capbtabboxtable[][]

# X-Ray Shapes of Elliptical Galaxies and Implications for Self-Interacting
Dark Matter

A. McDaniel,11footnotetext: Corresponding author. T., Jeltema  S., Profumo

###### Abstract

Several proposed models for dark matter posit the existence of self-
interaction processes that can impact the shape of dark matter halos, making
them more spherical than the ellipsoidal halos of collisionless dark matter.
One method of probing the halo shapes, and thus the strength of the dark
matter self-interaction, is by measuring the shape of the X-ray gas that
traces the gravitational potential in relaxed elliptical galaxies. In this
work we identify a sample of 11 relaxed, isolated elliptical galaxies and
measure the ellipticity of the gravitating matter using X-ray images from the
XMM-Newton and Chandra telescopes. We explore a variety of different mass
configurations and find that the dark matter halos of these galaxies have
ellipticities around $\epsilon\approx 0.2-0.5$. While we find non-negligible
scatter in the ellipticity distribution, our results are consistent with some
degree of self-interaction at the scale of $\sigma/m\sim 1$ cm2/g, yet they
also remain compatible with a cold dark matter scenario. We additionally
demonstrate how our results can be used to directly constrain specific dark
matter models and discuss implications for current and future simulations of
self-interacting dark matter models.

##  1 Introduction

The cold dark matter (CDM) paradigm has been immensely successful in
explaining many aspects of the universe and is particularly successful at
describing large-scale structure [1]. This has naturally led to CDM as the
benchmark for dark matter (DM) studies and the most extensively investigated
class of DM candidates. However, the lack of indirect or direct observational
evidence of collisionless cold dark matter (e.g. WIMPs) [2], along with
purported shortcomings of CDM at small scales [3, 4, 5, 6, 7, 8, 9] has led to
interest in alternative DM frameworks. In regards to the latter point,
challenges at small scales such as the missing satellites [3, 4], “Too-Big-To
Fail” [5, 6], and core-cusp [7, 8, 9] problems are potentially troubling for
the CDM paradigm (see however [10, 11, 12, 13, 14, 15]). Solutions to these
problems based on baryonic effects alone have been explored [13, 11, 16, 17],
though it is not clear that these effects are sufficient to account for the
discrepancies between observations and simulations [18, 19, 20]. An intriguing
solution to these problems is to consider self-interacting dark matter (SIDM)
that is not fully collisionless, which was first proposed in [21] and recently
reviewed in [22].

N-body simulations as well as analytical arguments suggest that DM self-
interactions can yield observable effects on the macroscopic properties of
halos that can impact and potentially even address some of the problems that
arise in a collisionless CDM scenario. Specifically, self-interactions can
flatten the centrally peaked cusps in the inner regions of galaxies and are
capable of disrupting the development of the dense satellite galaxies expected
from simulations that are at the crux of the core-cusp and TBTF problems [23,
24, 25]. SIDM is also predicted to affect the shapes of DM halos through
isotropized particle scattering [24, 21, 26], producing more spheroidal halos
than seen in CDM. These predictions for the macroscopic effects of SIDM
provide an opportunity to probe the microscopic properties of DM-DM
interactions. The figure of merit for SIDM is the ratio of the collisional
cross-section to the DM mass $\sigma/m$ , since the scattering rate for DM
self-interactions is proportional to the number density of dark matter
particles $n_{\rm DM}=\rho_{\rm DM}/m$ times the thermally averaged self-
interaction cross section times the relative velocity, $\langle\sigma v_{\rm
rel}\rangle$, thus $\Gamma_{\rm DM-DM}=n_{\rm DM}\langle\sigma v_{\rm
rel}\rangle$. In order to alleviate the small-scale challenges discussed
above, several groups have found that the necessary interaction strength is
roughly $\sigma/m\approx 0.5-10$ cm2/g [23, 24, 25, 21, 26, 15]. This is in
line with the simple expectation that for typical relative DM velocities the
rate $\Gamma_{\rm DM-DM}\sim H$, with $H$ the Hubble rate.

Studies of the predicted effects of SIDM in comparison with observations place
some constraints on the ratio of the DM self-interaction cross section to the
dark matter mass. These observational probes include cluster lensing [27, 24,
28], mergers [29, 30], and X-ray ellipticities [31]. While early constraints
on SIDM in clusters suggested low cross section values of $\sigma/m\lesssim
0.02$ cm2/g [27], subsequent higher resolution simulations indicated that
these limits were overly optimistic and that cross sections of
$\sigma/m\lesssim 1$ cm2/g are consistent with simulations across a variety of
mass scales [32, 24]. Furthermore, observations at a wide range of halo masses
indicate that a velocity dependent cross-section is needed in order to
alleviate small-scale issues while also being consistent with cluster
constraints [33], thus providing compelling motivation for studies at a range
of halo mass scales. Specifically, the interaction cross-section must decrease
with larger relative velocity (e.g in clusters). This can naturally be
achieved in SIDM models wherein the interactions are governed by a Yukawa
potential (see for instance the models described in Ref. [34, 35, 36, 37, 38].

An interesting class of targets for SIDM studies is elliptical galaxies with
halo masses of order $\sim 10^{12}-10^{13}M_{\odot}$. Elliptical galaxies are
interesting to DM studies in part because the interstellar X-ray emitting gas
fills the gravitational potential, acting as a tracer of the underlying
gravitational potential, well beyond the range where stellar dynamics can be
used as a probe of the mass content of the system [39]. However, this relies
on the assumption that the gas is in a state of hydrostatic equilibrium with
the potential. If the elliptical galaxies in question can reasonably be
treated as being in hydrostatic equilibrium, the gas traces the gravitating
potential, and determination of the shape of the X-ray gas allows for probes
of the DM dominated mass distribution. Thus, by observing the X-ray shapes of
relaxed (i.e. in thermal equilibrium) elliptical galaxies, the shape of the DM
halo can be inferred and compared with predictions of CDM and SIDM halos
shapes from simulations. This “X-ray Shape Theorem” first was developed by
[40] and has been applied to studies of elliptical galaxies and clusters [41,
42, 43, 44, 31].

Applying this method to the NGC 720 elliptical galaxy using Chandra X-ray
data, Ref. [31] determined the ellipticity of the DM halo was roughly
$\epsilon\approx 0.35-0.4$. Simulations for cross-sections of
$\sigma/m=0,0.1,1$ cm2/g showed that the NGC 720 ellipticity was consistent
with an interaction cross-section over mass of $\sigma/m\sim 0.1$ cm2/g, as
well as with CDM ($\sigma/m=0$ cm2/g) [24]. While these results presented
convincing evidence that $\sigma/m=1$ cm2/g was incompatible with the NGC 720
observation, strong assertions for lower cross-sections are difficult to make
given the singular observation and significant scatter in the ellipticities of
the simulated halos. Expanding on the NGC 720 results by performing a shape
analysis of an ensemble of elliptical galaxies can potentially lead to more
concrete statements. Still, the results of the X-ray ellipticity measurements
of NGC 720 have been useful in applications to a wide range of DM models
including hidden sector hydrogen [45], dissipative and “double-disk” dark
matter [46], charged DM [47, 48, 49], as well as alternative gravity theories
for DM phenomena such as MOND [50].

In the present study, we expand on the scope of the analysis of Ref. [31] by
analyzing a sample of elliptical galaxies while additionally leveraging the
capabilities of both the Chandra and XMM telescopes. Here we aim to build upon
the previous X-ray shape measurements and provide data on the ellipticities of
$M\sim 10^{12}-10^{13}M_{\odot}$ mass DM halos that will impact future studies
of SIDM models especially in connection with future, higher-resolution N-body
simulation of SIDM cosmologies.

