# Enhancing Path Selections with Interference Graphs in Multihop Relay
Wireless Networks

Cao Vien Phung, Andre Drummond, Admela Jukan  Technische Universität
Braunschweig, Germany
Email: {c.phung, andre.drummond<EMAIL_ADDRESS>
###### Abstract

The multihop relay wireless networks have gained traction due to the emergence
of Reconfigurable Intelligent Surfaces (RISs) which can be used as relays in
high frequency range wireless network, including THz or mmWave. To select
paths in these networks, the transmission performance plays the key network in
these networks. In this paper, we enhance and greatly simplify the path
selection in multihop relay RIS-enabled wireless networks with what we refer
to as interference graphs. Interference graphs are created based on SNR model,
conical and cylindrical beam shapes in the transmission and the related
interference model. Once created, they can be simply and efficiently used to
select valid paths, without overestimation of the effect of interference. The
results show that decreased ordering of conflict selections in the graphs
yields the best results, as compared to conservative approach that tolerates
no interference.

###### Index Terms:

Interference, mesh networks, Reconfigurable Intelligent Surface (RIS),
transmission scheduling.

##  I Introduction

Today, wireless communications within the mmWave/sub-Terahertz frequency bands
are evolving into relay wireless systems, due to the emergence of
Reconfigurable Intelligent Surfaces (RISs) [1] which can be deployed as
relays. RISs alone are not designed as relays, and are often used passively.
On the other hand, under the conditions of far-field path-losses, an achieved
channel gain of a RIS does not suffice [2]. Therefore, Relay Nodes (RNs) can
be deployed in combination with RIS to overcome the transmission impairments
[2], i.e., acting as a repeater of the transmission signals traversing RISs.

Since the high-frequency transmission range is restricted, frequency reuse is
inevitable which creates interference [3]. We find that since mmWave/sub-THz
typically uses shorter paths, and due to transmission impairments, the
interference detection alone is not determinant of the path selected, but it
also need to consider the effects of beam modelling, SNR computation, and
spacial geometry of beams, etc. This implies that the path selected can be
valid even in presence of interference [4]. Thus, we are interested in
enhancing path computation by considering the effect of interference
overestimation, comprehensively taking into consideration the quality of
transmission (QoT), which has not been studied yet.

In this paper, we propose to enhance path selection with what we refer to as
interference graphs, which are created in consideration of: (a) SNR model; (b)
Transmission beam model; and (c) Interference model. We then use the
interference graphs to enhance the space of path selection solutions, by
finding valid paths that also guarantee the network overall QoT. We implement
four algorithms based on interference graphs, and provide a comparative
analysis, i.e., (i) zero interference mapping (ZIM), (ii) interference mapping
by random conflict selection (RCS), (iii) interference mapping by decreasing-
ordered conflict selection (DCS), and (iv) interference mapping by increasing-
ordered conflict selection (ICS). We show that the proposed algorithms are
feasible, and can be a valid practical solutions. This is especially critical
due to the complexity of mmWave/sub-THz systems, where an optimal solution may
neither be trivial nor feasible. The results show that zero interference
mapping is rather inefficient while the best performance is achieved by the
ICS and we show its suitability for high-speed THz systems.

The rest of this paper is organized as follows. Section II presents the
analytical model. Section III proposes the interference graphs for path
selection. Section IV evaluates the performance numerically. Section V
concludes the paper.

##  II Multihop Relay Wireless Networks Modelling

In this section, we present a reference network as well the basic models of
SNR, transmission beam and interference, which are later used as input to the
creation of the interference graphs. While these models are here based on [5],
it should be noted that any SNR, beam shape or interference model can be used,
instead, or replaced herewith, such as [6, 7, 3].

###  II-A Reference scenario

Figure 1: Reference network.

The reference network is shown in Fig. 1. It includes $B$ base stations BSs
(as source devices), $R$ passive RISs with $N$ reflecting elements (as
intermediate reflecting devices, where $N$ can be up to thousands of
reflecting elements [8]), $E$ RNs (as intermediate half-duplex repeaters used
to improve SNR), and $U$ UEs (as destination devices). We assume that BSs
always act as transmitters, while UEs as receivers. RISs and RNs on the other
hand can both transmit and receive data. In this paper, the path finding
method defines valid paths for every transmission pair as the shortest path
considering the total distance covered by the transmissionand with the
sufficient quality of transmission. Moreover, we use relay nodes if $SNR\leq
T$ at UEs, where $T$ is the SNR threshold calculated based on the channel
model. For instance, in Fig. 1, the path between (BS $0$ $\rightarrow$ UE
$0$), BS $0$ $\rightarrow$ RN $0$ $\rightarrow$ RIS $4$ $\rightarrow$ UE $0$,
requires RN $0$ because the shorter path BS $0$ $\rightarrow$ RIS $4$
$\rightarrow$ UE $0$ would lead to an $SNR\leq T$ at UE $0$.

###  II-B SNR model

The $SNR_{(be,eu)}$ between one BS $b$/RN $e$ and one RN $e$/UE $u$ via $I$
RISs is given by:

$SNR_{(be,eu)}=\frac{P_{eu}\cdot G_{be}\cdot G_{eu}}{k\cdot\tau\cdot W},$ (1)

where $k$: Boltzmann constant, $\tau$: absolute temperature, and $W$:
bandwidth. $P_{eu}$ is the signal power received at the receiving node. As
illustrated in Fig. 2, $G_{be}$ with beamwidth $\alpha$ is the antenna gain of
BS $b$/RN $e$ (e.g.,, BS $0$ towards RIS $1$), i.e.,

$G_{be}=\frac{2}{1-cos(\frac{\alpha}{2})}.$ (2)

$G_{eu}$ is the antenna gain of RN $e$/UE $u$ ($G_{eu}=G_{be}$). The signal
power $P_{eu}$ at RN $e$/UE $u$ (for $I>1$) is calculated as:

$\begin{split}&P_{eu}=P_{be}\left|(H_{(r_{I},eu)}\cdot
N^{\prime})\times\right.\\\
&\left.\left(\prod_{i=2}^{I-1}H_{(r_{i},r_{i+1})}\cdot
N^{\prime}\right)\times(H_{(be,r_{1})}\cdot H_{(r_{1},r_{2})}\cdot
N^{\prime})\right|^{2},\end{split}$ (3)

where $P_{be}$ denotes the power of BS $b$/RN e; $N^{\prime}$ represents the
number of illuminated RIS reflecting elements (see $N^{\prime}$ from the
illuminated area of RIS $1$ and RIS $2$ in Fig. 2: $N^{\prime}\leq N$), as
analyzed later in Eq. (11); $H$ is the channel transfer function, e.g.,
$H_{(be,r_{1})}$ between BS $b$/RN $e$ and the first RIS of any transmission
between BS $b$/RN $e$ and RN $e$/UE $u$ (this first RIS is denoted as $r_{1}$,
e.g., RIS $1$ is the first RIS of the transmission BS $0$ $\rightarrow$ RIS
$1$ $\rightarrow$ RIS $2$ $\rightarrow$ UE $0$ in Fig. 2), calculated as:

$\begin{split}H=\left(\frac{c}{4\cdot\pi\cdot f\cdot
d}\right)e^{(-\frac{1}{2}k(f)\cdot d)},\end{split}$ (4)

where $c$ is the light speed, $\pi$ is Pi constant, $f$ denotes mmWave/sub-THz
frequency, $d$ represents the transmission distance between two devices, and
$k(f)$ denotes the overall molecular absorption coefficients. The signal power
at RN $e$/UE $u$ in case of $I=1$ is given by:

$\begin{split}P_{eu}=P_{be}\cdot|H_{(be,r_{1})}\cdot N^{\prime}\cdot
H_{(r_{1},eu)}|^{2}.\end{split}$ (5)

Figure 2: An example of three-hop transmission via two RISs.

The signal power at RN $e$/UE $u$ for $I=0$ (no RIS) is:

$P_{eu}=P_{be}|H_{(be,eu)}|^{2}.$ (6)

###  II-C Transmission beam model

Figure 3: Beam shapes: Conical beam from BS and cylindrical beam from RIS.
Figure 4: An example of interference analysis.

We assume that BS $b$/RN $e$ emits signals with conical beams, whereas the
illuminated RIS area is circular, while RIS reflects signals with cylindrical
beams (see Fig. 3). For circular illuminated RIS areas, we assume that they
maintain the same radius $\phi_{IRA}$ between RISs, e.g., RIS $3$ and RIS $1$
of the transmission path BS $2$ $\rightarrow$ RIS $3$ $\rightarrow$ RIS $1$
$\rightarrow$ UE $2$ have the same radius of actual illuminated areas. At the
same time, the actual size of the illuminated area on RIS depends solely on
the distance between the BS $2$ and the next RIS $3$ due to the conical beam
assumptions on base stations. As previously mentioned, the beams from RIS $1$
and RIS $2$ are cylindrical. Thus, the actual illuminated radius does not
change along the path. The footprint radius of conical beam can be given by:

$\phi_{fp}=tan\left(\frac{\alpha}{2}\right)\cdot d_{1},$ (7)

where $d_{1}$ denotes transmission distance of the first hop between BS $e$/RN
$e$ and the first RIS $r_{1}$ (see $d_{1}$, $\alpha$, and $\phi_{fp}$ in Fig.
3). The footprint area of conical beam is expressed:

$S_{fp}\approx\pi\cdot\phi_{fp}^{2}.$ (8)

Let $S_{RIS}$ and $\phi_{RIS}$ be the area and radius of RIS, respectively.
The actual illuminated RIS area can be given by:

$S=min(S_{fp},S_{RIS}),$ (9)

and the actual radius of illuminated area on RIS is expressed:

$\phi_{IRA}=min(\phi_{fp},\phi_{RIS}).$ (10)

The number of actual illuminated elements on RIS is given:

$N^{\prime}=\frac{S}{dx\cdot dy}\leq N,$ (11)

where $dx$ ($dy$) is the $x$ ($y$) dimension of RIS reflecting element. The
conical and cylindrical beam coverage is calculated by:

$\displaystyle V=$ $\displaystyle\frac{1}{3}\cdot\pi\cdot\phi_{fp}^{2}\cdot
d_{i}$ , if $i=1$, (12a) $\displaystyle V=$
$\displaystyle\pi\cdot\phi_{IRA}^{2}\cdot d_{i}$ , if $1<i<h$, (12b)
$\displaystyle V=$ $\displaystyle\pi\cdot\phi_{IRA}^{2}\cdot d_{h}$ , if
$i=h$. (12c)

The conical volume of the first hop with the distance $d_{i}$ is represented
in Eq. (12a), e.g., the first hop with conical volume between BS $2$ and RIS
$3$ of transmission BS $2\rightarrow$ RIS $3\rightarrow$ RIS $1\rightarrow$ UE
$2$ in Fig. 3 has the transmission distance $d_{1}$. The cylindrical volume
between two RISs of transmission hop $i$ is given by Eq. (12b), whereby $h$ is
the total number of hops between BS $b$/RN $e$ and RN $e$/UE $u$, e.g., the
second hop with cylindrical volume between RIS $3$ and RIS $1$ of BS
$2\rightarrow$ RIS $3\rightarrow$ RIS $1\rightarrow$ UE $2$ in Fig. 3 has the
transmission distance $d_{2}$. The cylindrical volume of the last hop is given
by Eq. (12c), whereby the distance of the last hop is given by:

$d_{h}=d_{th}-\sum_{j=1}^{h-1}d_{j},$ (13)

whereby $d_{th}$ denotes the threshold distance (longest one) where any
transmission over the longest distance still satisfies $SNR>T$ ($T$ is the SNR
threshold). For instance, if UE $2$ in Fig. 3 is at the the threshold distance
$d_{th}=d_{1}+d_{2}+d_{3}$, then the transmission between BS $2$ and UE $2$
still satisfies $SNR>T$, where the distance $d_{3}$ of last hop is larger than
or equal to the one between RIS $1$ and UE $2$. Based on the threshold $T$ and
(1), we can calculate $d_{th}$.

