# IXPE observations of the quintessential wind-accreting X-ray pulsar Vela X-1

Sofia V. Forsblom Department of Physics and Astronomy, FI-20014 University of
Turku, Finland  Juri Poutanen Department of Physics and Astronomy, FI-20014
University of Turku, Finland  Space Research Institute of the Russian Academy
of Sciences, Profsoyuznaya Str. 84/32, Moscow 117997, Russia Juri Poutanen
<EMAIL_ADDRESS>Sergey S. Tsygankov Department of Physics and Astronomy,
FI-20014 University of Turku, Finland  Space Research Institute of the Russian
Academy of Sciences, Profsoyuznaya Str. 84/32, Moscow 117997, Russia  Matteo
Bachetti INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy  Alessandro Di Marco INAF Istituto di Astrofisica e
Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
Victor Doroshenko Institut für Astronomie und Astrophysik, Universität
Tübingen, Sand 1, D-72076 Tübingen, Germany  Jeremy Heyl Department of Physics
and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada
Fabio La Monaca INAF Istituto di Astrofisica e Planetologia Spaziali, Via del
Fosso del Cavaliere 100, 00133 Roma, Italy  Christian Malacaria International
Space Science Institute, Hallerstrasse 6, 3012 Bern, Switzerland  Herman L.
Marshall MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  Fabio Muleri INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, Italy  Alexander A. Mushtukov
Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson
Building, Keble Road, Oxford OX1 3RH, UK  Leiden Observatory, Leiden
University, NL-2300RA Leiden, The Netherlands  Maura Pilia INAF Osservatorio
Astronomico di Cagliari, Via della Scienza 5, 09047 Selargius (CA), Italy
Daniele Rogantini MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA
02139, USA  Valery F. Suleimanov Institut für Astronomie und Astrophysik,
Universität Tübingen, Sand 1, D-72076 Tübingen, Germany  Roberto Taverna
Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Via
Marzolo 8, 35131 Padova, 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  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,
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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  Fiamma Capitanio INAF Istituto di Astrofisica e Planetologia Spaziali,
Via del Fosso del Cavaliere 100, 00133 Roma, 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  Enrico Costa INAF Istituto di Astrofisica e Planetologia Spaziali, Via
del Fosso del Cavaliere 100, 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  Michal Dovčiak Astronomical Institute of the Czech Academy
of Sciences, Boční II 1401/1, 14100 Praha 4, Czech Republic  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  Sergio Fabiani 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  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  Philip Kaaret NASA Marshall Space Flight Center, Huntsville, AL 35812,
USA  Department of Physics and Astronomy, University of Iowa, Iowa City, IA
52242, USA  Vladimir Karas Astronomical Institute of the Czech Academy of
Sciences, Boční II 1401/1, 14100 Praha 4, Czech Republic  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 Pisa, Largo B. Pontecorvo 3, 56127 Pisa, 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  Giorgio Matt Dipartimento di Matematica e Fisica, Università degli
Studi Roma Tre, Via della Vasca Navale 84, 00146 Roma, 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  Stephen L. O’Dell NASA Marshall Space Flight
Center, Huntsville, AL 35812, USA  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
Alessandro Papitto INAF Osservatorio Astronomico di Roma, Via Frascati 33,
00040 Monte Porzio Catone (RM), 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  John Rankin INAF Istituto di
Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133
Roma, Italy  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  Paolo Soffitta INAF Istituto di Astrofisica e
Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133 Roma, Italy
Gloria Spandre Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Largo
B. Pontecorvo 3, 56127 Pisa, Italy  Rashid A. Sunyaev Max Planck Institute for
Astrophysics, Karl-Schwarzschild-Str 1, D-85741 Garching, Germany  Space
Research Institute of the Russian Academy of Sciences, Profsoyuznaya Str.
84/32, Moscow 117997, Russia  Doug 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  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  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  Martin C. Weisskopf NASA
Marshall Space Flight Center, Huntsville, AL 35812, USA  Kinwah Wu Mullard
Space Science Laboratory, University College London, Holmbury St Mary,
Dorking, Surrey RH5 6NT, UK  Silvia Zane Mullard Space Science Laboratory,
University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK

