# Mobility and Coverage Evaluation of New Radio PDCCH for Point-to-Multipoint
Scenario

Hongzhi Chen1, De Mi1, Belkacem Mouhouche2, Pei Xiao1 and Rahim Tafazolli1
Email:{hongzhi.chen, d.mi, p.xiao<EMAIL_ADDRESS><EMAIL_ADDRESS>1Institute for Communication Systems, University of
Surrey, United Kingdom  2Samsung Electronics R&D UK, United Kingdom

###### Abstract

This paper provides the mobility and coverage evaluation of New Radio (NR)
Physical Downlink Control Channel (PDCCH) for Point-to-Multipoint (PTM) use
cases, e.g., eMBMS (evolved Multimedia Broadcast Multicast Services). The
evaluation methodology is based on analyses and link level simulations where
the channel model includes AWGN, TDL-A, TDL-C as well as a modified 0dB echo
to model different PTM scenarios. The final version of this work aims to
provide insightful guidelines on the delay/echo tolerance of the NR PDCCH in
terms of mobility and coverage. In this paper, it is observed that under eMBMS
scenario, i.e. SFN channel, due to the time domain granularity of pilots
distributed inside the PDCCH region, the system can support very high user
movement speed/Doppler with an relatively low requirement on the transmit
Signal/Carrier-to-Noise Ratio (SNR/CNR). On the other hand however, the system
falls short on its coverage due to the low frequency domain granularity of
pilots that effects the channel estimation accuracy.

###### Index Terms:

LTE Advanced Pro, New Radio, 0dB Echo Channel, Point-to-Multipoint, eMBMS,
PDCCH

††We would like to acknowledge the support of the University of Surrey 5GIC
(www.surrey.ac.uk/5gic) members for this work. This work was also supported in
part by the European Commission under the 5GPPP project 5G-Xcast
(H2020-ICT-2016-2 call, grant number 761498). The views expressed in this
contribution are those of the authors and do not necessarily represent the
project.

##  I Introduction

In the 3GPP (3rd Generation Partnership Project) standardization of 5G (5th
Generation) New Radio (NR), the design of control channels has been through
significant changes compared with that in LTE (Long Term Evolution) Advanced
Pro, in order to better handle NR data streams. Most changes of the main type
of the physical control channels, i.e., Physical Downlink Control Channel
(PDCCH), have been captured in our prior work [1, 2, 3]. In this work,
leveraging on our current project [4, 5], we aim at evaluating the performance
of the PDCCH in Point-to-Multipoint (PTM) scenarios, as a potential 5G
broadcast and multicast solution. This work utilizes as reference the Key
Performance Indicators (KPI) and evaluation methodology defined by the ITU-R
(International Communications Union - Recommendation) for the IMT-2020
(International Mobile Telecommunication) evaluation process [6], in which it
only takes into account the Point-to-Point (PTP) scenarios. By focusing on the
multimedia broadcast and multicast services, the performance of the New Radio
specified Physical Downlink Shared Channel (PDSCH) was studied in [7]. As well
as the Block Error Rate (BLER) of the Downlink Control Information (DCI) for
the PDCCH in the PTP scenario was studied in [1] which assumes static receiver
and AWGN channel. Methods/designs that aiming to enhance the
performance/reliability of the PDCCH under PTP environment were also studied
in the literature. For example, a potential power-based optimization for the
control information transmission based on LTE has been discussed in [8] and
can be extended into the NR scenario. In [9, 10], the first one studies the
BLER of the PDCCH with two types of transmit diversity schemes and different
diversity orders. The second one demonstrates that SFBC based transmit
diversity outperforms per-RE precoder cycling scheme. The second one presents
the exponential effective SNR mapping (EESM) results of the configurable
distributed PDCCH, and studies the the trade-off between achieved frequency
diversity and channel estimation gain with different adjusting resource
bundling level.

In this work, we present a comprehensive performance evaluation of both
mobility and coverage for point-to-multipoint (PTM) systems in order to exam
the delay/coverage tolerance for the PDCCH transmission on the current control
channel configuration. The analysis and simulation follow the physical layer
chain defined by 3GPP in [11] and [12]. And the results are compared with the
mobility and coverage requirement that are defined in [6] and [13]. The
simulation results show that based on the current pilot distribution in the
PDCCH area, control information can be reliably transmitted under a very wide
range of user mobility under a 5G channel i.e., TDL-A. Both ideal and real
channel estimation cases have been covered in the coverage evaluation. It
shows that finding a suitable interpolation method is very important in order
to reconstruct the DCI due to the lack of pilots compare to the channel delay
spread.

This paper will be organized as follows. First, it describes the NR PDCCH
framing, Transmitter side processing chain and the modified channel modeling
to model the PTM scenarios. Next, it presents the analysis as well as link
level simulation results, focusing on the delay/echo tolerance of the NR PDCCH
in terms of mobility and coverage. Finally, it summarizes the key findings of
the investigations carried out and discusses the potential improvements
towards the development of the next generation PTM technologies.

##  II Point-to-Multipoint PDCCH Initialization

###  II-A PDCCH Frame Structure

Figure 1: Resource Block structure in LTE-MBSFN (left) and NR (right), with 15
kHz carrier spacing.

The control information specifies the data scheduling and allocation for each
user equipment (UE) by means of Downlink Control Information (DCI). In NR, the
DCI is mapped to Control Resource Set (CORESET), in which its content can be
distributed at most over three consecutive Orthogonal Frequency Division
Multiplexing (OFDM) symbols, depending on high layer parameters. CORESET also
includes Demodulation Reference Signal (DMRS) for the correct demodulation of
the PDCCH. The right-side figure of Fig. 1 illustrates one possible DMRS/pilot
pattern.

###  II-B Transmitter Side Block Diagram

Figure 2: NR physical layer PDCCH transmit block diagram.

The PDCCH processing chain/transmit block diagram of the NR PDCCH processing
chain is shown in Fig.2. Therein, the generator polynomial
$g_{\text{CRC24C}}(D)$ is used for the Cyclic Redundancy Check Attachment.
Polar codes are used for channel coding, where the details of polar encoding
in NR can be found in [12]. It is worth mention that, the length of the
encoded bits is $N=2^{n}$, where $n$ is a integer between 5 and 9, and the
maximum length of the encoded bits is fixed at $2^{9}=512$. Rate matching
consists of sub-block interleaving, bit collection, and bit interleaving. Same
as in 4G LTE the modulation scheme that is operated on the PDCCH is QPSK.

###  II-C 0dB Echo Channel Modelling

TABLE I: Power Delay Profile for Modified Echo Channel Path | Amplitude(dB) | Delay($\mu s$)
---|---|---
1 | 0 | 0
2 | 0 |  $\alpha$*$T_{cp}$

Table I includes the power delay profile (PDP) of the 0dB echo channel, where
tuning the value of variable $\alpha\geq 0$ is equivalent to changing the
coverage of Single Frequency Network (SFN) area, such that we can evaluate the
performance for different PTM scenarios, and consequently the delay/echo
tolerance of the NR PDCCH in terms of mobility and coverage.

