# X-ray Polarization Reveals the Precessions of the Neutron Star in Hercules
X-1

Jeremy Heyl1 Victor Doroshenko2 Denis González-Caniulef3 Ilaria Caiazzo4 Juri
Poutanen5 Alexander Mushtukov6 Sergey S. Tsygankov5 Demet Kirmizibayrak1
Matteo Bachetti7 George G. Pavlov8 Sofia V. Forsblom5 Christian Malacaria9
Valery F. Suleimanov2 Iván Agudo10 Lucio Angelo Antonelli11,12 Luca
Baldini13,14 Wayne H. Baumgartner15 Ronaldo Bellazzini13 Stefano Bianchi16
Stephen D. Bongiorno15 Raffaella Bonino17,18 Alessandro Brez13 Niccolò
Bucciantini19,20,21 Fiamma Capitanio22 Simone Castellano13 Elisabetta
Cavazzuti23 Chien-Ting Chen24 Stefano Ciprini25,12 Enrico Costa22 Alessandra
De Rosa22 Ettore Del Monte22 Laura Di Gesu23 Niccolò Di Lalla26 Alessandro Di
Marco22 Immacolata Donnarumma23 Michal Dovčiak27 Steven R. Ehlert15 Teruaki
Enoto28 Yuri Evangelista22 Sergio Fabiani22 Riccardo Ferrazzoli22 Javier A.
Garcia4 Shuichi Gunji29 Kiyoshi Hayashida30 Wataru Iwakiri31 Svetlana G.
Jorstad32,33 Philip Kaaret15 Vladimir Karas27 Fabian Kislat34 Takao
Kitaguchi28 Jeffery J. Kolodziejczak15 Henric Krawczynski35 Fabio La Monaca22
Luca Latronico17 Ioannis Liodakis36 Simone Maldera17 Alberto Manfreda37
Frédéric Marin38 Andrea Marinucci23 Alan P. Marscher32 Herman L. Marshall39
Francesco Massaro17,18 Giorgio Matt16 Ikuyuki Mitsuishi40 Tsunefumi Mizuno41
Fabio Muleri22 Michela Negro42,43,44 C.-Y. Ng45 Stephen L. O’Dell15 Nicola
Omodei26 Chiara Oppedisano17 Alessandro Papitto11 Abel Lawrence Peirson26
Matteo Perri12,11 Melissa Pesce-Rollins13 Pierre-Olivier Petrucci46 Maura
Pilia7 Andrea Possenti7 Simonetta Puccetti12 Brian D. Ramsey15 John Rankin22
Ajay Ratheesh22 Oliver J. Roberts24 Roger W. Romani26 Carmelo Sgrò13 Patrick
Slane47 Paolo Soffitta22 Gloria Spandre13 Douglas A. Swartz24 Toru Tamagawa28
Fabrizio Tavecchio48 Roberto Taverna49 Yuzuru Tawara40 Allyn F. Tennant15
Nicholas E. Thomas15 Francesco Tombesi50,25,51 Alessio Trois7 Roberto
Turolla49,52 Jacco Vink53 Martin C. Weisskopf15 Kinwah Wu52 Fei Xie54,22
Silvia Zane52

###### Abstract

In an accreting X-ray pulsar, a neutron star accretes matter from a stellar
companion through an accretion disk. The high magnetic field of the rotating
neutron star disrupts the inner edge of the disc, funneling the gas to flow
onto the magnetic poles on its surface. Hercules X-1 is in many ways the
prototypical X-ray pulsar; it shows persistent X-ray emission and it resides
with its companion HZ Her, a two-solar-mass star, at about 7 kpc from
Earth[1]. Its emission varies on three distinct timescales[2]: the neutron
star rotates every 1.2 seconds, it is eclipsed by its companion each 1.7 days,
and the system exhibits a superorbital period of 35 days which has remained
remarkably stable since its discovery[3]. Several lines of evidence point to
the source of this variation as the precession of the accretion disc[4, 5],
the precession of the neutron star[6, 7] or both[8]. Despite the many hints
over the past fifty years, the precession of the neutron star itself has yet
not been confirmed or refuted. We here present X-ray polarization measurements
with the Imaging X-ray Polarimetry Explorer[9] (IXPE) which probe the spin
geometry of the neutron star. These observations provide direct evidence that
the 35-day-period is set by the free precession of the neutron star crust,
which has the important implication that its crust is somewhat asymmetric
fractionally by a few parts per ten million[10]. Furthermore, we find
indications that the basic spin geometry of the neutron star is altered by
torques on timescale of a few hundred days.

University of British Columbia, Vancouver, BC V6T 1Z4, Canada

Institut für Astronomie und Astrophysik, Universität Tübingen, Sand 1, 72076
Tübingen, Germany

Institut de Recherche en Astrophysique et Planétologie, UPS-OMP, CNRS, CNES, 9
avenue du Colonel Roche, BP 44346 31028, Toulouse CEDEX 4, France

Division of Physics, Mathematics and Astronomy, California Institute of
Technology, Pasadena, CA 91125, USA

Department of Physics and Astronomy, 20014 University of Turku, Finland

Department of Physics, University of Oxford, Oxford OX1 3RH, UK

INAF Osservatorio Astronomico di Cagliari, Via della Scienza 5, 09047
Selargius (CA), Italy

Department of Astronomy and Astrophysics, Pennsylvania State University,
University Park, PA 16802, USA

International Space Science Institute, Hallerstrasse 6, 3012 Bern, Switzerland

Instituto de Astrofísica de Andalucía—CSIC, Glorieta de la Astronomía s/n,
18008 Granada, Spain

INAF Osservatorio Astronomico di Roma, Via Frascati 33, 00078 Monte Porzio
Catone (RM), Italy

Space Science Data Center, Agenzia Spaziale Italiana, Via del Politecnico snc,
00133 Roma, Italy

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

NASA Marshall Space Flight Center, Huntsville, AL 35812, USA

Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Via
della Vasca Navale 84, 00146 Roma, Italy

