# First detection of X-ray polarization from the accreting neutron star 4U
1820$-$303

Alessandro Di Marco INAF Istituto di Astrofisica e Planetologia Spaziali, Via
del Fosso del Cavaliere 100, 00133 Roma, Italy  Fabio La Monaca INAF Istituto
di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  Juri Poutanen Department of Physics and Astronomy, 20014
University of Turku, Finland  Thomas D. Russell INAF, Istituto di Astrofisica
Spaziale e Fisica Cosmica, Via U. La Malfa 153, I-90146 Palermo, Italy
Alessio Anitra Università degli Studi di Palermo, Dipartimento di Fisica e
Chimica, Emilio Segrè, via Archirafi 36 I-90123 Palermo, Italy  Ruben
Farinelli INAF—Osservatorio di Astrofisica e Scienza dello Spazio di Bologna,
Via P. Gobetti 101, I-40129 Bologna, Italy  Guglielmo Mastroserio INAF
Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius
(CA), Italy  Fabio Muleri INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Fei Xie Guangxi
Key Laboratory for Relativistic Astrophysics, School of Physical Science and
Technology, Guangxi University, Nanning 530004, China  INAF Istituto di
Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  Matteo Bachetti INAF Osservatorio Astronomico di Cagliari, Via
della Scienza 5, 09047 Selargius (CA), Italy  Luciano Burderi Dipartimento di
Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7,
I-09042 Monserrato, Italy  Francesco Carotenuto Department of Physics,
Astrophysics, University of Oxford, Denys Wilkinson Building, Keble Road,
Oxford OX1 3RH, UK  Melania Del Santo INAF, Istituto di Astrofisica Spaziale e
Fisica Cosmica, Via U. La Malfa 153, I-90146 Palermo, Italy  Tiziana Di Salvo
Università degli Studi di Palermo, Dipartimento di Fisica e Chimica, Emilio
Segrè, via Archirafi 36 I-90123 Palermo, Italy  Michal Dovčiak Astronomical
Institute of the Czech Academy of Sciences, Boční II 1401/1, 14100 Praha 4,
Czech Republic  Andrea Gnarini Dipartimento di Matematica e Fisica, Università
degli Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy  Rosario
Iaria Università degli Studi di Palermo, Dipartimento di Fisica e Chimica,
Emilio Segrè, via Archirafi 36 I-90123 Palermo, Italy  Jari J. E. Kajava
Department of Physics and Astronomy, 20014 University of Turku, Finland  Serco
for the European Space Agency (ESA), European Space Astronomy Centre (ESAC),
Camino Bajo del Castillo s/n, E-28692 Villanueva de la Cañada, Madrid, Spain
Kuan Liu Guangxi Key Laboratory for Relativistic Astrophysics, School of
Physical Science and Technology, Guangxi University, Nanning 530004, China
Riccardo Middei INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00040
Monte Porzio Catone (RM), Italy  Space Science Data Center, Agenzia Spaziale
Italiana, Via del Politecnico snc, 00133 Roma, Italy  Stephen L. O’Dell NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Maura Pilia INAF
Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius
(CA), Italy  John Rankin INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Andrea Sanna Dipartimento
di Fisica, Università degli Studi di Cagliari, SP Monserrato-Sestu km 0.7,
I-09042 Monserrato, Italy  Jakob van den Eijnden Department of Physics,
University of Warwick, Coventry, CV4 7AL, UK  Martin C. Weisskopf NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Anna Bobrikova
Department of Physics and Astronomy, 20014 University of Turku, Finland
Fiamma Capitanio INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy  Enrico Costa INAF Istituto di
Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  Philip Kaaret NASA Marshall Space Flight Center, Huntsville, AL
35812, USA  Alessio Marino Institute of Space Sciences (ICE-CSIC), Barcelona,
Spain  Paolo Soffitta INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Francesco Ursini
Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via
della Vasca Navale 84, 00146 Roma, Italy  Filippo Ambrosino INAF Osservatorio
Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone (RM), Italy
Massimo Cocchi INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5,
09047 Selargius (CA), Italy  Sergio Fabiani INAF Istituto di Astrofisica e
Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
Herman L. Marshall MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  Giorgio Matt Dipartimento di Matematica e Fisica, Università degli
Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy  Sara Elisa Motta
INAF, Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate
(LC), Italy  Alessandro Papitto INAF Osservatorio Astronomico di Roma, Via
Frascati 33, 00040 Monte Porzio Catone (RM), Italy  Luigi Stella INAF
Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone
(RM), Italy  Antonella Tarana  Liceo Scientifico F. Enriques, via F. Paolini,
196, I-00122, Ostia (Rome), Italy  Silvia Zane Mullard Space Science
Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5
6NT, UK  Iván Agudo Instituto de Astrofísicade Andalucía – CSIC, Glorieta de
la Astronomía s/n, 18008 Granada, Spain  Lucio A. Antonelli INAF Osservatorio
Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio Catone (RM), Italy
Space Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy  Luca Baldini Istituto Nazionale di Fisica Nucleare, Sezione
di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Dipartimento di Fisica,
Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Wayne H.
Baumgartner NASA Marshall Space Flight Center, Huntsville, AL 35812, USA
Ronaldo Bellazzini Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy  Stefano Bianchi Dipartimento di
Matematica e Fisica, Università degli Studi Roma Tre, Via della Vasca Navale
84, 00146 Roma, Italy  Stephen D. Bongiorno NASA Marshall Space Flight Center,
Huntsville, AL 35812, USA  Raffaella Bonino Istituto Nazionale di Fisica
Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria 1,
10125 Torino, Italy  Alessandro Brez Istituto Nazionale di Fisica Nucleare,
Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Niccolò Bucciantini
INAF Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze,
Italy  Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze,
Via Sansone 1, 50019 Sesto Fiorentino (FI), Italy  Istituto Nazionale di
Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto Fiorentino
(FI), Italy  Simone Castellano Istituto Nazionale di Fisica Nucleare, Sezione
di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Elisabetta Cavazzuti
Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy  Chien-
Ting Chen Science and Technology Institute, Universities Space Research
Association, Huntsville, AL 35805, USA  Stefano Ciprini Istituto Nazionale di
Fisica Nucleare, Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica
1, 00133 Roma, Italy  Space Science Data Center, Agenzia Spaziale Italiana,
Via del Politecnico snc, 00133 Roma, Italy  Alessandra De Rosa INAF Istituto
di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  Ettore Del Monte INAF Istituto di Astrofisica e Planetologia
Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Laura Di Gesu
Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy  Niccolò
Di Lalla Department of Physics and Kavli Institute for Particle Astrophysics
and Cosmology, Stanford University, Stanford, California 94305, USA
Immacolata Donnarumma Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy  Victor Doroshenko Institut für Astronomie und Astrophysik,
Universität Tübingen, Sand 1, 72076 Tübingen, Germany  Steven R. Ehlert NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Teruaki Enoto RIKEN
Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan
Yuri Evangelista INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy  Riccardo Ferrazzoli INAF Istituto
di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  Javier A. Garcia California Institute of Technology, Pasadena, CA
91125, USA  Shuichi Gunji Yamagata University,1-4-12 Kojirakawa-machi,
Yamagata-shi 990-8560, Japan  Kiyoshi Hayashida  Deceased  Osaka University,
1-1 Yamadaoka, Suita, Osaka 565-0871, Japan  Jeremy Heyl University of British
Columbia, Vancouver, BC V6T 1Z4, Canada  Wataru Iwakiri International Center
for Hadron Astrophysics, Chiba University, Chiba 263-8522, Japan  Svetlana G.
Jorstad Institute for Astrophysical Research, Boston University, 725
Commonwealth Avenue, Boston, MA 02215, USA  Department of Astrophysics, St.
Petersburg State University, Universitetsky pr. 28, Petrodvoretz, 198504 St.
Petersburg, Russia  Vladimir Karas Astronomical Institute of the Czech Academy
of Sciences, Boční II 1401/1, 14100 Praha 4, Czech Republic  Fabian Kislat
Department of Physics and Astronomy and Space Science Center, University of
New Hampshire, Durham, NH 03824, USA  Takao Kitaguchi  RIKEN Cluster for
Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan  Jeffery J.
Kolodziejczak NASA Marshall Space Flight Center, Huntsville, AL 35812, USA
Henric Krawczynski Physics Department and McDonnell Center for the Space
Sciences, Washington University in St. Louis, St. Louis, MO 63130, USA  Luca
Latronico Istituto Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro
Giuria 1, 10125 Torino, Italy  Ioannis Liodakis Finnish Centre for Astronomy
with ESO, 20014 University of Turku, Finland  Simone Maldera Istituto
Nazionale di Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125
Torino, Italy  Alberto Manfreda Istituto Nazionale di Fisica Nucleare, Sezione
di Napoli, Strada Comunale Cinthia, 80126 Napoli, Italy  Frédéric Marin
Université de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR
7550, 67000 Strasbourg, France  Andrea Marinucci Agenzia Spaziale Italiana,
Via del Politecnico snc, 00133 Roma, Italy  Alan P. Marscher Institute for
Astrophysical Research, Boston University, 725 Commonwealth Avenue, Boston, MA
02215, USA  Francesco Massaro Istituto Nazionale di Fisica Nucleare, Sezione
di Torino, Via Pietro Giuria 1, 10125 Torino, Italy  Dipartimento di Fisica,
Università degli Studi di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
Ikuyuki Mitsuishi  Graduate School of Science, Division of Particle and
Astrophysical Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi
464-8602, Japan  Tsunefumi Mizuno Hiroshima Astrophysical Science Center,
Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526,
Japan  Michela Negro University of Maryland, Baltimore County, Baltimore, MD
21250, USA  NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA  Center
for Research and Exploration in Space Science and Technology, NASA/GSFC,
Greenbelt, MD 20771, USA  Chi-Yung Ng Department of Physics, University of
Hong Kong, Pokfulam, Hong Kong  Nicola Omodei Department of Physics and Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California 94305, USA  Chiara Oppedisano Istituto Nazionale di
Fisica Nucleare, Sezione di Torino, Via Pietro Giuria 1, 10125 Torino, Italy
George G. Pavlov Department of Astronomy and Astrophysics, Pennsylvania State
University, University Park, PA 16801, USA  Abel L. Peirson Department of
Physics and Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, Stanford, California 94305, USA  Matteo Perri Space Science Data
Center, Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy
INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00040 Monte Porzio
Catone (RM), Italy  Melissa Pesce-Rollins Istituto Nazionale di Fisica
Nucleare, Sezione di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy  Pierre-
Olivier Petrucci Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France
Andrea Possenti INAF Osservatorio Astronomico di Cagliari, Via della Scienza
5, 09047 Selargius (CA), Italy  Simonetta Puccetti Space Science Data Center,
Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy  Brian
D. Ramsey NASA Marshall Space Flight Center, Huntsville, AL 35812, USA  Ajay
Ratheesh INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso
del Cavaliere 100, 00133 Roma, Italy  Oliver J. Roberts Science and Technology
Institute, Universities Space Research Association, Huntsville, AL 35805, USA
Roger W. Romani Department of Physics and Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California 94305,
USA  Carmelo Sgrò Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy  Patrick Slane Center for
Astrophysics, Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA
Gloria Spandre Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo
B. Pontecorvo 3, 56127 Pisa, Italy  Douglas A. Swartz Science and Technology
Institute, Universities Space Research Association, Huntsville, AL 35805, USA
Toru Tamagawa RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan  Fabrizio Tavecchio INAF Osservatorio Astronomico di
Brera, via E. Bianchi 46, 23807 Merate (LC), Italy  Roberto Taverna
Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Via
Marzolo 8, 35131 Padova, Italy  Yuzuru Tawara  Graduate School of Science,
Division of Particle and Astrophysical Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya, Aichi 464-8602, Japan  Allyn F. Tennant NASA Marshall
Space Flight Center, Huntsville, AL 35812, USA  Nicholas E. Thomas NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Francesco Tombesi
Dipartimento di Fisica, Università degli Studi di Roma “Tor Vergata”, Via
della Ricerca Scientifica 1, 00133 Roma, Italy  Istituto Nazionale di Fisica
Nucleare, Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica 1,
00133 Roma, Italy  Department of Astronomy, University of Maryland, College
Park, Maryland 20742, USA  Alessio Trois INAF Osservatorio Astronomico di
Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy  Sergey S.
Tsygankov Department of Physics and Astronomy, 20014 University of Turku,
Finland  Roberto Turolla Dipartimento di Fisica e Astronomia, Università degli
Studi di Padova, Via Marzolo 8, 35131 Padova, Italy  Mullard Space Science
Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5
6NT, UK  Jacco Vink Anton Pannekoek Institute for Astronomy & GRAPPA,
University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands
Kinwah Wu Mullard Space Science Laboratory, University College London,
Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