##  III Analysis and Link-Level Simulation Evaluation

This section studies the speed tolerance of the New Radio PDCCH channel with
practical channel estimation algorithms as well as the corresponding link
level simulation results to valid the analysis.

###  III-A Doppler Effect

####  III-A1 Theoretical Doppler Limit

Figure 3: BLER/BER vs. CNR for Aggregation Level 1&2 with different user speed
Figure 4: PDCCH CNR vs. Doppler Shift (user speed) for 5G New Radio in TDL-A
mobile channel with real channel estimation.

In order to perform an effective channel estimation, a two-dimensional (i.e.,
frequency and time) sampling should satisfy:

  * •

In the frequency domain, the sampling rate must be faster than or equal to the
maximum delay spread of the channel;

  * •

In the time domain, the sampling rate must be greater than or equal to the
maximum Doppler spread of the channel.

Based on the above-mentioned condition, the maximum distance between two PDCCH
DMRS symbols in the time domain, $n_{max}$, is given by

$n_{max}\leq\frac{1}{2*(T_{U}+T_{cp})*d_{max}},$ (1)

where $T_{U}$ is the useful symbol duration, $T_{cp}$ is the cyclic prefix
(CP) length in time, and $d_{max}$ represents the maximum Doppler spread of
the channel. Due to the fact that the PDCCH DMRS symbols cover all resource
elements (REs) in time domain on selected subcarriers, thus the number of
PDCCH DMRS symbols is sufficient enough to capture the time variation of the
channel, i.e., to be able to potentially support a wide range of user speeds,
in the sense that

$f_{p}=\frac{1}{2D_{y}(T_{U}+T_{cp})}\text{Hz},$ (2)

where $f_{p}$ is the maximum frequency shift supported, $D_{y}$ is the length
of the PDCCH DMRS in OFDM symbols, and $D_{y}=1$ in this case. Hence,

$f_{p}=\frac{1}{2*1*10^{-6}*(66.67+5.2)}=6963.6\text{Hz},$ (3)

which corresponds to a user speed of 10743 km/h at 700 MHz and 1880 km/h at 4
GHz frequency. Note that the higher numerologies will increase the supported
UE speed, since lower $T_{U}+T_{cp}$ is obtained.

####  III-A2 Link-level Evaluation

To evaluate the performance of PDCCH in the mobile environments, link level
simulations are performed to, first, verify if any error floor occurs in the
Bit Interleaving, Coding and Modulation (BICM) performance when the estimated
channel response applies and different user speeds are considered. If not,
then the required Carrier-to-Noise Ratio (CNR) to achieve Block Error Rate
(BLER) $<$ 0.1% against the Doppler shift or user speeds can be evaluated. The
two-dimensional pilot-based estimation with the linear interpolation is used.
Different aggregation levels have been considered.

In Fig.3, the BLER/BER vs. CNR results for aggregation level 1 and 2 with user
speed of 3km/h and 120 km/h are presented. From Fig.3 we can see that the
required CNR to achieve 1e-3 BLER just slightly right shifted for the 120km/h
(i.e., about 4.5dB CNR) case compare to the 3km/h case (4.4dB CNR) and no
error floor occurs, which verify the above-discussed doppler effect based on
the sampling rate requirement. Therefore, the required CNR vs. a wide range of
Doppler Shift (equivalent to a wide range of user movement speed) are shown in
Fig. 4. From Fig. 4, we can see that the PDCCH channel can handle all required
user speeds with a slightly increased CNR requirement for all aggregation
levels, as in [13] for all the considered frequency bands. Also, the higher
the aggregation level, the lower the required CNR, due to the better coding
rate used (half with the next level).

###  III-B Coverage

In this sub-section, the performance of the NR-PDCCH channel with different
coverage settings has been evaluated.

####  III-B1 With Perfect Channel Estimation

First, we take a look at the BLER/BER results for both AWGN and 0dB each
channel ($\alpha=0.3$ and this represents the SFN receiver) with perfect
channel knowledge, and the results are shown in Fig. 5. From Fig. 5, we
conclude that a higher aggregation level generally gives more protection level
to the codewords, which is reflected on the required CNR for both AWGN and 0dB
echo channel, but with less spectrum efficiency. Breaking down to each
aggregation level, to achieve a BLER level of 1e-3, for AWGN channel, it
requires CNR values of -0.9, -3.8, -7, -10dB for aggregation level 1, 2, 4, 8,
respectively. While for 0dB each channel, the corresponding CNR values are
-2.3, -6.2, -9.3, -11.9dB, respectively. However, ideal channel estimation
cannot reflect the pilot granularity when different channel delay is implied.
So in the next subsection, we will introduce the results with real channel
estimation.

Figure 5: BER/BLER vs. CNR for AWGN and 0dB echo channel with perfect channel
estimation

####  III-B2 With real Channel Estimation

In theory, with a specific numerology $\mu$ and therefore a fixed useful
symbol duration $T_{U}$, the maximum channel delay spread is highly dependent
on the DMRS pattern, or to be specific, the pilot granularity in the frequency
domain $D_{x}$. The maximum distance between two frequency domain PDCCH DMRS
symbols, $m_{max}$, is given by:

$m_{max}\leq\frac{T_{U}}{\tau_{max}}$ (4)

where $\tau_{max}$ represents the maximum delay spread of the channel. Take
the 15kHz subcarrier spacing as an example, also as shown in the frame
structure in Fig.1, $D_{x}=4$, one can derive
$T_{U}=\frac{1}{\Delta_{f}}=66.7\mu s$. Therefore, the maximum channel delay
spread that can be tolerated would be
$\tau_{max}\leq\frac{T_{U}}{D_{x}}\approx 16.67\mu s$, greater than the one
from the channel such as TDL-A and TDL-C. Thus, similar to the NR-PDSCH, the
0dB echo channel has been modified and extended to include different delays.
Regarding the evaluation, in terms of interpolations after channel estimation
on the pilot, there is no need to perform interpolations in time domain due to
the corresponding pilot distribution in the PDCCH region. For the frequency
domain, the DFT-based interpolation is applied, which is a widely-used channel
estimation method and shows promise in [14], compared with the estimator with
the other types of interpolations such as the linear interpolation. Two set of
simulations are considered coverage capability:

  * •

Normal CP length with 7% Fast Fourier Transform (FFT) size (only for $\mu=0$).

  * •

Extended CP with 25% of FFT size.

Figure 6: BLER vs. CNR with different echo delay (PDCCH, AL=2). Figure 7:
Required CNR vs. corresponding echo delay (PDCCH, AL=2).

The pilot granularity remains the same in both scenarios, and the aggregation
level 2 is considered. First, we investigate the BLER vs. CNR for aggregation
level 2 with different echo delay i.e. $\alpha=0.1,1.5,1.6,1.7,1.8,2$ and the
frame structure for numerology $\mu=0$ is used. From Fig. 6 we can see when
the $\alpha$ go beyond 1.6, the required CNR dramatically increased (from
5.6dB to more than 8dB in order to achieve 1e-3 BLER level) and the link
becomes unreliable when $\aleph=2$. Same simulations have been executed for
extended CP with different numerologies. In Fig.7 we present the corresponding
results which show the trend of required CNR (to achieve 1e-3 BLER) following
the increasing of echo channel delay.