###### Abstract

The radiation from accreting X-ray pulsars was expected to be highly
polarized, with some estimates for the polarization degree of up to $80\%$.
However, phase-resolved and energy-resolved polarimetry of X-ray pulsars is
required in order to test different models and to shed light on the emission
processes and the geometry of the emission region. Here we present the first
results of the observations of the accreting X-ray pulsar Vela X-1 performed
with the Imaging X-ray Polarimetry Explorer (IXPE). Vela X-1 is considered to
be the archetypal example of a wind-accreting high-mass X-ray binary system,
consisting of a highly magnetized neutron star accreting matter from its
supergiant stellar companion. The spectro-polarimetric analysis of the phase-
averaged data for Vela X-1 reveals a polarization degree (PD) of 2.3$\pm$0.4%
at the polarization angle (PA) of $-47\fdg 3\pm 5\fdg 4$. A low PD is
consistent with the results obtained for other X-ray pulsars and is likely
related to the inverse temperature structure of the neutron star atmosphere.
The energy-resolved analysis shows the PD above 5 keV reaching 6–10%, and a
$\sim 90\arcdeg$ difference in the PA compared to the data in the 2–3 keV
range. The phase-resolved spectro-polarimetric analysis finds a PD in the
range 0–9% with the PA varying between $-80\arcdeg$ and $40\arcdeg$.

accretion – magnetic fields – pulsars: individual: Vela X-1 – stars: neutron –
X-rays: binaries

††facilities: IXPE, Swift (XRT and UVOT), NuSTAR††software: astropy (Astropy
Collaboration et al., 2013, 2018), xspec (Arnaud, 1996), ixpeobssim (Baldini
et al., 2022).

##  1 Introduction

Accreting X-ray pulsars (XRPs) harbor some of the strongest magnetic fields in
the entire Universe, which can be as large as several times $10^{12}$ G. The
strong magnetic field channels accreting matter onto the polar regions at the
neutron star surface, where it produces hot-spots that are bright in the
X-rays; these spots rotate in and out of the observer’s line-of-sight,
resulting in the appearance of pulsed X-ray emission. Highly magnetized XRPs
represent unique laboratories and much information is embedded in the
interplay between the immense magnetic fields and the accretion flow.
Observations of emission from accreting XRPs therefore constitute a
substantial area of interest for theoretical models of matter interactions
with ultrastrong magnetic fields, which cannot be replicated in terrestrial
laboratories (see Mushtukov & Tsygankov, 2022, for a recent review).

The magnetic field of the neutron star is the main cause for the polarized
X-ray emission from accreting XRPs. The scattering of photons in a highly
magnetized plasma is expected to result in a large degree of polarization of
the emerging X-ray emission, up to 80% (Meszaros et al., 1988; Caiazzo & Heyl,
2021). Meszaros et al. (1988) showed that linear X-ray polarization is
strongly dependent on the geometry of the emission region, and also that it
varies with energy and pulse phase. X-ray polarimetric observations of
accreting XRPs can therefore be used to probe the geometry of the emission
region. The phase-resolved polarimetry can be used as a tool to constrain the
viewing geometry and to distinguish between the models of their radiation.

A large window of opportunity to achieve significant observations of polarized
X-ray emission opened up recently, thanks to the launch of the Imaging X-ray
Polarimetry Explorer (IXPE, Weisskopf et al., 2022). IXPE is the first imaging
X-ray polarimetric mission. For observations of XRPs, the strategy is to
detect pulsations in the polarized emission, as well as to measure the
polarization degree (PD) and polarization angle (PA) as a function of the
pulse phase, which allows for the system geometry to be determined.

Vela X-1 (associated with the Uhuru source 4U 0900–40) is a high-mass X-ray
binary (HMXB) discovered as one of the first X-ray sources at the early years
of X-ray astronomy (Chodil et al., 1967) and remains one of the best studied
objects among neutron star HMXBs. It is one of the brightest, persistent XRPs
chosen to be observed by IXPE. Vela X-1, located at a distance of about 2 kpc
(Kretschmar et al., 2021), is often considered the quintessential wind
accretor. It displays strong X-ray pulsations with a pulse period of
$\mathrm{283\;s}$ (McClintock et al., 1976), variations with the orbital
period of $\mathrm{8.964\;days}$ (Ulmer et al., 1972; van Kerkwijk et al.,
1995), and eclipses lasting for about two days per orbit. The lower limit on
the orbital inclination was obtained at $i=73\arcdeg$ (van Kerkwijk et al.,
1995).

Persistent wind-accreting XRPs are expected to have a different emission
region geometry as opposed to the disk-accreting Be/X-ray binaries.
Additionally, polarization signatures are expected to be introduced by the
scattering in the dense asymmetric wind. The main goal is therefore to study
the accretion geometry for wind-accretion and the properties of the dense
stellar wind.

Quaintrell et al. (2003) have shown that the separation between the neutron
star and its stellar companion, the B0.5Ib supergiant HD 77581 (also known as
GP Vel), is only about 1.7 stellar radii and therefore the neutron star is
deeply embedded in the stellar wind of its companion star. The stellar
companion has a mass loss rate of $\sim 10^{-6}\;M_{\odot}$ yr-1 (Nagase et
al., 1986). The average X-ray luminosity of the source is $\sim 4\times
10^{36}$ erg s-1. The luminosity is, however, strongly variable on all time-
scales, varying up to a factor of at least 20–30 (Staubert et al., 2004;
Kreykenbohm et al., 2008).

