# A formation mechanism for ‘Wrong Way’ Radio Relics

Ludwig M. Böss Universitäts-Sternwarte, Fakultät für Physik, Ludwig-
Maximilians-Universität München, Scheinerstr.1, 81679 München, Germany
Excellence Cluster ORIGINS, Boltzmannstr. 2, D-85748, Garching, Germany
Ulrich P. Steinwandel Center for Computational Astrophysics, Flatiron
Institute, 162 5th Avenue, New York, NY 10010  Klaus Dolag  Universitäts-
Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München,
Scheinerstr.1, 81679 München, Germany  Max Planck Institute for Astrophysics,
Karl-Schwarzschildstr. 1, D-85748, Garching, Germany

###### Abstract

Radio Relics are typically found to be arc-like regions of synchrotron
emission in the outskirts of merging clusters. They typically show synchrotron
spectra that steepen towards the cluster center, indicating that they are
caused by relativistic electrons being accelerated at outwards traveling
merger shocks. A number of radio relics break with this ideal picture and show
morphologies that are bent the opposite way and spectral index distributions
which do not follow expectations from the ideal picture. We propose that these
‘Wrong Way’ Relics can form when an outwards travelling shock wave is bent
inwards by an in-falling galaxy cluster or group. We test this in an ultra-
high resolution zoom-in simulation of a massive galaxy cluster with an on-the-
fly spectral Cosmic Ray model. This allows us to study not only the
synchrotron emission at colliding shocks, but also their synchrotron spectra
to adress the open question of relics with strongly varying spectral indices
over the relic surface.

††journal: ApJ††facilities:  The simulations were carried out at the CCA
clusters “rusty” located in New York City as well as the cluster “popeye”
located at the San Diego Supercomputing center (SDSC). Additional computations
were carried out at the c2pap cluster at the Leibnitz Rechenzentrum under the
project pn36ze.††software: Simulations are carried out with OpenGadget3
(Springel, 2005; Dolag & Stasyszyn, 2009; Beck et al., 2016b; Groth et al.,
2023). Post-processing was done with julia (Bezanson et al., 2014) based on
the packages that can be found here: https://github.com/LudwigBoess. The
analysis scripts to reproduce this work will be provided under
https://github.com/LudwigBoess/<ArXiV-ID> and simulation data will be shared
on reasonable request to the corresponding author.

##  1 Introduction

Radio Relics are up-to Mpc-size regions of radio emission in galaxy clusters,
typically with an arc-like morphology, which shows strong polarisation, steep
integrated radio spectra ($L_{\nu}\propto\nu^{\alpha}$, where
$\alpha\sim-1.1$) and a steepening of these spectra towards the cluster center
(see van Weeren et al., 2019, for a recent review). They are typically
associated with ongoing mergers between massive galaxy clusters (see e.g.
Ensslin et al., 1998; Roettiger et al., 1999; Enßlin & Brüggen, 2002; Brüggen
et al., 2012; Brunetti & Jones, 2014). These mergers dissipate a large
fraction of their potential energy in the form of shocks which heat the intra-
cluster medium (ICM) to $\sim 10^{8}$ K which can be observed as thermal X-ray
emission of the fully ionized plasma (e.g. Böhringer & Werner, 2010, for a
review). A smaller part of the shock energy is dissipated into the
acceleration of Cosmic Ray (CR) electrons and protons in a process called
“diffusive shock acceleration” (DSA, see e.g. Bell, 1978a, b; Blandford &
Ostriker, 1978; Drury, 1983, the latter for a review). In this process
(supra-)thermal particles cross a shock front and are scattered by MHD
turbulence from the downstream of the shock back into the upstream. They gain
energy at every crossing until their gyro-radii are large enough to escape
from the acceleration region or they are advected away in the downstream of
the shock. Hybrid and PIC plasma simulations of shock fronts show that this
process can efficiently accelerate protons in low-$\beta$ supernova shocks
(e.g. Caprioli & Spitkovsky, 2014; Caprioli et al., 2018, 2020; Pohl et al.,
2020, the latter for a review) and high-$\beta$ structure formation shocks
(e.g. Ryu et al., 2019). For electrons it is found that this process is harder
to trigger, as their gyro-radii are smaller at equivalent magnetic field
strength and with that it is more difficult to start a cyclical DSA process
and with that efficient acceleration of thermal electrons to the GeV energies
expected from synchrotron emission by radio relics. They require an efficient
pre-acceleration process such as (stochastic) shock-drift acceleration (SDA),
or a seed population stabilized against cooling, to efficiently part-take in a
DSA process (see e.g. Guo et al., 2014; Park et al., 2015; Kang et al., 2019;
Tran & Sironi, 2020; Kobzar et al., 2021; Amano & Hoshino, 2022; Tran et al.,
2023). On top of that the acceleration efficiency is found to be dependent on
the shock obliquity, the angle between shock propagation and magnetic field
vector. Typically it is found that protons are more efficiently accelerated at
quasi-parallel shocks (see e.g. Kang & Ryu, 2013; Caprioli & Spitkovsky, 2014;
Ryu et al., 2019), while electrons are more efficiently accelerated at quasi-
perpendicular shocks (e.g. Guo et al., 2014; Kang et al., 2019; Ha et al.,
2021; Amano & Hoshino, 2022). The results from small-scale plasma simulations
have been adopted in cosmological simulations to model emission originating
from structure formation shocks (see e.g. Hoeft et al., 2008; Pfrommer, 2008;
Pfrommer et al., 2007, 2008, 2017; Skillman et al., 2013; Vazza et al., 2012,
2016; Wittor et al., 2017; Banfi et al., 2020; Wittor, 2021). However the
efficiencies found in plasma simulations are not sufficient to explain the
high synchrotron brightness of radio relics (see Botteon et al., 2020, for a
recent discussion). Recent observations of radio relics show not only bright
arc-like structures, but also more complex morphologies such as S-shapes (de
Gasperin et al., 2022), flat relics with varying thickness (e.g. van Weeren et
al., 2016; Rajpurohit et al., 2020a, b) and filamentary stuctures (e.g.
Trasatti et al., 2015; Rajpurohit et al., 2022; Chibueze et al., 2023). There
is also a small set of radio relics whose morphological curvature points in
the ”wrong” direction, instead of the typical outward-bent (concave) shape of
the relic, away from the cluster center, they show an inward-bent (convex)
morphology. Examples of these relics can be found in the Ant Cluster
(PSZ2G145.92-12.53) (Botteon et al., 2021), Source D1 in Abell 3266 (Riseley
et al., 2022), SPT-CL J2023-5535 (HyeongHan et al., 2020), Abell 168
(Dwarakanath et al., 2018) and the southern relic in Ciza J2242.8+5301 (e.g.
van Weeren et al., 2010; Stroe et al., 2013, 2016; Hoang et al., 2017; Di
Gennaro et al., 2018). They can show steep synchrotron spectra which would
indicate Mach numbers of the underlying shocks that are in disagreement with
the critical Mach numbers found to be required to efficiently accelerate CR
electrons (e.g. Kang et al., 2019) and some of their spectra are better fit by
broken power-laws rather than a single one hinting towards overlapping
(re-)acceleration processes (e.g. Owen et al., 2014; Trasatti et al., 2015;
Parekh et al., 2022). The southern relic of Ciza J2242.8+5301 shows additional
strong variations of the synchrotron slope, which makes it hard to explain in
the context of DSA at a single shock front (see discussion in Di Gennaro et
al., 2018). Cosmological simulations of galaxy clusters show that mergers
between clusters are not isolated events and that merger shocks can deform as
they expand into highly complex and turbulent ICM (e.g. Hoeft et al., 2008;
Skillman et al., 2013; Wittor et al., 2017; Nuza et al., 2017). In this work
we propose that a possible formation mechanism for these ‘Wrong Way’ relics
(as they are referred to in Riseley et al., 2022) is the collision of an
outwards travelling shock front with an in-falling substructure. We
investigate this scenario in the sibling simulation of an ultra-high
resolution MHD simulation of a $M_{\mathrm{vir}}\approx 1.3\times
10^{15}M_{\odot}$ galaxy cluster introduced in Steinwandel et al. (2023),
where we attached a population of CR protons and electrons to every resolution
element of our simulation. This effectively turns every particle into a tracer
particle for CRs, while also accounting for feedback by the CR component on
the thermal gas. We resolve these populations with a spectral resolution of 12
bins for protons, and 48 bins for electrons over a range of 6 orders of
magnitude in momentum. The distribution function of the CRs is updated on-the-
fly at every timestep of the simulation according to the method presented in
Böss et al. (2023). This allows us to study CR electron injection at colliding
shocks and the subsequent cooling of the relativistic electron population. To
the best of our knowledge this simulation is the first of its kind. This work
is structured as follows: In Sec. 2 we describe the simulation code, CR model
and initial conditions used in this work. In Sec. 3 we study the ‘Wrong Way’
Relic (WWR) found in the simulation and its origin. Sec. 4 contains a
discussion of our findings and a comparison to observed systems. Finally Sec.
5 contains our conclusion and outlook to future work.