With the numerology $\mu=0$, both normal and extended CPs can work properly
with the time delay equivalent to the normal CP duration. Then the larger the
time delay, the higher the required CNR for normal CP, due to the increased
inter-symbol interference (ISI). However, the required CNR for the extended CP
is increased even within the CP duration. Regarding the maximum delay spread
to be tolerated, both normal CP and extended CP can support up to around
16$\mu s$, aligned with the theoretical limit as calculated previously.
Focusing on the numerology $\mu=0$, the usage of extended CP does not help to
increase the PTM coverage and the required CNR to achieve the same BLER is
increased with extended CP. Two insights can be highlighted:

  * •

CNR loss due to increased CP length

  * •

Low pilot granularity in frequency domain that affect the channel estimation
accuracy

Figure 8: Normal & extended CP with 0.1*$T_{cp}$ as second path delay

For the first point, as we know, the insertion of CP disperses the transmitter
energy (the amount of consumed power depends on how large the CP length),
where the signal-to-noise (SNR) lost due to the CP introduction indicates the
loss of transmission energy. The loss factor is given as:

$\text{CNR}_{loss}=-10\log_{10}(1-\frac{T_{cp}}{T_{U}+T_{cp}}),$ (5)

which is equivalent to the increase of noise variance (if fixed symbol
energy.), thus yields an increased channel estimation error at the pilot
position. As we can see the comparison in Figure 29, with a relatively short
2nd path delay, both normal and extended CP scenarios can still reconstruct
the channel within an acceptable offset range. However, in Figure 30, with
higher 2nd path delay (but still inside the CP duration), with normal CP, the
channel can be reconstructed, which is not the case for the extended CP.
Combining with the second aspect, the setting with the extended CP results in
that the channel is almost unable to be reconstructed. Especially for the
second point, when comparing the two figures at the left side of Fig. 8, with
normal CP, the interpolation nearly perfectly captures the channel variant.
However, with extended CP, even with perfect channel estimation at the pilot
position, the interpolation cannot reconstruct the channel due to the low
pilot granularity, and the gap is further amplified by the equivalently
increased noise variance, as shown in the right two figures in Fig. 9. To
elaborate more on the relationship between the pilot granularity in the
frequency domain and the supported maximum delay spread of the channel, we
consider the requirement of the DFT-based channel estimation/interpolation. As
discussed in [14], it requires the number of pilots to be much greater than
the channel delay spread (counted as the number of the channel delay taps in
the time domain). The frequency domain pilot granularity in the current PDCCH
DMRS pattern cannot meet this requirement in certain scenarios, which can
cause the imperfection of the DFT-based channel estimation, thus the
performance degradation of the system.

Figure 9: Normal & extended CP with 0.6*$T_{cp}$ as second path delay

##  IV Possible Enhancement

One solution to enhance the coverage capability is to use the negative
numerology as proposed previously. Another possible solution is to design a
new PDCCH DMRS pattern, especially for the extended CP. Recall the
relationship between the distance between two frequency domain PDCCH DMRS
symbols and the supporting delay spread of the channel, one can design the
control channel frame or the DMRS pattern by decreasing $D_{x}$. This,
however, can result in the significantly decreased number of available REs for
the control channel payload transmission, considering the limited number of
available radio resources and very low spectral efficiency in the control
channel. Besides, only QPSK can be used, which capped the peak SE to 2 bits
per channel user. Shown in Fig. 10 is the spectral efficiency of the current
standard, as we can see even in AWGN channel, the spectral efficiency is
already far away from capacity.

Figure 10: BICM Spectral Efficiency vs. CNR for AWGN channel, Capacity vs.
PDCCH

##  V Conclusion

In this paper, the Doppler effect limitation is analyzed based on the PDCCH
pilot distribution as well as the coverage which is covered by using the 0dB
echo channel to represent the SFN receiver. The mobility and coverage
simulation of PDCCH performance has been evaluated for AWGN, as well as for
TDL-A channel models. The discussions and simulation results obtained in this
paper shows that based on the current pilot distribution inside the PDCCH
area, it can support a very wide range of user mobility (beyond the
requirement for a PTM scenario) under TDL-A channel. Regarding coverage, both
ideal and real channel estimation cases have been covered. It shows that
finding a suitable interpolation method is very important in order to
reconstruct the DCI due to the lack of pilots compare to the channel delay
spread. Besides, as a potential future work, the imperfect PDCCH recovery at
the receiver may lead to a performance loss for the PDSCH performance, so
these two channels can be combined together to get a more completed link level
simulation. A comparison with the signaling of the state-of-the-art DTT
standard ATSC 3.0 would be of interest too.

## References

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  * [6] International Telecommunications Union (ITU), Draft New Report ITU-R M., “Guidelines for evaluation of radio interface technologies for IMT-2020,” Oct. 2017.
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  * [11] 3GPP TS 36.212 v15.1.0, “Evolved Universal Terrestrial Radio Access (E-UTRA); Multiplexing and channel coding,” April 2018.
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11institutetext: Department of Physics and Astronomy, FI-20014 University of
Turku, Finland
11email<EMAIL_ADDRESS>22institutetext: Nordita, KTH Royal Institute of
Technology and Stockholm University, Hannes Alfvéns väg 12, SE-10691
Stockholm, Sweden 33institutetext: Instituto de Astrofísica de Canarias,
E-38205 La Laguna, Tenerife, Spain 44institutetext: Departamento de
Astrofísica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
55institutetext: Astrophysics Research Institute, Liverpool John Moores
University, 146 Brownlow Hill, Liverpool L3 5RF, United Kingdom
66institutetext: Serco for the European Space Agency (ESA), European Space
Astronomy Centre, Camino Bajo del Castillo s/n, E-28692 Villanueva de la
Cañada, Madrid, Spain 77institutetext: Graduate School of Sciences, Tohoku
University, Aoba-ku, 980-8578 Sendai, Japan 88institutetext: Istituto Ricerche
Solari Aldo e Cele Daccò (IRSOL), Faculty of Informatics, Università della
Svizzera italiana, 6605 Locarno, Switzerland 99institutetext: Institut für
Sonnenphysik (KIS), Georges-Köhler-Allee 401a, 79110 Freiburg, Germany

# Black hole X-ray binary A0620$-$00 in quiescence: hints of Faraday rotation
of near-infrared and optical polarization?

Vadim Kravtsov 11 Alexandra Veledina 1122 Andrei V. Berdyugin 11 Sergey
Tsygankov 11 Tariq Shahbaz 33 4 4 Manuel A. P. Torres 33 4 4 Helen Jermak 55
Callum McCall 55 Jari J. E. Kajava 11 6 6 Vilppu Piirola 11 Takeshi Sakanoi 77
Masato Kagitani 77 Svetlana V. Berdyugina 88 9 9 Juri Poutanen 11