Observations of the cyclotron resonance scattering features (CRSFs) in the
spectra of XRPs provide a direct measurement of the magnetic field strength in
the line-forming region. These features were first discovered in Vela X-1 by
Kendziorra et al. (1992) by utilizing Mir-HEXE data, reporting a fundamental
line around 25 keV and a first harmonic close to 50 keV. Evidence for these
features was also given by Makishima & Mihara (1992) and they were further
detailed by Kretschmar et al. (1996). Early observations with RXTE also
confirmed this detection (Kretschmar et al., 1997). More recent observations
of Vela X-1 by NuSTAR clearly detected the fundamental line at 25 keV together
with a more prominent first harmonic at 55 keV and revealed a positive
correlation between the harmonic line energy and the observed flux (Fürst et
al., 2014). La Parola et al. (2016) confirmed a flux dependence of the first
harmonic line energy and discovered its secular variation with time in the
long-term data collected by Swift/BAT (see also Ji et al., 2019).

Figure 1: Background-corrected light curves of Vela X-1 in the 2–8 keV energy
bands summed over the three DUs of IXPE for the first and second observation,
shown in the left and right panels, respectively. The light curve time bin
value was set to $\sim$250 s. The time of the eclipse is marked by the blue-
shaded area.

In this paper, we present the first results of X-ray polarimetric observations
of Vela X-1 by IXPE carried out on two separate occasions during 2022. In
Sect. 2, the data used in the paper are described. Sect. 3 is devoted to the
description of the results: the analysis of the phase-averaged, phase-
resolved, and energy-resolved polarimetric data are given. Finally, the
discussion and a short summary are presented in Sect. 4.

##  2 Data

Table 1: Orbital parameters for Vela X-1 adopted from the Fermi Gamma Ray Burst Monitor project (dated 2021 January 30). Parameter | Value | Unit
---|---|---
Orbital period | 8.9642140 | d
$T_{\pi/2}$ | 2459115.02085 | JED
$a_{\rm X}\sin i$ | 113.105 | light-sec
Longitude of periastron | 162.33 | deg
Eccentricity | 0.0872 |
Eclipse egress | 0.12 |
Eclipse ingress | 0.92 |

IXPE is an observatory launched on December 9, 2021, as a NASA mission in
partnership with the Italian space agency (ASI). IXPE consists of three
telescope-detector systems which provide imaging polarimetry over a nominal
2–8 keV band at $\sim$ 30″ angular resolution (half-power diameter). Each one
of the three grazing incidence telescopes is comprised of a mirror module
assembly (MMA), that focuses the X-rays onto a corresponding focal plane
polarization-sensitive gas pixel electron tracking detector unit (DU). The
detection principle is based on the photoelectric effect. All characteristics
of each detected photon (sky coordinates, time of arrival, energy, and
direction of the photo-electron) are measured simultaneously. A comprehensive
description of the observatory, the instruments, and their performance is
given by Soffitta et al. (2021) and Weisskopf et al. (2022).

IXPE observations of Vela X-1 were carried out between 2022 April 15–21 and
November 30 – December 6, with the total effective exposure of $\simeq$280 ks
and $\simeq$270 ks, respectively. The data have been processed with the
ixpeobssim package version 30.2.1 (Baldini et al., 2022) using the CalDB
released on November 17, 2022. The position offset correction and energy
calibration were applied before the scientific analysis of the data. Source
photons were collected using a circular region with a radius $R_{\rm
src}=70\arcsec$. The background region was chosen in the form of an annulus
with inner and outer radii equal to 2$R_{\rm src}$ and 4$R_{\rm src}$,
respectively. Data from the first observation were cleaned from events due to
solar events, which have been identified by comparing the IXPE light curve
with the one from the Geostationary Operational Environmental Satellite
(GOES), then removing time intervals where the IXPE count rate in the
background annular region was higher than the mean background value plus three
times the RMS of this count rate. The background makes up $\sim$3.6% and
$\sim$2.2% of the total count rate of the source region in the 2–8 keV energy
range for the first and second observation, respectively.

Figure 2: Pulse profiles of Vela X-1 as seen by IXPE in two different energy
bands for the first and second observations, combined for DU1–3.

The barycorr tool from the ftools package was used to correct the event
arrival times to the barycenter of the Solar System. This was followed by a
correction of arrival times as it relates to the effects of binary motion
using the orbital parameters obtained by the Fermi Gamma Ray Burst
Monitor111https://gammaray.nsstc.nasa.gov/gbm/science/pulsars.html for Vela
X-1 and given in Table 1.