###### Abstract

This paper reports the first detection of polarization in the X-rays for
atoll-source 4U 1820$-$303, obtained with the Imaging X-ray Polarimetry
Explorer (IXPE) at 99.999% confidence level (CL). Simultaneous polarimetric
measurements were also performed in the radio with the Australia Telescope
Compact Array (ATCA). The IXPE observations of 4U 1820$-$303 were coordinated
with Swift-XRT, NICER, and NuSTAR aiming to obtain an accurate X-ray spectral
model covering a broad energy interval. The source shows a significant
polarization above 4 keV, with a polarization degree of $2.0\%\pm 0.5\%$ and a
polarization angle of $-55\arcdeg\pm 7\arcdeg$ in the 4–7 keV energy range,
and a polarization degree of $10\%\pm 2\%$ and a polarization angle of
$-67\arcdeg\pm 7\arcdeg$ in the 7–8 keV energy bin. This polarization also
shows a clear energy trend with polarization degree increasing with energy and
a hint for a position-angle change of $\simeq 90\deg$ at 96% CL around 4 keV.
The spectro-polarimetric fit indicates that the accretion disk is polarized
orthogonally to the hard spectral component, which is presumably produced in
the boundary/spreading layer. We do not detect linear polarization from the
radio counterpart, with a $3\sigma$ upper limit of 50% at 7.25 GHz.

accretion, accretion disks – polarization – stars: individual: 4U 1820$-$303 –
stars: neutron – X-ray binaries

††facilities: IXPE, Swift–XRT, NICER, NuSTAR††software: ixpeobssim (Baldini et
al., 2022), xspec (Arnaud, 1996), HEASoft (Blackburn, 1995)

revtex4-1.clsrevtex4-2.cls

##  1 Introduction

Accreting weakly magnetized neutron stars (NSs) in low-mass X-ray binaries
(LMXBs) are among the brightest X-ray sources; they accrete matter via Roche
lobe overflow from a stellar companion of mass typically lower than a solar
mass (Bahramian & Degenaar, 2022). They can be classified according to the
shape of their tracks in the X-ray hard-color/soft-color diagram (CCD), or
their hard-color/intensity diagram (HID), and their correlated timing in the
1–10 keV band (van der Klis, 1989; Hasinger & van der Klis, 1989). The
following states are known: (i) the high soft state (HSS) of Z-sources with a
luminosity $>10^{37}$ erg s-1 (near-Eddington X-ray luminosities) exhibiting a
wide Z-like shape in the CCD; (ii) the low hard state (LHS) of atoll sources
with a luminosity $\simeq$ 1036 erg s-1 showing a single rounded spot in the
CCD (island state) typically and having a harder spectrum than HSS; (iii) the
HSS of bright atoll sources, having intermediate luminosities
($10^{36}$–$10^{37}$ erg s-1), typically following a banana shape figure in
the CCD. The banana state has been further divided on the basis of their
luminosity into left-lower (lower luminosity), upper (higher luminosity), and
lower banana in the middle.

The spectral and timing properties of these sources provide clues to their
emission mechanisms, and their X-ray emission generally is described by two
main spectral components: (i) a soft thermal component (blackbody-like
emission from the NS surface or from the accretion disk); (ii) a harder
component associated with the interaction of the accretion disk with the NS
surface. The interaction region that is coplanar to the accretion disk is
called the boundary layer (BL; Shakura & Sunyaev, 1988; Popham & Sunyaev,
2001), while the gas layer at the NS surface, extending up to high latitudes,
is called the spreading layer (SL) (Suleimanov & Poutanen, 2006; Inogamov &
Sunyaev, 1999). The properties of the SL/BL region (temperature and optical
depth) clearly distinguish sources in the LHS from the ones in the HSS, with
the former SL/BL much hotter and more transparent with respect to the latter
ones (Gnarini et al., 2022). The frequent observation of a reflection
component, whose most prominent feature is an iron emission line at $\sim$ 6.5
keV, suggests Compton reflection by a colder medium (e.g., the outer accretion
disk itself) as a further component to take into account, especially in the
HSS (see e.g., Cackett et al., 2008; Titarchuk et al., 2013; Mondal et al.,
2016; Egron et al., 2013).

The launch of the Imaging X-ray Polarimetry Explorer (IXPE) in December 2021
(Weisskopf et al., 2022; Soffitta et al., 2021), gave us the opportunity to
use X-ray polarization, in addition to spectral and timing information, to
disentangle several scenarios. The X-ray polarization of weakly magnetized NSs
strongly depends on the geometry of the emission region. In the LHS sources,
it can be produced by a non-spherical slab-like corona (as in the case of BH-
LMXB, Haardt & Matt, 1993; Poutanen & Vilhu, 1993; Poutanen & Svensson, 1996;
Schnittman & Krolik, 2010), or by the accretion disk (Chandrasekhar, 1960;
Loskutov & Sobolev, 1982). In the HSS sources, the BL and the SL (Lapidus &
Sunyaev, 1985) emission can be polarized; also, Comptonization of any seed
soft photons in a corona and reflection of the SL radiation off the accretion
disk are potential sources of polarized emission (Lapidus & Sunyaev, 1985;
Suleimanov & Poutanen, 2006; D’Aì et al., 2010).

