#  Beamforming Design for Integrated Sensing and Wireless Power Transfer
Systems

Qianyu Yang, , Haiyang Zhang, , and Baoyun Wang,

###### Abstract

This letter proposes a new concept of integrated sensing and wireless power
transfer (ISWPT), where radar sensing and wireless power transfer functions
are integrated into one hardware platform. ISWPT provides several benefits
from the integrating operation such as system size, hardware cost, power
consumption, and spectrum saving, which is envisioned to facilitate future 6G
wireless networks. As the initial study, we aim to characterizing the
fundamental trade-off between radar sensing and wireless power transfer, by
optimizing transmit beamforming vectors. We first propose a semi-definite
relaxation-based approach to solve the corresponding optimization problem
globally optimal, and then provide a low-complexity sub-optimal solution.
Finally, numerical results verify the effectiveness of our proposed solutions,
and also show the trade-off between radar sensing and wireless power transfer.

Index terms— Radar sensing, wireless power transfer, integrated sensing and
wireless power transfer (ISWPT).

##  I Introduction

The capability of Internet of Everything (IoE) services is expected to be an
important evolution metric of Beyond-5G networks [1], for supporting many new
emerging applications such as intelligent vehicle networks [2]. These
ambitious targets rely on sensing ability of wireless netwoks with radar
sensing [3] and massive low-power sensors [4]. For example, intelligent
vehicle networks that require sensors to know the road and vehicle status
through radar beams emitted by roadside units [5]. Meanwhile, due to the size
and hardware cost constraints, these low-power sensors are usually powered by
a limited battery or even be battery-less [6, 7]. Radio frequency-based
wireless power transfer (WPT) is regarded as a promising technology to solve
the energy shortage problem of low-power devices, which is capable of charging
wireless devices in a wireless manner [8, 9, 10].

It is worth pointing out that conventional radar sensing and WPT are studied
separately as in the works mentioned above. In this letter, we propose a novel
concept that integrates radar sensing and WPT into one hardware platform, and
call the considered scenario as integrated sensing and wireless power transfer
(ISWPT). The proposed ISWPT system has many promising application scenarios
such as intelligent vehicle networks [11], enabling the implementation of
radar sensing and charge sensors simultaneously. With such integrated
operation, ISWPT has several advantages compared to conventional separately
designs: i) the integration conserves the device cost and spectrum resources,
which is significant to future 6G networks; ii) the radar beam usually
provides continuous coverage of a range of areas, which facilitates a stable
energy supply to equipment in the area, and meanwhile, the collecting energy
from radar signals also reduces power waste of radars.

We consider a novel ISWPT scenario, where both the radar sensing and WPT are
integrated into one hardware platform. The multi-antenna transmitter generates
one type of waveforms to achieve the two functions simultaneously. As these
waveforms share the same antenna and transmit power resources, there exists a
fundamental performance trade-off between radar sensing and WPT. We
characterize this fundamental performance trade-off by optimizing the
transmitted signal beamforming vectors for the ISWPT system. The main
contributions of this letter are summarized as follows.

  * •

To the best of our knowledge, we are the first to propose the concept of
ISWPT, which enables the radar sensing and wireless power transfer sharing the
same hardware platform, spectrum and power resources. Specifically, we
consider a general ISWPT system, where a multiple-antenna transmitter serves
multiple energy receivers and senses an area simultaneously.

  * •

Under the considered scenario, we characterize the fundamental performance
trade-off between radar sensing and WPT, by optimizing the transmit
beamforming vectors. Though the corresponding beamforming design problem is
non-convex, we solve it globally optimally by utilizing the SDR approach.
Moreover, we also propose a low-complexity sub-optimal beamforming design to
further reduce the complexity of the optimal design.

  * •

We provide numerical results to demonstrate the fundamental performance trade-
off between radar sensing and WPT. Numerical results also verify the
effectiveness of the proposed beamforming designs for the novel ISWPT system.

Notation: Scalar variables, vectors and matrices are represented with lower
letters, lower bold letters, and capital bold letters, respectively (e.g.,
$x$, $\bf x$, and $\bf X$, respectively). The term $\mathbb{C}^{N\times N}$
denotes a complex space of dimension ${N\times N}$. $\left(\cdot\right)^{H}$
denotes the Hermittian operators. For a matrix $\bf R$, ${\bf R}\succeq{\bf
0}$ means that $\bf R$ is positive semidefinite, while ${\bf
R}_{\left[i:j,i:j\right]}$ denotes the extraction operation for corresponding
rows and columns of $\bf R$. ${\bf 0}_{M\times N}$ denotes the $M\times N$
dimension zero matrix.

##  II System Model

###  II-A Transmitted Signal Model

Figure 1: Illustration of one ISWPT scenario, where one multi-antenna array
simultaneously senses a range of area and charges multiple energy receivers.

We consider a novel ISWPT system, where one transmitter equipped with $N$
antennas serves $M$ energy receivers and sense an area simultaneously, as
shown in Fig. 1. Unlike conventional integrated radar and communication setups
where both the radar and information signals are required, here we consider
utilizing only radar signals to realize sensing and WPT. This is because
energy receivers can also harvest energy from radar signals, and there is no
need to distinguish radar signals and WPT signals in our considered ISWPT
systems. Thus, the transmitted base-band signal can be expressed as

${\bf x}=\sum_{n=1}^{N}{\bf w}_{n}s_{n},$ (1)

where ${\bf w}_{n}\in\mathbb{C}^{N\times 1}$, denotes the beamforming vector
for radar sensing and energy receivers; $s_{n}$ denotes the transmitted radar
symbols, which are independently and uncorrelated with each other and satisfy
$\mathbb{E}\left[|s_{n}|^{2}\right]=1$.

Define the covariance matrix of the transmitted signal by ${\bf
R}=\mathbb{E}\left[{\bf x}{\bf x}^{H}\right]$, according to (1), which is
given by

${\bf R}=\sum_{n=1}^{N}{\bf w}_{n}{\bf w}_{n}^{H}.$ (2)

Consider the per-antenna transmit power constraints, the transmit covariance
matrix ${\bf R}$ should satisfy

${\bf R}_{n,n}=P_{\rm t}/N,$ (3)

where $P_{\rm t}$ denotes the total transmit power budget.

###  II-B Radar Sensing and WPT Metrics

For radar sensing, the baseband signal at direction $\theta$ is expressed as
$y\left[\theta\right]={\bf a}^{H}\left(\theta\right){\bf x}$, where ${\bf
a}\left(\theta\right)=\left[1,e^{j2\pi\Delta\theta},\cdots,e^{j2\pi({\rm
N-1})\Delta\theta}\right]^{T}\in\mathbb{C}^{N\times 1}$ is the array steering
vector of direction $\theta$ with $\Delta$ being the spacing between adjacent
antenna elements. The goal of radar beamforming design is to match the desired
beam pattern. Consequently, similar to [12], we can use the following loss
function to evaluate the radar performance

$\displaystyle L_{\rm r}\left({\bf R}\right)$
$\displaystyle=\frac{1}{L}\sum_{l=1}^{L}\left|\alpha
d\left(\theta_{l}\right)-{\bf a}^{H}\left(\theta_{l}\right){\bf R}{\bf
a}\left(\theta_{l}\right)\right|^{2},$ (4)

where $d\left(\theta\right)$ is the given desired beam pattern, and
$\left\\{\theta_{l}\right\\}_{l=1}^{L}$ denote $L$ sampled angle grids, and
$\alpha$ is a scaling factor.

For WPT operation, the received power of the $m$-th user can be represented as
[13]

$\displaystyle P_{m}\left({\bf R}\right)=\zeta{{\bf g}_{m}}{\bf R}{\bf
g}^{H}_{m},$ (5)

where $0<\zeta<1$ is a constant energy conversion efficiency, and ${\bf
g}_{m}\in\mathbb{C}^{1\times N}$ denotes the wireless channel between the
transmitter and the $m$-th energy receiver.

To unify units with radar metric, the performance metric of WPT in the ISWPT
system is given by

$\displaystyle L_{\rm e}\left({\bf
R}\right)=\frac{1}{M}\sum_{m=1}^{M}\left|{P_{m}^{*}}-P_{m}\left({\bf
R}\right)\right|^{2},$ (6)

where $\\{P_{m}^{*}\\}_{m=1}^{M}$ denotes the desired received power of each
energy receiver. Without loss of generality, we set $N\geq M$. Particularly,
$\\{P_{m}^{*}\\}_{m=1}^{M}$ are received power of each energy receiver,
associated with the optimal solution of the following sum-received power
maximization problem

$\displaystyle\max_{{\bf R}\succeq{\bf 0}}~{}\sum_{m=1}^{M}P_{m}\left({\bf
R}\right)\quad s.t.~{}~{}\eqref{powerlimit}.$ (7)

Problem (7) is convex that can be directly solved using existing convex
optimization tools such as CVX [14].

###  II-C Problem Formulation

From (4) and (6), we know that both $L_{\rm r}\left({\bf R}\right)$ and
$L_{\rm e}\left({\bf R}\right)$ are functions of the transmit covariance
matrix $\bf R$. In general, the optimal $\bf R$ that minimize the radar metric
$L_{\rm r}\left({\bf R}\right)$ is not optimal for WPT metric $L_{\rm
e}\left({\bf R}\right)$, and vice versa. Therefore, there exists a fundamental
trade-off between the radar sensing and WPT performance in ISWPT systems. In
this letter, we would like to characterize this fundamental performance trade-
off between radar sensing and power transfer by optimizing the transmitted
beamforming vectors (or equivalently transmit covariance matrix).
Mathematically, our interested beamforming design problem is formulated as

$\displaystyle\min_{\\{{\bf w}_{n}\\}}{}$
$\displaystyle\left(1-\rho\right)L_{\rm r}\left({\bf R}\right)+\rho L_{\rm
e}\left({\bf R}\right),$ (8) $\displaystyle s.t.{}{}$ $\displaystyle{\bf
R}=\sum_{n=1}^{N}{\bf w}_{n}{\bf w}_{n}^{H},$ $\displaystyle{\bf
R}_{n,n}=P_{\rm t}/N,$ $\displaystyle{\bf R}\succeq{\bf 0},$

where $0\leq\rho\leq 1$ is a weighting factor that denotes the performance
priority for radar and power transfer tasks in the ISWPT system. The extreme
case, e.g., $\rho=0$ or $\rho=1$ corresponds to the scenario of only radar or
only WPT.

