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Does the Gamma-Ray Binary LS I +61°303 Harbor a Magnetar?
Arthur G. Suvorov
1,2
and Kostas Glampedakis
1,3
1
Theoretical Astrophysics, Eberhard Karls University of Tübingen, Tübingen, D-72076, Germany
2
Manly Astrophysics, 15/41-42 East Esplanade, Manly, NSW 2095, Australia; arthursuvorov@manlyastrophysics.org
3
Departamento de Física, Universidad de Murcia, Murcia, E-30100, Spain; kostas@um.es
Received 2022 August 17; revised 2022 September 28; accepted 2022 October 17; published 2022 November 29
Abstract
The high-mass X-ray binary LS I +61°303 is also cataloged as a gamma-ray binary as a result of frequent
outbursts at TeV photon energies. The system has released two soft-gamma flares in the past, suggesting a
magnetar interpretation for the compact primary. This inference has recently gained significant traction following
the discovery of transient radio pulses, detected in some orbital phases from the system, as the measured rotation
and tentative spin-down rates imply a polar magnetic field strength of B
p
10
14
G if the star is decelerating via
magnetic dipole braking. In this paper, we scrutinize magnetic field estimates for the primary in LS I +61°303 by
analyzing the compatibility of available data with the system’s accretion dynamics, spin evolution, age limits,
gamma-ray emissions, and radio pulsar activation. We find that the neutron star’s age and spin evolution are
theoretically difficult to reconcile unless a strong propeller torque is in operation. This torque could be responsible
for the bulk of even the maximum allowed spin-down, potentially weakening the inferred magnetic field by more
than an order of magnitude.
Unified Astronomy Thesaurus concepts: Magnetars (992);Magnetic fields (994);Accretion (14);Pulsars (1306)
1. Introduction
Weng et al. (2022)recently detected radio pulsations from the
gamma-ray binary LS I +61°303 (hereafter LSI)using the Five-
hundred-meter Aperture Spherical radio Telescope (FAST).
Barycentric correction and pulse alignment established a milli-
second rotation period for the primary, P=269.15508(16)ms,
together with a spin-down rate of =´
--
P4.2 1.2 10 ss
10 1
()
(though this measurement should be viewed as tentative; see
Section 2). The unambiguous detection of radio pulsations from
LSI settles the long-held debate about the nature of the primary
object as a neutron star, which was inconclusive from other,
multiwavelength data (Torres et al. 2010 and references therein).
As a couple of soft-gamma flares have been detected in the
direction of LSI (Dubus & Giebels 2008; Burrows et al. 2012),it
is tempting to theorize that the primary is a magnetar (Papitto
et al. 2012;Torresetal.2012). This proposition is now supported
by magnetic dipole braking theory, which predicts a polar
fieldstrengthof »´ ~´
B
PP6.410 G710
p19 14
G, which
ranks quite highly among the Galactic magnetar population
4
(Olausen & Kaspi 2014).
In this work, it is our goal to collect the various pieces of
evidence from observations of LSI to reexamine the magnetar
hypothesis. In this context, we define “magnetar”specifically
through an ultrastrong magnetic field, rather than using an
empirical definition involving outburst activity. One piece
of information concerns viable scenarios for explaining the
Pand
P
values obtained from radio timing. Although subject
to (super)orbital variability likely owing to the high eccentricity
of the binary (e=0.63 ±0.11, Casares et al. 2005; though
see Kravtsov et al. 2020, who argued e<0.2), the X-ray
luminosity in the 3–10 keV band is relatively modest,
10
33
L
X
/erg s
−1
10
34
, as the system cycles between
apastron (low)and periastron (high; Esposito et al. 2007;
Romero et al. 2007; Hadasch et al. 2012). This suggests a sub-
Eddington accretion rate even at periastron and therefore that a
propeller torque may have a hard time contributing to the bulk
of the spin-down (Ghosh & Lamb 1979; Wang 1995; Frank
et al. 2002). However, as recently shown by us in Glampedakis
& Suvorov (2021), large torques can be achieved when
relaxing various assumptions about the magnetospheric
geometry and the strength of induction-generated toroidal
fields (see also Das et al. 2022). Using the models developed
therein, we show that the spin-down data do not necessarily
point toward a magnetar in LSI, even if we accept such a
large
P
.