We present simultaneous high-precision optical polarimetric and near-infrared
(NIR) to ultraviolet (UV) photometric observations of low-mass black hole
X-ray binary A0620$-$00 in the quiescent state. Subtracting interstellar
polarization, estimated from a sample of field stars, we derive the intrinsic
polarization of A0620$-$00\. We show that the intrinsic polarization degree
(PD) is variable with the orbital period with the amplitude of $\sim 0.3\%$ at
least in the $R$ band, where the signal-to-noise ratio of our observations is
the best. It implies that some fraction of the optical polarization is
produced by scattering of stellar radiation off the matter that follows the
black hole in its orbital motion. In addition, we see a rotation of the orbit-
average intrinsic polarization angle (PA) with the wavelength from $164\degr$
in the $R$ to $180\degr$ in the $B$ band. All of the above, combined with the
historical NIR to optical polarimetric observations, shows the complex
behavior of average intrinsic polarization of A0620$-$00 with the PA making
continuous rotation from infrared to blue band by $\sim 56\degr$ in total,
while the PD $\sim 1\%$ remains nearly constant over the entire spectral
range. The spectral dependence of the PA can be described by Faraday rotation
with the rotation measure of RM=$-0.2$ rad $\mu$m-2, implying a few Gauss
magnetic field in the plasma surrounding the black hole accretion disk.
However, our preferred interpretation for the peculiar wavelength dependence
is the interplay between two polarized components with different PAs.
Polarimetric measurements in the UV range can help distinguishing between
these scenarios.

###### Key Words.:

accretion, accretion disks – polarization – stars: black holes – stars:
individual: A0620$-$00 – X-rays: binaries

##  1 Introduction

A0620$-$00 is the prototypical low-mass X-ray binary (LMXB) system containing
a black hole (BH), discovered during its 1975 outburst (Elvis et al., 1975).
Its luminosity increased by more than a million times compared to the
quiescent levels reaching the Eddington luminosity. Since then, the object has
resided in the quiescent state, where its spectrum is dominated by the
emission of $0.4M_{\odot}$ K-type star orbiting a $\sim 6M_{\odot}$ BH
(McClintock & Remillard, 1986; van Grunsven et al., 2017) with the period of
$P_{\rm orb}=7.75$ h. Nevertheless, an additional source of non-stellar origin
is needed to describe the excess of radio, near-infrared (NIR), and
ultraviolet (UV) emission of A0620$-$00 near quiescence (McClintock et al.,
1995; Froning et al., 2011; Gallo et al., 2019; Cherepashchuk et al., 2019).
There is still no agreement on the nature of the additional component: it
could be the jet, the accretion disk, the inner accretion flow, the gas
stream, or some combination of the above. Different components may give
similar contributions to the final spectrum, therefore additional information
is needed to distinguish between them. Polarization of the optical radiation
may provide such information.

Indeed, optical and NIR radiation produced in various physical processes,
including electron scattering of stellar radiation off the accretion disk/flow
and synchrotron emission in the presence of the ordered magnetic field, can be
polarized. The polarization degree (PD), polarization angle (PA), and their
spectral properties are different for different processes, which makes
polarimetry a powerful technique for studies of the physical mechanisms
responsible for the optical and NIR emission production in BH X-ray binaries.
A recent systematic study has shown, however, that in many quiescent BHs, the
intrinsic optical polarization (corrected for interstellar contribution) is
very small – in most cases $P_{\rm int}\lesssim 0.5\%$ (Kravtsov et al.,
2022). On the other hand, several LMXBs show significant polarization during
(or near) the quiescence: MAXI J1820+070 demonstrated high (up to $5\%$) PD
with the blue spectrum and PA different from the jet direction, suggesting BH
spin-orbit misalignment (Poutanen et al., 2022); A0620$-$00 showed $\sim$1%
optical and NIR polarization in the quiescent state (Dubus et al., 2008;
Russell et al., 2016).

Residing in the X-ray quiescence, A0620$-$00 shows optical state changes:
according to the Cantrell et al. (2008) classification, there are two
different states of the optical activity –– passive and active. In the passive
state, variations of the optical flux are consistent with the ellipsoidal
variations, produced by the rotation of the tidally distorted optical
companion. In the active state, the source is $20\%$ brighter and shows an
aperiodic high-frequency variation usually called “flickering”.

In this paper, we present the results of quasi-simultaneous high-precision
optical polarimetric and multiwavelength (NIR to UV) photometric observations
of A0620$-$00 during its passive quiescent state. The paper is organized as
follows. In Sect. 2, we describe the observational data. In Sect. 3, we
present the main results of our study: determination of the intrinsic optical
polarization of A0620$-$00, its significant orbital variability, and rotation
of the average intrinsic polarization angle with wavelength. In Sect. 4, we
discuss possible physical mechanisms that can reproduce the observed behavior.
Finally, in Sect. 5 we summarize our findings.

##  2 Data acquisition and analysis

Table 1: Log of polarimetric and photometric observations of A0620$-$00 in December 2022. Telescope | UT Date 2022 | Filters | $N_{\textrm{obs}}$
---|---|---|---
NOT | Dec 24–26 | B, V, R | 2
GTC | Dec 25 |  $K_{\textrm{s}}$, H, J | 1
LT | Dec 20–24 |  u, g, r, i, z, B, V | 3
UVOT | Dec 23–26 |  B, V, u, w1, m2, w2 | 4
Figure 1: Polarization map for A0620$-$00 (at the origin) and field stars in
$R$ band. The lines correspond to the observed polarization with the length of
the bars showing the PD, and the direction indicating the PA (measured from
north to east). Figure 2: Observed orbit-average Stokes parameters of
A0620$-$00 (squares) and field stars in the $B$, $V$, and $R$ bands (blue,
green, and red markers, respectively).