The remainder of this paper is organized as follows. In section 2 we describe
the calculations needed to relate the mass distribution to the X-ray
ellipticities, including discussion of the hydrostatic equilibrium condition.
In section 3 we present the criteria we utilized to create our galaxy sample
selection and describe the Chandra and XMM data reduction procedures. We
describe the analysis of the processed data including the ellipticities and
surface brightness profile calculations and fitting procedures in section 4.
We present the results of these procedures in section 5 and discuss the
implications of them for the DM self-interaction cross-section. Finally we
present our conclusions in section 6.

##  2 X-ray Emissivity as a Tracer of the Mass Distribution

###  2.1 Gravitational Potential of an Ellipsoidal Mass Distribution

For this study we use the “X-ray Shape Theorem” [40, 41, 42, 43, 44, 31] in
order to determine the shape of the gravitating mass through observations of
the X-ray emitting gas. This approach relies on the assumption that the gas is
in hydrostatic equilibrium and therefore traces the gravitational potential.
Thus, it is necessary to determine the gravitational potential produced by the
total assumed mass distribution. The gravitational potential for an
ellipsoidal mass distribution, $\rho(a)$, is given by the expression [51, 52]

$\Phi(a)=-\pi
Gpq\int_{0}^{\infty}\frac{\psi\left(a^{2}(\tau,\mathbf{x})\right)\>d\tau}{\sqrt{(\tau+1)(\tau+p^{2})(\tau+q^{2})}}$
(2.1)

where,

$a(\tau,\mathbf{x})^{2}=\frac{x^{2}}{\tau+1}+\frac{y^{2}}{\tau+p^{2}}+\frac{z^{2}}{\tau+q^{2}}$
(2.2)

and

$\psi\left(a^{2}(\tau,\mathbf{x})\right)=\int^{\infty}_{a^{2}(\tau,\mathbf{x})}\rho(\tilde{a}^{2})d\tilde{a}^{2}.$
(2.3)

The elliptical radius $a$ is defined as

$a^{2}=x^{2}+\frac{y^{2}}{p^{2}}+\frac{z^{2}}{q^{2}}.$ (2.4)

In this notation, the principal axes $a$, $b$, and $c$ are assumed to be
aligned along the $x$, $y$, and $z$ axes respectively with the values $p=b/a$
and $q=c/a$ being the axis ratios. The ellipticity is defined as
$\epsilon=1-p=1-b/a$. As we are interested in the flattening of the halo
profile, we consider the two cases of oblate and prolate spheroids defined by
$q=p$ (oblate) and $q=1$ (prolate) with the axis of symmetry along the line of
sight. This in effect brackets the range of projected ellipticities of a
triaxial ellipsoid. We take into consideration the following three mass
distribution densities in our analysis: (i) a Navarro-Frenk-White (NFW)
profile [53], (ii) a Hernquist profile [54], and (iii) a pseudo-isothermal
profile (pIso). The functional form of the aforementioned profiles are
explicitly given by

NFW: $\displaystyle\rho\propto\frac{1}{a(a_{s}+a)^{2}}$ (2.5) Hernquist:
$\displaystyle\rho\propto\frac{1}{a(a_{s}+a)^{3}}$ (2.6) pIso:
$\displaystyle\rho\propto\frac{1}{(a_{s}^{2}+a^{2})}$ (2.7)

where $a_{s}$ is the scale radius.

###  2.2  Hydrostatic Equilibrium — Gas Density and X-ray Emissivity

The isolated, relaxed elliptical galaxies are assumed to be in a state of
hydrostatic equilibrium so that we may treat the X-ray gas as a tracer of the
potential. This is expressed as a balance of the forces from internal gas
pressure and gravitation given by the relation:

$\mbox{\boldmath$\nabla$}P_{gas}=-\rho_{gas}\mbox{\boldmath$\nabla$}\Phi,$
(2.8)

where $P_{gas}$ is the gas pressure, $\rho_{gas}$ is the gas mass density and
$\Phi$ is the total gravitational potential. Taking the curl of both sides
yields
$(\mbox{\boldmath$\nabla$}\rho_{gas})\times(\mbox{\boldmath$\nabla$}\Phi)=0$,
implying that surfaces of constant gas density are also surfaces of constant
gravitational potential. For approximately isothermal gas distributions, the
X-ray emissivity $(j_{X})$ is related to the gas density as
$j_{X}\propto\rho_{gas}^{2}$ [42, 43]. Since surfaces of constant $\rho_{gas}$
are isopotential surfaces, it is also true that surfaces of constant
$\rho_{gas}^{2}$, and consequently, surfaces of constant $j_{X}$ are
isopotential surfaces as well [40, 41, 43, 44, 42, 31]. In practice, the
observable quantity is not the 3-D emissivity itself, but rather the X-ray
surface brightness, $\Sigma_{X}$, which is the 2-D projection of the
emissivity along the line of sight and is given in terms of the gas density by
the relation:

$\Sigma_{X}\propto\int_{los}j_{X}\propto\int_{los}\rho_{gas}^{2}.$ (2.9)

For an isothermal gas, Eq. (2.8) can be solved for the the gas density [51,
42]:

$\rho_{gas}(a)=\rho_{gas,\>0}\exp\left[-\frac{\mu
m|\Phi_{0}|}{k_{b}T}\left(1-\frac{\Phi(a)}{\Phi_{0}}\right)\right].$ (2.10)

We express the equation above more compactly as

$\tilde{\rho}_{gas}(a)=\exp\left[-\Gamma\left(1-\tilde{\Phi}(a)\right)\right],$
(2.11)

where the tildes denote the dimensionless form for the expression normalized
at the galaxy center and $\Gamma=\mu m|\Phi_{0}|/k_{b}T$.

Once a model for the mass distribution (Eqs. 2.5-2.7) has been chosen, the
parameters of interest in modeling the X-ray emission of a galaxy are
$a_{s},\>\Gamma$, and $\epsilon$, along with an appropriate normalization of
the surface brightness. In each panel of figures 1 and 2 we show the the
individual effects of varying each of these parameters on the surface
brightness profiles and ellipticity profiles respectively. For illustrative
purposes only, we adopt base parameter values of
$a_{s}=30^{{}^{\prime\prime}},\>\Gamma=5$, and $\epsilon=0.4$ and assume an
NFW profile. While the surface brightness profile unsurprisingly has a strong
dependence on $a_{s}$ and $\Gamma$, the effects of varying the ellipticity of
the halo are less pronounced. In the X-ray ellipticity profile, the $a_{s}$
and $\Gamma$ parameters have almost no discernible effects aside from a slight
decrease in observed $\epsilon_{X}$ at the lowest values ($a_{s}\sim 20$”,
$\Gamma\sim 4$). Naturally, the ellipticity of the matter distribution used in
the model has a major impact on the X-ray ellipticity profile. In addition,
for more elliptical mass distributions the X-ray profile also tends to exhibit
a radial dependence.

Figure 1: Normalized surface brightness profiles for a fiducial NFW dark
matter density profile with $a_{s}=30^{{}^{\prime\prime}},\>\Gamma=5$, and
$\epsilon=0.4$ while varying $a_{s}$ (left), $\Gamma$ (middle), and $\epsilon$
(right).

Figure 2: Ellipticity profiles for a fiducial NFW galaxy with
$a_{s}=30^{{}^{\prime\prime}},\>\Gamma=5$, and $\epsilon=0.4$ while varying
$a_{s}$ (left), $\Gamma$ (middle), and $\epsilon$ (right).