Another consideration concerns the switch-on of the object
as a radio pulsar. Conventional wisdom suggests that electron–
positron pair production occurring in unscreened magneto-
spheric “gaps”is a necessary ingredient to excite radio
pulsations in a neutron star magnetosphere (Goldreich & Julian
1969; Ruderman & Sutherland 1975; though see Melrose et al.
2021 for a critique). Depending on the multipolarity of the
neutron star’s magnetic field and the structure of these gaps, a
broadly defined “death valley”separates inactive pulsars
from those able to pair produce (see Section 5; Chen &
Ruderman 1993; Hibschman & Arons 2001). As LSI is rotating
quite rapidly, the magnetic field required to avoid the death
valley is moderate (at most ∼10
13
G with an outer gap), and the
radio activity of the source is unsurprising. An obvious
question then is why have radio pulsations only been observed
now? This is discussed in Section 2.2.
Bednarek (2009a,2009b; see also Khangulyan et al. 2007;
Papitto et al. 2014)argued that the gamma-ray shining of the
binary mandates a minimum surface B-field strength of order
∼10
14
G; the acceleration of charges near the magnetospheric
boundary via Fermi processes at the rate necessary to produce
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 https://doi.org/10.3847/1538-4357/ac9b48
© 2022. The Author(s). Published by the American Astronomical Society.
Original content from this work may be used under the terms
of the Creative Commons Attribution 4.0 licence. Any further
distribution of this work must maintain attribution to the author(s)and the title
of the work, journal citation and DOI.
4
A list of known magnetars, together with their observed properties, is
maintained at http://www.physics.mcgill.ca/∼pulsar/magnetar/main.html.
1
the TeV radiation frequently observed from the source (Albert
et al. 2006; Acciari et al. 2009)requires strong Poynting fluxes.
This estimate, however, depends on the location of the Alfvén
surface, among other things, which is sensitive to the accretion
geometry. Another possibility involves colliding winds; see
Section 2.6.
The above considerations are discussed in this work, which
is organized as follows. Section 2provides a detailed overview
of the observed properties of LSI, preparing the ground for the
theoretical analysis of the subsequent sections. Section 3forms
the main part of the paper and is devoted to the spin evolution
and accretion dynamics of LSI. The physics related to gamma-
band emissions from LSI are discussed in Section 4. The
specifics of radio pulsar activation for LSI are the subject of
Section 5. Section 6is a digression to the paper’s main topic,
discussing the gravitational-wave (GW)observability of LSI by
present and near-future detectors. Concluding remarks can be
found in Section 7.
Throughout, we use an asterisk to denote stellar parameters
like mass, radius, and moment of inertia. We also adopt the
following normalizations: M
1.4
=M
å
/1.4 M
e
,R
6
=R
å
/10
6
cm,
=--
MM M10 yr
nn1
,L
n
=L
X
/10
n
erg s
−1
,B
n
=B/10
n
G,
and P
−1
=P/0.1 s.
2. An Overview of the Observed Properties of LSI
Weng et al. (2022)detected 42 single pulses from LSI using
FAST data from 2020 January that, after alignment and
barycentric correction, revealed a source pulsating with period
P=0.269 s. The best-fit value for the period derivative was
found to be =´
--
P4.2 1.2 10 ss
10 1
() . This value, quoted
from Supplementary Figure 1 in Weng et al. (2022), was
obtained by folding data to maximize the signal-to-noise ratio
using the prepfold pipeline within the PulsaR Exploration and
Search TOolkit (PRESTO)package (Ransom et al. 2002). The
cited period derivative and its uncertainties are therefore not
obtained from direct timing. As noted by Weng et al. (2022),
there are only 3 hr worth of observations spanning a tight orbital
phase (∼0.58); thus, the authors were unable to recover the
Doppler-shifted signals necessary to determine the intrinsic
period derivatives. We can estimate the extent to which (orbitally
averaged)Doppler modulation imprints on the period derivative
through pwD+PPPaie2sin1cos
Dop orb 2
∣
()∣ ( ) ( )
(see, e.g.,
Israel et al. 2017).Here
a
isin and ωare the projected semimajor
axis in units of light-seconds and longitude of periastron,
respectively. Under the most favorable orbital solution of Casares
et al. (2005; see Table 3 therein),wefind D-
P10
Dop 10
∣
()∣ ;
thus,
P
can be trusted to within a factor of 2. At worst, though,
the maximum error could exceed the reported
P
.