###  II-D Interference model

We distinguish between conical and cylindrical beam interference. The conical
beam interference is illustrated in Fig. 4, whereby the primary path BS
$0\rightarrow$ RIS $1\rightarrow$ RIS $2\rightarrow$ UE $0$ is subject to
interference on UE $0$ by the secondary path by BS $3\rightarrow$ UE $3$ with
conical beam. Similarly, the primary path BS $2\rightarrow$ RIS $3\rightarrow$
RIS $1\rightarrow$ UE $2$ is subject to interference on RIS $1$ by the
secondary path BS $0\rightarrow$ RIS $1\rightarrow$ RIS $2\rightarrow$ UE $0$
with conical beam from BS $0$, or the primary path BS $0\rightarrow$ RIS
$1\rightarrow$ RIS $2\rightarrow$ UE $0$ is subject to interference on RIS $2$
by the secondary path by BS $1\rightarrow$ UE $1$ with conical beam from BS
$1$. With cylindrical beam interference, the cylindircal bean interferes with
the conical beam, e.g., the primary path BS $0$ $\rightarrow$ RIS $1$
$\rightarrow$ RIS $2$ $\rightarrow$ UE $0$ is subject to interference on RIS
$1$ by BS $2\rightarrow$ RIS $3\rightarrow$ RIS $1\rightarrow$ UE $2$ with
cylindrical beam between RIS $3$ and RIS $1$.

RN is modeled as a half-duplex transceiver repeater. In the receiver mode, the
interference RN is modeled as UE, while in the transmitter model it can be
modeled as BS. Therefore, in case which includes RN, the primary path between
BS and RN (RN as receiver) or the primary path between RN and UE (RN as
transmitter), or the primary path between RN (as a transmitter) and another RN
(as a receiver) is modeled as the examples analyzed above, i.e., for the
primary path the between BS and UE. For instance, for BS $0\rightarrow$ RN $0$
in Fig. 4, RN $0$ is modeled as receiver, while for RN $0\rightarrow$ RIS
$4\rightarrow$ UE $0$, RN $0$ it is modeled as transmitter. Note that as RN is
a half-duplex transceiver repeater, RN $0$ in Fig. 4 cannot transmit and
receive at the same time.

As analyzed above, equations (12a), (12b), and (12c) can be used to consider
the areas of interference from any secondary paths causing it. In case of
interference on RISs, the set of illuminated RIS elements affected by
interference is given by the intersection between the set
$N_{\omega}^{\prime}$ of illuminated RIS elements of the primary transmission
and the set $N^{\prime}_{c}$ of illuminated RIS elements of the secondary
transmission:

$N_{\imath}^{\prime}=N_{\omega}^{\prime}\cap N_{c}^{\prime},$ (14)

where $|N^{\prime}_{\omega}|$ and $|N^{\prime}_{c}|$ can be calculated from
Eqs. (7), (8), (9), (11). Since the radius $\phi_{IRA}$ of illuminated RIS
areas remains constant, by the same principle, the illuminated areas affected
by the interference by different RISs remains constant as well. The Signal-to-
Noise-plus-Interference Ratio (SNIR) of the primary path can be calculated by:

$SNIR_{(be,eu)}=\frac{P_{eu}\cdot G_{be}\cdot G_{eu}}{k\cdot T\cdot
W+\Delta_{(s,p)}},$ (15)

whereby the interference $\Delta_{(s,p)}$ of the secondary path causing on the
primary path is given by:

$\Delta_{(s,p)}=\delta_{eu}^{p}\cdot G_{be}^{s}\cdot G_{eu}^{p},$ (16)

where $\delta_{eu}^{p}$ denotes the interference received at RN $e$ (as
receiver mode) or UE $u$ of the primary path, $G_{be}^{s}$ is the transmitting
antenna gain of BS $b$ or RN $e$ (as transmitter mode) of the secondary path,
and $G_{eu}^{p}$ is the receiving antenna gain of RN $e$ (as receiver mode) or
UE $u$ of the primary path. We can calculate $\delta_{eu}^{p}$ from (3), (5),
if the secondary path causes the interference from RISs, whereby
$N^{\prime}=N^{\prime}_{\imath}$ from (14). If RN $e$ (as receiver mode) or UE
$u$ of the primary path obtains directly the interference from the secondary
path, then we can similarly calculate $\delta_{eu}^{p}$ from (3), (5), or (6).

TABLE I: Summary of potential interference of secondary paths causing primary paths from Fig. 4 Primary paths Secondary paths |  $\\{$BS $0$,UE $0\\}$ |  $\\{$BS $1$,UE $1\\}$ |  $\\{$BS $2$,UE $2\\}$ |  $\\{$BS $3$,UE $3\\}$ |  $\\{$BS $0$,RN $0\\}$ |  $\\{$RN $0$,UE $0\\}$
---|---|---|---|---|---|---
| $\Delta$ | $\Delta_{(s,p)}$ | $\Delta$ | $\Delta_{(s,p)}$ | $\Delta$ | $\Delta_{(s,p)}$ | $\Delta$ | $\Delta_{(s,p)}$ | $\Delta$ | $\Delta_{(s,p)}$ | $\Delta$ | $\Delta_{(s,p)}$
$\\{$BS $0$,UE $0\\}$ |  | $0$ | $\checkmark$ | $\Delta_{(s,p)}^{2}$ | $\checkmark$ | $\Delta_{(s,p)}^{1}$ | $\checkmark$ | $\Delta_{(s,p)}^{3}$ |  | $0$ |  | $0$
$\\{$BS $1$,UE $1\\}$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$
$\\{$BS $2$,UE $2\\}$ | $\checkmark$ | $\Delta_{(s,p)}^{4}$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$
$\\{$BS $3$,UE $3\\}$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$
$\\{$BS $0$,RN $0\\}$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ |  | $0$ | $\checkmark$ | $\Delta_{(s,p)}^{6}$
$\\{$RN $0$,UE $0\\}$ |  | $0$ |  | $0$ |  | $0$ | $\checkmark$ | $\Delta_{(s,p)}^{5}$ |  | $0$ |  | $0$

##  III Interference graphs

We define an interference graph as an undirected graph $G(V,E)$ on which the
vertex set $V$ represents all the communication pairs in the network and the
edge set $E$ indicate the existence of interference among the communication
pairs. We refer to the latter as a _conflict_. The creation of the
interference graphs is illustrated in Fig. 5. First, based on the SNR
calculation in Section II-B, we find set of paths of the communication pairs
(as primary paths) with SNR$>T$ in the network, e.g., all primary paths in
Fig. 4 are summarized in the rows of Table I, where each vertex in the
interference graph is equivalent to a primary path. Based on Eqs. (12a),
(12b), and (12c), we can consider the areas of interference from secondary
paths, see Section II-D. The primary paths in Fig. 4 are subject to
interference by secondary paths (columns of Table I) which are indicated in
the fields $\Delta$ of Table I, where if ($\checkmark$), then the primary path
is subject to interference by the marked secondary paths; otherwise there is
no interference. In Fig. 4, $\\{$BS $0$,RN $0\\}$+$\\{$RN $0$,UE $0\\}$ is
assumed to be the backup path for the primary one $\\{$BS $0$,UE $0\\}$, and
it is only used if the main one has higher interference. Thus, there is no
interference between $\\{$RN $0$,UE $0\\}$ and $\\{$BS $0$,UE $0\\}$, which we
denoted as blank in the respective $\Delta$ field on Table I.

The next step is to calculate the interference $\Delta_{(s,p)}$ of any
secondary path $s$ caused on any primary path $p$ by using Eq. (16). Assume
that the interference values $\Delta_{(s,p)}$ calculated in Eq. (4) is shown
in Table I. For each primary path $P_{i}$, we create an ordered set of
interfering paths (secondary paths) with the corresponding interference values
$\Delta_{(s,p)}$ according to different ordering methods which are discussed
below. For instance, if the primary path $P_{i}$ $\\{$BS $0$,UE $0\\}$ in
Table I is considered, then its secondary paths $\\{$BS $1$,UE $1\\}$, $\\{$BS
$2$,UE $2\\}$, and $\\{$BS $3$,UE $3\\}$ are considered for that ordered set.
Based on the impact of the ordered set of secondary paths on each primary path
$P_{i}$, we connect that primary path with its secondary paths, i.e., in the
interference graph, edges are added connecting the vertex (primary path) to
the other vertices (secondary paths).

Figure 5: ZIM/DCS/ICS/RCS simple flowchart. Figure 6: Interference graphs
generated by ZIM DCS and ICS methods.

We propose four interference mapping methods, i.e., Zero Interference (ZIM),
Decreasing-ordered Conflict Selection (DCS), Increasing-ordered Conflict
Selection (ICS) and Random Conflict Selection (RCS). The first method, Zero
Interference Mapping (ZIM), represents the baseline interference mapping as
typically used with omnidirectional antennas in [9, 10]. The flowchart of ZIM
is presented on Fig. 5 with dashed lines. As a result, ZIM builds an
interference graph solely based on the conflict (overlapping) for all path
pairs without specific interference calculation. For the remining three
methods, given an ordered set of interfering paths (secondary paths) $P$, for
each primary path $P_{i}$, the interference is calculated and accumulated in
respect to $P_{j}$ from the subset $P^{\prime}\in P$, composed of secondary
paths. At each iteration, the accumulated SNIR is verified and, if SNIR$\leq
T$, then the algorithm considers that the current secondary path $P_{j}$ and
all the remaining ones on the ordered subset $P^{\prime}$ have conflict with
the primary path $P_{i}$, and thus the corresponding edges are added to the
interference graph. Since ZIM builds its interference graph solely based on
the communication pairs overlapped. there is no specific interference
calculation. Thus, considering the marked fields $\Delta$ in Table I, the
interference graph of ZIM in Fig. 6 (a) can be directly obtained.

To better understand the actual complexity of the process of interference
graph creation, we give an example that generates the interference graphs
depicted in Fig. 6 for DCS and ICS. Note that RCS is not represented in the
figure given its random output nature. To simplify this example explanation,
please consider Fig. 7 that graphically show the information from Table I.