##  2 Methods

The simulation used in this work was carried out with the Tree-SPMHD code
OpenGadget3. OpenGadget3 is a derivative of Gadget2 (Springel, 2005) with
improvements to the hydro and gravity solvers as well as additional physics
modules. The SPH solver is updated as described in Beck et al. (2016b) to
include higher order kernels and their bias correction (see Dehnen & Aly,
2012) and artificial viscosity as well as artificial conduction to improve the
mixing behavior and shock capturing of SPH (e.g. Price, 2012; Hopkins, 2013;
Hu et al., 2014). Magnetohydrodynamics (MHD) have been implementation by Dolag
& Stasyszyn (2009) with updates to include non-ideal MHD in the form of
constant (physical) diffusion and dissipation presented in Bonafede et al.
(2011). Conduction is modelled via a conjugate gradient solver (Petkova &
Springel, 2009; Arth et al., 2014; Steinwandel et al., 2020), with a
suppression factor of the Spitzer value for conduction of 5 per cent. We adopt
a Wendland C4 kernel (Wendland, 1995, 2004) with 200 neighbors and bias
correction as suggested by Dehnen & Aly (2012).
We employ the on-the-fly spectral CR model Crescendo introduced in Böss et al.
(2023) to model the time evolution of CR protons and electrons in every
resolution element of our simulation. The time evolution of distributions of
CRs in the absence of CR transport, diffusion in momentum space and
catastrophic losses can be described by

$\displaystyle\frac{Df(p,\mathbf{x},t)}{Dt}$
$\displaystyle=\left(\frac{1}{3}\nabla\cdot\mathbf{u}\right)p\frac{\partial
f(p,\mathbf{x},t)}{\partial p}$ (1)
$\displaystyle+\frac{1}{p^{2}}\frac{\partial}{\partial
p}\left(p^{2}\sum_{l}b_{l}f(p,\mathbf{x},t)\right)$ (2)
$\displaystyle+j(\mathbf{x},p,t),$ (3)

where we used the convective derivative $\frac{Df}{Dt}=\frac{\partial
f}{\partial t}+\mathbf{u}\cdot\nabla f$ due OpenGadget3 being a Lagrangian
code. The right side of Eq. 1 describes changes due to adiabatic compression
or expansion of the gas the CRs are confined in, Eq. 2 describes energy losses
and Eq. 3 is the source term. We represent $f(p,\mathbf{x},t)$ as piece-wise
powerlaws in momentum space with 2 bins/dex for protons and 8 bins/dex for
electrons in the dimensionless momentum range
$\hat{p}\equiv\frac{p_{i}}{m_{i}c}\in[0.1,10^{5}]$, where $p_{i}$ and $m_{i}$
refer to the momentum and mass for protons and electrons, respectively. The
distribution function is updated at every timestep following the two-moment
approach as introduced in Miniati (2001) by computing CR number and energy
changes per bin. Adiabatic changes are accounted for at every timestep via the
density change within a SPH particle. We model energy losses of electrons due
to synchrotron emission and inverse Compton scattering off CMB photons. As a
source term we employ the DSA parametrisation by Kang & Ryu (2013) for the
dependency on sonic Mach number ($\eta(\mathcal{M}_{s})$), which allows for
DSA at shocks beyond a critical Mach number $\mathcal{M}_{s}>2$ and saturates
at a maximum efficiency of $\eta_{\mathrm{max}}\approx 0.2$. In addition to
that we use the model by Pais et al. (2018) for the dependency of CR
acceleration efficiency on shock obliquity ($\eta(\theta_{B})$). Ultimately we
divert a fraction

$\eta_{\mathrm{tot}}=\eta(\mathcal{M}_{s})\times\eta(\theta_{B})$ (4)

of the entropy change over the shock into the CR component. We detect the
shock properties on-the-fly in the simulation with the shock finder introduced
by Beck et al. (2016a) with improvements to compute the shock obliquity as the
angle between the pressure gradient within the kernel (which we treat as the
shock normal $\hat{\mathbf{n}}$) and the magnetic field vector upstream of the
shock $\mathbf{B}_{u}$. The slope of the injected CR spectrum follows linear
DSA theory and we use a fixed electron to proton injection ratio of
$K_{e/p}=0.01$. The CR component exerts feedback on the thermal gas by solving
the pressure integral