Throughout this work, we operate under the assumption of
an upper limit ´--
P5.6 10 ss
10 1
and intend to show that,
even if the extreme value is used, much of it could be attributed
to accretion torques. In the remainder of this section, we recap
which observational arguments support either a magnetar or
nonmagnetar interpretation.
2.1. Radio Pulsations: Spin-down
A naive application of standard magnetic braking in vacuum
implies a polar field strength of B
p
7×10
14
G, which is
clearly of magnetar level. The nonvacuum Spitkovsky (2006)
expression reduces this by only a factor of 2. A similar
estimate for B
p
is obtained if we use the particle-wind modified
formula of Harding et al. (1999)and Thompson et al. (2000),
even if we assume a wind luminosity as high as the maximum
spin-down energy rate. As we show in detail in Section 3,
however, a strong propeller torque may be able to relieve the
magnetic field from the spin-down demands to some degree.
For reasonable values of the torque particulars, we find that
values of B
p
∼10
13
G could be a viable alternative, even
assuming such a large
P
.
2.2. Radio Pulsations: Death Valley
The existence of LSI as a radio pulsar implies that the object
should reside outside of the “graveyard”(Ruderman &
Sutherland 1975), which implies a minimum B-field strength.
As we show in Section 5, however, for polar-gap configura-
tions, this minimum is only of order ∼10
11
G, though it could
reach ∼10
13
G for an outer gap (Chen & Ruderman 1993;
Hibschman & Arons 2001).
Given that pulsations were only observed recently, however
(2020 January 7; Weng et al. 2022), one may argue that
magnetic substructures atop the crust (“starspots”; Zhang et al.
2007)or in the magnetosphere (“twists”; Beloborodov 2009)
may have only just developed or (Hall)drifted into regions that
are conducive to radio activity. Magnetospheric twists, injected
following quake activity (that may have sparked the soft-
gamma flares in 2008 and 2012), can survive on long diffusion
timescales (years; Parfrey et al. 2013)and temporarily
reconfigure the geometry of the emission zone. Furthermore,
many of the known “radio magnetars”experience accelerated
spin-down following flare activity. Archibald et al. (2015)
reported that the spin derivative of 1E 1048.1–5937 varied by a
factor of ∼5 starting ∼100 days after each of its outbursts,
oscillating for ∼years before stabilizing. A magnetar inter-
pretation may therefore simultaneously explain a large
P
at the
time radio pulsations were observed and why the source
switched on at all.
Alternative explanations involve the mode of accretion or
dynamics related to the companion. The ram pressure of infalling
material may have temporarily subsided, allowing for the source to
activate as a radio pulsar, similar to what is thought to happen for
the “swinging”pulsars PSR J1023+0038 and IGR J18245–2452
(Tam et al. 2010; Papitto et al. 2013). The absence of X-ray
pulsations casts doubt on this interpretation, however. Another
possibility is that the pulsar beam is most often directed through
the wind from the companion and regularly quenched because the
region is optically thick to free–free absorption (Chernyakova &
Malyshev 2020). Zdziarski et al. (2010)estimated the optical
depth as t»´ ´ -
a510 310cm
ff 3123
()
,whichis?1evenat
apastron if e0.6.
Regardless, the role of the vast aperture available to FAST is
unquestionable, as the mean pulsed signal was at the μJy
level (Weng et al. 2022). This means we cannot conclude that
the source is only now radio-loud. Future observations will
help to determine whether the source is always “on”but only
visible at certain orbital cycles, which would favor the latter
explanations above, or has since shut off, which would point
toward dynamical phenomena in the magnetosphere.
2.3. X-Ray/Soft-gamma Emissions: Bursts
Bursts with luminosities exceeding 10
37
erg s
−1
, likely
though not definitely associated with LSI (see Muñoz-Arjonilla
et al. 2009), were detected by the Swift Burst Alert Telescope
2
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
in 2008 (Dubus & Giebels 2008)and again in 2012 (Burrows
et al. 2012). These short (0.3 s)bursts are characteristic of
those observed in magnetars (e.g., 1E 2259+586, with polar
field strength ∼10
14
G; Gavriil et al. 2004). Torres et al. (2012)
suggested that the low intensity of the bursts favors an
interpretation of a modest Bmagnetar with B∼5×10
13
G, and
that they are at variance with known type I X-ray bursts and
unlikely to be accretion-driven. Numerical simulations of
magnetically induced stresses in neutron stars suggest that local
field strengths of order 2×10
14
G are necessary to fracture
the crust (Lander et al. 2015), which is a popular model for
driving flares in soft-gamma repeaters (e.g., Pons & Perna
2011).