Figure 7: Visual representation of the interference scenario from Table I

RCS, DCS, and ICS require the calculation of the interference values
$\Delta_{(s,p)}$ in Fig. 7. Let us assume that the relation between the
interference values of the secondary paths on the primary path $\\{$BS $0$,UE
0$\\}$ is $0<\Delta_{(s,p)}^{1}\leq\Delta_{(s,p)}^{2}\leq\Delta_{(s,p)}^{3}$,
and that the accumulated interference of
$\Delta_{(s,p)}^{1}+\Delta_{(s,p)}^{2}$ and
$\Delta_{(s,p)}^{2}+\Delta_{(s,p)}^{3}$ causes SNIR $>T$ and SNIR $\leq T$,
respectively. Moreover, $\Delta_{(s,p)}^{4}$ and $\Delta_{(s,p)}^{5}$ are not
enough, thus SNIR $>T$. Finally, assume that $\Delta_{(s,p)}^{6}$ is
sufficient, i.e., SNIR $\leq T$.

In DCS, the set of interfering paths (secondary paths) with corresponding
interference values $\Delta_{(s,p)}$ is decreasing-ordered. Thus, the
computation of the cumulative interference for the path $\\{$BS $0$,UE 0$\\}$
starts with $\Delta_{(s,p)}^{3}$, down to $\Delta_{(s,p)}^{1}$. In the second
iteration (Fig. 5), $\Delta_{(s,p)}^{3}+\Delta_{(s,p)}^{2}$ leads to SNIR
$\leq T$, thus edges between vertex $\\{$BS $0$,UE 0$\\}$ and vertices $\\{$BS
$2$,UE $2$$\\}$ and $\\{$BS $1$,UE $1$$\\}$ will be added to the interference
graph. Another edge will also be added between vertex $\\{$RN $0$,UE 0$\\}$
and vertex $\\{$BS $0$,RN $0$$\\}$. As a result, we achieve the interference
graph for the DCS algorithm in Fig 6 (b).

For ICS algorithm, we apply the same process like in DCS, but with an
increasing-ordered set of the interference $\Delta_{(s,p)}$ (Fig 6 (c)). In
this case, the computation of the cumulative interference for the path $\\{$BS
$0$,UE 0$\\}$ starts with $\Delta_{(s,p)}^{1}$, up to $\Delta_{(s,p)}^{3}$.
Only in the third iteration (Fig. 5),
$\Delta_{(s,p)}^{1}+\Delta_{(s,p)}^{2}+\Delta_{(s,p)}^{3}$ leads to SNIR $\leq
T$, thus one edge between vertex $\\{$BS $0$,UE 0$\\}$ and vertex $\\{$BS
$3$,UE $3$$\\}$ is added to the interference graph. Similarly DCS, another
edge is be added between vertices $\\{$RN $0$,UE 0$\\}$ and $\\{$BS $0$,RN
$0$$\\}$.

The asymptotic time complexity of all the proposed methods lies in the search
for the interference. In case of ZIM, it is always quadratic with the number
of paths, thus $\Theta(P^{2})$, where $P$ is the cardinality of the path set.
For the other three methods, the time complexity for the worst case is also
$O(P^{2})$ but it can run faster in the best and average cases, depending on
the network scenario, specially for DCS.

##  IV Performance evaluation

(a) Conflict complexity (less is better)

(b) Ratio of conflict complexity reduction from ZIM

(c) Fraction of time

Figure 8: Performance analysis of transmission scheduling methods. TABLE II: Parameters used in simulations. Values | Meaning
---|---
$|B|=28$ | Number of BSs.
$|U|=28$ | Number of UEs.
$|R|=28$ | Number of RISs.
$|E|=14$ | Number of RNs.
$\alpha=10$ degrees | Antenna directivity angle.
$k(f)=0.0016$ $m^{-1}$ | Molecular absorption coefficient.
$W=3$ GHz | Bandwidth.
$f=1$ THz | Operational frequency.
$P_{be}=0.1$ Watt | Power of BS or RN.
$T_{o}=300$ Kelvin | Temperature.
$T=10$ dB | SNR threshold.
$d_{x},d_{y}=\frac{c}{2f}$ |  $x$ and $y$ dimensions of RIS reflecting element.
$N=10000$ | Number of RIS reflecting elements.

The studied RIS/relay-assisted mesh network evaluation scenario is generated
as a 3D area of size $32\times 32\times 32$ m. The parameters used for the
simulations are summarized in Table II and are based on an indoor sub-THz
campus network scenario [11]. We assume that the line of sight signal
reachability is at most $20$ m between devices. We perform $100$ tests cases
to evaluate all four interference mapping methods. For each test, our
simulation randomly generates $200$ pairs of transmitting devices (BSs) and
receiving devices (UEs). The position coordinates of BSs, RNs, UEs, and RISs
are also randomly generated.

Fig. 8a shows the performance of four interference mapping methods in terms of
metric of the conflict complexity, where the conflict complexity is defined to
be the number of conflicts among devices in the interference graphs, e.g., in
Fig. 6 the number of edges of the graphs multiplied by 2 gives the total
number of conflicts. Hence, the conflict complexity in the example is counted
as $10$, $6$ and $4$ units for the methods ZIM, DCS and ICS, respectively. In
Fig. 8a, as the ZIM accounts immediately for the potential transmission paths
causing interference, while redundant transmission conflicts cause the network
resources waste, leading to the highest conflict complexity, for all tests.
The remaining interference mapping methods present rather similar values, but
always with less complexity than the ZIM ones. This is due to the capacity of
these methods of evaluating the interpath interference in a more efficient
way, taking into account the shape and reach of the transmitted beans.

Fig. 8b confirms the results from Fig. 8a by the reduced ratio of conflict
complexity, defined to be the conflict complexity of the method ZIM divided by
methods RCS, DCS and ICS. We see that the reduced ratio of conflict complexity
of all three methods are always more than $1$, i.e., their interference
mapping performance is better than ZIM by up to 12%. In other words, we would
be 12% to finding valid paths. Moreover, we observe that the method ICS has
slightly better performance. This happens because this method consider the
paths causing interference from the smallest values to the greatest values, so
the SNIR values will not increase quickly, leading to removing the unnecessary
transmissions.

Fig. 8c evaluates the impact of the interference graph on the network
performance. Let $C$ be the conflict complexity, and $N_{p}$ the number of
communication pairs between BSs and UEs. Each communication pair between BS
and UE conflicts with $A=\frac{C}{N_{p}}$ other communication pairs between
BSs and UEs on average. Thus, the fraction of time when any communication pair
between BS and UE can occupy the transmission spectrum is given by:
$F=\frac{1}{A}$. Observe that since the ZIM has the highest conflict
complexity, its fraction of time is the lowest, whereas the fraction of time
of IDS/ICS/RCS with requiring the calculation of the interference values is
better due to the QoT of devices which results in simpler interference graphs.
The larger the fraction of time, the higher the network throughput.

##  V Conclusion

As the complexity of mmWave/sub-THz mesh networks rises, transmission paths
will likely always experience interference. On the other hand, the
interference values will largely vary depending on the positions of the
network devices. Therefore, the prediction towards the actual performance of
the four interference mapping methods proposed is not trivial. On the other
hand, one needs to weigh in the feasibility of optimizations versus
simulations efforts in such prediction and their feasiblity in complex
THz/mmWave systems. Our analytical results conclude that the interference
mapping by increasing-ordered conflict selection (ICS) method performs best,
resulting in a better network throughput. This study opens pathways for
further work, such as network routing optimization based on interference
graphs.

## References

  * [1] S. Dash, C. Psomas, I. Krikidis, I. F. Akyildiz, and A. Pitsillides, “Active control of thz waves in wireless environments using graphene-based ris,” _IEEE Transactions on Antennas and Propagation_ , pp. 1–1, 2022.
  * [2] S. Arzykulov, G. Nauryzbayev, A. Celik, and A. M. Eltawil, “Ris-assisted full-duplex relay systems,” _IEEE Systems Journal_ , vol. 16, no. 4, pp. 5729–5740, 2022.
  * [3] J. Ye, S. Dang, B. Shihada, and M.-S. Alouini, “Modeling co-channel interference in the thz band,” _IEEE Transactions on Vehicular Technology_ , vol. 70, no. 7, pp. 6319–6334, 2021.
  * [4] M. Yu, A. Tang, X. Wang, and C. Han, “Joint scheduling and power allocation for 6g terahertz mesh networks,” in _2020 International Conference on Computing, Networking and Communications (ICNC)_ , 2020, pp. 631–635.
  * [5] C. V. Phung, A. Drummond, and A. Jukan, “Maximizing throughput with routing interference avoidance in ris-assisted relay mesh networks,” _accepted at 47th International Convention on Information and Communication Technology, Electronics and Microelectronics (MIPRO)_ , 2024. [Online]. Available: https://doi.org/10.48550/arXiv.2402.08825
  * [6] C.-C. Wang, W.-L. Wang, and X.-W. Yao, “Interference and Coverage Modeling for Indoor Terahertz Communications with Beamforming Antennas,” _The Computer Journal_ , vol. 63, no. 10, pp. 1597–1606, 07 2020. [Online]. Available: https://doi.org/10.1093/comjnl/bxaa083
  * [7] V. Petrov, M. Komarov, D. Moltchanov, J. M. Jornet, and Y. Koucheryavy, “Interference and sinr in millimeter wave and terahertz communication systems with blocking and directional antennas,” _IEEE Transactions on Wireless Communications_ , vol. 16, no. 3, pp. 1791–1808, 2017.
  * [8] K. Ntontin, A.-A. A. Boulogeorgos, D. G. Selimis, F. I. Lazarakis, A. Alexiou, and S. Chatzinotas, “Reconfigurable intelligent surface optimal placement in millimeter-wave networks,” _IEEE Open Journal of the Communications Society_ , vol. 2, pp. 704–718, 2021.
  * [9] S. Sengupta, S. Rayanchu, and S. Banerjee, “Network coding-aware routing in wireless networks,” _IEEE/ACM Transactions on Networking_ , vol. 18, no. 4, pp. 1158–1170, 2010.
  * [10] K. Jain, J. Padhye, V. Padmanabhan, and L. Qiu, “Impact of interference on multi-hop wireless network performance,” vol. 11, 09 2003, pp. 66–80.
  * [11] C. Chaccour, M. N. Soorki, W. Saad, M. Bennis, and P. Popovski, “Risk-based optimization of virtual reality over terahertz reconfigurable intelligent surfaces,” in _ICC 2020 - 2020 IEEE International Conference on Communications (ICC)_ , 2020, pp. 1–6.


left in a binary mixture of stresslets. The profiles of the compartments are
asymmetric; propagation direction is towards the higher slope of the density
profiles. The parameters are set at $B=4$, $C=1$, $D=1$, $k_{1}=1$, $k_{2}=0$,
$k^{b}_{\text{rel}}=0$, $k_{3}=1$, $\zeta_{\text{avg}}=2.4$,
$\zeta_{\text{rel}}=-2$, $k_{4}=3$, $\chi(\rho_{a}^{0})=1$,
$\chi^{\prime}(\rho_{a}^{0})=0.01$, $\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$.