$P_{\mathrm{CR},c}=\frac{4\pi\>c}{3}\>a^{4}\>\int\limits_{p_{\mathrm{min}}}^{p_{\mathrm{cut}}}dp\>p^{3}f(p)$
(5)

between the minimum momentum $p_{\mathrm{min}}$ represented by the CR
population and the cutoff of the distribution function $p_{\mathrm{cut}}$. We
start the CR injection at $z=4$ to avoid too strong time-constraints due to
very efficient high-momentum energy losses of CR electrons.
We use a zoomed-in initial condition of a massive galaxy cluster with a virial
mass of $M_{\mathrm{vir}}\approx 1.3\times 10^{15}~{}M_{\odot}$ from the
sample presented in Bonafede et al. (2011). The cluster is up-sampled to 250x
base resolution, which corresponds to a mass resolution of
$M_{\mathrm{gas}}\approx 8.7\times 10^{5}M_{\odot}$ and
$M_{\mathrm{DM}}\approx 4.7\times 10^{6}M_{\odot}$ for gas and dark matter
particles, respectively. The cluster was selected from a lower-resolution dark
matter-only simulation of a Gpc volume, which is large enough to provide a
large sample of systems above a few $10^{15}M_{\odot}$. We reach a maximum
resolution for a gas particle of $h_{\mathrm{sml,min}}\approx 1$ kpc with a
gravitational softening of $\epsilon=0.48\>h^{-1}c$ kpc. The parent simulation
used a WMAP7 cosmology with $\Omega_{0}=0.24$, $\Omega_{\Lambda}=0.76$,
$\Omega_{\mathrm{baryon}}=0.04$, $h=0.72$ and $\sigma_{8}=0.8$, which we also
adopt for the present simulation. We start the simulation at redhsift $z=310$
and seed a constant magnetic field in x-direction with $B_{0}=$10-14 G (see
Steinwandel et al., 2021, for a study of the impact of the choice of $B_{0}$).
The initial conditions of this cluster at this resolution have been used to
study the interaction between internal- and accretion shocks in Zhang et al.
(2020a, b) and its magnetic field has been studied in Steinwandel et al.
(2023).

##  3 Results

###  3.1 Merger Geometry

Figure 1: A simplified schematic of the merger geometry that produces the
’Wrong Way’ relic. The initial merger between $M_{1}$ and $M_{2}$ drives two
shocks, the weaker of which is subsequently impacted by a third substructure
$M_{3}$. This impact deforms parts the outwards traveling shock $S_{1}$ and
produces the WWR $S_{3}$.

The ’Wrong Way’ relic in our simulation originates from a triple-merger at
$z\sim 0.35-0.2$. We show the schematic of the merger geometry in Fig. 1. A
high-velocity merger with a 1:10 mass ratio between impactor ($M_{2}\approx
10^{14}M_{\odot}$) and target ($M_{1}\approx 10^{15}M_{\odot}$) with a large
impact parameter of $b\approx 500$ kpc drives two shock waves. These shocks
follow the canonical picture (e.g. fig 7 in van Weeren et al., 2019) of the
lighter merging partner ($M_{2}$) driving a strong bow-shock ($S_{2}$ in our
schematic), while the heavier merging partner ($M_{1}$) drives a weaker
counter shock ($S_{1}$) in the in-fall direction of the lighter partner. This
counter shock is subsequently impacted by a third merger partner ($M_{3}$), a
group of galaxies with a total mass of $M_{3}\approx 2\times
10^{13}M_{\odot}$, which ultimately passes through the shock surface and falls
into the larger merger partner ($M_{1}$) in a low-impact parameter merger. The
impact of the group deforms the weaker counter shock ($S_{1}$) first from a
concave shape at $z=0.32$ to a convex shape at $z=0.29$ and subsequently to a
v-like shape pointing towards the cluster center at $z=0.27$, which also leads
to a complex superposition of the different parts original shock surface with
different mach numbers as well as differently aged cosmic ray electron
population.
Due to our system being a single, isolated cluster we can’t make any
predictions for the minimum critical mass of an in-falling sub-structure that
is able to deform such a shock front, or the statistical frequency of such an
event. We leave this question for future work with cosmological boxes, to
allow for a statistical analysis.

###  3.2 The Simulated ‘Wrong Way’ Radio Relic

Figure 2: From left to right: X-ray surface brightness, CR electron energy of
electrons with $E>1$ GeV, synchrotron surface brightness at 1.4 GHz and the
slope of the synchrotron spectrum between 144 MHz and 1.4 GHz. The upper three
rows show the time evolution of the in-falling group, the lowest row shows the
same relic at $z=0.27$ under a different projection. Figure 3: Histograms of
the sonic Mach number $\mathcal{M}_{s}$ for the three output times shown in
Fig. 2. The colors correspond to the different times and the dotted line
indicate the critical Mach number beyond which CR (re-)acceleration can occur
in our model. Figure 4: Time evolution of the synchrotron spectrum. Colors
correspond to the times in Fig. 2 and 3. Dashed lines and labels show the
spectral slope in the indicated frequency ranges.