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

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

INAF Istituto di Astrofisica e Planetologia Spaziali, Via del Fosso del
Cavaliere 100, 00133 Roma, Italy

ASI - Agenzia Spaziale Italiana, Via del Politecnico snc, 00133 Roma, Italy

Science and Technology Institute, Universities Space Research Association,
Huntsville, AL 35805, USA

Istituto Nazionale di Fisica Nucleare, Sezione di Roma ”Tor Vergata”, Via
della Ricerca Scientifica 1, 00133 Roma, Italy

Department of Physics and Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, Stanford, California 94305, USA

Astronomical Institute of the Czech Academy of Sciences, Boční II 1401/1,
14100 Praha 4, Czech Republic

RIKEN Cluster for Pioneering Research, 2-1 Hirosawa, Wako, Saitama 351-0198,
Japan

Yamagata University,1-4-12 Kojirakawa-machi, Yamagata-shi 990-8560, Japan

Osaka University, 1-1 Yamadaoka, Suita, Osaka 565-0871, Japan

International Center for Hadron Astrophysics, Chiba University, Chiba
263-8522, Japan

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

Department of Physics and Astronomy and Space Science Center, University of
New Hampshire, Durham, NH 03824, USA

Physics Department and McDonnell Center for the Space Sciences, Washington
University in St. Louis, St. Louis, MO 63130, USA

Finnish Centre for Astronomy with ESO, 20014 University of Turku, Finland

Istituto Nazionale di Fisica Nucleare, Sezione di Napoli, Strada Comunale
Cinthia, 80126 Napoli, Italy

Université de Strasbourg, CNRS, Observatoire Astronomique de Strasbourg, UMR
7550, 67000 Strasbourg, France

MIT Kavli Institute for Astrophysics and Space Research, Massachusetts
Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA

Graduate School of Science, Division of Particle and Astrophysical Science,
Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Aichi 464-8602, Japan

Hiroshima Astrophysical Science Center, Hiroshima University, 1-3-1
Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan

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

Department of Physics, The University of Hong Kong, Pokfulam, Hong Kong

Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France

Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA
02138, USA

INAF Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC),
Italy

Dipartimento di Fisica e Astronomia, Università degli Studi di Padova, Via
Marzolo 8, 35131 Padova, Italy

Dipartimento di Fisica, Università degli Studi di Roma ”Tor Vergata”, Via
della Ricerca Scientifica 1, 00133 Roma, Italy

Department of Astronomy, University of Maryland, College Park, Maryland 20742,
USA

Mullard Space Science Laboratory, University College London, Holmbury St Mary,
Dorking, Surrey RH5 6NT, UK

Anton Pannekoek Institute for Astronomy & GRAPPA, University of Amsterdam,
Science Park 904, 1098 XH Amsterdam, The Netherlands

Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China

Hercules X-1 was observed by IXPE[9] on 2022 February 17-24 (255 ks), at the
beginning of the 35 d precession cycle, the so-called “main-on” state[11] and
again in 2023 January and February during the “short-on” (148 ks) and the
“main-on” state (245 ks) as shown in Fig. 1. The gas-pixel detector on IXPE
registers the arrival time, sky position, and energy for each X-ray photon and
uses the photoelectric effect to provide an estimate of the position angle of
each photon[12]. During each observation, photon arrival events registered
between energies of two and eight keV, within 52 arcseconds of the position of
source were extracted for analysis. We do not subtract background for the
analysis as a bright source such as Hercules X-1 dominates the background over
a very large region of the detector[13]. The times of the photon arrivals were
corrected for the motion of IXPE around the barycenter of the Solar System,
and the orbit of the neutron star about its companion to be in a frame of
reference moving with the neutron star. The details of the data reduction are
outlined in the methods section of Doroshenko et al.[11].

The analysis focuses on the geometry of the spinning neutron star over the
pulsar rotation period and the superorbital period. We use a maximum
likelihood technique[14] to determine the mean polarization properties
averaged over the pulsar orbit as shown in Fig. 1, and as a function of pulse
phase during the different superorbital states of the system in Fig. 2. We
track the observed polarization angle as the star rotates to determine the
geometry of the spinning neutron star and study the geometry as the system
evolves through the superorbital period. We fix the phase of zero in the two
main-on epochs to the peak of the X-ray light curve. For the short-on, we fix
the phase of one half to the peak of the light curve in agreement with
previous work[5]. Our results do not depend on the choice of phasing among the
epochs. As already apparent in Fig. 1, the source appears much fainter during
the short-on than during the main-on (this is often attributed to occultation
by the accretion disk[5]), so the count rate for the short-on in Fig. 2 must
be inflated by a factor of ten to be visible clearly. The polarization degree
in the two main-on epochs is about five to fifteen percent and in the short-on
it is about two times larger. The pulse profile and polarization degree are
consistent between the two main-on epochs[15]. More dramatic changes are
apparent when one examines the polarization angle as a function of spin phase.
The evolution of the polarization angle over the spin period during the two
main-on epochs are nearly identical up to an offset. Over the course of the
year, between the two main-on epochs, the mean value of the polarization angle
has decreased by $9^{\circ}\\!\\!.2\pm 1^{\circ}\\!\\!.8$. The variation of
the polarization angle during the short-on, on the other hand, is much smaller
than during the main-on.