In order to investigate the geometry of the accretion flow in LMXBs, IXPE has
so far targeted several sources: the Z-source Cyg X-2 (Farinelli et al.,
2023), the Z-atoll transient source XTE J1701$-$462 (Jayasurya et al., 2023;
Cocchi et al., 2023), and two atoll sources, GS 1826$-$238 (Capitanio et al.,
2023) and GX 9+9 (Chatterjee et al., 2023; Ursini et al., 2023). These
observations found a higher polarization in the Z-sources than in the atoll
ones; moreover, the polarization angle (PA) of Cyg X-2, for the hardest part
of the energy spectrum, appears to be aligned with the radio jet. Also, a
marginal detection of polarization available for Sco X-1 from OSO-8 (Long et
al., 1979) and PolarLight (Long et al., 2022) appears to show such an
alignment between the radio jet and the PA. An attempt to measure the X-ray
polarization of 4U 1820$-$303 was performed by OSO-8 (Hughes et al., 1984),
but only 99% confidence level (CL) upper limits of 4.7% and 10.8% were
obtained at 2.6 keV and 5.2 keV, respectively.

4U 1820$-$303 is an ultra-compact LMXB consisting of a NS, accreting matter
via Roche lobe overflow from a He white dwarf. It is located at 0$\farcs$66
from the center of the globular cluster NGC 6624 (Rappaport et al., 1987;
Shaposhnikov & Titarchuk, 2004), and its distance was estimated from the GAIA
EDR3 to be $D=8.0\pm 0.1$ kpc (Baumgardt & Vasiliev, 2021). 4U 1820$-$303 has
an orbital period of 685 s (Stella et al., 1987) and a peculiar behavior with
intrinsic luminosity variation by a factor of at least 2 along a superorbital
$\sim$170 d period (Zdziarski et al., 2007). Chou & Grindlay (2001) found the
flux modulation to be stable in the period 1969–2000 with periodicity
171.0$\pm$0.3 d, compatible with the RXTE All Sky Monitor (ASM) data
(Zdziarski et al., 2007). These variations are also seen in the MAXI and
Swift-BAT light curves (see Figure 1).

Figure 1: Light curves from Swift-BAT monitor in the 15–50 keV energy band
(top) and MAXI in 2–20 keV energy band (bottom). The dashed vertical lines
identify the 170 day superorbital period.

4U 1820$-$303 is the first identified source of type-I X-ray bursts (Grindlay
et al., 1976), observed mainly around the flux minima, proving that the
observed variability is indeed due to intrinsic accretion rate changes; this
is also supported by the strong correlations between the observed flux
variations with the source spectral state (in a way typical of atoll-type NS
binaries) and the kHz frequency of quasi-periodic oscillations (QPOs) (Smale
et al., 1997).

Figure 2: Light curves from Swift-BAT in the 15–50 keV energy band (blue
points) and MAXI in the 2–20 keV energy band (red points), overlaid in dark
red are the IXPE intervals of observation. For the first observation (top
panel), where coordinated observations were not strictly simultaneous,
vertical bands show the observing period for NICER (orange), NuSTAR (green),
and Swift-XRT (blue), while for the second (bottom panel) they were strictly
simultaneous and included in the IXPE band.

4U 1820$-$303, as typically observed in weakly magnetized NS-LMXBs, shows a
spectrum mainly composed of two components: a blackbody or multicolor disk
that describes the softer spectral component, and a Comptonization to describe
the harder spectral one. Moreover, these sources can show a component due to
reflection, which has, as a main feature, the presence of a Gaussian due to
the iron K$\alpha$ lines at $\sim$6.5 keV (Cackett et al., 2008; Titarchuk et
al., 2013; Mondal et al., 2016). 4U 1820$-$303 shows a broad iron line (Mondal
et al., 2016), which Cackett et al. (2008) fitted with a diskline obtaining
the disk inclination of $\sim 20\arcdeg$.

4U 1820$-$303 is also a known radio emitter (e.g., Migliari et al., 2004; Díaz
Trigo et al., 2017; Russell et al., 2021). At low X-ray fluxes, the radio
spectrum is typically observed to be relatively flat, consistent with a
compact radio jet, while, at higher X-ray fluxes (Russell et al., 2021), the
radio spectrum becomes steep, consistent with either a quenching of the
compact jet emission or emission from a transient jet ejecta (Russell et al.,
2021). Assuming a uniform and ordered magnetic field, linear polarization from
a compact (self-absorbed) radio jet is expected to have a maximum degree of
$\sim$10%, while the optically thin ejecta can exhibit values up to $\sim$70%
(see pp. 217–222 in Longair, 2011); however, due to disorder in the magnetic
fields, lower values are typically observed (see, e.g., Curran et al., 2014).

##  2 X-Ray Observations

Figure 3: Hardness–intensity diagram, where the different states of the source
are identified: the island state (IS), where the source has a harder spectrum;
the softer banana state with Upper (UB), Lower (LB), and Left Lower (LLB)
banana. The red points report the source state during the first and the second
IXPE coordinated observations at the times discussed in the text. Figure 4:
Swift-XRT HR in time bins of 100 s during the first (left, we used ObsIDs:
00014980028, 00014980029, 00014980030, and 00014980031) and the second (right,
we used ObsIDs: 00014980055, 00014980056, and 00014980057) IXPE coordinated
observations. The horizontal gray band reports the average 99% CL HR interval
during the two observations, the vertical bands and/or horizontal arrows
report the time intervals when observations by other observatories were
performed.

IXPE observed 4U 1820$-$303 in two different periods: on 2022 October 11 from
13:53 to 22:55 UTC for a total exposure of $\simeq$16 ks per Detector Unit
(DU) and from 2023 April 15 01:33 to April 16 23:17 UTC for a total exposure
of $\simeq$86 ks per DU. During these periods, other observatories performed
simultaneous observations. To determine the state of 4U 1820$-$303 during the
IXPE observations, we analyzed the daily count rates in the 2–20 keV energy
band from the Monitor of All-sky X-ray Image (MAXI) telescope111
http://maxi.riken.jp/top/index.html (Matsuoka et al., 2009), and in the 15–50
keV band from the Swift Burst Alert Telescope (Swift-BAT) Hard X-ray Transient
Monitor222https://swift.gsfc.nasa.gov/results/transients/ (Krimm et al.,
2013). The 4U 1820$-$303 light curves, covering since August 2009 from both
observatories, are shown in Figure 1. IXPE observations were performed near
the maximum of the flux along the superorbital period, as shown in Figure 2.

In Figure 3, we show the corresponding hardness-intensity diagram (HID), where
the hardness is given by the ratio between Swift-BAT count rate in the 15–50
keV band and the MAXI count rate in the 2–4 keV band and the flux is
represented by the MAXI count rate in the 2–20 keV band. The red points report
the state of the source along our coordinated observations: 1 – on 2022
October 11 by NICER, IXPE, and Swift-XRT; 2 – on 2022 October 12 by Swift-XRT
and NuSTAR; 3 – on 2022 October 13 by Swift-XRT; 4 – 2023 on April 15 by IXPE,
Swift-XRT, NICER, and NuSTAR and ATCA; 5 – on 2023 April 16 by IXPE, NICER,
NuSTAR, and Swift-XRT. From this HID we observe that the source was in the
Lower Banana state in all of them. To further confirm this result, we used
coordinated Swift-XRT data to monitor the hardness ratio (HR) along the two
periods of observations; we defined the Swift-XRT HR as follows:

$\text{HR}=\frac{\text{countsin4--10\,keV}-\text{countsin0.3--
4\,keV}}{\text{countsin0.3--10\,keV}}.$ (1)

Figure 4 shows this HR as a function of time near the first IXPE pointing –
when, because of the brightest ever gamma-ray burst (Burns et al., 2023), the
IXPE observations were stopped, and other observatories were not able to
perform strictly simultaneous observations – and near the second IXPE pointing
when all the observations with other telescopes were strictly simultaneous.

From this analysis, we can confirm that the HR values fluctuate around an
average HR band as shown in Figure 4 thus the source has not changed its
state; only the data from ObsID 00014980055 (on April 15 since 00:41) show a
slightly harder spectrum, but this hardening corresponds to a slightly higher
counting rate, and it can be explained as due to pile-up (Romano et al.,
2006). Also from the NuSTAR HR, assuming the soft component in 3–10 keV and
the harder one in 10–30 keV, we see that HR does not vary much. Thus, for the
following spectral and spectro-polarimetric analysis, we join the data from
the different observations to improve the sensitivity.

Figure 5: Polar plot of the X-ray polarization in 4U 1820$-$303\. (a) Contours
at 99% CL in the 2–4 keV energy bin, for the first (red) and the second
(green) observations. (b) Same as (a), but in the 4–8 keV energy bin. (c)
Combined data in the three energy bins 2–4, 4–7, and 7–8 keV. The contours
correspond to 50%, 95%, and 99% CL.

##  3 Polarimetric analysis

IXPE observed 4U 1820$-$303 twice, both times during its lower banana state;
this suggests that the two observations have the same spectral properties,
allowing us to merge them. From the point of view of the polarization, due to
the lack of significant measurements before IXPE, this is something that we
need to confirm by comparing the polarization in the first and in the second
observation. At this aim, we used the ixpeobssim software (Baldini et al.,
2022) with the PCUBE algorithm to have a preliminary estimate of polarization.
In the IXPE data, the background is negligible (roughly two orders of
magnitude below the source in the whole energy band); thus, the prescription
reported in Di Marco et al. (2023) for bright sources have been applied.