Fig. 2 from top to bottom shows the time evolution of the counter shock
$S_{1}$ and its phasing through morphologies matching various ’Wrong Way’
relics in the $xz$-plane of the simulation. The bottom row shows the same
relic as the row above in the $yz$-plane. From left to right we show the X-ray
surface brightness, CR electron energy contained in the part of the
potentially synchrotron bright population with $E>1$ GeV, the synchrotron
surface brightness at 1.4 GHz and the slope of the synchrotron spectrum
obtained by fitting a powerlaw to the surface brightness at 144 MHz and 1.4
GHz. Accompanying to that we show the distribution of sonic Mach number
$\mathcal{M}_{s}$ of the different panels of Fig. 2 in Fig. 3 and the
synchrotron spectra in Fig. 4. Initially, in the top row of Fig. 2, we see the
acceleration of CR electrons at the counter shock of the main merger event.
Fig. 3 shows that only a fraction of the shocked particles are above the
critical Mach number $\mathcal{M}_{s,\mathrm{crit}}=2$ and with that can
accelerate CRs. We can readily identify the part of the shock surfce that
accelerates CRs in the center of the images, as it is the most synchrotron
bright part and shows a relatively flat synchrotron spectrum. The steeper
parts of the spectrum in the upper right corner of the images indicate that
these electrons have been accelerated at ealier times of the shock propagation
and have been freely cooling since. This is also evident in the synchrotron
spectrum in Fig. 4, which shows a strong break above $\nu\sim 200$ MHz. The
counter shock is initially not very synchrotron bright, akin to the counter-
shock in Abell 3667 (e.g. de Gasperin et al., 2022) or CIZA J2242.8+5301 (Di
Gennaro et al., 2018). The collision between outwards traveling counter shock
and the bow-shock of the in-falling group ($M_{3}$) increases the sonic Mach
number and with that the acceleration efficiency of the shock (see e.g. Kang,
2021; Inchingolo et al., 2022, for studies of multi-shock scenarios), which
significantly increases the synchrotron surface brightness at the contact
surface of the shocks. Fig. 3 shows that while the majority of the shocked
particles remain sub-critical, the shock develops a second Mach number peak
around Mach 3. The infalling sub-structure deforms the outwards traveling
shock towards a relic pointing ”the wrong way”, similar to the source D1
observed by Riseley et al. (2022). In the case of our relic the flat spectrum
part is further extended, which we attribute to the projection of the relic.
Here the surface of the collision between outwards and inwards traveling shock
is pointed roughly $45^{\circ}$ out of the image plane, while D1 seems to be
viewed almost perfectly edge-on. The collision between shock waves is also
visible in our X-ray image (left panel, second row in Fig. 2), which matches
the detection of a shock in X-ray by Sanders et al. (2022). The infall
scenario proposed here also produces a radio relic-like structure within
$r_{500}$, which is unlikely in the classical picture of radio relics (e.g.
Vazza et al., 2012). Another convex radio relic detected in PSZ2G145.92-12.53
similarly shows an increase in X-ray flux with convex morphology in close
proximity to the relic (see fig. 1 in Botteon et al., 2021). We note that
there is also a detected peak in X-ray surface brightness akin to the one
observed in the Rim region in PSZ2G145.92-12.53 (Botteon et al., 2021)
indicating that similar effects may be at play there, as briefly discussed by
the authors. As the in-falling halo passes through the outwards traveling
shock its own bow-shock collides with the older shock, causing the relic to
deform further into a v-shaped morphology, such as in the counter shock to the
sausage relic (e.g. Stroe et al., 2013; Di Gennaro et al., 2018), or the relic
in Abell 2256 (Rajpurohit et al., 2022). The Mach number distribution over the
shock surface has become smoother at this point, with the bulk of the shock
being sub-critical, however the total number of particles with
$\mathcal{M}_{s}>2$ has increased compared to the relic at $z=0.29$. This
leads to efficient acceleration at a part of the shock surface, visible in
increased synchrotron surface brightness and flatter synchrotron spectra. This
indicates that in the case of Abell 2256 the steep spectrum of the relic can
be explained in a scenario similar to the one here, where electrons
accelerated at an outwards traveling shock have already undergone cooling and
a collision with an infalling substructure adiabatically compresses the
population and some regions experience efficient re-acceleration by the re-
energized shock. Morphologically a best match for the relic in Abell 2256 is
expected to lie between $z=0.29-0.27$ shown here, however the simulation
output for this time is not available. For the lower panels of Fig. 2 we
rotate the image, as this projection more closely resembles the observations
of Di Gennaro et al. (2018). The collision of two shocks as shown here leads
to a superposition of multiple DSA-like events due to strong variations of the
Mach number over the shock surface. This leads to strong variations of
synchrotron surface brightness and spectral shape between the regions of the
shock surface where efficient (re-) acceleration can take place and the
regions that are dominated by cooling and adiabatic compression. These
variations can also be seen in the integrated spectrum in Fig. 4, where the
lower end of the spectrum is strongly injection dominated and the high
frequency end of the spectrum shows a significant steepening beyong $\nu\sim
1$ GHz in the cooling dominated part.

##  4 Discussion

To discuss our findings we will compare the morphologies in chronological
order to similar observed systems. Albeit the number of observed WWRs is still
small, the recent discoveries due to ASKAP and MeerKAT indicate that with
increased sensitivity a number of new WWRs can be detected over time.

###  4.1 Abell 520

Before the onset of the WWR morphology our cluster undergoes an internal
merger with an infalling group in the cluster periphery. This group falls into
the cluster at a similar trajectory as the cluster driving the current shock
waves and is therefore in the path of the weaker counter-shock of the ongoing
merger. A similar setup is observed in Abell 520 by Hoang et al. (2019). They
detect a shock with Mach number $\mathcal{M}_{\mathrm{SW}}=2.6_{-0.2}^{+0.3}$
propagating in SW direction with a weaker counter-shock moving with
$\mathcal{M}_{\mathrm{SW}}=2.1\pm 0.2$ in NE direction. Along the NE diagonal
Chandra observations by Andrade-Santos et al. (2017) indicate in-falling
matter along a similar path as the ongoing merger. This indicates that the
geometric setup is possible, albeit rare and Abell 520 could host a WWR in the
future.

###  4.2 Abell 3266

At $z=0.27-0.29$ or WWR resembles the one observed by Riseley et al. (2022),
at a distance of $\sim$ 1 Mpc from the center of Abell 3266. Their relic is
very faint and shows a very steep spectral index of $\alpha=2.76$ in the part
that is observable in frequencies above 1043 MHz. The lower frequency end of
the relic spectrum is significantly flatter, with a (for radio relics still
steep) spectral index of $\alpha\approx 1.4$. This indicates that there’s a
re-acceleration process which is superimposed on an older cooling spectrum,
even though the very steep spectrum still poses problems under this assumption
(see discussion in Riseley et al., 2022). Xray observations with eROSITA
(Sanders et al., 2022) show a number of discrete sources in close proximity to
D1, but no extended sources that could indicate an infalling group. The
extended sources $X4$ and $X6$ that lie in (projected) close proximity to D1
have significantly higher photometric redshifts ($z=0.532$ for $X4$ and
$z=0.907$ for $X6$) than Abell 3266 ($z=0.0589$, Struble & Rood, 1999), which
shows that these are background sources and not infalling groups. However, as
can be seen in the left panel of Fig. 2 at $z=0.27$, depending on the
projection it is not necessarily easy to distinguish the infalling structure
in the Xray emission.