##  III  Beamforming Design

###  III-A Optimal Beamforming Design

The optimization problem (8) is not convex because of the quadratic equality
constraint, and thus it is difficult to solve. To deal with this problem, we
use the semidefinite relaxation (SDR) approach [15] to transform (8) into a
convex semidefinite programming (SDP) problem. For this purpose, we define

${\bf R}_{i}={\bf w}_{i}{\bf w}_{i}^{H}~{}~{}i=1,\cdots,N.$ (9)

By substituting (9) into (8), we can rewritten it as an equivalent quadratic
semidefinite programming (QSDP) with rank-one constraints

$\displaystyle\min_{\\{{\bf R}_{i}\\}}$
$\displaystyle~{}\left(1-\rho\right)L_{\rm r}\left({\bf R}\right)+\rho L_{\rm
e}\left({\bf R}\right),$ (10a) s.t. $\displaystyle{\bf R}=\sum_{i=1}^{N}{\bf
R}_{i},$ (10b) $\displaystyle{\bf R}_{n,n}=P_{\rm t}/N,$ (10c)
$\displaystyle{\bf R}_{i}\succeq{\bf 0},\text{rank}\left({\bf
R}_{i}\right)=1.$ (10d)

In order to make (10) a convex problem, we adopt the SDR approach to omit the
non-convex rank-1 constraint in (10d). The resulting problem is convex which
can be solved directly using existing convex optimization solvers such as CVX.

Let $\\{{\bf\hat{R}}_{i}\\}$ denote the optimal solution to the relaxed
problem of (10). We note that $\\{{\bf\hat{R}}_{i}\\}$ may not satisfy the
rank-1 constraint due to the relaxation operation. To deal with this problem,
commonly used approaches are to find a rank-1 approximation solution by using
Randomization procedure [15], which will inevitably degrade the performance.
Instead of using these approximation methods, we propose an efficient approach
that is capable of directly constructing the optimal solution of (10) from the
relaxed solution, which is summarized as the following theorem.

######  Theorem 1.

Define $\\{{\bf\hat{R}}_{i}\\}$ as the optimal solution of the relaxed problem
of (10). Then the rank-1 optimal solution of (10), denoted by
${\bf\widetilde{R}}_{i}$, can be constructed by

${\bf\widetilde{R}}_{i}={\bf\widetilde{w}}_{i}{\bf\widetilde{w}}_{i}^{H},~{}i=1,\cdots,N,$
(11)

where ${\bf\widetilde{w}}_{i}$ is the $i$-th column vector of the matrix
${\bf\widetilde{W}}={\bf\widetilde{U}}{\widetilde{\boldsymbol{\Lambda}}}^{\frac{1}{2}}{\bf\widetilde{U}}^{H}$,
with ${\bf\widetilde{U}}$ and $\widetilde{\boldsymbol{\Lambda}}$ denoting the
eigenvector matrix and the eigenvalue diagonal matrix of
${\bf\hat{R}}=\sum_{i=1}^{N}{\bf\hat{R}}_{i}$, respectively.

###### Proof:

Define ${\bf\hat{R}}=\sum_{i=1}^{N}{\bf\hat{R}}_{i}$. ${\bf\hat{R}}$ is a
$N$-by-$N$ positive semi-definite matrix, and its eigenvalue decomposition can
be expressed as

${\bf\hat{R}}={\bf\widetilde{U}}{\widetilde{\boldsymbol{\Lambda}}}{\bf\widetilde{U}}^{H},$
(12)

where ${\bf\widetilde{U}}$ and ${\widetilde{\boldsymbol{\Lambda}}}$ are the
eigenvector matrix and the eigenvalue diagonal matrix of ${\bf\hat{R}}$,
respectively.

Define
${\bf\widetilde{W}}={\bf\widetilde{U}}{\widetilde{\boldsymbol{\Lambda}}}^{\frac{1}{2}}{\bf\widetilde{U}}^{H}$
and
${\bf\widetilde{W}}=[{\bf\widetilde{w}}_{1},\cdots,{\bf\widetilde{w}}_{N}]$.
Since ${\bf\widetilde{U}}$ is a unitary matrix, we have

${\bf\hat{R}}={\bf\widetilde{W}}{\bf\widetilde{W}}^{H}=\sum_{i=1}^{N}{\bf\widetilde{w}}_{i}{\bf\widetilde{w}}_{i}^{H}.$
(13)

Define a new set of rank-1 and positive semi-definite matrices,
$\left\\{{\bf\widetilde{R}}_{i}\right\\}$, given by

${\bf\widetilde{R}}_{i}={\bf\widetilde{w}}_{i}{\bf\widetilde{w}}_{i}^{H},~{}i=1,\cdots,N.$
(14)

From (13) and (14), we can verify that
${\bf\hat{R}}=\sum_{i=1}{\bf\widetilde{R}}_{i}$. This implies that
$\left\\{{\bf\widetilde{R}}_{i}\right\\}$ can achieve the same objective
function as the SDR optimal solution. Meanwhile,
$\left\\{{\bf\widetilde{R}}_{i}\right\\}$ can satisfy all the constraints in
(10). Therefore, $\left\\{{\bf\widetilde{R}}_{i}\right\\}$ are the globally
optimal solution to (10).

∎

According to Theorem 1, the corresponding optimal beamforming vectors
$\left\\{{\bf w}_{i}\right\\}$ to (8) can be recovered from
${\bf\widetilde{R}}_{i}$ directly.

The worst case complexity to solve the QSDP (10) is
$\mathcal{O}\left(N^{6.5}N^{6.5}\log(1/\epsilon)\right)$ with the primal-dual
interior-point algorithm [16], where $\epsilon$ is the solution accuracy.

###  III-B Sub-optimal Beamforming Design

In order to reduce the high complexity of optimal beamforming design from SDR
method, we further propose a sub-optimal beamforming design. Specifically, we
still adopt radar beamforming vectors to implement radar sensing and WPT
simultaneously, but consider the radar beamforming matrix has the following
structure:

${\bf W}=\left[{\bf W}_{\rm MRT},{\bf W}^{\bot}_{\rm MRT}\right]$ (15)

where ${\bf W}\in\mathbb{C}^{N\times N}$ denotes the beamforming matrix; ${\bf
W}_{\rm MRT}\in\mathbb{C}^{N\times M}$ is the maximum ratio transmission (MRT)
beams towards energy receivers, i.e., ${\bf W}_{\rm MRT}={\bf
G}^{H}{\bf\Lambda}$, where ${\bf G}^{H}=\left[{\bf g}^{H}_{1},\cdots,{\bf
g}^{H}_{M}\right]$ and ${\bf\Lambda}\in\mathbb{C}^{M\times M}$ is a diagonal
matrix with diagonal element $\lambda_{m}$ denoting the transmit power
allocated for the corresponding beamforming vector; ${\bf W}^{\bot}_{\rm MRT}$
is the orthogonal complement matrix of ${\bf W}_{\rm MRT}$.

By substituting (15) into problem (8), we then have

$\displaystyle\min_{{\bf W}}~{}~{}$ $\displaystyle\left(1-\rho\right)L_{\rm
r}\left({\bf R}\right)+\rho L_{\rm e}\left({\bf R}\right),$ (16a)
$\displaystyle s.t.~{}~{}$ $\displaystyle{\bf R}={\bf W}{\bf W}^{H},$ (16b)
$\displaystyle{\bf R}_{n,n}=P_{\rm t}/N,{\bf R}\succeq{\bf 0},$ (16c)
$\displaystyle{\bf G}{\bf W}=\left[{\bf G}{\bf G}^{H}{\bf\Lambda},{\bf
0}_{M\times(N-M)}\right],$ (16d)

where the (16d) comes from the fact that ${\bf G}{\bf W}^{\bot}_{\rm MRT}={\bf
0}$.

Though problem (16) is non-convex due to the quadratic equality constraint, we
can solve it globally optimal. To this end, we need the following theorem.

######  Theorem 2.

There always exists the covariance matrix ${\bf R}\in\mathbb{C}^{N\times N}$
that satisfies

${\bf G}{\bf R}{\bf G}^{H}={\bf G}{\bf G}^{H}{\bf\Lambda}^{2}{\bf G}{\bf
G}^{H},$ (17)

and for any ${\bf R}$ satisfies (17), there exists ${\bf
W}\in\mathbb{C}^{N\times N}$ that satisfies (16d) with ${\bf R}={\bf W}{\bf
W}^{H}$ and has the structure

${\bf W}={\bf DU}^{H}{\bf U}_{\rm H},$ (18)

where $\bf D$ is the Cholesky decomposition of $\bf R$, $\bf U$ is the unitary
matrix obtained from the the QR decomposition of ${\bf G}{\bf D}$, ${\bf
U}_{\rm H}$ is the unitary matrix obtained from the the QR decomposition of
$\left[{\bf G}{\bf G}^{H}{\bf\Lambda},{\bf 0}_{M\times(N-M)}\right]$.

###### Proof:

Please refer to Appendix A.

∎

From Theorem 2, we know the constraint (16d) is equivalent to (17), and once
$\bf R$ is determined, $\bf W$ can be recovered from (18) directly. Thus, the
optimal solution to (16) can be obtained by solving the problem below

$\displaystyle\min_{\bf R,\Lambda}\quad$
$\displaystyle\left(1-\rho\right)L_{\rm r}\left({\bf R}\right)+\rho L_{\rm
e}\left({\bf R}\right),$ (19a) $\displaystyle s.t.~{}~{}$ $\displaystyle{\bf
G}{\bf R}{\bf G}^{H}={\bf G}{\bf G}^{H}{\bf\Lambda}^{2}{\bf G}{\bf G}^{H},$
(19b) $\displaystyle{\bf R}_{n,n}=P_{\rm t}/N,$ (19c) $\displaystyle{\bf
R}\succeq{\bf 0}.$ (19d)

Problem (19) is a convex QSDP of $\bf R$. The worst case complexity of solving
(19) is $\mathcal{O}\left(N^{6.5}\log(1/\epsilon)\right)$, which is much lower
than that of the optimal beamforming design.

##  IV Numerical Evaluations

In this section, we provide numerical results to demonstrate the fundamental
trade-off between radar sensing and WPT under an novel ISWPT scenario. In our
experiments, we set $P_{t}=1$ W, $N=10$, $M=3$, and $\zeta=0.5$. The wireless
channels are generated according to the standard Complex Gaussian
distribution. The channel attenuation from the transmitter to energy receivers
is identical and set to be $30$ dB. We assume the ideal radar beam pattern
consisting of three main beams with directions
$\left[-40^{\circ},0^{\circ},40^{\circ}\right]$, and the direction grids in
(4) are set as a uniform sample in the range of $-90^{\circ}$ to $90^{\circ}$
with $0.1^{\circ}$ interval.

In Fig. 2, we compare the performance of our proposed schemes in terms of
minimizing the objective function value of problem (10) with two benchmark
schemes (SDR optimal solution and Rank-1 approximation solution [15]), under
different values of weight coefficient $\rho$. From Fig. 2, we can see that
our proposed optimal solution achieves the same performance as the SDR optimal
solution, which implies that our proposed optimal solution is the globally
optimal solution to problem (10). Moreover, as expected, our proposed optimal
solution performs better than the Rank-1 approximation solution and the
proposed low-complexity sub-optimal solution.

Figure 2: The objective function value of (10) versus different weight
coefficient.