Although the magnetic field structure in the emission region of the pulsar is
expected to be complicated[16], the polarization angles of photons coming from
different parts of the emission region are expected to follow the magnetic
field direction as they propagate in the highly magnetized plasma surrounding
the X-ray pulsar. Even at large distances from the neutron star, where the
plasma does not affect the radiation, vacuum birefringence causes the
polarized radiation in the magnetosphere to propagate in the ordinary (O) and
extraordinary (X) modes which represent oscillations of the electric field
parallel and perpendicular to the plane formed by the local magnetic field and
the photon momentum[17, 18], and propagation in the normal modes continues
within the so-called polarization limiting radius[19]. For typical X-ray
pulsars, this radius is estimated to be about thirty stellar radii[20], where
the field is dominated by the dipole component, so the polarization measured
at the telescope is expected to be either parallel or perpendicular to the
instantaneous projection of the magnetic dipole axis of the star onto the
plane of the sky. For this reason, the modulation of the polarization angle
(PA) with phase is decoupled from the evolution in polarization degreee (PD)
and intensity and should follow the rotating vector model (RVM)[21, 22, 14]

$\tan(\textrm{PA}-\chi_{\rm p})=\frac{\sin\theta\sin(\phi-\phi_{0})}{\cos
i_{\rm p}\sin\theta\cos(\phi-\phi_{0})-\sin i_{\rm p}\cos\theta},$ (1)

where $i_{\rm p}$ is the inclination of the angular velocity with respect to
the line of sight, $\chi_{\rm p}$ the position angle of the rotation axis in
the plane of the sky, $\theta$ is the inclination of the magnetic dipole to
the spin axis, $\phi$ is the spin phase, and $\phi_{0}$ is the phase of the
rotation when the magnetic dipole axis is closest to the line of sight.

Neutron stars probably do not rotate as rigid bodies[23]. The magnetic field,
whose geometry we measure through the polarization, is anchored to the rigid
crust of the neutron star and passes through the core of the neutron star
where it interacts with the core superfluid; consequently, the measured
polarization angle through the RVM probes the instantaneous rotation axis of
the neutron-star crust through the parameters $\chi_{\rm p}$ and $i_{\rm p}$,
and the location of the magnetic pole relative to the instantaneous rotation
axis through the parameters $\theta$ and $\phi_{0}$. If the neutron-star crust
is prolate, oblate, or triaxial and the spin axis does not coincide with a
symmetry axis of the crust, the neutron star can precess even in the absence
of torques, so the values of $\chi_{\rm p}$, $i_{\rm p}$, $\theta$ and
$\phi_{0}$ can change with time as well, over the precession period of the
neutron star. This would change the evolution of the polarization angle with
spin phase. We can infer that the neutron star is nearly symmetric to within a
few parts per ten million from the ratio of the spin period of 1.2 seconds to
the potential precession period of 35 days (the superorbital period)[24]; if
the angular momentum of the crust is conserved, the angles $\chi_{\rm p}$ and
$i_{\rm p}$ would remain constant also to within a few parts per ten million,
and therefore any observed changes in these angles are a hallmark of a net
torque on the neutron-star crust. The values of $\theta$ and $\phi_{0}$, on
the other hand, can change measurably even in the absence of torques as the
spin axis moves across the surface of the neutron star.

To quantify the observed changes, we fit the photon arrival times and photo-
electron angles directly to an RVM of the polarization angle[14]. The colored
curves in Fig. 2 depict the best-fitting RVM models to the measured photo-
electron trajectories. The best-fitting parameters for these fits are given in
Table 1, and the covariances are shown in Extended Data Figures 1 through 5.
Fig. 3 depicts the varying geometry schematically. The values for the first
main-on epoch agree with those measured by Doroshenko et al.[11] within the
uncertainties. To examine whether the parameters change over the course of a
single main-on phase, we fit only the data during the first two orbital
periods (Early Epoch 1) and the final orbit (Late Epoch 1), and present these
results as well. During the first main-on, the RVM parameters do not vary
substantially. However, the differences are significant among the three
epochs: first main-on, short-on, and second main-on. In particular, as
indicated by the phase-resolved polarimetry itself, the position angle of the
spin axis on the sky ($\chi_{\rm p}$) has changed by nine degrees
($8^{\circ}\\!\\!.6\pm 2^{\circ}\\!\\!.2$) between the two main-on epochs
while the other parameters remain the same within the uncertainties. This
indicates the angular momentum of the crust has changed from 2022 February to
2023 February.

Furthermore, between the short-on and the latter main-on, there is strong
evidence ($p\approx 0.008$) that the value of the angle between the spin axis
and the magnetic dipole ($\theta$) is larger during the main-on than during
the short-on: $16^{\circ}\\!\\!.0_{-4^{\circ}\\!\\!.3}^{+3^{\circ}\\!\\!.1}$
versus $3^{\circ}\\!\\!.7_{-1^{\circ}\\!\\!.9}^{+2^{\circ}\\!\\!.6}$. The
larger value of $\theta$ is reflected in the larger amplitude of modulation in
the polarization angle that we noted earlier. In principle, a larger
contribution of scattered X-rays could reduce the swing in polarization during
the short-on as well, but this would also reduce the polarization degree below
what is observed; furthermore, observations of the winds and corona through
the superorbital period[25, 26] support the conclusion that scattering does
not contribute much to the observed radiation during this epoch. The change in
the value of $\theta$ between the short-on and the second main-on indicates
that the spin axis of the crust has moved relative to the magnetic axis of the
neutron star; as we do not see evidence of a change of $i_{M}$ or $\chi_{M}$
over this short eighteen-day period, we argue that a net torque is not
required to account for this precession; it is approximately free.
Furthermore, the torque that would be required to cause the spin axis to move
through a symmetric crust by twelve degrees over eighteen days exceeds that
supplied by the disk by a factor of three[27, 28] and would very likely also
cause the orientation of the spin axis to change with the respect to the sky
by a comparable amount which is not observed. The approximately free
precession of modestly asymmetric crust over the 35-day superorbital period
remains as the most conservative explanation for the observed short-term
changes in the geometry.

Several models have been proposed for the free precession of the neutron star
in Hercules X-1 to explain the superorbital period. Postnov et al.[16] propose
a prolate model to account for the variation in pulse profiles, and
Kolesnikov, Shakura and Postnov[10] use a triaxial model to explain the
variation in the spin frequency of the star[10]. We first focus on these
specific models. The key observables are the minimum and maximum angles
between the instantaneous spin axis and the magnetic pole ($\theta$) which
determine broadly how the polarization will evolve through the precessional
cycle. In all cases, we assume that the angular momentum of the star makes an
angle $i_{M}=110^{\circ}$ with respect to the line of sight as this is the
approximate inclination of the orbit[11, 29]. We varied this angle from
$60^{\circ}$ to $120^{\circ}$ and verified that changes in this angle only
make minor changes to the predicted polarization angles.