###  4.3 Abell 2256

As previously discussed, at $z=0.27$ in the $xz$ plane our WWR closely
resembles the steep radio relic found in Abell 2256. Rajpurohit et al. (2022)
note an association between the relic and the source F, without an X-ray
counterpart (see also Owen et al., 2014; Ge et al., 2020). This could hint
towards the group having passed the shock before in a similar process as
discussed here. The superposition of injected and cooling parts of the shock
surface can also be seen in the color-color plots in Rajpurohit et al. (2022),
which indicate that the relic consists of a number of overlapping
substructures. The (re-)acceleration of particles in the turbulent downstream
of $S_{1}$, which becomes the upstream of $S_{3}$, also produces filamentary
structures seen in the relativistic electron component (second panels from
left in Fig. 2) as observed in Abell 2256 (see e.g. Domínguez-Fernández et
al., 2021; Wittor et al., 2023, for detailed studies of surface structures in
radio relics). The relic shows very little steepening, making it difficult to
discern if it was bent against its propagation direction. The little
steepening that is being detected however points towards the cluster center
akin to our simulated relic, which can indicate a similar process to the one
we discussed here.

###  4.4 CIZA J2242.8+5301

In the case of the counter shock to the sausage relic in CIZA J2242.8+5301 the
reason for the strong variations of synchrotron spectral index is still under
debate (see discussion in Di Gennaro et al., 2018). In the context of our
merger scenario these variations can be understood as follows: As the outwards
traveling shock ($S_{1}$) collides with the bow-shock of the infalling
substructre ($M_{3}$) and is deformed, the resulting shock surface ($S_{3}$)
shows strong variations in sonic Mach number. Wherever the sonic Mach number
is $\mathcal{M}_{s}<2$ our DSA model allows no CR (re-)acceleration and the
pre-existing population is simultaniously cooling due to synchrotron and IC
losses and being adiabatically compressed by the subcritical shock. This leads
to a continuously steepening synchrotron spectrum, while the adiabatic
compression leads to an increase in synchrotron surface brightness. In regions
of the shock surface where $\mathcal{M}_{s}>2$ there is ongoing
(re-)acceleration of CR electrons, which lead to a flatter spectrum than for
the cooled population. This superposition of cooling- and acceleration
dominated areas on the shock surface leads to a strong variation of
synchrotron spectral index.

##  5 Conclusion

In this work we showed the first results of a high-resolution simulation of a
massive galaxy cluster with an on-the-fly spectral Fokker-Planck solver to
study the acceleration, advection and aging of CR electrons in cosmological
zoom-in simulations. We applied this simulation to study a rare form of radio
relics that show convex, instead of the typical concave morphologies. Our
results can be summarized as follows:

  * •

In complex merging systems with multiple ongoing mergers collisions between
bow-shocks of infalling substructures and outwards traveling merger shocks can
deform the outwards traveling shocks in a way that is morphologically very
similar to the currently reported ’Wrong Way’ relics.

  * •

These collisions between shocks increase the Mach number at the contact
surface of the shocks and with that boost the (re-)acceleration efficiency of
CR electrons. This makes their detection easier than that of the cooled,
outwards moving shock.

  * •

The inclusion of an on-the-fly spectral treatment of CR electrons allows to
reproduce the large variance of synchrotron spectral slope across the relic
surface. This variance stems from the co-existance of an aged CR electron
population in the outwards traveling shock and newly injected CRs at the high
Mach number regions of the shock surface.

Future work will expand our sample size of radio relics by performing further
zoom-in simulations of the cluster set presented in Bonafede et al. (2011) at
250x base resolution and will study surface structure and polarisation
properties of these relics, as well as $\gamma$-ray emission by the
accelerated protons.

## Acknowledgments

We kindly thank Nadia Böss for the illustration of Fig. 1. LMB would like to
thank Bärbel Koribalski, Sebastian Nuza and Harald Lesch for helpful
discussion. LMB and KD are acknowledging support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germanys
Excellence Strategy – EXC-2094 – 390783311 and support for the COMPLEX project
from the European Research Council (ERC) under the European Union’s Horizon
2020 research and innovation program grant agreement ERC-2019-AdG 860744. UPS
is supported by the Simons Foundation through a Flatiron Research Fellowship
at the Center for Computational Astrophysics of the Flatiron Institute. The
Flatiron Institute is supported by the Simons Foundation.

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On leave of absence from: ]National Research Center “Kurchatov Institute”,
Petersburg Nuclear Physics Institute, St. Petersburg 188300, Russia

# Non-thermal photons and a Fermi-Dirac spectral distribution

Evgenii Ievlev<EMAIL_ADDRESS>[  Physics Department & Energetic
Cosmos Laboratory, Nazarbayev University,
Astana 010000, Qazaqstan  Almaty University of Power Engineering and
Telecommunications,
Almaty 050013, Qazaqstan  Michael R.R. Good<EMAIL_ADDRESS>Physics
Department & Energetic Cosmos Laboratory, Nazarbayev University,
Astana 010000, Qazaqstan  Leung Center for Cosmology and Particle
Astrophysics, National Taiwan University,
Taipei 10617, Taiwan

###### Abstract

Although non-intuitive, an accelerated electron along a particular trajectory
can be shown to emit classical electromagnetic radiation in the form of a
Fermi-Dirac spectral distribution when observed in a particular angular
regime. We investigate the relationship between the distribution, spectrum,
and particle count. The result for the moving point charge is classical, as it
accelerates along an exactly known trajectory. We map to the semi-classical
regime of the moving mirror model with a quantized spin-0 field. The scalars
also possess a $\beta$ Bogoliubov coefficient distribution with Fermi-Dirac
form in the respective frequency regime.

moving mirrors, black hole evaporation, acceleration radiation, Fermi-Dirac
statistics

###### pacs:

41.60.-m (Radiation by moving charges), 04.70.Dy (Quantum aspects of black
holes)

##  I Introduction

It is well-known that bosons obey Bose-Einstein statistics, and fermions obey
Fermi-Dirac statistics. Interestingly, Haro and Elizalde Haro and Elizalde
(2008) found a result for the $\beta$-Bogoluybov coefficient of a
semitransparent mirror demonstrating a flux of scalar particles obeying a
Fermi-Dirac distribution form in the large $\omega^{\prime}$ limit.
Nicolaevici Nicolaevici (2009) confirmed the Fermi-Dirac form with respect to
the energy $\omega$ but made special note that this does not establish the
number of particles since it only applies in the large $\omega^{\prime}$
limit. In a follow-up, Elizalde and Haro Elizalde and Haro (2010) recommended
further investigation into the Fermi-Dirac form and its relationship to the
sign in the $\beta$-Bogoliubov coefficient; in particular, the connection with
the number of particles emitted per mode.