##  3 Galaxy Sample and Data Reduction

###  3.1 Galaxy Selection

The selection of galaxies used in this analysis is guided by a few conditions
that justify the assumption of hydrostatic equilibrium. We focus our attention
on relaxed, isolated elliptical galaxies. Thus, we seek candidates that
exhibit regular, circular, or elliptical X-ray morphologies as expected for a
suitably relaxed galaxy [55, 39]. In particular, we avoid galaxies with bright
central AGN or significant AGN feedback. The elliptical galaxy NGC 4374 (M84)
[56] provides an example case where this relaxation condition does not hold
due to significant disruption of the X-ray gas from AGN feedback. The
isolation criteria include

(i) no close neighbors,

(ii) no signs of interaction, and

(iii) no evidence for a recent merger.

Additionally, since the stellar mass dominates the overall mass distribution
in the central regions, we seek galaxies for which existing observational data
extends far beyond the effective stellar radius. This is necessary in order to
probe the underlying mass distribution in the outer regions where the dark
matter component is the dominant contribution to the gravitational potential.
The galaxies that fit these criteria and are used in this analysis are listed
in Table 1 along with some basic properties including their distance and
K-Band effective radius taken from the Two Micron All Sky Survey (2MASS)
Extended Source Catalog (XSC) [57].

Galaxy | Dist. (Mpc) | ”/kpc |  $r_{e}$ (kpc)
---|---|---|---
IC4451 | 55.8 [58] | 0.27 | 4.07
IC4956 | 70.1 [59] | 0.34 | 4.59
NGC1521 | 69.5 [60] | 0.34 | 5.53
NGC4125 | 24.0 [60] | 0.12 | 3.65
NGC4555 | 110 [61] | 0.53 | 5.97
NGC57 | 70.1 [59] | 0.34 | 4.98
NGC6482 | 59.2 [62] | 0.29 | 3.47
NGC7785 | 48.4 [59] | 0.23 | 4.05
NGC7796 | 51.3 [60] | 0.25 | 4.03
NGC953 | 66.3 [63] | 0.32 | 2.91
NGC720 | 25.7 [64] | 0.12 | 3.14
Table 1: Sample of galaxies used in this analysis. References for the
distances are provided. The effective radius $r_{e}$ is the K-band effective
radius taken from the 2MASS XSC catalog [57].

| XMM | Chandra
---|---|---
| MOS1 + MOS2 |  |  |
Galaxy | ObsID | Exp1+2 (ks) | ObsID | Exp. (ks)
IC4451 | 0503480501 | 4.26 + 6.02 | 13808 | 25.05
0673080101 | 66.04 + 66.58 | – | –
IC4956 | 0503480801 | 0.96 + 3.38 | – | –
0693190401 | 81.44 + 86.74 | – | –
NGC1521 | 0552510101 | 98.15 + 100.82 | 10539 | 40.79
NGC4125 | 0141570201 | 98.15 + 100.82 | 2071 | 54.76
NGC4555 | 0403100101 | 62.64 + 62.26 | 1884 | 24.44
NGC57 | 0202190201 | 21.17 + 21.23 | 10547 | 8.69
NGC6482 | 0304160401 | 7.92 + 7.9 | 3218 | 18.94
0304160801 | 5.21 + 5.39 | 19584 | 23.50
– | – | 19585 | 17.00
– | – | 20850 | 17.97
– | – | 20857 | 20.00
– | – | 20978 | 17.97
– | – | 20979 | 8.44
– | – | 20980 | 67.77
NGC7785 | 0206060101 | 16.5 + 17.08 | – | –
NGC7796 | 0693190101 | 47.64 + 56.74 | 7401 | 16.91
– | – | 7061 | 44.20
NGC953 | 0722360201 | 69.49 + 75.84 | 11262 | 5.92
0762220101 | 95.55 + 95.04 | 14899 | 37.03
NGC720 | 602010101 | 81.76 + 81.71 | 492 | 21.69
Table 2: Sample of galaxies and their corresponding ObsIDs and cleaned
exposure times from XMM and Chandra (see section 3.2 for details). For the XMM
exposure time, we show the individual cleaned exposure for each of the MOS1
and MOS2 EPIC cameras.

###  3.2 Observations and Data Reduction

####  3.2.1 XMM-Newton

Each galaxy in our sample has at least one observation with the XMM-Newton
telescope. We use archival data from the EPIC MOS1 and MOS2 cameras in the
soft X-ray energy band $0.5-2$ keV. The processing of the data is performed
with the Science Analysis Software (SAS222”Users Guide to the XMM-Newton
Science Analysis System”, Issue 15.0, 2019 (ESA: XMM-Newton SOC). v16.1.0) and
the Extended Source Analysis Software (ESAS) [65] following the steps outlined
in the ESAS
Cookbook333https://heasarc.gsfc.nasa.gov/docs/xmm/esas/cookbook/xmm-esas.html
for diffuse emission. Specifically, we run the emchain program to prepare the
events list products for use with the ESAS tasks. The mos-filter routine
(which in turn calls the SAS routine espfilt) is used to remove periods of
high background and determine the good time intervals (GTI). Light curves for
each observation and MOS camera were inspected manually to ensure quality and
observations that were not adequate (see e.g. the examples in the ESAS
cookbook) are not included in this work. Nearly all observations exhibited
some periods of high background that were subsequently removed. The remaining
clean exposure times can be found in table 2. Images and exposure maps are
then created for each observation and MOS camera using the mos-spectra routine
and then combined. During the ESAS processing, images are binned to pixel
sizes of $\sim 2.5^{\prime\prime}\times 2.5^{\prime\prime}$.

####  3.2.2 Chandra

For nine out of the 11 galaxies in our sample there exists archival Chandra
data that is suitable for this analysis. We restrict our analysis to the ACIS
S3 chip and again consider the $0.5-2$ keV energy band. The data are processed
with the CIAO 4.11444https://cxc.cfa.harvard.edu/ciao/ [66] software along
with the corresponding Chandra calibration database (CALDB) v4.8.2 following
the standard procedure for Chandra data. The Chandra data are binned into
pixels of size $\sim 1$”$\times 1$”. For flare removal we use the lc_clean
routine with default parameters and again manually inspect the light curves
before proceeding. The remaining cleaned exposure times are shown for each
ObsID in 2.

####  3.2.3 Point Source Removal

Since we are interested in the extended diffuse emission of the galaxies, it
is necessary to remove bright point sources. Further, simply removing or
masking the point sources is insufficient as the empty source regions in the
image can significantly affect the ellipticity measurements. We therefore need
to both identify the point sources and reasonably model the true diffuse
emission in their place. For the identification of the point sources we use
the wavdetect CIAO routine which provides a wavelet function source detection
method. We supply this routine with the point spread function (PSF) map built
by running fluximage (and thus, mkpsfmap) for the Chandra data. In the XMM
data we use a psfmap with a constant size of 5”. The exposure maps created by
fluximage and mos-spectra are used for the Chandra and XMM images,
respectively. Identified source regions were then removed and filled using the
CIAO dmfilth task which takes as input source and background regions produced
by the roi routine. When running dmfilth we use the POISSON method, which
replaces the source region by sampling from the Poisson distribution whose
mean is that of the pixel values in the background region. In figure 3 we show
example Chandra and XMM images of NGC 6482 before source detection, after
source detection and replacement, as well as an image smoothed using the CIAO
task csmooth.