###  10D Swapping in a binary mixture of stresslets

Movie S4. In this movie we see swapping of the two stresslets in a mixture.
The profiles of the pulses are symmetric; there is a small to and fro
oscillation of the pulsatile profiles. The parameters are set at $B=4$, $C=1$,
$D=1$, $k_{1}=1$, $k_{2}=0$, $k^{b}_{\text{rel}}=0$, $k_{3}=1$,
$\zeta_{\text{avg}}=2.2$, $\zeta_{\text{rel}}=-2$, $k_{4}=3$,
$\chi(\rho_{a}^{0})=1$, $\chi^{\prime}(\rho_{a}^{0})=0.01$,
$\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$.

###  10E Temporal coexistence of swap and travelling wave in a binary mixture
of stresslets

Movie S5. In this movie we see a temporal coexistence of the swap and the
travelling wave phase. The slow to and fro oscillations of the symmetric
pulses observed at earlier times, transitions to a travelling state where the
whole wave train moves to the left (in the direction set by the higher slope
of the density profile). This gives rise to a pulsatile travelling wave phase.
The parameters are set at $B=4$, $C=1$, $D=1$, $k_{1}=1$, $k_{2}=0$,
$k^{b}_{\text{rel}}=0$, $k_{3}=1$, $\zeta_{\text{avg}}=2.35$,
$\zeta_{\text{rel}}=-2$, $k_{4}=3$, $\chi(\rho_{a}^{0})=1$,
$\chi^{\prime}(\rho_{a}^{0})=0.01$, $\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$.

###  10F Segregation of single stresslet on elastomer

Movie S6. In this movie we see segregation between high density and low
density phases of a single species of stresslets on an elastomer. The high
density regime eventually forms singular structures. The parameters are set at
$B=6$, $C=1$, $D=1$, $k_{u}=1$, $\zeta=2$, $\chi(\rho_{a}^{0})=1$,
$\chi^{\prime}(\rho_{a}^{0})=1$, $\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$, $\eta=1$, $\alpha=3$.

###  10G Segregation of single stresslet in fluid

Movie S7. In this movie we see segregation between high density and low
density phases of a single species of stresslets in a fluid. The high density
regime eventually forms singular structures. The parameters are set at $D=1$,
$\zeta_{\text{avg}}=19.76$, $\zeta_{\text{rel}}=-0.54$, $\zeta_{3}=0.64$,
$k_{1}^{b}=2$, $k_{1}^{u}=0.5$, $\eta=1$.

###  10H Segregation of contractile stresslets with the stronger wetting a
substrate

Movie S8. In this movie we see segregation of two species of contractile
stresslets with stronger stresslet interacting with the substrate with a
Lennard-Jones wetting potential ($f(x)$). Because of the catch-bond response
of the two stresslets, their density peaks co-localize. The parameters are set
at $B=10$, $C=1$, $D=1$, $k_{1}=1$, $k_{2}=0$, $k^{b}_{\text{rel}}=0$,
$k_{3}=1$, $\zeta_{\text{avg}}=3$, $\zeta_{\text{rel}}=2$, $k_{4}=2$,
$\chi(\rho_{a}^{0})=1$, $\chi^{\prime}(\rho_{a}^{0})=0.01$,
$\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$, $\eta=1$, and
$f(x)=\frac{1}{2}(\frac{1}{x^{12}}-\frac{1}{x^{6}})$.

###  10I Segregation of contractile stresslets with the weaker wetting a
substrate: stratification

Movie S9. In this movie we see segregation of two species of contractile
stresslets with weaker stresslet interacting with the substrate with a
Lennard-Jones wetting potential ($f(x)$). Because of the slip-bond response of
the two stresslets, their density peaks separate. The parameters are set at
$B=10$, $C=1$, $D=1$, $k_{1}=1$, $k_{2}=0$, $k^{b}_{\text{rel}}=0$,
$k_{3}=-2$, $\zeta_{\text{avg}}=2.5$, $\zeta_{\text{rel}}=-4.7$, $k_{4}=-4.8$,
$\chi(\rho_{a}^{0})=1$, $\chi^{\prime}(\rho_{a}^{0})=0.01$,
$\chi^{\prime\prime}(\rho_{a}^{0})=0.001$,
$\chi^{\prime\prime\prime}(\rho_{a}^{0})=-0.001$, $\eta=1.1$, and
$f(x)=\frac{1}{2}(\frac{1}{x^{12}}-\frac{1}{x^{6}})$.


# The Discovery of a Remnant Radio Galaxy in A2065 Using GMRT

Dharam V. Lal  National Centre for Radio Astrophysics - Tata Institute of
Fundamental Research, Post Box 3, Ganeshkhind P.O., Pune 411007, India
<EMAIL_ADDRESS>
###### Abstract

The upgraded Giant Metrewave Radio Telescope (GMRT) has been used to map the
cluster A2065 at $z$ = 0.0726. We report the discovery of a remnant radio
galaxy at the peripheral cluster region. The spatially resolved radio emission
from the remnant radio galaxy shows an elongated, bar-shaped structure, whose
size is $\approx$ 52′′ $\times$ 110′′ ($\simeq$ 72 $\times$ 152 kpc2). Our
study with the multiwavelength GMRT data and Chandra data shows that across
the remnant radio galaxy there is a hint of a surface-brightness edge in the
hot X-ray gas. We detect tentative flattening of the radio spectral index as
the old plasma at the near end of the surface-brightness edge is reinvigorated
by the passage of possible shock front and shows the expected change in radio
emission characteristics. We suggest that the remnant radio galaxy has been
seeded by the lobes of the active galactic nucleus (AGN), hosted by the WISEA
J152228.01$+$274141.3 source, demonstrating the connection between AGNs and
remnant radio sources. Although the number of known remnant radio sources is
beginning to increase, we emphasize the need for better data to understand the
physics and nature of poorly understood remnant radio sources.

###### Subject headings:

galaxies: active — galaxies: jets — galaxies: nuclei — galaxies: structure —
radio continuum: galaxies — galaxies: cluster

##  1\. Introduction

The hierarchical structure formation scenario suggests that cluster mergers
are the mechanisms by which galaxy clusters form and these mergers are the
most energetic phenomena in the universe since the Big Bang. A large number of
clusters of galaxies are known to contain large-scale, low-surface-brightness
diffuse radio emission whose origin is not often related to the active
galactic nuclei (AGNs) but to the intracluster medium (ICM). According to the
location with respect to the cluster center, these sources are of two key
types, radio halos and relics. Radio halos show a regular, amorphous emission
around the cluster center of a size from a few hundred kiloparsecs (minihalo)
to megaparsecs (halo), whereas radio relics show an irregular emission located
in peripheral cluster regions of a size from several hundreds of kiloparsecs
to megaparsecs. Another third broad type of sources comprises of revived
fossil plasma sources and phoenices. These do not have any preferred location
in the ICM are of size from several tens of kpc to a few hundred kpc and trace
AGN radio plasma that has somehow been reenergised through the processes in
the ICM (van Weeren et al., 2019; Kempner et al., 2004; Giovannini & Feretti,
2002).

The radio jets emanating from an AGN arise from the accretion onto a
supermassive black hole. These form synchrotron-emitting radio lobes in a
radio-loud AGN in the environments of their host galaxies. The active phase of
an AGN can last several tens of megayears, following this the radio jets stop
and their features, namely an unresolved radio core, radio jets, hotspots, and
radio lobes, all start to fade away due to synchrotron cooling. This phase
once the jets have switched off is called as the remnant phase of a radio
galaxy, or merely the remnant radio galaxy. The remnant radio galaxy falls
into the third broad type of sources in the ICM, which is poorly understood
and presently very few sources are known (Quici et al., 2021; Brienza et al.,
2017; Parma et al., 2007).

The high spatial resolution of the Chandra and upgraded Giant Metrewave Radio
Telescope (GMRT) allow us to study the interplay between X-ray-emitting hot
gas and nonthermal radio emission. We have obtained new upgraded GMRT data,
and have used archive GMRT and Chandra data of the nearby $z$ = 0.0726
(Struble & Rood, 1999) richness class 2 cluster, A2065, which is one of the 10
galaxy clusters that make up the Corona Borealis supercluster. Postman, Geller
& Huchra (1988) noted that two cD galaxies dominate the cluster center in the
optical and their radial velocities differ by $\sim$600 km s-1.

Table 1Summary of the radio observations and image properties. Obs. Band | Obs_ID | Obs. Date | Flux Cal. | Phase Cal. | $\nu$ | $\Delta\nu$ | $t_{\rm int.}$ | FWHM, P.A. | rms
---|---|---|---|---|---|---|---|---|---
|  |  |  |  | (MHz) | (MHz) | (hr) | (${}^{\prime\prime}\times^{\prime\prime},^{\circ}$) | (mJy beam-1)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10)
“classic” GMRT
150 MHz | 12RKA02 | 2007 Aug 10 | 3C 286 | 1459$+$716 | 152 |  6 | 0.9 | 26.90$\times$15.19, 136.15 | 4.98
240 MHz | 14RMD01 | 2008 Jul 26 | 3C 286 | 1513$+$236 | 240 |  8 | 5.4 | 13.95$\times$10.94, 98.89 | 0.56
610 MHz | 14RMD01 | 2008 Jul 26 | 3C 286 | 1513$+$236 | 606 | 16 | 5.4 |  4.93$\times$3.97, 65.80 | 0.06
Upgraded GMRT
250–500 MHz | 32_084 | 2017 Jun 27 | 3C 147 | 3C 286 |  400 | 200 | 1.3 |  9.48$\times$6.38, 69.18 | 0.06
1050–1450 MHz | 32_084 | 2017 Jun 28 | 3C 147 | 1609$+$266 | 1274 | 400 | 0.9 | 11.13$\times$5.90, 95.45 | 0.03

Previous radio study using 100 m Green Bank Telescope at 1.4 GHz (Farnsworth
et al., 2013) detected a smooth diffuse structure $\simeq$1 Mpc in extent, a
giant radio halo, whereas a recent study showed a slightly smaller extent for
the radio halo using Low-Frequency Array (LOFAR) at 153 MHz (in Drabent et
al., 2017). Similarly, previous X-ray studies led to detections of two
surface-brightness peaks coincident with the two cD galaxies observed in the
cluster’s center. The data indicated that the cluster is an unequal-mass
merger and only one of the two cooling cores has survived the merger. The
northern cluster seems to have fallen into the more massive southern cluster
from the southeast, and having lost its gas content, it has now moved to its
present location to the northwest. Evidence of shock with $M$ $\approx$ 1.7 at
$\sim$140 kpc from the southern cD galaxy too is noted and the cooling flow is
displaced to the southeast of the southern cD galaxy (Chatzikos, Sarazin &
Kempner, 2006; Markevitch, Sarazin & Vikhlinin, 1999). In this study, we
report the discovery of a remnant radio galaxy in cluster A2065 using the
GMRT. We also present an X-ray analysis of archive Chandra data, which
enhances the interpretation of the current discovery.

We define spectral index, $\alpha$ as, $S_{\nu}$ $\propto$ $\nu^{\alpha}$,
where Sν is the flux density at frequency, $\nu$. To estimate the intrinsic
parameters, we use a $\Lambda$CDM cosmology with $\Omega_{\rm m}$ = 0.27,
$\Omega_{\Lambda}$ = 0.73, and H0 = 70 km s-1 Mpc-1. At the redshift of the
cluster, 1 arcsec corresponds to 1.384 kpc. All uncertainties are at 1$\sigma$
error bars unless otherwise stated. The position angles (P.A.s) are measured
from north to east. Throughout, positions are given in J2000 coordinates.