In Fig. 3, we evaluate the radar transmit beam patterns for the optimal
beamforming scheme and sub-optimal beamforming scheme. The transmit beam
patterns are depicted in Fig. 3(a) for $\rho=0.1$ and Fig. 3(b) for
$\rho=0.5$, respectively. The radar only beam pattern is also provided as the
optimal beam for radar sensing for comparison, which is obtained by solving
(10) with $\rho=0$. From Fig. 3, we can see that when the weighting factor is
small, e.g., $\rho=0.1$, the beam patterns of optimal and sub-optimal approach
that of the only radar optimal beamforming, whereas the peak value of main
beams of optimal scheme and sub-optimal scheme decreases significantly
compared to that of only radar optimal beam pattern when $\rho=0.5$. This
illustrates the effect of WPT on radar sensing. Moreover, we notice that the
average beam pattern of the sub-optimal scheme is comparable to that of the
optimal scheme, which demonstrates the effectiveness of our proposed low-
complexity sub-optimal design.

(a) $\rho=0.1$

(b) $\rho=0.5$

Figure 3: Radar beam pattern in different weight $\rho$: (a) $\rho=0.1$; (b)
$\rho=0.5$. Figure 4: Trade-off between radar beam pattern MSE and received
power.

In Fig. 4, we explicitly demonstrate the trade-off between WPT and radar
sensing for different beamforming schemes, where points ”Radar only” and ”WPT
only” indicate the performance boundaries of radar and WPT, respectively.
Meanwhile, we can observe that the radar beam pattern MSE of both optimal and
sub-optimal designs increases with the increment of received power of users,
which implies that the increase of received power is at the expense of radar
performance. Moreover, we can see that the trade-off performance gap between
the optimal scheme and sub-optimal scheme increases as the received power
increases. This is because as the received power requirement increases, more
transmit power is required to allocate to MRT beamforming that doesn’t take
the radar performance into account in sub-optimal scheme. Besides, we also
compare the performance of our proposed optimal solution with the commonly
applied, and our proposed optimal solution is better than that of the rank-1
approximation solution in terms of beam pattern MSE, while achieve the same
performance as the SDR optimal solution, this is consistent with our numerical
results in Fig. 2.

##  V Conclusions

In this letter, we studied a new ISWPT setup, in which the radar sensing and
wireless power transfer are integrated into one hardware platform. We
characterized the fundamental trade-off between radar sensing and wireless
power transfer by optimizing transmitted beamforming vectors. Both globally
optimal solution and low-complexity sub-optimal solution are provided.
Simulation results demonstrated the performance trade-off for integrated radar
sensing and wireless power transmission, and also verified the effectiveness
of our proposed beamforming designs.

##  Appendix A Proof of Theorem 2

Define ${\bf G}^{*}=\left({\bf G}{\bf G}^{H}\right)^{-1}{\bf G}$. (17) is then
rewritten as

${\bf G}^{*}{\bf R}{\bf G}^{*H}={\bf\Lambda}^{2}.$ (20)

The eigenvalue decomposition of ${\bf R}$ is given by ${\bf R}={\bf U}_{\rm
r}{\bf\Lambda}_{\rm r}{\bf U}_{\rm r}^{H}$, where ${\bf U}_{\rm
r}\in\mathbb{C}^{N\times N}$ is the eigenvector matrix and ${\bf\Lambda}_{\rm
r}\in\mathbb{C}^{N\times N}$ is the eigenvalue dialog matrix. The column
vectors of ${\bf U}_{\rm r}$ form a set of $N$-dimensional orthogonal bases,
thus there must exist ${\bf U}_{\rm r}={\bf U}_{\rm r}^{*}$ that satisfy ${\bf
G}^{*}{\bf U}_{\rm r}^{*}=\left[{\bf I}_{M},{\bf 0}_{M\times(N-M)}\right]$.
Meanwhile, if we set ${\bf\Lambda}_{{\rm
r}\left[1:M,1:M\right]}={\bf\Lambda}^{2}$, then (20) will hold. Therefore,
there always exists ${\bf R}={\bf U}_{\rm r}^{*}{\bf\Lambda}_{\rm r}{\bf
U}_{\rm r}^{*H}$ satisfies

${\bf G}{\bf R}{\bf G}^{H}={\bf G}{\bf G}^{H}{\bf\Lambda}^{2}{\bf G}{\bf
G}^{H}.$ (21)

Next we compute $\bf W$ from a given $\bf R$. Denote the Cholesky
decomposition of $\bf R$ as ${\bf R}={\bf D}{\bf D}^{H}$. Then, by performing
the QR decomposition to $\left({\bf G}{\bf D}\right)^{H}$, we have ${\bf
G}{\bf D}=\left[{\bf L},{\bf 0}_{M\times(N-M)}\right]{\bf U}$. Hence, we have

${\bf G}{\bf D}{\bf D}^{H}{\bf G}^{H}={\bf LL}^{H}={\bf G}{\bf R}{\bf G}^{H}.$
(22)

Similarly, by defining ${\bf H}=\left[{\bf G}{\bf G}^{H}{\bf\Lambda},{\bf
0}_{M\times(N-M)}\right]$ and writing the QR decomposition of ${\bf H}^{H}$ as
${\bf H}=\left[{\bf L}_{\rm H},{\bf 0}_{M\times(N-M)}\right]{\bf U}_{\rm H}$,
we have

${\bf H}{\bf H}^{H}={\bf L}_{\rm H}{\bf L}^{H}_{\rm H}={\bf G}{\bf R}{\bf
G}^{H}.$ (23)

Comparing (22) and (23), we find that ${\bf L}={\bf L}_{\rm H}$. Thus, we can
set

${\bf W}={\bf DU}^{H}{\bf U}_{\rm H},$ (24)

which satisfy

$\displaystyle~{}~{}~{}~{}~{}~{}{\bf W}{\bf W}^{H}={\bf DU}^{H}{\bf U}_{\rm
H}{\bf U}_{\rm H}^{H}{\bf U}{\bf D}^{H}={\bf R},$ (25a) $\displaystyle{\bf
G}{\bf W}={\bf G}{\bf DU}^{H}{\bf U}_{\rm H}=\left[{\bf G}{\bf
G}^{H}{\bf\Lambda},{\bf 0}_{M\times(N-M)}\right].$ (25b)

Thus, the proof is completed.

## References

  * [1] E. Dahlman, S. Parkvall, J. Peisa, and H. Tullberg, “5G evolution and beyond,” in _2019 IEEE 20th International Workshop on Signal Processing Advances in Wireless Communications_ , 2019, pp. 1–5.
  * [2] M. Giordani, M. Polese, M. Mezzavilla, S. Rangan, and M. Zorzi, “Toward 6G networks: Use cases and technologies,” _IEEE Communications Magazine_ , vol. 58, no. 3, pp. 55–61, 2020.
  * [3] O. B. Akan and M. Arik, “Internet of radars: Sensing versus sending with joint radar-communications,” _IEEE Communications Magazine_ , vol. 58, no. 9, pp. 13–19, 2020.
  * [4] S. Li, Q. Ni, Y. Sun, G. Min, and S. Al-Rubaye, “Energy-efficient resource allocation for industrial cyber-physical IoT systems in 5G era,” _IEEE Transactions on Industrial Informatics_ , vol. 14, no. 6, pp. 2618–2628, 2018.
  * [5] A. Fascista, G. Ciccarese, A. Coluccia, and G. Ricci, “A localization algorithm based on V2I communications and AOA estimation,” _IEEE Signal Processing Letters_ , vol. 24, no. 1, pp. 126–130, 2017.
  * [6] X. Lu, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RF energy harvesting: A contemporary survey,” _IEEE Communications Surveys Tutorials_ , vol. 17, no. 2, pp. 757–789, 2015.
  * [7] H. Zhang, N. Shlezinger, F. Guidi, D. Dardari, M. F. Imani, and Y. C. Eldar, “Near-field wireless power transfer for 6G internet of everything mobile networks: Opportunities and challenges,” _IEEE Communications Magazine_ , vol. 60, no. 3, pp. 12–18, 2022.
  * [8] B. Clerckx, R. Zhang, R. Schober, D. W. K. Ng, D. I. Kim, and H. V. Poor, “Fundamentals of wireless information and power transfer: From RF energy harvester models to signal and system designs,” _IEEE Journal on Selected Areas in Communications_ , vol. 37, no. 1, pp. 4–33, 2019.
  * [9] S. Li and C. C. Mi, “Wireless power transfer for electric vehicle applications,” _IEEE Journal of Emerging and Selected Topics in Power Electronics_ , vol. 3, no. 1, pp. 4–17, 2015.
  * [10] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” _IEEE Transactions on Wireless Communications_ , vol. 12, no. 5, pp. 1989–2001, 2013.
  * [11] F. Liu, W. Yuan, C. Masouros, and J. Yuan, “Radar-assisted predictive beamforming for vehicular links: Communication served by sensing,” _IEEE Transactions on Wireless Communications_ , vol. 19, no. 11, pp. 7704–7719, 2020.
  * [12] F. Liu, C. Masouros, A. Li, H. Sun, and L. Hanzo, “MU-MIMO communications with MIMO radar: From co-existence to joint transmission,” _IEEE Transactions on Wireless Communications_ , vol. 17, no. 4, pp. 2755–2770, 2018\.
  * [13] J. Xu, L. Liu, and R. Zhang, “Multiuser MISO beamforming for simultaneous wireless information and power transfer,” _IEEE Transactions on Signal Processing_ , vol. 62, no. 18, pp. 4798–4810, 2014.
  * [14] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex programming, version 2.1,” 2014.
  * [15] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” _IEEE Signal Processing Magazine_ , vol. 27, no. 3, pp. 20–34, 2010.
  * [16] K.-C. Toh, “An inexact primal–dual path following algorithm for convex quadratic SDP,” _Mathematical programming_ , vol. 112, no. 1, pp. 221–254, 2008.


