feature of jitter radiation. The small residual X-ray polarization suggest
that a fraction of the non thermal radiation is ascribable to the synchrotron
regime, and hence the synchrotron cutoff energy $E_{\rm ph,cut}$ should be in
the soft X-ray band, as indeed seems to be the case according to Table 3. A
consequence that is potentially measurable should be that at larger photon
energies—i.e. moving away from the synchrotron part of the spectrum—the
polarization fraction should decrease. This is opposite to what is expected
for pure synchrotron radiation, where at higher energies the spectrum steepens
and originates from smaller regions, giving rise to a polarization fraction
increasing with energy (see Bykov et al., 2011).

As for the narrow width of the nonthermal X-ray filaments: these are usually
interpreted as due to strong synchrotron losses of the high-energy electrons,
giving rise to a width of $l_{\rm syn}=\frac{1}{4}V_{\rm s}\tau_{\rm syn}$,
with $V_{\rm s}$ the shock velocity and $\tau_{\rm syn}\approx 624/B^{2}E_{\rm
e}$ s, the synchrotron loss timescale, or they are thought to be indicative of
the electron diffusion length scale, $l_{\rm diff}\approx 4D_{2}/V_{s}$, with
$D_{2}=\eta cE/eB$ the diffusion coefficient and $\eta=1$ in the Bohm limit.
We see that for $l_{\rm syn}$ the synchrotron filaments width should decrease
with electron energy, and increase for $l_{\rm diff}$. However, near the
cutoff electron energy one expects $l_{\rm syn}\approx l_{\rm diff}$ (e.g.
Vink, 2020).

Nevertheless, there have been attempts to measure the filament widths as a
function of photon energy. Most recently this was done by Picquenot et al.
(2023), who reported that there is a ” narrowing with energy of the
synchrotron filaments in Cassiopeia A”, and that the ”energy dependency of
this narrowing seems stronger at high energy (Picquenot et al., 2023). In
light of the synchrotron model this is somewhat surprising, as it suggest that
filaments widths are purely defined by $l_{\rm syn}$ and not $l_{\rm diff}$.
However, these measurements can be explained by jitter radiation, considering
that the small-scale magnetic-field fluctuations generated near the shock
dampen further downstream of the shock. This argument is somewhat reminiscent
of the work of Pohl et al. (2005), but that argued for a decay of the overall
magnetic field, rather than a decay of the magnetic-field turbulence. So
jitter radiation offers a natural explanation for the narrowing of the
filament widths with energies. In the context of Cas A it would imply again
that at softer X-ray energies the width is partially determined by the
combination of $l_{\rm diff}$ and $l_{\rm syn}$, but at higher energies, where
the spectrum becomes more dominated by the jitter component, the width starts
to be determined by the length scale over which small-scale turbulence is
dampened. In principle this suggest that the narrowing of the filaments should
only become prominent for energies beyond $E_{\rm break}$, consistent with the
findings of Picquenot et al. (2023).

##  5 Summary and conclusions

In this paper we presented our results on the analysis of multi-instrument
X-ray data of the SNR Cas A aiming at characterizing the shape of nonthermal
emission in a wide energy band and in investigating its origin. We analyzed
Chandra/ACIS-S, NuSTAR/FPMA,B, SWIFT/BAT and INTEGRAL/ISGRI observations of
Cas A in the 4-100 keV energy band and performed a spatially resolved spectral
analysis adopting various nonthermal spectral models. Our findings can be
summarized a follows.

  1. 1.

While DSA intrinsically requires high magnetic turbulence, the spectral models
used to fit nonthermal spectra of SNRs do not include the effect of turbulence
in the shaping of the spectra. The standard approach to fit SNRs spectra in
the 4-6 keV energy band to investigate its synchrotron radiation is
insensitive to the actual shape of the nonthermal model adopted.

  2. 2.

The 15-100 keV spectra of the whole Cas A can be adequately fitted only
through a power-law with a photon index $\Gamma\sim 3.3$. The 15-100 keV
spectra are not well fitted either by the srcut (Reynolds & Keohane, 1999) or
the zira(Zirakashvili & Aharonian, 2007) model, as there is no evidence for an
exponential cutoff.

  3. 3.