###  2.1 Optical polarimetric observations

Table 2: Polarization of field stars. |  | $B$ |  | $V$ |  | $R$
---|---|---|---|---|---|---
Field star | Parallax (mas) |  $q$ (%) |  $u$ (%) |  |  $q$ (%) |  $u$ (%) |  |  $q$ (%) |  $u$ (%)
Ref 1 | $0.72\pm 0.04$ | 0.71 $\pm$ | 0.03 |  $-1.45$ $\pm$ | 0.03 |  | 0.56 $\pm$ | 0.03 |  $-1.45$ $\pm$ | 0.03 |  | 0.64 $\pm$ | 0.02 |  $-1.40$ $\pm$ | 0.02
Ref 2 | $0.87\pm 0.02$ | 0.64 $\pm$ | 0.07 |  $-1.32$ $\pm$ | 0.07 |  | 0.45 $\pm$ | 0.04 |  $-1.29$ $\pm$ | 0.04 |  | 0.45 $\pm$ | 0.03 |  $-1.20$ $\pm$ | 0.03
Ref 3 | $0.56\pm 0.02$ | 0.50 $\pm$ | 0.08 |  $-1.33$ $\pm$ | 0.08 |  | 0.49 $\pm$ | 0.08 |  $-1.06$ $\pm$ | 0.08 |  | 0.38 $\pm$ | 0.06 |  $-1.26$ $\pm$ | 0.06
Ref 4 | $0.55\pm 0.02$ | 0.84 $\pm$ | 0.20 |  $-1.32$ $\pm$ | 0.20 |  | 0.36 $\pm$ | 0.12 |  $-0.68$ $\pm$ | 0.12 |  | 0.56 $\pm$ | 0.07 |  $-0.92$ $\pm$ | 0.07
Table 3: One-orbit-average observed and intrinsic PD and PA of A0620$-$00 together with interstellar polarization estimate.  | Observed |  | Interstellar |  | Intrinsic
---|---|---|---|---|---
Filter |  $P_{\text{obs}}$ (%) |  $\theta_{\text{obs}}$ (deg) |  |  $P_{\text{is}}$ (%) |  $\theta_{\text{is}}$ (deg) |  |  $P_{\text{int}}$ (%) |  $\theta_{\text{int}}$ (deg)
B | $1.96\pm 0.11$ | $156\pm 2$ |  | $1.61\pm 0.03$ | $148\pm 1$ |  | $0.62\pm 0.11$ | $180\pm 5$
V | $2.34\pm 0.07$ | $155\pm 1$ |  | $1.56\pm 0.03$ | $146\pm 1$ |  | $1.01\pm 0.07$ | $170\pm 2$
R | $2.27\pm 0.04$ | $153\pm 1$ |  | $1.54\pm 0.02$ | $147\pm 1$ |  | $0.83\pm 0.04$ | $164\pm 2$

High-precision optical polarimetric observations of A0620$-$00 were performed
using broad-band BVR polarimeter DIPol-UF (Piirola et al., 2021), a visitor
instrument installed at the 2.56 m Nordic Optical Telescope (NOT),
Observatorio del Roque de los Muchachos, La Palma, Spain. Field stars used for
the interstellar polarization estimation were observed with DIPol-2 (Piirola
et al., 2014), mounted on the remotely-controlled 60 cm Tohoku telescope (T60)
at Haleakala Observatory, Hawaii. Both polarimeters utilize a “double-image”
design that effectively eliminates the polarization of the sky, even if it
varies throughout the observations. The instrumental polarization is small
($<10^{-4}$) and well-calibrated by observing 20 unpolarized standard stars.
The zero points of the PAs were determined by observing highly polarized
standards HD 236928 and HD 25443. A more detailed description of the methods
and calibrations can be found in Piirola et al. (2020) and Kravtsov et al.
(2022), and references therein.

A0620$-$00 was observed during two nights between 2022 December 24–26 and 140
and 52 individual measurements of the Stokes parameters were made during the
two nights, respectively. The first observation was continuous, 8 h long, and
hence covered the whole orbital period of the binary, while the second
observation covered only $40\%$ of the period. Hereafter we will refer to the
average polarization measured during the first observation as orbit-averaged
polarization. Each individual linear polarization measurement was obtained
from four consecutive images with 50-s exposures taken at different half-wave
plate positions, resulting in one polarization measurement per $\sim$3.3 min.
To increase the signal-to-noise ratio, we split the data into bins so that
each bin contains 10 individual measurements of the Stokes parameters. The
errors of the normalized Stokes parameters $q$ and $u$ were computed as the
standard errors of the weighted mean values within the bin. The Stokes
parameters $(q,u)$ then were translated into the PD $P$ and PA $\theta,$

$P=\sqrt{q^{2}+u^{2}},\qquad\theta=\frac{1}{2}\operatorname{atan2}(u,q).$ (1)

The uncertainty on the PD is equal to the uncertainty of the individual Stokes
parameters, and uncertainty on the PA in radians was estimated as
$\sigma_{\theta}=\sigma_{P}/(2P)$ (Serkowski, 1962; Kosenkov et al., 2017).

###  2.2 Multiwavelength photometry

Quasi-simultaneous multiwavelength photometric observational campaign was
organized on several telescopes (see Table 1). Near-infrared JHK photometry
was made using EMIR wide-field imager (Garzón et al., 2022), installed on the
10.4 m Gran Telescopio Canarias (GTC), La Palma, Spain. Observations in the
broadband SDSS-ugriz, Bessel V, and Bessel B filters were performed using IO:O
instrument of the 2-m Liverpool Telescope (LT; Steele et al., 2004), La Palma,
Spain. For all the instruments, basic data reductions such as bias and dark
subtraction and flat fielding are done via the internal common pipelines.

###  2.3 Swift/UVOT

The Neil Gehrels Swift Observatory (Gehrels et al., 2004) observed A0620$-$00
four times from 2022 December 23 to 26 (see Table 1) with total exposure of
about 7 ks. The image analysis has been done following the procedure provided
by the UK Swift Science Data Centre.111https://www.swift.ac.uk/analysis/uvot/
Photometry in all available filters ($V$, $B$, $U$, $UVW1$, $UVW2$, and
$UVM2$) was performed using the tool uvotsource from the heasoft package
version 6.32 and the latest calibration files. The source and background
photons were extracted from the apertures with radii of 5″ and 10″,
respectively. The background was chosen with the center about 18″ away from
the source for all filters.

For the spectral fitting, all available data were converted to the spectral
files. For the Swift/UVOT data the uvot2pha tool was applied using the
corresponding response files in the CALDB. All other data were converted to
the spectral files using tool ftflx2xsp from the ftools package. The following
spectral fitting was performed using xspec version 12.13.1 (Arnaud, 1996).

##  3 Results

###  3.1 Average intrinsic polarization

Figure 3: Intrinsic Stokes parameters of the polarization of A0620$-$00 in
quiescence measured in different filters from the $K_{\rm s}$ to $B$ as
indicated by letters near each data point. The light blue circle at the origin
illustrates the uncertainty of the interstellar polarization. The red and blue
arrows indicate the intrinsic polarization vector directions in $K_{\rm s}$
and $B$ bands, respectively. The dashed curved arrow shows the track left by
the intrinsic polarization vector during its rotation from the IR to the blue
part of the spectrum. An orange star shows the polarization of A0620$-$00
measured during the 1975 outburst by Dolan (1976).

The observed optical polarization of a distant star is a combination of the
intrinsic polarization of the object and interstellar (IS) polarization. The
main source of the interstellar polarization ($q_{\rm is}$, $u_{\rm is}$) is
the optical dichroism of non-spherical interstellar dust particles aligned by
Galactic magnetic field. To extract the intrinsic polarization ($q_{\rm int}$,
$u_{\rm int}$) of A0620$-$00 from the observed polarization ($q_{\rm obs}$,
$u_{\rm obs}$), we first estimated the IS polarization. We measured the
polarization of the four close field stars (see Fig. 1 and Tab. 2) with
similar parallaxes according to Gaia DR3 (Gaia Collaboration et al., 2021)
and, taking into account the proximity of the star both in parallax and in
angular distance on the image, we choose the polarization of the star Ref 1 as
the best IS polarization estimate in the direction of A0620$-$00\. Our
estimate of the IS polarization of $P\sim 1.5\%$ and $\theta\sim 147\degr$
(see Table 3) is consistent with the previous estimates in the $V$ band of
$P\sim 1.5\%$ and $\theta\sim 153\degr$ by Dubus et al. (2008) and $P\sim
1.7\%$ and $\theta\sim 142\degr$ by Dolan (1976).