If the neutron star occasionally accretes, a stipulation that is
defensible because ∼kilosecond-long X-ray bursts have been
observed (Li et al. 2011), its magnetic field may be “buried.”
Burial reduces the global dipole moment while tangling the
field near the surface. Simulations suggest that strong (10
14
G)patches within accreted mountains are consistently
generated even if the global field, controlling magnetospheric
radii and spin-down, is 10
13
G(Suvorov & Melatos 2020;
Fujisawa et al. 2022). Buried fields may persist on ohmic
(10
5
yr)timescales (Vigelius & Melatos 2009).
2.4. X-Ray Emissions: Persistent, Hard
Hadasch et al. (2012)found that the 3–10 keV flux for LSI is
of order 4×10
−12
erg cm
−2
s
−1
at apastron and 2×
10
−11
erg cm
−2
s
−1
at periastron. The recent Gaia survey
indicates a distance of d
LSI
=2.65(9)kpc (Lindegren et al.
2021), implying that the overall X-ray luminosity is of order
10
33
L
X
/erg s
−1
10
34
. Esposito et al. (2007)cited a
column density of N
H
=5.7(3)×10
21
cm
−2
, while Frail &
Hjellming (1991)gave N
H
10
22
cm
−2
, implying a factor of
2 uncertainty in the unabsorbed luminosity. These persistent,
nonthermal X-ray emissions could be accretion-powered, implying
that L
X
can be converted into an estimate for the mass accretion
rate M
(see Section 3.2). An alternative reservoir for L
X
could be
the colliding wind (see Section 2.6), in which case, any inversions
for the accretion rate from L
X
will be overestimates. It is important
to note, however, that the relevant L
X
at the time the source was
radio visible is unknown. Weng et al. (2022)detected pulses when
the system had an orbital phase of ∼0.58 (see their Table 1), but
given the uncertainties in the orbital modeling itself (see Kravtsov
et al. 2020 for a detailed discussion), it is nontrivial to deduce the
orbital separation at the observation time.
2.5. X-Ray (Non)Emissions: Persistent, Soft
Despite long-term monitoring campaigns (e.g., Esposito
et al. 2007; Paredes et al. 2007), there is little evidence for a
thermal component to LSI’s persistent emissions. Isolated
magnetars display relatively high surface luminosities (Olausen
& Kaspi 2014), thought to be provided by the secular decay of
their internal magnetic fields (Thompson & Duncan 1993,
1996). A conservative upper limit is set by L
X,therm
L
X
∼
10
34
erg s
−1
; unfortunately, this is of little use to the present
analysis, as it is comparable to the typical magnetar quiescent
luminosity (Olausen & Kaspi 2014). Magnetothermal simula-
tions carried out by Anzuini et al. (2022a)show that the surface
luminosity of a heavy (M∼1.8 M
e
)but highly magnetized
(B10
14
G)neutron star can drop below 10
34
erg s
−1
even
after ∼centuries if the nucleon and hyperon direct Urca
processes are active (see their Figures 5 and B1 and also
Anzuini et al. 2022b). As such, in order for thermal emission
limits to have a true impact as concerns the nature of LSI,
L
X,therm
would need to be pushed downward by (at least)an
order of magnitude. Although a more precise statement can
be made using spectral fits, it is likely that even fields of
strength 10
15
G cannot be definitively ruled out if fast Urca
mechanisms operate (Lattimer et al. 1991).
2.6. Gamma-Ray Emissions
As argued by Bednarek (2009a,2009b; see also Khangulyan
et al. 2007), surrounding the line where magnetic pressure
balances the gravitational pressure of infalling matter (i.e., the
Alfvén surface), there exists a turbulent region where electrons are
accelerated via Fermi processes, especially if the neutron star
velocity exceeds the local speed of sound at the inner boundary of
the corona (supersonic propeller state; Papitto et al. 2012).
Synchrotron processes and inverse-Compton scattering of radia-
tion from the Be star companion result in the electrons achieving
a Lorentz factor of order
g
»´ -
BM1.8 10
max 614
514 10
37
(see
Equation (14)in Bednarek 2009b). Requiring that
g
10
max
6
therefore demands a minimum Bfield.
