# Black hole spin–orbit misalignment in the x-ray binary MAXI J1820+070

Juri Poutanen1,2,3∗, Alexandra Veledina1,3,2, Andrei V. Berdyugin1,
Svetlana V. Berdyugina4, Helen Jermak5, Peter G. Jonker6,7, Jari J. E.
Kajava1,8,
Ilia A. Kosenkov1, Vadim Kravtsov1, Vilppu Piirola1, Manisha Shrestha5,9,
Manuel A. P. Torres10,11, Sergey S. Tsygankov1,2

1: Department of Physics and Astronomy, FI-20014 University of Turku, Finland
2: Space Research Institute (IKI) of the Russian Academy of Sciences, 117997
Moscow, Russia
3: Nordic Institute for Theoretical Physics (Nordita), KTH Royal Institute of
Technology and Stockholm University, SE-10691 Stockholm, Sweden
4: Leibniz-Institut für Sonnenphysik, 79104 Freiburg, Germany
5: Astrophysics Research Institute, Liverpool John Moores University, L3 5RF
Liverpool, United Kingdom
6: Department of Astrophysics, Institute for Mathematics, Astrophysics and
Particle Physics (IMAPP), Radboud University, NL-6500 GL Nijmegen, The
Netherlands
7: Space Research Organisation of the Netherlands (SRON), Netherlands
Institute for Space Research, NL-2333, CA Leiden, The Netherlands
8: Centro de Astrobiología, Villanueva de la Cañada, S-28692 Madrid, Spain
9: Department of Physics & Astronomy, University of Denver, Denver, CO 80208,
USA
10: Instituto de Astrofísica de Canarias, E-38205 La Laguna, Tenerife, Spain
11: Departamento de Astrofísica, Universidad de La Laguna, E-38206 La Laguna,
Tenerife, Spain

∗Corresponding author. E-mail<EMAIL_ADDRESS>
> The observational signatures of black holes in x-ray binary systems depend
> on their masses, spins, accretion rate and the misalignment angle between
> the black hole spin and the orbital angular momentum. We present optical
> polarimetric observations of the black hole x-ray binary MAXI J1820+070,
> from which we constrain the position angle of the binary orbital axis.
> Combining this with previous determinations of the relativistic jet
> orientation axis, which traces the black hole spin, and the inclination of
> the orbit, we determine a lower limit of $40\hbox{${}^{\circ}$}$ on the
> spin-orbit misalignment angle. The misalignment has to originate from either
> the binary or black hole formation stage. If other x-ray binaries have
> similarly large misalignments, these would bias measurements of black hole
> masses and spins from x-ray observations.

Black holes can be characterized with just two parameters: mass and spin. When
a black hole resides in a binary system, accreting material from a companion
donor star through the accretion disk, there are additional parameters that
determine its observational signatures: the mass accretion rate and the
misalignment angle between the black hole spin and the orbital axis. Standard
methods to measure black hole spin from x-ray observations – iron line
spectroscopy (?) or modeling of the accretion disk spectrum (?) – assume that
the misalignment angle is small. Conversely, the standard interpretation of
low-frequency quasi-periodic oscillations in x-ray and optical observations of
black hole x-ray binaries as precession of the accretion disk (?, ?, ?),
requires the assumption that the misalignment angle is non-negligible.
Substantial misalignment is theoretically predicted for x-ray binaries that
received high velocities during formation (?). The misalignment angle must be
inherited from the formation process, because it can only decrease when the
black hole is accreting (?). Gravitational wave observations of merging black
holes have detected signatures of orbital precession (?) indicating non-zero
misalignment in these systems (?), though they might not be representative of
the wider population.

Measuring the misalignment angle in x-ray binaries requires determining the
three-dimensional orientation of the black hole spin and orbital axis.
Accreting black holes often show relativistic jets, which are launched along
an axis determined by the black hole spin direction (?). The jet inclination
angle can be directly obtained in some cases from radio observations (?),
whereas the jet position angle can be measured using either radio or x-ray
imaging. Combining these two angles allows the black hole spin orientation to
be determined. Orbital parameters, such as period and orbital inclination, can
be determined using spectroscopic measurements of radial velocities of the
donor star taken during quiescence, the stage at which accretion to the black
hole is reduced and optical emission is not dominated by the accretion disk,
through orbital modulation of the optical photometry, and using constraints
from the presence or absence of x-ray and optical disk eclipses (?).

The black hole x-ray binary MAXI J1820+070 was discovered as a transient x-ray
source on 2018 March 11 (?). X-ray quasi-periodic oscillations, detected
shortly after the discovery, were observed for more than 100 days (?).
Ejections of material traveling at relativistic velocities have been observed
from this source in both radio and x-rays, indicating the jet inclination
(measured from the line-of-sight) is $i_{\rm jet}=63\hbox{${}^{\circ}$}\pm
3\hbox{${}^{\circ}$}$ and the position angle (measured on the plane of the sky
from North to East) is $\theta_{\rm jet}=25\mbox{$.\\!\\!^{\circ}$}1\pm
1\mbox{$.\\!\\!^{\circ}$}4$ (?, ?, ?). Both angles were determined to be
stable over the observed duration of the outburst. The orbital inclination has
been constrained to the range $66\hbox{${}^{\circ}$}<i_{\rm
orb}<81\hbox{${}^{\circ}$}$ by the lack of x-ray eclipses and the detection of
grazing optical eclipses (?). To determine the orientation of the orbital axis
requires one further parameter: the orbital position angle $\theta_{\rm orb}$.

We monitored MAXI J1820+070 in the optical $B$, $V$, and $R$ photometric bands
using the Double Image Polarimeters (?, ?) during the 2018 outburst and in
quiescence. We obtained the source intrinsic linear polarization by
subtracting the foreground interstellar polarization, measured from nearby
field stars. During the outburst, when the relativistic jets were detected at
radio frequencies, the intrinsic linear polarization degree (PD) in the $V$
and $R$ bands reached 0.5% at a polarization angle (PA, also measured from
North to East) of 23∘–24∘, which coincides with the jet position angle within
the uncertainties (?, ?). After the source faded in the x-rays, the PD
increased by a factor of 5–10 and the PA changed by 40∘$\pm$4∘ to
$-$17∘$\pm$4∘ (Table S1 and Fig. 1) (?). This increase in PD is most prominent
in the $B$-band, which also has the highest PD in the range 1.5–5%; the
$R$-band polarization changes from 0.4% to 2%. The PA is most precisely
determined in the $B$-band, which also shows the least variability, with the
mean being $\langle\mbox{PA}\rangle=-19\mbox{$.\\!\\!^{\circ}$}7\pm
1\mbox{$.\\!\\!^{\circ}$}2$.

Fig. 1. Observed optical polarization properties of MAXI J1820+070. (A)
Intrinsic polarization degree (PD) and (B) polarization angle (PA) of MAXI
J1820+070 in quiescence are shown as a function of orbital phase [using a
published ephemeris (?)]. The intrinsic values were obtained from the observed
ones by subtracting the foreground interstellar polarization, measured from
nearby field stars. Blue circles, green triangles and red squares correspond
to $B$, $V$, and $R$ bands, respectively, with the error bars showing the 68%
confidence level. Polarization is strongest in the $B$ band and weakest in the
$R$ band, while the angle does not change substantially.

We identify three properties of the quiescent-state polarization: it is
strongest in the blue part of the optical, with approximate dependence on
frequency $\nu$ as PD$({\nu})\propto\nu^{3}$ (Table S1, Fig. 2), the PD
remains high in the range 0.5–5% and the PA is stable. The PA undergoes
apparently stochastic variations with an amplitude of $<$10∘ with no
dependence on the orbital phase. These properties constrain the mechanism of
the polarized emission. We modeled broadband photometric data obtained with
the Liverpool Telescope and the Swift Ultraviolet and Optical Telescope (UVOT)
together with the polarized fluxes (Fig. 2). We decompose the total spectral
energy distribution into three components: a companion star (contributing
$\sim 25\%$ to the $R$-band flux), an accretion disk with inner temperature
$T_{\rm d}\approx 6,200$ K and inner radius $R_{\rm d}\approx 6\times 10^{10}$
cm, and an additional ultraviolet (UV) component with blackbody temperature
$T_{\rm bb}\approx 15,000$ K and radius $R_{\rm bb}\approx 9\times 10^{9}$ cm
(Table S4). The properties of the polarized flux are consistent with being
produced by the UV component with constant PD of 5–8%.

Fig. 2. Spectral energy distribution (SED). The average SED of MAXI J1820+070
(red diamonds) as observed by the Liverpool Telescope and Swift/UVOT telescope
in July 2020 and corrected for reddening with color excess $E(B-V)=0.29$. The
photometric bands are indicated at the top of the figure. The black dotted
lines give the lower/upper limits on the flux for lower/higher extinction with
$E(B-V)=0.25$ and 0.325, respectively. The polarized flux divided by the best-
fitting model polarization degree $P_{\rm UV}=0.055$ (i.e. multiplied by a
factor of $\sim$18) is shown with blue triangles. Error bars show 68%
confidence level. The black line gives the total model flux consisting of the
companion star modeled as a blackbody (pink dot-dashed line), accretion disk
(red dotted) and a hot blackbody (blue dashed). The spectrum of a K7 star (?)
is shown as the solid green line for comparison.