In the prolate model, the ratio of the spin period to the superorbital period
is simply the relative difference between the moments of inertia, and the two
additional parameters are the time during the precession when the spin axis
passes closest to the magnetic axis, and the angle between the spin and the
symmetry axis of the star (the spin misalignment angle). Two prolate models
are depicted in the upper two panels of Fig. 4. The upper-left one has a spin
misalignment of $\alpha=50^{\circ}$ and an angle of $\theta_{B}=30^{\circ}$
between the symmetry axis and the magnetic pole (magnetic obliquity), as
proposed by Postnov et al.[16]. These angles, along with the precession and
rotation angles ($\psi$ and $\varphi$), are depicted in Extended Data Figure
6. This model, in blue in the figure, predicts a variation in the polarization
angle throughout the precession that is larger than observed (superimposed
colored regions). To achieve better agreement with the observed polarization
angles, the spin misalignment and magnetic obliquity must be reduced to
$11^{\circ}$ and $8^{\circ}$ respectively (upper-right panel). These angles
depend modestly on the assumed value of $i_{M}$: both are smaller if
$i_{M}<50^{\circ}$ or $i_{M}>130^{\circ}$.

Although the triaxial models cannot be solved using elementary functions, the
evolution can be expressed in closed form[24, 10, 30] (see the Methods
section). In particular, the precession period is determined by the relative
difference between the largest and smallest eigenvalues of the moment of
inertia[10], taken to be $6.7\times 10^{-7}$. The relative difference between
the second largest and smallest eigenvalues is taken[10] to be $2.7\times
10^{-7}$. We examine two pairings for the minimum spin misalignment angle and
the magnetic obliquity: $50^{\circ}$ and $30^{\circ}$, as postulated by
Kolesnikov, Shakura and Postnov[10], in the bottom-left panel, and
$9.5^{\circ}$ and $7.5^{\circ}$ in the bottom-right panel. In both cases, the
magnetic axis lies toward the intermediate axis of the neutron star. In the
bottom-left panel, we see that the variation in polarization angle is larger
throughout the entire precession period than observed during either the short-
on or the main-on. As with the prolate model, better agreement is achieved by
reducing the misalignment angle and the magnetic obliquity which results in
the models in the bottom-right panel which can account for the observed
evolution in the polarization over the superorbital period. Furthermore,
frequency variations reported by Kolesnikov, Shakura and Postnov[10] are
consistent with free precession models, and the models that we present here
are also consistent with those variations as depicted in Extended Data Fig.7.

The neutron star crust and the core are strongly coupled and, in the absence
of torques, relative motion between them dissipates on a timescale of minutes
to hours[23]. The coupling between the crust and the superfluid within the
crust is mediated through the pinning and unpinning of vortices on nuclei in
the crust in a process called vortex creep. Typically, the lag between the
crust and the crust superfluid is larger than between the crust and the core
superfluid. These dissipative processes will damp the wobbling of the crust
over a timescale of a few hundred years in the absence of external torques. On
the other hand, the external torques, supplied by accretion in the case of
Hercules X-1, can support an equilibrium where the external torques on the
crust balance both the torques between the crust and the core superfluid and
the larger torque between the crust and the crust superfluid, so the crust
itself can exhibit a precession approximately free of torques. In fact an
external torque, as the one exerted by accretion, can drive the crust to
precess and the amplitude of the precession depends on the long-term behaviour
of the external torque. This torque increases with the sine of the
misalignment angle between the spin axis and the figure axis of the neutron
star crust, so larger misalignments require larger torques to maintain, and a
stable equilibrium with approximately free precession of the crust can be
achieved. For Hercules X-1, the total precessional torque exerted magnetically
by the accretion disk on the neutron-star crust is[27, 28]

$N^{{\textrm{\scriptsize prec}}}_{\textrm{\scriptsize tot}}\approx
2\sin^{2}\theta\sin 2\beta\left(\frac{B_{p}}{3\times
10^{12}\textrm{G}}\right)^{2}\left(\frac{R_{m}}{1000\textrm{km}}\right)^{-3}\times
10^{36}~{}\textrm{dyne cm}$

where $B_{p}$ is the magnetic field strength at the pole of the neutron star,
$R_{m}$ is radius of the inner edge of the accretion disk and $\beta$ is the
misalignment angle between the angular momentum of the spin of the neutron
star and the angular momentum of the orbit. Furthermore, the magnetic field
exerts an additional torque on the disk that tends to warp it[28]

$N^{{\textrm{\scriptsize warp}}}_{\textrm{\scriptsize tot}}\approx
0.2\cos^{2}\theta\sin 2\beta\left(\frac{B_{p}}{3\times
10^{12}\textrm{G}}\right)^{2}\left(\frac{R_{m}}{1000\textrm{km}}\right)^{-3}\times
10^{36}~{}\textrm{dyne cm}.$

Both of these torques vanish if the accretion disk lies above the rotational
equator of the neutron star ($\beta=0$); however, the misalignment between the
neutron-star spin and the orbit is measured to be at least 24 degrees[11] so
$\sin 2\beta>0.7$, so both the precessional and warping torques are finite and
vary with the angle between the spin axis and the magnetic axis ($\theta$).

As the crust executes its precessional motion, the torque exerted on the disk
by the magnetic field of the neutron star will also vary on the same timescale
because the angle between the spin axis and the magnetic axis ($\theta$)
changes. These torques excite the precession and warping of the disk that are
invoked to explain the evolution of the system through the superorbital phase,
and their variation through the changing value of $\theta$ over the precession
of the neutron-star crust sets the superorbital timescale. From the point of
view of the neutron star, these torques serve to maintain the approximate free
precession of the crust against internal dissipation which would damp the
precession within hundreds of years[23].

