# The auto and cross angular power spectrum of the Cas A supernova remnant in
radio and X-ray

Preetha Saha,1 Somnath Bharadwaj,1 Susmita Chakravorty, Nirupam Roy,2 Samir
Choudhuri,3 Hans Moritz Günther,4 Randall K. Smith5

1 Department of Physics and Centre for Theoretical Studies, Indian Institute
of Technology, Kharagpur 721302, India
2 Department of Physics, Indian Institute of Science, Bangalore 560012, India
3 Astronomy Unit, Queen Mary University of London, Mile End Road, London E1
4NS, United Kingdom
4 MIT, Kavli Institute for Astrophysics and Space Research, 77 Massachusetts
Avenue, Cambridge, MA 02139, USA
5 Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA
02138, USA preetha<EMAIL_ADDRESS>
(Accepted yyyy month dd. Received yyyy month dd; in original form yyyy month
dd)

###### Abstract

The shell type supernova remnant (SNR) Cas A exhibits structures at nearly all
angular scales. Previous studies show the angular power spectrum $(C_{\ell})$
of the radio emission to be a broken power law, consistent with MHD
turbulence. The break has been identified with the transition from 2D to 3D
turbulence at the angular scale corresponding to the shell thickness.
Alternatively, this can also be explained as 2D inverse cascade driven by
energy injection from knot-shock interactions. Here we present $C_{\ell}$
measured from archival VLA $5$GHz (C band) data, and Chandra X-ray data in the
energy ranges ${\rm A}=0.6-1.0\,\,{\rm keV}$ and ${\rm B}=4.2-6.0\,\,{\rm
keV}$, both of which are continuum dominated. The different emissions all
trace fluctuations in the underlying plasma and possibly also the magnetic
field, and we expect them to be correlated. We quantify this using the cross
$C_{\ell}$ between the different emissions. We find that X-ray B is strongly
correlated with both radio and X-ray A, however X-ray A is only very weakly
correlated with radio. This supports a picture where X-ray A is predominantly
thermal bremsstrahlung whereas X-ray B is a composite of thermal
bremsstrahlung and non-thermal synchrotron emission. The various $C_{\ell}$
measured here, all show a broken power law behaviour. However, the slopes are
typically shallower than those in radio and the position of the break also
corresponds to smaller angular scales. These findings provide observational
inputs regarding the nature of turbulence and the emission mechanisms in Cas
A.

###### keywords:

ISM: supernova remnants - methods: data analysis - methods: statistical -
turbulence - (magnetohydrodynamics) MHD - radiation mechanisms: general

††pagerange: The auto and cross angular power spectrum of the Cas A supernova
remnant in radio and X-ray–References††pubyear: 2021

##  1 Introduction

A supernova explosion is characterized by the release of a vast amount of
kinetic energy from a point in space. This energy in the form of a shock wave
travels ahead of the ejected stellar material from the explosion. Bounded by
the shock wave expanding through the surrounding interstellar medium (ISM), a
supernova remnant (SNR) consists of the stellar ejecta and the interstellar
material swept along its way. The temporal evolution of the remnant is
described in terms of four evolutionary stages (Woltjer, 1972). In the first
phase the mass swept up by the shock is much less than the mass of the stellar
ejecta and the stellar ejecta expands at the initial shock speed. This stage
is referred to as the ‘ejecta-driven’ (ED) phase. Since the energy radiated is
still very small compared to the kinetic energy released during the explosion,
the remnant undergoes adiabatic expansion and enters its next ‘Sedov-Taylor’
(ST) phase. In this phase, the mass swept up by the shock begins to dominate
the ejecta. Based on a simple spherically symmetric model, the self-similarity
solutions (Sedov, 1946; Taylor, 1950) can explain only the adiabatic phase of
the evolution. However, the one-dimensional self-similarity does not remain
valid for the later phases of the adiabatic evolution due to the interaction
of the ejecta with the inhomogeneities in the ISM (Chevalier, 1977). The
evolution ceases to be adiabatic once the radiative loss become significant.
We now have the ‘pressure driven snowplow’ (PDS) phase where the radiative
shock is driven by the interior pressure. Here the blast wave decelerates and
the dense shell surrounding the hot interior cools down. In the fourth phase,
the interior gas starts losing energy by ‘snowplowing’ or moving the shell
through the ISM. This is referred to as the ‘momentum driven snowplow’ (MDS)
phase. In an uniform density circumstellar medium (CSM), the temporal
evolution of the forward shock radius follows a power-law $t^{m}$, with values
$m=1,\frac{2}{5},\frac{2}{7},\frac{1}{4}$ for these four phases respectively
(Cioffi et al., 1988). For Cas A SNR, it is found that CSM has a density
varying inversely with the square of the shock radius (Hwang & Laming, 2009).
This indicates that the remnant is in its ST phase of evolution as the swept
up mass is much greater than that of the ejecta, but most of the X-ray
emission comes from the ejecta.

Besides the expansion studies, SNRs are studied over the entire
electromagnetic spectrum from radio to high energy gamma bands. Each frequency
band of observation offers a different physical insight to the understanding
of supernovae and SNRs. Considering any particular frequency, every SNR
exhibits a variety of overall morphology as well as rich structures over a
wide range of scales. SNRs can be used to probe the global properties of the
galaxy as well as the local environment in different regions. Unlike the SNRs
which are unresolved at extragalactic distances, SNRs in our Galaxy provide us
an opportunity to investigate their structures in finer details (Green, 2019).

Although there have been many multi wavelength observations of the Galactic
SNRs, there have been very few systematic studies to quantify the statistical
properties of the fine-scale angular structures associated with the remnant.
Roy et al. (2009) have estimated the angular power spectrum of the observed
intensity fluctuations for two Galactic SNRs, Cas A and Crab using the
visibility data from Very Large Array (VLA) at frequencies $~{}1.5$ GHz (L
band) and $~{}5$ GHz (C band). The resultant angular power spectra of both the
SNRs show a power law with index $-3.24\pm 0.03$. However, for the Cas A SNR a
break in the power law was observed at large angular scales where the index
was found to change from $-3.2$ to $-2.2$. This break was attributed to a
transition from 3D to 2D turbulence on scales larger than the shell thickness
of Cas A. A similar transition was not observed for the Crab SNR which has a
filled centre structure. In Roy et al. (2009), the power spectrum was
estimated using pairwise correlations of the measured visibilities (the Bare
Estimator of Choudhuri et al. 2014). In a recent work Saha et al. (2019) have
estimated the angular power spectrum of the Kepler SNR using an improved
estimator the Tapered Gridded Estimator (TGE) (Choudhuri et al., 2016b). This
estimator reduces the residual point source contamination from regions of the
sky away from the target source by tapering the primary beam response through
an appropriate convolution in the visibility domain. The angular power
spectrum was measured in both L and C bands using VLA data. In both cases the
power spectrum measured across the angular multipole $\ell$ range
$(1.9-6.9)\times 10^{4}$ was found to be a broken power law with a break at
$\ell_{b}=3.3\times 10^{4}$, and power law index of $-2.84$ and $-4.39$ before
$(\ell<\ell_{b})$ and after $(\ell>\ell_{b})$ the break respectively. The C
and L band results were found to differ at $\ell>6.91\times 10^{4}$. In this
$\ell$ range the L band power spectrum was found to flatten out at a value
which is consistent with model predictions for the expected residual point
source foreground contribution. However, the C band power spectrum was found
to follow a power law with index $-3.07$. Considering the synchrotron emitting
plasma of the SNR, the slope of $-2.84$ found at large angular scales is
consistent with 2D Kolmogorov turbulence, whereas at small angular scales the
C band power law slope $-3.07$ is roughly consistent with 3D Kolmogorov
turbulence. The intermediate $\ell$ range with a steep $-4.39$ power law
behavior was attributed to the complex morphology of the SNR. Furthermore, the
angular power spectrum estimates of Cas A and Crab obtained using TGE were
found to be similar with the earlier results reported in Roy et al. (2009). In
another recent work Shimoda et al. (2018) have analyzed VLA L band image of
the Tycho SNR to estimate the two-point correlation function of the specific
intensity fluctuations. They have reported a Kolmogorov-like magnetic energy
spectrum showing a scaling $r^{2/3}$. Vishwakarma & Kumar (2020) have carried
out a similar analysis for Cas A using $410-460$ MHz Giant Metrewave Radio
Telescope (GMRT) data, their results are consistent with the above mentioned
Tycho SNR results.

In this paper we revisit Cas A considering not only radio but also X-ray
observations of this SNR. For the present analysis we have analysed archival
Chandra X-ray data (Chandra observation ID: $4638$). Cassiopeia A or Cas A
(G111.7$-$2.1) is located in the constellation Cassiopeia in the second
Galactic quadrant. Cas A is a relatively young Galactic SNR which has been
extensively studied since its discovery. At the radio frequencies, Cas A SNR
shows a clear shell like structure with compact emission knots. It is one of
the strongest radio sources in the sky with a flux density of $2720\pm 50\,$Jy
at $1\,$GHz. The shell structure subtends an angular diameter of
$5^{{}^{\prime}}$ at a distance of approximately $3.4\,$kpc (Reed et al.,
1995). The shell thickness estimated from the radial brightness profile at
radio frequencies is found to be approximately $30^{{}^{\prime\prime}}$. It is
most likely a remnant of a late 17th century supernova (Fesen et al., 2006).
Krause et al. (2008) suggests that CasA was a type IIb supernova. The
spectroscopic observations of Cas A SNR at X-ray frequencies reveal that it is
a strong X-ray source with the presence of different elements in the supernova
ejecta. Each of these elements produces X-rays within narrow energy ranges. In
X-ray waveband, the overall emission structure is an approximately circular
region of diffuse emission which surrounds an inner ring of clumpy emission
(Charles et al., 1977; Fabian et al., 1980; Jansen et al., 1988). The circular
region has a protrusion at the northeast.