Figure 3: Chandra (left column) and XMM-Newton (right column) images of
NGC6482. The top row is prior to point source detection. The middle row is
after running the wavdetect and dmfilth routines to remove points sources.
Finally, the bottom row shows the point source cleaned, smoothed, and exposure
corrected images.

##  4 X-ray Ellipticity and Brightness Profiles

The ellipticities of the X-ray images for each galaxy in our data set are
calculated using the image moments method as outlined in [67, 68, 42, 31, 69]
which we briefly review here. We begin by finding the centroid of a circular
aperture at the desired semi-major radius $a$. The centroid of the radius is
then calculated from the first order moment and the aperture shifted to that
point. This process is repeated until the centroid shift changes less than
some tolerance (roughly $\sim 1$ pixel). With the centroid found, we again
start with a circular aperture of radius $a$, and calculate the second order
image moments. These are effectively the elements of the inertia tensor of the
image within the aperture, and allow the determination of the ellipticity
within the aperture and orientation angle of the semi-major axis [67, 68, 42,
69]. This process is performed iteratively until the measured ellipticities
and orientations converge. For the error estimation of the ellipticity
profile, we follow the procedure of [31] and use a Monte Carlo approach. The
pixel counts are assumed to follow Poisson statistics, so we create a
simulated image by sampling the pixel values from a Poisson distribution with
the original pixel count as the mean. We then calculate the ellipticity for
each radii $a$ as in the case of the original data image, repeating this for
100 instances. The standard deviation of the samples generated through this
process is taken to be the $1\sigma$ error of our ellipticity.

In addition to the ellipticity radial profile, we also require the radial
surface brightness profile in order to characterize the shape of the
gravitating mass distribution. This is calculated by adding the counts within
several annuli and dividing by the area of each annulus. Errors are calculated
assuming Gaussian statistics (i.e. $\sigma_{i}\sim\sqrt{N_{i}}$ for $N_{i}$
counts in annulus $i$) based on the lowest counts per bin in our samples being
on the order of $\gtrsim 100$.

To perform the fitting procedure we begin by generating a model image for each
assumed mass distribution and either prolate or oblate configuration based on
the calculations in sections 2.1 and 2.2. The ellipticity and surface
brightness profiles are calculated for the model image and a $\chi^{2}$
statistic is used to determine the fit to the data. We then minimize the
$\chi^{2}$ statistic with $a_{s},\>\Gamma,$ and $\epsilon$ as free parameters
using a Nelder-Mead simplex algorithm [70, 71] with a dimension dependent
implementation of the expansion, contraction, and shrink parameters [72].

In the process described above, we treat the background modeling in a slightly
different manner for the ellipticity and brightness profiles. The constant
background model is estimated from the flattening of the brightness profile at
large radii. This is illustrated in figure 4. For the ellipticity
measurements, we subtract this background from the data and fit to the model
generated profiles.

Figure 4: Surface brightness profile of IC 4451 without background
subtraction. The red line shows the background model used in later portions of
the analysis.

We note that problems can arise in the larger radius bins if the profile is
calculated from the background subtracted image due to over subtraction.
Essentially, some annular bins may have average counts per pixel slightly less
than the background model. To avoid these problems we model the background
simultaneously with the calculated model image rather than subtracting from
the data. For the ellipticity profiles, over-subtracting the observed counts
is not an issue due to the elliptical aperture that contains the bright
central regions and ensures that the average counts per pixel are greater than
the background model.

##  5 Results

In figures 5 and 6 we show the best fit profiles for an NFW oblate mass
configuration overlaid on the data for each galaxy. Figure 7 shows the best-
fit profiles for each configuration overlaid on the IC 4956 XMM data. The
best-fit results for the XMM and Chandra data are summarized in tables A.1 and
A.2 respectively, along with the reduced $\chi^{2}$ values. In most cases, the
$\chi^{2}$ statistic does not provide a quantitatively “good” fit to the data.
In the case of the surface brightness profile this can at least partially be
attributed to small error bars that amplify the $\chi^{2}$ value despite what
appears to be a fairly strong qualitative agreement between the data and the
model. This is similar to what was found in the study of NGC 720 from Ref.
[31]. However, it is clear from looking at figure 5 that a large contributor
to the $\chi^{2}$ is the fact that the data for several of the galaxies do not
follow the relatively flat ellipticity profiles of the model. This is not
entirely surprising, since in this analysis we are assuming either prolate or
oblate spheroidal mass-distributions with a constant ellipticity. More
realistically, many of these galaxies are likely to be better represented by a
triaxial distribution. They would therefore be expected to exhibit an
isophotal “twist” and variations in their X-ray ellipticity profile [73, 74].
Rotating cooling flows may also alter the X-ray isophotes and contribute to
variations in the radial profiles [73]. Nevertheless, since we have used the
prolate and oblate configurations as a way to bracket the triaxial case we
treat the results as relatively representative of the underlying mass
distribution, though keeping in mind the potential need for more sophisticated
modeling of the halo shape and possible astrophysical activity.

Another point of note is the radial range chosen for the fitting procedure. In
our basic approach the minimum of the range was chosen to be $10$ pixels for
stability of the iterative moment method when calculating the ellipticity. The
maximum radial limit was taken to be the point at which the emission region
became background dominated (see figure 4) or the chip edge was reached.
However, there are good reasons to restrict this range from both ends on a
galaxy-by-galaxy basis. In the inner regions, the stellar component
contributes significantly to the overall mass distribution. For some of the
galaxies in our sample, previous studies have been able to model the mass
distribution to give a detailed description of how the mass profile and its
constituent components change with radius (for example, see [75, 76, 77, 78,
79, 55]). In cases for which the full mass modeling has not been performed, it
can be roughly assumed that the stellar mass has a non-negligible contribution
within the effective radius $r_{e}$ [80, 81]. In choosing a more restrictive
minimum radius these characteristics of the galaxies should be kept in mind.

At larger radii, it is less clear that there is a natural maximum limit
smaller than the background limit. In the NGC 720 study from Ref. [31], the
upper radial limit $\sim 150^{\prime\prime}$ was chosen because of strange
behavior observed at larger radii wherein the profile diverged from the values
at smaller radii. Applying this to our sample there are potentially six
galaxies (IC 4451, NGC 1521, NGC 57, NGC 6482, NGC 953, NGC 720) that exhibit
this behavior to some extent for at least one of the observations. The origin
of this divergence is unclear, although one physically motivated possibility
is that the assumption of hydrostatic equilibrium does not accurately apply at
these radii.

However, Ref. [31] also points out that for their observation the region
$\gtrsim 150^{\prime\prime}$ is near the CCD edge which may be causing the
strange behavior, and that observations with a wider field of view such as XMM
could give insight into whether this is an observational or physical effect.
In some of our galaxies, we find that in these large radius regimes the
Chandra data exhibits this behavior to a much greater degree and diverges from
the XMM profile. For example, the Chandra ellipticity diverges from the XMM
values for NGC 6482, NGC 1521, and NGC 720 at $a\gtrsim 40,20,18$ kpc
respectively (note that for NGC 720, 18 kpc $\approx 150^{\prime\prime}$).
This suggests the strange behavior at large radii is likely due to
observational effects rather than a failure of hydrostatic equilibrium, and
furthermore that removing regions at large radii exhibiting this behavior
could improve the accuracy of the fits.