The jet cannot be the source of the polarized emission because its optically
thin synchrotron spectrum is red, inconsistent with the observed blue spectrum
of polarized light. Moreover, the PA is offset by about
$40\hbox{${}^{\circ}$}$ from the jet position angle. The absence of detectable
orbital variations in the PA excludes a hot spot origin. An optically thick
accretion disk is excluded by the high PD and blue spectrum. A potential
source of the polarized emission is scattering of the accretion disk radiation
in the hot optically thin and geometrically thick accretion flow close to the
disk inner radius (?, ?), which can also be responsible for the observed UV
excess. This mechanism would produce polarization parallel to the meridional
plane, i.e. the plane formed by the orbital axis and the direction towards the
observer. Another possibility is dust scattering, thought to be responsible
for blue polarized spectra observed from accretion disks around some
supermassive black holes (?). The presence of dust in quiescent-state black
hole x-ray binaries has been inferred from the detection of the mid-IR excess
in two systems (?). If dust is located within a flattened envelope, in the
wind around the accretion disk, or in a circumbinary disk, the resulting
polarization vector would also be parallel to the meridional plane. However,
if dust forms an extended, approximately spherical structure at high elevation
above the accretion disk, the polarization would be perpendicular to the
meridional plane. We consider the latter scenario to be implausible, as a
nearly spherical envelope cannot produce the high observed PD. A dust
scattering mechanism would not explain the UV excess, because the disk does
not emit in that range and hence there are no photons to be scattered by the
dust.

Fig. 3. Geometry of the system from the observer’s perspective. The grey plane
is the plane of the sky, labelled with North and East axes, perpendicular to
the line of sight towards the observer $\hat{o}$. The angles between the line
of sight and the vectors of the orbital angular momentum $\hat{\Omega}$ and
the black hole spin $\hat{s}$ are the inclinations $i_{\rm orb}$ and $i_{\rm
bh}$. The corresponding position angles $\theta_{\rm orb}$ and $\theta_{\rm
bh}$ are the azimuthal angles projected onto the sky, measured from North to
East. The misalignment angle $\beta$ is defined as the angle between $\hat{s}$
and $\hat{\Omega}$. The red cone indicates the jet and the blue ellipse
indicates the companion star orbit around the black hole, which is at the
coordinate center.

Independent of the spectral modeling and geometry of the emission, the
stability of the PA (most evident in the $B$-band, Fig. 1) over the orbital
phase suggests that the polarization is related to the orbital axis, either
parallel or perpendicular to it. Hence, the observed PA provides information
about the position angle of the orbital axis. The misalignment angle $\beta$
can be determined from

$\cos\beta=\cos i_{\rm bh}\cos i_{\rm orb}+\sin i_{\rm bh}\sin i_{\rm
orb}\cos\Delta,$ (1)

where $i_{\rm bh}$ is the inclination of the black hole spin vector (measured
from the line-of-sight) and $\Delta=\theta_{\rm bh}-\theta_{\rm orb}$ is the
difference between the position angles of the black hole spin vector
$\theta_{\rm bh}$ and the orbital angular momentum $\theta_{\rm orb}$ (the
geometry is illustrated in Fig. 3). If the black hole spin vector is directed
along the southern approaching jet, then its inclination $i_{\rm bh}=i_{\rm
jet}=63\hbox{${}^{\circ}$}\pm 3\hbox{${}^{\circ}$}$ and its position angle is
$\theta_{\rm bh}=180\hbox{${}^{\circ}$}+\theta_{\rm
jet}=205\mbox{$.\\!\\!^{\circ}$}1\pm 1\mbox{$.\\!\\!^{\circ}$}4$ (?, ?, ?).
The smallest misalignment $\beta\approx 42\hbox{${}^{\circ}$}$ is achieved
when the orbital spin is also directed South at $\theta_{\rm
orb}=\langle\mbox{PA}\rangle$+180∘=$160\mbox{$.\\!\\!^{\circ}$}3\pm
1\mbox{$.\\!\\!^{\circ}$}2$ (because the PA has an ambiguity of 180∘), at the
inclination $i_{\rm orb}\approx 73\hbox{${}^{\circ}$}$. The probability
distribution for $\beta$ in this case is shown in Fig. 4. The radial velocity
measurements (?) do not differentiate between orbital inclinations $i_{\rm
orb}$ and $180\hbox{${}^{\circ}$}-i_{\rm orb}$, so there is a second solution
with $i_{\rm orb}\approx 107\hbox{${}^{\circ}$}$ and $\beta\approx
63\hbox{${}^{\circ}$}$. If either the orbital angular momentum or the black
hole spin is instead directed to the North, the black hole rotation is then
retrograde, resulting in $\beta\approx 117\hbox{${}^{\circ}$}$ or 138∘ for the
same two solutions for the orbital inclination as above.

If the polarization vector is perpendicular to the meridional plane, the
orbital position angle can take values $\theta_{\rm
orb}=\langle\mbox{PA}\rangle+90\hbox{${}^{\circ}$}$ or
$\langle\mbox{PA}\rangle+270\hbox{${}^{\circ}$}$. This geometrical arrangement
leads to nearly identical values for $\beta$ because the difference between
jet position angle and $\langle\mbox{PA}\rangle$ is about 45∘. All possible
cases for the orientations of the black hole and orbital spins, the resulting
values for $\beta$ and the azimuthal angle of the black hole spin in the
orbital plane are listed in Table S5. Corresponding probability distributions
are shown in Figs. S4 and S5.

The difference of $\approx 45\hbox{${}^{\circ}$}$ between the jet position
angle and the PA indicates $\gtrsim 40\hbox{${}^{\circ}$}$ misalignment
between the black hole spin and the orbital angular momentum. This result is
independent of modeling or geometric ambiguities, because it relies only on
the observed difference between the polarization angle and jet position angle.

During outbursts, when the matter reaches the black hole, this misalignment
affects the innermost regions of the accretion disk. For a non-zero spin,
particles moving around the black hole in orbits tilted with respect to the
black hole equatorial plane undergo precession at a rate that decreases with
radius (?). Hence, a tilted disk is subject to twist and warp. A high
misalignment adds complications to the models of quasi-periodic oscillations
observed in black hole x-ray binaries, which rely on precession of the inner
parts of the accretion flow, implying the whole flow is misaligned by $2\beta$
from the orbital axis in some phases (?). For $\beta\sim
40\hbox{${}^{\circ}$}$ the inner parts of the accretion disk would need to
become almost perpendicular to its outer parts. Most models assume smaller
misalignment angles, typically, $\beta\sim
10\hbox{${}^{\circ}$}-20\hbox{${}^{\circ}$}$ (?, ?, ?), although highly
inclined possibilities with $\beta\sim
45\hbox{${}^{\circ}$}-65\hbox{${}^{\circ}$}$ have sometimes been considered
(?).

Fig. 4. Probability distribution function for the misalignment angle. The
distribution normalized to the peak value is shown for the smallest
misalignment angle possible. This case corresponds to the black hole spin
directed along the southern approaching jet and the orbital spin being
directed South at a position angle $\theta_{\rm
orb}=\langle\mbox{PA}\rangle+180\hbox{${}^{\circ}$}$ and inclination $i_{\rm
orb}\approx 73\hbox{${}^{\circ}$}$. The red hatched region corresponds to the
68% confidence interval (i.e. between 16th and 84th percentiles of the
posterior probability distribution). Distributions of $\beta$ for the other
seven possible combinations of $\theta_{\rm orb}$, $i_{\rm orb}$ and $i_{\rm
bh}$ are shown in Fig. S4.

High misalignment has previously been suggested based on observations of the
gamma-ray light curves produced by the jet in Cyg X-3 (?), and differences
between orbital and jet inclination angles are $15\hbox{${}^{\circ}$}$ in GRO
J1655–40 (?) and $50\hbox{${}^{\circ}$}$ in V4641 Sgr (?) though the latter is
highly uncertain. Misalignment has also been theorized, based on the inferred
high kick velocities of x-ray binaries acquired during formation (?). For the
black hole x-ray binary MAXI J1820+070, the high misalignment was identified
only after obtaining the constraints on the position angle of the orbital
angular momentum $\theta_{\rm orb}$. Without information on the binary plane
orientation, we would have obtained only a lower limit on the misalignment
angle in MAXI J1820+070 of $\gtrsim$5∘, because the orbital inclination is
only marginally different from the jet inclination.

Our results demonstrate the need to treat the misalignment angle as a free
parameter when measuring black hole masses and spins. Assuming that the black
hole spin and the orbital angular momentum are aligned introduces a systematic
bias on measurements (?, ?, ?). A large misalignment angle is expected to
drive precession of the binary orbital plane, altering the gravitational waves
emitted during a subsequent merger event (?). Evidence for orbital precession
has been found from population properties of black hole mergers observed using
gravitational waves (?).