Subtracting the interstellar component from the observed polarization, we find
the average intrinsic polarization of A0620$-$00 to be $P=0.6$–1.0% with
$\theta$ changing from $164\degr$ in the $R$ filter to $180\degr$ in $B$ (see
Table 3). Combining NIR polarimetric observations (Russell et al., 2016)
corrected for IS polarization with our optical measurements, we see the
continuous rotation of the intrinsic polarization vector with the wavelength
from NIR to $B$ (see Fig. 3). The amplitude of the polarization vector
rotation on the sky is $\Delta\theta=\theta(B)-\theta(K_{\rm s})\approx
56\degr$ (see Fig. 4). In contrast to the PA, PD barely changes with
wavelength remaining at $\sim$1% level from $K_{\rm s}$ to $B$ band.

Figure 4: Rotation of the polarization on the sky. Red and blue arrows
correspond to the direction of the polarization in the $B$ and $K_{\rm s}$
filters, respectively. The blue region shows the direction of the radio
ejections as measured in Kuulkers et al. (1999).

We note that although the determination of the intrinsic PA strongly depends
on the accuracy of the IS polarization estimate, there are several additional
reasons to believe that the intrinsic polarization estimate is close to its
true value. First of all, our estimation of the IS polarization is consistent
with the previous independent measurements (Dubus et al., 2008; Dolan, 1976).
Furthermore, although the observed PD of A0620$-$00 in Dubus et al. (2008)
$V$-band observations was significantly higher than in this paper, subtracting
our estimate of the IS polarization from their observed Stokes parameters, we
get the PA of intrinsic polarization matching our value within a few degrees –
this can be seen in Fig. 3, if one connects the origin to the corresponding
observational points and compare the directions of the resulting vectors. This
alignment of the intrinsic polarization vectors is unlikely to be coincidental
and confirms our IS polarization estimate.

Figure 5: Orbital profiles of the flux and of the normalized Stokes parameters
of A0620$-$00\. Blue solid and red empty circles correspond to the data from
two different orbits. Panel (a): photometric $R$ magnitudes of A0620$-$00
folded with the orbital period. Panels (b) and (c): observed normalized Stokes
parameters. Panel (d): intrinsic PD of A0620$-$00 in the $R$ filter folded
with the orbital period. Each circle with the 1$\sigma$ error bar shows the
35-min average polarization.

###  3.2 Orbital variability of the polarization

We folded the photometric and polarimetric observations of A0620$-$00 in
$R$-band (for which the S/N is the best) with the orbital period using the
recent ephemeris of Cherepashchuk et al. (2019). We find statistically
significant orbital variability of intrinsic polarization of the source in the
$R$ filter (Fig. 5b,c). We plot the orbital profile of the observed
polarization together with the optical light curve of A0620$-$00, obtained
simultaneously (Fig. 5a). We see that the polarization degree is constant
within the measurement errors during the first half of the orbit, and varies
significantly in the second half. Variations of PD follow closely the
variations of the R-band magnitudes. The absence of flickering in the
photometric observations together with the visual magnitude $V=18.3\pm 0.1$ of
A0620$-$00 suggests that the source was in the passive quiescent state during
our campaign (Cantrell et al., 2008).

###  3.3 Broadband spectrum

Figure 6: Spectral energy distribution of A0620$-$00, corrected for reddening
assuming $E(B-V)=0.35$. Red squares with error bars correspond to the data
from this paper. Blue crosses, black triangles and purple diamonds show data
from McClintock & Remillard (2000), Froning et al. (2011), and Dinçer et al.
(2018), respectively. Solid blue, dotted blue, and dashed orange lines show
the spectra of the K5V star (Pickles, 1998), the blackbody of $T_{*}=4500$ K
and an additional blackbody of $T_{\rm bb}=10,000$ K, respectively.

The observed NIR-to-UV spectral energy distribution (SED) of A0620$-$00
corrected for reddening assuming $E(B-V)=0.35$ (Wu et al., 1983) is shown in
Fig. 6. The key feature of the SED is the excess of UV photons, previously
observed in the quiescent state (Froning et al., 2011; McClintock et al.,
1995) – an extra source of UV radiation in addition to the optical companion
is needed to reproduce the observed shape of the spectrum. The spectrum alone,
however, does not allow unambiguous separation of the second component: even
the simplest model with one additional blackbody is degenerate due to the
uncertainty in the stellar spectrum normalization. Various different methods
were used to estimate the contribution of non-stellar emission to the
quiescent spectrum of A0620$-$00 (Marsh et al., 1994; Gelino et al., 2001;
Harrison et al., 2007; Froning et al., 2011; Dinçer et al., 2018). Despite the
fact that there is no complete consensus, it can be cautiously noted that most
authors agree that the companion star dominates the entire NIR-to-optical
range, with the contribution of additional component increasing towards the
blue part of the spectrum. The additional component becomes comparable with
the stellar flux only near $B$ band, where its contribution reaches 20–50%.
Non-stellar emission spectrum can be described with the blackbody of
temperature $T_{\rm bb}=$ 9,000–11,000 K depending on the state of activity.
The nature of the additional component is still under discussion, but most
likely it corresponds to the brightest part of the accretion disk (either to
the hot inner disk regions or to the bright spot formed at the impact point of
the accretion stream). The bright spot is clearly present in the Doppler
tomograms (Marsh et al., 1994; Shahbaz et al., 1994, 2004; Neilsen et al.,
2008) and it is needed to explain the asymmetric light curves (Froning &
Robinson, 2001; Cantrell et al., 2010; van Grunsven et al., 2017;
Cherepashchuk et al., 2019). The contribution of the non-stellar radiation is
variable on short (Haswell et al., 1993) and long (Cantrell et al., 2008)
timescales. Our new observations are consistent with both interpretations (see
Fig. 6 and Sect. 4).

##  4 Discussion

###  4.1 The source of the polarization

The polarization measurements of optical light from the binary were obtained
both in the quiescent state and in the outburst (Dolan, 1976, see the orange
star in Fig. 3). The observed PD, PA, and their spectral properties were found
to be consistent with the interstellar origin. Our IS polarization estimate is
very close to the outburst polarization level of A0620$-$00, which confirms
the above statement. Sub-per cent intrinsic optical polarization of A0620$-$00
is in line with that of the other X-ray binaries – most known sources observed
during the outburst show low intrinsic polarization levels (XTE J1118+480,
Schultz et al. 2004; MAXI J0637$-$430, Kravtsov et al. 2019; MAXI J1820+070,
Veledina et al. 2019; Kosenkov et al. 2020; SwiftJ 1727.8$-$1613, Kravtsov et
al. 2023a). However, some BH LMXBs show a significant increase of intrinsic PD
as they approach quiescence (e.g., MAXI J1820+070, Poutanen et al. 2022, and
A0620$-$00, Dubus et al. 2008).