An alternative location for the production of high-energy
emissions is in a colliding wind (e.g., Maraschi & Treves 1981).
The continuous outflow of relativistic particles from the pulsar
encounters the stellar wind from the Be companion, resulting in
a termination shock at the point where the pressures balance
(Dubus 2006). Particles are scattered at the shock front and
accelerated. Given the high maximum spin-down luminosity of
the primary, the pulsar wind is easily energetic enough to
accommodate TeV emissions (Dubus 2013). Using smoothed
particle hydrodynamic (SPH)simulations in 3D, Romero et al.
(2007)found that the geometry of the termination front does not
match the morphology of the radio observations in LSI, though
their simulations adopt wind luminosities of 10
36
erg s
−1
,far
lower than the maximum spin-down luminosity. Torres et al.
(2012)separately argued that it is difficult to explain the
anticorrelation between GeV and TeV emissions from LSI with a
colliding wind. The external torques associated with this model
may be minimal due to a high degree of spherical symmetry, thus
making magnetic spin-down the bulk contributor to
P
.A
magnetar conclusion for a large
P
would therefore be hard to
escape.
2.7. Age Limits
The lack of an associated supernova remnant for LSI (Frail
et al. 1987)implies a likely age of at least a few kiloyears
(Papitto et al. 2012). By contrast, if magnetic braking
dominates over any accretion-related torques, and the braking
index of the source is held constant at n=3 over the lifetime of
LSI, one predicts an age of 10 yr. This is obviously excluded
by observations dating back to the late 1950s (Hardorp et al.
1959). It should be noted, however, that magnetars, and pulsars
more generally (Shaw et al. 2022), exhibit noisy spin-down.
For example, the magnetar Swift J1834.9–0846 resides at the
center of supernova remnant W41 (Tian et al. 2007), strongly
suggesting their association (Granot et al. 2017).Ann=3
history for the former implies a star of age τ
sd
≈4.9 kyr
(Olausen & Kaspi 2014), though the true age of W41 was
estimated by Tian et al. (2007)to be between 60 and 200 kyr,
3
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
depending on expansion assumptions (e.g., the extent of
radiative cooling).
In any case, it is difficult to imagine a scenario where the
magnetic field gets significantly stronger over time; therefore,
magnetic braking today should occur at a similar or weaker rate
than has occurred historically. An alternative explanation to
avoid this age-related problem is that the object possesses a
weaker field, though is presently spinning down at a high rate
due to a large propeller torque.
Based on its kinematic velocity relative to the Heart Nebula
cluster IC 1805, Mirabel et al. (2004)argued that LSI may have
been ejected from the young complex of massive stars from
whence it came ≈1.7 ±0.7 Myr ago. Such an age is not
unusual for binaries involving neutron stars, though is virtually
impossible to accommodate with a (present-day)subsecond
magnetar scenario, as spin-down and field decay prevent old
objects from being both fast and strongly magnetized
simultaneously.
2.8. Magnetars in Binaries?
King & Lasota (2019)suggested that magnetars in binaries
should be rare. They argued that previous suggestions that
some high-mass X-ray binaries (HMXBs), and ultraluminous
sources in particular, contain magnetars are problematic (see
Israel et al. 2017; Igoshev & Popov 2018). The most natural
evolutionary scenario involves one member from a binary star
(e.g., of class OB)undergoing core collapse and eventually
leaving behind an X-ray binary with a magnetar primary. Large
angular momentum reserves may be necessary, however, to
entice dynamo activity in the protostar to generate fields
exceeding ∼10
15
G(Thompson & Duncan 1993). King &
Lasota (2019)argued in this case that the supernova will be
superluminous (though see Vink & Kuiper 2006), likely
destroying the companion and leaving only an isolated
magnetar; see Popov (2016)and White et al. (2022).
If instead, a magnetar captures a companion, such a process
must occur within ∼a megayear after birth, or else the field is
likely to have decayed sufficiently to depose the star of its
magnetar status. Though the probability of a capture happening
within this time is low, the issue of magnetars in binaries
becomes one of semantics to some degree, as there are some
“low-field”magnetars (most notably SGR 0418+5729, which
houses a surface field of only ∼6×10
12
G; Rea et al. 2013).