The organization of the paper is as follows. We present the X-ray and radio
observations and their data analyses in Sec. 2. Our results, i.e., radio
morphology, spectral structure, and X-ray gas properties are presented in Sec.
3. In Sec. 4, we discuss our observational results, source diagnostics, and
dynamical interpretation of the remnant radio galaxy. Sec. 5 summarizes our
conclusions.

##  2\. Observations

###  2.1. Chandra data

A2065 was observed by Chandra with the acis-i detector on 2002 August 18 and
November 24 for 28 and 22 ks (Obs_ID 3182) respectively, and on 2007 September
13 for 5 ks (Obs_ID 7689). The data reduction of these three archival data
sets has been performed independently. We used the calibration databases,
caldb version 4.7.6 and ciao version 4.9. After the initial calibration, i.e.,
standard filtering and removal of bad pixels, the event files of the three
observations were projected to a common reference point and then merged to
create Gaussian-smoothed, background-subtracted, exposure-corrected image in
the broadband 0.5–7.0 keV energy range.

We excised the central AGN and background point sources. Using the ciao tool
specextract, we extracted the source, background spectra, and the response
files of selected regions in the Chandra images separately for the three data
sets. The spectra for each data set were grouped so that each bin contained at
least 20 counts. We restricted our spectral extraction in the energy range
0.5–7.5 keV, and fitted for the thermal gas emission with xspec (version
12.10.1, Arnaud, 1996) using the isothermal apec model. The abundance was
fixed to 0.3 $\times$ solar (Chatzikos, Sarazin & Kempner, 2006) and all
spectral fits include absorption by the Galactic column (= 2.9 $\times$ 1020
cm-2, Dickey & Lockman, 1990).

---
Figure 1.— Exposure-corrected, background-subtracted, broadband (0.5–7.0 keV)
Chandra image of A2065. The image was binned (bin = 16) and smoothed using a
Gaussian kernel of 10′′. The color bar shows the surface-brightness, in 10-6
counts s-1 per 8 $\times$ 8 pixel binning. The X-ray peak is located at R.A.
15:22:29.18, decl. 27:42:21.05. The peak surface-brightness and the faintest
regions transitioning between blue and green have a surface-brightness of 6.0
$\times$ 10-5 counts s-1 arcsec-2 and 1.6 $\times$ 10-6 counts s-1 arcsec-2
respectively. The surface-brightness is displayed in logarithmic scales to
emphasize low-surface-brightness emission.

###  2.2. GMRT data

---
Figure 2.— Chandra X-ray contours (see Figure 1) and 250–500 MHz band radio
contours overlaid on the exposure-corrected, background-subtracted broadband
(0.5–7.0 keV) Chandra image (left panel) and Digitized Sky Survey image (right
panel) of A2065. The yellow X-ray contour levels are 3.18, 6.36, 12.72 and
25.44 $\times$ 10-6 counts s-1 arcsec-2 and the red radio contour levels are
$-$3, 3, 6, 12, 24 and 48 $\times$ 77.5 $\mu$Jy beam-1 on the Chandra image
(left panel). The yellow X-ray contour levels are 3.18, 6.36 and 12.72
$\times$ 10-6 counts s-1 arcsec-2 and the red radio contour levels are $-$3,
3, 6, 9, 15, 42 and 84 $\times$ 0.1 mJy beam-1 on the Digitized Sky Survey
image (right panel). The radio emission from the two cD galaxies (the brighter
southern galaxy 2MASS J152229.17$+$274227.5 and the fainter northern galaxy
2MASX J152228.92$+$274244.1) of the cluster is shown on the east of the
remnant radio galaxy emission (denoted with two blue plus marks). The emission
from the remnant radio galaxy corresponds to the 19.19 mag, WISEA
J152228.01$+$274141.3 source, denoted by a red plus mark that is 7$\farcs$8
away from V1CG 136 galaxy located at R.A.: 15:22:28.062, decl.:
$+$27:41:41.51, $z$ = 0.072 (Griffin, Dai & Kochanek, 2014) denoted by a black
cross. The color scale was chosen to emphasize the X-ray surface-brightness
edge near the remnant radio galaxy. The images are displayed in logarithmic
scales to emphasize a faint optical host associated with the remnant radio
galaxy. Note the X-ray peak surface-brightness seems to be associated with the
radio peak corresponding to the southern cD galaxy.

A summary of the radio observations is presented in Table 1. We inspected the
data for bad antennas, bad time stamps and data impacted by radio frequency
interference. The data were analyzed using the standard reduction
methodologies (see also Lal, 2020). After the initial steps of flux density
and bandpass calibration, all corrupted data were removed using standard
routines. The bandpass calibrated data for wide-bandwith GMRT data, i.e., the
250–500 MHz band and 1050–1450 MHz band data were split into five 40 MHz (from
300–500 MHz) and eight 50 MHz (from 1050–1450 MHz) subbands respectively.
Subsequently, each subband data was analyzed separately, similar to narrowband
“classic” GMRT data. Several cycles of phase-only self-calibration and imaging
were performed and the calibration solutions were applied to the target data
to correct for residual phase errors. To take care of the wide-field imaging
at low GMRT frequencies, we performed faceting imaging. Problematic bright
sources in the field of view whose side lobes affected the area of the target
were subtracted, a.k.a. peeled; this (direction-dependent) calibration step
improved the dynamic range of images. The calibrated data for five 40 MHz
250–500 MHz band data and eight 50 MHz 1050–1450 MHz band data were joined to
form full 200 MHz and 400 MHz, respectively, wide-bandwidth-calibrated
visibility data sets. The combined wide-bandwidth-calibrated data (and
similarly the narrowband-calibrated “classic” GMRT data) were imaged using the
casa task tclean. We used 3D imaging (gridder = ‘widefield’), two Taylor
coefficients (nterms = 2, tt0 and tt1) and Briggs weighting (robust = 0.5) in
task tclean. A final amplitude-and-phase self-calibration with solution
interval equal to the length of observation was then carried out using the
gaincal and applycal tasks in casa. The two Stokes polarizations, RR and LL,
were combined to obtain the final total intensity Stokes I image, and
corrected for the GMRT primary beam response. This provides images at center
frequencies (tt0) and the corresponding in-band spectral index maps (tt1/tt0).
The amplitude errors are estimated to be within 5% or less for archival
narrow-bandwidth and wide-bandwidth GMRT data. Table 1 also provides image
properties, including the restoring beams and root mean square (rms) noise
levels at half-power points of the final images. The uncertainty in the flux
density measurements are estimated as

$\left[(S_{\nu}\times f)^{2}+(\textsc{rms})^{2}\times N_{\rm
beams}\right]^{0.5},$

where $S_{\nu}$ is the flux density, $f$ is an absolute flux density
calibration uncertainty, rms is the noise, and $N_{\rm beams}$ is the number
of synthesized beams. We use these error estimates for the flux density
measurements when computing the error estimates for spectral index
measurements.

---
Figure 3.— The central 3′ $\times$ 3′ cluster region with 250–500 MHz band
radio contours overlaid on the Digitized Sky Survey image of A2065. Note the
northern cD galaxy registers well with the northern radio peak, but the
southern cD galaxy is displaced by $\sim$2$\farcs$4 north of the radio peak.
The X-ray peak at the cluster center is coincident with the southern radio
peak (see also Figure 2). The radio emission from the two cD galaxies (the
brighter southern galaxy 2MASS J152229.17$+$274227.5 and the fainter northern
galaxy 2MASX J152228.92$+$274244.1) of the cluster is shown on the east of the
remnant radio galaxy emission. The radio contour levels are $-$3, 3, 6, 9, 15,
42 and 84 $\times$ 80 $\mu$Jy beam-1. The cluster-member galaxies are marked
(NED positions, and “diffuse radio emission in the Corona Borealis
supercluster field”, Drabent et al., 2017) and a majority ($\approx$60%) show
radio emission. The two bright cluster galaxies, SDSS $+$230.6$+$27.7$+$0.08
(R.A.: 15:22:24.0, decl. $+$27:42:51; $z$ = 0.0723) and 2MASS
J152229.17$+$274222.75 ($z$ = 0.074875) are marked in blue colour (Postman,
Geller & Huchra, 1988).

##  3\. Results

The exposure-corrected background-subtracted X-ray binned (bin = 16) and
Gaussian-smoothed (10′′ kernel) image in the 0.5–7.0 keV band is shown in
Figure 1. Figure 2 shows Chandra X-ray contours from Figure 1 and 250–500 MHz
band radio contours overlaid on the exposure-corrected, background-subtracted
broadband (0.5–7.0 keV) Chandra (bin = 8) image (left panel) and Digitized Sky
Survey image (right panel) of A2065. The central 3′ $\times$ 3′ cluster region
with 250–500 MHz band radio contours overlaid on the Digitized Sky Survey
image of A2065 is shown in Figure 3. Figure 3 also marks the cluster member
galaxies, including the two bright cluster galaxies in marked in blue colour,
SDSS $+$230.6$+$27.7$+$0.08 (NED; R.A.: 15:22:24.0, decl. $+$27:42:51; $z$ =
0.0723, $m$ = 16.1) and 2MASS J152229.17$+$274222.75 (NED; $z$ = 0.074875, $m$
= 14.65 Postman, Geller & Huchra, 1988), and a majority of them, $\approx$60%,
show radio emission.

Figure 4 shows clear detections for the southern cD galaxy and remnant radio
galaxy at the GMRT bands, namely the 240 MHz band (top-left panel), 250–500
MHz band (top-right panel), 610 MHz band (bottom-left panel), and 1050–1450
MHz band (bottom-right panel). The northern cD galaxy is only detected in our
250–500 MHz band data. The images have rms noise values in the range 0.56–0.06
mJy beam-1 at the half-power points. The 150 MHz band and 1050–1450 MHz band
data sets had very short on-source integration times, $\lesssim$0.9 hr, and
therefore these images have relatively high rms noise values (see Table 1).
Note that we also made images at several angular resolutions, including
imaging data sets using the same range of baseline lengths, and do not find
the presence or absence of additional morphological features. The integrated
flux densities computed thus are also consistent, within error bars with those
reported in Table 1.

###  3.1. Southern and northern cD galaxies

---
Figure 4.— Images of the cD galaxy and the remnant radio galaxy are shown
using the GMRT bands, i.e., 240 MHz band (top-left panel), 250–500 MHz band
(top-right panel), 610 MHz band (bottom-left panel), and 1050–1450 MHz
(bottom-right panel) data. Two peaks (marked as blue cross-marks in the
250–500 MHz band image), the brighter southern and the fainter northern peaks
correspond to the two cD galaxies, 2MASS J152229.17$+$274222.75 and 2MASX
J152228.92$+$274244.1, respectively. The radio contour levels plotted in each
panel are $-$3, 3, 6, 9, 12, 24, 36, 48, 60 and 72 times the rms noise
(denoted at upper-left corner in each panel) in units of mJy beam-1. The black
plus and the red cross marks are, respectively, the positions of the WISEA
J152228.01$+$274141.3 source and optical host galaxy V1CG 136 (R.A.:
15:22:28.062, decl.: $+$27:41:41.51, $z$ = 0.072; Griffin, Dai & Kochanek,
2014).

