11institutetext: Université de Strasbourg, CNRS, Observatoire Astronomique de
Strasbourg, UMR 7550, 67000 Strasbourg, France 22institutetext: ASI - Agenzia
Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy 33institutetext:
Dipartimento di Fisica, Università di Roma “Tor Vergata”, Via della Ricerca
Scientifica 1, 00133 Roma, Italy 44institutetext: Space Science Data Center,
SSDC, ASI, Via del Politecnico snc, 00133 Roma, Italy 55institutetext: INAF
Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere
100, I-00133 Roma, Italy 66institutetext: Dipartimento di Fisica, Università
degli Studi di Roma “La Sapienza”, Piazzale Aldo Moro 5, I-00185 Roma, Italy
77institutetext: Dipartimento di Matematica e Fisica, Università degli Studi
Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy 88institutetext: MIT
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
99institutetext: Science and Technology Institute, Universities Space Research
Association, Huntsville, AL 35805, USA 1010institutetext: Université Grenoble
Alpes, CNRS, IPAG, 38000 Grenoble, France 1111institutetext: School of
Mathematics, Statistics, and Physics, Newcastle University, Newcastle upon
Tyne NE1 7RU, UK 1212institutetext: Center for Astrophysics, Harvard &
Smithsonian, 60 Garden St., Cambridge, MA 02138, USA 1313institutetext:
Astronomical Institute of the Czech Academy of Sciences, Boční II 1401/1,
14100 Praha 4, Czech Republic 1414institutetext: Istituto Nazionale di Fisica
Nucleare, Sezione di Roma ”Tor Vergata”, Via della Ricerca Scientifica 1,
00133 Roma, Italy 1515institutetext: Department of Astronomy, University of
Maryland, College Park, Maryland 20742, USA 1616institutetext: Instituto de
Astrofísica de Andalucía—CSIC, Glorieta de la Astronomía s/n, 18008 Granada,
Spain 1717institutetext: INAF Osservatorio Astronomico di Roma, Via Frascati
33, 00078 Monte Porzio Catone (RM), Italy 1818institutetext: INAF Osservatorio
Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy
1919institutetext: Istituto Nazionale di Fisica Nucleare, Sezione di Pisa,
Largo B. Pontecorvo 3, 56127 Pisa, Italy 2020institutetext: Dipartimento di
Fisica, Università di Pisa, Largo B. Pontecorvo 3, 56127 Pisa, Italy
2121institutetext: NASA Marshall Space Flight Center, Huntsville, AL 35812,
USA 2222institutetext: Istituto Nazionale di Fisica Nucleare, Sezione di
Torino, Via Pietro Giuria 1, 10125 Torino, Italy 2323institutetext:
Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria 1,
10125 Torino, Italy 2424institutetext: INAF Osservatorio Astrofisico di
Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy 2525institutetext:
Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via
Sansone 1, 50019 Sesto Fiorentino (FI), Italy 2626institutetext: Istituto
Nazionale di Fisica Nucleare, Sezione di Firenze, Via Sansone 1, 50019 Sesto
Fiorentino (FI), Italy 2727institutetext: Department of Physics and Kavli
Institute for Particle Astrophysics and Cosmology, Stanford University,
Stanford, California 94305, USA 2828institutetext: 1Institut für Astronomie
und Astrophysik, Universität Tübingen, Sand 1, 72076 Tübingen, Germany
2929institutetext: RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako,
Saitama 351-0198, Japan 3030institutetext: California Institute of Technology,
Pasadena, CA 91125, USA 3131institutetext: Yamagata University, 1-4-12
Kojirakawamachi, Yamagatashi 990-8560, Japan 3232institutetext: University of
British Columbia, Vancouver, BC V6T 1Z4, Canada 3333institutetext:
International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan 3434institutetext: Institute for Astrophysical Research,
Boston University, 725 Commonwealth Avenue, Boston, MA 02215, USA
3535institutetext: Department of Astrophysics, St. Petersburg State
University, Universitetsky pr. 28, Petrodvoretz, 198504 St. Petersburg, Russia
3636institutetext: Department of Physics and Astronomy and Space Science
Center, University of New Hampshire, Durham, NH 03824, USA 3737institutetext:
Physics Department and McDonnell Center for the Space Sciences, Washington
University in St. Louis, St. Louis, MO 63130, USA 3838institutetext: Finnish
Centre for Astronomy with ESO, 20014 University of Turku, Finland
3939institutetext: Department of Physics and Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, Stanford, California 94305,
USA 4040institutetext: Graduate School of Science, Division of Particle and
Astrophysical Science, Nagoya University, Furocho, Chikusaku, Nagoya, Aichi
464-8602, Japan 4141institutetext: Hiroshima Astrophysical Science Center,
Hiroshima University, 1- 3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima
739-8526, Japan 4242institutetext: University of Maryland, Baltimore County,
Baltimore, MD 21250, USA 4343institutetext: NASA Goddard Space Flight Center,
Greenbelt, MD 20771, USA 4444institutetext: Center for Research and
Exploration in Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771,
USA 4545institutetext: Department of Physics, The University of Hong Kong,
Pokfulam, Hong Kong 4646institutetext: Department of Astronomy and
Astrophysics, Pennsylvania State University, University Park, PA 16802, USA
4747institutetext: Department of Physics and Astronomy, 20014 University of
Turku, Finland 4848institutetext: INAF Osservatorio Astronomico di Brera, Via
E. Bianchi 46, 23807 Merate (LC), Italy 4949institutetext: Dipartimento di
Fisica e Astronomia, Università degli Studi di Padova, Via Marzolo 8, 35131
Padova, Italy 5050institutetext: Dipartimento di Fisica e Astronomia,
Università degli Studi di Padova, Via Marzolo 8, 35131 Padova, Italy
5151institutetext: Mullard Space Science Laboratory, University College
London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 5252institutetext: Anton
Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam, Science
Park 904, 1098 XH Amsterdam, The Netherlands 5353institutetext: Guangxi Key
Laboratory for Relativistic Astrophysics, School of Physical Science and
Technology, Guangxi University, Nanning 530004, China