Between the first and second main-on observed by IXPE, the position angle of
the spin axis on the sky has shifted by about nine degrees. If we take the
moment of inertia of the neutron-star crust to be $10^{43}$ g cm2 (about one
hundredth of the entire neutron star)[31], the mean torque over the 355 days
between the observations is $2.6\times 10^{35}$ dyne cm, a factor of ten
smaller than the precessional torque supplied by the disk with an inner radius
of 1000 km. Although in equilibrium the net torque on the crust vanishes, the
evolution of the spin of Hercules X-1 is complicated. The amplitude of the
precession and therefore the magnitude of the internal torques necessary to
maintain the precession depend on the mean value of the accretion torques over
the damping time of several hundred years; whereas the external torques depend
on the current accretion; if these two differ by about ten percent, the
resulting net torque on the crust could result in the observed net precession
of the spin axis of the crust on the sky. In fact continued observations of
Hercules X-1 could constrain the mean accretion rate over the past few hundred
years as well as details of the crust-superfluid coupling.

Polarization measurements with IXPE reveal that the polarization angle as a
function of spin phase changes over the superorbital period and also over
longer timescales. We interpret these observations as signatures of the free
precession and forced precession of the crust of the neutron star in Hercules
X-1, confirming that the neutron star itself provides the clock for the
superorbital period[32] and revealing that torques also change the angular
momentum of the crust on a year-long timescale. This forced-precession may
account for the anomalous lows in Hercules X-1 and modulation of the
superorbital period that occur on a five-year timescale[8]. Further
polarization observations of Hercules X-1 will probe the interior of the
neutron star, in particular the coupling between the crust and the superfluid,
as well as the accretion dynamics, and polarization observations of other
X-ray binaries with superorbital periods, a common phenomenon in these
sources, may verify their underlying clocks.

Table 1: Best-fitting RVM Parameters. The observations for the first main-on were from 2022 February 17 to 24, and the early portion corresponds to the first four days of this segment. The short-on was observed from 2023 January 18 to 21, and the second main-on was from 2023 February 3 to 8. The second-to-last column gives the median precession phase for the observation. The final column indicates the quality of the fit. When the data is consistent with the model, the log-likelihood ($\log L$) is normally distributed, and the fit quality in the last column is given in terms the best-fitting log-likelihood compared the expected value where positive values indicate better than expected values. | Mean PD | $\chi_{\rm p}$ | $\theta$ | $i_{\rm p}$ | $\phi_{0}/2\pi$ | Prec. Phase | $\Delta\log L$
---|---|---|---|---|---|---|---
| [%] | [deg] | [deg] | [deg] |  |  |  [$\sigma$]
First Main-On | $9.5\pm 0.5$ | $55.4\pm 1.6$ | $14.5^{+3.0}_{-4.0}$ | $58^{+28}_{-22}$ | $0.19^{+0.03}_{-0.02}$ | $0.088$ | $-1.52$
Early | $8.6\pm 0.6$ | $57.9\pm 2.1$ | $16.3^{+3.5}_{-4.1}$ | $64^{+25}_{-22}$ | $0.19^{+0.03}_{-0.02}$ | $0.073$ | $+0.07$
Late | $9.3\pm 0.7$ | $52.2\pm 2.7$ | $15.9^{+3.6}_{-4.0}$ | $85^{+35}_{-37}$ | $0.22^{+0.05}_{-0.05}$ | $0.162$ | $+0.05$
Short-On | $17.8\pm 1.4$ | $41.9\pm 2.2$ | $3.7^{+2.6}_{-1.9}$ | $90^{+30}_{-30}$ | $0.85^{+0.18}_{-0.20}$ | $0.687$ | $-0.19$
Second Main-On | $9.1\pm 0.5$ | $46.8\pm 1.5$ | $16.0^{+3.1}_{-4.3}$ | $56^{+24}_{-20}$ | $0.20^{+0.02}_{-0.02}$ | $0.159$ | $+0.48$
Figure 1:  Evolution of the observed flux, PA and PD from Her X-1. Flux
depicted by grey curve from IXPE and olive from Swift, PD by green triangles
using the left $y-$axis) and PA by red circles using the right $y-$axis, time
and phase of the 35-day super-orbital precession cycle on the two $x-$axes.
The turn-on time 2022 February 18 is estimated from the IXPE data, and those
of 2022 December 26 and 2023 January 30 are estimated from the Swift data. The
main-on lasts from precessional phase 0 to 0.3 (2022 December 26 to 8 January
2023), and the short-on lasts from 0.5 to 0.8 (15-25 January 2023). These
phases repeat every 34.84 days (the superorbital period). The reported values
and the uncertainties correspond to the mean values and $1\sigma$ (68%)
confidence intervals and agree with the previously reported results[11, 33].
The shaded regions depict the intervals when the companion star and its
accretion stream eclipse the neutron star. We define the superorbital phase of
zero to coincide with the beginning of the main-on[11, 5]. Figure 2: IXPE
Observations of Hercules X-1 as a Function of Spin Phase. The top, middle and
bottom rows show with black errorbars: the IXPE count rate, the polarization
degree, and the polarization angle, respectively. Left, middle and right
columns are for the first main-on (2022 February), the short-on (2023
January), and the second main-on (2023 February), respectively. Error bars
indicate $\Delta\log L=1/2$ and are equivalent to one-sigma. The solid red,
dashed green and dot-dashed blue curves in the bottom row depict the best-
fitting RVM models for the first main-on, the short-on and the second main-on
respectively. Figure 3: Schematic showing the varying geometry through the
precession of Hercules X-1. The dark spot on the surface of the neutron star
indicates the location of the magnetic pole. Figure 4: Four models for the
precession. The extent of the polarization angle swing as a function of
precessional phase for the four precession models is depicted by the light-
blue bands, and the extent of best-fitting RVM PAs from the three epochs is
superimposed in red, green and dark-blue bands from left to right. Top Left: a
prolate model[16] with spin misalignment of $50^{\circ}$ and magnetic
obliquity of $30^{\circ}$. Top Right: a prolate model with spin misalignment
of $11^{\circ}$ and magnetic obliquity of $8^{\circ}$. Bottom Left: a triaxial
model[10] with a magnetic obliquity of $30^{\circ}$. Bottom Right: a triaxial
model[10] with a magnetic obliquity of $7.5^{\circ}$. The blue dot-dashed
curve traces the spin misalignment angle $\alpha$, and the orange dashed curve
traces $\theta$, the angle between the spin axis and the magnetic axis.