Stokes $I$ energy spectra have been binned to have at least 30 counts per
energy channel, and the same energy binning was applied to the energy spectra
of Stokes parameters $Q$ and $U$. The energy spectra were fitted in the xspec
package (Arnaud, 1996) using $\chi^{2}$ statistics, using the version 12
instrument response functions. The reported uncertainties are at the 68.3%
confidence level ($1\sigma$), unless stated otherwise.

##  3 Results

###  3.1 Light curve and pulse profile

The light curves from the first and second observation of Vela X-1 in the 2–8
keV energy range obtained with the IXPE observatory are shown in Figure 1. For
the first observation of Vela X-1, the observing window can be separated into
three parts: pre-eclipse (MJD 59684.7–59687.7), eclipse (MJD 59687.7–59689.5),
and post-eclipse (MJD 59689.5–59690.5). The post-eclipse count rate of the
source was about one order of magnitude greater than the pre-eclipse source
count rate. During the eclipse, which lasted about two days, the count rate
dropped by an order of magnitude compared to the pre-eclipse value. For the
following analysis, only pre- and post-eclipse data were included, i.e., only
data outside of the eclipse.

A spin period of $P_{\rm spin}=283.488(7)$ s and $P_{\rm spin}=283.437(6)$ s
were measured for Vela X-1 for the first and second observation, respectively,
using phase connection technique. The pulsed fraction in the 2–8 energy band,
defined as $PF=(F_{\max}-F_{\min})/(F_{\max}+F_{\min})$ where $F_{\max}$ and
$F_{\min}$ are the maximum and minimum count rates in the pulse profile,
respectively, was determined as $PF=53.3\pm 0.7\%$ for the first observation
and as $PF=48.1\pm 0.9\%$ for the second observation. The resulting pulse
profiles for Vela X-1 in two separate energy bands are shown in Figure 2.

###  3.2 Polarimetric analysis

Table 2: Measurements of the normalized Stokes parameters $q$ and $u$, PD, and PA for the phase-averaged data of Vela X-1 in different energy bins using the PCUBE algorithm for the combined data set.  Energy | $q$ | $u$ | PD | PA
---|---|---|---|---
(keV) | (%) | (%) | (%) | (deg)
2–3 | $\phantom{-}0.5\pm 1.2$ | $\phantom{-}3.8\pm 1.2$ | 3.9$\pm$1.2 | $\phantom{-}41.4\pm 9.1$
3–4 | $\phantom{-}0.2\pm 0.8$ | $-1.4\pm 0.8$ | 1.4$\pm$0.8 | $-40.7\pm 16.1$
4–5 | $-0.1\pm 0.8$ | $-2.7\pm 0.8$ | 2.7$\pm$0.8 | $-46.4\pm 8.8$
5–6 | $-1.2\pm 1.0$ | $-5.6\pm 1.0$ | 5.7$\pm$1.0 | $-50.9\pm 4.8$
6–7 | $-0.3\pm 1.2$ | $-4.1\pm 1.2$ | 4.1$\pm$1.2 | $-47.3\pm 8.3$
7–8 | $-2.7\pm 2.0$ | $-9.4\pm 2.0$ | 9.7$\pm$2.0 | $-53.1\pm 6.0$
2–8 | $-0.6\pm 0.5$ | $-3.7\pm 0.5$ | 3.7$\pm$0.5 | $-49.9\pm 4.1$
Figure 3: Energy dependence of the normalized Stokes parameters $q$ and $u$
for the phase-averaged data of the combined data set, obtained with the PCUBE
algorithm.  Figure 4: Phase-resolved analysis of Vela X-1 for the combined
data set in the 2–7 keV range, combining data from all DUs. (a) Pulse profile.
Panels (b) and (c) display the dependence of the Stokes $q$ and $u$
parameters, respectively, on the pulse phase, obtained for the phase-resolved
polarimetric analysis utilizing the PCUBE algorithm. Panels (d) and (e) show
the dependence of the PD and PA, respectively, on the pulse phase, obtained
from the phase-resolved spectro-polarimetric analysis using xspec. Upper
limits (arrows) to the PD are at 99.73% (3$\sigma$) confidence level and are
computed using a $\chi^{2}$ with one degree of freedom.

First, the analysis of the polarimetric properties of Vela X-1 was carried out
by using the PCUBE algorithm (xpbin tool) in the ixpeobssim package, which is
implemented according to the formalism by Kislat et al. (2015). The unweighted
analysis has been used. We compute the normalized Stokes parameters, $q=Q/I$
and $u=U/I$, and the polarization degree using the formula
PD=$\sqrt{q^{2}+u^{2}}$ and ignoring the bias at low signal-to-noise ratios
(Serkowski, 1958; Simmons & Stewart, 1985; Maier et al., 2014; Mikhalev, 2018)
and the PA=$\frac{1}{2}\arctan(u/q)$ (measured counterclockwise on the sky
from north to east).