# X-ray polarization measurement of the gold standard of radio-quiet active
galactic nuclei : NGC 1068

F. Marin E-mail<EMAIL_ADDRESS>A. Marinucci 22 M. Laurenti
33441717 D. E. Kim 556633 T. Barnouin 11 A. Di Marco 55 F. Ursini 77 S.
Bianchi 77 S. Ravi 88 H.L. Marshall 88 G. Matt 77 C.-T. Chen 99 V. E. Gianolli
101077 A. Ingram 1111 W. P. Maksym 1212 C. Panagiotou 88 J. Podgorny 1313 S.
Puccetti 44 A. Ratheesh 55 F. Tombesi 3314141515 I. Agudo 1616 L. A. Antonelli
441717 M. Bachetti 1818 L. Baldini 19192020 W. Baumgartner 2121 R. Bellazzini
1919 S. Bongiorno 2121 R. Bonino 22222323 A. Brez 1919 N. Bucciantini
242425252626 F. Capitanio 55 S. Castellano 1919 E. Cavazzuti 22 S.Ciprini
441414 E. Costa 55 A. De Rosa 55 E. Del Monte 55 L. Di Gesu 22 N. Di Lalla
2727 I. Donnarumma 22 V. Doroshenko 2828 M. Dovčiak 1313 S. Ehlert 2121 T.
Enoto 2929 Y. Evangelista 55 S. Fabiani 55 R. Ferrazzoli 55 J. Garcia 3030 S.
Gunji 3131 J. Heyl 3232 W. Iwakiri 3333 S. Jorstad 34343535 P. Kaaret 2121 V.
Karas 1313 F. Kislat 3636 T. Kitaguchi 2929 J. Kolodziejczak 2121 H.
Krawczynski 3737 F. La Monaca 553366 L. Latronico 2222 I. Liodakis 3838 G.
Madejski 3939 S. Maldera 2222 A. Manfreda 1919 A. Marscher 3434 F. Massaro
22222323 I. Mitsuishi 4040 T. Mizuno 4141 F. Muleri 55 M. Negro 424243434444
S. Ng 4545 S. O’Dell 2121 N. Omodei 3939 C. Oppedisano 2222 A. Papitto 1717 G.
Pavlov 4646 M. Perri 441717 M. Pesce-Rollins 1919 P.-O. Petrucci 1010 M. Pilia
1818 A. Possenti 1818 J. Poutanen 4747 B. Ramsey 2121 J. Rankin 55 O. Roberts
99 R. Romani 3939 C. Sgrò 1919 P. Slane 1212 P. Soffitta 55 G. Spandre 1919 D.
Swartz 99 T. Tamagawa 2929 F. Tavecchio 4848 R. Taverna 4949 Y. Tawara 4040 A.
Tennant 2121 N. Thomas 2121 A. Trois 1818 S. Tsygankov 4747 R. Turolla
50505151 J. Vink 5252 M. Weisskopf 2121 K. Wu 5151 F. Xie 535355 S. Zane 5151

(Received February X, 2024; accepted Month Day, 2024)

###### Abstract

Context. NGC 1068 is the radio-quiet active galactic nucleus (AGN) which has
been the most observed in photo-, spectro-, imaging and timing polarimetry. It
led to the unified scheme of AGNs in the 90’s, a global, zeroth-order picture
of AGNs that is still widely used.

Aims. However, high energy polarimetry has never been probed before due to the
lack of dedicated polarimeters. This is critical because X-rays allow us to
probe the orientation and geometric arrangement of (sub)parsec-scale matter
without being polluted by the host emission.

Methods. We used the Imaging X-ray Polarimetry Explorer (IXPE) satellite to
measure, for the first time, the 2-8 keV polarization of NGC 1068. We pointed
IXPE for a net exposure time of 1.15 Ms on the target, in addition to two
$\sim$ 10 ks each Chandra snapshots in order to account for the potential
impact of several ultraluminous X-ray source (ULXs) within IXPE’s field-of-
view.

Results. We measured a 2 – 8 keV polarization degree of 12.4% $\pm$ 3.6% and
an electric vector polarization angle of 101∘ $\pm$ 8∘ at 68% confidence
level. If we exclude the spectral region containing the bright Fe K lines and
other soft X-ray lines where depolarization occurs, the polarization fraction
rises up to 21.3% $\pm$ 6.7% in the 3.5 – 6.0 keV band, with a similar
polarization angle. The observed polarization angle is found to be
perpendicular to the parsec scale radio jet. Using a combined Chandra and IXPE
analysis plus multi-wavelength constraints, we estimated that the
circumnuclear ”torus” may sustain a half-opening angle of 50 – 55∘ (from the
vertical axis of the system).

Conclusions. Thanks to IXPE, we have measured the X-ray polarization of NGC
1068 and found comparable results, both in terms of polarization angle
orientation with respect to the radio-jet and torus half-opening angle, to the
X-ray polarimetric measurement achieved for the other archetypal Compton-thick
AGN : the Circinus galaxy. Probing the geometric arrangement of parsec-scale
matter in extragalactic object is now feasible thanks to X-ray polarimetry.

###### Key Words.:

polarization – X-rays: galaxies – X-rays: individuals: NGC 1068 – galaxies:
active – galaxies: Seyfert

##  1 Introduction

NGC 1068, also known as Messier 77 (M77), is an optically bright (V = 11.8 mag
in a 4.9′′ aperture, Sandage 1973) type-2 active galactic nuclei (AGN),
located in the nearby Universe ($z$ = 0.00379). Because of its proximity and
brightness, it is among the AGNs that have been the most studied over the past
century. In particular, it is the source that is at the origin of the Unified
Scheme of AGNs such as established by Antonucci (1993) and Urry & Padovani
(1995). In this zeroth-order model, all AGNs are similar in morphology, but
their apparent observational properties can differ depending on the
inclination of the AGN core with respect to the observer. This is mainly due
to the fact that, along the equatorial plane of the AGN, lies an optically-
thick circumnuclear reservoir of gas and dust that creates a strong geometric
asymmetry in emission and absorption. The model has evolved since then,
encompassing evolution and accretion to explain several observational
differences that orientation cannot explicate alone (Dopita, 1997), but
determining the real inclination and morphology of AGNs remains a challenge.