Figure 1 shows a diffuse X-ray tail extending to the north at the center and
that the cluster emission is elongated along northwest-southeast direction.
The hot gas emission (i) shows a compact, bright region and a diffuse tail
extending to the north at the center, and (ii) is extended along the
northwest–southeast direction showing pieces of evidence for surface-
brightness edges. Though the X-ray peak emission and peak of the radio
emission register well (see Figure 2, left), the position of the optical host
of the southern cD galaxy (R.A.: 15:22:29.17, decl.: $+$27:42:27.5) is
displaced by $\sim$2$\farcs$4 from the compact, bright, X-ray peak emission at
the cluster center (see Figure 2, right, and also Figure 3). A low-surface-
brightness diffuse tail, both in X-ray as well as in radio images, extending
to the north at the center of the image, reaches out to the second cD galaxy
(R.A.: 15:22:28.92, decl.: $+$27:42:44.1; Figures 3 and 4, top-right panels).
The X-ray peak corresponding to the second northern cD galaxy and the peak of
the second weaker component of radio emission at this location also register
well. The southern cD galaxy seems to have retained a cool core residing at
the center of its subcluster and there is no evidence for a cool core
corresponding to the northern cD galaxy (Chatzikos, Sarazin & Kempner, 2006;
Markevitch, Sarazin & Vikhlinin, 1999). The closely associated radio and X-ray
morphology at the center and its connection with the southern cD galaxy
suggest that both X-ray and radio emissions are due to ram pressure stripping
as the southern cD galaxy moves relative to the ICM.

The two blue plus signs in the 250–500 MHz band image marked in Figure 4 (top-
right panel) correspond to the hosts of the brighter southern cD galaxy, 2MASS
J152229.17$+$274222.75 at $z$ = 0.074875 and fainter northern cD galaxy, 2MASX
J152228.92$+$274244.1 at $z$ = 0.072854 (see also Figure 3). The northern cD
galaxy is only detected in our 250–500 MHz band image. However, we not only
detected the southern cD galaxy in our long observations, we also detected the
peak radio emissions at the location of the southern cD galaxy at R.A.:
15:22:29.17, decl.: $+$27:42:27.5 in our short, 150 MHz band, and 1050–1450
MHz band data sets. The two cD galaxies are barely resolved using LOFAR at 153
MHz (“diffuse radio emission in the Corona Borealis supercluster field”,
Drabent et al., 2017). The southern cD galaxy also had a 2.8$\sigma$ (upper
limit) detection at 74 MHz (VLA Low-Frequency Sky Survey, VLSS: Cohen et al.,
2007). It seems that around the southern cD radio galaxy there is some
extended radio emission, which is more clearly seen in our 250–500 MHz band
image (Figure 4, top-right panel). Since several cluster-member galaxies also
show radio emission, it is unclear if this extended radio emission is indeed
associated with the southern cD radio galaxy or with the two adjacent cluster-
member galaxies southeast of it (see Figure 3).

Table 2Radio flux densities for the remnant radio galaxy and the southern cD galaxy | $S_{\rm 74\,MHz}$ | $S_{\rm 150\,MHz}$ | $S_{\rm 240\,MHz}$ | $S_{\rm 400\,MHz}$ | $S_{\rm 610\,MHz}$ | $S_{\rm 1250\,MHz}$ | $S_{\rm 1400\,MHz}$
---|---|---|---|---|---|---|---
(mJy) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
Remnant radio galaxy |  |  | 46.52 $\pm$0.90 | 26.76 $\pm$0.08 | 2.60 $\pm$0.22 | 0.73 $\pm$0.15 |
Southern cD galaxy | 395.2 $\pm$141.1 | 46.52 $\pm$0.90 | 36.26 $\pm$0.90 | 21.97 $\pm$0.08 | 11.12 $\pm$0.22 | 3.83 $\pm$0.09 | 3.09 $\pm$0.15

###  3.2. Remnant radio galaxy

Figures 3 and 4 also suggest that the remnant radio galaxy is hosted by the
19.19 magnitude WISEA J152228.01$+$274141.3 source that is 7$\farcs$8 away
from the optical galaxy V1CG 136 (R.A.: 15:22:28.5, decl.: $+$27:41:46) at
redshift $z$ = 0.072 (Griffin, Dai & Kochanek, 2014). The cluster contains gas
not only with an extremely wide range of temperatures and complex structure
but that also shows a cold front at 30′′ ($\simeq$ 41.5 kpc) and a shock-like
bow-shaped feature beyond $\sim$100′′ ($\simeq$ 140 kpc) on the southeast of
the southern cD (Chatzikos, Sarazin & Kempner, 2006). Figure 2 shows that the
northeastern boundary of the remnant radio galaxy coincides with the 1.3
$\times$ 10-7 counts s-1 arcsec-2 surface-brightness contour level; gas in
false color transitioning from the green region to the blue region at a
distance of $\sim$33′′ ($\simeq$ 45.7 kpc) from the southern cD galaxy. All
four panels of Figure 4 suggest that the radio emission from the remnant radio
galaxy has an elongated, bar-shaped structure. The remnant radio galaxy,
buried in the radio halo emission that is barely detected using LOFAR at 153
MHz seems to possess an elongated, bar-shaped structure (“diffuse radio
emission in the Corona Borealis supercluster field, Drabent et al., 2017),
consistent with our results. This suggests that the remnant radio galaxy has a
steep spectrum, and its consistency in radio morphology gives us additional
confidence in our image fidelity, imaging, calibration and data reduction
processes. The best sensitivity image for the low surface-brightness was
obtained from the 250–500 MHz band data and we now discuss below the
morphology of the remnant radio galaxy.

The remnant radio galaxy broadly consists of a high-surface-brightness
southeastern part and a fainter low-surface-brightness extended northwestern
part. The two oppositely directed radio jets emanating from the apex of the
WISEA J152228.01$+$274141.3 source initially traverse toward the southeast and
northwest directions. As the galaxy plows through the dense intracluster gas,
these jets traversing in opposite directions form a trail, after sharp bends
in the jets, behind the WISEA J152228.01$+$274141.3 source due to interaction
with the ICM. The width of the remnant radio galaxy increases from $\sim$17′′
($\simeq$ 23.5 kpc) at the southeastern end to $\sim$27′′ ($\simeq$ 37.4 kpc)
and reduces again before flaring to reach a maximum width of $\sim$35′′
($\simeq$ 48.4 kpc) and finally fading completely. The average width of the
narrower southeastern part is a factor of $\sim$2.3 smaller than the broader
northwestern part. It seems that a pinch is present at $\sim$35′′ ($\simeq$
48.4 kpc) of the $\sim$16$\farcs$2 ($\simeq$ 22.4 kpc) width from the
southeastern end, i.e., between the two narrower and the broader parts. The
northeastern boundary toward the two cD galaxies of the remnant radio galaxy
is sharp, while the emission fades more slowly at the southwestern boundary.
The overall source size, assuming elongation, of the bar-shaped structure is
$\approx$ 52′′ $\times$ 110′′, corresponding to 72 $\times$ 152 kpc2 at the
assumed distance.

---
Figure 5.— Integrated radio spectrum of the remnant radio galaxy using the
data in Table 2. Peak flux density is reported from our low-resolution, 15′′
image at the 1050–1450 MHz band for the remnant radio galaxy (see also Sec.
3.2.1 for a discussion). We also show the spectra of the southern cD galaxy
for comparison; note that the (nearly) straight spectra of it provides a good
check of our calibration. A 2.8$\sigma$ limit is reported at the 74 MHz (VLSS;
Cohen et al., 2007), peak flux densities at the 150 MHz band, and 1050–1450ṀHz
band are our measurements; the 1400 MHz data is from the VLA FIRST survey
(Chatzikos, Sarazin & Kempner, 2006) for the southern cD galaxy.

####  3.2.1 Radio spectra

The total flux densities were determined from integration in polygons with
measured in areas encompassing the remnant radio galaxy, i.e., we determined
the surface-brightness and divided it by the number of pixels in the polygon
at each frequency. These results are presented in Table 2 along with the total
intensity spectra of the remnant radio galaxy, and the southern cD galaxy is
shown in Figure 5. A 2.8$\sigma$ upper limit at the 74 MHz (VLSS at 80′′
angular resolution; Cohen et al., 2007) and the peak flux density from our 150
MHz band data for the southern cD galaxy are also reported. We see that all
these radio measurements, within the error bars, fall nearly along with a
‘straight’ power law, $\alpha$ = $-$1.2 $\pm$0.2 with some hints of energy
losses by synchrotron cooling for the dominant southern cD galaxy. This
provides confidence in our data calibration and its reduction methodologies.

The remnant radio galaxy is not detected in our 150 MHz band short
observation, and is barely detected using LOFAR at 153 MHz (see also Sec.
3.2). However in the 1050–1450 MHz band short observation, we see hints of
radio emission from the remnant radio galaxy. We made a low-resolution, 15′′
image at the 1050–1450 MHz band and report peak flux density for the radio
emission from the remnant radio galaxy in Figure 5. Since this 1050–1450 MHz
band measurement comes from a short $\lesssim$0.9 hr duration observation, has
relatively high rms noise (see also Sec. 3), and the observation does not
sample similar ($u,v$)-spacings as the rest of the low-frequency observations,
we henceforth no longer include it in our spectral index analysis. It appears
that the integrated spectrum for the remnant radio galaxy cannot be
represented by a single power law. The segment of the spectrum with the best
data is $<$ 1250 MHz, and the low-frequency spectral index $\alpha$(240–610
MHz) = $-$1.4 $\pm$0.2. Our data cannot rule out the possibility of spectral
shape to be due to two populations of the relativistic electrons: a low-
frequency component characterized by $\alpha$(240–610 MHz) $\approx$ $-$1.4
and a second component having $\alpha$($>$ 610 MHz) $<$ $-$1.4 with a cutoff
or a spectral break between 400 and 1250 MHz.

---
Figure 6.— The plot shows the radio spectra between the 240 MHz and 250–500
MHz band data sets (red upper triangles) and between the 250–500 MHz band and
610 MHz band data sets (blue lower triangles) as a function of distance from
the head. The location of the pinch is shown as a vertical line.
---
Figure 7.— The averaged in-band spectral index image showing spectra between
300 MHz and 500 MHz using 250–500 MHz band data. The total intensity surface-
brightness contours are at 0.23 and 0.46 mJy beam-1 from the 300–500 MHz radio
image (Figure 4, top-right panel). We used three irregular polygon regions,
marked in the image, and determined the mean (and rms) spectral indices that
are denoted at the upper-right corner of the image. The averaged integrated
in-band spectral index for the remnant radio galaxy is denoted at the lower-
left corner of the image. The cross marks correspond to the positions of two
cD galaxies and the position of the optical host galaxy, discussed in Sec. 3.1
and 3.2. The dashed line indicates the best-fitting position of the X-ray
surface-brightness edge discussed in Sec. 3.3.
---
Figure 8.— Background-subtracted and exposure-corrected Chandra image at
$\sim$4′′ angular resolution in the broad (0.5–7.0 keV) energy band. The color
bar shows the surface-brightness, in photon cm-2 s-1 pixels-1. The black
surface-brightness contours are displayed using the 250–500 MHz band image.
The inset shows the enlargement of the inner region. Also shown are two sets
of sectors (in cyan and blue colors) that were used for the surface-brightness
and temperature measurements.