In the entire IXPE energy band (2–8 keV), the average PD and PA are found to
be 3.9$\pm$0.9% and $-51\fdg 5\pm 6\fdg 5$, respectively, for the first
observation. For the second observation, the average PD and PA are found to be
3.7$\pm$0.7% and $-48\fdg 9\pm 5\fdg 2$, respectively.

Considering the similarities between the first and second observations, the
data were combined into one single set of data in order to increase the
statistics and further study the polarization properties of Vela X-1. In order
to correctly phase-tag each event, the phase difference between the pulse
profiles from the first and second observations was determined from cross-
correlation (using the implementation provided by the Python library NumPy).
Using the PCUBE algorithm (xpbin tool) in the ixpeobssim package, the average
PD and PA are found to be 3.7$\pm$0.5% and $-49\fdg 9\pm 4\fdg 1$,
respectively, in the entire IXPE energy band.

We then studied the energy dependence of polarization by dividing the data
into six energy bins. The PD is above the minimum detectable polarization at a
99% confidence level, $\mathrm{MDP_{99}}$ (Weisskopf et al., 2010), in all of
the energy bins, except for the 3–4 keV bin, where the PD is below the
$\mathrm{MDP_{99}}$ and in this case, the PA is not well constrained (Table 2
and Figure 3). The energy-resolved analysis shows that at higher energies
(above 5 keV) the PD reaches 6–10% with the PA differing by $\sim$90° from
that below 3 keV.

Figure 5: Stokes $I$, $Q$, and $U$ energy distributions for the combined data set of Vela X-1 with the best-fit model superimposed for IXPE’s DUs (upper panels). The residuals between the best-fit model and the data are shown in the lower panels. Upper and lower plots correspond to the first and second observations, respectively, and are shown separately for clarity.  Table 3: Pulse phase dependence of the normalized Stokes $q$ and $u$ parameters for the combined data set from the polarimetric analysis (2–7 keV) using the PCUBE algorithm and the spectral parameters, PD and PA obtained by the spectro-polarimetric analysis. An upper limit to the PD at 99.73% (3$\sigma$) confidence level is computed using a $\chi^{2}$ with one degree of freedom.  Phase | $q$ | $u$ | $N_{\mathrm{H}}$ | $N_{\mathrm{H,tbpcf}}$ | $f_{\mathrm{cov}}$ | Photon index | PD | PA |  $\chi^{2}$/d.o.f.
---|---|---|---|---|---|---|---|---|---
| (%) | (%) | ($10^{22}\mathrm{\;cm^{-2}}$) | ($10^{22}\mathrm{\;cm^{-2}}$) |  |  | (%) | (deg) |
0.000–0.100 | $\phantom{-}1.7\pm 1.4$ | $-1.2\pm 1.4$ | $6.0_{-0.8}^{+0.7}$ | $23.0_{-2.2}^{+1.8}$ | $0.82\pm 0.02$ | $1.55_{-0.16}^{+0.13}$ | $2.3\pm 1.2$ | $-22.0\pm 16.9$ | 2160/2152
0.100–0.150 | $\phantom{-}0.8\pm 1.8$ | $-4.4\pm 1.8$ | $4.4_{-1.4}^{+1.0}$ | $18.5_{-2.0}^{+2.1}$ | $0.83_{-0.03}^{+0.04}$ | $1.26\pm 0.16$ | $4.2\pm 1.6$ | $-33.0\pm 11.3$ | 2162/2029
0.150–0.200 | $\phantom{-}0.6\pm 1.9$ | $-8.7\pm 1.9$ | $4.9_{-2.1}^{+1.2}$ | $18.4_{-3.4}^{+3.2}$ | $0.77_{-0.04}^{+0.07}$ | $0.96_{-0.22}^{+0.19}$ | $7.3\pm 1.7$ | $-42.4\pm 6.8$ | 2081/2011
0.200–0.300 | $\phantom{-}0.9\pm 1.4$ | $-5.1\pm 1.4$ | $3.8_{-0.9}^{+0.8}$ | $19.9_{-1.9}^{+2.1}$ | $0.80\pm 0.02$ | $0.75_{-0.13}^{+0.14}$ | $5.0\pm 1.2$ | $-36.6\pm 7.2$ | 2284/2179
0.300–0.400 | $-0.2\pm 1.3$ | $-1.5\pm 1.3$ | $4.0_{-1.1}^{+0.9}$ | $16.5_{-1.4}^{+1.7}$ | $0.81_{-0.03}^{+0.04}$ | $1.11\pm 0.11$ | $<4.6$ | — | 2304/2185
0.400–0.500 | $\phantom{-}0.0\pm 1.7$ | $\phantom{-}0.1\pm 1.7$ | $4.2_{-1.2}^{+0.8}$ | $20.5_{-2.9}^{+2.5}$ | $0.78\pm 0.03$ | $1.08\pm 0.2$ | $<4.6$ | — | 2057/2086
0.500–0.600 | $\phantom{-}1.3\pm 1.6$ | $\phantom{-}3.2\pm 1.6$ | $2.1_{-1.2}^{+0.9}$ | $17.7_{-1.7}^{+1.7}$ | $0.83\pm 0.03$ | $0.63\pm 0.13$ | $2.6\pm 1.5$ | $37.7\pm 17.2$ | 2230/2116
0.600–0.650 | $-7.1\pm 2.2$ | $-3.5\pm 2.2$ | $3.8_{-1.5}^{+1.0}$ | $21.3_{-3.0}^{+2.6}$ | $0.82\pm 0.03$ | $0.98_{-0.22}^{+0.19}$ | $8.3\pm 2.0$ | $-77.3\pm 6.9$ | 1885/1858
0.650–0.700 | $-7.0\pm 2.2$ | $-8.7\pm 2.2$ | $2.5_{-1.8}^{+1.2}$ | $18.2_{-2.1}^{+2.4}$ | $0.84\pm 0.04$ | $0.73\pm 0.18$ | $9.2\pm 2.0$ | $-64.2\pm 6.1$ | 1934/1912
0.700–0.775 | $-3.4\pm 1.9$ | $-6.4\pm 1.9$ | $0.6_{-0.6}^{+1.5}$ | $15.3_{-1.2}^{+2.0}$ | $0.85_{-0.05}^{+0.02}$ | $0.30_{-0.13}^{+0.15}$ | $5.6\pm 1.6$ | $-60.6\pm 8.7$ | 2025/2038
0.775–0.875 | $-2.0\pm 1.6$ | $-4.3\pm 1.6$ | $4.0_{-1.0}^{+0.7}$ | $20.6_{-2.2}^{+2.0}$ | $0.81\pm 0.02$ | $1.05_{-0.15}^{+0.14}$ | $4.4\pm 1.4$ | $-58.7\pm 9.2$ | 2117/2134
0.875–1.000 | $\phantom{-}3.7\pm 1.6$ | $\phantom{-}1.6\pm 1.6$ | $3.1_{-2.1}^{+1.2}$ | $14.9_{-1.7}^{+1.9}$ | $0.81_{-0.05}^{+0.07}$ | $0.95_{-0.15}^{+0.13}$ | $4.4\pm 1.4$ | $21.4\pm 9.5$ | 2271/2092