In the case of NGC 1068, there seems to be a consensus about its nucleus
inclination. Studies based on the modelling of the spectral energy
distribution usually find a (torus) inclination of the order of 70 – 90∘
(Hönig et al., 2007; Hönig, Prieto, & Beckert, 2008; Lopez-Rodriguez et al.,
2018). Observations and modeling of the polar outflows kinematics bring more
extreme inclination values, $\sim$ 85∘, under the assumption of biconical
outflow models (Das et al., 2006; Fischer et al., 2013; Miyauchi & Kishimoto,
2020). Interestingly, Fig. 8b in Miyauchi & Kishimoto (2020) shows the 0.1′′
($\sim$ 7 pc) scale distribution of the gas clouds in NGC 1068 that has been
essentially confirmed by the VLTI/MATISSE reconstructed image obtained by
Gámez Rosas et al. (2022), who also estimate that the circumnuclear, optically
thick, dusty region is seen nearly edge-on. Finally, the presence of water
masers coexisting with the central engine in NGC 1068 can be used to trace the
structure and velocity field of the torus, provided that the dusty molecular
torus is indeed responsible for the maser emission (Greenhill et al., 1996).
VLA observations and Effelsberg 100m monitoring allowed to estimate that the
maser disk is rather thin and inclined by 80 – 90∘ (Gallimore et al., 2001).
VLBI observations of the same water maser also allowed to estimate the central
black hole mass ($\sim$ 107 M⊙, Greenhill et al. 1996).

Because polarization is sensitive to the morphology of a source, it can be
used as an independent tool to determine the orientation of AGNs. Using three
dimensional models of the system, Packham et al. (1997) and Kishimoto (1999)
attempted to reproduce their near-infrared and near-ultraviolet polarization
maps of NGC 1068, respectively. Both found that the northern outflowing region
is inclined towards the observer, while the southern wind is directed away,
but could not determine the AGN inclination nor the torus geometry. The main
problem with this approach is that the ultraviolet, optical and infrared bands
are heavily polluted by unpolarized starlight emission from the host and/or
dust reemission by the torus. To get around this problem, it is necessary to
go to higher energies, where there is neither stellar emission nor re-emission
by hot dust. Thanks to the very low contribution of polluting X-ray sources,
and therefore to a rather clean observational window, X-ray polarimetry offers
a powerful probe to narrow down the uncertainties in determining the AGN
inclination, together with its torus half-opening angle and density (Goosmann
& Matt, 2011; Marin et al., 2018).

It is now possible to measure the X-ray polarization of extragalactic objects
thanks to the Imaging X-ray Polarimetry Explorer (IXPE), NASA’s first mission
to study the polarization of astronomical X-rays that was launched in December
2021 (Weisskopf et al., 2022). IXPE is capable of measuring the linear
polarization of electromagnetic waves from 2 to 8 keV, and it was already
pointed towards several radio-quiet AGNs : NGC 4151 (Gianolli et al., 2023),
MCG-05-23-16 (Marinucci et al., 2022; Tagliacozzo et al., 2023), IC 4329A
(Ingram et al., 2023) and the Circinus galaxy (Ursini et al., 2023). The later
observation revealed that the neutral reflector in Circinus produces a
polarization degree of 28% $\pm$ 7% at 68% confidence level, with a
polarization angle roughly perpendicular to the radio jet. According to Monte
Carlo simulations, the half-opening angle of the torus was determined to be of
the order of 50∘ $\pm$ 5∘ (Ursini et al., 2023). In the hopes to achieve
similar results, we hereby report the first pointing of NGC 1068 by IXPE,
together with two Chandra snapshots, to enable a spatially resolved spectro-
polarimetric study. AFor the remainder of this paper, a $\Lambda$CDM cosmology
with $H_{0}=70$ km s-1 Mpc-1, $\Omega_{\mathrm{m}}=0.3$ and
$\Omega_{\Lambda}=0.7$ is adopted. Also, unless specified otherwise, errors
are given at 68% confidence level for one parameter of interest and upper
limits are given at 99% confidence level.

##  2 Observations and data reduction

###  2.1 IXPE

IXPE observed NGC 1068 (02h42m40.711s, -00d00m47.81s in J2000.0 equatorial
coordinates) from January 3 to January 29, 2024 (ObsID 02008001), for a net
exposure time of 1.15 Ms. Cleaned level 2 event files were firstly treated
with the background rejection procedure described in Di Marco et al.
(2023)111The adopted script is available on
https://github.com/aledimarco/IXPE-background. Then they were produced and
calibrated using standard filtering criteria with the dedicated FTOOLS tasks
and the latest calibration files available in the IXPE calibration data base
(CALDB 20230526).

I, Q, and U spectra were extracted according to the weighted analysis method
described in Di Marco et al. (2022) by setting the parameter stokes=neff in
XSELECT. The background spectra were extracted from source-free circular
regions with a radius of 78′′. Extraction radii for the I Stokes spectra of
the source were computed via an iterative process which leads to the
maximization of the signal-to-noise ratio (SNR) in the 2 -– 8 keV energy band
(similarly to Piconcelli et al., 2004), which led to radii of 47′′, 52′′ and
47′′ for the Detector Units (DUs) 1, 2 and 3, respectively. The same circular
regions (centered on the source) were used for all three DUs for I, Q and U.
This results in a 2-5.5 keV background of 10.4%, 14.9% and 13.9% for the total
DU1, DU2, and DU3 I spectra. These values rise to 23.7%, 28.4% and 25.7% in
the 5.5 – 8 keV band. We used a constant energy binning of 0.2 keV for Q and U
Stokes spectra and required an SNR higher than 3 in the intensity spectra. The
choice of dividing the full energy band into two bins below and above 5.5 keV
is driven by the need to explore the influence of the iron line complex that
dominates the emission above 5.5 keV.