The spectral index measurements between the 240 MHz band and 250–500 MHz band
data sets, and between the 250–500 MHz band and 610 band MHz band data sets
show that the radio spectrum steepens with distance from the head, from $-$1.2
$\pm$0.1 to $-$1.4 $\pm$0.2, and from $-$1.3 $\pm$0.1 to $-$1.6 $\pm$0.2,
respectively, due to synchrotron and inverse Compton losses (Figure 6). The
location after the pinch, where the radio plasma from the AGN connects to the
remnant radio galaxy, the spectrum, between the 240 MHz band and 250–500 MHz
band data, remains flattened, $\alpha$ = $-$1.3 $\pm$0.1.

The wide-bandwith data sets were imaged to make multiscale multifrequency
synthesis images, which provides images at center frequencies and the
corresponding in-band spectral index maps (see also Sec. 2.2). The in-band
spectral index map averaged over a region of interest, to increase signal-to-
noise ratio, for the 250–500 MHz band data is shown in Figure 7. We divided
the remnant radio galaxy into three distinct regions: (1) the high-surface-
brightness narrower southeastern part ahead of the pinch (in cyan). The latter
low-surface-brightness broader northwestern part is divided into two; (2) the
northeastern region (in yellow), and (3) the southwestern far end region (in
magenta). The size of the regions is chosen such that the signal in the total
intensity image is (approximately) more than five times the rms noise. The
southern cD galaxy has an in-band spectral index of $-$0.9 $\pm$0.2.
Similarly, the spectra of three regions of the remnant radio galaxy are $-$1.2
$\pm$0.2, $-$1.3 $\pm$0.2, and $-$1.4 $\pm$0.2 for regions (1), (2), and (3)
respectively. This is consistent with the averaged in-band spectral index of
$-$1.3 $\pm$0.2 within the error bars for the remnant radio galaxy. It is also
consistent with the crude estimates of two 50 MHz subband images, 300–350 MHz
and 450–500 MHz, at two extreme ends of the 250–500 MHz band data. The region
ahead of the pinch (in cyan) and the northeastern region close to two cD
galaxies (in yellow) have comparable spectral indices. It appears that the
high-surface-brightness narrower region (in cyan) ahead of the pinch, is
flatter than the two regions, together forming low-surface-brightness broader
northwestern region (in yellow and magenta) after the pinch, consistent with
our result discussed above. Furthermore, the area between the two regions of
the broader northwestern part of the source, i.e., the southwestern far-end
region (in magenta) have a significantly steeper spectrum than the
northeastern region close to two cD galaxies (in yellow).

###  3.3. ICM gas properties

Chatzikos, Sarazin & Kempner (2006) reported that the cluster contains gas
with an extremely wide range of temperatures and a wealth of structure. They
also reported a discontinuity at $\sim$30′′ ($\simeq$ 41.5 kpc) in the
southeast region from the southern cD galaxy. It is at nearly this same
distance where we report an enhancement of radio surface-brightness, toward
the two cD galaxies, showing a sharp feature. We aim to search an X-ray
surface-brightness edge, i.e., the shock front that is coincident with the
northeastern sharp edge toward two cD galaxies of the remnant radio galaxy. In
order to understand the gas physics at this region of the cluster, we have
extracted the surface-brightness profile from two box-shaped regions, one
encompassing the remnant radio galaxy (called “outside” here) and the other
adjacent to it toward cD galaxies (called “inside” here) as shown in Figure 8
(in magenta). We extracted the surface-brightnesses and the spectra from these
regions, which are displayed in Figure 9. A single temperature model with
Galactic absorption and the abundance fixed provided a good fit to the
spectrum.

We noticed the X-ray surface-brightness edge, where the surface-brightness
drops (see Figures 8 and 9) and therefore implies a discontinuity, or jump, in
the gas density. The surface-brightnesses are $S_{\rm x}$(inside) and $S_{\rm
x}$(outside) $\approx$ 8.2 $\pm$0.3 $\times$ 10-6 counts s-1 arcsec-2 and 3.7
$\pm$0.3 $\times$ 10-6 counts s-1 arcsec-2, respectively. Consequently, our
modeled densities in the two regions, $\rho$(inside) and $\rho$(outside) are
approximately 2.9${}^{+0.2}_{-0.1}$ $\times$ 10-3 cm-3 and
1.9${}^{+0.2}_{-0.2}$ $\times$ 10-3 cm-3, respectively. Our corresponding
derived temperatures for the two regions, $kT_{\rm(inside)}$ and
$kT_{\rm(outside)}$ are approximately 6.1${}^{+0.9}_{-0.8}$ keV ($\chi^{2}$ =
0.93 for 221 degrees of freedom) and 5.5${}^{+0.5}_{-0.4}$ keV ($\chi^{2}$ =
1.08 for 462 degrees of freedom), respectively. We also repeated this exercise
for a pie-circular aperture (shown in blue in Figure 8) centered on the
southern cD galaxy and lying toward the radio emission from the remnant radio
galaxy with two regions containing the radio emission from the remnant radio
galaxy and the smaller adjacent region toward the cD galaxy. The quantitative
estimates from this are nearly identical and the fit statistics are at 90%
confidence intervals. Although with large error bars, these density and
temperature measurements are hinting that the surface-brightness edge toward
the two cD galaxies is possibly a shock front. Consequently, the observed
density jump and the temperature jump across the edge,
$\rho_{\rm(inside)}$/$\rho_{\rm(outside)}$ = 1.5${}^{+0.2}_{-0.2}$ and
$kT_{\rm(inside)}$/$kT_{\rm(outside)}$ = 1.1${}^{+0.7}_{-0.5}$ suggest the
Mach numbers, $M$ = 1.3${}^{+0.2}_{-0.2}$ and 1.1${}^{+0.4}_{-0.3}$,
respectively under the Rankine-Hugoniot shock conditions for the intracluster
gas, $\gamma$ = 5/3, which are all obviously in agreement, are within the
error bars of Chatzikos, Sarazin & Kempner (2006). We also believe that a
smaller measure for the temperature jump is due to the contribution of cool
thermal emission from the core to the adjacent region toward two cD galaxies,
$kT_{\rm(inside)}$. Note, it is equally possible that the surface-brightness
edge is a cold front. Here, we assume it to be a shock front because of
consistencies between our measurements and those of Chatzikos, Sarazin &
Kempner (2006) and Markevitch, Sarazin & Vikhlinin (1999).

---
Figure 9.— Plot of the radial surface-brightness profile and the temperature
profile for the two red sectors (shown in Figure 8) as a function of distance
from the southern cD galaxy. The error bars represent 90% confidence
intervals.

##  4\. Discussion

In passing, we note that we do not detect the radio halo of A2065 in our
current $\approx 8^{\prime\prime}$ resolution 250–500 MHz band image. This is
likely to be due to the fairly short duration of our observing run (just
$\approx 1.3$ hr), which results in poor ($u,v$)-coverage at the short
spacings that are critical to detect $\approx 10^{\prime}$-sized structures
like the radio halo (e.g. Farnsworth et al., 2013). However, the large
fractional bandwidth at 250–500 MHz band implies that it has excellent
($u,v$)-coverage at the intermediate and long baselines that are needed to
accurately map the arcminute-sized structures like our target, the remnant
radio galaxy (angular size $\approx 1^{\prime}\times 2^{\prime}$). In
addition, we have performed deep clean, where we have used the multiscale
multifrequency synthesis process of combining data from multiple spectral
channels onto the same spatial-frequency grid during imaging to take advantage
of the increased ($u,v$)-coverage and imaging sensitivity, and hence we do not
foresee any loss of flux density that could be linked to the remnant radio
galaxy in the data presented above. Below we discuss the nature of this newly
discovered remnant radio galaxy in A2065.

As the remnant radio galaxy ages, its electrons will lose so much energy that
their spectra will eventually steepen. This steepening would reach to a point
where they do not emit significant radiation at high frequencies, instead, the
radio emission may only be observable at low frequencies. However, Brienza et
al. (2017) argues that in addition to the aged population of remnant radio
galaxies, there should be younger population as well in which lobes have not
had time to steepen over the observed frequency range (e.g., 3C 28 and MIDAS
J230304$-$323228 in Feretti et al., 1984; Quici et al., 2021, respectively).
It is possible that the remnant phase of a radio galaxy is governed by a
Sedov-like expansion, which will cause the dimming of radio emission due to a
decrease in the magnetic field strength and a decrease in particle energies
due to adiabatic expansion losses. Unfortunately, the models of the spectra of
remnant radio galaxy are highly degenerate, and the modeling of an individual
remnant radio galaxy cannot constrain the remnant phase of radio galaxy
(Kaiser & Cotter, i2002). Therefore, a possible way to constrain the remnant
phase is via a statistical approach, and sadly, due to their small number, the
physics of remnant radio galaxies remains poorly understood. Of course,
alongside their must exist a significant population of remnant radio galaxies,
which show the absence of detectable radio emission in many clusters
associated with merger activity (see also Godfrey, Morganti & Brienza, 2017).
Thus, searching for low-surface-brightness profiles, say $\lesssim$ 50 mJy
arcmin-2, with an absence of radio cores and hotspots, and with no well-
defined radio morphologies, is a possible way to identify remnant radio
sources in large survey images.

Radio morphology of the remnant radio galaxy in A2065 shows that it is an
elongated, bar-shaped structure (52′′ $\times$ 110′′ $\simeq$ 72 $\times$ 152
kpc2), which is buried within a radio halo emission of size $\lesssim$ 1 Mpc
in size (Drabent et al., 2017) and (Farnsworth et al., 2013). The remnant
radio galaxy is located in the near vicinity of the southern cD galaxy; the
northeastern boundary of the remnant radio galaxy is merely at a distance of
$\sim$33′′ ($\simeq$ 45.7 kpc) from the southern cD galaxy. The possible radio
core of the remnant radio galaxy is coincident with the WISEA
J152228.01$+$274141.3 source that is 7$\farcs$8 away from the optical galaxy
V1CG 136. Our spectral index measurements show that the radio spectrum
steepens with distance from the possible radio core from $\alpha$ = $-$1.2
$\pm$0.1 to $\alpha$ = $-$1.4 $\pm$0.2 in our low-frequency (between the 240
MHz band and the 250–500 MHz band) spectra, and from $\alpha$ = $-$1.3
$\pm$0.1 to $\alpha$ = $-$1.6 $\pm$0.2 in our high-frequency (between the
250–500 MHz band and the 610 MHz band) spectra (Figure 6), as is expected for
synchrotron and inverse Compton losses. We also find evidence for spectral
steepening across the remnant radio galaxy in our in-band spectral index
measurements, from $\alpha$ = $-$1.3 $\pm$0.2 to $\alpha$ = $-$1.4 $\pm$0.2
(Figure 7), in the direction away from the two cD galaxies. Our X-ray
measurements show a surface-brightness edge that is coincident with the
northeastern boundary of the remnant radio galaxy and its nature (shock or
cold front) is not clear from the data. If it is a shock front, it is rather
weak, with $M$ being between 1.1 and 1.3; however it is unclear if an
(inward?) traveling shock is responsible for spectral steepening across the
remnant radio galaxy. Below we discuss the possible progenitor of the remnant
radio galaxy in A2065.