To model the polarization direction as a function of rotational and
precessional phase we use the formalism of Euler angles[24] to yield the
following expression

$\displaystyle\tan(\textrm{PA}-\chi_{M})$ $\displaystyle=$
$\displaystyle\Biggr{[}{\sin{\theta_{B}}\sin{\varphi}\cos{\left(\psi-\psi_{B}\right)}+\sin{\theta_{B}}\sin{\left(\psi-\psi_{B}\right)}\cos{\varphi}\cos{\alpha}-}$
$\displaystyle{\sin{\alpha}\cos{\theta_{B}}\cos{\varphi}}\Biggr{]}\Biggr{[}{-\sin{i_{M}}\sin{\theta_{B}}\sin{\alpha}\sin{\left(\psi-\psi_{B}\right)}-}$
$\displaystyle{-\sin{i_{M}}\cos{\theta_{B}}\cos{\alpha}-\sin{\theta_{B}}\sin{\varphi}\sin{\left(\psi-\psi_{B}\right)}\cos{i_{M}}\cos{\alpha}+}$
$\displaystyle{\sin{\theta_{B}}\cos{i_{M}}\cos{\varphi}\cos{\left(\psi-\psi_{B}\right)}+\sin{\varphi}\sin{\alpha}\cos{i_{M}}\cos{\theta_{B}}}\Biggr{]}^{-1},$

where $\chi_{M}$ is the position angle of the angular momentum on the plane of
the sky, $\theta_{B}$ is the angle between the symmetry axis with the largest
moment of inertia and the magnetic axis (the magnetic obliquity), $\varphi$ is
the rotation angle, $\psi$ is the precession angle, $\psi_{B}$ is longitude of
the magnetic axis using the symmetry axis with the smallest moment of inertia
as a reference, $\alpha$ is the angle between the angular velocity and the
symmetry axis with the largest moment, and $i_{M}$ is the angle between the
angular momentum and the line of sight. The angles are depicted in Extended
Data Figure 6.

Because the asymmetry of the moment of inertia is small, the values of
$\chi_{M}$ and $i_{M}$ are nearly constant in the absence of torques;
furthermore, the evolution of the angles in the symmetric case where two of
the moments of inertia are equal is straightforward

$\psi=2\pi\nu_{\textrm{\scriptsize
prec}}t,\qquad\varphi=2\pi\nu_{\textrm{\scriptsize
rot}}t,\qquad\alpha=\textrm{constant}.$ (3)

In the case where the star is oblate the two frequencies have opposite signs,
and the precession is retrograde. Even in the biaxial case, the observed
rotation frequency defined as the inverse of the timescale between the
crossings of PA through $\chi_{M}$, for example, is not constant through the
precession (see Extended Data Figure 7)[10]. The value of
$\nu_{\textrm{\scriptsize rot}}$ is the mean rotation frequency over the
precession period.

The situation for a triaxial body is somewhat more complicated but also
straightforward[24]. Let us define the three eigenvalues of the moment of
inertia tensor to be $I_{3}>I_{2}>I_{1}$. If we let $\alpha_{0}$ to the be
minimum spin misalignment which occurs when the spin axis is nearest the axis
with the smallest moment of inertia ($I_{1}$), we can define the following
parameters

$k^{2}=\frac{I_{3}\left(I_{2}-I_{1}\right)}{I_{1}\left(I_{3}-I_{2}\right)}\tan^{2}\alpha_{0},\qquad
n=\frac{I_{1}(I_{3}-I_{2})}{I_{2}(I_{3}-I_{1})},$ (4)

where the time parameter through the precession is given by

$\tau=4K(k)\nu_{\textrm{\scriptsize prec}}t,$ (5)

where $K(k)$ is a complete elliptic integral of the first kind (ellipk in
scipy). The precession rate is

$\nu_{\textrm{\scriptsize
prec}}=\frac{\Omega_{3,0}}{4K(k)}\sqrt{\frac{(I_{3}-I_{2})(I_{3}-I_{1})}{I_{1}I_{2}}},$
(6)

where $\Omega_{3,0}$ is the angular velocity about the minor axis at the time
of minimum spin misalignment.

Using the Jacobi elliptic functions (ellipj in scipy) we have

$\tan\psi(\tau)=\sqrt{n}\
\frac{\textrm{cn}(\tau|k^{2})}{\textrm{sn}(\tau|k^{2})},~{}~{}\cos\alpha(\tau)=\textrm{dn}(\tau|k^{2})\cos\alpha_{0},$
(7)

where $\tan\psi$ is inverted to take account for the sign of the numerator and
denominator so the range is zero to $2\pi$ (e.g. by using arctan2). The Jacobi
elliptic functions are periodic with a period of $K(k)$ so the functions
$\psi(\tau)$ and $\alpha(\tau)$ are also periodic. The expression for the
angle $\psi$ ensures that $\psi$ decreases in time, so the precession again is
retrograde as expected. The final parameter is the rotational angle $\varphi$
which is given by[24]