###  2.2 Chandra

Similarly to the Circinus case (Ursini et al., 2023), there are several
ultraluminous X-ray (ULX) sources around NGC 1068 (Smith & Wilson, 2003; Matt
et al., 2004; Zaino et al., 2020). To monitor the flux levels of the off-
nuclear point like sources, two Chandra observations were performed at the
beginning and at the end of the IXPE pointings, on 2024, January 4 (ObsID
29071) and on January 28 (ObsID 29072), with the Advanced CCD Imaging
Spectrometer (ACIS, Garmire et al., 2003). To reduce pileup effects, the frame
time was set to 0.5 s and custom CCD subarrays were used. Data were reduced
with the Chandra Interactive Analysis of Observations (CIAO, Fruscione et al.,
2006) 4.14 and the Chandra Calibration Data Base 4.11.0, adopting standard
procedures. We generated event files for the two observations with the CIAO
tool chandra-repro and, after cleaning for background flaring, we got net
exposure times of 10.3 ks and 9.3 ks for the first and for the second
observation, respectively.

We used a circular extraction region with a 15′′ radius for the AGN and
background spectra were extracted from source-free circular regions with a
radius of 10′′. Spectra from the two observations were added and then binned
in order to oversample the instrumental resolution by a factor of 3 and to
have no less than 30 counts in each background-subtracted spectral channel.
This allows the applicability of the $\chi^{2}$ statistic. We ignored channels
between 8 and 10 keV due to pileup, which was calculated via the CIAO tool
pileup-map. We estimated an average pileup fraction of 7.5% in the central
$3\times 3$ pixels region, ranging from 5% to 10% (in the central pixel) for
both observations.

Spectra from other point-like sources within the field of view were extracted
from circular regions with 2.5′′ radii. In the 2-8 keV band, the brightest
ones are: SN 2018ivc (Bostroem et al., 2020; Maeda et al., 2023), CXOU
J024241.4-000013, CXOU J024238.9-000055 and CXOU J024241.0-000125 (Smith &
Wilson, 2003). Spectral fit of the other point-like sources showed 2-8 keV
fluxes lower than $\sim 2.5\times 10^{-14}$ erg cm-2 s-1, which is
approximately 0.5% of the total flux of the AGN. These sources were not
considered in the spectro-polarimetric fit, since their overall contribution
is negligible. Spectra were then binned requesting at least 5 counts in each
spectral bin and the Cash statistics (Cash, 1979) was used for the data
analysis.

Figure 1: The 2.′$\times$ 2.′central region of NGC 1068 is shown in the
0.3-7.0 keV band (Chandra ObsID 29071). The dashed white circle corresponds to
the IXPE DU1 source extraction region. A Gaussian smoothing kernel of $3\times
3$ pixels has been applied for the for the sake of visual clarity.

##  3 Measures of NGC 1068’s X-ray polarization

###  3.1 IXPE polarization cubes

We estimated the X-ray polarization properties using event-based Stokes
parameter analysis methods, as implemented in Kislat et al. (2015). The model-
independent analysis was performed using the unweighted method employing the
PCUBE algorithm in ixpeobssim, as detailed in Baldini et al. (2022). The
polarization degree and angle were derived from normalized $q$ and $u$ Stokes
parameters, calculated as $P$ = $\sqrt{({q})^{2}+({u})^{2}}$ and $\Psi$ =
$1/2\tan^{-1}({u/q})$ (measured from North to East), with the background
polarization subtracted. Consequently, we estimated the time-averaged X-ray
polarization properties of the source within the 2 -– 8 keV range as $P$ = 9%
$\pm$ 5% and $\Psi$ = 113∘ $\pm$ 16∘. As justified in Sect. 2.1, we divided
the nominal energy band into two bins and measured $P$ = 14% $\pm$ 4% and
$\Psi$ = 105∘ $\pm$ 8∘ (99.7% confidence level) for the energy range 2 – 5.5
keV. The remaining IXPE band, 5.5 – 8 keV, only shows an upper limit with $P<$
20%.

Furthermore, to explore the variability of polarization over time and energy,
we segmented the entire observation into subsets based on time and energy, as
described in Kim et al. (2024). Specifically, we divided the $q$ and $u$ data
into identical time spans, depending on the selected number of bins (e.g., 2
bins with 100 ks gives 50 ks/bin) from 2 to 15 bins. In all cases, the
probability of the hypothesis that the data are constant was greater than 28%,
implying no significant evidence of time variability. Energy-resolved analysis
was performed in a similar manner. In this case, we divided the 2–8 keV range
into 2, 3, 4, 6, 8, and 12 bins (e.g., 2 bins = 2 -– 5 keV, 5 –- 8 keV). The
polarization is consistent with being constant with energy, with a probability
greater than 24% in all cases.

###  3.2 IXPE spectropolarimetry

The X-ray spectropolarimetric analysis was carried out with XSPEC v12.13.1e
(Arnaud, 1996). The Galactic absorption along the line of sight amounts to
$N_{\mathrm{H,Gal}}=2.59\times 10^{20}$ cm-2 (HI4PI Collaboration et al.,
2016) and is accounted for by using the tbabs model with the ISM abundances
from Wilms, Allen, & McCray (2000). We fit the I, Q, and U spectra
simultaneously while including a cross-calibration constant (const) between
the three different DUs. For what concerns the sole IXPE data, we focus on a
phenomenological modeling while a more physical approach is described in the
following section. In this case, the X-ray continuum is modeled as a simple
power law (powerlaw) throughout the whole feasible 2 – 8 keV energy interval.