# The peculiar variable X-ray spectrum of the active galactic nucleus PKS
2005$-$489

Owen Chase Department of Astronomy & Astrophysics, The Pennsylvania State
University, University Park, PA 16802, USA  Felicia McBride Department of
Physics and Astronomy, Bowdoin College, Brunswick, Maine 04011, USA  Andrea
Gokus Department of Physics, Washington University in St. Louis, One Brookings
Drive, St. Louis, MO 63130, USA  McDonnell Center for the Space Sciences,
Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130,
USA  Dr. Karl Remeis-observatory & ECAP, Friedrich-Alexander Universität
Erlangen-Nürnberg, Sternwartstr. 7, 96049 Bamberg, Germany  Lehrstuhl für
Astronomie, Universität Würzburg, Emil-Fischer-Strasse 31, 97074 Würzburg,
Germany  Matteo Lucchini MIT Kavli Institute for Astrophysics and Space
Research, Cambridge, MA 02139, USA  Haocheng Zhang NASA Postdoctoral Program
Fellow  NASA Goddard Space Flight Center Greenbelt, MD 20771, USA  Roopesh
Ojha  NASA HQ, 300 E St SW, Washington DC, 20546-0002, USA  Derek B. Fox
Department of Astronomy & Astrophysics, The Pennsylvania State University,
University Park, PA 16802, USA

(Received 09 Jan, 2023; Revised 07 Mar, 2023; Accepted 16 Mar, 2023)

###### Abstract

PKS 2005$-$489 is a well-known, bright southern BL Lac object that has been
detected up to TeV energies. In a low-flux state it exhibits the expected
multiwavelength double-peaked spectrum in the radio – $\gamma$-ray band. The
high-flux state shows extreme flux variations in the X-ray band with a
hardening as well as a peculiar curved feature in the spectrum. Thus far, PKS
2005$-$489 is the only source to exhibit such a feature. To study the X-ray
variability further, we obtained the first hard X-ray spectrum of the source
with NuSTAR. We compare quasi-simultaneous radio, optical, UV, soft and hard
X-ray, and $\gamma$-ray data of PKS 2005$-$489 to archival data in order to
study its broadband behavior. We find a very consistent quiet state in the
SED, with little variation in spectral shape or flux between the 2012 and 2020
data. A possible explanation for the peculiar X-ray spectrum in the flaring
state is an additional component in the jet, possibly accelerated via magnetic
reconnection, that is not co-spatial to the low-flux state emission region.

galaxies: active — BL Lacertae objects: general — BL Lacertae objects:
individual (PKS 2005-489) — galaxies: jets

††journal: ApJ

##  1 Introduction

Active galactic nuclei (AGN) are among the most luminous persistent sources we
observe. They can be divided into radio-quiet and radio-loud sources based on
the absence or presence of a relativistic outflow of plasma, called a jet. A
radio-loud AGN whose jet is oriented at small angles with respect to our line
of sight is called a “blazar” (Blandford et al. 1978) and often exhibits rapid
variability at all wavelengths. Blazars can further be subdivided into two
classes, flat-spectrum radio quasars (FSRQs) and BL Lacertae (BL Lac) objects
(Urry & Padovani 1995). The two classes likely differ in jet power, accretion
rate and magnetic field strength. Observationally, FSRQs are identified by
their strong emission lines, which are weak or absent in BL Lac type objects.
BL Lac type objects (at weaker jet power) are typically observed as high-
energy peaked objects — i.e., their spectral energy distributions (SEDs) show
synchrotron and high-energy peaks that are observed at $\gtrsim 10^{15}$ Hz
and $\gtrsim 10^{23}$ Hz, respectively. This makes these sources ideal targets
for Very-High Energy (VHE) observatories, including H.E.S.S., MAGIC, VERITAS,
FACT, and the Cherenkov Telescope Array (CTA). In fact, of the 279 sources
listed in TeVCat, 84 are AGN, including 57 high-energy peaked blazars (HBLs)
or BL Lac type sources, one of which is PKS 2005$-$489 (Wakely & Horan 2008).

PKS 2005$-$489 is a BL Lac type blazar at coordinates $\alpha=302\fdg
3558,\delta=-48\fdg 8316$ (Gaia Collaboration 2018), and redshift $z=0.07067$
(Keeney et al. 2018). Its host galaxy is a giant elliptical galaxy (Scarpa et
al. 2000). PKS 2005$-$489 was detected by Parkes in the 2.7 GHz survey in 1972
(Wall et al. 1975). EGRET (20 MeV – 30 GeV band) was the first telescope to
make a $\gamma$-ray detection of the source (Lin et al. 1999), which was later
confirmed in the 30 GeV – 100 TeV VHE band by the H.E.S.S. telescopes in
Namibia (H. E. S. S. Collaboration et al. 2010) . The extreme X-ray
variability of the source was first seen in EXOSAT data by Giommi et al.
(1990), who describe flux changes of a factor of 35$\pm$7 in the 0.7–8 keV
band, as well as variability on 4–5 hour timescales and a hardening of the
spectrum. Sambruna et al. (1995) study the broadband behavior of PKS
2005$-$489 and find similar variability in ROSAT observations. The X-ray
spectrum is analyzed in more detail by Perlman et al. (1999) using RXTE data.
They report on a flare that occurred in 1998, during which the 2–10 keV flux
of the source increased to $3.33\times 10^{-10}$ erg cm-2 s-1, an increase in
flux by a factor of 30. Perlman et al. (1999) discuss the origin of the flare
as particle injection into the jet or in situ particle acceleration with a
hardened X-ray spectrum resulting from synchrotron cooling. BeppoSAX
observations taken during the 1998 flare reveal a slightly curved spectrum
(Tagliaferri et al. 2001).

X-ray observations with XMM-Newton and RXTE in 2009 revealed an unexpected
spectral break at 4 keV. Strong X-ray variability was observed (flux increased
by a factor of $\sim 50$), while the TeV $\gamma$-rays do not show strong
variability (flux increase by a factor of $\sim 2$). A possible explanation is
electron scattering during high-flux states occurring primarily in the Klein-
Nishina regime, resulting in less efficient photon production at TeV energies.
The H. E. S. S. Collaboration & Fermi-LAT Collaboration (2011) provided
multiwavelength data from the 2009 flare in which the flux decreased after the
flare in both the Fermi/LAT and optical bands. The SED was modeled with a one-
zone SSC model, where the X-ray spectrum was explained as a break in the
synchrotron emission. The authors also reported that the VHE $\gamma$-ray flux
hardly changed from quiescence while the X-ray flux varied drastically (H. E.
S. S. Collaboration & Fermi-LAT Collaboration 2011).

No other multiwavelength campaigns of this object has been published since
2011, excluding the work by (Krauß et al. 2016), and no other BL Lac object
show a comparable X-ray flaring behavior as far as we know.

Krauß et al. (2016) presented quasi-simultaneous data for the source that
included a period of low emission ($\sim$ 2012) and a period of high-emission
(2009). The SED shows a deviation from the expected parabola shape, with a
hardened X-ray spectrum in the high state that is incompatible with the
expected synchrotron peak. The origin of the peculiar X-ray feature remains
unknown. It is possible that this is an additional jet component, an extreme
peak shift of the synchrotron peak, an additional hadronic component (proton
synchrotron or a pion cascade), or a signature of magnetic reconnection.
However, the question remains why this seems to be the only source to exhibit
such a peculiar feature.

In this paper, we analyze newly obtained Nuclear Spectroscopic Telescope Array
(NuSTAR) data for PKS 2005$-$489 and compare it with archival data in order to
study the source’s extreme variability in the X-ray band and to constrain the
emission mechanisms at play during the quiescent and flaring states. The paper
is organized as follows. In Section 2, we discuss the observations and the
methods used to analyze the multiwavelength data. In Section 3, we describe
and discuss the results of our observations. In Section 4, we present our
conclusions.

Throughput the paper, we use a cosmological model with $\Omega_{m}=0.3$,
$\Lambda=0.7$, $H_{0}=70$ km s-1Mpc-1 (Beringer et al. 2012).

##  2 Multiwavelength Observations

In this section, we describe the multiwavelength data we extracted and
analyzed, including archival non-simultaneous data that were added to the SED.
We analyze data in three different time periods: 2008–2009 ($\alpha$),
2011–2012 ($\beta$), and 2020 ($\gamma$). Archival data was then added to the
SED. The time ranges designated $\alpha$ and $\beta$ are identical to those
defined in Krauß et al. (2016). The details of the epochs are listed in Table
1. Archival data refers to all other data sets included in the SED that are
not quasi-simultaneous with the other three time ranges.

Table 1: Table of observation time ranges used throughout the paper. The 2020 ($\gamma$) time range is centered on the NuSTAR time range, and the length depends on the instrument used. The $\alpha$ and $\beta$ time ranges are defined by Krauß et al. (2016) by using a Bayesian Blocks analysis with Fermi/LAT light curves. Label | Time [UTC] | Time [MJD] | time criteria
---|---|---|---
$\alpha$ | 2008-08-04 – 2009-08-03 | 54682.6552778 – 55046.6552778 |  Fermi/LAT light curve
$\beta$ | 2011-03-14 – 2012-03-15 | 55634.6552778 – 56264.6552778 |  Fermi/LAT light curve
$\gamma$ | 2020-11-28 07:06:09 | 59181.2959375 |  NuSTAR observation

###  2.1 NuSTAR

NuSTAR is a hard X-ray telescope with focusing optics that is sensitive in the
3 keV-79 keV energy range (Harrison et al. 2013). NuSTAR observed PKS
2005$-$489 on 2020-11-28 07:16:57 UTC (59181.3034 MJD) for about 64 ks (ID
60601016002). Data sets from focal plane modules A and B are grouped and
treated as one data set unless otherwise noted.

The raw data were reprocessed using the nupipeline tool (version 0.4.8) in the
HEASoft data analysis package (version 6.28). Spectra were extracted from the
resulting event files using the nuproducts tool. Focal plane module A source
data were extracted from within a circle of radius 77.364′′ centered on
coordinates 302$\fdg$3605, $-$48$\fdg$8330\. A background region was defined
with a circle of radius 211.153′′ centered on coordinates 302$\fdg$2699,
$-$48$\fdg$9151\. Focal plane module B source data were extracted from within
a circle of radius 86.224′′ at coordinates 302$\fdg$3541, $-$48$\fdg$8326,
while the circular background region was centered at coordinates
302$\fdg$2576, $-$48.9028 with a radius of 234.367′′. The final spectra were
analyzed in the Interactive Spectral Interpretation System (ISIS; Houck &
Denicola 2000). Data were binned to a minimum signal-to-noise ratio of 20. As
the background dominates the data above 25 keV, we only used data in the 3–25
keV range.