Fields of this order could persist over megayear timescales
without significant decay, depending on the electrical con-
ductivity of the crust, which could be hampered by impurities
(Igoshev & Popov 2018). Field decay can also be stalled by
plastic flows in a highly magnetized crust, as such flows tend to
oppose the existing electron fluid motions (Gourgouliatos &
Lander 2021), thus preventing the formation of small-scale,
tangled fields that are most at risk of ohmic decay.
The following sections offer a more detailed discussion of
the above topics. Each one concludes with a “final verdict”as
to the magnetar versus no-magnetar nature of LSI.
3. Spin History
In this section, we show how combinations of electro-
magnetic braking with various field geometries, together with
accretion-induced braking torques, can be applied to investigate
the spin history of the primary in LSI.
3.1. Electromagnetic Spin-down
Neutron stars intrinsically decelerate through electro-
magnetic and gravitational torques, the relative impacts of
which can be quantified in terms of a braking index, n. For a
simple, centered dipole, one has n=3, while for a general ℓ-
polar field, one has n=2ℓ+1. Braking indices n<3 are also
theoretically possible in cases where particle winds make up
the bulk of the spin-down torque (Maraschi & Treves 1981;
Thompson et al. 2000)or if the star is precessing and/or
oblique (Melatos 1997; Pétri 2019).
Including corrections induced by a force-free magnetosphere
according to the Spitkovsky (2006)formulae, the spin
evolution of an inclined rotator may be approximately
described by (see also, e.g., Mastrano et al. 2011)
pa»+
-
-+
PBP R
Ic k261sin, 1
npnn
n
1
223
2
() ( ) ()
for inclination angle α, magnetospheric factor k≈1, and
moment of inertia I
å
. Throughout this work, we adopt
»
I
MR0.38 2
as pertains to a Tolman-VII equation of state
(Lattimer & Prakash 2001)for stellar mass M
å
. Equation (1)
can be used to infer B
p
as a function of nand α, given values of
Pand
P
. More generally, however, Equation (1)can be solved
for some (time-dependent)choices of nand B
p
, and one can
infer the star’s age. Late-time solutions to Equation (1)are
relatively insensitive to the exact value of the initial birth period,
P(0)=P
0
. In particular, the spin-down age of the object is
approximated by t»-Pn P1
sd
[( )
]
. Demanding τ
sd
>60 yr
at minimum (owing to the object’s discovery by Hardorp et al.
1959), one then obtains n1.4, which in turn implies, in fact,
that B
p
≈10
12
G for an orthogonal rotator–nonmagnetar. The
absence of a detectable supernova remnant, however, suggests
that τ
sd
?10
3
yr, and, unless the braking index
5
is very close to
unity, additional physics is needed to explain the object’s
behavior.
3.2. Accretion Torques
Although accretion disks are rare in HMXB systems,
Hayasaki & Okazaki (2004)found with SPH simulations that
a persistent disk comes to surround the neutron star in Be/X-
ray binaries with periods P
orb
∼24 days, independent of the
model specifics. We are therefore justified in describing LSI as
an accreting system with a thin disk threaded by the neutron
star’s magnetic field (see Ghosh & Lamb 1979; Frank et al.
2002). This working hypothesis comes with a certain degree of
approximation; the geometry of the actual accretion flow could
depart from the well-ordered axisymmetric structure of a thin
equatorial disk, especially if there is a strong quasi-spherical
wind component. To some extent, some of the key parameters
discussed below, like the magnetospheric radius and
L
M
X
()
relation, are robust enough to remain accurate even if the
accretion flow would not take the form of a thin disk. On the
other hand, if we were to consider the opposite limit of
spherical Bondi accretion, say, then the resulting torque would
5
In what follows, the braking index here applies only to intrinsic torques in
the sense of Equation (1). Even if external torques modify the overall spin
evolution so that the true observational braking index is different from what we
call nin Equation (1), the same symbol is adopted for ease of presentation.