In this paper we have extended the line of investigation started in Roy et al.
(2009) and continued in Saha et al. (2019) to present the angular power
spectrum of Cas A SNR both at low radio frequency and at high X-ray frequency.
The radio power spectrum arises mainly from the fluctuations in the non-
thermal synchrotron radiation. A convective instability due to the interaction
of the ISM and the ejecta makes enough turbulent energy available to account
for the observed synchrotron emission (Gull, 1973). The emissivity of the
synchrotron radiation depends on $\sim\,n_{e}\,B^{\frac{p+1}{2}}$ where
$n_{e}$ is the electron number density, $B$ is the magnetic field and $p$ is
the power-law index of the electron spectrum. Considering the emission from
the SNRs at X-ray frequency, this can be either thermal bremsstrahlung or non-
thermal synchrotron or a combination of both. For Cas A SNR Allen et al.
(1997) have shown evidence for a non thermal high energy tail in the X-ray
continuum spectrum which is likely to be produced by synchrotron radiation. In
addition to the synchrotron component, the high energy tail can also have
contribution from non-thermal bremsstrahlung (Asvarov et al., 1990; Bleeker et
al., 2001; Grefenstette et al., 2015). Later, Helder & Vink (2008) argued that
about $54\%$ of the overall continuum emission ($4-6$ keV) of Cas A is non
thermal synchrotron radiation, while the rest is thermal bremsstrahlung. While
the radio power spectrum from the earlier works already sheds some light on
the statistics of the small scale turbulent fluctuations in the electron
density and magnetic field, we expect the X-ray angular power spectrum to
provide independent additional inputs on this. Further, we may expect the
fluctuations in the radio and non thermal X-ray emission to be correlated as
both are related to the electron density and also the magnetic field. In this
paper we have also estimated the cross-correlation angular power spectrum
between the fluctuations in the radio and the X-ray emission.

Cross-correlation is a commonly used methodology in signal and image
processing. The noise and other artefacts like extraneous background or
foreground emission affecting the two different signals are expected to be
uncorrelated, and their contribution can be avoided if we cross-correlate the
two signals. Considering the radio observations used for our analysis, the
contribution from various foregrounds like the extragalactic point sources and
the diffuse Galactic synchrotron emission (DGSE) both increase if the
observing frequency is reduced. Guided by this we have chosen the VLA C band
visibility data of Cas A SNR over that of the L band for the analysis
presented here. A brief outline of the paper follows.

The details of the archival VLA radio data and Chandra X-ray data used here
are briefly outlined in Section 2. The methodology of the power spectrum
estimation and its error are described in Section 3. We report the results in
Section 4 and modeling of the radio power spectrum in 5, followed by its
discussion and conclusion in Section 6.

##  2 Data

We have used the Very Large Array (VLA) archival C band ($5\,$GHz) multi
configuration uv data (Project code: AR0435) of the Cas A supernova remnant
(SNR). Roy et al. (2009) have calibrated and used this data to estimate the
angular power spectrum of the Cas A SNR, and the reader is referred to Table 1
of the above mentioned paper for further details of the data. To visualise the
source structure the left panel of Figure 1 here shows a CLEANed radio image
made from the calibrated visibilities using standard AIPS111NRAO Astrophysical
Image Processing System, a commonly used software for radio data processing
tasks. We see that the Cas A SNR subtends approximately $5^{{}^{\prime}}\times
5^{{}^{\prime}}$ in the radio image. We note that the radio image here shows
the specific intensity $I(\bm{\theta})$ in units of $\mathrm{Jy\,beam^{-1}}$.
This can be converted to brightness temperature $T(\bm{\theta})$ through
$T(\bm{\theta})=(2\,k_{B}/\lambda^{2})^{-1}\,I(\bm{\theta})$ in the Rayleigh-
Jeans limit which is applicable here.

Figure 1: Images of Cas A SNR in Radio (left) from the VLA archival visibility
data in C band D configuration, and X-ray from Chandra archival data spanning
$0.6-1.0$ keV (X-Ray A, middle) and $4.2-6.0$ keV (X=ray B, right). The images
are all in J2000 coordinates. Figure 2: Log-log plot of X-ray spectrum of Cas
A SNR using Chandra archival data (observation ID: $4638$). The background has
been substracted here. We have marked the lower (X-ray A, marked with red
lines) and the higher (X-ray B, marked with blue lines) energy ranges that we
have used to derive our images for the X-ray analysis, in this paper.

For X-ray, we have analysed Advanced CCD Imaging Spectrometer (ACIS-S) non-
grating archival Chandra data of Cas A. The data was observed on $14$th April,
$2004$ as a part of observation ID $4638$ and has a long exposure time of
$164.5\,$ks. The field of view (FoV) corresponds to a large region of angular
extent $16.9\arcmin\times 8.3\arcmin$. We analyze the downloaded data using
Chandra Interactive Analysis of Observations (CIAO) version $4.10$ (Fruscione
et al., 2006). We create a reprocessed event$2$ file from the downloaded
event$1$ file using chandra_repro. Using the event$2$ file, we generate the
X-ray spectrum which is shown in Figure 2. For the subsequent analysis we have
identified two different energy ranges which are free of any strong emission
lines, namely X-ray A and X-ray B (or simply A and B) which respectively spans
$0.6-1.0$ keV and $4.2-6.0$ keV (Figure 2). We expect A to be dominated by
thermal emission while B is expected to be a mixture of thermal and non-
thermal emission. The middle and right panels of Figure 1 respectively show
the X-ray images corresponding to A and B. Here from the X-ray images, we
consider $J(\bm{\theta})$ which refers to the X-ray photon flux per unit
energy interval per solid angle. We note that $J(\bm{\theta})$ is analogous to
the specific intensity $I(\bm{\theta})$ which refers to the energy flux per
frequency interval per solid angle (Bradt, 2008). As seen by comparing the
different images in Figure 1, the angular size of Cas A is similar in radio as
well as X-ray A and B. While the radio and X-ray B images are relatively more
symmetric, the circular structure in X-ray A is incomplete with the appearance
of an arc on the southwestern side. The inner ring and the protrusion of the
remnant in the northeast are also more prominent in the X-ray A image.

##  3 Methodology

###  3.1 Radio power spectrum

In radio interferometric observations, the measured quantity is a set of
complex visibilities $\mathcal{V}_{i}$ which are sensitive to the angular
fluctuations in the sky signal. In addition to the sky signal component
$\mathcal{S}(\textbf{{U}}_{\rm i})$, the measured visibility $\mathcal{V}_{i}$
has a system noise $\mathcal{N}$ component

$\mathcal{V}_{i}=\mathcal{S}(\textbf{{U}}_{\rm i})+\mathcal{N}_{i}\,.$ (1)

The sky signal component $\mathcal{S}(\textbf{{U}}_{\rm i})$ is the Fourier
transform of the product of the telescope’s primary beam pattern
$\textit{A}(\bm{\theta})$ and the brightness temperature fluctuations
$\delta\textit{T}(\bm{\theta})$ on the sky

$\mathcal{S}(\textbf{{U}}_{\rm i})=Q_{\nu}\int d^{2}\bm{\theta}\,e^{2\pi
i\textbf{{U}}_{\rm
i}\cdot\bm{\theta}}\,\textit{A}(\bm{\theta})\,\delta\textit{T}(\bm{\theta})$
(2)

where $Q_{\nu}=2k_{B}/\lambda^{2}$ is the conversion factor from brightness
temperature to specific intensity in the Rayleigh Jeans limit and $k_{B}$ is
the Boltzmann constant. Here we assume that $\delta\textit{T}(\bm{\theta})$ is
a particular realisation of a statistically homogeneous and isotropic Gaussian
random field whose statistical properties are completely quantified by the
angular power spectrum $C^{TT}_{\ell}$. In the flat sky approximation adopted
here, $C^{TT}_{\ell}$ which is the angular power spectrum of brightness
temperature fluctuations is defined using

$\langle\Delta\tilde{T}(\textbf{{U}})\,\Delta\tilde{T}^{*}(\textbf{{U}}^{{}^{\prime}})\rangle=\delta_{D}^{2}(\textbf{{U}}-\textbf{{U}}^{{}^{\prime}})\,C^{TT}_{\ell=2\pi
U}$ (3)

where $\Delta\tilde{T}(\textbf{{U}})$ is the Fourier transform of
$\delta\textit{T}(\bm{\theta})$,
$\delta_{D}^{2}(\textbf{{U}}-\textbf{{U}}^{{}^{\prime}})$ is the 2D Dirac
delta function and the angular brackets $\langle...\rangle$ denote an ensemble
average with respect to different realizations of the Gaussian random field.

The angular extent of the target source is usually smaller compared to that of
$\textit{A}(\bm{\theta})$. In addition to the sky signal from the target
source, the measured visibilities $\mathcal{V}_{i}$ may have significant
contributions from other sources which lie within the angular extent of
$\textit{A}(\bm{\theta})$. Therefore, it is advantageous to restrict the sky
response to a small region around the target source. The Tapered Gridded
Estimator (TGE) (Choudhuri et al., 2014, 2016a) achieves this by tapering the
sky response with a suitable window function $\textit{W}(\bm{\theta})$. TGE
restricts the sky response by convolving the measured visibilities with
$\tilde{w}(\textbf{{U}})$ which is the Fourier transform of
$\textit{W}(\bm{\theta})$. This visibility based estimator for the radio
frequency data allow us to bypass any additional complications due to imaging
and deconvolution, which are quite challenging for strong extended source like
Cas A SNR. TGE also grids the visibilties in order to reduce the volume of
computations. The convolved gridded $\mathcal{V}_{cg}$ are given by

$\mathcal{V}_{cg}=\sum_{i}\tilde{w}(\textbf{{U}}_{\rm g}-\textbf{{U}}_{\rm
i})\mathcal{V}_{i}$ (4)

where $\textbf{{U}}_{\rm g}$ refers to the baseline corresponding to the
different grid points (labeled $g$) of a rectangular grid in the uv plane. The
size of the rectangular uv grid is set by the choice of the baseline range
$U_{max}$ and $U_{min}$. This is chosen such that the grid can incorporate the
visibility data of both the largest configuration and the shortest
configuration of VLA. We have chosen a grid spacing within which the
convolution in eq. (4) is well represented. The uniform grid size and spacing
allows us to collapse the visibility data of the different VLA configurations
into a single grid. The angular power spectrum estimator is defined as

$\hat{E}_{g}=(M_{g})^{-1}\times\bigg{(}|\mathcal{V}_{cg}|^{2}-\sum_{i}|\tilde{w}(\textbf{{U}}_{\rm
g}-\textbf{{U}}_{\rm i})|^{2}|\mathcal{V}_{i}|^{2}\bigg{)}$ (5)

where $M_{g}$ is a normalization constant and the second term in the r.h.s. is
introduced to subtract out the noise bias arising from the noise contribution
present in each visibility. Here $\langle\hat{E}_{g}\rangle$ gives an unbiased
estimate of the angular auto power spectrum $C^{TT}_{\ell}$ at the angular
multipole $\ell_{g}=2\pi\mid\textbf{{U}}_{\rm g}\mid$.