In order to probe the polarization properties of NGC 1068 we adopted the
polconst model, that assumes an energy-independent polarization and has two
free parameters: $P$ and $\Psi$. Thus, we first start with a model that can be
written as
$\textsc{const}\times\textsc{tbabs}\times\textsc{polconst}\times\textsc{powerlaw}$
in XSPEC. The spectral fit is clearly unsatisfactory
($\chi^{2}/\mathrm{dof}=931/476$), since we observe large positive residuals
at energies larger than 6 keV, likely owing to the broad iron line complex. We
thus add a Gaussian emission line (zgauss) to account for this component, and
this leads to a significant fit improvement ($\chi^{2}/\mathrm{dof}=537/473$),
implying $\Delta{\chi^{2}}=394$ for three degrees of freedom. We fixed the
line width to the value of $\sigma$ = 500 eV returned by the best-fit, since
the centroid energy cannot be constrained otherwise. Such a broad emission
feature is the result of the blend of several emission lines from both neutral
and ionized gas (Matt et al., 2004), that we cannot individually resolve with
IXPE. The emission line is centered at $E$ = 6.58${}_{-0.04}^{+0.05}$ keV. We
find a rather soft X-ray continuum, with the photon index having a value of
$\Gamma$ = 2.35${}_{-0.04}^{+0.05}$.

Nonetheless, we still observe some residuals at softer energies which we try
to account for by including an additional power-law component. This leads to a
further improvement of the spectral fit ($\chi^{2}/\mathrm{dof}=474/472$).
Thus, our best-fit phenomenological model consists of:

$\textsc{const}\times\textsc{tbabs}\times\textsc{polconst}\times(\textsc{powerlaw}_{1}+\textsc{powerlaw}_{2}+\textsc{zgauss})$

The line centroid shifts towards $E$ = 6.67 $\pm$ 0.06 keV and the two power-
law photon indices are $\Gamma_{1}$ = $0.9\pm 0.4$ and $\Gamma_{2}$ =
4.3${}_{-0.6}^{+0.8}$. The best-fit values of the 2 – 8 keV polarization
degree and angle are $P$ = 12.4% $\pm$ 3.6% and $\Psi$ = 100.7∘ $\pm$ 8.3∘,
respectively.

Figure 2: Contour plot of the polarization angle vs polarization degree
accounting for the 68%, 90% and 99% confidence levels, computed by taking into
account the IXPE I, Q and U spectra in the 2 – 5.5 keV band (the energy band
with the highest significance in terms of polarization detection). The orange
cross indicates the best-fit values of $P$ and $\Psi$, that are also reported
in the lower left corner. The dashed blue line describes the direction of the
parsec-scale radio jet (Wilson & Ulvestad, 1983). The dashed brown line
indicates the orientation of the molecular torus derived by García-Burillo et
al. (2019) with ALMA, see Sect. 4.1.

Finally, we divided the broad energy band into smaller intervals to probe the
polarization properties of NGC 1068. The 2 – 5.5 keV band shows the highest
significance in terms of polarization detection. Fig. 2 shows the contour plot
of the polarization angle versus the polarization degree in such interval. At
energies higher than 5.5 keV, there is a clear rise of background contribution
in addition to the emission lines from the iron K complex that are blended and
widened due to the modest IXPE spectral resolution. The results are listed in
Tab. 1 and are generally in good agreement with the values reported from the
PCUBE model-independent analysis described in the previous section.

###  3.3 IXPE+Chandra fitting

Thanks to the Chandra data, we are able to take into account the contribution
of the ULXs in the spectral analysis (Bauer et al., 2015). We co-add the
Chandra spectra of extranuclear point sources that are significantly detected
above 2 keV, and we model this combined ULX emission with a power law having a
fixed slope of 1. We obtain a 2–8 keV flux of $2.2\times 10^{-13}$ erg cm-2
s-1 for the ULX power law. The AGN, on the other hand, has a 2 – 8 keV flux of
$4.3\times 10^{-12}$ erg cm-2 s-1.

The 2–8 keV continuum is well described by two components: a warm reflector,
mostly contributing below 4 keV, and a cold reflector, dominating above 5 keV
(see also Matt et al., 2004). In both cases, the reflected continuum is
expected to be significantly polarized, while the associated emission lines
are expected to be unpolarized. We fit jointly the Chandra and IXPE $I$, $Q$,
$U$ Stokes spectra with a phenomenological model consisting of: a power law to
describe the warm reflector continuum, a pexrav (Magdziarz & Zdziarski, 1995)
component for the cold reflector continuum, and six Gaussian emission
lines222We use pexrav plus emission lines instead of models which self-
consistently include both the continuum and the lines because the polarization
of the reflected continuum and of the emission lines is expected to be
different, see also Ursini et al. (2023).. The photon indices of the warm and
cold reflection components are not well constrained by the Chandra + IXPE fit,
therefore we adopt values consistent with those reported by Matt et al.
(2004), Marinucci et al. (2016) and Zaino et al. (2020). We fix the photon
index of pexrav at 2, and that of the warm reflector at 3. To fit the IXPE
spectra, we also include the power law describing the ULX emission as derived
from Chandra, fixing both the photon index and the flux. Finally, we multiply
the different components by the polconst model. We assume the emission lines
and the ULXs to be unpolarized333ULXs are likely polarized, with $P$ depending
on their true nature, orientation and geometry (Veledina et al., 2023).
However, the ULXs around NGC 1068 contribute so weakly to the total flux that
assuming no polarization is a reasonable hypothesis.. The xspec model is as
follows:

$\displaystyle\textsc{c\\_cal}\times\textsc{tbabs}\times$
$\displaystyle[\textsc{polconst}^{(0)}\times(\textstyle\sum\textsc{zgauss}^{(i)}$
lines $\displaystyle+\textsc{powerlaw})$ ULXs
$\displaystyle+\textsc{polconst}^{(w)}\times\textsc{powerlaw}$ warm refl.
$\displaystyle+\textsc{polconst}^{(c)}\times\textsc{pexrav}]$ cold refl.

where c_cal is the cross-calibration constant. We fix the polarization degree
of polconst(0) at zero, which corresponds to an unpolarized component in
xspec. The data and best-fitting model are shown in Figs. 3 and 4.