Next, a phase-resolved polarimetric analysis was performed utilizing the
PCUBE-algorithm. The results in the 2–7 keV energy band are given in Table 3
and are shown in Figure 4 for the combined data set.

Then, the spectro-polarimetric analysis was performed according to the
following steps. Source $I$, $Q$, and $U$ Stokes spectra were produced via the
xpbin tool’s PHA1, PHA1Q, and PHA1U algorithms, producing a full data set
comprised of nine spectra per observation, three for each DU. As the
background region is dominated by source events, background subtraction is not
applied (Di Marco et al., 2023). Here we also used unweighted analysis. The
xspec package (version 12.12.1) (Arnaud, 1996), which is a part of the
standard high-energy astrophysics software suite HEASOFT, was used to study
polarization as a function of energy. All 18 spectra were fitted
simultaneously in xspec.

There are several phenomenological spectral models used to describe the
spectral continuum of Vela X-1. However, it is well known that except for a
soft excess below 3 keV, the X-ray emission below 10 keV can be well described
by a simple absorbed power law with an iron line at 6.4 keV. Due to the
restricted energy range covered by IXPE and the energy resolution of the
instrument (Weisskopf et al., 2022), we used a simple model consisting of a
power law affected by interstellar absorption (model tbabs with the abundances
from Wilms et al. 2000) combined with the polconst polarization model, which
assumes energy-independent PD and PA. In order to account for the soft excess
below 3 keV, a partial covering fraction absorption (model tbpcf) was
introduced as well, which applies an added column density to a fraction of the
power law. The re-normalization constant, const, was used to account for the
possibility of discrepancies between the different DUs, and for DU1 it was
fixed to unity. The final model

tbabs$\times$tbpcf$\times$polconst$\times$powerlaw$\times$const

was subsequently applied to both the phase-averaged and phase-resolved data.
The spectral analysis was confined to the 2–7 keV energy band, ignoring
photons above 7 keV due to remaining calibration uncertainties.