These considerations along with the ability to compare our results with other
studies motivate us to also explore a more physically motivated range over
which to perform the fitting procedure. We select the range
$a=r_{e}\rightarrow 5r_{e}$, which roughly corresponds with the range used in
both the NGC 720 analysis [31] as well as simulation results [24]. In some
cases data is not available at this range either due to the presence of the
CCD edge or being background limited. In such cases we only fit to the extent
of available data. The results of fitting in this range for the XMM and
Chandra datasets are shown in tables A.3 and A.4 respectively. Broadly
speaking, the results over this range tend to perform similarly, both in terms
of fit parameters and $\chi^{2}_{red}$ though this varies on a galaxy-to-
galaxy basis. While the $a_{s}$ and $\Gamma$ are quite consistent between the
two fitting ranges, the ellipticities tend toward slightly lower values in the
range-restricted setup. This may be attributable to avoiding the potentially
anomalous behavior at large radii discussed above; however, it is difficult to
draw strong conclusions given the smaller statistics of the limited fit range.
A direct comparison between the two choices of radial range can be seen in the
right panel of figure 11. since the results of the two choices are not highly
disparate and the background limited setup provides somewhat better fits, the
remainder of this section will focus on these results. However, see section
5.1 for more discussion comparing the two choices.

Fits to the data were fairly consistent across the various mass configurations
for a given galaxy. The slope parameter $\Gamma$ varies from galaxy to galaxy
ranging from $\sim 6-9$ but is otherwise roughly consistent across mass
configurations. An interesting note is that the scale radius $a_{s}$ in the
pIso models is almost always considerably smaller than for the NFW or
Hernquist profiles. As the scale radius characterizes the core in the pIso
profile, this seems to suggest that the X-ray emission prefers a small-core,
nearly isothermal $\rho\propto a^{-2}$ profile. In figure 8 we show the
ellipticities for each galaxy and mass distribution configuration. For
comparison, we also show the ellipticity for NGC 720 determined in [31] as
horizontal colored lines. In general, their measured ellipticity falls
comfortably within the range of the measured ellipticities of our ensemble.
The values found in our analysis for NGC 720 tend to be slightly lower,
although this could potentially be due to exposure corrections (see section 2
of [31]) or calibration effects. In addition, there does not appear to be any
significant relationship between the ellipticity and the mass configuration
for a given galaxy (e.g. NFW halos do not consistently result in the lowest
ellipticities) .

Figure 5: Ellipticity profiles of the best-fit models for each galaxy assuming
an oblate NFW mass distribution plotted along with the profiles from the
observational data. Figure 6: Surface brightness profiles of the best-fit
models for each galaxy assuming an oblate NFW mass distribution plotted along
with the profiles from the observational data.

Figure 7: Best-fit ellipticity and surface brightness profiles for the XMM
observations of IC4956. Here we show the best fitting profiles for each mass
distribution configuration. Colors correspond to different profiles, while the
solid and dashed lines refer to the prolate and oblate configurations
respectively.

Figure 8: Best-fit DM halo ellipticities $\epsilon$ for each galaxy in our
sample determined from the XMM (X’s) and Chandra (circles) observations in the
background limited range. The top panel shows the prolate configuration, while
in the bottom panel we show the oblate configuration. The horizontal lines
show the values found for NGC720 in [31] (Note that in the prolate
configuration the ellipticities for the $\rho\sim a^{-2}$ and NFW halos are
overlapping.)

###  5.1 Implications for Self-Interacting Dark Matter

In line with early analytic arguments concerning SIDM [21], simulations have
demonstrated that the DM self-interactions produce halos with greater
sphericity than in CDM [24, 26]. In Ref. [24], DM halos were simulated for
cross-sections of $\sigma/m=$ 0 cm2/g (CDM), 0.1 cm2/g, and 1 cm2/g. These
simulations were originally motivated by the results of the NGC 720 study
[31], and therefore use halos models based on NGC 720 with a mass range of
$(3-10)\times 10^{12}M_{\odot}$ and ellipticities calculated within 8.5 kpc
$<r<$ 14 kpc (roughly corresponding to $2.7r_{e}<r<4.5r_{e}$ based on the
effective radius given in table 1). The resulting ellipticities of the
simulated halos were then compared with the observed ellipticities of NGC 720
[31]. It was found that the NGC 720 DM halo ellipticity ($\epsilon\approx
0.35-0.4$) was consistent with both the CDM and 0.1 cm2/g interaction regimes
(cf. Figure 8 of [24]). However the distribution in ellipticity of the
simulated halos exhibited considerable scatter, making concrete conclusions
based on the one observed galaxy difficult. With the supplement provided in
this analysis we can consider a similar comparison, though now with a sample
of galaxies rather than the singular case of NGC 720 from Ref. [31].

In figure 11 we show the normalized distribution of halo ellipticities of our
sample in the background-limited and range-restricted scenarios for an NFW
profile overlaid with the results of the simulations from [24]. For the
background limited setup it can be seen the halo ellipticities of our sample
are consistent with some non-negligible DM interactions. Particularly for the
Chandra data, the distribution of ellipticities is in strong agreement with
$\sigma/m=0.1$ cm2/g, and the Buote et. al. (2002) [31] analysis of NGC 720
aligns close to the peak of the distribution. The ellipticities taken from the
XMM data tend toward lower ellipticities, though also having greater spread
and essentially overlapping to some degree the histogram for each interaction
strength. This may to some extent be attributable to the larger PSF of XMM
producing a somewhat more “smoothed” image compared to the Chandra data,
although further investigation would be needed to determine whether this is a
meaningful effect. The distributions for the range restricted case where we
fit over $r_{e}\rightarrow 5r_{e}$ tend to favor more spherical halos, but
also have much more scatter than the background limited radial range.

To quantify the level of agreement between the results of our fits and the
simulated data we perform a two sample $\chi^{2}$ test [71] between each of
the distributions. The results are recorded in Table 10 and plotted in Figure
10. Here we see that generally the results tend to favor some degree of self-
interaction, with the lowest $\chi^{2}_{red}$ for the full sample
corresponding to $\sigma/m=0.1$ cm2/g.

$\chi^{2}_{red.}$
---
| BG Limited | $r_{e}\rightarrow 5r_{e}$
$\sigma/m_{\chi}$ | Chandra | XMM | Chandra | XMM
0.0 | 3.80 | 5.87 | 3.77 | 7.33
0.1 | 1.98 | 4.07 | 4.06 | 3.62
1.0 | 5.44 | 4.78 | 4.46 | 3.25

Figure 9: Reduced $\chi^{2}$ values for each of the distributions shown in the
left panel of figure 11.

Figure 10: Visual representation of the data in table 10. Colors represent
results from XMM (blue) and Chandra (red) data, while the circles and squares
represent the BG limited and range-restricted setups respectively.

Figure 11: (Left) Normalized distributions of the best fit galaxy
ellipticities from the Chandra (red) and XMM (blue) data for an NFW profile.
Solid lines are for the BG limited results, while dash-dot lines are for the
range-restricted results. The histograms show the simulated ellipticities for
various interaction cross sections from Figure 8 of [24]. We also show for
reference the best-fit model and errors for the analysis of NGC 720 preformed
in Ref. [31]. (Right) Comparison of the ellipticity histograms between the BG
limited results and the range restricted $(r_{e}\rightarrow 5r_{e})$ results.
These include the results for the NFW halos from the XMM observations (i.e.
the larger sample).