###  2.2 Fermi/LAT

The Fermi satellite observes the sky at $\gamma$-ray energies. One of the two
instruments onboard is the Large Area Telescope (LAT), a pair-conversion
telescope, which is sensitive to energies of 20 MeV up to more than 300 GeV
(Atwood et al. 2009). The observing strategy of Fermi/LAT is to monitor the
whole sky, which is achieved on average after two orbits (3.2 h). As a result,
continuous $\gamma$-ray data have been gathered since the launch of the
satellite in 2008. The Fermi/LAT analysis has been performed using the
ScienceTools version 1.2.23 and fermipy version 0.20.0. The time range chosen
for the analysis is 120 days, i.e., from 2020 September 29, 00:00 UTC to 2021
January 27, 00:00 UTC (MJD 59121 to MJD 59241), which is centered on the date
of the NuSTAR observation. We extracted SOURCE class events for an energy
range between 100 MeV and 300 GeV in a region of interest (ROI) of
$10^{\circ}$ around the position of PKS 2005$-$489 reported in the Fourth
Fermi/LAT source catalog (4FGL; Abdollahi et al. 2020). We included events
that fulfilled the flags DATA_QUAL>0 and LAT_CONFIG==1 and excluded events
with a zenith angle $z\geq 90^{\circ}$ in order to dispose of $\gamma$-rays
produced by Earth-limb effects. For the instrument response, we use the post-
launch function P8R2_SOURCE_V2. The model we used to describe the ROI includes
all known $\gamma$-ray sources from the 4FGL within $15^{\circ}$, as well as
the Galactic diffusion model gll_iem_v07 and the isotropic diffusion emission
model iso_P8R3_SOURCE_V2_v1. For our source of interest, PKS 2005$-$489, we
choose a log-parabola model for the spectrum as defined in the 4FGL. The model
parameters are optimised via a maximum likelihood analysis. The significance
of the modeled $\gamma$-ray signal is then determined via the test statistic
$\mathrm{TS}=2\Delta\log(\mathcal{L})$, with $\mathcal{L}$ being the
likelihood function representing the difference between two models — one with
a point source at the given source coordinates and one without(Mattox et al.
1996). We kept all spectral parameters free for PKS 2005$-$489 and all sources
within $3^{\circ}$. We allow the normalization to be free for sources within
$5^{\circ}$ and for those with a TS value higher than 500. In addition, the
parameters in the models for isotropic and Galactic diffusion are kept free.
Other sources are modelled with the spectral parameters reported in the 4FGL.
The TS value of PKS 2005$-$489 in the chosen time range is 170, which is $\sim
13\sigma$, and we determine the flux to be $1.2\pm 0.3\times
10^{-8}\,\mathrm{ph}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$. The best-fit results
for the spectral parameters are $\alpha=1.44\pm 0.17$ and $\beta=1.5\pm 0.8$.

###  2.3 Swift

The Neil Gehrels Swift Observatory is a multiwavelength mission with three
instruments: the X-ray Telescope (XRT), the UltraViolet Optical Telescope
(UVOT) and the Burst Alert Telescope (BAT). BAT, XRT, and UVOT are sensitive
in the following wavelength ranges: 15–150 keV, 0.2–10 keV, and 170–600 nm,
respectively (Gehrels et al. 2004).

Swift has observed PKS 2005$-$489 irregularly since its launch in 2004 through
a TANAMI111https://pulsar.sternwarte.uni-erlangen.de/tanami/ fill-in program,
additional Guest Observer (GO) programs, and Target of Opportunity (ToO)
observations. We used observations taken quasi-simultaneously with NuSTAR, as
well as archival data from the BAT 104-month catalog to compile the SED. The
XRT and UVOT data were reduced and extracted using HEASOFT
v.6.30.1222https://heasarc.gsfc.nasa.gov/docs/software/heasoft/. BAT archival
data has been taken from the 104-month catalog (Baumgartner et al. 2013).

####  2.3.1 Swift/XRT

XRT data have been extracted using an annulus centered on R.A.$=302\fdg 3558$,
dec.$=-48\fdg 8316$ with an inner radius of $11.787^{\prime\prime}$ and an
outer radius of $47.146^{\prime\prime}$. The innermost region up to
$11.787^{\prime\prime}$ has been excluded to correct for pile-up. The
background region has been extracted from an annulus surrounding the source
with inner and outer radii of 1.886′ and 2.947′, respectively. We reduced and
extracted the data using xrtpipeline, XSELECT, and xrtmkarf. For spectral
modeling, all X-ray data were binned with a minimum signal-to-noise ratio of
10. We used data in the 0.3–9 keV energy band.

####  2.3.2 Swift/UVOT

Data has been extracted using the uvotimsum and uvot2pha tools. The data were
extracted using a $5^{\prime\prime}$ region centered on the source
coordinates. We extracted the background from an annulus centered at the
source coordinates with radii of 25′′ and 50′′, while ensuring that no other
sources are present in the region.

###  2.4 RXTE/PCA

The Rossi X-Ray Timing Explorer (RXTE) was an X-ray satellite designed to
study the 2–200 keV band up until its mission end in January 2012. We analyzed
all available RXTE data from its Proportional Counter Array (PCA), which is
sensitive to the 3–60 keV band. RXTE observed PKS 2005$-$489 164 times between
1997 and 2009. All spectra and backgrounds were re-analyzed and extracted
using the standard tools in HEASOFT v6.28 provided through the publicly-
available, python-based Chromos333https://github.com/davidgardenier/chromos
pipeline developed by Gardenier & Uttley (2018). This pipeline also provides a
detailed description of the data reduction process. The spectra were extracted
using only the standard-2 events in the proportional counter unit 2 (PCU2) as
these data are best calibrated. We exclude data above 15 keV due to an issue
with the background modeling. Data were binned to a minimum signal-to-noise
ratio of 15 for display purposes. In Figs. 1, 2, and 3, we show the three
different RXTE spectra. The highest flux state is observation 30432-01-03-00
(1998-11-10), the medium flux state is observation 94340-01-01-00 (2009-05-22;
quasi-simultaneous with our $\alpha$ epoch), and the lowest flux state is
20342-03-01-02 (1997-07-30).

###  2.5 ATCA

The Australian Telescope Compact Array observes PKS 2005$-$489 regularly as a
calibrator object, which is available through the ATCA calibrator
database444https://www.narrabri.atnf.csiro.au/calibrators/calibrator_database.html.
Flux measurements at 2.1, 5.5, 9.0, 17.0, 33.0, and 93.0 GHz bands are
available. Only the 93.0 GHz band lacks data that are contemporaneous to the
NuSTAR observation (2020-11-28). The data we chose for our analysis were taken
between 2020-09-14 and 2021-02-21 and are considered quasi-simultaneous with
the NuSTAR observation.

###  2.6 H.E.S.S.

The High Energy Stereoscopic System (H.E.S.S.) telescopes are TeV Cherenkov
$\gamma$-ray detectors located in Namibia. Archival data is available through
the SSDC SED tool555https://tools.ssdc.asi.it/SED/. While the available data
are generally not simultaneous to our observing periods, it is the only data
set publicly available in the TeV band. One H.E.S.S. observation took place in
2009, which we consider quasi-simultaneous with the $\alpha$ state. We
included H.E.S.S data in our modeling in order to constrain the SED at the
highest energies (H. E. S. S. Collaboration et al. 2010). As the data are not
simultaneous, the model is potentially not accurate. However, when comparing
the SED to the data used by Krauß et al. (2016), which lacks the TeV
information, Fermi/LAT data alone are not able to constrain the high-energy
peak due to its position. Constraining the peak with the H.E.S.S. data is
therefore preferable to having no constraint.

###  2.7 Other multiwavelength data

All remaining multiwavelength and archival data are identical to the data used
by Krauß et al. (2016), which includes information regarding the data
reduction and analysis. Archival data (with the exception of the H.E.S.S.
data) are not included in multiwavelength modeling of the SED.

###  2.8 Multiwavelength data modeling

Details on the treatment of multiwavelength data are described by Krauß et al.
(2016, Sections 2.4 and 2.5). We treat multiwavelength data in a detector-
space approach using ISIS v.1.6.2-48. We combine data with an assigned
response (ARF/RMF; e.g. Swift/XRT) with data sets that are only available as
fluxes or flux densities. These data sets are assigned a diagonal matrix.
Unfolding of data is used for display purposes only; fluxes are calculated
independently from the fit model (for a detailed discussion, see Nowak et al.
2005). Furthermore, we use a $\chi^{2}$ approach to modeling the SED and
determining the best-fit. This is not statistically reliable because the data
have been obtained by different instruments with different statistical
backgrounds (e.g., likelihood analyses are required for Fermi/LAT data), and
the flux measurements obtained with the different instruments have not been
cross calibrated. Therefore, $\chi^{2}$ values do not indicate an absolute
model goodness-of-fit (i.e. interpreted as a probability); however, the
relative reduced $\chi^{2}$ values between models of the same state and source
are useful for finding a good empirical description.

Data in the X-ray, UV, optical, and infrared bands are affected by absorption
and reddening effects. Photoelectric absorption is treated in the model by
using the tbabs model (Wilms et al. 2000) with the vern cross-sections (Verner
et al. 1996) and the wilm abundances (Wilms et al. 2000). The $N_{\mathrm{H}}$
value for deabsorption and dereddening corrections ($3.63\times 10^{20}$ cm-2)
has been taken from the HI4PI survey (HI4PI Collaboration et al. 2016). We
further apply the absorbing column to deredden the IR/optical/UV data by
converting it to $A_{\mathrm{V}}$ from dust scattering halo measurements
(Predehl & Schmitt 1995) with the corrections to the abundances of the ISM by
Nowak et al. (2012).

Systematic uncertainties are taken into account and added to the statistical
uncertainties as described by Krauß et al. (2016). For fitting the SED, we
largely follow the procedure outlined in Krauß et al. (2016, Section 2.5). We
employ a double logarithmic parabola model that is characterized in ISIS as
(logpar(1)+logpar(2))*tbabs(1)*redden(1), which includes two parabolas, one
for the synchrotron and one for the high-energy peak, as well as the
absorption and reddening models. For the $\beta$ state, we include a blackbody
model that indicates the presence of a weak host galaxy signature. This
feature may be present in the 2020 $\gamma$ data, but due to the lack of
optical information (with contemporary observations from only one UVOT
filter), this cannot be confirmed. Therefore, we do not add a blackbody
component to the model. We reconstruct the SEDs first shown by Krauß et al.
(2016) with the 2009 and 2012 epochs, labeled $\alpha$ and $\beta$,
respectively. Then, we add our new 2020 data (dubbed $\gamma$) using the
NuSTAR and quasi-contemporaneous data discussed in Section 2. Previously, the
source exhibited high and low fluxes during the $\alpha$ and $\beta$ states,
respectively. The following section describes the results from the modeling of
the SED and X-ray data and discusses the implications.

##  3 Results and Discussion

###  3.1 The multiwavelength SED

We compiled the multiwavelength SED for the three epochs (2008–2009, $\alpha$;
2011–2012, $\beta$; and 2020, $\gamma$) with good quasi-simultaneous
multiwavelength coverage (see Fig. 1). The Swift data for $\alpha$ and $\beta$
have been reanalyzed with the updated calibrations and HEASoft tools, but the
SED states consist of identical observations and data to those shown by Krauß
et al. (2016).

In 2008–2009 ($\alpha$), the source was in a high-emission state with the
X-ray flux reaching $\sim$20 times the flux of the quiet state. The 2009
flaring epoch is not as bright as the 1998 flare detected by RXTE, which
reached an X-ray flux of nearly 50 times higher than the quiet state (see
Table 5).