## Acknowledgements

We thank K. Belczynski and A. Zdziarski for comments and suggestions. A.V.
thanks the International Space Science Institute (ISSI) in Bern, Switzerland,
for providing the environment for stimulating discussions. This paper is 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. We thank the NOT staff for
their excellent support. DIPol-UF is a joint effort between University of
Turku (Finland) and Leibniz Institute for Solar Physics (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. Funding: A.V. acknowledges support
from the Academy of Finland grant 309308. J.P., A.V., V.K., and S.S.T.
received funding from the Russian Science Foundation grant 20-12-00364. I.A.K.
and V.K. thank Magnus Ehrnrooth foundation for support. S.V.B. acknowledges
support from the ERC Advanced Grant HotMol ERC-2011- AdG-291659. M.A.P.T.
acknowledges support from the State Research Agency (AEI) of the Spanish
Ministry of Science, Innovation and Universities (MCIU) and the European
Regional Development Fund (FEDER) under grant AYA2017-83216-P and via a Ramón
y Cajal Fellowship (RYC-2015-17854). Author contributions: J.P. and A.V.
initiated the project, performed modeling and led writing of the text. A.V.B.,
V.P., and I.A.K. planned and performed polarimetric observations. H.J., M.S.
and M.A.P.T. planned and executed observations at the Liverpool Telescope and
V.K. reduced the photometric data. J.J.E.K. obtained the Swift observations
and analysed the data together with S.S.T. P.G.J. and S.V.B. contributed to
the interpretation of the results. All authors provided input and comments on
the manuscript. Competing interests: We declare no competing interests. Data
and materials availability: The raw DIPol-UF data are available at Zenodo
https://zenodo.org/record/5767398 (?). The Liverpool Telescope data are
available from the data archive at https://telescope.livjm.ac.uk/DataProd/
(proposal ID JQ20A01). The Swift observatory data used in this study are
available through the NASA data archive at https://heasarc.gsfc.nasa.gov
(ObsIDs 000106272**, with ** being 19–22, 24–26). Our software for computation
of probability distribution of the misalignment angle and for modelling
polarization properties of the hot accretion flow is available at Zenodo
https://zenodo.org/record/5837857 (?).

Supplementary Materials for

Black hole spin–orbit misalignment in the x-ray binary MAXI J1820+070

Juri Poutanen∗, Alexandra Veledina, Andrei V. Berdyugin, Svetlana V.
Berdyugina,

Helen Jermak, Peter G. Jonker, Jari J. E. Kajava, Ilia A. Kosenkov, Vadim
Kravtsov,

Vilppu Piirola, Manisha Shrestha, Manuel A. P. Torres, Sergey S. Tsygankov

∗ Corresponding author. Email<EMAIL_ADDRESS>
This PDF file includes:

Materials and Methods
Figs. S1 to S5
Tables S1 to S5
References

## Materials and Methods

### DIPol-UF/NOT observations

We have performed polarimetric observations of the black hole x-ray binary
MAXI J1820+070 (?, ?, ?) both during the bright stages of the outburst in 2018
as well as in the quiescence. The data during the outburst (?, ?) were
obtained with the Double Image Polarimeter-2 (DIPol-2) polarimeter (?) mounted
on the Tohoku 60-cm telescope (T60) at Haleakala observatory, Hawaii. During
the quiescence MAXI J1820+070 was observed with the DIPol Ultra Fast (DIPol-
UF) (?) at the 2.56-m Nordic Optical Telescope (NOT) in July 2019, April and
July 2020, and July 2021 (Table S1) (?). Both polarimeters have similar
design: incoming light passes through a modulator (superachromatic half-
wavelength plate, HWP), it is then split by the plane-parallel calcite plate
into two orthogonally polarized rays (ordinary and extraordinary) and further
is reflected by two dichroic beam splitters to produce o- and e-images on
three charge-coupled devices (CCDs) simultaneously in the $B,V,R$ bands. This
design optically eliminates the sky polarization at the registration stage
(even if it is variable), reaching up to $10^{-5}$ polarization precision (?,
?, ?). The instrumental polarization is below $10^{-4}$ level and can be
accounted for using zero-polarization standards (?) but is negligible for our
study. The zero point of the PA was determined by observing high polarization
standards HD 161056 and HD 204827 (?).

For each independent measurement of linear polarization, four sequential
images with the HWP rotated by $22\mbox{$.\\!\\!^{\circ}$}5$ in each step are
obtained. Each measurement cycle consists of 16 images recorded simultaneously
by three CCDs for one full rotation of the HWP modulator (0∘–360∘) giving four
independent measurements of the normalized Stokes parameters $(q,u)$ in the
$B$, $V$, and $R$ bands. This algorithm helps eliminate effects arising from
dust particles on the retarder, nonparallelism of rotating components, etc.
The images produced by DIPol-UF require standard CCD calibrations (?).

The average normalized Stokes parameters $(q,u)$ are then obtained from 16 to
40 individual measurements using a $2\sigma$ weighting algorithm (?, ?). The
typical duration of an observational window where the data were averaged was
one hour. The polarization produced by the interstellar medium (ISM) has been
estimated from numerous (more than 400) observations of sample of field stars
(stars #2, 3, 6, 7, 9 from fig. 9 of ref. (?)), which are close in distance to
the target as indicated by their parallaxes (?). These normalized Stokes
parameters $(q_{\rm ISM},u_{\rm ISM})$ were subtracted from the measured
values of the normalized Stokes parameters to obtain the intrinsic
polarization information of the source $(q_{\rm intr},u_{\rm intr})$. We then
define the complex linear polarization quantity as ${\cal P}=q_{\rm
intr}+iu_{\rm intr}$. Intrinsic polarization degree (PD) $P$ and polarization
angle (PA) $\theta$ are then obtained from the formulae

$P=|{\cal P}|=\sqrt{q_{\rm intr}^{2}+u_{\rm
intr}^{2}},\quad\theta=\frac{1}{2}\arg({\cal P}).$ (S1)

Because our PD measurements typically have significance 5–10$\sigma$, the bias
in the measured PD is negligible and the uncertainty in PD $\Delta P$ is the
same as the uncertainty on individual Stokes parameters. The uncertainty of
the PA is estimated as $\Delta\theta=\Delta P/(2P)$ (?). The intrinsic PD and
PA computed following this procedure are reported in Table S1.

Polarization observed during quiescence shows no clear dependence on the
orbital phase (Fig. 2), apart from some spread of the points in close orbital
phases. The PD has a blue spectrum depending on frequency $\nu$ as
$\propto\nu^{3}$, which is inconsistent with the red spectra expected from a
jet or accretion disk. The PA is very stable in the $B$-band, where the PD is
the highest and the relative uncertainty is smallest. Therefore, we computed
the average PA in that band as a inverse-variance weighted mean of individual
PAs. Since the deviations of individual measurements exceed the measurement
uncertainties, the standard error of the weighted mean was corrected for
overdispersion by the square root of the reduced $\chi^{2}$, resulting in
$\langle\mbox{PA}\rangle=-19\mbox{$.\\!\\!^{\circ}$}7\pm
1\mbox{$.\\!\\!^{\circ}$}2$. We also computed the mean and its uncertainty
using a bootstrap (sampling with replacement) method (?) arriving at an
identical value. The mean PA in the $V$\- and $R$-bands have similar values,
but larger uncertainties (Table S1).