We find significant degeneracy among the polarization parameters of the warm
and cold reflectors. The contour plot (not shown here) between the
polarization degrees of the two components indicates that even at 1-sigma,
$P_{\rm cold}$ is barely above zero and $P_{\rm warm}$ has only an upper limit
of 28%. To reduce such degeneracy, we need to make some assumptions. We thus
fix the polarization parameters of the warm component using the far-UV
polarimetric measurements by Antonucci, Hurt, & Miller (1994), see also Sect.
4.3. Those authors have shown that, within the first arcseconds, electron
scattering in the winds is mainly responsible for the far-UV polarization. We
postulate that the X-ray counterpart should be similar. We thus fix the
polarization degree and angle of the warm reflector at 16% and 97∘,
respectively. With this assumption, we obtain a good fit with
$\chi^{2}/\mathrm{dof}=534/527$ : the cold reflector has a polarization degree
of 20% $\pm$ 10% (non-zero at merely two sigma) and a polarization angle of
102∘ $\pm$ 15∘. The best-fitting paramters are reported in Table 2. Finally,
we note that the warm reflection component is expected to be still
significantly contaminated by unresolved emission lines, in which case the
polarization degree could be much lower than our initial assumption. In the
extreme hypothesis of null polarization of the warm reflector, we obtain a
statistically equivalent fit, and a polarization degree of the cold reflector
of $36\%\pm 10\%$.

Figure 3:  Top panel: Chandra/ACIS and IXPE $I$ spectra with best-fitting model and residuals. Bottom panel: IXPE $Q$ and $U$ Stokes spectra with best-fitting model and residuals.  Figure 4:  Best-fitting total model (black solid line) for the Chandra+IXPE data. The plot shows the different components, i.e. cold reflection (black dotted line), warm reflection (blue dashed line), ULXs (red dotted line).  Energy (keV) |  $P$ (%) |  $\Psi$ (∘)
---|---|---
2.0 – 8.0 | 12.4 $\pm$ 3.6 | 100.7 $\pm$ 8.3
2.0 – 3.5 | 9.8 $\pm$ 4.9 | 104.2 $\pm$ 14.9
3.5 – 6.0 | 21.3 $\pm$ 6.7 | 102.2 $\pm$ 9.2
6.0 – 8.0 | $<38.5$ | –
2.0 – 5.5 | 14.6 $\pm$ 3.9 | 101.1 $\pm$ 7.6
5.5 – 8.0 | $<26.8$ | –
Table 1: xspec measured polarization degree $P$ and angle $\Psi$ for various energy bands. Errors are given at 68% confidence level for one parameter of interest. Upper limits are given at 99% confidence level. Parameter | Value
---|---
Lines (zgauss)
$E_{1}$ (keV) | $1.96\pm 0.02$
$N_{1}$ | $5.6\pm 0.8\times 10^{-5}$
$E_{2}$ (keV) | $2.42\pm 0.07$
$N_{2}$ | $3.5\pm 0.3\times 10^{-5}$
$E_{3}$ (keV) | $3.09^{+0.02}_{-0.04}$
$N_{3}$ | $0.55^{+0.19}_{-0.17}\times 10^{-5}$
$E_{4}$ (keV) | $6.41^{+0.01}_{-0.02}$
$\sigma_{4}$ (eV) | $76\pm 12$
$N_{4}$ | $7.4^{+0.5}_{-0.6}\times 10^{-5}$
$E_{5}$ (keV) |  $6.7$(f)
$N_{5}$ | $2.3\pm 0.4\times 10^{-5}$
$E_{6}$ (keV) |  $6.966$(f)
$N_{6}$ | $1.1\pm 0.5\times 10^{-5}$
ULXs (powerlaw)
$\Gamma$ |  $1$(f)
$N$ |  $2.3\times 10^{-5}$(f)
Cold reflector (pexrav)
$\Gamma$ |  $2$(f)
$N$ | $(1.97\pm 0.08)\times 10^{-2}$
P.D. (%) | $20\pm 10$
P.A. (deg) | $102\pm 15$
Warm reflector (powerlaw)
$\Gamma$ |  $3$(f)
$N$ | $2.36^{+0.06}_{-0.12}\times 10^{-3}$
P.D. (%) |  $16$(f)
P.A. (deg) |  $97$(f)
Cross-calibration constants
$C_{\rm DU1-ACIS}$ | $0.68\pm 0.01$
$C_{\rm DU2-ACIS}$ | $0.64\pm 0.01$
$C_{\rm DU3-ACIS}$ | $0.60\pm 0.01$
Observed flux
$F_{\rm 2-8keV}$ | $(4.32\pm 0.15)\times 10^{-12}$
$\chi^{2}$/d.o.f. | $534/527$
Table 2: Best-fitting parameters (68 per cent confidence level for one
parameter of interest) of the joint Chandra and IXPE fit. Normalizations are
in units of photons keV-1 cm-2 s-1, while the flux is in unit of erg cm-2 s-1.
(f) denotes a fixed parameter. The energies of the two lines $E_{5}$ and
$E_{6}$ are fixed at the values of the Fe XXV and Fe XXVI K$\alpha$ lines. The
width of all lines is fixed at zero, with the only exception of the neutral Fe
K$\alpha$ line.

##  4 Analysis

###  4.1 The geometrical distribution of scatterers

The observed X-ray polarization angle can be compared to the position angle PA
of the parsec-scale radio-jet seen in NGC 1068 to determine whether scattering
occurs along the equatorial plane ($\Psi$ \- PA $\approx$ 0∘) or in the polar
direction ($\Psi$ \- PA $\approx$ 90∘). The resolved parsec-scale jet on the
4.9 GHz map of Wilson & Ulvestad (1983) sustains a PA $\sim$ 34∘ (see also
Mutie et al. 2024 for a very precise cartography of the radio emission in NGC
1068). We find that the observed X-ray polarization angle of NGC 1068 is
perpendicular to the radio structure axis, implying that the observed X-ray
polarization arises from scattering onto material that is preferentially
situated well above the equatorial plane.