The value of $M_{g}$ is calculated using simulated visibilities. We have
simulated visibilities corresponding to an unit angular power spectrum (UAPS,
$C^{UAPS}_{\ell}=1$) and having the same baseline distribution as the
observational data. These simulations incorporate only the sky signal
contribution, with no system noise. We use an ensemble of such simulations to
evaluate the normalization constant

$M_{g}=\bigg{\langle}\,\big{(}|\mathcal{V}_{cg}|^{2}-\sum_{i}|\tilde{w}(\textbf{{U}}_{\rm
g}-\textbf{{U}}_{\rm
i})|^{2}|\mathcal{V}_{i}|^{2}\big{)}\,\bigg{\rangle}UAPS\,.$ (6)

We apply the visibility based TGE to the radio visibility data of the SNR to
estimate the angular auto power spectrum $C^{TT}_{\ell}$. We have binned the
estimated $C^{TT}_{\ell}$ values into $\ell$ bins of equal logarithmic
interval.

The details of the TGE angular power spectrum estimator are discussed in
Choudhuri et al. (2014, 2016a), whereas the detailed methodology for the TGE
based $C^{TT}_{\ell}$ estimation for several supernova remnants (Cas A, Crab
and Kepler) is given in Section $3.1$ of Saha et al. (2019).

###  3.2 X-ray power spectrum

For the X-ray data of the Cas A SNR we consider $J(\bm{\theta})$ which, as
discussed in Section 2, is analogous to the specific intensity
$I(\bm{\theta})$. Here we use $J^{A}(\bm{\theta})$ and $J^{B}(\bm{\theta})$ to
refer to X-ray A and B respectively. Considering A, the X-ray image (middle
panel of Figure 1) presents us with values of $J^{A}(\bm{\theta})$ on a
pixelized image of angular dimensions $L\times L=\Omega$ subtending a solid
angle $\Omega$. We calculate the Fourier components $\Delta
J^{A}(\textbf{{U}})$ of the image, and calculate the X-ray auto angular power
spectrum $C^{AA}_{\ell}$ using

$C^{AA}_{\ell=2\pi U}=\Omega^{-1}\,|\Delta J^{A}(\textbf{{U}})|^{2}$ (7)

The power spectrum $C^{BB}_{\ell}$ was estimated in exactly the same way using
$J^{B}(\bm{\theta})$.

We have also estimated the cross-power spectrum between the two different
X-ray energy bands using

$C^{AB}_{\ell=2\pi U}=\Omega^{-1}\,\mid[\Delta J^{A}(\textbf{{U}})]\,[\Delta
J^{B}(\textbf{{U}})]^{*}\mid\,.$ (8)

The estimated power spectra were all individually binned in $\ell$ bins of
equal logarithmic interval.

###  3.3 Radio X-ray Cross power spectrum

We now discuss how we have cross-correlated the radio visibility data with the
X-ray A and the X-ray B image data to estimate the cross power spectrum
$C^{TA}_{\ell}$ and $C^{TB}_{\ell}$ respectively. Considering the X-ray A, we
have an exposure corrected image which provides us with $\Delta
J^{A}(\textbf{{U}})$. This is available at uniformly sampled U values, and we
may interpret the measured $\Delta J^{A}(\textbf{{U}})$ as directly pertaining
to different Fourier components of the X-ray structure of the Cas A SNR. In
contrast, the visibility data $\mathcal{V}_{i}$ is only available at the
$\textbf{{U}}_{i}$ values corresponding to the VLA baseline distribution.
Typically, the baseline distribution $\textbf{{U}}_{i}$ does not span the
entire Fourier domain of our interest. Further, we cannot interpret the
measured visibilities directly as the Fourier components of
$\delta\textit{T}(\bm{\theta})$ due to the primary beam pattern
$\textit{A}(\bm{\theta})$ which appears in eq. (2). Note that field of view of
the Chandra X-ray image is larger than the angular extent of the VLA primary
beam pattern $\textit{A}(\bm{\theta})$. In addition, we have to also consider
the window function $\textit{W}(\bm{\theta})$ if we deal with the convolved,
gridded visibilities $\mathcal{V}_{cg}$ (eq. 4). It is necessary to account
for these aspects of the radio data when estimating the cross power spectrum.
In order to overcome these issues, we have converted the X-ray A data into
X-ray A visibilities $\mathcal{V}_{{A}_{i}}$ using

$\mathcal{V}_{{A}_{i}}=\int d^{2}\bm{\theta}\,e^{2\pi i\textbf{{U}}_{\rm
i}\cdot\bm{\theta}}\,\textit{A}(\bm{\theta})\,\delta\textit{J}^{A}(\bm{\theta}).$
(9)

which are exactly analogous to the radio visibility data. Here
$\textit{A}(\bm{\theta})$ is the VLA primary beam pattern, and the X-ray A
visibilities have been computed at the $\textbf{{U}}_{i}$ values corresponding
to the VLA baselines distribution. We have used the X-ray A visibilities for
the cross-correlation analysis.

We have used the TGE to estimate the cross-angular power spectrum
$C^{TA}_{\ell}$. In order to maintain a similar sky response from the source
in both the radio and the X-ray A, the X-ray A visibilties
$\mathcal{V}_{{A}_{i}}$ were convolved with the same window function as the
radio data (eq. 4)

$\mathcal{V}_{{A}_{cg}}=\sum_{i}\tilde{w}(\textbf{{U}}_{\rm
g}-\textbf{{U}}_{\rm i})\mathcal{V}_{{A}_{i}}\,,$ (10)

and the grid also is identical to that used for the radio data. We generalize
eq. (5) to define an estimator for the cross power spectrum

$\hat{E}_{g}=(M_{g})^{-1}\,\mid\mathcal{V}_{A_{cg}}\,\mathcal{V}^{*}_{cg}\mid$
(11)

where $M_{g}$ is the same proportionality constant as in eq. (6). The noise in
the radio and the X-ray A visiblities are uncorrelated, and it is not
necessary to consider the noise bias here. We follow exactly same procedure to
calculate the cross spectrum $C^{TB}_{\ell}$ using the measured radio
visibilties $\mathcal{V}_{i}$ and the estimated X-ray B visibilities
$\mathcal{V}_{{B}_{i}}$. We present the bin averaged values of the angular
cross power spectra $C^{TA}_{\ell}$ and $C^{TB}_{\ell}$.

###  3.4 Interpreting the power spectra

To interpret the radio sky signal, the observed brightness temperature
fluctuation $\delta\textit{T}(\bm{\theta})$ of the remnant can be represented
as

$\delta\textit{T}(\bm{\theta})=\textit{R}(\bm{\theta})[\bar{T}_{s}+\delta\textit{T}_{s}(\bm{\theta})]$
(12)

where $\bar{T}_{s}$ is the mean and $\delta\textit{T}_{s}(\bm{\theta})$ is the
fluctuating component. $\textit{R}(\bm{\theta})$ is a dimensionless profile
function of the remnant which captures the angular profile of the remnant.
This $\textit{R}(\bm{\theta})$ modulates both $\bar{T}_{s}$ and
$\delta\textit{T}_{s}(\bm{\theta})$ and cuts off the emission beyond the
finite size of the SNR. We assume $\delta\textit{T}_{s}(\bm{\theta})$ to be
the outcome of a statistically homogeneous and isotropic Gaussian random
process whose statistical properties are completely specified by the angular
power spectrum $C_{\ell}$. A similar statistical interpretation is described
in Chandrasekhar & Münch (1952) in the context of turbulent ISM of our Galaxy.

The angular power spectrum $C^{TT}_{\ell}$ estimated from
$\delta\textit{T}(\bm{\theta})$ is related to $C_{\ell}$ (which corresponds to
$\delta\textit{T}_{s}(\bm{\theta})$ ) through a convolution

$C^{TT}_{2\pi\mid\textbf{{U}}\mid}=\int
d^{2}\textbf{{U}}^{{}^{\prime}}\,\mid\tilde{r}(\textbf{{U}}-\textbf{{U}}^{{}^{\prime}})\mid^{2}\,C_{2\pi\mid\textbf{{U}}^{{}^{\prime}}\mid}$
(13)

where $\tilde{r}(\textbf{{U}})$ is the Fourier transform of
$\textit{R}(\bm{\theta})$. Considering a power law of the form
$C_{\ell}\propto A\,\ell^{\beta}$ with a negative power law index $(\beta<0)$,
we find that the estimated $C^{TT}_{\ell}$ has the same slope as $C_{\ell}$ at
large $\ell$ $(\gg\ell^{T}_{m})$. In this range, $C^{TT}_{\ell}$ and
$C_{\ell}$ differ only by

$C_{2\pi\mid\textbf{{U}}\mid}=B^{-1}\,C^{TT}_{2\pi\mid\textbf{{U}}\mid}$ (14)

the proportionality constant $B=\left[\int
d^{2}\bm{\theta}\,\mid\textit{R}(\bm{\theta})\mid^{2}\right]=\left[\int
d^{2}\textbf{{U}}^{{}^{\prime}}\,\mid\tilde{r}(\textbf{{U}}^{{}^{\prime}})\mid^{2}\right]$
(Dutta et al., 2009). However, at small $\ell$ $(\leq\ell^{T}_{m})$,
$C^{TT}_{\ell}$ flattens out and the convolution (eq. 13) introduces a break
at $\ell^{T}_{m}$. The value of $\ell^{T}_{m}$ is inversely proportional to
the angular extent of the SNR. Depending on the shape of
$\textit{R}(\bm{\theta})$ and power law index $\beta$, the value of
$\ell^{T}_{m}$ can vary.