The sample of measured ellipticities can be applied to comparisons between
additional existing and future halo simulations, and can additionally be used
to constrain a variety of SIDM models. To illustrate the utility of these
results in constraining specific dark matter models, we follow the theoretical
framework of [49, 48] and consider a dark matter candidate governed by a dark
sector Coulomb interaction. We constrain the dark coupling constant
$(\alpha_{\chi})$ and DM particle mass $(m_{\chi})$ by enforcing that the
relaxation time be greater $\tau_{r}>10$ Gyr. The relaxation time is given in
[48] by the expression

$\tau_{r}=E_{k}/\dot{E_{k}}=\simeq\frac{m_{\chi}^{3}v_{0}^{3}}{4\sqrt{\pi}\alpha_{\chi}^{2}\rho_{\chi}}\left(\ln\left(\frac{(b_{max}m_{\chi}v_{0}^{2}\alpha_{\chi}^{-1})^{2}+1}{2}\right)\right)^{-1},$
(5.1)

where $b_{max}$ is the impact parameter and $v_{0}$ corresponds to the
velocity dispersion of the dark matter halo. We can rewrite this expression
as:

$\tau_{r}=\frac{m_{\chi}^{3}v_{0}^{3}}{4\sqrt{\pi}\alpha_{\chi}^{2}\rho_{\chi}\ln\Lambda},$
(5.2)

where the “Coulomb logarithm ‘’ has a value of $\ln\Lambda\sim 90$. In Ref.
[49] the ellipticity of the DM halos is characterized as a function of time
with the following form:

$\epsilon(t)=\frac{\epsilon_{0}}{\frac{t}{\tau_{r}}\epsilon_{0}+1}$ (5.3)

Combining Eq 5.2 and Eq 5.3, we determine a constraining relation between the
dark sector coupling and DM mass given by:

$\alpha_{\chi}^{2}=\frac{m_{\chi}^{3}v_{0}^{3}}{\sqrt{\pi}\rho_{\chi}\ln\Lambda}\frac{\frac{\epsilon_{0}}{\epsilon}-1}{t\epsilon_{0}}$
(5.4)

We adopt nominal values for the velocity dispersion and density profiles of
250 km/s and 1 GeV/cm3 over the spatial extent of the region in which we
perform our fits — essentially the region where the dark matter dominates the
mass contribution but that still provides reliable x-ray data. The choice of
these values is based on the results of mass and concentration parameter
measurements in NGC 720 (see e.g. [77, 55]) and are taken to be roughly
representative of our sample. In future works, modeling each of the galaxies
in a more detailed manner could provide slightly different results.

The resulting combined constraints using the measured ellipticities are shown
in figure 12 for both the Chandra and XMM data sets. The constraints are quite
similar between the two and mostly overlap. The XMM data prefers slightly
weaker limits, likely attributable to the preference for rounder halos found
in the fits (see figure 11 and the related discussion). Ultimately our
constraints found here are in good agreement with those of Ref. [48], with the
benefit in our work of having used a full sample of measured ellipticities
rather than the sole measurements of NGC 720.

Figure 12: Constraints on the $\alpha_{\chi}-m_{\chi}$ space from our sample
of galaxies ellipticities. The excluded region is shown in grey, and the blue
and red bands give the range of limits from the combined results of the XMM
and Chandra data respectively. Although the two bands mostly overlap, the
upper and lower limits of the XMM band (blue) are slightly higher than those
of the Chandra band (red). We also show for reference where the electron
(magenta star) and proton (green star) would be located on a Standard Model
version of this plot.

##  6 Conclusions

In this analysis we have studied a sample of isolated elliptical galaxies as a
probe of the dark matter halo shape. Using archival data from the XMM-Newton
and Chandra telescopes we analyzed the X-ray surface brightness and
ellipticity profiles of 11 elliptical galaxies. By selecting galaxies that
meet the criteria of being relaxed, isolated, and approximately isothermal, we
made the assumption plausibly well-justified assumption of hydrostatic
equilibrium with the gravitational potential. Under this assumption we showed
the relation between the shape of the 3D matter distribution and the 2D
projected X-ray image, which allowed us to model the 3D matter distribution
and fit to the data, thereby determining the underlying shape for the assumed
matter distribution.

We considered three separate spheroidal mass distributions including an NFW,
Hernquist, and pseudo-isothermal profile. Additionally, we considered both
prolate and oblate configurations for each profile, effectively bracketing the
range of triaxial models. The best fitting ellipticities did not appear to
have a significant relation to the profile. We also considered two choices of
radial range over which we performed our fits: one that extended until a chip
edge was reached or the background became dominant, and another range
restricted setup motivated by previous studies and simulations. While the
range restricted setup generally favored lower ellipticities and had higher
scatter, the differences between the two choices do not significantly alter
our conclusions.

For most of the galaxies in the sample the ellipticities fell roughly within
the range of $\epsilon\approx 0.2-0.5$. Comparing the measured ellipticities
with simulations of DM halos that have varying degrees of self-interactions
shows consistency and a marginal statistical perference with interaction
cross-sections of $\sigma/m=0.1$ cm2/g. However, the simulations of Ref. [24]
used for this comparison reported significant scatter. This is also apparent
in our observational results, and the ellipticity distribution also overlaps
significantly with the distribution for CDM halos. We note that these findings
are highly consistent with the comparison between the simulations and the
ellipticity analysis of NGC 720 performed in Ref. [31].

While the work of Ref. [31] and subsequent studies have shown that X-ray
shapes of elliptical galaxies can have powerful constraining ability on SIDM,
most of these analyses have used only the singular case of NGC 720. The
results presented here now allow for a comparison using a statistically
meaningful sample in order to produce constraints as demonstrated in section
5.1. While these results can be used constrain specific models, our sample
also now allows detailed comparison with simulations containing only DM as
well as future simulations that also seek to incorporate baryonic effects.

## Acknowledgments

This work is funded by NASA award number 80NSSC20K0434. A.M. is partially
supported by a Department of Education GAANN fellowship.

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##  Appendix A Results of Model Fitting Procedure