A jitter model describes the 4-40 keV spectra of all the regions considered
much better than any other model showing a clear steepening of the spectrum
non-compatible with a cutoff and with a best-fit photon index $\Gamma_{2}\sim
3-3.4$.

  4. 4.

The best-fit photon index $\Gamma_{2}\sim 3-3.4$ imply a spectral index for
the turbulence spectrum $\nu_{B}\sim 2-2.4$, higher than common values such as
Kolmogorov $\nu_{\rm{Kol}}=5/3$ and Kraichnan $\nu_{\rm{Kra}}=3/2$. This
applies to magnetic-field fluctuations on scales of $\sim 100$ km, much
smaller than the scales normally invoked for the DSA mechanism, and more in
line with fluctuations near collionless shocks as induced by the Weibel
instability.

  5. 5.

We estimated upper limits on the minimum turbulence scale which are typically
smaller than $100$ km, whereas the estimated transition energies from
synchrotron to jitter regimes imply cutoff electron energies of typically 15
TeV, as in the standard synchrotron scenario.

  6. 6.

If indeed the nonthermal X-ray emission is due to both synchrotron and jitter
components this offers an alternative natural explanation for the low
polarization fraction for Cas A and the narrowing of nonthermal X-ray
filaments with increasing energy. So, potentially, with nonthermal X-ray
emission we are zooming into regions very close to the collisionless shocks
themselves.

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## Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation program under grant agreement No. 101004131 (SHARP).
We thank the XSPEC helpdesk and R. Kristin for the help in the implementing of
the jitter model. We thank Brian Reville for helpful comments on jitter
radiation.

##  Appendix A Zira and jitter models in XSPEC

###  A.1 Zira

We implemented the Eq. 37 in Zirakashvili & Aharonian (2007) through the
mdefine task available within XSPEC, defining a new zira model. The
corresponding equation in the notation used in this paper can be read as:

$(E_{\rm{rad}}/E)^{2}(1+0.38\sqrt{E/E_{\rm{cut}}})^{11/4}exp[-(E/E_{\rm{cut}})^{1/2}]$
(A1)

where $E_{\rm{rad}}$ is always kept fixed to 1. The exponential 2 in
$(E_{\rm{rad}}/E)^{2}$ is added to match the default XSPEC units.

###  A.2 Jitter

The total jitter model does not have an analytical expression and cannot be
included within XSPEC through the mdefine task. We followed the approach
described in Load_model_XSPEC to include our jitter model in the library of
the available models. The synchrotron dominated and jitter dominated regimes
are respectively modeled by two power laws with an additional exponential
cutoff for the low energy component. However the simple addition of both
components in a single model does not replicate the complex shape of the total
emission, and advocates for additional energy and/or flux dependent
constraints. For this purpose the flux given by the two components is computed
iteratively in each energy bin (starting from low energy) and the highest is
kept as the model flux. The ratio of both component flux, labeled jratio, is
added as a parameter to the model and acts as a normalization. For continuity
and self-consistency reasons, the jitter flux is set to 0 for energies lower
than the synchrotron break, preventing any ill behavior in situations of
steeper jitter slope resulting in high flux at low energies. We tested our
model by comparing it with other well-known models such as pow, bknpower, i.e.
a broken power-law, and cutoffpl, a power-law with an exponential cutoff, in
regimes in which the jitter radiation simply reduces to one of these.

The jitter model is finally described by six parameters: PhoIndex1:
synchrotron power-law slope ($\Gamma_{1}$), Ebreak1: synchrotron break energy
($E_{\rm{break}}=\hbar\omega_{break}$, with $\omega_{break}$ defined in Fig.
1), PhoIndex2: jitter-radiation power-law slope ($\Gamma_{2}$), jratio: ratio
between flux in regime 1 and regime 2 ($P_{2}/P_{1}$ in TF87), beta1: shape of
the exponential cutoff around Ebreak1
($\propto\exp[-(\omega/\omega_{break})^{\beta}]$), norm: total normalization
of the model.