4
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
be negligible, and, as we discuss below, it is difficult to explain
the P
P
,data of LSI. The thin disk model includes two basic
length scales; the first one is the corotation radius
pn=
R
GM 4
co 2213
()
, which marks the radial distance where
the local Keplerian angular frequency of the orbiting gas
matches the stellar angular frequency. The magnetospheric
radius
x=RBR
GM M 2
A
p
47 127
17 27
()
()
is defined by the energy balance between the gas and the stellar
poloidal magnetic field. Note that while Equation (2)implicitly
assumes the field is a dipole, multipolar corrections are small,
as an ℓ-pole decays as -+
r
ℓ21()
and is therefore weak at radii
r?R
å
(see Section 4.1 of Glampedakis & Suvorov 2021).At
this radius, the accretion flow becomes dominated by the
magnetic field, and the disk is effectively truncated. The
parameter ξ, typically taking values in the range 0.1–1, is a
useful phenomenological device for describing the complex
(and highly uncertain)physics taking place at the boundary
layer around R
A
(see Glampedakis & Suvorov 2021 for details).
The recent general-relativistic simulations conducted by Das
et al. (2022)show that Equation (2)adequately describes the
magnetospheric radius in most cases, though with the general
trend that the effective ξdecreases as the dipole field strength
increases (see Table 1 therein).
In addition to the above two length scales, the light cylinder
radius, R
lc
=c/2πν for speed of light c, is a key quantity for the
pulsar mechanism. One finds that
x»
--
R
R
RB
MMP
289 , 3
A
co
6
12 7
14
47
1.4
10 21
10
27
1
23
()
x»
--
R
R
RB
MM P
22 . 4
A
lc
6
12 7
14
47
1.4
17
10
27
1
()
The R
A
>R
co
regime, likely the relevant one for LSI, is the so-
called propeller regime; the disk is truncated so far away from
the star that the inward motion of the gas is centrifugally
inhibited by the much faster rotating magnetic lines. This
arrangement results in a net spin-down for the neutron star.
An R
A
>R
lc
arrangement means that the magnetic field is
strong enough (and the accretion rate is weak enough)to
provide the neutron star with a clean, gas-free magnetosphere,
thereby allowing for radio emission as in ordinary pulsars.
Based on the above estimate, LSI could sit on either side of the
fence, especially if ξ=1. Given its high orbital eccentricity,
the transition from radio emission to (pulsational)radio silence
could even take place during an orbital period provided M
is
sufficiently variable from apastron to periastron, as in the “flip-
flop”model of Torres et al. (2012).
The standard formula =
L
GM M R
X
for estimating M
is
not applicable to a propellering system, where instead
»LGMMR.5
AX
()
A simple inversion then leads to
x»-
M LBRM0.71 . 6
10 79 33
79
12
49
6
43
1.4
89
()
This expression, when combined with our estimate for the
nonthermal luminosity L
X
, leads to an accretion rate range,
--
M
M
10 10 , 7
4
Edd
2
()
for ξ0.1 (see below)and 1 B
12
100, where »MEdd
´--
M1.5 10 yr
81
is the Eddington accretion rate.
The most basic torque associated with the thin disk model
comes from infalling matter at the lever-arm distance R
A
,
/
=NMGMR .8
AA
12
˙() ()
This is easily modified to account for the relative rotation
between the disk and the magnetic lines, viz.
w=-NN1, 9
mA A
() ()
where the so-called fastness parameter is defined as
//
w
=RR
AAco
32
()
. This torque allows for spin equilibrium,
N
m
=0, as well as a propeller regime, N
m
<0.
A more rigorous modeling of the disk–magnetic field
coupling should account for the induced azimuthal magnetic
field component; the outcome is an accretion torque that
comprises N
A
and a radially integrated contribution from the
entire disk (Wang 1995). This torque has been revised by us in
Glampedakis & Suvorov (2021)to include the boundary layer
parameter ξ, with the result being
//
xxw=+-
-
NN
1
313 2, 10
disk 72A72 A
()()
which combines with the torque(s)implied by Equation (1).
Once the torque is obtained, the spin-up/-down rate of the
neutron star is easily found with the help of
n
p=NI2
(
)
.
The spin-down rate associated with Equation (10)is shown
in Figure 1. Each band is delimited by a mass accretion rate
specified by the range of Equation (7), where the lower limit
(depicted with solid curves in Figure 1)roughly corresponds to
LSI’s revised persistent X-ray luminosity at apastron; see
Equation (5). The dashed curves in Figure 1correspond to
periastron-like M
values, more typical for low-mass X-ray
binaries. The ξ=1 spin-down band is compatible with the
upper limits on
n
provided B
p
5×10
14
G. The ξ=0.1 band
Figure 1. Propeller spin-down rate for LSI, n
, calculated with the torque
(Equation (10)), as a function of the polar field, B
p
/10
12
G. Each of the two
shaded bands corresponds to an
M
range varying from =-
M
10 2
(leftmost
dashed curves)to -M
1
04Edd
(rightmost solid curves). The former (latter)value
corresponds to the estimated quiescent L
X
luminosity of LSI at periastron
(apastron). The left (right)band corresponds to ξ=0.1 (ξ=1). The horizontal
band marks the maximum spin-down rate of LSI, viz. n-< <
--
6.04 10 s
92
-4.39 (Weng et al. 2022).