The orbit-averaged intrinsic polarization of A0620$-$00 in the quiescent state
has nearly the same PD $\sim 1\%$ at all wavelengths from $K_{\rm s}$ to $B$
band, yet PA shows a significant rotation over those wavelengths (Fig. 3).
This makes A0620$-$00 the only source known to date for which such a strong PA
dependence on the wavelength is observed. Such polarization behavior cannot be
explained in terms of simple mechanisms of polarization production. Indeed, if
the polarization is produced mostly by scattering, its PA should not depend on
the wavelength.

We first study the case where Faraday rotation causes the observed dependence
of PA on wavelength. We consider the scenario illustrated in Fig. 7. The
unpolarized radiation of the companion star gets scattered by the accretion
disk and the polarized scattered radiation has initial PA independent of the
wavelength. After the scattering, the linearly polarized radiation propagates
toward the observer through the magnetized plasma surrounding the accretion
disk. The PA of the light propagating along the magnetic field lines
experiences the Faraday rotation, resulting in the observed PA dependence on
wavelength.

Figure 7: Schematic illustration of the scattering geometry (not to scale).
Figure 8: Dependence of the intrinsic PD and PA of A0620$-$00 on
$\lambda^{2}$. The circles and squares correspond to optical observations from
this paper and NIR polarimetric observations from Russell et al. (2016),
respectively. The blue line shows the best-fit Faraday rotation model given by
Eq. (2).

The PA of linearly polarized radiation propagating through the magnetized
plasma rotates with the wavelength $\lambda$ as:

$\theta(\lambda)=\theta_{0}+{\rm RM}\,\lambda^{2},$ (2)

where the rotation measure RM is defined through the integral along the line
of sight

$\mbox{RM}=\frac{e^{3}}{2\pi m_{\rm e}^{2}c^{4}}\int{n_{\rm
e}(l)B_{||}(l)dl},$ (3)

where $n_{\rm e}$ is the electron number density, $B_{||}$ (in G) is the line
of sight magnetic field strength, $e$ is the electric charge, $m_{\rm e}$ is
the electron mass and $c$ is the speed of light. If $B_{||}$ is constant, the
RM can be expressed through the Thomson optical depth $\tau_{\rm T}=\int
n_{\rm e}\sigma_{\rm T}dl$. In this case, RM$\approx 0.4\tau_{\rm T}B_{||}$
rad $\mu$m-2.

We use Eq. (2) with parameters $\theta_{0}$ and RM to find the best-fit
solution for the Faraday rotation of PA in our optical points and NIR points
reported in Russell et al. (2016). We find the best-fit parameters
$\theta_{0}=175\degr\pm 4\degr$ and RM=$-0.20\pm 0.01$ rad $\mu$m-2 (see Fig.
8). The latter value gives the relation between the line-of-sight optical
depth and magnetic field (in Gauss) $\tau_{\rm T}B_{||}\approx 0.5$. For the
realistic values of matter density in quiescence $\tau_{\rm T}\lesssim 0.1$,
we find $B\gtrsim 5$ G. The latter value is roughly consistent with the
magnetic fields expected in the quiescent-state optically thin accretion flows
(Wallace & Pe’er, 2021), which may be similar to those of optically thin
plasma surrounding the disk (Fig. 7).

We note, however, that the NIR data have been taken almost ten years prior to
our optical observations, hence potential magnetic field variations may lead
to the inconsistency of trends between the optical and NIR PA points. We
indeed observe that our recent measurements (leftmost three points in Fig. 8)
lie on a straight line with the slope differing from the general trend. If we
apply Eq (2) only to our data, we obtain $\theta_{0}=186\degr\pm 6\degr$ and
RM=$-1.3\pm 0.3$ rad $\mu$m-2 (see Fig. 8). This translates to $\tau_{\rm
T}B_{||}\approx 3.2$, requiring almost seven times higher magnetic field
strength for the Faraday rotation to occur in the optically thin plasma
($\tau_{\rm T}\lesssim 1$). We also note that the Thomson scattering and
Faraday effect cannot explain the complex dependence of PD on energy, with
$P_{V}$ being higher than $P_{R}$ and $P_{B}$. The sharp rise of polarization
in a narrow wavelength range may be related to the atomic features.

Figure 9: Polarimetric properties of A0620$-$00\. Panel (a): Energy dependence
of flux $F_{\nu}$ as in Fig. 6 but only the data from this paper are shown.
Panels (b) and (c): energy dependence of intrinsic PD and PA of A0620$-$00\.
Circles and squares correspond to optical observations from this paper and NIR
polarimetric observations from Russell et al. (2016), respectively. Blue line
corresponds to the model with two polarized components described in Sect. 4.1.

Next, we consider the scenario where the observed polarization arises from two
components: one coming from the scattering of the stellar light and the other
from the polarized contribution of the additional UV component. Stellar
polarization is assumed to have constant PD and PA which implies a constant
fraction of the scattered radiation, and spectrum as that of the optical
companion (blue solid line in Fig. 9a). The additional UV component (dashed
orange line), that was used to describe the UV part of the observed SED, is
set to have constant PD and PA, whose values were free to vary. The blue lines
in Fig. 9b,c show the dependence of PD and PA for this two-component scenario.
The PD dependence on energy is noteworthy: $P_{V}$ appears higher than $P_{B}$
and $P_{R}$ due to the narrow absorption of the stellar spectrum (and its
scattering) in this band. We find the parameters $\rm{PA_{sc}}\approx
150\degr$, $\rm{PD_{sc}}\approx 1\%$ for the stellar scattering and
$\rm{PA_{UV}}\approx 215\degr$ with $\rm{PD_{UV}}\approx 2\%$ for the UV
component reproduce the energy dependence of PD and PA quite well (see Fig.
9b,c), especially in the optical band, where two components have comparable
contribution to the observed flux. The model is also generally consistent with
the NIR points (taken decades earlier); some deviation in the $K_{\rm s}$ band
might also arise from the Faraday effect or from the different treatment of
the interstellar contribution to polarization.

The polarization of the UV component may arise from the first (single) Compton
up-scattering of the disk or synchrotron photons by electrons in the hot
accretion flow, similar to the case of MAXI J1820+070 (Poutanen et al., 2022).
The difference of PA in the scattered and UV components can then indicate
misalignment by $\sim 65\degr$ (or $\sim 25\degr$) between the axes of these
components. This scenario predicts rise of PD and further rotation of PA
towards the UV band (see Fig. 9). It can be tested in the future, by measuring
polarization of A0620$-$00 simultaneously from NIR up to UV.