IXPE in its nominal energy band reports an average polarization that is not
significant in the first observation and has a low significance in the second.
Also combining the two data sets, we obtain a polarization below the 99% CL.
The polarimetric analysis was performed using two energy bins: 2–4 and 4–8
keV, as reported in Figure 5(a) and (b). The contour regions resulting from
the two IXPE observations in both the energy bins are compatible allowing to
combine the two data sets.

We performed a study of the polarization using the combined data from the two
IXPE observations by using the PCUBE analysis to measure the normalized Stokes
parameters as a function of the energy with bins of 1 keV; the results are
shown in Figures 6, and reported in Table 1. We obtain that only the energy
bin 7–8 keV has a highly significant polarization at 99.99% CL. On the other
hand, taking into account each energy bin as an independent data set, the
statistical significance of this observation in any one bin has been
evaluated; then, the CL for the polarization detection was tested assuming the
ensemble of bins against the null hypothesis (i.e. assuming a null
polarization in every bin). The result of this test allowed us to obtain a
probability of null polarization of 5.32$\times 10^{-6}$, which correspond to
a polarization detection at level of 99.995% CL. Moreover, we see that the
Stokes parameters show a clear trend of the polarization degree (PD)
increasing with energy. We identify three energy ranges showing similar
polarization from Figure 6: 2–4, 4–7, and 7–8 keV. By grouping the data within
these wider bins, we can increase the significance of detection both in each
bin (2–4 keV and 4–7 keV have a polarization detection at 96% and 99.97% CL,
respectively) and for the whole set of bins reaching a 99.99997% CL. The
resulting PD and PA contour plots for the three wider bins are shown in Figure
5(c), while the numerical results are reported in Table 1. The most
significant polarization is measured at CL $>99.99$% in the 7–8 keV energy
bin, where the PD reaches a value of $\sim$10%, never observed in other weakly
magnetized sources (Farinelli et al., 2023; Capitanio et al., 2023; Chatterjee
et al., 2023; Jayasurya et al., 2023; Cocchi et al., 2023; Ursini et al.,
2023). We also observe a position-angle change by nearly 90° between 2–4 keV,
and the higher energy bins (significant at 96% CL).

Table 1: Polarization properties obtained with the PCUBE algorithm for the
merged data from both the IXPE observations.

Energy bin | PD | PA | $Q/I$ | $U/I$
---|---|---|---|---
(keV) | (%) | (deg) | (%) | (%)
2–3 | $0.8\pm 0.4$ | $24\pm 15$ | $0.5\pm 0.4$ | $0.6\pm 0.4$
3–4 | $0.8\pm 0.4$ | $42\pm 16$ | $0.1\pm 0.4$ | $0.8\pm 0.4$
4–5 | $1.9\pm 0.6$ | $-54\pm 10$ | $-0.6\pm 0.6$ | $-1.8\pm 0.6$
5–6 | $2.7\pm 0.9$ | $-56\pm 10$ | $-1.1\pm 0.9$ | $-2.5\pm 0.9$
6–7 | $1.3\pm 1.3$ | $-57\pm 29$ | $-0.6\pm 1.3$ | $-1.2\pm 1.3$
7–8 | $10.3\pm 2.4$ | $-67\pm 7$ | $-7.1\pm 2.4$ | $-7.5\pm 2.4$
2–4 | $0.75\pm 0.30$ | $30\pm 11$ | $0.37\pm 0.30$ | $0.66\pm 0.30$
4–7 | $2.0\pm 0.5$ | $-55\pm 7$ | $-0.7\pm 0.5$ | $-1.9\pm 0.5$

Note. — Errors are reported at 68% CL.

Figure 6: Top: normalized Stokes parameters $Q/I$ and $U/I$ in different
energy bins as measured with PCUBE in ixpeobssim. Bottom: polar plot of the
X-ray polarization in 4U 1820$-$303 when 1 keV binning is applied. Contours
are reported at 68% CL. A trend of polarization growing with energy is visible
in both plots.

##  4 Spectral analysis

Using the data from NICER (in 0.7–12 keV) and NuSTAR (in 3–30 keV) – see
Appendix A for details about data extraction – and IXPE (in 2–8 keV) weighted
spectra (Di Marco et al., 2022a) and on the basis of the spectral models
present in the literature (Titarchuk et al., 2013; Tarana et al., 2007), we
applied the model tbabs*(diskbb+comptb) in xspec v.12.13.0c (Arnaud, 1996). To
estimate the absorption from the interstellar medium, we set the abundances at
the wilm values (Wilms et al., 2000). The NuSTAR residuals show an excess, as
reported in Figure 7, compatible with the presence of a broad iron line, such
an excess is not appreciable in the IXPE and NICER data.

Figure 7: Energy spectra residuals in the energy range 5.5–7.5 keV from the 4U
1820$-$303 observations with NuSTAR (top), NICER (center), and IXPE (bottom),
when the spectral model does not include the iron line. We see that NuSTAR
data show an excess compatible with a broad iron line, not appreciable in IXPE
and NICER data.

Because of the excess in the NuSTAR residuals, we included in the model a
Gaussian line: tbabs*(diskbb+comptb+gauss). The $\chi^{2}$/dof is 1955/1785
without a line, to compare with 1835/1780 when a line is included.
Summarizing, the adopted best-fit model includes diskbb for the disk emission,
gauss for the Fe K${\alpha}$ line, and comptb (with bulk parameter $\delta=0$)
for thermal Comptonization with the seed photons following a blackbody
distribution ($\Gamma=3$) and a large Comptonized fraction $A\gg 1$ that is
obtained by freezing $\log A=5$. We tested also bbody for the soft component
and the compTT for the harder one, obtaining similar results, but we opted for
this spectral model that provides a better reduced $\chi^{2}$.

This model has been applied to fit a broadband spectrum of the combined
simultaneous dataset by NICER, NuSTAR, and IXPE. We also used the energy-
independent cross-normalization factors. The best-fit parameters are
summarized in Table 2. The $EF_{E}$ spectrum and its residuals are reported in
Figure 8.

Table 2: Best-fit parameters of the spectral model tbabs(diskbb + comptb +
gauss) applied to the simultaneous data from NICER, NuSTAR, and IXPE. The
estimated flux in 2–8 keV is $5.23\times 10^{-9}$erg s-1 cm-2.

Model | Parameter | Value
---|---|---
tbabs |  $N_{\rm H}$ ($\times 10^{22}$ cm-2) | $0.14\pm 0.01$
diskbb |  $kT_{\rm in}$ (keV) | $0.52^{+0.03}_{-0.04}$
|  norm ($[R_{\rm in}/D_{10}]^{2}\cos\theta$) | $1900^{+500}_{-400}$
|  $R_{\rm in}$aaThe inner radius for the diskbb component is estimated assuming an inclination at 40°, as reported in Anderson et al. (1997), and a distance of 8 kpc (Baumgardt & Vasiliev, 2021). (km) | $40\pm 20$
comptb |  $kT_{\rm s}$ (keV) | $0.72\pm 0.05$
| $\alpha$ | $0.93\pm 0.01$
|  $kT_{\rm e}$ (keV) | $2.919^{+0.016}_{-0.008}$
|  $L_{\rm X}$ bbThe source luminosity for $D=8$ kpc. ($\times 10^{39}$ erg s-1) | $0.0347\pm 0.0004$
gauss |  $E_{\rm line}$ (keV) | $6.63\pm 0.09$
|  $\sigma$ (keV) | $0.56\pm 0.01$
|  norm (photon cm-2 s-1) | $0.0020^{+0.0006}_{-0.0005}$
| Equivalent width (eV) | 34.2$\pm$1.2
$\chi^{2}$/dof = 1835/1780 = 1.03
Cross normalization factors
| $C_{\rm NICER}$ | 1.0
| $C_{\rm{NuSTAR}-A}$ | $1.063\pm 0.004$
| $C_{\rm{NuSTAR}-B}$ | $1.057\pm 0.004$
| $C_{\rm{IXPE}-DU1}$ | $1.034\pm 0.007$
| $C_{\rm{IXPE}-DU2}$ | $0.990\pm 0.007$
| $C_{\rm{IXPE}-DU3}$ | $0.932\pm 0.007$
Photon flux ratios in 2–8 keV
| $F_{\rm diskbb}/F_{\rm tot}$ | 0.114
| $F_{\rm comptb}/F_{\rm tot}$ | 0.884
| $F_{\rm gauss}/F_{\rm tot}$ | 0.002

Note. — Errors are reported at 90% CL.

Figure 8: Spectral energy distribution of 4U 1820$-$303 in $EF_{E}$
representation. The points show the data from NICER (brown), NuSTAR (blue),
and IXPE (red). The different spectral model components are reported in black
lines for diskbb (dashed), comptb (dotted-dashed), and gauss (dotted). The
bottom panel shows the residuals between the data and the best-fit model.