Here we investigate the situation using an ordinary moving point charge in
classical electrodynamics Jackson (1999). We demonstrate the phenomenon
without appealing to quantum field theory; i.e., one does not need to use
moving mirrors or semi-transparency to understand the situation. Nevertheless,
we find the perfectly reflecting accelerating boundary corresponding to the
moving point charge and examine its spectral statistics for clarity.

The functional mapping between moving mirrors and moving point charges Ford
and Vilenkin (1982); Unruh and Wald (1982); Ritus (2003, 2002, 1998); Nikishov
and Ritus (1995); Zhakenuly  _et al._ (2021); Ritus (2022) is leveraged to
understand the problem in both contexts. The situation has different physical
meanings when examined for the classical electromagnetic field or quantized
scalar field Ievlev and Good (2023); Ievlev  _et al._ (2023); and thus,
different implications for the classical radiation in ordinary 3+1 dimensions
and quantum radiation in 1+1 dimensions.

We use natural units, setting $\hbar=c=k_{B}=\mu_{0}=1$; the electron’s charge
is then a dimensionless number $e^{2}=4\pi\alpha_{\textrm{fs}}\approx 0.092$.

##  II Fermi-Dirac Trajectory

###  II.1 Dynamics and total energy

Let us start with a simple illustration of the situation. Consider an electron
moving in a straight line along the $z$-axis. We take the trajectory defined
implicitly as

$t(z)=\frac{\kappa}{4}z^{2}+\frac{2}{\kappa}\ln(\kappa z)+z\zeta,$ (1)

where $\kappa>0$ is the acceleration scale and $-1<\zeta<1$. The inverse
velocity along the trajectory and the maximum velocity are, respectively,

$\frac{1}{v}=\frac{\mathop{}\\!\mathrm{d}t(z)}{\mathop{}\\!\mathrm{d}z}=\frac{\kappa
z}{2}+\frac{2}{\kappa z}+\zeta\,,\quad v_{\text{max}}=\frac{1}{2+\zeta}\,.$
(2)

From Eq. (2), it is evident that for $\zeta>-1$ this trajectory travels along
a time-like, relativistic worldline. This trajectory is asymptotically static;
see Fig. 1 for a spacetime diagram and Fig. 2 for a Penrose diagram.

Figure 1: The position $z(t)$ trajectories, plotted to demonstrate various
$\zeta$ values of Eq. (1). The key takeaway is the motions are asymptotic
resting and are limited to the half-plane $z>0$.  Figure 2: The position
$z(t)$ trajectories, plotted in a Penrose diagram to demonstrate the
$\zeta=(-1,0,+1)$ values of Eq. (1). The key takeaway is the motions are
asymptotic resting and are limited to the half-plane $z>0$. The same color
scheme of Fig. 1 is used; here, $\kappa=2$ for illustration.

The total energy emitted can be calculated with the Larmor formula; this
energy is finite for $\zeta>-1$. For example, when $\zeta=0$ it takes the
analytic form:

$E=\frac{e^{2}\kappa}{36}\left(\frac{1}{3\sqrt{3}}-\frac{1}{4\pi}\right).$ (3)

For other values of the parameter $\zeta$, the analytic formula for the total
energy exists but is complicated; nevertheless, it is simple to illustrate
numerically, see Fig. 3.

Figure 3: Total radiated energy $E/e^{2}\kappa$ as a function of parameter
$\zeta$. The energy blows up as $\zeta\to-1$; as this corresponds to the speed
of light for the trajectory’s maximum velocity, e.g. see the red worldline in
Fig. 1 and Fig. 2.

One can use the total energy, e.g. Eq. (3), to check the consistency of the
spectral results (see the Appendix for detail).

###  II.2 Spectral distribution of the accelerating electron’s radiation

To find the spectral distribution for Eq. (1), we use the standard approach in
classical electrodynamics Jackson (1999). Here the energy $E$ can be found by
the spectrum $I(\omega)$, or spectral distribution
$\mathop{}\\!\mathrm{d}{I}/\mathop{}\\!\mathrm{d}{\Omega}$ Schwinger (1949),

$E=\int\mathop{}\\!\mathrm{d}{\omega}I(\omega)=\int\mathop{}\\!\mathrm{d}{\omega}\int\mathop{}\\!\mathrm{d}{\Omega}\frac{\mathop{}\\!\mathrm{d}{I(\omega)}}{\mathop{}\\!\mathrm{d}{\Omega}}.$
(4)

For example, by the use of Eq. 13 in Ievlev and Good (2023),

$\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}=\frac{e^{2}\omega^{2}}{16\pi^{3}}\left|\sin\theta\int\displaylimits_{0}^{\infty}\mathop{}\\!\mathrm{d}ze^{i\phi(z)}\right|^{2},$
(5)

where $\phi=\omega(t-z\cos\theta)$, we write

$\frac{1}{\sin^{2}\theta}\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}=\frac{e^{2}\omega^{2}}{16\pi^{3}}\left|\int\displaylimits_{0}^{\infty}\mathop{}\\!\mathrm{d}ze^{i\phi(z)}\right|^{2}.$
(6)

This integral can be solved exactly as is (see the Appendix), but for
simplicity, consider the spectral distribution at a particular angle
$\theta_{0}$ instead. Specialize to $\theta\to\theta_{0}$ such that the phase
is

$\phi(z)=\frac{\kappa\omega}{4}z^{2}+\frac{2}{\kappa}\omega\ln z+\omega
z(\zeta-\cos\theta_{0}).$ (7)

It is now straightforward to integrate Eq. (6) when $\cos\theta_{0}=\zeta$.
One obtains the spectral distribution,

$\frac{1}{\sin^{2}\theta_{0}}\left.\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}\right|_{\theta_{0}}=\frac{e^{2}}{8\pi^{2}}\frac{\omega/\kappa}{e^{2\pi\omega/\kappa}+1},$
(8)

where the spectral frequency content of the radiation has a Fermi-Dirac form.
Before examining this form and its relationship to its particle spectrum
$N(\omega)$, let us first look at its quantum dual in the moving mirror model
DeWitt (1975); Fulling and Davies (1976); Davies and Fulling (1977), in the
spirit of previous moving mirror studies on the Fermi-Dirac result Haro and
Elizalde (2008); Nicolaevici (2009); Elizalde and Haro (2010).