The above interpretation has been extended to the rest of the estimated power
spectra $C^{AA}_{\ell}$, $C^{BB}_{\ell}$, $C^{TA}_{\ell}$, $C^{TB}_{\ell}$ and
$C^{AB}_{\ell}$ as well. We have modeled the angular profile of the SNR as a
Gaussian of the form
$\textit{R}(\bm{\theta})=e^{-{\theta^{2}/\theta_{r}^{2}}}$. For each data the
value of $\theta_{r}$ was chosen separately so as to correctly reproduce the
observed values of $\ell_{m}$.

###  3.5 Error estimation

The error estimates $\delta C_{\ell}$ for $C^{TT}_{\ell}$ is relatively
straight forward, and this has been discussed in Saha et al. (2019) and
references therein. Here the sample variance which is the statistical
fluctuation inherent to the sky signal as well as the system noise both
contribute to the errors. We have used simulations to estimate these errors.
For these simulations we use a model input angular power spectrum
$C^{M}_{\ell}$ which is defined as $C^{M}_{\ell}=B^{-1}C^{TT}_{\ell}$ (eq. 14)
for $\ell>\ell^{T}_{m}$. In this $\ell$ range we interpolate the estimated
$C^{TT}_{\ell}$ to obtain $C^{M}_{\ell}$ as a continuous function of $\ell$.
For $\ell<\ell^{T}_{m}$, we fit a power law to $C^{TT}_{\ell}$ in the vicinity
of $\ell\approx\ell^{T}_{m}$ and extrapolate this to calculate $C^{M}_{\ell}$.
We construct several statistically independent realizations of Gaussian random
fields corresponding to $C^{M}_{\ell}$ and multiply these with the SNR angular
profile function $\textit{R}(\bm{\theta})$ discussed in the previous section.
The images of the simulated SNR were multiplied with the VLA primary beam
pattern and used to calculate (eq. 2) the visibilities corresponding to the
observed baseline distribution.

We have assumed that the system noise contribution to both the real and
imaginary parts of the visibilities are Gaussian random variables of zero mean
and variance $\sigma^{2}_{N}$. We have used

$\sigma_{N}=\frac{SEFD}{\eta_{c}\sqrt{2t_{int}\Delta\nu}}$ (15)

where SEFD, $\eta_{c}$
222https://science.nrao.edu/facilities/vla/docs/manuals/oss/performance/sensitivity
are the system equivalent flux density (Jy) and the correlator efficiency
respectively, $t_{int}$ is the integration time per visibility in seconds and
$\Delta\nu$ is the channel width in Hz. These values are different for
different VLA configurations for a source and are shown in Table $3$ of Saha
et al. (2019). The resulting simulated visibilities, which have the simulated
SNR signal and simulated system noise, were used to determine $C_{\ell}$. We
have constructed several statistically independent realizations of these
simulations for which the average $C_{\ell}$ matches $C^{TT}_{\ell}$ at
$\ell\geq\ell^{T}_{m}$. The variance determined from these realizations of
these simulated $C_{\ell}$ is used to estimate $\delta C_{\ell}$ for the
measured $C^{TT}_{\ell}$.

Considering $C^{AA}_{\ell}$, in addition to the sample variance the Poisson
noise also contributes to the errors. The Poisson noise contribution at any
pixel depends on the number of photons that contribute to the value of
$J^{A}(\bm{\theta})$ at that pixel. Considering the Chandra observation, we
have a complicated mapping from the photon counts to $J^{A}(\bm{\theta})$
which is carried out by the CIAO software. This poses a difficulty for
incorporating the Poisson noise in our analysis, and we have only considered
the sample variance contribution to the errors. The sky signal was simulated
following exactly the same procedure as for the radio, except that we have
used $C^{AA}_{\ell}$ to determine the input model angular power spectrum. The
width of the radial profile $\textit{R}(\bm{\theta})$ was also set to a value
consistent with the X-ray observation. The simulated $J^{A}(\bm{\theta})$
values were obtained on an image of the same size and resolution as the
Chandra image, and these were used to estimate the simulated $C^{AA}_{\ell}$.
We generate $50$ statistically independent realizations of the simulated
Chandra images and the variance calculated from these multiple estimates of
the simulated $C^{AA}_{\ell}$ were used to compute error $\delta C_{\ell}$ for
the estimated $C^{AA}_{\ell}$. The same technique is followed to calculate the
error $\delta C_{\ell}$ for the estimated X-ray B $C^{BB}_{\ell}$ and X-ray
cross power spectrum $C^{AB}_{\ell}$.

For the radio X-ray cross power spectrum $C^{TA}_{\ell}$, the error variance
for a measurement at a single U mode is given by

$(\delta
C^{TA}_{\ell})^{2}=\frac{1}{2}\,\bigg{[}(C^{TA}_{\ell})^{2}+(C^{TT}_{\ell})(C^{AA}_{\ell})\bigg{]}$
(16)

where the radio and X-ray A auto power spectra include the respective signal
and the noise contributions. The total error variance for an $\ell$ bin is
also inversely proportional to the number of independent modes which
contribute to the bin. Here it is difficult to incorporate the individual
contributions in the simulations to estimate the errors. For the present
purpose we have made a few simplifying assumptions to estimate these errors.
We assume that the errors $\delta C_{\ell}$ are primarily dominated by the
sample variance which is determined by the number of independent measurements
that contribute to the estimated signal. Since the cross-correlation estimator
(eq. 11) has exactly the same sky coverage and baseline distribution as the
radio observations, it is then reasonable to assume that

$\delta C^{TA}_{\ell}=\frac{C^{TA}_{\ell}}{C^{TT}_{\ell}}\,\times\,\delta
C^{TT}_{\ell}$ (17)

which we use in this analysis. Similarly, we estimate the errors $\delta
C_{\ell}$ for $C^{TB}_{\ell}$ except that $C^{TA}_{\ell}$ is now replaced by
$C^{TB}_{\ell}$ in eq. (16) and (17).

##  4 Results

We first consider the three power spectra $C^{TT}_{\ell}$, $C^{AA}_{\ell}$ and
$C^{BB}_{\ell}$ which are shown in Figure 3. Throughout the rest of the paper,
in addition to $C_{\ell}$ we have shown the scaled angular power spectrum
$\ell(\ell+1)\,C_{\ell}/2\pi$. For radio we can interpret
$\ell(\ell+1)\,C^{TT}_{\ell}/2\pi$ as the variance of the observed brightness
temperature fluctuations at the angular scale corresponding to $\ell$, the
other power spectra may be similarly interpreted in terms of photon number
counts, etc. Considering $C^{TT}_{\ell}$ first, the $\ell$ range shown in the
left panel corresponds to the entire baseline range of the radio observations.
The $1-\sigma$ error bars shown in the figure are quite large at small $\ell$
where the sample variance dominates. The error bars are smaller at large
$\ell$ where they are not noticeable in the figure. We see that
$C^{TT}_{\ell}$ flattens out at $\ell\leq\ell_{m}^{T}(=1.88\times 10^{4})$.
This flattening can be attributed to the convolution (eq. 13) due to the
finite angular extent of the SNR. We may interpret $C^{TT}_{\ell}$ in the
$\ell$ range $\ell>\ell_{m}^{T}$ as arising from specific intensity
fluctuations within the SNR, and we restrict the subsequent analysis to this
$\ell$ range. We see that in this $\ell$ range $C^{TT}_{\ell}$ shows a power
law like behaviour, however a single power law does not provide a good fit for
the entire $\ell$ range as there is a break at $\ell^{T}_{b}=6.60\times
10^{4}$. We find that separately fitting two different power laws of the form
$C_{\ell}=A\,\ell^{\beta}$, one for $\ell<\ell^{T}_{b}$ and another for
$\ell>\ell^{T}_{b}$ provides a reasonably good fit which is shown in the right
panel of Figure 3. We obtain the best fit power law index $\beta=-2.28\pm
0.08$ and $\beta=-3.13\pm 0.01$ for $\ell<\ell^{T}_{b}$ and
$\ell>\ell^{T}_{b}$ respectively. The power law index steepens by $-0.85$ as
we go across the break from small $\ell$ to large $\ell$. The goodness of fit
and the fitting parameters are summarized in Table 1. We have also tried
fitting a broken power law with $\ell^{T}_{b}$ as an additional parameter,
however this does not improve the reduced $\chi^{2}$. These findings have been
reported previously in Saha et al. (2019). We note that an earlier work (Roy
et al., 2009) has analyzed the same visibility data to estimate
$C^{TT}_{\ell}$ for the Cas A SNR, however the power spectrum estimator used
there was different. The earlier work also found a break at $\ell^{T}_{b}\sim
6.60\times 10^{4}$, with a power law having $\beta=-2.22\pm 0.03$ at the small
$\ell$ values in the range $1.00\times 10^{4}-6.28\times 10^{4}$ and a
different power law having $\beta=-3.23\pm 0.09$ in the $\ell$ range
$\ell>6.28\times 10^{4}$. We find that our results here are consistent with
the results reported in the earlier work.

band | $\ell_{m}$ | $\ell_{b}$ | power index $\beta$ | power index $\beta$ |  $\ell$ range | No.of points | $\frac{\chi^{2}}{DOF}$
---|---|---|---|---|---|---|---
|  |  | $(\ell\leq\ell_{b})$ | $(\ell>\ell_{b})$ | for fitting |  |
radio T | $1.88\times 10^{4}$ | $6.60\times 10^{4}$ | $-2.28\pm 0.08$ |  | $\ell_{m}-\ell_{b}$ | $7$ | $0.86$
|  |  |  | $-3.13\pm 0.01$ | $8.17\times 10^{4}-1.88\times 10^{6}$ | $18$ | $3.44$
X-ray A | $1.55\times 10^{4}$ | $(2.67\pm 0.36)\times 10^{5}$ | $-2.33\pm 0.05$ | $-2.81\pm 0.10$ | $\ell_{m}-6.00\times 10^{5}$ | 16 | 0.89
X-ray B | $7.50\times 10^{3}$ | $(1.93\pm 0.11)\times 10^{5}$ | $-1.83\pm 0.05$ | $-2.62\pm 0.03$ | $\ell_{m}-6.00\times 10^{5}$ | $20$ | $1.23$
cross TA | $1.80\times 10^{4}$ | $(2.67\pm 0.43)\times 10^{5}$ | $-2.27\pm 0.04$ | $-2.83\pm 0.17$ | $\ell_{m}-6.00\times 10^{5}$ | $20$ | $9.17$
cross TB | $1.80\times 10^{4}$ | $(1.16\pm 0.08)\times 10^{5}$ | $-2.11\pm 0.05$ | $-2.76\pm 0.03$ | $\ell_{m}-6.00\times 10^{5}$ | $20$ | $3.69$
cross AB | $7.50\times 10^{3}$ | $(2.11\pm 0.23)\times 10^{5}$ | $-2.11\pm 0.06$ | $-2.70\pm 0.04$ | $\ell_{m}-6.00\times 10^{5}$ | $20$ | $2.32$
Table 1: The values of the parameters obtained by fitting $C^{TT}_{\ell}$,
$C^{AA}_{\ell}$, $C^{BB}_{\ell}$, $C^{TA}_{\ell}$, $C^{TB}_{\ell}$ and
$C^{AB}_{\ell}$ of Cas A SNR.