In 2011-2012 ($\beta$) and 2020 ($\gamma$) PKS 2005$-$489 was in a quiet
state. It is worth noting that both data sets are perfectly compatible in flux
and spectral shape, i.e., the data suggest a stable period of quiescence at
similar flux levels and physical conditions in the jet. We discuss the X-ray
data in detail in Sect. 3.2

The SEDs in the two quiet states ($\beta$ and $\gamma$) are very well
described with logarithmic parabolas (see Table 4 for a complete list of fit
results). The best-fit empirical parabola models for each epoch are shown in
Fig. 1. The residuals are shown in the bottom panel, while the reduced
$\chi^{2}$ values are also listed in the left side. We further examine the
high-flux state ($\alpha$) of the source in Fig. 2 with the fit results
provided in Table 4. In all plots, the $\alpha$ data are shown in light blue,
and the RXTE data are shown in dark blue in order to distinguish between the
two data sets. The low-state data ($\beta$ and $\gamma$) are shown in dark
green and turquoise, respectively. Archival data are shown in orange, while
the reddened and absorbed data are shown in light gray. Error bars of all data
sets are shown in dark gray. Instrumental energy ranges are shown at the top
of the plot with the position of the line along the vertical axis
corresponding to the position of the labels (i.e., higher lines correspond to
higher labels).

Fig. 1 shows the best-fit empirical logarithmic parabola models. In the
$\beta$ state, a host galaxy blackbody has been added in order to describe the
optical data. The quiescent states are reasonably well described by the
empirical model, including the newer 2020 $\gamma$ epoch. It is worth noting
that both the $\beta$ and the $\gamma$ data are compatible in flux and
spectral shape. As the data were taken $\sim$ eight years apart, it suggests a
stable mode of quiescence. We confirm that the high state ($\alpha$) is
incompatible with a simple empirical model. The X-ray band shows a peculiar,
narrowly curved shape that is incompatible with the expected wide synchrotron-
peak behavior. As far as we are aware, this is the only source to exhibit this
peculiar feature in the X-ray band, as well as the extreme X-ray variability
not accompanied by strong variability in other wavebands.

Figure 1: Left: The three panels show the three epochs $\alpha$ (2008–2009;
top panel), $\beta$ (2011–2012; middle panel) and $\gamma$ (2020; bottom
panel) in separate panels. Archival data (non-contemporaneous to the epoch
shown) is displayed in gray. The instruments and their respective energy
ranges are shown at the top of the plot, where gray indicates non-simultaneous
data. In order to distinguish X-ray data, NuSTAR data (simultaneous with the
2020 data, bottom left) is shown in purple. RXTE data taken simultaneous with
the $\alpha$ SED are shown in light blue (top panel). Observed IR, optical,
UV, and X-ray data are shown in green, except when dereddened and deabsorbed,
where they are shown in blue. Data used for fitting include the quasi-
simultaneous data for each epoch (shown in color) and the archival H.E.S.S.
data. Reduced $\chi^{2}$ values are shown in the top left side of the plot.
The bottom panel shows the residuals for the fits to the parabolas.
Right: The simultaneous data has been combined into one SED, where the three
epochs have been separated into three different colors: $\alpha$ (2008–2009,
blue), $\beta$ (2011–2012, green) and $\gamma$ (2020,turquoise). The reddened
and absorbed data is shown in gray. The data and models are identical to the
panels presented on the left side, but are shown in one plot for easy
comparison of fluxes and models.
The only clear deviation from the model occurs in the $\alpha$ state in both
the Swift/XRT and RXTE/PCA data, which exhibit a narrow curved feature
incompatible with the broad SED peak expected from synchrotron emission. The
quiescent state in the $\beta$ and $\gamma$ SEDs is virtually identical in
flux.

We investigate the broadband behavior of PKS 2005$-$489 in the $\alpha$ state
in more detail in Fig. 2, where we model the flaring state with a variety of
empirical models in order to constrain the shape of the feature. We discuss
the X-ray band in detail in Sect. 3.2 and possible origins of the peculiar
feature in Sect. 3.3. Data used for fitting include the quasi-simultaneous
data for $\alpha$ in addition to the archival H.E.S.S. data to constrain the
high-energy peak. The H.E.S.S. data can be considered quasi-simultaneous with
the $\alpha$ state. The first model is identical to the empirical SED model
shown in Fig. 1 and is not able to explain the shape of the X-ray spectrum.
The second model (M2, residuals shown in the third panel from the top) adds a
blackbody model in an attempt to describe the narrowly curved feature with a
thermal component. The blackbody itself is also shown as a dashed line. It is
not able to adequately describe the shape as indicated by the residuals. The
third model (M3, residuals shown in the fourth panel from the top) adds a
third parabola to the two describing the synchrotron and high-energy peaks in
order to describe the X-ray data. As it is most flexible in terms of how
narrow the curvature is, it is able to describe the feature well as indicated
by the residuals and reduced $\chi^{2}$ values. The fourth model (M4, bottom
panel) adds a three-segment broken powerlaw, which is also able to explain the
X-ray data well. However, it requires five parameters that are free to vary.
This is likely too many, as indicated by the reduced $\chi^{2}$ value below 1
for all three datasets. The best-fit values of the logarithmic parabola model
are given in Table 4. Additional model features, such as the broken powerlaw,
are described in Table 4.

Figure 2: The broadband SED of PKS 2005$-$489 during the $\alpha$ (2008–2009)
epoch is shown. Archival H.E.S.S. data (not contemporaneous) are shown in
gray. Multiple models and their residuals are given in each of the four panels
and are shown with their associated residual panel below each SED. A simple
logarithmic parabola (top left panel) or an added blackbody are not able to
model the strong X-ray flux with its break. Both a third parabola (bottom
left) or a broken powerlaw component (bottom right panel) are able to explain
the data.

Table 2: Best-fit results for the logarithmic parabola fits to the various
epochs of the PKS 2005$-$489 SED shown in Fig. 1. If no uncertainty is given,
the parameter was fixed to the given value. The absorbing column for the
reddening and absorption model was frozen to the Galactic value of
$N_{\mathrm{H}}=3.93\times 10^{20}\,\mathrm{cm}^{-2}$. A blackbody fit was
only used to describe a host galaxy signature in the $\beta$ state. The
blackbody temperature has been corrected for the redshift of the source. The
$E_{p}$ of the synchrotron peak is 0.05 keV in all models. The uncertainties
are statistical only.

| $N_{\mathrm{sync}}$ | $a_{\mathrm{sync}}$ | $b_{\mathrm{sync}}$ | $N_{\mathrm{HE}}$ | $a_{\mathrm{HE}}$ | $b_{\mathrm{HE}}$ | $E_{\mathrm{p,HE}}$ | $N_{\mathrm{blackbody}}$ | $T_{\mathrm{blackbody}}$ | $\chi^{2}$ | $\chi^{2}_{r}$ | $F_{\mathrm{bol}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  | [keV] |  [$10^{-10}$] |  |  |  [$10^{-4}$] |  [$10^{-3}$keV] |  |  |  [erg s-1 cm-2]
$\alpha$ | $27.7\pm 0.7$ | $1.954\pm 0.005$ | $0.0751^{+0.0018}_{-0.0017}$ | $0.24^{+0.08}_{-0.07}$ | $0.503^{+0.025}_{-0.026}$ | $0.185$ | 500 | – | – | 460.18 | 6.39 | $\left(1.173\pm 0.019\right)\times 10^{-9}$
$\beta$ | $8.1\pm 0.4$ | $2.310\pm 0.013$ | $0.1269^{+0.0030}_{-0.0029}$ | $1.7\pm 0.4$ | $0.579^{+0.021}_{-0.019}$ | 0.15 | 100 | $3.2\pm 0.7$ | $1.14^{+0.13}_{-0.10}$ | 86.73 | 2.80 | $\left(4.94\pm 0.10\right)\times 10^{-10}$
$\gamma$ | $8.4\pm 0.4$ | $2.306^{+0.008}_{-0.009}$ | $0.1231\pm 0.0023$ | $0.088\pm 0.01$ | 0.0582 | 0.197 | 100 | – | – | 146.95 | 2.77 | $\left(4.53\pm 0.21\right)\times 10^{-10}$

Table 3: Best-fit results for the logarithmic parabola fits to the various
models of the $\alpha$ SED shown in Fig. 2. If no uncertainty is given, the
parameter was fixed to the given value. The absorbing column for the reddening
and absorption model was frozen to the Galactic value of
$N_{\mathrm{H}}=3.93\times 10^{20}\,\mathrm{cm}^{-2}$. The $E_{p}$ of the
synchrotron peak is 0.05 keV in all sources. The uncertainties are statistical
only.

| $N_{\mathrm{sync}}$ | $a_{\mathrm{sync}}$ | $b_{\mathrm{sync}}$ | $N_{\mathrm{HE}}$ | $a_{\mathrm{HE}}$ | $b_{\mathrm{HE}}$ | $E_{\mathrm{p,HE}}$ | $\chi^{2}$ | $\chi^{2}_{r}$ | $F_{\mathrm{bol}}$
---|---|---|---|---|---|---|---|---|---|---
|  |  |  |  [$10^{-10}$] |  |  | [keV] |  |  |  [erg s-1 cm-2]
M1 | $27.7\pm 0.7$ | $1.954\pm 0.005$ | $0.0751^{+0.0018}_{-0.0017}$ | $0.24^{+0.08}_{-0.07}$ | $0.503^{+0.025}_{-0.026}$ | $0.185$ | 500 | 460.18 | 6.39 | $\left(1.173\pm 0.019\right)\times 10^{-9}$
M2 | $26.2\pm 0.7$ | $2.007\pm 0.009$ | $0.0844^{+0.0026}_{-0.0024}$ | $3.6^{+1.3}_{-1.0}$ | $0.618\pm 0.024$ | 0.15 | 100 | 195.34 | 2.79 | $\left(1.075\pm 0.022\right)\times 10^{-9}$
M3 | $19^{+4}_{-5}$ | $2.19^{+0.13}_{-0.08}$ | $0.113^{+0.018}_{-0.011}$ | $0.25^{+0.08}_{-0.07}$ | $0.505\pm 0.025$ | 0.185 | 500 | 81.62 | 1.18 | $\left(1.007\pm 0.022\right)\times 10^{-9}$
M4 | $14.1^{+2.9}_{-4.6}$ | $2.34^{+0.21}_{-0.20}$ | $0.17^{+0.04}_{-0.07}$ | $0.24^{+0.08}_{-0.07}$ | $0.501\pm 0.026$ | 0.185 | 500 | 77.62 | 1.18 | $\left(1.01^{+0.08}_{-0.06}\right)\times 10^{-9}$

Table 4: Best-fit results for the additional components of the various models
of the $\alpha$ SED shown in Fig. 2. If no uncertainty is given, the parameter
was fixed to the given value. The absorbing column for the reddening and
absorption model was frozen to the Galactic value of
$N_{\mathrm{H}}=3.93\times 10^{20}\,\mathrm{cm}^{-2}$. The uncertainties are
statistical only.