Figure 10: Flux and PD variations of A0620$-$00\. Panel (a): the normalized
flux, folded with the orbital phase (grey circles). Red solid line corresponds
to the best fit of the data with the Fourier series, while the blue dashed
line corresponds to the sinusoidal variations, assumed to be produced by the
tidally distorted optical companion. Panel (b): intrinsic PD of A0620$-$00
(red circles) folded with the orbital period together with the model of a
scattering cloud on a circular orbit from the Appendix of Kravtsov et al. 2020
(blue line).

###  4.2 The source of the polarization variations

In addition to the changes in the polarization between different spectral
states, we also see the orbital variations of the polarization. Although the
$2\%$ orbital modulation of the polarization observed by Dolan & Tapia (1989)
has not been confirmed (Dubus et al., 2008), we see the variations at the
level of $0.3\%$ at the timescales of hours (see Fig. 5). To confidently claim
that the polarization of A0620$-$00 is variable with the orbital phase one
needs to cover at least two consecutive orbital periods with the high-
precision polarimetric observations (which is a rather difficult task given
the brightness of the object in quiescence and its short orbital period), we
can still cautiously state that the observed polarization variability is
related to the orbital phase rather than having a stochastic nature. Indeed,
the polarization of A0620$-$00, folded with the orbital phase shows the
pattern, typical for binary stars – double-sinusoidal variations with the PD
reaching minima in the conjunctions (phases 0 and 0.5) and maxima in the
quadratures (phases 0.25 and 0.75), arising from Thomson scattering of the
stellar radiation on the matter, following the compact object in its orbital
motion (Brown et al., 1978; Kravtsov et al., 2020), and similar to these
observed in another BH X-ray binary Cyg X-1 (Kemp et al., 1979; Kravtsov et
al., 2023b). In Fig. 10, we see fairly good agreement of the PD variations
with the expected behavior (blue line shows the model of a scattering cloud on
a circular orbit, see Appendix of Kravtsov et al. 2020). In addition, our
first and second observations, performed during two different orbits are
consistent with each other (see Fig. 5).

The optical light curve shown in Fig. 10 has an asymmetric profile – the peak
and the dip at the first half of the orbit are brighter than the other peak
and dip. The so-called ellipsoidal variations of the flux, produced by the
tidally distorted optical star (shown in blue dashed line at Fig. 10) cannot
produce such asymmetry, therefore an additional source of flux variations is
needed. There are two alternatives: either there is an additional bright
component visible only at the first part of the orbit and generating
additional flux (e.g. phase-dependent disk/bright spot, see e.g. Haswell et
al. 1993), or some object blocks the light of the optical companion, reducing
the total flux at phases from 0.5 to 1.0 (e.g. dark spots at the surface of
the star, e.g. Cherepashchuk et al. 2019). We see similar asymmetry in the
orbital polarization profile of A0620$-$00 – while around orbital phases from
0.5 to 1.0 the PD shows good agreement with the Thomson scattering model, at
the first part of the orbit the PD is consistent with being constant. The
additional bright unpolarized component, visible only at phases from 0.0 to
0.5 could reduce the polarization at these phases and explain both the
asymmetric PD and flux variations, therefore we argue that the bright
spot/phase-dependent disk model is more likely than the dark star spot model.
Another alternative is that the asymmetric PD profile is produced by
scattering of stellar radiation off slightly tilted accretion disk. In this
scenario, the illuminated part of the accretion disk is visible to the
observer only for half of the orbit (see Kravtsov et al., 2023b). However,
scattering alone is not enough to reproduce $\sim 10\%$ increase of flux
observed at the first half of the orbit, and the quality of our polarimetric
data is not sufficient to discriminate between the complex models.

##  5 Conclusions

In this paper, we presented new high-precision phase-resolved optical
polarimetric and quasi-simultaneous NIR-to-UV photometric observations of BH
X-ray binary A0620$-$00 in a passive quiescent state. We determined and
subtracted the interstellar polarization, which allowed us to derive the
intrinsic polarization of A0620$-$00\. Combined with the NIR polarimetric
observations from Russell et al. (2016), we found that the orbit-average
intrinsic PA rotates with the wavelength, changing from $124\degr$ in the
$K_{\mathrm{s}}$ filter to $180\degr$ in $B$, while the PD remains at $\sim
1\%$ level throughout this spectral range. Folding our polarimetric
observations with the orbital period, we found the significant orbital
variability of polarization properties in the $R$ band. The shape of the
variations with two minima and two maxima per period suggest that the
polarization is most probably produced by Thomson scattering of the companion
star emission off the matter which follows the BH in its orbital motion (e.g.
scattering off the accretion disk/stream). The lack of variations at the first
part of the orbit suggests that either we see a depolarization effect caused
by the bright spot visible only at these phases or (less probable) the
scattering material producing the polarization is obscured or tilted relative
to the orbital axis. However, more high-precision polarimetric observations
during at least several consecutive orbital periods are needed to draw
unambiguous conclusions about the geometry of the scattering medium.

The flat spectrum of the polarization and the presence of the orbital
variations suggest that NIR-to-optical polarization has a scattering origin.
However, in that case, the PA is not expected to change with the wavelength as
observed. We first considered the possibility that stellar radiation scattered
off the accretion disk experiences the Faraday rotation while traveling toward
the observer through the magnetized plasma surrounding the accretion disk. The
estimated values of magnetic field and optical depth can be consistent with
realistic estimates if we consider our optical points in combination with the
NIR points taken almost ten years ago. On the other hand, the steep trend of
our optical points favour higher Faraday rotation rate, resulting in higher
values for the line-of sight magnetic field.

We considered the scenario of two polarized components having different PDs
and PAs. One component is coming from the stellar scattered light and the
other associated with the additional UV component seen in the spectrum. The
second component may potentially arise from Compton up-scattering of the disk
or synchrotron photons in the hot inner flow, similar to that in the
quiescent-state low-mass X-ray binary MAXI J1820+070. The PAs of these
components differ by $\sim 65\degr$, which translates to either $65\degr$ or
$25\degr$ misalignment between their axes of symmetry. Future simultaneous
polarimetric observations covering NIR-to-UV range would be extremely helpful
in making an unambiguous choice between the models.

###### Acknowledgements.

Based on observations made with the Nordic Optical Telescope, owned in
collaboration by the University of Turku and Aarhus University, and operated
jointly by Aarhus University, the University of Turku, and the University of
Oslo, representing Denmark, Finland, and Norway, the University of Iceland and
Stockholm University at the Observatorio del Roque de los Muchachos, La Palma,
Spain, of the Instituto de Astrofisica de Canarias. The DIPol-2 and DIPol-UF
polarimeters were built in cooperation between the University of Turku,
Finland, and the Leibniz-Institut für Sonnenphysik, Germany. The Liverpool
Telescope is operated on the island of La Palma by Liverpool John Moores
University in the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofisica de Canarias with financial support from the UK
Science and Technology Facilities Council. MAPT acknowledge support from the
Agencia Estatal de Investigación (MCIN/AEI) and the European Regional
Development Fund under grant PID2021-124879NB-I00. This research has been
supported by the Finnish Cultural Foundation (VK) and by the Academy of
Finland grants 355672 (AV). Nordita is supported in part by NordForsk.

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