Figure 3:  Left panel: Estimated angular auto power spectra $C^{TT}_{\ell}$,
$C^{AA}_{\ell}$ and $C^{BB}_{\ell}$ with their respective $1-\sigma$ error
estimates. Right panel: The scaled angular auto power spectra for the $\ell$
range used for fitting, along with the respective best fit broken power-law.
The values of the power law index $\beta$ are shown. These values and the
break $\ell_{b}$ are also provided in Table 1.

We next consider the X-ray A angular power spectrum $C^{AA}_{\ell}$ shown in
the left panel of Figure 3. We see that $C^{AA}_{\ell}$ flattens out for
$\ell\leq\ell^{A}_{m}(=1.55\times 10^{4})$ whereas it is a declining function
of $\ell$ for $\ell>\ell^{A}_{m}$. Similar to radio, we attribute this
flattening to the convolution (eq. 13) due to the finite angular extent of the
SNR. We see that the value of $\ell^{A}_{m}$ is close to the value of
$\ell^{T}_{m}=1.88\times 10^{4}$ obtained for radio which indicates that the
Cas A SNR is expected to have the same angular extent in radio and X-ray A
(Figure 1). To verify this we consider the respective radial profiles
$\textit{R}(\bm{\theta})$ shown in Figure 4. These have been determined by
averaging the intensity along various radii emanating from the centers of the
respective images in Figure 1. The radial profiles are all normalised using
$\int\textit{R}(\bm{\theta})\,d^{2}\theta=1$. We find that the radial profiles
of the radio and X-ray A images both peak at $\theta\approx 1.8^{{}^{\prime}}$
confirming that they have very similar angular extents. Note that the angular
extent estimated from Figure 4 is possibly larger than the values predicted by
one-dimensional self-similar hydrodynamics models (Hamilton & Sarazin, 1984;
Laming & Hwang, 2003; Hwang & Laming, 2012) due to projection effects. We may
interpret $C^{TT}_{\ell}$ in the $\ell$ range $\ell>\ell_{m}^{T}$ as arising
from specific intensity fluctuations within the SNR, and we restrict the
subsequent analysis to this $\ell$ range. As can be seen from Figure 3, the
error estimates are large at $\ell<9.00\times 10^{4}$ due to the large sample
variance, the error decreases thereafter. From the left panel of Figure 3,
$C^{AA}_{\ell}$ appears to hit a floor at the three largest $\ell$ values. We
have omitted these data points from the subsequent analysis which is
restricted to the $\ell$ range $\ell^{A}_{m}-6.00\times 10^{5}$. Like radio,
here also a single power law does not provide a good fit to the data. We have
obtained a good fit using a broken power law considering the position of the
break $\ell^{A}_{b}$ as a free parameter. The best fit values and the goodness
of fit are summarized in Table 1. We obtain $\ell^{A}_{b}=(2.67\pm 0.36)\times
10^{5}$ with the power law index values $\beta=-2.33\pm 0.05$ and
$\beta=-2.81\pm 0.10$ for $\ell<\ell^{A}_{b}$ and $\ell>\ell^{A}_{b}$
respectively. The power law slope $\beta$ steepens by $-0.48$ across
$\ell^{A}_{b}$ from low to high $\ell$.

We next consider the X-ray B angular power spectrum $C^{BB}_{\ell}$ shown in
the left panel of Figure 3. Similar to $C^{TT}_{\ell}$ and $C^{AA}_{\ell}$, we
see that $C^{BB}_{\ell}$ also flattens out at
$\ell\leq\ell_{m}^{B}(=7.50\times 10^{3})$. The values of $\ell_{m}^{T}$ for
radio and $\ell_{m}^{A}$ for X-ray A are respectively $2.5$ and $2.1$ times
larger than $\ell_{m}^{B}$ for X-ray B. Taking this at face value, this
indicates that the angular extent of the SNR in the radio C Band and X-ray A
are approximately $2.5$ and $2.1$ times smaller than that in X-ray B
respectively. By comparing the three radial profiles (Figure 4), and also by
directly comparing the images in Figures 1, we see that the angular extents
are nearly the same in radio, X-ray A and X-ray B. The origin of the
difference between the values of $\ell_{m}^{T}$, $\ell_{m}^{A}$ and
$\ell_{m}^{B}$ is not clear at present and is discussed in Section 6.
Considering $C^{BB}_{\ell}$ in Figure 3, we see that the estimated error bars
of $C^{BB}_{\ell}$ are quite large at smaller $\ell$ ($\ell<5.00\times
10^{4}$) where we have a relatively large sample variance, the errors
decreases at larger $\ell$. Like $C^{AA}_{\ell}$, $C^{BB}_{\ell}$ appears to
hit a floor at the four largest $\ell$ values (left panel of Figure 3). To
understand the origin of this floor, we have simulated images of Cas A using
the radial profile function $\textit{R}(\bm{\theta})$ (Figure 4) and
$C^{BB}_{\ell}$ as the input and having mean similar to that of the observed
X-ray B image. Next, we have applied a detailed Chandra instrument model
(marx, Davis et al. (2012)) to these simulated images to incorporate the
instrumental effects like point spread function and Poisson noise from a
finite number of photons. We find that both the effects affect the angular
power spectra computed from these images only at large $\ell$ i.e.
$\ell>6.00\times 10^{5}$ where it appear flatter than the angular power
spectra calculated from the simulated images before the application of marx.
For $\ell>6.00\times 10^{5}$ corresponding to angular scales less than
$2.16^{\arcsec}$, the point spread function $(\sim 0.5^{\arcsec}-3^{\arcsec})$
away from the center can smooth out spatial structures. We have excluded this
range from the subsequent analysis. We have used $\chi^{2}$ minimisation to
determine the best fit power law for $C^{BB}_{\ell}$ in the $\ell$ range
$\ell^{B}_{m}-6.00\times 10^{5}$. Like $C^{TT}_{\ell}$ and $C^{AA}_{\ell}$, we
find that a single power law does not provide a good fit to the data. We have
obtained a good fit to the data using a broken power law with the position of
the break $\ell^{B}_{b}$ also included as a free parameter. We have used this
to identify a break at $\ell^{B}_{b}=(1.93\pm 0.11)\times 10^{5}$. The fit to
the data is shown in the right panel of Figure 3, and the values of the slopes
and the reduced $\chi^{2}$ are shown in Table 1. We have obtained the slopes
$\beta=-1.83\pm 0.05$ and $\beta=-2.62\pm 0.03$ for $\ell<\ell^{B}_{b}$ and
$\ell>\ell^{B}_{b}$ respectively. We find that the power law index steepens by
$-0.79$ as we go across the break from small $\ell$ to large $\ell$.

Figure 4: The normalised radio and X-ray radial profiles
$\textit{R}(\bm{\theta})$ estimated from the respective images in Figure 1.
The 2D profile for the simulation model used in Section 6 is also shown.

Figure 5: Cross power spectra $C_{\ell}^{TA}$ (left); $C_{\ell}^{TB}$
(center); $C_{\ell}^{AB}$ (right) scaled with $\ell(\ell+1)/(2\pi)$ along with
$1-\sigma$ error bars. Respective best fit broken power laws are also shown
with an offset in amplitude.

For the radio X-ray cross power spectra $C^{TA}_{\ell}$ and $C^{TB}_{\ell}$
the respective X-ray data were converted to have exactly the same baseline
distribution as the radio data, and the binned $\ell$ values exactly match
those of $C^{TT}_{\ell}$. The left and middle panels of Figure 5 show
$C^{TA}_{\ell}$ and $C^{TB}_{\ell}$ respectively. We find that $C^{TA}_{\ell}$
and $C^{TB}_{\ell}$ both flatten out due to the convolution (eq. 13) at small
$\ell$ below $\ell^{TA}_{m}=\ell^{TB}_{m}=1.80\times 10^{4}$. This value
matches that of $\ell^{T}_{m}$ and is also comparable to $\ell^{A}_{m}$.
Similar to $C^{AA}_{\ell}$ and $C^{BB}_{\ell}$, for both $C^{TA}_{\ell}$ and
$C^{TB}_{\ell}$ we observe the presence of a floor at large $\ell$. The
occurrence of this flat region (floor) in both $C^{TA}_{\ell}$ and
$C^{TB}_{\ell}$ indicates that this feature is not due to the X-ray photon
Poisson noise.The Poisson noise due to the finite number of X-ray photons is
not expected to be correlated with the radio signal. For both $C^{TA}_{\ell}$
and $C^{TB}_{\ell}$ we have restricted the subsequent fitting to the $\ell$
range $\ell^{TA}_{m}-6.00\times 10^{5}$. A single power law does not provide a
good fit for any of these two power spectra. Considering a broken power law
with the position of the break as an additional free parameter, for
$C^{TA}_{\ell}$ we obtain the best fit parameters $\beta=-2.27\pm 0.04$
($\ell\leq\ell^{TA}_{b}$) and $\beta=-2.83\pm 0.17$ ($\ell>\ell^{TA}_{b}$)
with $\ell^{TA}_{b}=(2.67\pm 0.43)\times 10^{5}$. We however note that this
does not provide a good fit to $C_{\ell}^{TA}$ as there appears to be several
breaks and different slopes in the $\ell$ range $\ell^{TA}_{m}-\ell^{TA}_{b}$.
Considering $C^{TB}_{\ell}$, we have obtained the break at
$\ell^{TB}_{b}=(1.16\pm 0.08)\times 10^{5}$ with slopes $\beta=-2.11\pm 0.05$
and $\beta=-2.76\pm 0.03$ for $\ell<\ell^{TB}_{b}$ and $\ell>\ell^{TB}_{b}$
respectively. The best fit curves for $C^{TA}_{\ell}$ and $C^{TB}_{\ell}$ are
shown in the respective panels of Figure 5, whereas the goodness of fit and
the best fit parameters are summarized in Table 1. We note that our errors for
$C_{\ell}^{TB}$ are possibly underestimated leading to a $\chi^{2}$ which is
somewhat on the higher side.