Figure S1: Variation of flux and polarization. (A) The dependence of the observed flux (not corrected for reddening), (B) intrinsic source PD and (C) PA on time during the July 2020 observations of MAXI J1820+070 with DIPol-UF in three filters: $B$ (blue circles), $V$ (green triangles), and $R$ (red squares). (D) Dependence of the PD on flux in the same three filters. An anticorrelation is evident.  Table S1: Intrinsic polarization of MAXI J1820+070. The intrinsic values of the PD and PA derived from observations of MAXI J1820+070 during the outburst with DIPol-2 (?, ?) and during quiescence with DIPol-UF at NOT. Uncertainties are 1$\sigma$. The Stokes parameters have been corrected for the interstellar polarization. The dates for observations during quiescence correspond to the middle of observing windows of a typical duration of one hour.  Date | $B$ |  | $V$ |  | $R$
---|---|---|---|---|---
(MJD) | PD (%) | PA (deg) |  | PD (%) | PA (deg) |  | PD (%) | PA (deg)
Outburst
58195–58222 | $0.28\pm 0.01$ | $9.2\pm 1.0$ |  | $0.36\pm 0.01$ | $22.9\pm 1.0$ |  | $0.30\pm 0.01$ | $29.0\pm 0.9$
58223–58234 | $0.34\pm 0.02$ | $8.8\pm 1.4$ |  | $0.51\pm 0.02$ | $23.4\pm 1.4$ |  | $0.53\pm 0.02$ | $23.9\pm 1.1$
58312–58344 | $0.16\pm 0.01$ | $-15.8\pm 1.6$ |  | $0.15\pm 0.01$ | $13.4\pm 2.3$ |  | $0.02\pm 0.01$ | $39.1\pm 11.0$
58406–58428 | $0.06\pm 0.04$ | $-3.0\pm 15.4$ |  | $0.13\pm 0.06$ | $2.8\pm 12.4$ |  | $0.09\pm 0.04$ | $44.8\pm 12.5$
Quiescence
58686.93 | $3.86\pm 0.29$ | $-23.8\pm 2.1$ |  | $1.81\pm 0.24$ | $-23.8\pm 3.8$ |  | $0.70\pm 0.14$ | $-24.1\pm 5.7$
58688.01 | $2.35\pm 0.38$ | $-25.9\pm 4.7$ |  | $1.09\pm 0.25$ | $8.0\pm 6.5$ |  | $1.70\pm 0.10$ | $-42.4\pm 1.7$
58961.15 | $1.47\pm 0.28$ | $-26.8\pm 5.4$ |  | $0.52\pm 0.24$ | $-4.2\pm 13.3$ |  | $0.63\pm 0.19$ | $8.5\pm 8.7$
58964.16 | $4.31\pm 0.28$ | $-18.5\pm 1.8$ |  | $3.10\pm 0.29$ | $-17.5\pm 2.7$ |  | $1.05\pm 0.18$ | $-2.0\pm 4.8$
58965.11 | $2.81\pm 0.24$ | $-23.5\pm 2.5$ |  | $2.12\pm 0.21$ | $-35.1\pm 2.9$ |  | $0.84\pm 0.18$ | $-26.3\pm 6.2$
58967.12 | $4.53\pm 0.28$ | $-24.1\pm 1.8$ |  | $2.09\pm 0.39$ | $-11.0\pm 5.3$ |  | $1.09\pm 0.31$ | $-27.0\pm 8.3$
59050.94 | $4.58\pm 0.34$ | $-15.3\pm 2.1$ |  | $3.57\pm 0.31$ | $-16.5\pm 2.5$ |  | $1.54\pm 0.18$ | $-14.5\pm 3.4$
59050.98 | $5.03\pm 0.30$ | $-11.0\pm 1.7$ |  | $3.38\pm 0.34$ | $-11.7\pm 2.9$ |  | $1.62\pm 0.16$ | $0.3\pm 2.9$
59051.05 | $3.35\pm 0.26$ | $-23.5\pm 2.2$ |  | $2.04\pm 0.23$ | $-17.1\pm 3.2$ |  | $1.08\pm 0.18$ | $-12.1\pm 4.9$
59051.10 | $3.14\pm 0.30$ | $-10.7\pm 2.7$ |  | $1.95\pm 0.29$ | $-4.8\pm 4.3$ |  | $0.95\pm 0.16$ | $5.6\pm 4.9$
59051.92 | $4.08\pm 0.27$ | $-18.1\pm 1.9$ |  | $3.40\pm 0.25$ | $-12.9\pm 2.1$ |  | $1.39\pm 0.15$ | $-18.0\pm 3.1$
59051.97 | $4.67\pm 0.30$ | $-16.5\pm 1.8$ |  | $3.27\pm 0.30$ | $-17.5\pm 2.6$ |  | $1.96\pm 0.15$ | $-17.9\pm 2.2$
59052.03 | $4.83\pm 0.31$ | $-16.7\pm 1.9$ |  | $3.37\pm 0.24$ | $-10.3\pm 2.1$ |  | $1.68\pm 0.15$ | $-20.0\pm 2.6$
59052.09 | $3.78\pm 0.23$ | $-21.5\pm 1.7$ |  | $2.47\pm 0.22$ | $-28.0\pm 2.5$ |  | $1.06\pm 0.16$ | $-12.4\pm 4.3$
59052.92 | $4.14\pm 0.23$ | $-24.2\pm 1.6$ |  | $2.46\pm 0.23$ | $-16.6\pm 2.7$ |  | $1.21\pm 0.14$ | $-18.4\pm 3.3$
59052.98 | $3.30\pm 0.28$ | $-20.5\pm 2.4$ |  | $2.18\pm 0.23$ | $-23.6\pm 3.0$ |  | $1.18\pm 0.17$ | $-14.0\pm 4.0$
59053.92 | $4.02\pm 0.25$ | $-12.8\pm 1.8$ |  | $2.22\pm 0.26$ | $-3.5\pm 3.4$ |  | $1.24\pm 0.15$ | $0.2\pm 3.4$
59053.97 | $4.30\pm 0.27$ | $-16.5\pm 1.8$ |  | $3.18\pm 0.24$ | $-10.4\pm 2.1$ |  | $1.51\pm 0.15$ | $-13.1\pm 2.9$
59400.99 | $1.62\pm 0.12$ | $-30.5\pm 2.0$ |  | $0.77\pm 0.10$ | $-17.1\pm 3.9$ |  | $0.41\pm 0.07$ | $6.4\pm 5.1$
59401.94 | $2.86\pm 0.21$ | $-25.4\pm 2.1$ |  | $2.04\pm 0.20$ | $-29.9\pm 2.8$ |  | $0.36\pm 0.14$ | $-6.1\pm 10.7$
59051–59054 | $4.00\pm 0.19$ | $-17.5\pm 1.3$ |  | $2.65\pm 0.19$ | $-14.7\pm 1.9$ |  | $1.32\pm 0.08$ | $-12.8\pm 2.3$
58686–59402 | $3.18\pm 0.22$ | $-19.7\pm 1.2$ |  | $1.89\pm 0.19$ | $-16.8\pm 1.9$ |  | $0.94\pm 0.11$ | $-18.6\pm 3.3$
Interstellar polarization
58195–59054 | $0.81\pm 0.03$ | $64.0\pm 1.1$ |  | $0.71\pm 0.03$ | $68.4\pm 1.2$ |  | $0.60\pm 0.02$ | $64.4\pm 0.8$
Table S2: Observed flux of MAXI J1820+070. Mean fluxes and their $1\sigma$ uncertainties derived from observations of MAXI J1820+070 during quiescence in July 2020 with DIPol-UF at NOT.  Date | Flux (mJy)
---|---
(MJD) | $B$ | $V$ | $R$
59050.94 | $0.0732\pm 0.0014$ | $0.1391\pm 0.0019$ | $0.1858\pm 0.0020$
59050.98 | $0.0747\pm 0.0022$ | $0.1495\pm 0.0030$ | $0.1966\pm 0.0029$
59051.05 | $0.1000\pm 0.0022$ | $0.2002\pm 0.0034$ | $0.2491\pm 0.0035$
59051.10 | $0.1090\pm 0.0018$ | $0.2142\pm 0.0030$ | $0.2772\pm 0.0028$
59051.92 | $0.0819\pm 0.0015$ | $0.1577\pm 0.0021$ | $0.2082\pm 0.0023$
59051.97 | $0.0700\pm 0.0014$ | $0.1420\pm 0.0022$ | $0.1901\pm 0.0022$
59052.03 | $0.0681\pm 0.0015$ | $0.1395\pm 0.0024$ | $0.1941\pm 0.0025$
59052.09 | $0.0813\pm 0.0018$ | $0.1572\pm 0.0026$ | $0.2142\pm 0.0027$
59052.92 | $0.0902\pm 0.0015$ | $0.1745\pm 0.0023$ | $0.2255\pm 0.0023$
59052.98 | $0.0922\pm 0.0021$ | $0.1754\pm 0.0031$ | $0.2323\pm 0.0031$
59053.92 | $0.0861\pm 0.0017$ | $0.1759\pm 0.0027$ | $0.2275\pm 0.0028$
59053.97 | $0.0736\pm 0.0013$ | $0.1512\pm 0.0020$ | $0.1995\pm 0.0020$

Using the images from DIPol-UF we also measured photometry of MAXI J1820+070
relative to two nearby stars (#1 and 2 from table 2 in ref. (?)). Absolute
photometry during July 2020 observations was obtained using simultaneous
observations by the Liverpool Telescope (LT), see Fig. S1A and Table S2. The
flux of MAXI J1820+070 in all bands is anti-correlated with the PD (Fig. S1D).
This anti-correlation together with the spectral dependence of the PD can be
interpreted as an interplay between two components: one, polarized, with
contribution growing towards blue and another, unpolarized or, possibly,
weakly polarized, variable component dominating the flux in the red. Following
the spectral decomposition (Fig. 2), the UV component produces only 20% of the
total flux in the $R$ band. This makes PA measurements of the UV component
less reliable in this band. Stronger variations of the PA in $R$ may be caused
by the presence of a red and polarized at a level of 0.1–0.3% component, whose
PA is different from that of the UV component. Such red and polarized
component, with the PA consistent with the jet position angle, has been
observed during the outburst (Table S1).

### Liverpool Telescope observations

Photometric observations in the optical band were performed using the optical
imaging component of the Infrared-Optical suite of instruments (IO:O) with the
Sloan Digital Sky Survey (SDSS) $i^{\prime}r^{\prime}z^{\prime}$, Bessel $V$
and $B$ filters on the LT (?) on La Palma, Spain. The object was observed
during four nights from 2020 July 20 to 24, i.e. Modified Julian Dates (MJD)
59050–59054. LT image reduction was provided by the basic IO:O pipeline (?).
As the photometric standards, we used four stars with known Panoramic Survey
Telescope and Rapid Response System (PS1) point spread function magnitudes. To
convert PS1 magnitudes to SDSS and Johnson-Cousins magnitudes we used equation
(6) and table 6 from ref. (?). To obtain fluxes of the object from its
magnitudes we used standard zero-points for SDSS and Johnson-Cousins systems
(?, ?). The fluxes and their uncertainties are presented in Table S3.