Table 4: Spectral parameters for the best-fit model obtained from the phase-averaged spectro-polarimetric analysis of the combined data set. Parameter | Value | Unit
---|---|---
$N_{\mathrm{H}}$ | $3.8\pm 0.3$ | $10^{22}\mathrm{\;cm^{-2}}$
$N_{\mathrm{H,tbpcf}}$ | $18.5_{-0.6}^{+0.7}$ | $10^{22}\mathrm{\;cm^{-2}}$
$f_{\mathrm{cov}}$ | $0.80\pm 0.01$ |
$\mathrm{const_{DU2}}$ | $0.964\pm 0.003$ |
$\mathrm{const_{DU3}}$ | $0.923\pm 0.002$ |
Photon index | $0.93\pm 0.04$ |
PD | $2.3\pm 0.4$ | %
PA | $-47.3\pm 5.4$ | deg
$\mathrm{Flux_{2-8\;keV}}$ | $7.93_{-0.06}^{+0.02}$ | $\mathrm{10^{-10}\;erg\;cm^{-2}\;s^{-1}}$
$\mathrm{Luminosity_{2-8\;keV}}$ | $3.8\times 10^{35}$ |  $\mathrm{erg\;s^{-1}}$ at $d=2.0$ kpc
$\chi^{2}$ (d.o.f.) | 2640.19 (2222) |
Figure 6: Polarization vector of Vela X-1 from the results of the phase-
averaged spectro-polarimetric analysis of the combined data set. Contours at
68.3%, 95.45% and 99.73% confidence, are shown in blue, purple, and red,
respectively.

Figure 7: Polarization vectors of Vela X-1 from the results of the phase-
resolved spectro-polarimetric analysis of the combined data set. Contours at
68.3%, 95.45% and 99.73% confidence, are shown in blue, purple, and red,
respectively.

For the phase-averaged data, the results of the spectral fitting, including
the best-fit model, are shown in Figure 5 and the best-fit model parameters
are found in Table 4. The steppar command in xspec was used to create the
confidence contours for the polarization measurements, and the resulting
contour plots at 68.3%, 95.45% and 99.73% confidence levels are presented in
Figure 6. The results of the phase-averaged polarimetric analysis for the two
different approaches give compatible results.

A wavy structure of the Stokes $U$ parameter residuals in Figure 5 (see also
Fig. 3) indicates that polarization is energy dependent. Thus, we replaced the
polconst polarization model with the pollin model, corresponding to a linear
energy dependence of the PD and PA. We assumed an energy-independent PA
($\psi_{1}$ in xspec; we fixed $\psi_{\rm slope}=0$), and allowed the PD to
vary with photon energy $E$ (keV) as $\mbox{PD}(E)=A_{1}+A_{\rm slope}(E-1)$.
This results in an improved fit ($\chi^{2}/\mathrm{d.o.f.}=2608.51/2221$ with
an F-test probability of $2.3\times 10^{-7}$). The best-fit parameters are
$A_{1}=-5.3\pm 2.3$% and $A_{\rm slope}=2.2\pm 0.6$%, and PA=$\psi_{1}=-47\fdg
1\pm 6\fdg 1$. The negative $A_{1}$ means that the PA at lower energies is
rotated by 90° relative to the PA at higher energies and the PD is zero at
$\approx$3.4 keV. Such a model is able to describe the observed behavior of
Stokes $Q$ and $U$ parameters in Figures 3 and 5.

For the phase-resolved spectro-polarimetric analysis, $I$, $Q$, and $U$ Stokes
spectra were extracted for each phase bin individually, again utilizing the
xpbin tool’s PHA1, PHA1Q, and PHA1U algorithms. The $I$, $Q$, and $U$ Stokes
spectra were fitted with the same model as utilized for the phase-averaged
spectro-polarimetric analysis, with the cross-calibration constants for DU2
and DU3 fixed to the values obtained for the phase-averaged analysis (see
Table 4). The results of the phase-resolved spectro-polarimetric analysis are
summarized in Table 3 and confidence contours corresponding to each phase bin
are shown in Figure 7.

A spectro-polarimetric phase-averaged analysis done separately for the eclipse
data did not find a significant polarization, with an upper limit to the PD of
25.1% at 99.73% confidence level. An intensity-resolved spectro-polarimetric
analysis performed on the combined data did not reveal a significant
difference in polarization properties between different luminosity levels.