| $N_{\mathrm{bb}}$ | $T$ | $N_{\mathrm{lp3}}$ | $a_{\mathrm{lp3}}$ | $b_{\mathrm{lp3}}$ | $E_{\mathrm{p,lp3}}$ | $N_{\mathrm{bknpow}}$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $E_{\mathrm{break,1}}$ | $E_{\mathrm{break,2}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
|  [$10^{-4}$] | [keV] |  |  |  | [keV] |  |  |  |  | [keV] | [keV]
M1 | – | – | – | – | – | – | – | – | – | – | – | –
M2 | $5.8\pm 0.6$ | $1.33\pm 0.07$ | – | – | – | – | – | – | – | – | – | –
M3 | – | – | $0.059^{+0.035}_{-0.029}$ | $0.83^{+0.30}_{-0.46}$ | $0.74^{+0.23}_{-0.15}$ | 0.5 | – | – | – | – | – | –
M4 | – | – | – | – | – | – | $0.037^{+0.011}_{-0.017}$ | $1.596^{+0.015}_{-0.494}$ | $2.09^{+0.23}_{-0.50}$ | $2.55^{+0.09}_{-0.08}$ | $2.2^{+0.7}_{-1.6}$ | $4.5^{+1.3}_{-0.6}$

###  3.2 X-ray data

From the multiwavelength SED, it is clear that the measurements in most
wavebands are relatively consistent in flux and shape between low and high-
flux states. A strong deviation from quiescence is only observed in the X-ray
band. In this section, we present additional modeling of the X-ray data,
independent of the multiwavelength observations. Fig. 3 shows the X-ray band
with the data of all three epochs in the same colors as in the SED. Various
models are shown in shades of purple. All models include photoelectric
absorption fixed to the Galactic value (see Tables 5, 6, and 7 for details).
The first model is identical to the SED model for comparative purposes.
Secondly, we fit an absorbed powerlaw to all of the data. While it describes
the quiescence data well for both states ($\beta$, $\gamma$), it is not able
to adequately model the high-flux state $\alpha$. The third model adds a third
logarithmic parabola in order to describe the peculiar X-ray feature. It
describes the X-ray data well as indicated in Fig. 2. Despite also describing
the other states well, $\chi^{2}_{r}<1$ indicates too many degrees of freedom
in the model. The fourth model adds a three-segment broken powerlaw for the
$\alpha$ state and a two-segment broken powerlaw for the other two states. All
states are well described by the additional broken powerlaw, but the
quiescence SEDs indicate a large number of degrees of freedom
($\chi^{r}_{\mathrm{red}}<1$), and the $\beta$ state find the same photon
index for both sides, requiring one to be frozen due to the two parameters
being correlated.

Figure 3: The X-ray spectrum of PKS 2005$-$489\. We include simultaneous X-ray
data of PKS 2005$-$489 and the reddened and absorbed data in gray. The colors
of the SED data correspond to the colors in the left panel of Fig. 1. The
three epochs are shown in the three panels. Four models are shown in orange,
pink, magenta and purple in each of the panels with the corresponding reduced
$\chi^{2}$ values. Residual panels are shown below the plots and are shown in
the color of each model. The 0.5–10 keV data in all three panels (dark blue,
green, and turquoise) are Swift/XRT data. Light blue data in the left panel
are RXTE/PCA data, while the purple data in the right panel are NuSTAR data.

The models in Fig. 3 are given in Tables 5, 6, and 7. In Table 5, we show two
powerlaw fits to the $\alpha$ state, one with a frozen absorbing column and
one where $N_{\mathrm{H}}$ was left free to vary. A higher absorption is able
to partially mimic the observed curved shape, as indicated by the improved
$\chi^{2}$. From the residuals (not shown), it is clear that this is not a
feasible model for the high flux state. Further, significant obscuration of a
blazar jet is extremely unlikely.

Table 5: Best-fit results of a powerlaw fit to the X-ray data of all epochs, shown in Fig. 3. Residuals can be found in the second residual panel. We include two fits to the $\alpha$ state. The second fit leaves the absorbing column free to vary in the fit. The uncertainties are statistical only. Epoch | $N_{\mathrm{H}}$ | $N$ | $\Gamma_{\mathrm{powerlaw}}$ | $\chi$ | $\chi^{2}_{\mathrm{red}}$ | $F_{2-10\,\mathrm{keV}}$
---|---|---|---|---|---|---
| [$10^{20}$ cm-2] |  |  |  |  | [erg s-1 cm-2]
$\alpha$ | 3.93 | $0.0503\pm 0.0014$ | $2.102\pm 0.017$ | 354.94 | 6.34 | $\left(1.099\pm 0.016\right)\times 10^{-10}$
$\alpha$ (var. $N_{\mathrm{H}}$) | $17.5^{+2.5}_{-2.3}$ | $0.070\pm 0.004$ | $2.286\pm 0.030$ | 169.47 | 3.08 | $\left(1.170\pm 0.019\right)\times 10^{-10}$
$\beta$ | 3.93 | $\left(4.8\pm 0.4\right)\times 10^{-3}$ | $2.45\pm 0.14$ | 1.93 | 0.19 | $\left(6.3^{+1.4}_{-1.3}\right)\times 10^{-12}$
$\gamma$ | 3.93 | $\left(4.11\pm 0.23\right)\times 10^{-3}$ | $2.56\pm 0.04$ | 21.89 | 0.63 | $\left(4.68\pm 0.14\right)\times 10^{-12}$
1998 | 3.93 | $0.172\pm 0.007$ | $2.258\pm 0.023$ | 21.65 | 0.77 | $\left(2.99\pm 0.04\right)\times 10^{-10}$
Table 6: Best-fit results of a logarithmic parabola fit to the X-ray data of all epochs, shown in Fig. 3. Residuals can be found in the third residual panel. The absorbing column was frozen to the Galactic value of $N_{\mathrm{H}}=3.93\times 10^{20}\,\mathrm{cm}^{-2}$. Epoch | $N_{\mathrm{H}}$ | $N$ | $a$ | $b$ | $E_{p}$ | $\chi^{2}$ | $\chi^{2}_{\mathrm{red}}$ | $F_{2-10\,\mathrm{keV}}$
---|---|---|---|---|---|---|---|---
| [$10^{20}$ cm-2] |  |  |  | [keV] |  |  | [erg s-1 cm-2]
$\alpha$ | 3.93 | $0.0479^{+0.0018}_{-0.0019}$ | $1.68\pm 0.06$ | $0.48\pm 0.06$ | 1 | 44.73 | 0.81 | $\left(1.183\pm 0.018\right)\times 10^{-10}$
$\beta$ | 3.93 | $\left(4.8^{+0.6}_{-0.5}\right)\times 10^{-3}$ | $2.46\pm 0.14$ | $0.0^{+0.6}_{-0.5}$ | 1 | 1.91 | 0.21 | $\left(6.4^{+3.4}_{-2.3}\right)\times 10^{-12}$
$\gamma$ | 3.93 | $\left(4.17^{+0.25}_{-0.24}\right)\times 10^{-3}$ | $2.47\pm 0.08$ | $0.12\pm 0.09$ | 1 | 16.72 | 0.49 | $\left(4.80\pm 0.17\right)\times 10^{-12}$
Table 7: Best-fit results of a three/two-segment broken powerlaw fit to the X-ray data of all epochs, shown in Fig. 3. Residuals can be found in the fourth residual panel. The uncertainties are statistical only. Epoch | $N$ | $\Gamma_{1}$ | $\Gamma_{2}$ | $\Gamma_{3}$ | $E_{\mathrm{break,1}}$ | $E_{\mathrm{break,2}}$ | $\chi^{2}$ | $\chi^{2}_{\mathrm{red}}$ | $F_{2-10\,\mathrm{keV}}$
---|---|---|---|---|---|---|---|---|---
| [$10^{-3}$ cm-2] |  |  |  | [keV] | [kev] |  |  | [erg s-1 cm-2]
$\alpha$ | $44.7^{+1.6}_{-1.9}$ | $1.79\pm 0.09$ | $2.11^{+0.22}_{-0.15}$ | $2.56\pm 0.07$ | $2.4^{+1.3}_{-1.1}$ | 4.5 | 30.69 | 0.58 | $\left(1.177\pm 0.024\right)\times 10^{-10}$
$\beta$ | $4.8\pm 0.4$ | $2.46\pm 0.15$ | 2.46 | – | 2.5 | – | 1.90 | 0.21 | $\left(6.3\pm 1.0\right)\times 10^{-12}$
$\gamma$ | $4.2\pm 0.4$ | $2.40^{+0.14}_{-0.17}$ | $2.653^{+0.089}_{-0.0002}$ | – | $1.8^{+2.2}_{-0.9}$ | – | 15.08 | 0.46 | $\left(4.86^{+0.21}_{-0.20}\right)\times 10^{-12}$

From the SED and the X-ray spectrum, it is noteworthy how consistent
quiescence is in PKS 2005$-$489\. Between the 2011–2012 and 2020 epochs, the
source either did not vary in flux, or returned to quiescence with identical
fluxes and spectral indices. The 2020 data include our NuSTAR data, which is
consistent with Swift/XRT data in flux and spectral shape of both low-flux
epochs ($\beta$ and $\gamma$), but not consistent with the Swift/BAT data.

###  3.3 Possible explanations of the peculiar X-ray feature in the high-flux
state

X-ray observations of PKS 2005$-$489 reveal a peculiar feature in high-flux
states, while at other wavelengths, the source varies little, and the spectral
shape does not change significantly. During high-flux states, the X-ray band
is characterized by extreme flares with X-ray fluxes reaching 10–50 times
greater than fluxes during quiescent periods. We note that the host galaxy of
PKS 2005$-$489 is a giant elliptical galaxy (Scarpa et al. 2000), which is
unlikely to contribute to the X-ray emission through active star formation or
other processes that would outshine the relativistic jet.

####  3.3.1 Instrumental feature

Swift/XRT data have been examined and treated for pile-up, which cannot
sufficiently explain the curvature and high-flux state. Furthermore, RXTE data
are consistent with Swift/XRT, indicating that an instrumental artifact or
calibration issues can be ruled out. Additional archival data, such as the
EXOSAT data presented in Giommi et al. (1990), show extreme flaring periods,
strongly suggesting that this is not an instrumental issue.

####  3.3.2 Thermal feature

A single-temperature blackbody cannot adequately model the data (see Fig. 3,
Table 4), which indicates that the origin of the feature is likely nonthermal
in origin. Temperatures are higher than possible for a host galaxy or an
accretion disk (Malkan 1983).