Table S3: Log of observations with the Liverpool Telescope. The observed fluxes of MAXI J1820+070 and their $1\sigma$ uncertainties in various filters.  Date | Filter | Exposure | Flux |  | Date | Filter | Exposure | Flux
---|---|---|---|---|---|---|---|---
(MJD) |  | (s) | (mJy) |  | (MJD) |  | (s) | (mJy)
59050.92075 | $i$ | 100 |  0.2700$\pm$0.0014 |  | 59052.92842 | $z$ | 100 |  0.3944$\pm$0.0033
59050.92213 | $i$ | 100 |  0.2468$\pm$0.0015 |  | 59052.92980 | $z$ | 100 |  0.4037$\pm$0.0033
59050.92350 | $i$ | 100 |  0.2482$\pm$0.0015 |  | 59052.93193 | $V$ | 200 |  0.1605$\pm$0.0011
59050.92505 | $z$ | 100 |  0.3990$\pm$0.0028 |  | 59052.93463 | $B$ | 200 |  0.0974$\pm$0.0007
59050.92642 | $z$ | 100 |  0.4093$\pm$0.0029 |  | 59052.93637 | $r$ | 40 |  0.3063$\pm$0.0018
59050.92780 | $z$ | 100 |  0.3733$\pm$0.0029 |  | 59052.93705 | $r$ | 40 |  0.2142$\pm$0.0018
59050.93052 | $V$ | 300 |  0.1440$\pm$0.0009 |  | 59053.92120 | $i$ | 100 |  0.3071$\pm$0.0017
59051.01279 | $i$ | 100 |  0.3321$\pm$0.0014 |  | 59053.92257 | $i$ | 100 |  0.3600$\pm$0.0018
59051.01417 | $i$ | 100 |  0.3075$\pm$0.0014 |  | 59053.92395 | $i$ | 100 |  0.3450$\pm$0.0018
59051.01554 | $i$ | 100 |  0.3164$\pm$0.0014 |  | 59053.92550 | $z$ | 100 |  0.3917$\pm$0.0033
59051.01709 | $z$ | 100 |  0.4208$\pm$0.0027 |  | 59053.92687 | $z$ | 100 |  0.3598$\pm$0.0033
59051.01846 | $z$ | 100 |  0.5048$\pm$0.0028 |  | 59053.92825 | $z$ | 100 |  0.3736$\pm$0.0034
59051.01984 | $z$ | 100 |  0.4703$\pm$0.0027 |  | 59053.93038 | $V$ | 200 |  0.1369$\pm$0.0011
59051.02254 | $B$ | 300 |  0.1263$\pm$0.0005 |  | 59053.93308 | $B$ | 200 |  0.0762$\pm$0.0007
59051.09342 | $i$ | 100 |  0.3476$\pm$0.0016 |  | 59053.93482 | $r$ | 40 |  0.2281$\pm$0.0019
59051.09479 | $i$ | 100 |  0.3489$\pm$0.0016 |  | 59053.93550 | $r$ | 40 |  0.2174$\pm$0.0019
59051.09617 | $i$ | 100 |  0.3809$\pm$0.0015 |  | 59053.95474 | $i$ | 100 |  0.2752$\pm$0.0016
59051.09771 | $z$ | 100 |  0.4877$\pm$0.0032 |  | 59053.95612 | $i$ | 100 |  0.2927$\pm$0.0016
59051.09909 | $z$ | 100 |  0.4963$\pm$0.0033 |  | 59053.95750 | $i$ | 100 |  0.2633$\pm$0.0017
59051.10047 | $z$ | 100 |  0.4706$\pm$0.0033 |  | 59053.95904 | $z$ | 100 |  0.3587$\pm$0.0032
59051.10168 | $r$ | 40 |  0.2339$\pm$0.0019 |  | 59053.96042 | $z$ | 100 |  0.3824$\pm$0.0030
59051.10236 | $r$ | 40 |  0.2592$\pm$0.0018 |  | 59053.96180 | $z$ | 100 |  0.4097$\pm$0.0033
59051.92206 | $i$ | 100 |  0.3003$\pm$0.0015 |  | 59053.96393 | $V$ | 200 |  0.1467$\pm$0.0010
59051.92343 | $i$ | 100 |  0.2982$\pm$0.0015 |  | 59053.96663 | $B$ | 200 |  0.0799$\pm$0.0007
59051.92481 | $i$ | 100 |  0.2860$\pm$0.0015 |  | 59053.97116 | $i$ | 100 |  0.3032$\pm$0.0042
59052.01060 | $i$ | 100 |  0.2838$\pm$0.0016 |  | 59053.97254 | $i$ | 100 |  0.3390$\pm$0.0016
59052.01197 | $i$ | 100 |  0.2693$\pm$0.0015 |  | 59053.97391 | $i$ | 100 |  0.3308$\pm$0.0016
59052.01335 | $i$ | 100 |  0.2501$\pm$0.0016 |  | 59053.97546 | $z$ | 100 |  0.4278$\pm$0.0032
59052.09336 | $i$ | 100 |  0.2979$\pm$0.0015 |  | 59053.97684 | $z$ | 100 |  0.4017$\pm$0.0032
59052.09474 | $i$ | 100 |  0.2800$\pm$0.0016 |  | 59053.97821 | $z$ | 100 |  0.4193$\pm$0.0031
59052.09611 | $i$ | 100 |  0.2974$\pm$0.0015 |  | 59053.98035 | $V$ | 200 |  0.1535$\pm$0.0010
59052.92275 | $i$ | 100 |  0.3161$\pm$0.0017 |  | 59053.98305 | $B$ | 200 |  0.0601$\pm$0.0007
59052.92412 | $i$ | 100 |  0.3058$\pm$0.0016 |  | 59053.98479 | $r$ | 40 |  0.2427$\pm$0.0017
59052.92550 | $i$ | 100 |  0.3048$\pm$0.0017 |  | 59053.98546 | $r$ | 40 |  0.2100$\pm$0.0017
59052.92705 | $z$ | 100 |  0.3686$\pm$0.0034 |  |  |  |  |

###  Swift/UVOT observations

The Neil Gehrels Swift Observatory (Swift) (?) observed MAXI J1820+070 in the
low state with the X-ray Telescope (XRT) and Ultraviolet/Optical Telescope
(UVOT) instruments (partly quasi-simultaneously with DIPol-UF/NOT) between
2020 July 20 – September 6. All the XRT data were taken in the photon counting
mode with about 9 ks of total exposure. An averaged spectrum was extracted
using the online tools (?) provided by the UK Swift Science Data Centre. The
spectrum was rebinned to have at least 1 count per energy channel in order to
apply $W$-statistic (?, ?). The data were fitted with a model consisting of a
power law modified by the interstellar absorption tbabs*powerlaw in the 0.5–10
keV band using the xspec package (?). Fixing the hydrogen column density to
$N_{\rm H}=1.6\times 10^{21}$ cm-2 (?) we obtained a power-law photon index
$\Gamma=1.62\pm 0.24$ with $W$-statistics of 42.7 for 46 degrees of freedom.
We find flux corrected for absorption of $(3.3\pm 0.7)\times
10^{-13}\,\textrm{erg}\,\textrm{cm}^{-2}\,\textrm{s}^{-1}$ in the 0.5–10 keV
energy range.

The UVOT data were processed with the uvotproduct tool (?, ?, ?). We used a
$5\mbox{${}^{\prime\prime}$}$ source aperture around the known position of
MAXI J1820+070 (?), and a circular source-free $10\mbox{${}^{\prime\prime}$}$
background aperture about $40\mbox{${}^{\prime\prime}$}$ east of it. After
comparing the magnitudes obtained between July 20 – September 6 to other
quiescence data taken when the x-ray counting rate was low, we found that
during this period MAXI J1820+070 was roughly 0.5–1 magnitude fainter in all
the UV filters than in the data taken in 2019 or earlier in 2020. The same
trend was observed in the $V$-band and, to a lesser extent, in the $R$-band.
Therefore, we decided to stack all the 2020 July 20–September 6 UVOT data
together to obtain time averaged fluxes in the u, uvw1, uvm2 and uvw2 filters
for the spectral energy distribution (SED). Conversion of the background
corrected count rates obtained from the uvotproduct to the fluxes is non-
trivial, as it depends on the assumed spectral shape (?). Therefore, for
reliable estimate of the fluxes we performed direct spectral fitting with a
broad-band SED model using the xspec package (Fig. 2).

### Decomposition of the broadband spectrum

The broad-band average spectrum obtained with the LT and Swift/UVOT is shown
in Fig. 2. The spectrum was corrected for reddening using model of ref. (?)
with extinction $A_{V}=0.89$ (corresponding to the color excess $E(B-V)=0.29$)
obtained from the hydrogen column density $N_{\rm H}=(0.16\pm 0.02)\times
10^{22}\mbox{cm}^{-2}$ (?) using transformation $A_{V}=5.59\,N_{\rm
H}/10^{22}\,\mbox{cm}^{-2}$ (?). The optical/infrared SED obtained with the LT
is very red and can be described by a joined contribution of companion star
and multicolor accretion disk, but the UVOT data show a UV excess. The excess
is also seen in the non-dereddened data and is not an artefact of dereddening
procedure. A similar excess has been detected previously in quiescent-state
observations of the black hole x-ray binary A 0620$-$00 (?), but its nature
remained a mystery.