To take into account NuSTAR calibration uncertainties (Madsen et al., 2022),
we assumed a gain offset free in the fit, obtaining $-(0.085\pm 0.009)$ keV
and $-(0.068\pm 0.008)$ keV for the focal plane modules A and B (FPMA and
FPMB), respectively; similarly for IXPE calibration uncertainties (Di Marco et
al., 2022b) we left free the gain slope and offset obtaining a slope of
$0.981\pm 0.003$ keV-1, $0.973\pm 0.003$ keV-1, $0.980\pm 0.003$ keV-1 for the
DU1, DU2 and DU3 respectively, while the offset in each one is $0.003\pm
0.012$ keV, $0.032\pm 0.012$ keV and $0.020\pm 0.012$ keV. As reported also by
the photons fluxes of Table 2, the spectrum in the whole 2–8 keV IXPE energy
band is dominated by the Comptonization component, while the disk contributes
only at lower energies (see also Figure 8).

It is worth noting that the Fe line is typically found along with a broadband
reflection component (e.g., Iaria et al., 2016; Ursini et al., 2023). We
therefore tested several self-consistent models (such as relxillns) or
convolution ones (such as rdblur*rfxconv) to take into account the reflection
component. However, none of them produced a statistically significant
improvement to the fit, and the reflection fraction was negligible ($<$5%).
Since a more detailed treatment of the reflection goes beyond the scope of
this manuscript, we did not include any reflection component in the following
spectro-polarimetric modeling.

Table 3: Main results of the spectro-polarimetric analysis

Polarimetric |  | polconst*diskbb + | polconst*diskbb + | polconst*diskbb +
---|---|---|---|---
components |  | polconst*comptb | pollin*comptb | polpow*comptb
diskbb | PD (%) | 9.8$\pm 4.2$ | 8.1${}^{+7.1}_{-6.8}$ | 3.2${}^{+3.0}_{-2.9}$
| PA ($\deg$) | $32\pm 13$ | $-59^{+13}_{-26}$ | $43\pm 35$
comptb | PD / $A_{1}$ (%)aaFor the pollin and polpow models, $A_{1}$ and $\psi_{1}$ refer to the PD and PA values at 1 keV. | $5.31\pm 0.24$ | $5.7\pm 2.4$ | $0.46^{+9.63}_{-0.46}\times 10^{-3}$
|  $A_{\rm slope}$ (% keV-1) | – | $-1.9\pm 0.6$ | –
| $A_{\rm index}$ | – | – | $-4.9^{+1.6}_{-2.6}$
| PA / $\psi_{1}$ (deg)aaFor the pollin and polpow models, $A_{1}$ and $\psi_{1}$ refer to the PD and PA values at 1 keV. | $-63\pm 11$ | $38\pm 7$ | $-63\pm 7$
|  $\psi_{\rm slope}$ (deg keV-1) | – |  $0$ (f)bbWe fixed the slope and the index of the PA at zero. | –
| $\psi_{\rm index}$ | – | – |  $0$ (f)bbWe fixed the slope and the index of the PA at zero.
|  $\chi^{2}$/dof | 724/662 = 1.094 | 699/661=1.057 | 697/661=1.054

Note. — Errors are at 90% CL.

##  5 Spectro-polarimetric analysis

The IXPE Stokes parameters $I$, $Q$, and $U$ spectra have been fitted with
xspec v.12.13.0c (Arnaud, 1996) freezing the spectral model at the one
reported in Sect. 4 and summarized in Table 2, and applying the same gain
corrections to the response files of the $I$, $Q$, and $U$ spectra. As
discussed above, we tested different models, but for the spectro-polarimetric
analysis in this section, we opt for the simplest model satisfactorily fitting
the data, which is tbabs*(diskbb+gauss+comptb).

Figure 9: Spectral joint fit for the Stokes parameters $I$, $Q$, and $U$ in
the 2–8 keV energy band using three IXPE detectors and applying the model
tbabs*(gauss+polconst*diskbb+polpow*comptb). The best-fit gives $\chi^{2}$/dof
=1.054 (see also Table 3).

The $I$ spectrum alone gives a $\chi^{2}$/dof = 226/212 = 1.07. The first
spectro-polarimetric analysis, including $Q$ and $U$ spectra, has been
performed to confirm an increase of the polarization with energy; thus, we
tested two models, the first one assuming a constant polarization using the
polconst model from xspec for both the diskbb and comptb components, obtaining
a PD of 0.5%$\pm$0.3%, PA =$-75\arcdeg\pm 27\arcdeg$, and $\chi^{2}$/dof =
740/664 = 1.11. Hereafter, all the errors are reported at 90% CL. Then, we
considered the possibility that polarization linearly depends on energy using
the pollin model: $\mbox{PD}(E)=A_{1}+A_{\rm slope}(E-1)$,
$\mbox{PA}(E)=\psi_{1}+\psi_{\rm slope}(E-1)$ (where the energy $E$ is in
keV). We obtain the PD at 1 keV of $A_{1}=2.9\%\pm 1.0\%$ with a slope of
$A_{\rm slope}=-1.3\%\pm 0.3$% keV-1 and a PA of $\psi_{1}=43\arcdeg\pm
27\arcdeg$ at 1 keV with a slope $\psi_{\rm slope}=-3\arcdeg\pm 5\arcdeg$
keV-1, with a $\chi^{2}$/dof = 704/662 = 1.063. These results are compatible
with the PA being constant, while the PD has a negative slope. We note that
the best-fit PD at 1 keV and the PD slope imply that the PD becomes close to
zero at about 3 keV and reaches a value of $\sim$6% at 7–8 keV at a PA of
$-47\arcdeg$ (change of sign of PD is equivalent to a 90° swing of the PA).
This confirms the results obtained with PCUBE shown in Figures 6 and 5(c). The
F-test for these two fits gives a value of 17.13 (probability 5.6$\times
10^{-8}$), confirming that the energy-dependent polarization is favored by the
data.

Considering that the PD at low energy is not significant (see Figure 5(c) and
Table 1), we tested the hypothesis that only the comptb component is
constantly polarized, but in this scenario, $\chi^{2}$/dof reaches a value
higher than 4, confirming the need for a polarized diskbb component in the
spectro-polarimetric fit. At this point, we tried to disentangle the
polarization of the diskbb and comptb components by associating each component
with its own polarization. We assumed for the disk a constant polarization,
and for the Comptonization, we considered three different polarimetric models:
polconst with a constant polarization, pollin, i.e., polarization depends
linearly on energy, and polpow model that assumes that polarization is
changing as a power law of energy $\mbox{PD}(E)=A_{\rm norm}E^{-A_{\rm
index}}$ and $\mbox{PA}(E)=\psi_{\rm norm}E^{-\psi_{\rm index}}$. In order to
reduce the number of parameters, we fixed $\psi_{\rm slope}$ and $\psi_{\rm
index}$ at zero. The results from this analysis are reported in Table 3 and
Figure 9.

All considered models give similar results, with polarization at low energies
being orthogonal to that at higher energies. For the polconst model, the disk
polarization reaches $\sim$10%, the Comptonization component is significantly
polarized at a 5% level, with the PA being orthogonal to that of the disk. In
the case of the pollin model for the comptb component, the disk polarization
becomes compatible with smaller values, as expected in literature, while, at
the same time, the PD of the Comptonization component changes sign around 4
keV corresponding to the rotation of the PA from 28° at lower energies to
$-62\arcdeg$ at higher energies confirming the result of PCUBE. For the polpow
model, we obtain a PD strongly increasing with energy (note the negative PD
index $A_{\rm index}\approx-5$) from a PD at 1 keV of just $\sim 10^{-3}$%.
The uncertainties on the parameters are quite large due to the correlation
between PD and PA, but also because the statistical uncertainty of the data
does not allow us to clearly disentangle different models.

##  6 Radio observation

4U 1820$-$303 was observed with the Australia Telescope Compact Array (ATCA)
on 2021 April 15 from 12:49:40 and 21:51:20 UT (under project code CX530).
During this observation, ATCA was in a relatively compact H214 configuration.
The data were recorded simultaneously at two central frequencies, 5.5 and 9
GHz, with 2 GHz of bandwidth at each frequency. We used the unpolarized PKS
B1934$-$638 for primary calibration and to solve for antenna leakages. The
nearby calibrator B1817$-$254 was used for gain calibration. The calibration
and imaging followed standard procedures using the Common Astronomy Software
Applications for radio astronomy (casa, version 5.1.2; CASA Team et al. 2022).
The polarization calibration used the casa task atcapolhelpers.py and
qufromgain routine.333From https://github.com/radio-astro The imaging used a
Briggs robust parameter of 2 to maximize the image sensitivity. The isolated
antenna 6 (located 6 km from the array core) was used during imaging.

While 4U 1820$-$303 was detected at both frequency bands, the radio
counterpart was relatively faint. Fitting for a point source in the image
plane (Stokes $I$), we measured a flux density of $100\pm 12\,\mu$Jy at 5.5
GHz and $80\pm 10\,\mu$Jy at 9 GHz. This corresponds to a radio spectral index
of $-0.45\pm 0.30$. The X-ray brightness, state, and radio spectral index at
the time of the observations suggest that the radio emission originates from
either a quenched compact jet or transient ejecta (see, e.g., Russell et al.
2021). Imaging the field in both Stokes $Q$ and $U$, no significant linear
polarization was detected. We place a 3$\sigma$ upper limits on the fractional
linear polarization of 60% and 70% at both 5.5 and 9 GHz, respectively.
Stacking the two bands gives a 3$\sigma$ upper limit of 50% (at 7.25 GHz).