###  II.3 Corresponding Bogolubov Coefficients

While the above result is classical radiation from the 3+1 dimensional
electromagnetic field, we can investigate the quantum radiation from the 1+1
dimensional scalar field of the moving mirror model. The mapping recipe Ievlev
and Good (2023) between electron and mirror links the spectral distribution on
the electron side and the Bogolubov coefficient squared on the mirror side,
c.f. Ievlev  _et al._ (2023):

$\displaystyle\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}(\omega,\cos\theta)=\frac{e^{2}\omega^{2}}{4\pi}|\beta_{pq}|^{2},$
(9) $\displaystyle p+q=\omega\,,\quad p-q=\omega\cos\theta.$

Using the full spectral distribution from Eq. (19) one can use this recipe to
obtain the corresponding Bogolubov coefficients. Let us however consider
setting $\theta=\theta_{0}$, $\zeta=\cos\theta_{0}$. In this case, the
electron’s spectral distribution has a simple form Eq. (8). In terms of the
scalar frequencies $p,q$, the corresponding condition reads

$\theta=\theta_{0}\,,\ \zeta=\cos\theta_{0}\Longleftrightarrow
p=\omega\frac{1+\zeta}{2}\,,\ q=\omega\frac{1-\zeta}{2}$ (10)

So, the scalar frequencies $p$ and $q$ are not independent, but related to
each other through this condition. The recipe Eq. (9) gives particular beta
Bogolubov coefficients:

$|\beta_{pq}|^{2}=\frac{1-\zeta^{2}}{2\pi(p+q)\kappa}\frac{1}{e^{2\pi(p+q)/\kappa}+1}\,,\quad\frac{p}{q}\equiv\frac{1+\zeta}{1-\zeta}\,.$
(11)

This result demonstrates the Fermi-Dirac form for the $\beta$-Bogolubov
coefficients of the quantum scalars.

By tuning the value of $\zeta$ one can obtain a trajectory that gives the
Fermi-Dirac at any pre-assigned angle (or a bespoke frequency regime). For
instance, the ‘high-frequency’ regime Hawking (1975), $p\sim 0$ ($q\gg p$),
corresponds to $\zeta\sim-1$; Eq. (11) becomes to leading order

$|\beta_{pq}|^{2}=\frac{1+\zeta}{\pi q\kappa}\frac{1}{e^{2\pi q/\kappa}+1}\,.$
(12)

Using the duality to map back to the electron, the choice
$\zeta=\cos\theta_{0}=-1$ corresponds to a viewpoint behind the accelerating
electron $\theta_{0}\sim\pi$.

###  II.4 Connection to Particle Count

The notion of discrete radiation energy $\hbar\omega$ allows an introduction
of a particle spectrum $N(\omega)$. The connection between the spectral
distribution of electromagnetic waves and the particle spectrum is Jackson
(1999):

$N(\omega)=\frac{1}{\omega}I(\omega)=\frac{1}{\omega}\int
d\Omega\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}},$ (13)

which must be consistent with the total energy emission as computed by the
spectral distribution,

$E=\int\mathop{}\\!\mathrm{d}{\omega}\;\omega
N(\omega)=\int\mathop{}\\!\mathrm{d}{\omega}\int\mathop{}\\!\mathrm{d}{\Omega}\frac{\mathop{}\\!\mathrm{d}{I}}{\mathop{}\\!\mathrm{d}{\Omega}}.$
(14)

Therefore, the Fermi-Dirac distribution does not correspond to the particle
spectrum $N(\omega)$, or the energy spectrum $I(\omega)$. It does not even
correspond to the spectral distribution
$\mathop{}\\!\mathrm{d}{I}/{\mathop{}\\!\mathrm{d}{\Omega}}$ at an arbitrary
observation angle $\theta$; but only in a specific angular regime
$\theta\to\theta_{0}$:
$\left.\mathop{}\\!\mathrm{d}{I}/{\mathop{}\\!\mathrm{d}{\Omega}}\right|_{\theta_{0}}$
using the corresponding trajectory, Eq. (1). Nevertheless, the interesting
question remains if it is possible to observe such radiation measured in such
a specified angular regime $\theta_{0}$. Does an observer see Fermionic
electromagnetic radiation? Is the spectral content congruent with Fermi-Dirac
statistics?

We stress again that the trajectory is easily generalized in a number of
different ways to illustrate the robust Fermi-Dirac form of the spectral
distribution at particular angles. For instance, a particular choice of
$\zeta$ in Eq. (1) results in a new trajectory form, capable of a new
observation angle $\theta_{0}$, which gives the Fermi-Dirac result. This
means, depending on the particular bespoke trajectory of interest, the
relevant zeta-angle $(\zeta,\theta_{0})$ could be in any desired direction,
such as to the side $(0,\pi/2)$, in front $(+1,0)$, or behind $(-1,\pi)$ the
accelerating electron.

##  III Discussion

The particular physics of this result depends in subtle ways on dimension (3+1
vs 1+1), source (electron vs. mirror), and regime (angle vs. frequency). Two
other notable examples in the literature confirm this, namely, scalar charges
and even-odd dimensional dependence. Let us consider these examples now.

For an example of a source other than an electric charge or mirror, Nikishov
and Ritus found Nikishov and Ritus (1995) that in the case of a scalar charge,
the emitted scalar radiation along a particular trajectory will obey Fermi-
Dirac statistics; in contrast to an electric charge following the same
particular trajectory whose spin-1 radiation field obeys Bose-Einstein
statistics.

As an example of dimensional dependence, scalar field radiation measured by a
uniformly accelerated DeWitt detector obeys Bose-Einstein statistics when the
dimension of the spacetime is even, but when the dimension is odd, one obtains
Fermi-Dirac statistics Takagi (1986).

Taken as a whole, it is clear the subtleties involved make it especially
important to precisely define the context and the regime of applicability and
explicitly examine the form of the computed observables.

We note that Hawking radiation Hawking (1975) and its Schwarzschild moving
mirror analog Good  _et al._ (2016) utilize the high-frequency regime lending
support to the notion of thermality and Bose-Einstein distributed scalars at
late-times. However, in the Schwarzschild case (as opposed to extremal cases
Good (2020) or asymptotically inertial situations Good  _et al._ (2020)),
there is no finite total energy check corresponding to the Bogoliubov
coefficients. Moreover, the total particle count is infinite. This is
ultimately due in part to the horizon; and in the analog moving mirror
situation, the fact that the proper acceleration is asymptotically infinite.
The same goes for the eternal black hole analog of Carlitz-Willey Carlitz and
Willey (1987).

In this work, we have shown that the scalars can possess Fermi-Dirac
distributed $\beta$-Bogliubov coefficients. If one considers
$\beta$-Bogoliubov coefficients sufficient evidence of a thermal Bose-Einstein
distribution for the Schwarzschild or Carlitz-Willey trajectories; then the
result Eq. (12) is also sufficient evidence of a thermal Fermi-Dirac
distribution for the trajectories Eq. (1). Moreover, we have an additional
check of total finite energy and finite particle emission (see the Appendix
for more detail).