##  Appendix B Cross-calibration between X-ray telescopes

We investigated potential issues causing the significant discrepancy observed
in the Chandra and NuSTAR spectra in the 4-6 keV energy range. The distortion
in the Chandra/ACIS-S spectra could be produced by pileup effects. We
produced, through the pileup_map task within CIAO, a pileup map of Cas A,
finding that the average pileup fraction in the regions considered is of
roughly 3%, and it was therefore excluded as a possible origin for the
flattening. We checked that the flux of NuSTAR spectra in the $>$15 keV energy
band is consistent with those reported for BeppoSAX (Vink & Laming 2003) and
INTEGRAL (Renaud et al. 2006) considering the nominal cross-calibration
constant (Madsen et al. 2015, 2022). We used srcflux on the Chandra image in
the 4-6 keV band and checked that the observed counts matched the counts in
the same energy range in the spectra. We repeated this procedure also for the
NuSTAR data with the ftool countsinregion. Finally, we used adopted different
values for the parameter boxsize=5,10,20,30,40 obtaining identical spectra and
set flatflagarf=yes as in Grefenstette et al. (2017).

##  Appendix C SWIFT upper limit on flux

We estimated the upper limit on the X-ray flux of A Cas above 100 keV by
estimating, through the error command within XSPEC, the 90% and 99.7%
confidence level (corresponding to $\Delta\chi=2.706$ and $\Delta\chi^{2}=9$)
starting from the SWIFT/BAT spectra of Cas A. The best-fit model was a power-
law having $\Gamma=3.5$. The error on the flux was estimated through the cflux
component. The gaussian components, describing the lines of Ti44, are kept
fixed since these are not resolved by SWIFT but contribute to the global flux.
The corresponding upper limit on the flux between 100 and 200 keV were
$0.28\times 10^{-11}$ erg/s and $0.33\times 10^{-11}$ erg/s at the 90% and
99.7% confidence levels, respectively.

##  Appendix D Nonthermal to total flux ratio

In Sect. 3.2 we showed that, we cannot rule out or confirm the presence of a
cutoff in the 4-6 keV spectra. As already mentioned, a possible source of
uncertainty is the thermal emission from the shocked-heated plasma. Because of
the bad spatial resolution of NuSTAR and the need for high statistics up to 40
keV, we selected regions with a characteristic size at least comparable to the
NuSTAR PSF ($\sim 60^{\prime\prime}$ in Half Power Diameter). It was then
natural to expect significant shocked ejecta signatures in the extracted
spectra. This additional thermal contribution is added to the putative
nonthermal cutoff that is therefore blurred out.

A partial, solution for this issue is to measure the nonthermal and the
thermal flux from Chandra/ACIS-S best-fit models performed in the 0.5-8 keV
band. We modeled the Chandra/ACIS-S data with a model made of an absorbing
component, TBabs model in XSPEC, a power-law and a number of vnei components,
depending on the complexity of the plasma thermal emission. Given the
extremely high number of counts in the spectra extracted from regions I, W and
Whole, we included a systematic error of 3%, 3% and 5% respectively during the
fitting procedure. We measured the total and the nonthermal only flux through
the cflux component available within XSPEC. In Fig. 10 the ratios nonthermal
to total flux for all the regions considered are shown.

Figure 10: Ratio nonthermal over total flux in the 4-6 keV band obtained from
the fits in the 0.5-8 keV band as a function of the region.

All the regions but West and Whole showed a nonthermal to total flux ratio
higher than 80%, and consistent with 100%. Lower values found for West and
Whole regions were not surprising since in these regions significant emission
from shocked ejecta was revealed in previous works (e.g. Hwang & Laming 2012).
These results confirmed that not including a thermal component when fitting
4-6 keV energy range had a small to zero effect on the estimate of the slope
of the spectra. It is worth to mention that the values displayed in Fig. 10
did not provide information on the shape of the photon spectra because of the
degeneracy between emission measure and abundance (see Greco et al. 2020 for
details). In fact, by repeating the analysis with pure-ejecta components, i.e.
vnei models with abundances values fixed to $\gtrsim 10^{3}$, the ratio
increased by reaching values of 95$\%$, as also reported by Vink et al.
(2022b).

##  Appendix E Plots for all the regions

Figure 11: Same as Fig. 7 but for all the regions considered Figure 12: Same
as Fig. 8 but for all the regions Figure 13:  Same as Fig. 9 but for all the
regions