The blue PD spectrum differs from the red (even after correcting for the
reddening) spectrum of the infrared to UV continuum. This rules out accretion
disk as the source of polarized light. We infer that the component producing
the UV excess is responsible for polarization. To extract the average shape of
the polarized component, we calculated the polarized fluxes as

$(PF)_{k}=P_{k}F_{k},$ (S2)

where $k$ is the index corresponding to one of the $BVR$-bands, $P_{k}$ is the
average PD and $F_{k}$ is the average flux in this band. If PD of the
polarized component is wavelength-independent, then the computed $PF$ directly
replicates its total spectral shape, with the normalization being smaller by
the polarization fraction. The average PD was computed from the average Stokes
parameters $(q_{\rm intr},u_{\rm intr})$ obtained in 12 individual
measurements in July 2020 during quiescence with DIPol-UF (Table S1). Because
absolute photometry with DIPol-UF is not very reliable, the average flux was
obtained from the LT observations. The averages were computed as a inverse-
variance weighted mean of individual measurements and their standard errors
were corrected for overdispersion by the square root of the reduced
$\chi^{2}$.

Published spectroscopy obtained during quiescence indicates a K-type spectral
type for the companion star, which contributed about 20% to the flux in the
$r$-band (?, ?). The average observed $r$-band flux during the spectroscopy
was about 0.33 mJy, higher than the average flux of 0.24 mJy in our LT
observations. Thus, the contribution of the companion star to the $r$-band
flux rose to 27%. The radius of the companion star, which fills its Roche
lobe, can be established from the measured orbital period $P_{\rm
orb}=0.68549\pm 0.00001$ d, projected rotational velocity $v_{\rm rot}\sin
i_{\rm orb}=84\pm 5$ km s-1, and orbital inclination $i_{\rm
orb}=73\hbox{${}^{\circ}$}\pm 6\hbox{${}^{\circ}$}$ (?)

$R_{\rm c}=\frac{P_{\rm orb}(v_{\rm rot}\sin i_{\rm orb})}{2\pi\sin i_{\rm
orb}}=(1.19\pm 0.08)R_{\odot}=(8.3\pm 0.6)\times 10^{10}\ \mbox{cm}.$ (S3)

A moderately evolved late-type star of spectral class K7 can satisfy the
constraint on radius and contribute about 27% to the $r$-band flux. A star
with higher temperature overpredicts the companion star contribution in the
$r$-band. For the fitting purposes, we approximate the stellar spectrum (?) by
the blackbody with $T_{*}=4000$ K and $R_{*}=R_{\rm c}=8.3\times 10^{10}$ cm
(see Fig. 2).

Using xspec v.12.11 (?), we modelled the total SED from LT and Swift/UVOT and,
jointly, the three points of the $PF$. The total spectrum is described by the
model redden (bbodyrad1+diskbb+bbodyrad2). The redden model describes
interstellar extinction (?). Component bbodyrad1 corresponds to the spectrum
of the stellar companion, modeled as a blackbody of fixed temperature $T_{*}$
and radius $R_{*}$. The diskbb corresponds to the multicolor accretion disk.
It has two free parameters: normalization, which is related to the inner
radius $R_{\rm in}$, and temperature at that radius $T_{\rm in}$. Component
bbodyrad2 corresponds to the UV excess that is modeled by a blackbody with two
free parameters: temperature $T_{\rm bb}$ and radius $R_{\rm bb}$. The $PF$ is
modeled by the reddened second blackbody multiplied by the polarization
fraction $P_{\rm UV}$. Blackbody and disk normalizations were converted to
radii assuming the distance to the source of 2.96 kpc as determined by the
radio parallax (?) and disk inclination $i_{\rm orb}=73\hbox{${}^{\circ}$}$
(?).

The best-fitting model parameters with the corresponding uncertainties are
listed in Table S4. We find the accretion disk temperature of about 6200 K,
which is very close to that expected for the disk in quiescence (?). The inner
radius of the disk $R_{\rm in}\approx 6\times 10^{10}$ cm, which is 30%
smaller than the estimate of the circularization radius for the measured
$P_{\rm orb}$ and companion-to-black hole mass ratio $q=0.072\pm 0.012$ (?).
The outer radius has to be at least 2–3 times larger to avoid underpredicting
the red part of the spectrum. This is consistent with the expectation that the
accretion disk size does not exceed the tidal radius of about $2.4\times
10^{11}$ cm (?).

Table S4: Best-fitting parameters of the SED model. Distance of 2.96 kpc (?) and inclination $i_{\rm orb}=73\hbox{${}^{\circ}$}$ (?) are assumed. The temperature and the radius of the blackbody approximating the SED of the stellar companion were fixed.  Parameter | Value | Units
---|---|---
$T_{\rm in}$ | $6200^{+1400}_{-1100}$ | K
$R_{\rm in}$ | $(5.6^{+2.7}_{-1.8})\times 10^{10}$ | cm
$T_{\rm bb}$ | $14900^{+2300}_{-1400}$ | K
$R_{\rm bb}$ | $(8.9^{+2.3}_{-2.1})\times 10^{9}$ | cm
$T_{*}$ | $4000$ | K
$R_{*}$ | $8.3\times 10^{10}$ | cm
$P_{\rm UV}$ | $0.055^{+0.023}_{-0.011}$ |

The temperature of the additional blackbody is $T_{\rm bb}\sim 15,000$ K and
the characteristic size $R_{\rm bb}\sim 9\times 10^{9}$ cm. This reproduces
the shape of the polarized flux SED and the UV excess. The PD of this
component is $P_{\rm UV}=0.055^{+0.023}_{-0.011}$, corresponding to 5–8%
intrinsic PD (on average) of the polarized component. The broad-band SED and
the best-fitting spectral components are shown in Fig. 2. The computed values
of $PF$ were divided by the constant $P_{\rm UV}$ to demonstrate the fit with
the UV excess component.

For lower and higher values of the color excess of 0.25 and 0.325,
corresponding to the 1$\sigma$ uncertainties in $N_{\rm H}$, the best-fitting
spectral parameters change. The lower $E(B-V)$ leads to the 8% smaller disk
temperature $T_{\rm in}$ and 13% larger inner radius $R_{\rm in}$, 8% smaller
blackbody temperature $T_{\rm bb}$ and 3% larger radius $R_{\rm bb}$. For the
higher $E(B-V)$, the effect is opposite: $T_{\rm in}$ is 7% larger, $R_{\rm
in}$ is 9% smaller, $T_{\rm bb}$ is 7% larger and $R_{\rm bb}$ is smaller by
3%. This uncertainty does not affect any of our conclusions.

### The source of polarized light and the nature of the spectral components

In this section we seek to answer two questions: what is the nature of the UV
excess and what is the source of polarized radiation? The accretion disk
itself cannot be the source of the UV emission, because in quiescence its
temperature is lower as we see from the red optical spectrum. Another
possibility is a hotspot (or hot line), the place where the accretion stream
hits the disk. This component could be responsible for the UV excess. However,
whether it can also be the source of polarization is questionable. The high
temperature of this component implies that the matter is ionized. The PD from
the optically thick electron-scattering dominated atmosphere (?, ?) depends on
the cosine of the viewing angle $\mu=\cos i_{\rm orb}$ approximately as
$11.7\%\times(1-\mu)/(1+3.582\mu)$ (?). For inclination angles permitted by
the absence of eclipses (?), $i_{\rm orb}<81\hbox{${}^{\circ}$}$, the expected
PD is $\lesssim 6$%. A high implied PD, $P_{\rm UV}\approx$5–8%, is barely
consistent with that.

On the other hand, high PD can be produced by synchrotron radiation in the
ordered magnetic field. The blue spectrum indicates that we might see the
optically thick part of this radiation, with the possible transition to
optically thin part at $\nu\gtrsim 10^{15}$ Hz. Self-absorption becomes
important at the turn-over frequency (?, ?, ?), $\displaystyle\nu_{\rm
t}\approx 3\times 10^{15}B_{6}^{\frac{p+2}{p+4}}(\tau_{\rm t}\gamma_{\rm
t}^{p})^{\frac{2}{p+4}}$, where $B_{6}$ is the magnetic field in units of
$10^{6}$ G, $\tau_{\rm t}$ is the Thomson optical depth of electrons emitting
at the turn-over frequency, $\gamma_{\rm t}$ is their Lorentz factor and $p$
is the index of the power-law distribution of electron number density on the
Lorentz factor $\gamma$, $dn_{\rm e}/d\gamma\propto\gamma^{-p}$. For the
observed $\nu_{\rm t}\approx 10^{15}$ Hz, this requires highly opaque source
with $\tau_{\rm t}\gamma_{\rm t}^{p}\sim 1$ and/or high magnetic field with
$B_{6}\sim 1$, which would be expected in the bright hard state, but
inconsistent with the relevant values for the source in (near-)quiescence,
when both optical depth and the magnetic field drop by 2 to 4 orders of
magnitude (?). The highest theoretically possible PD below the self-absorption
frequency (?) of $P_{\max}=3/(6p+13)\sim 10\%$ requires highly ordered – and
constant – magnetic field during two years, July 2019–July 2021. Hence, we
find the synchrotron radiation to be an implausible source of the observed
polarized flux.