This result demonstrates that, although the high-frequency approximation (or
the low-frequency approximation, if one prefers) is frequently used in the
literature, one should clearly understand whether this approximation
represents the physical system under consideration. The result Eq. (12) does
not reveal e.g. the particle spectrum, as the high-frequency region does not
dominate the corresponding contribution from the beta Bogolubov coefficients.
In other words, one should be careful when applying the high-frequency (or the
low-frequency) approximation; in each case, this approximation should be well-
motivated; otherwise, peculiar results may arise.

##  IV Conclusion

We have shown that moving point charge radiation can possess a Fermi-Dirac
spectral distribution form. A particular trajectory and corresponding angular
regime demonstrate the result. By appealing to classical electrodynamics, we
have analyzed the physical reason for the resulting unexpected spectral-
statistics form.

The spectral-statistics (as explicitly derived from the spectral distribution
in a particular angular regime for the radiation from a moving point charge)
do not necessarily characterize the spin-statistics of the electromagnetic
field in question. Instead, they depend crucially on the observation angle and
the specific electron trajectory interaction with the radiation field.

##  V Acknowledgements

Funding comes in part from the FY2021-SGP-1-STMM Faculty Development
Competitive Research Grant No. 021220FD3951 at Nazarbayev University.

##  Appendix A Partial contribution from the FD particles

The partial energy contribution when $\zeta=\cos\theta_{0}$ can be found from
the Fermi-Dirac form Eq. (8) of the electron,

$E^{fd}_{\textrm{electron}}=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{\omega}\int_{0}^{2\pi}\mathop{}\\!\mathrm{d}{\varphi}\left.\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}\right|_{\theta_{0}}=\frac{e^{2}\kappa(1-\zeta^{2})}{192\pi}$
(15)

The corresponding contribution from the mirror can be derived from this result
in the following way. We insert into Eq. (15) the integral
$1=\int_{-1}^{1}\mathop{}\\!\mathrm{d}(\cos\theta)\,\delta(\cos\theta-\zeta)$
and change the variables according to Eq. (9). The Jacobian is $2/(p+q)$, and
the resulting contribution is

$\displaystyle E^{fd}_{\textrm{mirror}}$ (16)
$\displaystyle=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{p}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{q}\;\delta\left(\frac{p-q}{p+q}-\zeta\right)\,(p+q)|\beta_{pq}|^{2}$
$\displaystyle=\frac{\kappa(1-\zeta^{2})}{192\pi}.$

This gives the partial contribution to the energy emitted to both sides of the
corresponding mirror, counting only the FD particles.

Let us look at how the analogy extends to finite particle count. For the
moving mirror, the total number of scalars emitted to the right side of the
mirror is:

$\displaystyle N_{\textrm{mirror}}$ (17)
$\displaystyle=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{p}\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{q}\;\delta\left(\frac{p-q}{p+q}-\zeta\right)\,|\beta_{pq}|^{2}$
$\displaystyle=\frac{(1-\zeta^{2})\ln 2}{8\pi^{2}}\,.$

For the case of the accelerating electron, we may integrate over all
frequencies $\omega$ on Eq. (13) which gives

$\displaystyle\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{\omega}\;N(\omega)$
$\displaystyle=\int_{0}^{\infty}\frac{\mathop{}\\!\mathrm{d}{\omega}}{\omega}\int_{0}^{2\pi}\mathop{}\\!\mathrm{d}{\psi}\left.\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}\right|_{\theta_{0}},$
(18) $\displaystyle=e^{2}\frac{(1-\zeta^{2})\ln 2}{8\pi^{2}},$

in agreement with Eq. (17). This highlights the dual consistency between the
mirror and electron but also demonstrates the advantage of a finite energy and
finite particle count with an exact analytic solution, Eq. (1).

##  Appendix B Exact spectrum

To demonstrate the difference between a particular choice of observation angle
and the spectrum $I(\omega)$ which results from integration over the solid
angle, we briefly look at an exact answer for the integral of Eq. (6), setting
$\zeta=0$ and leaving $\theta$ unset. Integrating Eq. (6) gives

$\displaystyle\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}=\frac{e^{2}\omega\sin^{2}\theta}{16\pi^{3}\kappa}e^{-\frac{\pi\omega}{\kappa}}\times$
(19)
$\displaystyle\times\left|\Gamma\left(\frac{1}{2}-\frac{i\omega}{\kappa}\right)A+2\cos\theta\sqrt{\frac{i\omega}{\kappa}}\Gamma\left(1-\frac{i\omega}{\kappa}\right)B\right|^{2}\,,$

where

$\displaystyle A$
$\displaystyle=\,_{1}F_{1}\left(\frac{1}{2}-\frac{i\omega}{\kappa};\frac{1}{2};\frac{i\omega\cos^{2}\theta}{\kappa}\right)\,,$
(20) $\displaystyle B$
$\displaystyle=\,_{1}F_{1}\left(1-\frac{i\omega}{\kappa};\frac{3}{2};\frac{i\omega\cos^{2}\theta}{\kappa}\right)\,.$

This spectral distribution, Eq. (19), gives the total energy, Eq. (3) by
numerical integration,

$\displaystyle E$
$\displaystyle=\int_{0}^{\infty}\mathop{}\\!\mathrm{d}{\omega}\int_{0}^{2\pi}\mathop{}\\!\mathrm{d}{\phi}\int_{0}^{\pi}\mathop{}\\!\mathrm{d}{\theta}\sin\theta\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}$
(21)
$\displaystyle=\frac{e^{2}\kappa}{36}\left(\frac{1}{3\sqrt{3}}-\frac{1}{4\pi}\right)\,.$

We cannot integrate Eq. (19) exactly over the solid angle to obtain an
analytic form of $I(\omega)$. However, it is clear that Eq. (19) will not
result in a Fermi-Dirac form for the spectrum $I(\omega)$, even though at
$\theta_{0}=\pi/2$, Eq. (19) gives

$\left.\frac{\mathop{}\\!\mathrm{d}I(\omega)}{\mathop{}\\!\mathrm{d}\Omega}\right|_{\theta_{0}}=\frac{e^{2}}{8\pi^{2}}\frac{\omega/\kappa}{e^{2\pi\omega/\kappa}+1},$
(22)

which is the result Eq. (8). For this reason, one cannot say the particle
count, $N(\omega)$ rests in a Fermi-Dirac distribution, which is ultimately
consistent with the horizonless globally defined motion, Eq. (1), evolving to
an asymptotic stop.

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