####  3.3.3 Hadronic origin

In principle, the X-ray emission could result from synchrotron emission by
secondary pairs generated by hadronic cascades (e.g., Böttcher et al. 2013;
Keivani et al. 2018). Nonetheless, this scenario requires extreme physical
parameters for the observed spectrum here. The roughly comparable optical and
X-ray flux indicate very powerful secondary synchrotron emission, which
generally points to efficient photomeson production. However, the target
photons cannot be the result of synchrotron radiation from the primary
electrons located in the hadronic emission region. In such a scenario, the
primary electrons would upscatter the X-rays to $\gamma$-rays, which would
have been observable by Fermi/LAT and TeV telescopes. To avoid this, the
hadronic emission region could be located in the broad line region whose flux
may be lower than the primary electron synchrotron emission (so-called
masquerading BL Lacs; Giommi et al. 2013). In this situation, the kinetic
luminosity of protons must be very high to produce adequate secondary pairs to
match the observed X-ray flux. This scenario would likely be disfavored due to
the unrealistic jet powers necessitated by detailed spectral modeling.
Additionally, the proton cooling due to photomeson processes is generally
slow, which is inconsistent with the fast X-ray variability that were
previously observed in this source (Giommi et al. 1990; Sambruna et al. 1995).
If a hadronic model were plausible in a certain parameter space, then the
X-ray flux in the high state would likely be accompanied by a comparably high
neutrino flux. This secondary pair synchrotron component can be strongly
polarized, which may be detected by X-ray polarimeters such as IXPE (e.g.,
Zhang 2017).

####  3.3.4 Additional jet emission component and magnetic reconnection

Another possibility is that the X-ray spectrum originates from synchrotron
emission by a population of high-energy nonthermal electrons in an environment
with a high magnetic field. Since the ratio of Compton scattering flux over
synchrotron is proportional to the ratio of photon energy density over
magnetic energy density, a high magnetic field can suppress the inverse
Compton scattering, resulting in negligible contribution to $\gamma$-rays
(see, e.g., Rybicki & Lightman 1985). By comparing the X-ray flux and TeV
$\gamma$-ray flux, we expect that the magnetic energy density in this orphan
component should exceed $\gtrsim 100$ times the photon energy density. In
contrast, the same ratio derived from the optical and Fermi $\gamma$-ray
fluxes is $\sim 10$. This suggests that the high state may result from the
emergence of a new emission blob with very strong magnetic field, which is not
likely to be located in the emission region responsible for the low state.
Given the rather hard spectral shape in the X-ray component, these high-energy
electrons are probably accelerated via magnetic reconnection. Zhang et al.
(2021) have shown that reconnection in a highly magnetized environment can
lead to very hard X-ray synchrotron spectra. Both the flux and polarization
can be strongly variable in reconnection, which may be examined by IXPE and
future X-ray polarimetry instruments. Alternatively, turbulence in a highly
magnetized environment may also accelerate high-energy electrons. However, the
electron spectrum is usually much softer than reconnection, which may be
disfavored by spectral modeling (Comisso & Sironi 2018).

##  4 Conclusions

We discuss the broadband SED of the southern BL Lac object PKS 2005$-$489\.
While the SEDs during two epochs have already been shown by Krauß et al.
(2016), we present new data taken in 2020, including the first hard X-ray
spectrum with NuSTAR data. We find that quiescence is stable in the source,
since the 2020 data are perfectly compatible with the 2012 low-state data. The
most peculiar feature of PKS 2005$-$489 is its extreme X-ray variability that
is not accompanied by large flux changes in other wavebands and exhibits a
peculiar spectrum. The SED in the high state cannot be explained with a
typical logarithmic parabola that, though empirical, is able to describe
blazar spectra well (Krauß et al. 2016). Currently, the most likely
explanation seems to be an additional jet component that is not co-spatial
with the low-state emission region. The X-ray flaring activity can be
explained through magnetic reconnection, which also predicts strong X-ray
polarization variability. In order to understand the flaring in PKS
2005$-$489, additional data sets during the high state are crucial. NuSTAR and
X-ray polarimetry data in the high state and data in additional wavebands
(e.g., MeV) would allow us to better constrain the physical mechanisms at
play. As the extreme source variability is only present in the X-ray band and
does not exhibit frequent flaring, it is difficult to monitor its flux changes
and detect a bright state due to the lack of an all-sky monitor at X-ray
energies capable of providing frequent flux measurements.

##  5 Acknowledgements

We thank Tonia Venters and Matthew Kerr for their helpful comments that
improved the manuscript. We thank Victoria Grinberg for discussions and
comments. O.C. is supported by the NASA program NNH19ZDA001N-NUSTAR through
grant #19-NUSTAR19-0040. H.Z. is supported by an appointment to the NASA
Postdoctoral Program at Goddard Space Flight Center, administered by Oak Ridge
Associated Universities under contract with NASA. This research has made use
of a collection of ISIS functions (ISISscripts) provided by ECAP/Remeis
observatory and MIT (http://www.sternwarte.uni-erlangen.de/isis/). Part of
this work is based on archival data, software or online services provided by
the Space Science Data Center – ASI.

The Fermi/LAT Collaboration acknowledges generous ongoing support from a
number of agencies and institutes that have supported both the development and
the operation of the LAT as well as scientific data analysis. These include
the National Aeronautics and Space Administration and the Department of Energy
in the United States, the Commissariat à l’Energie Atomique and the Centre
National de la Recherche Scientifique / Institut National de Physique
Nucléaire et de Physique des Particules in France, the Agenzia Spaziale
Italiana and the Istituto Nazionale di Fisica Nucleare in Italy, the Ministry
of Education, Culture, Sports, Science and Technology (MEXT), High Energy
Accelerator Research Organization (KEK) and Japan Aerospace Exploration Agency
(JAXA) in Japan, and the K. A. Wallenberg Foundation, the Swedish Research
Council and the Swedish National Space Board in Sweden. Additional support for
science analysis during the operations phase is gratefully acknowledged from
the Istituto Nazionale di Astrofisica in Italy and the Centre National
d’Etudes Spatiales in France. This work performed in part under DOE Contract
DE- AC02-76SF00515.

## References

  * Abdollahi et al. (2020) Abdollahi, S., Acero, F., Ackermann, M., et al. 2020, ApJS, 247, 33, doi: 10.3847/1538-4365/ab6bcb
  * Atwood et al. (2009) Atwood, W. B., Abdo, A. A., Ackermann, M., et al. 2009, ApJ, 697, 1071, doi: 10.1088/0004-637X/697/2/1071
  * Baumgartner et al. (2013) Baumgartner, W. H., Tueller, J., Markwardt, C. B., et al. 2013, apjs, 207, 19
  * Beringer et al. (2012) Beringer, J., Arguin, J. F., Barnett, R. M., et al. 2012, prd, 86, 010001
  * Blandford et al. (1978) Blandford, R., Rees, M., & Wolfe, A. 1978, AM Wolfe, Ed, 328
  * Böttcher et al. (2013) Böttcher, M., Reimer, A., Sweeney, K., & Prakash, A. 2013, ApJ, 768, 54, doi: 10.1088/0004-637X/768/1/54
  * Comisso & Sironi (2018) Comisso, L., & Sironi, L. 2018, Phys. Rev. Lett., 121, 255101, doi: 10.1103/PhysRevLett.121.255101
  * Gaia Collaboration (2018) Gaia Collaboration. 2018, VizieR Online Data Catalog, I/345
  * Gardenier & Uttley (2018) Gardenier, D. W., & Uttley, P. 2018, MNRAS, 481, 3761, doi: 10.1093/mnras/sty2524
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, apj, 611, 1005
  * Giommi et al. (1990) Giommi, P., Barr, P., Garilli, B., Maccagni, D., & Pollock, A. M. T. 1990, apj, 356, 432
  * Giommi et al. (2013) Giommi, P., Padovani, P., & Polenta, G. 2013, mnras, 431, 1914
  * H. E. S. S. Collaboration et al. (2010) H. E. S. S. Collaboration, Acero, F., Aharonian, F., et al. 2010, aap, 511, A52
  * H. E. S. S. Collaboration & Fermi-LAT Collaboration (2011) H. E. S. S. Collaboration, & Fermi-LAT Collaboration. 2011, aap, 533, A110
  * Harrison et al. (2013) Harrison, F. A., Craig, W. W., & Christensen, F. E. 2013, ApJ, 770, 103
  * HI4PI Collaboration et al. (2016) HI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, aap, 594, A116
  * Houck & Denicola (2000) Houck, J. C., & Denicola, L. A. 2000, in Astronomical Society of the Pacific Conference Series, Vol. 216, Astronomical Data Analysis Software and Systems IX, ed. N. Manset, C. Veillet, & D. Crabtree, 591
  * Keeney et al. (2018) Keeney, B. A., Stocke, J. T., Pratt, C. T., et al. 2018, ApJS, 237, 11
  * Keivani et al. (2018) Keivani, A., Murase, K., Petropoulou, M., et al. 2018, ApJ, 864, 84, doi: 10.3847/1538-4357/aad59a
  * Krauß et al. (2016) Krauß, F., Wilms, J., Kadler, M., & Ojha, R. 2016, A&A, 591
  * Lin et al. (1999) Lin, Y. C., Bertsch, D. L., Bloom, S. D., et al. 1999, apj, 525, 191
  * Malkan (1983) Malkan, M. A. 1983, ApJ, 268, 582
  * Mattox et al. (1996) Mattox, J. R., Bertsch, D. L., Chiang, J., et al. 1996, ApJ, 461, 396, doi: 10.1086/177068
  * Nowak et al. (2005) Nowak, M. A., Wilms, J., Heinz, S., et al. 2005, apj, 626, 1006
  * Nowak et al. (2012) Nowak, M. A., Neilsen, J., Markoff, S. B., et al. 2012, apj, 759, 95
  * Perlman et al. (1999) Perlman, E. S., Madejski, G., Stocke, J. T., & Rector, T. A. 1999, apjl, 523, L11
  * Predehl & Schmitt (1995) Predehl, P., & Schmitt, J. H. M. M. 1995, aap, 293, 889
  * Rybicki & Lightman (1985) Rybicki, G. B., & Lightman, A. P. 1985, Radiative processes in astrophysics. (John Wiley & Sons)
  * Sambruna et al. (1995) Sambruna, R. M., Urry, C. M., Ghisellini, G., & Maraschi, L. 1995, apj, 449, 567
  * Scarpa et al. (2000) Scarpa, R., Urry, C. M., Padovani, P., Calzetti, D., & O’Dowd, M. 2000, ApJ, 544, 258
  * Tagliaferri et al. (2001) Tagliaferri, G., Ghisellini, G., Giommi, P., et al. 2001, aap, 368, 38
  * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, doi: 10.1086/133630
  * Verner et al. (1996) Verner, D. A., Ferland, G. J., Korista, K. T., & Yakovlev, D. G. 1996, apj, 465, 487
  * Wakely & Horan (2008) Wakely, S. P., & Horan, D. 2008, in International Cosmic Ray Conference, Vol. 3, International Cosmic Ray Conference, 1341–1344
  * Wall et al. (1975) Wall, J. V., Shimmins, A. J., & Bolton, J. G. 1975, Australian Journal of Physics Astrophysical Supplement, 34, 55
  * Wilms et al. (2000) Wilms, J., Allen, A., & McCray, R. 2000, apj, 542, 914
  * Zhang (2017) Zhang, H. 2017, Galaxies, 5, 32, doi: 10.3390/galaxies5030032
  * Zhang et al. (2021) Zhang, H., Li, X., Giannios, D., & Guo, F. 2021, apj, 912, 129