Substantial polarization could instead be produced by electron scattering of
radiation in an optically thin slab if the seed photons are injected along the
slab plane (?). The PD$=(1-\mu^{2})/(3-\mu^{2})$ reaches 33% edge-on and is a
weak function of orbital inclination (for $i_{\rm orb}\gtrsim
66\hbox{${}^{\circ}$}$, i.e. $\mu\lesssim 0.41$, the PD is larger than
$\sim$30%). Scattering in the Thomson regime means that the scattered
radiation does not gain a systematic shift in energy with respect to the
incident continuum, and hence the peak of the spectrum of the polarized
component directly probes the characteristic energy and spectral shape of the
incident radiation. Hence, the source of the incident light should have
narrow, blackbody-like shape, with characteristic temperature $T\sim 15,000$
K. Because this temperature is much higher than the disk inner temperature,
and the polarization angle is independent of orbital phase, the location and
physical properties of such a component are unclear.

Figure S2: Polarization properties of the hot accretion flow model. (A) Flux
from the spectral components: blackbody ($T_{\rm in}=6200$ K, black solid
line) and first Compton scattering of these photons by hot medium (red dashed
line) with $kT_{\rm e}=200$ keV and scale-height given by parameter
$\cos\alpha_{\rm crit}=0.5$, and spectrum of the polarized flux of the first
scattering component (blue dot-dashed line) that has been scaled up to match
the low-energy part of the intensity spectrum (divided by the PD at low
energies). This shows that the polarized flux has the same shape as the
scattered component flux, justifying our joint model fitting of the UV flux
together with the polarized flux (shown in Fig. 2). Only results for
inclination of $i_{\rm orb}=73\hbox{${}^{\circ}$}$ are shown, as the
dependence on the inclination in its allowed range is minor. (B) PD of the
scattered component as a function of frequency is shown for different
inclinations $i_{\rm orb}=81\hbox{${}^{\circ}$}$ (blue dashed line),
$73\hbox{${}^{\circ}$}$ (green dotted) and $66\hbox{${}^{\circ}$}$ (red
solid). PD for energies below the peak of the scattered component is nearly
frequency-independent. The observed strong dependence of the PD on frequency
is caused by the dilution of the polarized scattered radiation by the
unpolarized accretion disk.

The additional polarized component may instead arise from Compton up-
scattering of soft photons by hot electrons, which is related to a systematic
shift in energy. A hot, geometrically thick accretion flow can exist in low-
luminosity black hole binaries and active galactic nuclei (?, ?).
Alternatively, a hot layer (corona) atop of the cold disk, heated by the
unwinding stream of matter, could be present. In both cases, the hot medium is
expected to be optically thin, and may Compton up-scatter the photons coming
from the cold accretion disk. We calculate the spectra and polarization
properties of Compton scattering using iterative scattering method (?). We
assume the spectrum of incident, non-polarized photons is a blackbody with
temperature equal to the disk inner temperature $T_{\rm in}=6200$ K. For these
seed photons, to explain the peak of the first scattering order at $\sim
10^{15}$ Hz (as demanded by the spectral decomposition), we need to have
electrons with temperature of about 200 keV. For such a high temperature, the
PD is expected to drop with respect to the maximal, Thomson-regime PD by a
factor of 3 (?). In the case of inner hot flow, only the photons travelling at
inclinations $\alpha$ to the disk normal that are larger than some critical
value $\alpha_{\rm crit}$ can interact with the hot matter. This limiting
angle is related to the scale-height $H$ of the hot medium $\cot\alpha_{\rm
crit}\sim(H/R_{\rm in})$. Increasing the flow scale-height leads to a drop of
PD. In Fig. S2 we show the resulting flux, polarized flux and PD spectra at
different viewing angles $i_{\rm orb}=66\hbox{${}^{\circ}$}$,
$73\hbox{${}^{\circ}$}$ and $81\hbox{${}^{\circ}$}$. In this case the
polarization is parallel to the disk normal. We find that the PD of scattered
radiation is nearly independent of the photon frequency up to its peak at
$\nu\sim 10^{15}$ Hz and therefore the spectral shape of the flux and $PF$ of
the scattered component are nearly identical in the optical band.

For the slab-corona geometry, an appropriate parameter determining PD is the
Thomson optical depth of the hot slab $\tau_{\rm T}$. Only for $\tau_{\rm
T}\gtrsim 0.5$ a substantial polarization exceeding 5% can be produced and the
dominant electric-field oscillations are perpendicular to the projection of
the disk normal. However, Comptonization in a slab of $\tau_{\rm T}=0.1$ and
electron temperature $kT_{\rm e}=200$ keV overproduces the observed x-ray
flux. This would imply that a more probable model that is more consistent with
the observed spectral energy distribution and polarization properties is
scattering of the disk radiation in the inner hot flow. Such a geometry favors
the measured PA being parallel to the orbital axis.

Finally, the blue PD spectrum could be produced by dust scattering of the
accretion disk radiation. This process is thought to be responsible for
similarly blue polarization spectra observed from supermassive black holes in
Seyfert galaxies and quasars (?, ?, ?, ?, ?, ?). The dust would likely be
located in a flattened envelope (equatorial wedge) around the accretion disk
(?), where it is shielded from the inner disk radiation, or in a circumbinary
disk (?). In this case, the polarization vector lies in the meridional plane.
If instead the dust has a more spherical distribution, the PD is expected to
be smaller and the polarization to be perpendicular to the meridional plane.
The dust scattering model, however, does not explain the UV excess.

### Geometry

Here we define the coordinate systems and derive the formulae to compute the
misalignment angle between the black hole spin and the orbital angular
momentum, and the azimuthal angle of the black hole spin projection in the
orbital plane. We consider a Cartesian system with the $x-y$ plane coinciding
with the orbital plane. Thus the unit vector of the orbital angular momentum
is $\mbox{\boldmath$\hat{\Omega}$}=(0,0,1)$. We choose the direction to an
observer to lie in the $x-z$ plane at inclination angle $i_{\rm orb}$ as
measured from the orbital axis (see Fig. S3 for geometry), so the observer
unit vector is

$\mbox{\boldmath$\hat{o}$}=(\sin i_{\rm orb},0,\cos i_{\rm orb}).$ (S4)

We assume that the black hole spin is directed at an angle $\beta$ from the
$z$-axis at azimuthal angle $\Phi_{\rm bh}$, which is measured from the
$x$-axis in the counter-clockwise direction in the $x-y$ plane as viewed from
the top. The unit vector of the black hole spin is

$\mbox{\boldmath$\hat{s}$}=(\sin\beta\cos\Phi_{\rm bh},\sin\beta\sin\Phi_{\rm
bh},\cos\beta).$ (S5)

Figure S3: Geometry of the system from the point of view of the binary. The
orbital plane here coincides with the $x-y$ plane of the Cartesian system with
the orbital axis being along the $z$-axis. The observer is situated in the
$x-z$ plane at inclination angle $i_{\rm orb}$ as measured from the orbital
axis. The black hole spin is directed at angle $\beta$ from the $z$-axis at
azimuthal angle $\Phi_{\rm bh}$ as measured from the $x$-axis in the counter-
clockwise direction in the $x-y$ plane as viewed from the top. The red cones
indicate the jet, and the blue disk indicates the binary orbit. Fig. 3 shows
the same geometry from the observer’s perspective.