##  4 Discussion and summary

XRPs are prime targets for X-ray polarimetric missions, where a high PD has
been theoretically predicted because of the strong dependence of the primary
radiation processes on the polarization of X-ray photons. The birefringence of
highly magnetized plasma allows for the radiative transfer to be treated in
terms of two normal polarization modes: the ordinary (“O”) and extraordinary
(“X”) mode (Gnedin & Pavlov, 1974). These two modes have different
orientations in relation to the plane made up by the direction of the magnetic
field and the momentum of photons. For O-mode photons, the electric vector
oscillates mainly within the plane, while for X-mode photons the electric
vector oscillations are mainly oriented perpendicular to the plane. Below the
cyclotron energy the opacities of the polarization modes differ significantly,
where the opacity of the X-mode is largely reduced compared to that of the
O-mode (Lai & Ho, 2003), resulting in the predictions for PD as high as 80%
(Meszaros et al., 1988; Caiazzo & Heyl, 2021).

However, a significantly lower PD of only $\sim$2.3% detected for Vela X-1 in
the IXPE data is in line with recent measurements done for other accreting
XRPs (Her X-1, Doroshenko et al. 2022; Cen X-3, Tsygankov et al. 2022; 4U
1626–67, Marshall et al. 2022), where similar, relatively low PDs have been
reported. The X-ray polarization of accreting XRPs greatly depends on the
structure of the emission region (which is unknown), and most theoretical
models mentioned here do not account yet for the temperature structure of the
NS atmosphere, assuming instead a uniform temperature in the spectral forming
region. In the case of the above-mentioned XRPs, the contradiction between the
observed and the theoretically predicted values of the PD was explained with a
model of the neutron star atmosphere overheated by the accretion process.

The key factor for the depolarized emission in this model is a conversion of
modes at the so-called vacuum resonance. In strong magnetic fields, both the
plasma and the vacuum are birefringent, where vacuum birefringence is a
fundamental QED effect. Generally speaking, the two effects (plasma and vacuum
birefringence) tend to work against each other, and at vacuum resonance they
cancel out (Mészáros & Ventura, 1978). This leads to a transformation of the
normal modes of radiation and a loss of the linear polarization degree. The
vacuum resonance occurs at a plasma density $\rho_{\mathrm{V}}\approx
10^{-4}B_{12}^{2}E_{\mathrm{keV}}^{2}$ $\mathrm{[g\;cm^{-3}]}$ for a given
photon energy $E$ (in keV) and local magnetic field strength
$B_{12}=B/10^{12}$ G. This may result in a much smaller PD than normally
predicted when considering the specific temperature structure of the
atmospheres of accreting NSs. Doroshenko et al. (2022) have shown that a low
PD of the X-ray radiation can be achieved if the point of vacuum resonance is
located in an atmospheric transition layer with a strong temperature gradient.
If the transition region is located at the border of the overheated upper
atmospheric layer and the cooler underlying atmosphere, the low PD occurs as a
result of the fast mode conversion (Gnedin et al., 1978). For the specific
atmospheric thickness of $\sim$3 $\mathrm{g\;cm^{-2}}$, corresponding to the
Thomson optical depth around unity, a PD of the order of 10% can be achieved
(Doroshenko et al., 2022). As lower PDs than previously predicted seem to be a
staple of sub-critical XRPs, theoretical models may well have to take into
account the specific temperature structure of the NS atmosphere. On the other
hand, the average observed luminosity for Vela X-1 is $\sim 4\times
10^{35}\;\mathrm{erg\;s^{-1}}$, roughly two orders of magnitude lower than the
observed luminosities of Her X-1 and Cen X-3. Thus, it is rather puzzling that
this scenario suggests a similar thickness of the overheated layer for a much
smaller accretion rate; however, the key quantity here may be the proton
stopping depth (Zel’dovich & Shakura, 1969; Nelson et al., 1993; Zane et al.,
2000; González-Caniulef et al., 2019).

The low PD may also be a result of the strong variations of the PD and PA with
energy (see Figure 3), considering the evident $\sim 90\arcdeg$ difference in
the PA below and above 3.5 keV. However, a complete and detailed analysis of
the complicated energy dependence of the polarization properties of Vela X-1
is out of the scope of this paper and is subject to future, more extensive
work.

Finally, we can speculate that the observed small PD is a result of strong
variations of the PA with the pulsar phase. The observed pulse profile has a
very complicated shape, which is related either to the complex structure of
the surface magnetic field, or to the presence of a number of different
components (see Tsygankov et al., 2022, for discussion). The present photon
statistics allowed us to obtain significant detection of polarization in 9 out
of 12 phase bins, while to resolve the variations of the PA we likely needed
many more bins.

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). We acknowledge support
from the RSF grant 19-12-00423 (SST), the Academy of Finland grants 333112,
349144, 349373, and 349906 (JP, SST), the German Academic Exchange Service
(DAAD) travel grant 57525212 (VD, VFS), and the German Research Foundation
(DFG) grant WE 1312/53-1 (VFS).

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