The right panel of Figure 5 shows the cross power spectrum $C^{AB}_{\ell}$
between X-ray A and B evaluated using eq. (8). We see that $C^{AB}_{\ell}$
becomes flat for $\ell\leq\ell^{AB}_{m}(=7.50\times 10^{3})$, this value is
similar to $\ell^{B}_{m}$. The estimated error bars are quite large for
$\ell<7.00\times 10^{4}$, thereafter the errors decrease at larger $\ell$.
Like the other X-ray auto and cross power spectra, $C^{AB}_{\ell}$ too hits a
floor at the largest $\ell$ values. We have restricted the $\ell$ range to
$\ell^{AB}_{m}-6.00\times 10^{5}$ for subsequent fitting. We find that a
single power law does not provide a good fit across the entire $\ell$ range
considered here. We have fitted $C^{AB}_{\ell}$ with a broken power law with
the position of the break $\ell^{AB}_{b}$ as a free parameter. The best fit
broken power law is also shown in the right panel of Figure 5, with an offset
in the amplitude for clarity. We have obtained the break at
$\ell^{AB}_{b}=(2.11\pm 0.23)\times 10^{5}$ with slopes $\beta=-2.11\pm 0.06$
and $\beta=-2.70\pm 0.04$ for $\ell<\ell^{AB}_{b}$ and $\ell>\ell^{AB}_{b}$
respectively. The best fit parameters are presented in Table 1.

We next consider the dimensionless correlation coefficient

$c^{AB}_{\ell}=\frac{C^{AB}_{\ell}}{\sqrt{C^{AA}_{\ell}\,C^{BB}_{\ell}}}$ (18)

which, for a fixed $\ell$, quantifies the strength of the correlation between
the angular fluctuations in X-ray A with those in X-ray B. The correlation
coefficients $c^{TA}_{\ell}$ and $c^{TB}_{\ell}$ are similarly defined. The
correlation coefficients are expected to have values in the range $-1$ to
$+1$, with the value $1$ occurring if the two signals are perfectly
correlated. We expect the correlation to have a values $0$ and $-1$ if the two
signals are uncorrelated and anti-correlated respectively. The left, middle
and right panels of Figure 6 respectively show the correlation coefficients
$c^{TA}_{\ell}$, $c^{TB}_{\ell}$ and $c^{AB}_{\ell}$. The $1-\sigma$ error
bars shown in the respective panels of Figure 6 account for the errors in both
the cross-power spectra as well as the auto-power spectra. Considering
$c^{TA}_{\ell}$, we find that the values are positive but very small ($\sim
10^{-4}$). In contrast, both $c^{TB}_{\ell}$ and $c^{AB}_{\ell}$ also have
positive values but the values are relatively large $(0.4-0.8)$, with the
values of $c^{AB}_{\ell}$ being slightly larger than those of $c^{TB}_{\ell}$.
This shows that X-ray B is highly correlated with both radio and X-ray A.
However, X-ray A has extremely weak correlations with radio. The radio
frequency emission from SNRs like Cas A is predominantly non-thermal
synchrotron radiation. For Cas A, X-ray A is known to be predominantly thermal
emission (Vink et al., 1996; Willingale et al., 2002). The fact that radio and
X-ray A are very weakly correlated bears out the picture that these two
originate from two different emission mechanisms. Interestingly, X-ray B is
correlated with both radio and X-ray A, this supports a picture where X-ray B
is a mixture of both thermal and non-thermal emission.

Figure 6: Correlation coefficient ($c_{\ell}$) for the cross power spectra
$C_{\ell}^{TA}$ (left); $C_{\ell}^{TB}$ (center); $C_{\ell}^{AB}$ (right).

Table 1 summarizes the best fit parameters obtained for the different
estimated power spectra. We find that all the estimated power spectra, except
$C^{TA}_{\ell}$, can be represented by a broken power law with a break at
$\ell_{b}$ where the power law slope $\beta$ has a particular value for
$\ell\leq\ell_{b}$ and a different value for $\ell>\ell_{b}$. In all cases the
slope is negative ($\beta<0$), and it steepens (change $\Delta\beta<0$) from
small $\ell$ to large $\ell$ across the break. Considering the auto-power
spectra, for radio, X-ray A and B respectively the position of the break has
values $\ell_{b}=(0.66,2.67,1.93)\times 10^{5}$ with $\beta=-2.28,-2.33,-1.83$
for $\ell\leq\ell_{b}$ and $\beta=-3.13,-2.81,-2.62$ for $\ell>\ell_{b}$. We
see that for $\ell\leq\ell_{b}$ X-ray A has the steepest power spectrum
whereas radio is steepest for $\ell>\ell_{b}$ and also overall. Considering
the cross-power spectra, we exclude $C^{TA}_{\ell}$ where the correlation is
extremely weak and the broken power law does not provide a good fit. For
$C^{TB}_{\ell}$ and $C^{AB}_{\ell}$ the positions of the respective breaks are
$\ell_{b}=(1.16,2.11)\times 10^{5}$ with $\beta=-2.11,-2.11$ for
$\ell\leq\ell_{b}$ and $\beta=-2.76,-2.70$ for $\ell>\ell_{b}$. Considering
the change in slope, we have $\Delta\beta=-0.85,-0.48,-0.79,-0.65$ and $-0.59$
for $C^{TT}_{\ell},C^{AA}_{\ell},C^{BB}_{\ell},C^{TB}_{\ell}$ and
$C^{AB}_{\ell}$ respectively. Here we see that radio is somewhat unique as the
position of the break $\ell^{TT}_{b}$ is smallest, the slope $\beta=-3.13$ is
the steepest and the change in slope $\Delta\beta=-0.85$ is the largest among
all the power spectra estimated here. For X-ray B the slopes are approximately
$0.5$ less steeper than radio and the change in slope $\Delta\beta=-0.79$ is
comparable to that for radio. For X-ray A the slope is slightly steeper than
radio for $\ell\leq\ell_{b}$, however the change $\Delta\beta$ is much
smaller. Considering the cross-correlation $C^{TB}_{\ell}$, we see that
$\ell^{TB}_{b}$ is the geometric mean of $\ell^{T}_{b}$ and $\ell^{B}_{b}$,
while the value of $\beta$ is close to that of radio for
$\ell\leq\ell^{TB}_{\ell}$ and it is close to that of X-ray B for
$\ell>\ell^{TB}_{\ell}$. Considering the cross-correlation $C^{AB}_{\ell}$, we
see that $\ell^{AB}_{b}$ is slightly smaller than the mean of $\ell^{A}_{b}$
and $\ell^{B}_{b}$, however the values of $\beta$ are very close to those for
$C^{TB}_{\ell}$.

##  5 Modelling the radio power spectrum $C^{TT}_{\ell}$

Considering the angular power spectrum $C^{TT}_{\ell}$ of the Cas A SNR in
radio, we interpret this as arising from fluctuations in the synchrotron
radiation which in turn trace the fluctuations in the electron density and the
magnetic field both of which results from MHD turbulence in the ionized plasma
of the SNR. It is likely that the contribution from the fluctuations in the
electron density is subdominant to the contribution from the fluctuations in
the magnetic field. For $\ell>\ell^{T}_{m}$ we find that $C^{TT}_{\ell}$ is a
broken power law with a steepening of the power law index from $\beta=-2.28$
for $\ell^{T}_{m}\leq\ell<\ell^{T}_{b}$ to $\beta=-3.13$ for
$\ell\geq\ell^{T}_{b}$. Several earlier works (Roy et al., 2009; Saha et al.,
2019) have interpreted such a change in the power law index in terms of a
transition from 2D to 3D turbulence at an angular scale of
$\theta^{T}_{b}\approx\pi/\ell_{b}^{T}$, which turns out to have a value
$\theta^{T}_{b}=0.16\arcmin$ for Cas A. The angular scale $\theta^{T}_{b}$ can
be associated with the shell thickness of Cas A. Recently, Choudhuri et al.
(2020) have also shown that the angular scale of the break of the observed
power spectrum for a shell-type SNR is related to its shell thickness. Using
simulations, they present a systematic study of the effect of shell geometry
and the projection of SNR on to a 2D plane which can cause the turbulence to
change from 3D to 2D at angular scales greater than the shell thickness. At
scales smaller than the shell thickness, the SNR shell can have Fourier modes
of fluctuations in three independent spatial directions. However at scales
greater than the shell thickness, there will be no modes perpendicular to the
shell thickness. At these length-scales the SNR shell can have Fourier modes
of fluctuations in only two independent spatial directions. We expect this
transition to be manifested as a break in the power spectrum. Considering
incompressible 3D Kolmogorov turbulence (Kolmogorov, 1941), the energy
spectrum follows a power law with slope ${-5/3}$. The equivalent velocity
power spectrum is predicted to have slopes $-8/3$ and $-11/3$ for 2D and 3D
turbulence respectively (Frisch, 1995; Elmegreen & Scalo, 2004). Further, the
density power spectrum is also predicted to follow the velocity power spectrum
(Goldreich & Sridhar, 1995). The values $\beta=-2.28$ and $\beta=-3.13$ and a
difference of slope $\Delta\beta=-0.85$ obtained here for Cas A are roughly
consistent with these predictions. These findings strongly indicate that the
measured radio power spectrum originates from MHD turbulence in the plasma of
the SNR, and the break is associated with the geometrical effect of the shell
thickness of Cas A. It is interesting to note that a similar break is also
found in the the angular power spectrum of HI emission from the disk of
external spiral galaxies. The location of the break there is associated with
the scale height of the disk, and this has been used to estimate the scale
height of the galaxy NGC $628$ (Dutta et al., 2008).