Two components of the AGN can be responsible for this: the outflows or the
torus. Our measurements reported in Tab. 1 show that $\Psi$ is similar in the
2 – 3.5 keV band (where the warm component dominates the X-ray spectrum, see
Sect. 3.3) and in the 3.5 – 6 KeV band, where the cold component takes
precedence. Both components are thus likely sharing the polarization position
angle. While producing perpendicular polarization in a polar wind is trivial
(Goosmann & Gaskell, 2007), obtaining a similar polarization angle from an
equatorial torus implies that scattering must occur inside the funnel and/or
on the uppermost (outer) edges of this Compton-thick region. Such finding is
strongly supported by the Atacama Large Millimeter Array (ALMA) observation of
an extended patch of linear polarization arising from a spatially resolved
elongated nuclear disk of dust of $\sim$ 50 – 60 pc in diameter and oriented
along an averaged PA of $\sim$ 113∘ (García-Burillo et al., 2019).

The Chandra map at 0.5” resolution of the X-ray emission in NGC 1068 presented
by Ogle et al. (2003) also supports our conclusion that the X-ray polarization
we measured with IXPE mainly originates from the torus. The authors have shown
(see their Figs. 3 and 4) that the peak of the 6 – 8 keV (Fe K) emission is
coincident with the nucleus, as expected if produced in the inner wall of the
molecular torus. Surrounding the X-ray peak in the nuclear region, 3 – 6 keV
extended emission is detected, which is attributed to X-rays that have
scattered on the outer edge of the torus. The emission from the warm reflector
(the ionization cones) is seen in their Chandra 1.3 – 3 keV map and is
dominated by emission from highly ionized Mg, Si, and S. The warm component is
also appearing in their 3 – 6 keV emission map but it is mostly scattered
continuum. Using xspec to measure the associated X-ray polarization in those
three energy bands (see Tab. 1) gives only an upper limit to the 6 – 8 keV
emission, resulting from the combination of higher background levels and
strong, little-to-no polarized emission from the iron K complex. In the 3.5 –
6 keV band, where scattering off the torus is prevalent, $P$ rises up to
$\sim$ 21%. In the softest X-ray band (2 – 3.5 keV), the observed polarization
degree is much lower ($\sim$ 10%), mostly due to the forest of narrow emission
lines that depolarise the signal. Many of those lines are significantly
enhanced by photo-excitation, so they might be polarized, albeit at a lower
level from the hard continuum.

###  4.2 Estimating the AGN inclination and torus geometry

Even if nothing definitive can be said about the intrinsic polarization degree
and angle originating from the torus, we can try to evaluate the geometry of
this region thanks to Monte Carlo radiative transfer simulation of optically
thick tori. We used the radiative transfer Monte Carlo code stokes (Goosmann &
Gaskell, 2007; Marin et al., 2012; Marin, Goosmann, & Gaskell, 2015; Marin,
2018; Rojas Lobos et al., 2018) to simulate a central, isotropical, continuum
source with a power law index $\Gamma$ $\sim$ 2.04 (Matt et al. 2004,Pounds &
Vaughan 2006 and Sect. 3.3). The source emits a 2% parallel polarized primary
continuum, since it has been shown in NGC 4151 that the X-ray continuum is
parallelly polarized by a few percents (Gianolli et al., 2023). Around the
central source, we modeled a uniform-density torus using a circular cross-
section. The inner wall of the torus is set at a fixed distance of rin = 0.25
pc (Lopez-Rodriguez et al., 2018; Vermot et al., 2021) and the radius $a$ of
the torus is set by the region’s half-opening angle $\Theta$ such as $a$ = rin
$\cos$ $\Theta$/(1-$\cos$ $\Theta$). The neutral hydrogen density NH is set to
1025 atom cm-2 along the equatorial plane (Matt et al., 2004). The half-
opening angle of the torus and the system inclination $i$ are free parameters,
both measured from the vertical symmetry axis of the system. Type-2 AGNs are
thus found for $i$ $>$ $\Theta$. Such models have been examined in details in
Podgorný, Marin, & Dovčiak (2023) and we refer the reader to this publication
for in-depth details.

We present in Fig. 5 the results of the simulations for a wide variety of
$\Theta$ and $i$ angles. The polarization is integrated between 3.5 and 6 keV
to avoid as much contamination as possible from the fluorescent and
recombination lines (see Fig. 4). Superimposing the estimated neutral
reflector X-ray polarization (20% $\pm$ 10%) to the models allows us to put
strong constraints on the geometrical configuration of the system. Excluding
a) all torus half-opening angles smaller than that of the ionized polar winds
seen in NGC 1068, that sustain a half-opening angle of 40∘ as traced by [O
III] emission (Macchetto et al., 1994), b) all results for $i$ $<$ $\Theta$
and c) all non-perpendicular polarization angles (an assumption compatible
with IXPE data), we find that the range of permitted AGN inclinations is 42∘ –
87∘ and the range of torus half-opening angles is 40∘ – 57.5∘ (as illustrated
by the black dots on Fig. 5). Further observation with better statistics could
narrow down the parameter space but, if we rely on the range of inclinations
$i$ estimated from the literature (70∘ – 90∘, see Sect. 1), the permitted
range of $\Theta$ becomes 45∘ – 57.5∘, with a much higher probability to be in
the range 50∘ – 55∘. Interestingly, this is comparable to the range of
$\Theta$ found in the case of the Circinus galaxy (45∘ \- 55∘), the only other
type-2 AGN probed by IXPE (Ursini et al., 2023).