###  4.1. Remnant source diagnostic

Remnant radio sources are old, and are bound by model-dependent physical
parameters because of the required assumptions of volume-filling factors, etc.
We use the parameter $N$, which is the number of relativistic electrons found
by integrating the energy spectrum of radiating electrons between Lorentz
factors of 1000 and 10,000 for a typical equipartition magnetic field of 5
$\mu$G in the 2–200 MHz band for the remnant radio galaxy. The number
distribution of relativistic electrons drops rapidly with energy for a steep
spectrum remnant radio galaxy. Hence the number of low-energy electrons is a
good measure of the total acceleration of relativistic electrons during the
lifetime of $\sim$108 yr for the small redshift of the source, and indicates
the entire history of the source.

We thus determine $N$ $\approx$ 2.1 $\times$ 1060 and the total energy stored
in the magnetic field $\approx$ 1.1 $\times$ 1057 erg assuming cylindrical
geometry for the size of the remnant radio galaxy (Miley, 1980; Pacholczyk,
1970). Here, we have assumed the volume filling factor is unity, and hence the
volume and the number of relativistic electrons $N$ are thus the upper limits.
The emitting plasma will most likely occupy a smaller volume, leading to
larger energy densities and therefore larger values of the magnetic field
strength and will require a smaller number of electrons to produce the
observed luminosity. These basic energetics are similar to the Coma radio
relic source 1253$+$275 (Giovannini, Feretti & Stanghellini, 1991). Moreover,
the number of relativistic electrons, $N$, is similar to that of the typical
FR I radio galaxy 3C 31 but two orders of magnitude smaller than the FR II
radio galaxy Cygnus A (Harris et al., 1993). This suggests that the FR I radio
galaxy is possibly the progenitor of the remnant radio galaxy in A2065.

Thus, our high-resolution, high-sensitivity radio image at the 250–500 MHz
band along with the presence of the WISEA J152228.01$+$274141.3 source suggest
that the source is a wide-angle-tailed radio source (see also Sec. 3.2).
Furthermore, the position of the WISEA J152228.01$+$274141.3 source exactly
coincides with the likely radio core position of the wide-angle-tailed radio
source. The source diagnostic based on the number of low-energy relativistic
electrons also suggests that the remnant radio galaxy is fed by the small
radio tail of the wide-angle-tailed FR I radio galaxy. If this proposition is
correct, the two oppositely directed radio jets emanating from the apex of the
WISEA J152228.01$+$274141.3 source initially traverse toward the northwest and
southeast. As the galaxy plows through the dense ICM, these jets traversing in
opposite directions form a trail after a sharp bend in the southeastern jet
due to interaction with the ICM. The jets probably overlap and twist forming a
pinch and then expand, and the jet becomes weaker (e.g., NGC 4869; Lal,
2020a). The sharpness in surface-brightness toward the northeast, i.e., the
side facing two cD galaxies and diffuse extensions toward the far southwest
side, along with corresponding spectral gradients, suggests the passage of an
(inward?) traveling shock front.

##  5\. Summary and conclusions

The remnant radio source seems to trace AGN radio plasma that has somehow been
reenergised through processes in the ICM. It is possible that, similar to
radio relic sources, remnant radio sources are also believed to be tracers of
merger shocks, unfortunately, only a few clusters have shown to possess
corresponding features in hot X-ray gas. The remnant radio galaxy in A2065 may
belong to this (rare) class possessing an X-ray surface-brightness edge across
the remnant radio galaxy, indicating a possible connection with the shock
front. Using our deep high-resolution low-frequency observations, we
discovered a bar-shaped remnant radio galaxy, whose size is $\approx$ 52′′
$\times$ 110′′ (= 72 $\times$ 152 kpc2), remnant radio galaxy in the massive
unequal-mass-merger galaxy cluster A2065 at $z$ = 0.072. It has a steep
spectral index $\alpha$(240–610 MHz) = $-$1.4 $\pm$0.2 and $\alpha$($>$ 610
MHz) $<$ $-$1.4 $\pm$0.2, corresponding to the line representing the best-
fitting regression to the radio data. The Chandra data, and the upgraded and
“classic” GMRT data, show (i) the presence of an X-ray surface-brightness edge
attributed to a shock front, though a possibility of cold front cannot be
ruled out, at the remnant radio galaxy’s edge toward the two cD galaxies, (ii)
the tentative flattening of the radio spectral index, in our 250–500 MHz in-
band data, of the remnant radio galaxy at the near end of the X-ray surface-
brightness edge, and (iii) possibly a wide-angle-tailed, FR I radio galaxy is
the progenitor of the remnant radio galaxy. Thus, the remnant radio galaxy has
possibly been reinvigorated by the passage of the shock front and adiabatic
compression, and shows the expected change in radio emission. We also suggest
that the remnant radio galaxy was seeded by the AGN during its past active
phase, and has been hosted by the WISEA J152228.01$+$274141.3 source,
demonstrating the connection between AGNs and remnant radio galaxy, similar to
the connection between AGNs and radio relic sources (e.g., A3411-3412, van
Weeren et al., 2017).

Although low-frequency observations are starting to reveal more and more of
these type of sources, presently there are a very few known remnant radio
systems; clearly an outstanding difficulty is the small sample size. Although
the number of known remnant radio sources is increasing, better data are
necessary for several of them to allow a detailed study. Fortunately,
forthcoming radio surveys using the LOFAR, MeerKAT, Square Kilometre Array,
etc., will greatly increase the sample size, along with deep multifrequency
radio data, e.g., using the upgraded GMRT, which will constrain the physics
and nature of poorly understood remnant radio sources. We are undertaking an
upgraded GMRT study at the 125–250 MHz, 250–500 MHz, and 550–850 MHz bands of
several clusters in order to detect such ultra-steep-spectrum sources, map
radio halo emission, and provide exact flux density estimates, models, and
statistics.

## Acknowledgments

I would like to thank the anonymous referee for helpful remarks. D.V.L.
acknowledges the support of the Department of Atomic Energy, Government of
India, under project No. 12-R&D-TFR-5.02-0700, and thanks the staff of the
GMRT who made these observations possible. The GMRT is run by the National
Centre for Radio Astrophysics of the Tata Institute of Fundamental Research.
This research made use of the radio astronomical database GalaxyClusters.com,
maintained by the Observatory of Hamburg. This research has made use of the
NED, which is operated by the Jet Propulsion Laboratory, Caltech, under
contract with the NASA, and NASA’s Astrophysics Data System.

Facilities: Chandra, GMRT

Data Availability: The GMRT and Chandra data underlying this article are
available via the GMRT Online Archive, naps.ncra.tifr.res.in/goa and the
Chandra Data Archive, cxc.harvard.edu/cda. All data analyses packages used in
this work are publicly available.

## References

  * Arnaud (1996) Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, ASP Conf. Ser. Vol.101, Astronomical Data Analysis Software and Systems V. Astron. Soc. Pac., San Francisco, p. 17
  * Brienza et al. (2017) Brienza, M., Godfrey, L. E. H., Morganti, R., et al. 2017, A&A 606, A98
  * Chatzikos, Sarazin & Kempner (2006) Chatzikos, M., Sarazin, C. L., & Kempner, J. C., 2006, ApJ, 643, 751
  * Cohen et al. (2007) Cohen A.S., Lane W.M., Cotton W.D., et al. 2007, AJ, 134, 1245
  * Dickey & Lockman (1990) Dickey, J. M., & Lockman, F. J., 1990, ARA&A, 28, 215
  * Drabent et al. (2017) Drabent, A., Hoeft, M., Brüggen, M., et al., 2017, in “Diffuse synchrotron emission in clusters of galaxies - what’s next?”, https://indico.cern.ch/event/617524/contributions/2743186/
  * Farnsworth et al. (2013) Farnsworth, D., Rudnick, L., Brown, S., Brunetti, G. 2013, ApJ, 779, 189
  * Feretti et al. (1984) Feretti L., Gioia I. M., Giovannini G., Gregorini L., Padrielli L. 1984, A&A, 139, 50
  * Giovannini & Feretti (2002) Giovannini, G. & Feretti, L. 2002 in Merging Processes in Galaxy Clusters, eds. L. Feretti, I. M. Gioia, & G. Giovannini, ASSL, Kluwer Ac. Publish., p. 197
  * Giovannini, Feretti & Stanghellini (1991) Giovannini, G., Feretti, L., Stanghellini, C., 1991, A&A, 252, 528
  * Godfrey, Morganti & Brienza (2017) Godfrey, L. E. H., Morganti, R. & Brienza, M. 2017, MNRAS, 471, 891
  * Griffin, Dai & Kochanek (2014) Griffin, R. D., Dai, X., & Kochanek, C. S., 2014, ApJL, 795, L21
  * Harris et al. (1993) Harris, D. E., Stern, C. P., Willis, A. G., & Dewdney, P. E., 1993, AJ 105, 769
  * Hoeft et al. (2004) Hoeft, M., Brüggen, M., Yepes, G., et al. 2004, MNRAS, 347, 389
  * Kaiser & Cotter (i2002) Kaiser, C. R. & Cotter, G. 2002, MNRAS, 336, 649
  * Kempner et al. (2004) Kempner, J. C., Blanton, E. L., Clarke, T. E. et al. in proc. of The Riddle of Cooling Flows in Galaxies and Clusters of Galaxies, p. 335
  * Lal (2020) Lal, D. V., 2020, ApJS, 250, 22
  * Lal (2020a) Lal, D. V., 2020, AJ, 160, 161
  * Lal et al. (2019) Lal, D. V., Sebastian, B., Cheung, C. C., Rao, A. P. 2019, AJ, 157, 195
  * Markevitch, Sarazin & Vikhlinin (1999) Markevitch, M., Sarazin, C. L., & Vikhlinin, A., 1999, ApJ, 521, 526
  * Miley (1980) Miley, G. K., 1980, ARA&A, 18, 165
  * Pacholczyk (1970) Pacholczyk, A. G., 1970, Radio Astrophysics (San Francisco: W. E. Freeman & Co.)
  * Parma et al. (2007) Parma, P., Murgia, M., de Ruiter, H. R., et al. 2007, A&A, 470, 875
  * Postman, Geller & Huchra (1988) Postman M., Geller, M. J., & Huchra, J. P. 1988, AJ, 95 267
  * Quici et al. (2021) Quici, B., Hurley-Walker, N., Seymour, N., et al. 2021, PASA, 38, 8
  * Struble & Rood (1999) Struble, M. F., & Rood, H. J., 1999, ApJS, 125, 35
  * van Weeren et al. (2019) van Weeren, R. J., de Gasperin, F., Akamatsu, H., et al. 2019, Space Sci. Rev., 215, 16
  * van Weeren et al. (2017) van Weeren, R. J., Andrade-Santos, F., Dawson, W. A., et al. 2017, Nat. Astron. 1, 5