##  7 Discussion

In this paper, we report the first detection of polarization in the X-rays
with IXPE and a new constraint on radio polarization by ATCA for the atoll
source 4U 1820$-$303\. The spectral analysis performed using the data from
different X-rays observatories confirms the presence of a broad Fe line, as
reported in Cackett et al. (2008), Titarchuk et al. (2013), and Mondal et al.
(2016), in the NuSTAR data. In the radio, the source showed an emission that
is consistent with either a quenched compact jet or a transient ejecta.

IXPE results on the X-ray polarization obtained using the model-independent
ixpeobssim–PCUBE analysis show a different behavior of this source with
respect to the other atoll and Z-sources observed up to now (Farinelli et al.,
2023; Capitanio et al., 2023; Chatterjee et al., 2023; Jayasurya et al., 2023;
Cocchi et al., 2023; Ursini et al., 2023). In particular, this is the first
source showing a strong increase of the polarization with energy, up to 10% in
the 7–8 keV energy band. At low energy, there is no secure detection of
polarization, even if there is a hint of a $\sim$0.8% PD at 96% CL, with a PA
orthogonal to the ones at higher energies.

We also attempted a spectro-polarimetric analysis of the source emission.
Assuming simple polarimetric phenomenological models for the Comptonization
component, a need for non-zero polarization of the disk emission (which
dominates in the lower part of the IXPE band) emerged to account for the
overall low PD. In this scenario, leaving the parameters unconstrained, the
polarization of the disk tends to be orthogonal to the one associated with the
Comptonization. The PD varies, depending on the polarimetric models, from a
few percent up to $\sim$10%. In the lower boundary, these values are
compatible with the polarization expected in case of an electron-scattering
dominated optically thick accretion disk, which is below 2% (Chandrasekhar,
1960; Sobolev, 1963) considering 4U 1820$-$303 inclination – ranging from
$\sim 20\arcdeg$ to 55° (Cackett et al., 2008; Mondal et al., 2016; Anderson
et al., 1997). We also note that, in case of a disk whose opacity is dominated
by electron scattering, the PA is expected to be orthogonal to the disk axis.
Therefore, the PA of the disk component measured by IXPE, which is $\sim
30\arcdeg$ East of North, would indicate that the position angle of the disk
rotation axis is $\simeq-60\arcdeg$ (or 120°).

The Comptonized component is firmly associated with a high and energy
dependent polarization. For an optically thin corona, the PD can reach 10%–20%
in slab geometry (Poutanen & Svensson, 1996; Schnittman & Krolik, 2010;
Gnarini et al., 2022; Poutanen et al., 2023). However, the Thomson optical
depth $\tau_{\rm T}$, required to produce the observed spectrum with the
photon index of $\Gamma\approx 1.9$ and electron temperature $T_{\rm e}$ of
$\sim$3 keV, exceeds 10. For such a high optical depth, the PD is even smaller
than that given by the classical results of Chandrasekhar (1960) and Sobolev
(1963) for $\tau_{\rm T}\gg 1$ (see Figure 5 in Sunyaev & Titarchuk, 1985).
Thus, the $\sim$10% polarization observed by IXPE at highest energies is
difficult to explain for any reasonable inclination. Therefore, our results
require some non-standard coronal geometry.

One of the possibilities is the presence of a mildly relativistic outflow from
the inner part of the accretion flow where Comptonization takes place. Because
of relativistic aberration, the PD at a given inclination becomes larger than
that in the static corona with the PA parallel to the disk normal
(Beloborodov, 1998; Poutanen et al., 2023). For an optically thin outflow, the
PD of the scattered component can reach 15%, but the PD of the total emission
will be reduced by unscattered radiation. Thus, we are forced to assume that
the outflow is optically thick; in this case, the total PD can reach 10%–15%
even at moderate inclinations of 40°–60° (with the PA being still parallel to
the disk normal, see Figure 4 in Beloborodov, 1998). The remaining question is
why the PD is high only in the 7–8 keV band, but not below.

An alternative to the outflow would be the presence of a strong reflection
already claimed to be present in previous weakly magnetized neutron stars
observed by IXPE (Farinelli et al., 2023; Gnarini et al., 2022; Cocchi et al.,
2023; Ursini et al., 2023). As already discussed in Sect. 4, such a component
was investigated, and its relative flux in the IXPE band is estimated to be
below 5% even above 4 keV. Because a typical PD for the reflected component is
about 20% (Matt, 1993; Poutanen et al., 1996), it cannot alone easily explain
the observed PD of 10% above 7 keV and its fast rise with energy.
Notwithstanding, adding a polarized contribution from the reflection component
aligned with that of the corona could partially reduce the PD of the
Comptonized component to a level more easily explained with standard
geometries.

Overall, the spectro-polarimetric modeling support a geometry having a SL
perpendicular to the disk plane, already described in the context of previous
IXPE observations of atoll sources, with the important difference of the PD
for the Comptonized component which is about 10% for 4U 1820$-$303 instead of
a non-detection for GS 1826$-$238 (Capitanio et al., 2023), $\simeq 3$% for GX
9+9 and Cyg X-2 in the energy band 4–8 keV (Farinelli et al., 2023; Ursini et
al., 2023) and $\simeq 5$% for XTE J1701$-$462 in the 4–8 keV band in the
horizontal branch. This can be contrasted to a $\simeq$50% of Comptonized
fraction in GS 1826$-$238 (Capitanio et al., 2023) and $<70$% in GX 9+9, where
the reflection component is well determined (Ursini et al., 2023). This might
imply a correlation between the corona dominated spectrum and the high PD, but
further observations are needed to disentangle the polarization associated
with each spectral component.

Linearly polarized radio emission was not detected, with an upper limit at
99.73% CL of 50% on the PD in the radio band; unfortunately, this result does
not allow us to do a direct comparison between the PA in the radio and the
X-rays. As such, due to the radio faintness, we cannot compare our result to
the one obtained for Cyg X-2 (Farinelli et al., 2023) and Sco X-1 (Long et
al., 1979, 2022), where the X-ray PA was aligned with the direction of the
radio jets.

##  8 Summary

The IXPE result for 4U 1820$-$303 shows a different polarization behavior with
respect to other atoll sources, GX 9+9 (Ursini et al., 2023; Chatterjee et
al., 2023) and GS 1826$-$238 (Capitanio et al., 2023), but also with respect
to the Z-sources Cyg X-2 (Farinelli et al., 2023) and XTE J1701$-$462
(Jayasurya et al., 2023; Cocchi et al., 2023). In particular, the need to have
a polarized disk emission is more evident. Moreover, the model-independent
analysis gives an indication of a 90° rotation of the PA at lowest energies,
where the disk contributes the most. A very high PD of about 10% detected
above 7 keV did not find an obvious explanation within standard models of
X-ray emission of weakly magnetized NSs. One possibility is that the
polarization is produced in an optically thick outflow emanating from the
inner part of the accretion disk. Future studies of this source with a longer
exposure could help to disentangle the polarization of each spectral
component. Monitoring the source along its superorbital period or when it
moves along the banana and island states could help to better understand the
polarization variations and, thanks to this, verify possible geometry
variations as a function of the mass accretion rate.

## Acknowledgments

The Imaging X-ray Polarimetry Explorer (IXPE) is a joint US and Italian
mission. The US contribution is supported by the National Aeronautics and
Space Administration (NASA) and led and managed by its Marshall Space Flight
Center (MSFC), with industry partner Ball Aerospace (contract NNM15AA18C). The
Italian contribution is supported by the Italian Space Agency (Agenzia
Spaziale Italiana, ASI) through contract ASI-OHBI-2017-12-I.0, agreements ASI-
INAF-2017-12-H0 and ASI-INFN-2017.13-H0, and its Space Science Data Center
(SSDC) with agreements ASI-INAF-2022-14-HH.0 and ASI-INFN 2021-43-HH.0, and by
the Istituto Nazionale di Astrofisica (INAF) and the Istituto Nazionale di
Fisica Nucleare (INFN) in Italy. This research used data products provided by
the IXPE Team (MSFC, SSDC, INAF, and INFN) and distributed with additional
software tools by the High-Energy Astrophysics Science Archive Research Center
(HEASARC), at NASA Goddard Space Flight Center (GSFC).

This research has made use of the MAXI data provided by RIKEN, JAXA, and the
MAXI team and of the Swift/BAT transient monitor results provided by the
Swift/BAT team. We thank Jamie Stevens and ATCA staff for the scheduling of
these observations. ATCA is part of the Australia Telescope National Facility
(https://ror.org/05qajvd42), which is funded by the Australian Government for
operation as a National Facility managed by CSIRO. We acknowledge the Gomeroi
people as the Traditional Owners of the ATCA observatory site.

The authors acknowledge the prompt schedules of the simultaneous observations:
Keith G. Gendrau, Zaven Arzoumanian, and the NICER SOC team; Boris Sbarufatti,
Kim Page, Brad Cenko, and Swift-XRT SOC team; Karl Forster, Murray Brightman,
Fiona A. Harrison and NuSTAR SOC team. We also acknowledge support from the
Academy of Finland grants 333112, 349144, 349373, and 349906 (JP, SST) and the
German Academic Exchange Service (DAAD) travel grant 57525212 (VD).