XMM
---
|  | Prolate | Oblate
Galaxy | Profile |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$ |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$
IC4451 | NFW | 61.0 | 7.8 | 0.34 | 8.3 | 24.5 | 7.9 | 0.35 | 8.2
Hernq. | 126.2 | 7.0 | 0.31 | 9.8 | 45.8 | 6.8 | 0.34 | 9.6
pIso | 10.2 | 6.6 | 0.32 | 10.2 | 4.8 | 7.4 | 0.26 | 14.1
IC4956 | NFW | 40.7 | 8.4 | 0.17 | 0.9 | 15.8 | 8.4 | 0.19 | 0.9
Hernq. | 85.0 | 7.8 | 0.16 | 1.0 | 34.2 | 7.8 | 0.16 | 1.1
pIso | 6.1 | 7.0 | 0.15 | 0.9 | 2.4 | 7.4 | 0.12 | 1.2
NGC1521 | NFW | 56.9 | 7.9 | 0.43 | 13.1 | 21.7 | 8.0 | 0.44 | 12.7
Hernq. | 104.6 | 6.8 | 0.44 | 14.3 | 40.8 | 6.9 | 0.46 | 13.7
pIso | 8.7 | 6.6 | 0.42 | 14.0 | 2.7 | 6.9 | 0.49 | 13.6
NGC4125 | NFW | 77.0 | 7.4 | 0.48 | 1.3 | 29.0 | 7.6 | 0.5 | 1.2
Hernq. | 123.9 | 6.2 | 0.48 | 1.3 | 49.6 | 6.3 | 0.51 | 1.3
pIso | 11.2 | 6.2 | 0.48 | 1.3 | 4.2 | 6.3 | 0.52 | 1.2
NGC4555 | NFW | 24.2 | 6.9 | 0.28 | 5.2 | 8.0 | 6.9 | 0.28 | 4.9
Hernq. | 58.2 | 6.2 | 0.36 | 7.6 | 19.5 | 6.2 | 0.32 | 7.0
pIso | 1.8 | 7.6 | 0.27 | 2.3 | 0.6 | 7.7 | 0.31 | 2.2
NGC6482 | NFW | 46.3 | 8.0 | 0.21 | 7.9 | 16.6 | 7.9 | 0.23 | 7.8
Hernq. | 102.9 | 7.2 | 0.22 | 10.6 | 40.0 | 7.2 | 0.26 | 11.0
pIso | 5.6 | 7.6 | 0.2 | 6.6 | 2.2 | 7.6 | 0.23 | 6.4
NGC7785 | NFW | 52.0 | 7.6 | 0.23 | 4.4 | 20.6 | 7.8 | 0.25 | 4.2
Hernq. | 194.4 | 9.8 | 0.17 | 2.0 | 47.9 | 7.5 | 0.18 | 3.1
pIso | 29.8 | 9.7 | 0.16 | 1.2 | 7.3 | 7.8 | 0.21 | 1.8
NGC7796 | NFW | 47.9 | 7.3 | 0.45 | 3.0 | 22.9 | 8.0 | 0.43 | 3.6
Hernq. | 107.4 | 6.7 | 0.42 | 3.6 | 40.9 | 6.6 | 0.44 | 3.6
pIso | 4.8 | 6.6 | 0.42 | 2.7 | 5.9 | 7.6 | 0.32 | 7.4
NGC953 | NFW | 36.8 | 8.2 | 0.27 | 3.2 | 11.3 | 7.7 | 0.24 | 2.9
Hernq. | 73.4 | 7.6 | 0.22 | 3.9 | 28.0 | 7.6 | 0.29 | 4.0
pIso | 3.8 | 7.4 | 0.25 | 2.9 | 1.4 | 7.4 | 0.24 | 2.8
NGC720 | NFW | 79.1 | 7.6 | 0.34 | 8.9 | 30.7 | 7.6 | 0.35 | 8.4
Hernq. | 126.7 | 6.4 | 0.26 | 12.2 | 60.0 | 6.7 | 0.33 | 9.5
pIso | 13.2 | 6.5 | 0.3 | 10.7 | 4.6 | 6.6 | 0.34 | 9.9
Table A.1: Results of the best-fitting parameters for the XMM data. We show
the results for both the prolate and oblate configurations for each mass
profile. The reduced-$\chi^{2}$ is also provided and all fits have 37 degrees
of freedom.

Chandra
---
|  | Prolate | Oblate
Galaxy | Profile |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$ |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$
IC4451 | NFW | 12.4 | 6.3 | 0.46 | 4.1 | 11.8 | 6.3 | 0.44 | 4.0
Hernq. | 27.0 | 5.8 | 0.21 | 7.7 | 25.1 | 5.7 | 0.51 | 4.6
pIso | 0.8 | 7.0 | 0.36 | 3.4 | 0.9 | 6.8 | 0.37 | 3.3
NGC1521 | NFW | 13.8 | 6.2 | 0.41 | 2.8 | 13.2 | 6.1 | 0.41 | 2.9
Hernq. | 26.4 | 5.6 | 0.19 | 6.1 | 27.0 | 5.5 | 0.26 | 5.0
pIso | 1.1 | 6.5 | 0.29 | 3.2 | 1.4 | 6.3 | 0.45 | 2.5
NGC4125 | NFW | 43.4 | 6.8 | 0.48 | 5.8 | 32.4 | 6.5 | 0.59 | 4.3
Hernq. | 166.3 | 7.4 | 0.5 | 12.0 | 68.0 | 5.8 | 0.6 | 4.9
pIso | 9.9 | 5.8 | 0.5 | 7.2 | 4.5 | 6.0 | 0.41 | 9.2
NGC4555 | NFW | 7.2 | 7.5 | 0.36 | 1.4 | 7.2 | 7.7 | 0.24 | 1.1
Hernq. | 16.1 | 7.0 | 0.25 | 1.6 | 18.2 | 7.1 | 0.51 | 2.7
pIso | 0.8 | 8.4 | 0.41 | 1.8 | 0.5 | 8.7 | 0.38 | 1.1
NGC57 | NFW | 12.8 | 8.7 | 0.34 | 1.6 | 8.7 | 7.7 | 0.32 | 1.3
Hernq. | 17.7 | 6.9 | 0.28 | 1.4 | 15.5 | 6.4 | 0.46 | 1.6
pIso | 1.5 | 7.5 | 0.34 | 1.2 | 1.3 | 7.6 | 0.34 | 1.3
NGC6482 | NFW | 23.8 | 8.2 | 0.2 | 43.5 | 23.3 | 8.1 | 0.22 | 42.6
Hernq. | 51.8 | 7.2 | 0.32 | 61.0 | 47.0 | 7.3 | 0.22 | 53.7
pIso | 2.6 | 8.2 | 0.2 | 28.7 | 2.2 | 8.4 | 0.25 | 29.7
NGC7796 | NFW | 9.0 | 6.8 | 0.41 | 4.7 | 8.3 | 6.7 | 0.44 | 4.4
Hernq. | 18.0 | 6.2 | 0.22 | 7.8 | 19.5 | 6.1 | 0.43 | 5.9
pIso | 0.5 | 7.6 | 0.38 | 2.6 | 0.4 | 7.8 | 0.41 | 2.3
NGC953 | NFW | 6.1 | 7.0 | 0.3 | 1.9 | 6.2 | 7.1 | 0.34 | 1.9
Hernq. | 13.5 | 6.5 | 0.21 | 2.4 | 13.3 | 6.3 | 0.35 | 2.2
pIso | 0.5 | 7.6 | 0.27 | 1.6 | 0.7 | 7.5 | 0.36 | 1.6
NGC720 | NFW | 46.8 | 7.3 | 0.34 | 7.2 | 45.5 | 7.3 | 0.38 | 7.0
Hernq. | 94.2 | 6.4 | 0.35 | 8.2 | 88.8 | 6.4 | 0.36 | 8.1
pIso | 5.9 | 6.5 | 0.34 | 7.3 | 5.6 | 6.5 | 0.35 | 6.8
Table A.2: Results of the best-fitting parameters for the Chandra data. We
show the results for both the prolate and oblate configurations for each mass
profile. The reduced-$\chi^{2}$ is also provided and all fits have 37 degrees
of freedom. XMM
---
|  | Prolate | Oblate
Galaxy | Profile |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$ |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$
IC4451 | NFW | 62.9 | 7.8 | 0.26 | 10.0 | 24.1 | 8.0 | 0.32 | 13.8
Hernq. | 125.8 | 7.1 | 0.22 | 11.0 | 49.2 | 7.1 | 0.25 | 11.2
pIso | 8.3 | 6.7 | 0.3 | 12.1 | 2.7 | 6.8 | 0.38 | 14.9
IC4956 | NFW | 35.4 | 8.0 | 0.17 | 0.8 | 12.5 | 7.6 | 0.19 | 0.9
Hernq. | 96.7 | 8.2 | 0.16 | 1.2 | 35.2 | 8.0 | 0.17 | 1.1
pIso | 8.5 | 7.4 | 0.28 | 2.5 | 2.2 | 6.9 | 0.19 | 1.0
NGC1521 | NFW | 42.7 | 7.6 | 0.26 | 4.1 | 21.2 | 8.0 | 0.4 | 14.8
Hernq. | 111.7 | 7.3 | 0.24 | 5.8 | 33.4 | 6.8 | 0.24 | 4.5
pIso | 7.4 | 6.8 | 0.32 | 5.7 | 2.9 | 6.8 | 0.44 | 15.4
NGC4125 | NFW | 54.6 | 7.0 | 0.44 | 1.4 | 27.3 | 7.3 | 0.52 | 1.4
Hernq. | 121.2 | 6.3 | 0.46 | 1.4 | 48.6 | 6.3 | 0.49 | 1.3
pIso | 9.5 | 6.2 | 0.37 | 1.6 | 3.9 | 6.2 | 0.51 | 1.4
NGC4555 | NFW | 22.8 | 6.9 | 0.33 | 6.9 | 8.3 | 6.9 | 0.32 | 8.6
Hernq. | 74.3 | 6.2 | 0.55 | 14.3 | 21.2 | 6.2 | 0.39 | 10.1
pIso | 1.8 | 7.5 | 0.29 | 2.7 | 0.6 | 7.7 | 0.32 | 4.1
NGC57 | NFW | 24.4 | 8.1 | 0.24 | 1.1 | 12.1 | 8.7 | 0.37 | 3.1
Hernq. | 48.0 | 7.2 | 0.18 | 1.0 | 19.6 | 7.3 | 0.21 | 1.0
pIso | 3.9 | 8.0 | 0.25 | 1.2 | 1.4 | 8.1 | 0.2 | 1.1
NGC6482 | NFW | 51.4 | 8.4 | 0.29 | 13.6 | 19.4 | 8.5 | 0.17 | 15.4
Hernq. | 93.0 | 7.4 | 0.08 | 13.6 | 40.0 | 7.6 | 0.2 | 13.9
pIso | 7.6 | 7.9 | 0.25 | 11.7 | 2.7 | 8.0 | 0.16 | 14.4
NGC7785 | NFW | 59.1 | 8.1 | 0.23 | 4.7 | 36.0 | 10.0 | 0.28 | 3.9
Hernq. | 120.3 | 7.4 | 0.21 | 4.0 | 76.2 | 9.8 | 0.2 | 2.6
pIso | 23.6 | 8.4 | 0.18 | 1.8 | 11.6 | 9.7 | 0.18 | 1.7
NGC7796 | NFW | 44.4 | 7.2 | 0.41 | 3.5 | 13.5 | 7.0 | 0.48 | 3.9
Hernq. | 96.9 | 6.5 | 0.42 | 4.3 | 28.9 | 6.2 | 0.38 | 3.5
pIso | 4.4 | 6.6 | 0.32 | 2.9 | 1.8 | 6.6 | 0.48 | 3.5
NGC953 | NFW | 27.4 | 7.7 | 0.22 | 2.9 | 12.5 | 7.9 | 0.28 | 4.8
Hernq. | 92.4 | 8.0 | 0.26 | 6.5 | 22.3 | 6.9 | 0.29 | 4.3
pIso | 4.3 | 7.5 | 0.27 | 3.6 | 1.6 | 7.5 | 0.27 | 4.5
NGC720 | NFW | 75.6 | 7.7 | 0.23 | 11.8 | 32.4 | 7.8 | 0.32 | 12.0
Hernq. | 143.1 | 6.7 | 0.2 | 13.8 | 53.4 | 6.6 | 0.25 | 13.3
pIso | 13.4 | 6.5 | 0.33 | 15.2 | 4.7 | 6.6 | 0.33 | 14.3
Table A.3: Results of the best-fitting parameters for the XMM data in the
range restricted scenario. We show the results for both the prolate and oblate
configurations for each mass profile.