We define the polarization basis with the unit vector
$\mbox{\boldmath$\hat{e}$}_{1}$ directed on the sky in the direction of the
projection of the orbital spin, vector $\mbox{\boldmath$\hat{e}$}_{2}$ being
directed to the left on the sky, and the third vector coincides with the
observer direction:

$\displaystyle\mbox{\boldmath$\hat{e}$}_{1}$
$\displaystyle=\frac{\mbox{\boldmath$\hat{\Omega}$}-\cos i_{\rm orb}\
\mbox{\boldmath$\hat{o}$}}{\sin i_{\rm orb}}=(-\cos i_{\rm orb},0,\sin i_{\rm
orb}),$ (S6) $\displaystyle\mbox{\boldmath$\hat{e}$}_{2}$
$\displaystyle=(0,-1,0),$ $\displaystyle\mbox{\boldmath$\hat{e}$}_{3}$
$\displaystyle=\mbox{\boldmath$\hat{o}$}.$

In this vector basis, the black hole and the orbital spin vectors can be
represented as

$\hat{s}$ $\displaystyle=$ $\displaystyle(\sin i_{\rm bh}\cos\Delta,\sin
i_{\rm bh}\sin\Delta,\cos i_{\rm bh}),$ (S7) $\hat{\Omega}$ $\displaystyle=$
$\displaystyle(\sin i_{\rm orb},0,\cos i_{\rm orb}),$ (S8)

where $\Delta=\theta_{\rm bh}-\theta_{\rm orb}$ is the difference in position
angles of the black hole and the orbital spins. The misalignment angle $\beta$
between the black hole and the orbital axis is then given by the scalar
product:

$\cos\beta=\mbox{\boldmath$\hat{\Omega}$}\cdot\mbox{\boldmath$\hat{s}$}=\cos
i_{\rm bh}\cos i_{\rm orb}+\sin i_{\rm bh}\sin i_{\rm orb}\cos\Delta.$ (S9)

The direction cosines of the black hole spin vector $\hat{s}$ in polarization
basis are

$\displaystyle\mbox{\boldmath$\hat{s}$}\cdot\mbox{\boldmath$\hat{e}$}_{1}$
$\displaystyle=\sin i_{\rm bh}\cos\Delta=\cos\beta\sin i_{\rm
orb}-\sin\beta\cos i_{\rm orb}\cos\Phi_{\rm bh},$ (S10)
$\displaystyle\mbox{\boldmath$\hat{s}$}\cdot\mbox{\boldmath$\hat{e}$}_{2}$
$\displaystyle=\sin i_{\rm bh}\sin\Delta=-\sin\beta\sin\Phi_{\rm bh},$
$\displaystyle\mbox{\boldmath$\hat{s}$}\cdot\mbox{\boldmath$\hat{e}$}_{3}$
$\displaystyle=\cos i_{\rm bh}=\cos\beta\cos i_{\rm orb}+\sin\beta\sin i_{\rm
orb}\cos\Phi_{\rm bh},$

allowing us to obtain the azimuthal angle of the black hole spin:

$\displaystyle\cos\Phi_{\rm bh}$ $\displaystyle=\frac{\sin i_{\rm orb}\cos
i_{\rm bh}-\cos i_{\rm orb}\sin i_{\rm bh}\cos\Delta}{\sin\beta},$ (S11)
$\displaystyle\sin\Phi_{\rm bh}$ $\displaystyle=-\frac{\sin i_{\rm
bh}\sin\Delta}{\sin\beta}.$

We assume that the black hole spin is aligned with the jet. If the spin is
directed along the southern approaching jet, then its inclination $i_{\rm
bh}=i_{\rm jet}=63\hbox{${}^{\circ}$}\pm 3\hbox{${}^{\circ}$}$ and its
position angle is $\theta_{\rm bh}=180\hbox{${}^{\circ}$}+\theta_{\rm
jet}=205\mbox{$.\\!\\!^{\circ}$}1\pm 1\mbox{$.\\!\\!^{\circ}$}4$ (?, ?, ?).
If, on the other hand, the black hole spin points along the northern receding
jet, then $i_{\rm bh}=180\hbox{${}^{\circ}$}-i_{\rm
jet}=117\hbox{${}^{\circ}$}\pm 3\hbox{${}^{\circ}$}$ and its position angle is
$\theta_{\rm bh}=\theta_{\rm jet}=25\mbox{$.\\!\\!^{\circ}$}1\pm
1\mbox{$.\\!\\!^{\circ}$}4$.

Table S5: Geometrical parameters. Geometrical parameters for the 16 possible cases identified by letters A–P of relative orientation of the orbital and the black hole spins.  | $i_{\rm bh}$ | $i_{\rm jet}=63\hbox{${}^{\circ}$}\pm 3\hbox{${}^{\circ}$}$ |  | $180\hbox{${}^{\circ}$}-i_{\rm jet}=117\hbox{${}^{\circ}$}\pm 3\hbox{${}^{\circ}$}$
---|---|---|---|---
| $\theta_{\rm bh}$ | $\theta_{\rm jet}+180\hbox{${}^{\circ}$}=205\mbox{$.\\!\\!^{\circ}$}1\pm 1\mbox{$.\\!\\!^{\circ}$}4$ |  | $\theta_{\rm jet}=25\mbox{$.\\!\\!^{\circ}$}1\pm 1\mbox{$.\\!\\!^{\circ}$}4$
| $i_{\rm orb}$ |  | $73\hbox{${}^{\circ}$}\pm 6\hbox{${}^{\circ}$}$ |  | $107\hbox{${}^{\circ}$}\pm 6\hbox{${}^{\circ}$}$ |  |  | $73\hbox{${}^{\circ}$}\pm 6\hbox{${}^{\circ}$}$ |  | $107\hbox{${}^{\circ}$}\pm 6\hbox{${}^{\circ}$}$
Polarization parallel to the meridional plane
$\theta_{\rm orb}=\langle\mbox{PA}\rangle$ |  $\beta$ (deg) |  A | $117.3\pm 4.3$ |  B | $137.6\pm 2.4$ |  |  C | $62.7\pm 4.3$ |  D | $42.4\pm 2.4$
=$-19\mbox{$.\\!\\!^{\circ}$}7\pm 1\mbox{$.\\!\\!^{\circ}$}2$ |  $\Phi_{\rm bh}$ (deg) | $45.2\pm 3.4$ | $69.7\pm 7.0$ |  | $225.2\pm 3.4$ | $249.0\pm 7.0$
$\theta_{\rm orb}=\langle\mbox{PA}\rangle$+180∘ |  $\beta$ (deg) |  E | $42.4\pm 2.4$ |  F | $62.7\pm 4.3$ |  |  G | $137.6\pm 2.4$ |  H | $117.3\pm 4.3$
$\Phi_{\rm bh}$ (deg) | $289.8\pm 7.0$ | $314.8\pm 3.4$ |  | $110.3\pm 7.0$ | $134.8\pm 3.4$
Polarization perpendicular to the meridional plane
$\theta_{\rm orb}=\langle\mbox{PA}\rangle$+90∘ |  $\beta$ (deg) |  I | $117.0\pm 4.3$ |  J | $137.2\pm 2.4$ |  |  K | $63.0\pm 4.3$ |  L | $42.8\pm 2.4$
$\Phi_{\rm bh}$ (deg) | $314.5\pm 3.4$ | $290.3\pm 6.9$ |  | $134.5\pm 3.4$ | $110.3\pm 6.9$
$\theta_{\rm orb}=\langle\mbox{PA}\rangle$+270∘ |  $\beta$ (deg) |  M | $42.8\pm 2.4$ |  N | $63.0\pm 4.3$ |  |  O | $137.3\pm 2.4$ |  P | $117.0\pm 4.3$
$\Phi_{\rm bh}$ (deg) | $69.7\pm 6.9$ | $45.5\pm 3.4$ |  | $249.7\pm 6.9$ | $225.5\pm 3.4$

The polarimetric data provide us with the average polarization angle
$\langle\mbox{PA}\rangle=-19\mbox{$.\\!\\!^{\circ}$}7\pm
1\mbox{$.\\!\\!^{\circ}$}2$, which carries information about orientation of
the orbital axis on the sky. If dominant oscillations of the electric field
lie in the meridional plane formed by the orbital spin and photon propagation
direction, the position angle of the orbital spin can be either $\theta_{\rm
orb}=\langle\mbox{PA}\rangle$ or
$\langle\mbox{PA}\rangle+180\hbox{${}^{\circ}$}$. The electric field
oscillations can also be perpendicular to the meridional plane, then the
orbital spin position angle is $\langle\mbox{PA}\rangle+90\hbox{${}^{\circ}$}$
or $\langle\mbox{PA}\rangle+270\hbox{${}^{\circ}$}$. Furthermore, the radial
velocity measurements are not able to differentiate between inclinations
$i_{\rm orb}$ and $180\hbox{${}^{\circ}$}-i_{\rm orb}$. The possible
combinations result in 16 different geometrical arrangements of the black hole
and orbital spins that satisfy the observational constraints. These 16 cases
can be reduced to eight different values for the misalignment angle $\beta$:
four for misalignment less than $90\hbox{${}^{\circ}$}$ and four for
misalignment between $90\hbox{${}^{\circ}$}$ and $180\hbox{${}^{\circ}$}$ for
the retrograde rotation of the black hole (Table S5).

The probability distribution for the orbital inclination $i_{\rm orb}$ was
assumed to be a Gaussian with the peak at 73∘ with $1\sigma$ error of 6∘ and a
cutoff at 81∘ (?, ?). For an alternative case of inclination exceeding 90∘,
the distribution mirror reflected relative to 90∘ is considered. Other
parameters are assumed to follow a Gaussian distribution with corresponding
$1\sigma$ errors. Using Monte-Carlo simulations (?), we obtain the probability
distributions for $\beta$ and $\Phi_{\rm bh}$ using Equations (S9) and (S11),
respectively. Their mean and standard deviation are given in Table S5. In Fig.
S4 we show the posterior probability distribution for $\beta$ for the eight
different cases from Table S5. The probability distributions for $\Phi_{\rm
bh}$ for the 16 cases from Table S5 are shown in Fig. S5.

Figure S4: Probability distribution function for the misalignment angle.
Distributions normalized to the peak values are shown for the eight different
cases presented in Table S5.  Figure S5: Probability distribution function for
the black hole spin azimuthal angle. Distributions normalized to the peak
values are shown for the 16 different cases (A-P) presented in Table S5.

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