$\varphi=\left[2\pi\nu_{\textrm{\scriptsize
rot}}t-\frac{2\pi\nu_{\textrm{\scriptsize
rot}}}{\Omega_{3,0}}\frac{I_{2}-I_{1}}{\sqrt{(I_{3}-I_{2})(I_{3}-I_{1})}}\sqrt{\frac{I_{2}}{I_{1}}}\int_{0}^{\tau}\cos^{2}\psi\
d\tau^{\prime}\right]\left(1-\frac{I_{2}-I_{1}}{I_{1}}\langle\cos^{2}\psi\rangle\right)^{-1}.$
(8)

and in the approximation where we neglect the differences in the moments of
inertia beyond first order we have $\Omega_{3,0}\approx
2\pi\nu_{\textrm{\scriptsize rot}}\cos\alpha_{0}$ and

$\varphi=2\pi\nu_{\textrm{\scriptsize
rot}}t-\frac{1}{\cos\alpha_{0}}\frac{I_{2}-I_{1}}{\sqrt{(I_{3}-I_{2})(I_{3}-I_{1})}}\int_{0}^{\tau}\left(\cos^{2}\psi-\langle\cos^{2}\psi\rangle\right)d\tau^{\prime}.$
(9)

These two integrals can be expressed in closed form using Jacobi theta
functions with complex arguments[24]; however, it is straightforward to
integrate them numerically. Although the second term in this expression is
explicitly periodic in $\tau$ with a period of $K(k)$, the first term is not,
so the star does not necessarily return to the same configuration.

## References

  * [1] Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M. & Andrae, R.  Estimating Distances from Parallaxes. V. Geometric and Photogeometric Distances to 1.47 Billion Stars in Gaia Early Data Release 3\.  _AJ_ 161, 147 (2021).  2012.05220.
  * [2] Tananbaum, H. _et al._ Discovery of a Periodic Pulsating Binary X-Ray Source in Hercules from UHURU.  _ApJ_ 174, L143 (1972).
  * [3] Giacconi, R. _et al._ Further X-ray observations of Hercules X-1 from Uhuru.  _ApJ_ 184, 227 (1973).
  * [4] Petterson, J. A., Rothschild, R. E. & Gruber, D. E.  A Model for the 35 Day Variations in the Pulse Profile of Hercules X-1.  _ApJ_ 378, 696 (1991).
  * [5] Scott, D. M., Leahy, D. A. & Wilson, R. B.  The 35 Day Evolution of the Hercules X-1 Pulse Profile: Evidence for a Resolved Inner Disk Occultation of the Neutron Star.  _ApJ_ 539, 392–412 (2000).  astro-ph/0002327.
  * [6] Brecher, K.  Her X-1: A Precessing Binary Pulsar?  _Nature_ 239, 325–326 (1972).
  * [7] Truemper, J., Kahabka, P., Oegelman, H., Pietsch, W. & Voges, W.  EXOSAT Observations of the 35 Day Cycle of Hercules X-1: Evidence for Neutron Star Precession.  _ApJ_ 300, L63 (1986).
  * [8] Staubert, R. _et al._ Two ~35 day clocks in Hercules X-1: evidence for neutron star free precession.  _A &A_ 494, 1025–1030 (2009).  0811.4045.
  * [9] Weisskopf, M. C. _et al._ The Imaging X-Ray Polarimetry Explorer (IXPE): Pre-Launch.  _Journal of Astronomical Telescopes, Instruments, and Systems_ 8, 026002 (2022).  2112.01269.
  * [10] Kolesnikov, D., Shakura, N. & Postnov, K.  Evidence for neutron star triaxial free precession in Her X-1 from Fermi/GBM pulse period measurements.  _MNRAS_ 513, 3359–3367 (2022).  2204.06408.
  * [11] Doroshenko, V. _et al._ Determination of X-ray pulsar geometry with IXPE polarimetry.  _Nature Astronomy_ 6, 1433–1443 (2022).  2206.07138.
  * [12] Soffitta, P. _et al._ The Instrument of the Imaging X-Ray Polarimetry Explorer.  _AJ_ 162, 208 (2021).  2108.00284.
  * [13] Di Marco, A. _et al._ Handling Background in IXPE polarimetric data.  _AJ_ 165, 143 (2023).  2302.02927.
  * [14] González-Caniulef, D., Caiazzo, I. & Heyl, J.  Unbinned likelihood analysis for X-ray polarization.  _MNRAS_ 519, 5902–5912 (2023).  2204.00140.
  * [15] Soong, Y., Gruber, D. E., Peterson, L. E. & Rothschild, R. E.  Spectral Behavior of Hercules X-1: Its Long-Term Variability and Pulse Phase Spectroscopy.  _ApJ_ 348, 641 (1990).
  * [16] Postnov, K. _et al._ Variable neutron star free precession in Hercules X-1 from evolution of RXTE X-ray pulse profiles with phase of the 35-d cycle.  _MNRAS_ 435, 1147–1164 (2013).  1307.6026.
  * [17] Gnedin, Y. N., Pavlov, G. G. & Shibanov, Y. A.  The effect of vacuum birefringence in a magnetic field on the polarization and beaming of X-ray pulsars.  _Soviet Astronomy Letters_ 4, 117–119 (1978).
  * [18] Pavlov, G. G. & Shibanov, Y. A.  Influence of vacuum polarization by a magnetic field on the propagation of electromagnetic waves in a plasma.  _Soviet Journal of Experimental and Theoretical Physics_ 49, 741 (1979).
  * [19] Heyl, J. S. & Shaviv, N. J.  Polarization evolution in strong magnetic fields.  _MNRAS_ 311, 555–564 (2000).  astro-ph/9909339.
  * [20] Heyl, J. & Caiazzo, I.  Strongly Magnetized Sources: QED and X-ray Polarization.  _Galaxies_ 6, 76 (2018).  1802.00358.
  * [21] Radhakrishnan, V. & Cooke, D. J.  Magnetic Poles and the Polarization Structure of Pulsar Radiation.  _Astrophys. Lett._ 3, 225 (1969).
  * [22] Poutanen, J.  Relativistic rotating vector model for X-ray millisecond pulsars.  _A &A_ 641, A166 (2020).  2006.10448.
  * [23] Alpar, A. & Oegelman, H.  Neutron star procession and the dynamics of the superfluid interior.  _A &A_ 185, 196–202 (1987).
  * [24] Landau, L. D. & Lifshitz, E. M.  _Mechanics, Third Edition: Volume 1 (Course of Theoretical Physics)_ (Butterworth-Heinemann, 1976), 3 edn.  URL http://www.worldcat.org/isbn/0750628960.
  * [25] Shakura, N. I. _et al._ Observations of Her X-1 in low states during SRG/eROSITA all-sky survey.  _A &A_ 648, A39 (2021).  2102.12983.
  * [26] Kosec, P. _et al._ Vertical wind structure in an X-ray binary revealed by a precessing accretion disk.  _NatAst_ (2023).  2304.05490.
  * [27] Lipunov, V. M. & Shakura, N. I.  Interaction of the accretion disk with the magnetic field of a neutron star.  _Soviet Astronomy Letters_ 6, 14–17 (1980).
  * [28] Lai, D.  Magnetically Driven Warping, Precession, and Resonances in Accretion Disks.  _ApJ_ 524, 1030–1047 (1999).  astro-ph/9904110.
  * [29] Leahy, D. A. & Abdallah, M. H.  HZ Her: Stellar Radius from X-Ray Eclipse Observations, Evolutionary State, and a New Distance.  _ApJ_ 793, 79 (2014).  1406.6138.
  * [30] Shakura, N. I., Postnov, K. A. & Prokhorov, M. E.  On some features of free precession of a triaxial body: the case of HER X-1.  _A &A_ 331, L37–L40 (1998).  astro-ph/9801033.
  * [31] Steiner, A. W., Gandolfi, S., Fattoyev, F. J. & Newton, W. G.  Using neutron star observations to determine crust thicknesses, moments of inertia, and tidal deformabilities.  _Phys. Rev. C_ 91, 015804 (2015).  1403.7546.
  * [32] Suleimanov, V., Potekhin, A. Y. & Werner, K.  Models of magnetized neutron star atmospheres: thin atmospheres and partially ionized hydrogen atmospheres with vacuum polarization.  _A &A_ 500, 891–899 (2009).  0905.3276.
  * [33] Garg, A., Rawat, D., Bhargava, Y., Méndez, M. & Bhattacharyya, S.  Flux-resolved Spectropolarimetric Evolution of the X-Ray Pulsar Hercules X-1 Using IXPE.  _ApJ_ 948, L10 (2023).