##  Appendix A Data handling for X-ray observatories

In this study, as we reported in the previous sections, IXPE observations were
coordinated with NuSTAR, NICER, and Swift-XRT aiming to verify the state of
the source and also to obtain a better constraint on the spectral model thanks
to better energy resolution and broadband coverage. In this section, we
briefly report the data handling and extraction applied to the data we used.

###  A.1 IXPE

The Imaging X-ray Polarimetry Explorer (IXPE), a NASA mission in partnership
with the Italian Space Agency, was launched on 2021 December 9. A detailed
description of the observatory and its performance is given in Weisskopf et
al. (2022). IXPE consists of three identical grazing incidence telescopes,
providing imaging and spectral polarimetry over the 2–8 keV energy band with a
time resolution better than 10 $\mu$s. Each telescope comprises an X-ray
module of mirror assembly (MMA) and a polarization-sensitive detector unit
(DU) equipped with a gas-pixel detector (GPD) (Costa et al., 2001; Soffitta et
al., 2021).

IXPE observed 4U 1820$-$303 twice, in October 2022 for a short period and for
a longer one in April 2023 (ObsID 02002399) with a total effective exposure of
16 ks and 86 ks per DU, respectively. Level 2 data were processed with the
IXPE dedicated software ixpeobssim (Baldini et al., 2022) v. 30.3.0 and with
ftool released in HEASOFT v. 6.31.1 using the Calibration database released on
2022 November 17.

Source photons were selected from the IXPE telescopes images collected within
a circular region having radius 100″ centered on the source position, while
the background, following the prescription by Di Marco et al. (2023) for the
sources with high count rate ($\gtrsim 2$ cnt s-1), has not been subtracted
from the data, because it is negligible (in these observations, the background
is roughly two orders of magnitude below the source in all energy bins).

For the spectral analysis, the flux (Stokes parameter $I$) energy spectra from
the three DUs have been extracted following the weighted approach described in
Di Marco et al. (2022a) and data were binned to have at least 30 counts per
energy channel, while in the spectro-polarimetric analysis, we applied a
constant energy binning of 200 eV for all the three energy distributions of
Stokes parameters $I$, $Q$, and $U$.

###  A.2 NICER

The Neutron Star Interior Composition Explorer (NICER) is a soft X-ray
instrument onboard the International Space Station (ISS), launched in June
2017, and it consists of 56 co-aligned concentrator X-ray optics, each of
which is paired with a single silicon drift detector. NICER does not offer
imaging capabilities, but offers a large collecting area providing unmatched
time resolution in the soft X-ray bandpass, with a sensitive energy interval
0.2–12 keV.

Contemporaneous observations of 4U 1820$-$303 during both IXPE observations
were performed in the frame of the NICER GO Cycle 5 (proposal 6189 having PI
Alessandro Di Marco). During the first observation, NICER observed the source
for a total exposure of $\simeq$1.7 ks (Observation ID 5050300117), while
during the second observation, two data-sets were released (Observation ID
6689020101 and 6689020102) covering April 15 and 16 in several snapshots for a
total exposure time of $\simeq$10.3 ks and $\simeq$13.7 ks respectively.

The NICER data were processed with the NICER Data Analysis Software v010a
released on 2022 December 16 provided under HEASOFT v6.31.1 with the CALDB
version released on 2022 October 30. The background spectra have been
estimated by applying the new SCORPEON model in the 0.22–15 keV band. All the
obtained spectra were grouped to have at least 30 counts per bin.

###  A.3 Swift-XRT

The Neil Gehrels Swift Observatory (Gehrels et al., 2004) carries three
instruments to enable the most detailed observations of gamma-ray bursts to
date; the X-ray Telescope (XRT) is one of them, based on a sensitive,
flexible, autonomous X-ray CCD imaging spectrometer. Swift-XRT coordinated
observations with IXPE for 4U 1820$-$303 were performed; they are used in this
work to monitor the source status and to obtain spectral information during
both the IXPE observations. Given the source brightness, the Swift-XRT
observations were performed in Windowed Timing mode (WT). Four Swift-XRT
pointings covered the first IXPE observation in the period 2022 October 11–13
(ObsIDs: 00014980028, 00014980029, 00014980030, 00014980031), while the second
IXPE observation consisted of 5 pointings on 2023 April 15 and 16 grouped in 3
data-sets (ObsIDs: 00014980055, 00014980056 and 00014980057). These data have
been used to monitor the source HR along the two IXPE observations, as
reported in Figure 4. The data were extracted using xselect v2.5b released
with HEASOFT v6.31.1. The source and background extractions were performed
using the imaging capabilities of Swift-XRT. Given the source counting rate
$\geq$100 count s-1, data can be affected by pile-up as reported in the
appendix of Romano et al. (2006).

###  A.4 NuSTAR

NuSTAR, the Nuclear Spectroscopic Telescope Array, observatory (Harrison et
al., 2013) consists of two identical X-ray telescope modules, referred to as
FPMA and FPMB, providing broadband X-ray imaging, spectroscopy, and timing in
the energy range of 3–79 keV with an angular resolution of 18″ (FWHM) and
spectral resolution of 400 eV (FWHM) at 10 keV During both the IXPE
observations of 4U 1820$-$303, NuSTAR performed coordinated observations, one
(ObsID: 90802327002) from 2022 October 12 at 14:31 UTC to October 13 at 2:41
UTC with a total exposure of $\simeq$16.9 ks and other two on 2023 April 15
and 16 (ObsIDs: 90902308002 and 90902308004) with a total exposure of
$\simeq$16.8 ks and $\simeq$15.2 ks, respectively.

The NuSTAR data were processed with the standard Data Analysis Software
(nustardas 16Feb22 v2.1.2) provided under HEASOFT v6.31.1 with the CALDB
version released on 2023 April 4. A circular 150″ radius region has been used
for both source and background spectra extractions, with the source region
centered on the locations of 4U 1820$-$303 and the background one in an off-
center sourceless region in the detector image. All the obtained spectra were
grouped to have at least 30 counts per bin.