We have carried out 3D simulations where we have implemented a simple
geometrical model to illustrate the above interpretation of the observed
$C^{TT}_{\ell}$ of Cas A. The simulations are in terms of 3D angular unit
$\mathbf{r}$ and its Fourier conjugate $\mathbf{k}$ for which
$|\mathbf{k}|=\ell$. We have generated Gaussian random brightness temperature
fluctuations $\delta T(\mathbf{r})$ inside a 3D cube of size
$[20.48^{{}^{\prime}}]^{3}$ with $[1024]^{3}$ grid points and spacing
$0.02^{{}^{\prime}}$. The simulated $\delta T(\mathbf{r})$ corresponds to a 3D
power spectrum $P(k)=Ak^{-3.1}$ which matches the observed $C^{TT}_{\ell}$ at
small angular scales or large $\ell$. To replicate the emission structure as
seen from the radio observation (left panel of Figure 1), we first introduce a
single spherical shell of outer radius $2.5\arcmin$ and thickness
$0.16\arcmin$ which is the value of $\theta^{T}_{b}$. The shell radius is
approximately the observed size of the remnant. Here, there are two regions,
the shell and the core which is a sphere enclosed within the inner radius of
the shell. The simulated fluctuations $\delta T(\mathbf{r})$ only fill up the
shell, we have erased the fluctuations outside the shell as well as inside the
core. The resulting cube is projected to a 2D plane which corresponds to the
plane of the sky. We have estimated the angular power spectrum from the
projected 2D brightness temperature fluctuations. We find that the simulated
$C_{\ell}$ matches the observed $C^{TT}_{\ell}$ at large $\ell$, and the
position of the break also is close to the observed $\ell_{b}^{T}$. However,
we find that at $\ell<\ell_{b}^{T}$ the simulated $C_{\ell}$ is considerably
steeper that the observed $C^{TT}_{\ell}$ and the simulated amplitude also is
larger. The position of the break can be shifted by adjusting the shell
thickness, however we find that the mismatch in slope and amplitude persists
even when we vary the shell thickness in the range $0.125\arcmin-0.25\arcmin$.

It is interesting to note that the radio image in Figure 1 actually hints at a
double shelled structure with a bright inner shell corresponding to the
shocked ejecta and a fainter outer shell corresponding to the shocked CSM. In
order to emulate this we have tried out different double spherical shell
structures, here we focus on one for which the simulated $C_{\ell}$ is closest
to the observed $C^{TT}_{\ell}$. Figure 7 shows the radial structure of the 3D
double spherical shell consisting of an inner shell of radius $2.0\arcmin$ and
thickness $0.2\arcmin$ and an outer spherical shell of radius $2.5\arcmin$ and
thickness $0.5\arcmin$. We now have three distinct simulated regions namely
the empty core, the inner shell and the outer shell. The simulated brightness
temperature fluctuations fill the inner shell while the same fluctuations with
reduced amplitude fill up the outer shell. The fluctuations inside the core
and outside the outer shell are erased. Figure 4 also shows the radial profile
$\textit{R}(\bm{\theta})$ corresponding to the 2D projected image of the
double shelled structure. We see that the simulated $\textit{R}(\bm{\theta})$
closely matches the observed radio profile interior to the peak which occurs
at $\approx 1.8^{{}^{\prime}}$, however the simulated peak amplitude is larger
and the simulated profile drops faster at larger angles as compared to the
observed profile.

We have used several statistically independent realizations of the simulated
$\delta T(\mathbf{r})$ to obtain the mean $C_{\ell}$ shown in Figure 8. We
find that simulated power spectrum $C_{\ell}$ can be fitted with two different
power laws in two different $\ell$ ranges, one with $\beta=-2.45\pm 0.06$ in
the range $\ell=(2.0-7.0)\times 10^{4}$ and another with $\beta=-2.99\pm 0.04$
in the range $\ell=8.5\times 10^{4}-7.0\times 10^{5}$. At large $\ell$, the
slope $\beta=-2.99$ is roughly consistent (by construction) with the observed
slope of $\beta=-3.13$. At small $\ell$ the slope $\beta=-2.45$ is still
somewhat steeper than the measured value of $\beta=-2.28$, and the simulated
amplitude also is larger. However, we note that the double shelled simulation
is closer to the measured $C^{TT}_{\ell}$ than is possible with a single
shelled structure. The inner shell thickness $0.2\arcmin$ is reflected in the
break at $\ell\sim 7.00\times 10^{4}$ which is close to the measured value
$\ell^{T}_{b}=6.60\times 10^{4}$. Although the model considered here is not
able to accurately reproduce the observed $C^{TT}_{\ell}$, the similarity
between the simulated $C_{\ell}$ and the observed $C^{TT}_{\ell}$ may be
interpreted as indicating that what we have observed is indeed a geometrical
effect associated with the shell thickness. The difference between the two may
be attributed to the possibility that the actual SNR structure may be more
complex than the double shelled structure considered here. The fact that the
peak in the observed $\textit{R}(\bm{\theta})$ has a smaller amplitude and
larger angular extent at $\theta>2^{{}^{\prime}}$ as compared to the
simulation ( Figure 4) also points to the same possibility. A better match may
be possible with more detailed modelling of the 3D shell structure, however
this is beyond the scope of the present paper and we do not attempt this here.

Figure 7: The 3D radial profile used to simulate the double shelled SNR
structure considered here. Figure 8: Comparison of the observed
$C^{TT}_{\ell}$ with the model $C_{\ell}$ obtained from the simulations. The
red dotted and dashed lines respectively show the best fit power laws to the
model $C_{\ell}$ in two different $\ell$ ranges.

The double cascade theory of 2D turbulence proposed by Kraichnan (1967, 1971)
provides an alternative explanation for the broken power law power spectrum
observed here. This theory predicts an inverse energy cascade with an energy
spectrum of slope ${-5/3}$ on length-scales larger than the length-scale where
energy is injected into the system, and a direct enstrophy cascade with an
energy spectrum of slope ${-3}$ at smaller length-scales. Several subsequent
studies (e.g. Lindborg 1999, Boffetta & Musacchio (2010)) have validated this
scenario for 2D turbulence. Here we find that the corresponding slopes of
$-8/3$ and $-4$ predicted for the 2D velocity power spectrum roughly match the
slopes $\beta=-2.28$ and $\beta=-3.13$ obtained for Cas A. Knot-shock
interactions in the CSM provides a possible mechanism for energy and vorticity
injection into the SNR plasma (Beresnyak et al., 2009) at a length-scale which
is comparable to the Cas A shell thickness. The validation of the above
interpretation requires detailed simulations which we intend to address in
future work.

##  6 Discussion & Conclusion

Synchrotron radiation is the dominant emission from SNRs at radio frequencies.
We have interpreted the radio angular power spectrum $C^{TT}_{\ell}$ of Cas A
SNR as arising from fluctuations of MHD turbulence in the synchrotron emitting
plasma. It is known that the synchrotron emissivity depends on the electron
number density and the magnetic field component perpendicular to the line of
sight. The estimated $C^{TT}_{\ell}$ is expected to quantify the underlying
fluctuations of both the electron density and the magnetic field.
$C^{TT}_{\ell}$ shows a broken power law with the power law index
$\beta=-2.28$ at $\ell<\ell^{T}_{b}=6.60\times 10^{4}$ steepening to
$\beta=-3.13$ at $\ell\geq\ell^{T}_{b}$. The values of power law indices and
the amount of steepening of the power law slope $\Delta\beta=-0.85$ from small
$\ell$ to large $\ell$ are consistent with the predictions of incompressible
Kolmogorov turbulence (Kolmogorov, 1941). Like in earlier works (Roy et al.,
2009; Saha et al., 2019), we have identified the angular scale corresponding
to the break $\theta^{T}_{b}\approx\pi/\ell_{b}^{T}=0.16\arcmin$ with the
shell thickness of Cas A which sets the transition scale from 2D to 3D
turbulence. We have carried out simulations to validate the above
interpretation. We find that a double shelled structure is able to
qualitatively match some of the features in the observed $C^{TT}_{\ell}$,
however quantitative differences persist between the simulated and observed
$C^{TT}_{\ell}$. The rough match indicates that the break in the power
spectrum is possibly a geometrical effect, however a more complex shell
structure is needed for the model to quantitatively reproduce the observed
$C^{TT}_{\ell}$. Moreover, in a companion paper (Choudhuri et al., 2020),
detailed simulations are carried out to study various effects such as
remnant’s shell thickness, projection from $3$D structure onto a $2$D
observational plane, presence of diffuse foreground emission, on the observed
power spectrum of a SNR.

We now discuss the X-ray properties of Cas A. The spectral characteristics of
Cas A in X-ray varies on all angular scales due to fluctuations in the
underlying radiation processes and composition (Hughes et al., 2000), down to
the resolution limit of the telescope which corresponds to physical scales of
$0.02$ pc. Considering the global spectrum shown in Figure 2, we see that this
matches Helder & Vink (2008). Further, both X-ray A and B considered here are
devoid of any strong line emission. At X-ray frequencies, the emission from
SNRs can be broadly of two types, the thermal emission which usually refers to
a combination of bremsstrahlung and line emission and the non-thermal emission
which usually refers to X-ray synchrotron emission. For both radio and X-ray,
the synchrotron emissivity depends on the electron density and the magnetic
field component perpendicular to the line of sight. The fluctuations in the
synchrotron radiation are probably caused mainly by fluctuations in the
background magnetic field. In contrast, the bremsstrahlung emissivity depends
on the product of the electron and ion densities (Rybicki & Lightman, 1979).
Moreover, these density fluctuations are related to the magnetic field
fluctuations by flux freezing. The energy $0.5-10$ keV band which include both
X-ray A and B, can be well fitted with a thermal model (Vink et al., 1996;
Willingale et al., 2002). However, for X-ray B, a part of the continuum
emission is non-thermal synchrotron (Vink et al., 1999) and the contribution
of synchrotron was found to be less than one-third (Vink & Laming, 2003).
Later using 1Ms data of Cas A SNR of Hwang et al. (2004), Helder & Vink (2008)
claimed that non-thermal emission accounts for $54\%$ of the total emission of
X-ray B while the rest is thermal bremsstrahlung. Interestingly, the size and
the width of the forward shock and the inner ring of the X-ray continuum
reported in Helder & Vink (2008) are roughly similar to the value of the
radius and shell thickness of outer and inner shell respectively used in the
shell structure simulations that we have conducted in this paper. Particularly
for X-ray B, we understand that it is difficult to completely disentangle
thermal from non-thermal component, even with the best data. This is in
contrast with X-ray A which is inferred to be predominantly thermal.
Furthermore accounting for the non-thermal bremsstrahlung, Vink (2008) showed
that for Cas A it is restricted to very high energies above $100$ keV which is
much greater than X-ray A and B.