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). JH acknowledges support
from the Natural Sciences and Engineering Research Council of Canada (NSERC)
through a Discovery Grant, the Canadian Space Agency through the co-
investigator grant program, and computational resources and services provided
by Compute Canada, Advanced Research Computing at the University of British
Columbia, and the SciServer science platform (www.sciserver.org). D.G.-C.
acknowledges support from a CNES fellowship grant. JP and SST were supported
by the Academy of Finland grants 333112, 349144, 349373, and 349906 and the
Väisälä Foundation. VD and VFS thank the German Academic Exchange Service
(DAAD) travel grant 57525212. We used Astropy:111http://www.astropy.org a
community-developed core Python package and an ecosystem of tools and
resources for astronomy.

The data used for this analysis are available through HEASARC under IXPE
Observation IDs: 01001899, 02003801 and 02004001.

The software used for this analysis is available at
https://github.com/UBC-Astrophysics/IXPE-Analysis

J.H. analysed the data and wrote the draft of the manuscript. J.P. led the
work of the IXPE Topical Working Group on Accreting Neutron Stars and
contributed to the interpretation and the text. V.D., D.G.-C., I.C., A.M.,
S.S.T., D.M. and V.F.S. contributed to interpretation of the results and
writing of the text. A.M. created Fig. 3. M.B. and G.G.P. acted as internal
referees of the paper and contributed to interpretation. Other members of the
IXPE collaboration contributed to the design of the mission and its science
case and planning of the observations. All authors provided input and comments
on the manuscript

The authors declare that they have no competing financial interests.

Correspondence and requests for materials should be addressed to J.H. (email:
heyl@phas.ubc.ca).

Extended Data Figure 1: Posteriors for the RVM for the First Main-On (2022
January) The two-dimensional contours correspond to $68\%$, $95\%$ and $99\%$
confidence levels. The histograms show the normalized one-dimensional
distributions for a given parameter derived from the posterior samples.
Extended Data Figure 2: Posteriors for the RVM for the First Main-On (early)
Extended Data Figure 3: Posteriors for the RVM for the First Main-On (late)
Extended Data Figure 4: Posteriors for the RVM for the Short-On (2023 January)
Extended Data Figure 5: Posteriors for the RVM for the Second Main-On (2023
February) Extended Data Figure 6: Euler Angles as defined in the Methods. The
axes $\hat{1},\hat{2}$ and $\hat{3}$ denote the body axes of the neutron star
crust, and the axes $x,y$ and $z$ denote a frame of reference with the
direction to Earth fixed. The line of sight is indicated by the black line
labelled “Earth”, makes an angle $i_{M}$ with the $z$-axis and lies in the
$x-z-$plane. The magnetic pole is indicated by the purple line, makes an angle
$\theta_{B}$ with the $\hat{3}-$axis and lies in the $\hat{2}-\hat{3}-$plane.
The vector labeled N denotes the line of nodes about which the rotation by
$\alpha$ is performed. Extended Data Figure 7: Spin Frequency Deviation. After
the long-term trend of the spin frequency of Hercules X-1 is removed[10] from
the Fermi GBM frequency measurements, the variation with superorbital period
becomes apparent. The curves are not a fit to the data, but rather the
expectations from the two small-misalignment precession models outlined in the
text. The results from the prolate model in Fig. 4 are superimposed in green
(single hump), and those from triaxial model are shown in orange (double
hump). The xX-ray flux as observed by the Swift observatory is depicted by an
olive curve along the bottom of the plot.