Chandra
---
|  | Prolate | Oblate
Galaxy | Profile |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$ |  $a_{s}$ (”) | $\Gamma$ | $\epsilon$ | $\chi_{red}^{2}$
IC4451 | NFW | 12.1 | 6.6 | 0.43 | 4.3 | 11.5 | 6.5 | 0.47 | 4.5
Hernq. | 32.7 | 5.8 | 0.22 | 6.1 | 26.2 | 5.8 | 0.42 | 4.6
pIso | 1.0 | 6.9 | 0.36 | 3.3 | 1.1 | 6.7 | 0.45 | 3.3
NGC1521 | NFW | 19.8 | 7.0 | 0.37 | 2.8 | 18.3 | 6.9 | 0.4 | 3.7
Hernq. | 40.1 | 6.1 | 0.22 | 3.4 | 35.3 | 6.0 | 0.35 | 3.1
pIso | 2.5 | 6.6 | 0.26 | 2.4 | 1.9 | 6.6 | 0.37 | 3.2
NGC4125 | NFW | 41.2 | 7.0 | 0.52 | 5.4 | 39.4 | 7.0 | 0.56 | 5.4
Hernq. | 80.5 | 6.1 | 0.52 | 6.3 | 78.5 | 6.1 | 0.56 | 5.9
pIso | 5.5 | 6.1 | 0.5 | 6.4 | 4.9 | 6.1 | 0.58 | 5.5
NGC4555 | NFW | 8.0 | 7.8 | 0.12 | 2.9 | 8.3 | 8.0 | 0.33 | 2.3
Hernq. | 15.4 | 7.3 | 0.05 | 3.6 | 18.0 | 7.2 | 0.36 | 2.6
pIso | 0.7 | 8.7 | 0.29 | 1.4 | 0.8 | 8.8 | 0.28 | 1.7
NGC57 | NFW | 6.4 | 6.8 | 0.19 | 2.5 | 6.8 | 7.2 | 0.14 | 2.7
Hernq. | 26.9 | 8.2 | 0.22 | 2.2 | 15.7 | 6.7 | 0.36 | 1.2
pIso | 1.5 | 7.1 | 0.37 | 1.5 | 1.2 | 7.4 | 0.36 | 1.4
NGC6482 | NFW | 19.9 | 8.0 | 0.01 | 62.3 | 21.3 | 8.0 | 0.17 | 58.8
Hernq. | 44.7 | 7.2 | 0.09 | 77.8 | 44.8 | 7.2 | 0.16 | 75.7
pIso | 3.7 | 7.7 | 0.4 | 67.7 | 2.3 | 8.3 | 0.17 | 33.3
NGC7796 | NFW | 9.3 | 6.6 | 0.34 | 6.8 | 8.7 | 6.7 | 0.53 | 6.0
Hernq. | 16.0 | 6.1 | 0.05 | 11.8 | 20.0 | 6.0 | 0.54 | 7.0
pIso | 0.4 | 7.8 | 0.36 | 3.5 | 0.4 | 7.7 | 0.5 | 3.2
NGC953 | NFW | 6.4 | 6.9 | 0.24 | 3.0 | 6.7 | 7.3 | 0.13 | 3.0
Hernq. | 17.4 | 7.0 | 0.27 | 2.5 | 16.2 | 6.8 | 0.31 | 2.4
pIso | 1.0 | 7.9 | 0.3 | 2.0 | 0.8 | 7.8 | 0.35 | 2.0
NGC720 | NFW | 20.7 | 6.7 | 0.53 | 15.7 | 31.2 | 6.4 | 0.34 | 3.1
Hernq. | 60.7 | 5.9 | 0.41 | 5.6 | 65.3 | 5.8 | 0.3 | 3.5
pIso | 4.0 | 6.2 | 0.27 | 2.1 | 3.2 | 6.4 | 0.35 | 2.4
Table A.4:  Results of the best-fitting parameters for the Chandra data in the
range restricted scenario. We show the results for both the prolate and oblate
configurations for each mass profile.