## References

  * Anderson et al. (1997) Anderson, S. F., Margon, B., Deutsch, E. W., Downes, R. A., & Allen, R. G. 1997, ApJ, 482, L69, doi: 10.1086/310672
  * Arnaud (1996) Arnaud, K. A. 1996, in ASP Conf. Ser., Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. Jacoby & J. Barnes (San Francisco: Astron. Soc. Pac.), 17–20
  * Bahramian & Degenaar (2022) Bahramian, A., & Degenaar, N. 2022, Low-Mass X-ray Binaries, ed. C. Bambi & A. Santangelo (Singapore: Springer Nature Singapore), 1–62, doi: 10.1007/978-981-16-4544-0_94-1
  * Baldini et al. (2022) Baldini, L., Bucciantini, N., Lalla, N. D., et al. 2022, SoftwareX, 19, 101194, doi: 10.1016/j.softx.2022.101194
  * Baumgardt & Vasiliev (2021) Baumgardt, H., & Vasiliev, E. 2021, MNRAS, 505, 5957, doi: 10.1093/mnras/stab1474
  * Beloborodov (1998) Beloborodov, A. M. 1998, ApJ, 496, L105, doi: 10.1086/311260
  * Blackburn (1995) Blackburn, J. K. 1995, in Astronomical Society of the Pacific Conference Series, Vol. 77, Astronomical Data Analysis Software and Systems IV, ed. R. A. Shaw, H. E. Payne, & J. J. E. Hayes, 367
  * Burns et al. (2023) Burns, E., Svinkin, D., Fenimore, E., et al. 2023, ApJ, 946, L31, doi: 10.3847/2041-8213/acc39c
  * Cackett et al. (2008) Cackett, E. M., Miller, J. M., Bhattacharyya, S., et al. 2008, ApJ, 674, 415, doi: 10.1086/524936
  * Capitanio et al. (2023) Capitanio, F., Fabiani, S., Gnarini, A., et al. 2023, ApJ, 943, 129, doi: 10.3847/1538-4357/acae88
  * CASA Team et al. (2022) CASA Team, Bean, B., Bhatnagar, S., et al. 2022, PASP, 134, 114501, doi: 10.1088/1538-3873/ac9642
  * Chandrasekhar (1960) Chandrasekhar, S. 1960, Radiative Transfer (New York: Dover)
  * Chatterjee et al. (2023) Chatterjee, R., Agrawal, V. K., Jayasurya, K. M., & Katoch, T. 2023, MNRAS, 521, L74, doi: 10.1093/mnrasl/slad026
  * Chou & Grindlay (2001) Chou, Y., & Grindlay, J. E. 2001, ApJ, 563, 934, doi: 10.1086/324038
  * Cocchi et al. (2023) Cocchi, M., Gnarini, A., Fabiani, S., et al. 2023, A&A, 674, L10, doi: 10.1051/0004-6361/202346275
  * Costa et al. (2001) Costa, E., Soffitta, P., Bellazzini, R., et al. 2001, Nature, 411, 662, doi: 10.1038/35079508
  * Curran et al. (2014) Curran, P. A., Coriat, M., Miller-Jones, J. C. A., et al. 2014, MNRAS, 437, 3265, doi: 10.1093/mnras/stt2125
  * D’Aì et al. (2010) D’Aì, A., di Salvo, T., Ballantyne, D., et al. 2010, A&A, 516, A36, doi: 10.1051/0004-6361/200913758
  * Di Marco et al. (2022a) Di Marco, A., Costa, E., Muleri, F., et al. 2022a, AJ, 163, 170, doi: 10.3847/1538-3881/ac51c9
  * Di Marco et al. (2022b) Di Marco, A., Muleri, F., Fabiani, S., et al. 2022b, in Proc. SPIE, Vol. 12181, Space Telescopes and Instrumentation 2022: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, S. Nikzad, & K. Nakazawa, 121811C, doi: 10.1117/12.2629413
  * Di Marco et al. (2023) Di Marco, A., Soffitta, P., Costa, E., et al. 2023, AJ, 165, 143, doi: 10.3847/1538-3881/acba0f
  * Díaz Trigo et al. (2017) Díaz Trigo, M., Migliari, S., Miller-Jones, J. C. A., et al. 2017, A&A, 600, A8, doi: 10.1051/0004-6361/201629472
  * Egron et al. (2013) Egron, E., Di Salvo, T., Motta, S., et al. 2013, A&A, 550, A5, doi: 10.1051/0004-6361/201219675
  * Farinelli et al. (2023) Farinelli, R., Fabiani, S., Poutanen, J., et al. 2023, MNRAS, 519, 3681, doi: 10.1093/mnras/stac3726
  * Gehrels et al. (2004) Gehrels, N., Chincarini, G., Giommi, P., et al. 2004, ApJ, 611, 1005, doi: 10.1086/422091
  * Gnarini et al. (2022) Gnarini, A., Ursini, F., Matt, G., et al. 2022, MNRAS, 514, 2561, doi: 10.1093/mnras/stac1523
  * Grindlay et al. (1976) Grindlay, J., Gursky, H., Schnopper, H., et al. 1976, ApJ, 205, L127, doi: 10.1086/182105
  * Haardt & Matt (1993) Haardt, F., & Matt, G. 1993, MNRAS, 261, 346, doi: 10.1093/mnras/261.2.346
  * Harrison et al. (2013) Harrison, F. A., Craig, W. W., Christensen, F. E., et al. 2013, ApJ, 770, 103, doi: 10.1088/0004-637X/770/2/103
  * Hasinger & van der Klis (1989) Hasinger, G., & van der Klis, M. 1989, A&A, 225, 79
  * Hughes et al. (1984) Hughes, J. P., Long, K. S., & Novick, R. 1984, ApJ, 280, 255, doi: 10.1086/161992
  * Iaria et al. (2016) Iaria, R., Di Salvo, T., Del Santo, M., et al. 2016, A&A, 596, A21, doi: 10.1051/0004-6361/201628210
  * Inogamov & Sunyaev (1999) Inogamov, N. A., & Sunyaev, R. A. 1999, Astronomy Letters, 25, 269.  https://arxiv.org/abs/astro-ph/9904333
  * Jayasurya et al. (2023) Jayasurya, K. M., Agrawal, V. K., & Chatterjee, R. 2023, arXiv e-prints, arXiv:2302.03396, doi: 10.48550/arXiv.2302.03396
  * Krimm et al. (2013) Krimm, H. A., Holland, S. T., Corbet, R. H. D., et al. 2013, ApJS, 209, 14, doi: 10.1088/0067-0049/209/1/14
  * Lapidus & Sunyaev (1985) Lapidus, I. I., & Sunyaev, R. A. 1985, MNRAS, 217, 291, doi: 10.1093/mnras/217.2.291
  * Long et al. (1979) Long, K. S., Chanan, G. A., Ku, W. H. M., & Novick, R. 1979, ApJ, 232, L107, doi: 10.1086/183045
  * Long et al. (2022) Long, X., Feng, H., Li, H., et al. 2022, ApJ, 924, L13, doi: 10.3847/2041-8213/ac4673
  * Longair (2011) Longair, M. S. 2011, High Energy Astrophysics (Cambridge, UK: Cambridge University Press)
  * Loskutov & Sobolev (1982) Loskutov, V. M., & Sobolev, V. V. 1982, Astrofizika, 18, 81
  * Madsen et al. (2022) Madsen, K. K., Forster, K., Grefenstette, B., Harrison, F. A., & Miyasaka, H. 2022, JATIS, 8, 034003, doi: 10.1117/1.JATIS.8.3.034003
  * Matsuoka et al. (2009) Matsuoka, M., Kawasaki, K., Ueno, S., et al. 2009, PASJ, 61, 999, doi: 10.1093/pasj/61.5.999
  * Matt (1993) Matt, G. 1993, MNRAS, 260, 663, doi: 10.1093/mnras/260.3.663
  * Migliari et al. (2004) Migliari, S., Fender, R. P., Rupen, M., et al. 2004, MNRAS, 351, 186, doi: 10.1111/j.1365-2966.2004.07768.x
  * Mondal et al. (2016) Mondal, A. S., Dewangan, G. C., Pahari, M., et al. 2016, MNRAS, 461, 1917, doi: 10.1093/mnras/stw1464
  * Popham & Sunyaev (2001) Popham, R., & Sunyaev, R. 2001, ApJ, 547, 355, doi: 10.1086/318336
  * Poutanen et al. (1996) Poutanen, J., Nagendra, K. N., & Svensson, R. 1996, MNRAS, 283, 892, doi: 10.1093/mnras/283.3.892
  * Poutanen & Svensson (1996) Poutanen, J., & Svensson, R. 1996, ApJ, 470, 249, doi: 10.1086/177865
  * Poutanen et al. (2023) Poutanen, J., Veledina, A., & Beloborodov, A. M. 2023, ApJ, 949, L10, doi: 10.3847/2041-8213/acd33e
  * Poutanen & Vilhu (1993) Poutanen, J., & Vilhu, O. 1993, A&A, 275, 337
  * Rappaport et al. (1987) Rappaport, S., Nelson, L. A., Ma, C. P., & Joss, P. C. 1987, ApJ, 322, 842, doi: 10.1086/165778
  * Romano et al. (2006) Romano, P., Campana, S., Chincarini, G., et al. 2006, A&A, 456, 917, doi: 10.1051/0004-6361:20065071
  * Russell et al. (2021) Russell, T. D., Degenaar, N., van den Eijnden, J., et al. 2021, MNRAS, 508, L6, doi: 10.1093/mnrasl/slab087
  * Schnittman & Krolik (2010) Schnittman, J. D., & Krolik, J. H. 2010, ApJ, 712, 908, doi: 10.1088/0004-637X/712/2/908
  * Shakura & Sunyaev (1988) Shakura, N. I., & Sunyaev, R. A. 1988, Advances in Space Research, 8, 135, doi: 10.1016/0273-1177(88)90396-1
  * Shaposhnikov & Titarchuk (2004) Shaposhnikov, N., & Titarchuk, L. 2004, ApJ, 606, L57, doi: 10.1086/421015
  * Smale et al. (1997) Smale, A. P., Zhang, W., & White, N. E. 1997, ApJ, 483, L119, doi: 10.1086/310749
  * Sobolev (1963) Sobolev, V. V. 1963, A Treatise on Radiative Transfer (Princeton: Van Nostrand)
  * Soffitta et al. (2021) Soffitta, P., Baldini, L., Bellazzini, R., et al. 2021, AJ, 162, 208, doi: 10.3847/1538-3881/ac19b0
  * Stella et al. (1987) Stella, L., Priedhorsky, W., & White, N. E. 1987, ApJ, 312, L17, doi: 10.1086/184811
  * Suleimanov & Poutanen (2006) Suleimanov, V., & Poutanen, J. 2006, MNRAS, 369, 2036, doi: 10.1111/j.1365-2966.2006.10454.x
  * Sunyaev & Titarchuk (1985) Sunyaev, R. A., & Titarchuk, L. G. 1985, A&A, 143, 374
  * Tarana et al. (2007) Tarana, A., Bazzano, A., Ubertini, P., & Zdziarski, A. A. 2007, ApJ, 654, 494, doi: 10.1086/508744
  * Titarchuk et al. (2013) Titarchuk, L., Seifina, E., & Frontera, F. 2013, ApJ, 767, 160, doi: 10.1088/0004-637X/767/2/160
  * Ursini et al. (2023) Ursini, F., Farinelli, R., Gnarini, A., et al. 2023, A&A, in press, arXiv:2306.02740, doi: 10.1051/0004-6361/202346541
  * van der Klis (1989) van der Klis, M. 1989, ARA&A, 27, 517, doi: 10.1146/annurev.aa.27.090189.002505
  * Weisskopf et al. (2022) Weisskopf, M. C., Soffitta, P., Baldini, L., et al. 2022, JATIS, 8, 026002, doi: 10.1117/1.JATIS.8.2.026002
  * Wilms et al. (2000) Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914, doi: 10.1086/317016
  * Zdziarski et al. (2007) Zdziarski, A. A., Gierliński, M., Wen, L., & Kostrzewa, Z. 2007, MNRAS, 377, 1017, doi: 10.1111/j.1365-2966.2007.11688.x