Here we now discuss the estimated X-ray power spectra $C^{AA}_{\ell}$ and
$C^{BB}_{\ell}$. Like the radio power spectrum $C^{TT}_{\ell}$, we also model
both $C^{AA}_{\ell}$ and $C^{BB}_{\ell}$ as broken power laws in the $\ell$
range $\ell_{m}-6.00\times 10^{5}$. The close match between
$\ell^{A}_{m}=1.55\times 10^{4}$ and $\ell^{T}_{m}=1.88\times 10^{4}$
indicates similar angular extents for the Cas A SNR in radio and X-ray A,
whereas $\ell^{B}_{m}=7.5\times 10^{3}$ indicates a larger angular extent
(nearly twice) in X-ray B. However, comparing the various radial profiles
(Figure 4) we see that $\textit{R}(\bm{\theta})$ peaks at roughly the same
value $\bm{\theta}\approx 1.8\arcmin$ indicating very similar angular extents
in radio, X-ray A and B. However, the peak amplitude differs with X-ray A,
radio and X-ray B occurring in decreasing order. The width of the peak is
similar for all three, whereas at $\bm{\theta}<1.2\arcmin$ the amplitude of
$\textit{R}(\bm{\theta})$ for X-ray A is smaller than radio and X-ray B which
have comparable amplitude. We however note that for X-ray B
$\textit{R}(\bm{\theta})$ shows considerable fluctuations which indicates that
the sources responsible for this emission have a much more clumpy distribution
as compared to the radio and X-ray A. Considering $C_{\ell}^{AA}$, for
$\ell<\ell^{A}_{b}=2.67\times 10^{5}$ we have $\beta=-2.33$ which is slightly
steeper (by $-0.05$) as compared to $C^{TT}_{\ell}$, whereas for
$\ell\geq\ell^{A}_{b}$ we have $\beta=-2.81$ which is shallower than
$C^{TT}_{\ell}$ by $0.32$. Considering $C^{BB}_{\ell}$, we have $\beta=-1.83$
for $\ell<\ell^{B}_{b}=1.93\times 10^{5}$ and $\beta=-2.62$ for
$\ell>\ell^{B}_{b}$, which are both shallower by approximately $0.5$ as
compared to the corresponding slopes for $C^{TT}_{\ell}$. Comparing the two
X-ray power spectra we see that the slope of $C^{BB}_{\ell}$ is shallower than
$C^{AA}_{\ell}$ by approximately $0.5$ and $0.2$ at $\ell<\ell_{b}$ and
$\ell>\ell_{b}$ respectively. We note that
$\ell^{T}_{b}<\ell^{B}_{b}<\ell^{A}_{b}$. In the earlier discussion, for radio
we have associated the break in the power law as arising from a transition
from 3D to 2D turbulence at an angular scale comparable to the SNR shell
thickness. If we interpret the various $\ell_{b}$ in terms of the respective
shell thicknesses, we then have the largest shell thickness
$\theta^{T}_{b}=0.16\arcmin$ followed by $\theta^{B}_{b}=0.056\arcmin$ and
$\theta^{A}_{b}=0.040\arcmin$ respectively. The analysis in Section 5
indicates that such an interpretation may be valid for radio and also X-ray B
both of which have $\Delta\beta\approx-0.8$ which is comparable to
$\Delta\beta=-1$ expected in a transition from 2D to 3D. However for X-ray A
we have $\Delta\beta=-0.48$ which is considerably smaller, and this is
unlikely to arise from purely geometrical considerations.

The radio, X-ray A and B signals all originate from the same turbulent plasma
of the SNR, and we can expect the angular fluctuations in these three
different signals to be correlated. However, each may originate from a
different emission mechanism and therefore trace a different aspect of the
fluctuations in the turbulent plasma and possibly also the magnetic field. We
therefore expect the correlations between the angular fluctuations in the
different radiation fields to provide insights into the nature of turbulence
and also the emission mechanism in each of the three bands considered here. We
have considered the cross angular power spectra
$C^{TA}_{\ell}$,$C^{TB}_{\ell}$ and $C^{AB}_{\ell}$ to quantify the $\ell$
dependence of the correlations, and the dimensionless correlation coefficients
$c^{TA}_{\ell}$,$c^{TB}_{\ell}$ and $c^{AB}_{\ell}$ to quantify the strength
of the correlations. Considering $c^{TA}_{\ell}$, we see (Figure 6) that this
has very small positive values $(\sim 10^{-4})$ which indicates that radio and
X-ray A are extremely weakly correlated. Further, $C^{TA}_{\ell}$ shows
several breaks in the range $\ell^{TA}_{m}-\ell^{TA}_{b}$ and a single power
law or a power law with a single break does not provide a good fit to the
data. In contrast, $c^{TB}_{\ell}$ and $c^{AB}_{\ell}$ both have values in the
range $0.4-0.8$ indicating strong correlations between radio and X-ray B, and
X-ray A and B respectively. Considering $C^{TB}_{\ell}$ we see that this may
be fitted with a broken power law. In the range $\ell<\ell^{TB}_{b}=1.16\times
10^{5}$, the power law slope $\beta=-2.11$ is close to the power law
$\beta=-2.28$ observed for radio, whereas at $\ell\geq\ell^{TB}_{b}$,
$C^{TB}_{\ell}$ follows $\beta=-2.76$ which roughly matches with the power law
scaling $\beta=-2.62$ of $C^{BB}_{\ell}$. We may interpret this as indicating
that the emission sources for X-ray B roughly follow the angular distribution
of the radio sources at small $\ell$ or large angular scales whereas the radio
sources roughly follow the angular distribution of the X-ray B sources at
large $\ell$ or small angular scales. Furthermore, $\ell^{TB}_{b}$ is
approximately the geometric mean of $\ell^{T}_{b}$ and $\ell^{B}_{b}$.
Considering $C^{AB}_{\ell}$, this has a power law slopes $\beta=-2.11$ for
$\ell<\ell^{AB}=2.11\times 10^{5}$ and $\beta=-2.70$ for
$\ell\geq\ell^{AB}_{b}$. These $\beta$ values roughly match the mean values of
the slopes for X-ray A and X-ray B at low $\ell$ and high $\ell$ respectively.
Moreover, $\ell^{AB}_{b}$ is roughly a little smaller than the mean of
$\ell^{A}_{b}$ and $\ell^{B}_{b}$. To summarize the results of the correlation
analysis, we find that X-ray B is correlated with both radio and X-ray A while
the latter two are not correlated among themselves. It is well known that the
radio emission is predominantly non-thermal synchrotron radiation while X-ray
A is believed to be mostly thermal bremsstrahlung radiation. Our study shows
that the angular distribution of these two respective emission sources is
largely uncorrelated. This possibly originates from the fact that the
synchrotron radiation comes from the regions associated with the forward shock
while most of the thermal bremsstrahlung in X-ray A comes from ejecta shocked
by the reverse shock. Interestingly, X-ray B is well correlated with both
radio and X-ray A. This supports the possibility that X-ray B is a combination
of thermal bremsstrahlung and non-thermal synchrotron radiation.

In conclusion we note that nearly all of the power spectra measured here, both
auto and cross, can be modelled with broken power laws where the slope
steepens at $\ell$ larger than the break. The various power spectra measured
here provides a wealth of information regarding the nature of turbulence and
the structure of the SNR. For radio, we interpret this in terms of MHD
turbulence in the plasma of the SNR. Our simplistic model indicates that we
may possibly interpret the break in the radio power spectrum in terms of the
geometry of the SNR shell structure. A more detailed modeling of the SNR
taking into account inputs from all the various power spectra measured here is
beyond the scope of the present work. This requires detailed simulations and
theory which will be carried out in the future.

The theoretical prediction for 2D turbulence by Kraichnan (1967, 1971) gives
an alternative description for the observed break of the various power spectra
presented here. The change of the power law slope can be interpreted to occur
due to energy transfer in an inverse cascade to larger length scales and a
cascade of enstrophy to smaller length scales in response to the energy
injection at the break. The ejecta and CSM interacting with the shocks
possibly contributes to the input of energy and vorticity (Beresnyak et al.,
2009) at a length scale corresponding to the shell thickness of Cas A SNR. The
verification of the above conjectures needs detailed simulations which will be
attempted in a separate work.

## Acknowledgements

This research has made use of data obtained from the Chandra Data Archive and
the Chandra Source Catalog, and software provided by the Chandra X-ray Center
(CXC) in the application packages CIAO, and Sherpa. NR thanks Harvard-
Smithsonian Center for Astrophysics for hospitality during a visit and stay
when part of this work was done. Support for HMG was provided by the National
Aeronautics and Space Administration through the Smithsonian Astrophysical
Observatory contract SV3-73016 to MIT for Support of the Chandra X-Ray Center,
which is operated by the Smithsonian Astrophysical Observatory for and on
behalf of the National Aeronautics Space Administration under contract
NAS8-03060. We thank Martin Laming for a very useful review.

## Data Availability

The datasets used in this article were derived from sources in the public
domain: [radio: NRAO Science Data Archive,
https://archive.nrao.edu/archive/advquery.jsp and X-ray: Chandra Data Archive,
https://cxc.harvard.edu/cda/ ].

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