For the sake of completeness, we also tested the line dominated warm reflector
scenario ($P$ = 0%), that would lead to a neutral reflector polarization of
$36\%\pm 10\%$ (see Sect. 3.3). By doing so, the torus half-opening angle
estimated from Monte Carlo radiative transfer simulations becomes 40∘ – 50∘,
with a much higher statistical probability to be between 45∘ and 50∘. This is
also comparable to the range of $\Theta$ found in the case of the Circinus
galaxy.

Figure 5: stokes simulations of a circularly-shaped torus with various half-
opening angles $\Theta$ seen at several observer inclinations $i$. The color-
coded polarization degree is integrated from 3.5 to 6.0 keV to avoid
contribution from intense emission lines. A positive polarization degree
indicates that the photon electric field vector is parallel to the axis of
symmetry of the system. Thus, negative polarization stands for perpendicularly
polarized photons. Each colored dot represents a ($\Theta$,$i$) simulation. If
the polarization degree and angle of a dot is consistent with the X-ray
polarization of the torus as measured by IXPE, the dot is shown in black.

###  4.3 Comparison with optical data

The broadband polarization of NGC 1068 has been thoroughly examined from the
ultraviolet to the infrared band (see Marin 2018 and reference therein for a
review). Yet, due to strong starlight dilution by the host, the true
(intrinsic) polarization levels of the AGN cannot be measured. The addition of
the X-ray polarization measurement achieved by IXPE brings another piece to
the puzzle, since it is essentially free from diluting sources (albeit the
almost negligible ULX signal). A comparison between X and optical data can
thus lead to better constraints on the global geometry of the AGN.

To do so, 0.35 – 1.1 $\mu$m linear spectropolarimetry was obtained on December
1 – 6th, 2023, with the FOcal Reducer/low dispersion Spectrograph 2 (FORS2)
instrument mounted on the Very Large Telescope (VLT), quasi-simultaneously to
the IXPE observation. The observation, under program ID 112.26WL.001, will be
analyzed and published later on (Marin et al., in prep.), but the reduced
optical data confirm that the linear polarization from NGC 1068 remained
stable since the last decades, especially in the continuum. The polarization
degree rises from $\sim$ 1.5% in the red up to $\sim$ 12% in the blue band.
The polarization position angle is almost constant with wavelength, slowly
rotating from $\sim$ 100∘ in the red to $\sim$ 90∘ in the blue. The rise of
$P$ continues in the ultraviolet band up to 16% at a position angle of 97∘
shortward of 2700 Å (Antonucci, Hurt, & Miller, 1994). The polarization degree
then plateau, as the host starlight contribution becomes negligible,
indicating that electron scattering dominates in the winds.

The matching of the polarization angle between the near-infrared, optical,
ultraviolet and X-rays clearly indicates that the polarization results, in all
cases, from scattering by material well above the equatorial plane. But they
are not due to the same AGN component. X-ray spectropolarimetry tends to favor
light reflection off atoms and molecules from the cold component (the torus),
while optical imaging and spectropolarimetry advocates for electron (or dust)
scattering inside the polar outflows (Antonucci & Miller, 1985; Antonucci,
Hurt, & Miller, 1994). In the spectral region where emission from the ionized
winds dominates the X-ray signal (2 – 3.5 keV, see Fig. 4), we measured $P$ =
9.8% $\pm$ 4.9% at 104.2∘ $\pm$ 14.9∘. Although diluted by the emission lines,
the polarized X-ray continuum probably comes from the optically thin region
that scatters the optical and ultraviolet photons (Antonucci & Miller, 1985).

As things currently stand and with the statistics offered by our observation,
we note that the optical/ultraviolet and X-ray measurements are different in
$P$ (but not in $\Psi$), so we caution using ultraviolet polarization
measurements of Seyfert-2 galaxies as a basic proxy to predict the
polarization level of a source in the X-rays.

##  5 Conclusions

We have measured the 2 – 8 keV polarization of the archetypal Seyfert-2 AGN
NGC 1068 with IXPE. We found a linear polarization degree of 12.4% $\pm$ 3.6%
at 100.7∘ $\pm$ 8.3∘ (68% confidence level). The polarization is perpendicular
to the position angle of the parsec-scale radio structure, similarly to what
is observed for all Seyfert-2s in the optical band. A combined Chandra and
IXPE analysis indicated a significant degeneracy among the polarization
parameters of the warm and cold reflectors but, from multi-wavelength
constraints, we estimated that the cold reflector has a polarization degree of
20% $\pm$ 10% and polarization angle of 102∘ $\pm$ 15∘. By taking the result
at the face value, numerical simulations allowed us to derive a probable torus
half-opening angle of 50 – 55∘ (from the vertical axis of the system). This
morphological constraint is quite similar to the torus half-opening angle
derived for the Circinus galaxy thanks to X-ray polarimetry, the only other
type-2 AGN probed by IXPE so far. By doing so, X-ray polarimetry has shown its
powerful and unique capabilities of constraining the geometrical arrangement
of matter around supermassive black holes.

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-2022-13-I.0, agreements ASI-INAF-2022-19-HH.0 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). A.D.M., E.Co., R.F., S.F., F.L.M., F. Mu. P.So. are partially
supported by MAECI with grant CN24GR08 “GRBAXP: Guangxi-Rome Bilateral
Agreement for X-ray Polarimetry in Astrophysics”. F.T. acknowledges funding
from the European Union - Next Generation EU, PRIN/MUR 2022 (2022K9N5B4).
M.D., J.P., and V.K. thank GACR project 21-06825X for the support and
institutional support from RVO:67985815. I.L. was supported by the NASA
Postdoctoral Program at the Marshall Space Flight Center, administered by Oak
Ridge Associated Universities under contract with NASA. The work of RTa and
RTu is partially supported by grant PRIN-2022LWPEXW of the Italian MUR. The
French contribution (F.M., T.B., V.E.G., P.-O.P.) is partly supported by the
French Space Agency (Centre National d’Etude Spatiale, CNES) and by the High
Energy National Programme (PNHE) of the Centre National de la Recherche
Scientifique (CNRS).

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