5
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
does the same job using a somewhat moderate field,
10
13
GB
p
10
14
G. A value ξ=1 is perhaps the most
physically motivated; ξ0.1 is required in order to theoreti-
cally explain the observed spin-up rates of several accretion-
powered X-ray pulsars (Glampedakis & Suvorov 2021), most
notably SAX J1808.4–3658. Becker et al. (2012)also argued
for ξ∼0.1 in a number of systems from cyclotron line
considerations.
The above analysis suggests that LSI’s tentative
n
, or some
other value within an order of magnitude, could be entirely
or partially the result of propeller action, especially if the
instantaneous M
during the spin-down measurement is a factor
of 10–100 higher than the estimated persistent accretion rate. In
that case, the magnetic field inferred from the P
P
,data may be
an overestimation of the neutron star’s true magnetic field,
making LSI either a “low-B”magnetar or a “high-B”pulsar.
3.3. Spin Evolutions: Weaker Field
As demonstrated in Section 3.2, propeller torques alone with
B
p
10
13
G can theoretically explain the maximum
P
value of the
neutron star in LSI, provided that one uses the strong torque
profile (Equation (10)) together with ξ∼0.1 and »-
MM10 2Edd
.
However, the propeller torque will be aided by intrinsic braking
torques, as described in Section 3.1.
In general, if we wish to explain the present-day
P
with a
large Bfield, the historical
P
is also likely to be large, implying
a very young object. This can be avoided if the surface field is
highly multipolar (see Section 3.4)or propeller torques activate
sporadically, including just prior to when the radio pulses were
observed (i.e., at a recent “switching time”defined below).
Figure 2shows a variety of possible evolutionary tracks, with
the left (right)panel showing =--
MM10 10
23
Edd
() with two
different values of ξ(blue and red curves), where propeller
torques are considered small (i.e., MM
Edd
)at times t
5 kyr, though they become large at this fiducial transition point
t=5 kyr. At this switching time, the star enters a phase of rapid
spin-down to allow for an enormous
P
. This is illustrated
within the figure inset, which shows a smooth though rapid
increase in
P
at the switching time. In this simple picture, the
stellar magnetic field is of order B
p
10
13
G, as is typical for
HMXBs (Papitto et al. 2012). The switching time sets a floor to
the age of the system (kiloyears)but also cannot be too large
(10 kyr)for a given B
p
or the system will exceed the
measured spin period without matching the
P
value. As can be
seen in the x==
-
MM10 , 0.09
3Edd
case (right panel, blue
curve), pure n=3 braking already has the star approach the
present-day period value at t≈5 kyr. The key points described
above persist when involving multiple switching times; if the
system dances between propeller and accreting states, the
number switching times could be large, and the system could
be old. Furthermore, the evolution should be “bumpier”than
shown in reality, with periodic variations in M
due to the
orbital eccentricity.
3.4. Spin Evolutions: Stronger Field
One way that a magnetar-like field can be accommodated
while also permitting a relatively old system is if the historical
braking index n5. Such a value is appropriate for a
quadrupolar field. Similar to Figure 2, we suppose that the
propeller mechanism activates at some fiducial time t=5 kyr
with =-
M10
5
(red curve)or -M10 6Edd
(blue curve), as shown
in the left panel of Figure 3. Here we take n=5, which
prevents magnetar-level polar field strengths, B
p
10
14
G,
from slowing down the star too much at early times.
Alternatively, the magnetic geometry may have reconfigured
near this switching time (see Section 2.2), such that the
multipolarity of the field decreased. The right panel shows an
idealized situation, where a pure quadrupole (n=5)reconfi-
gures to a dipole-like (n≈3)field near the switching time
instead, thus increasing the instantaneous spin-down rate (see
Equation (1)). Ignoring the Hall effect, a reduction in the field
multipolarity as time progresses may be expected from ohmic
decay, as more tangled patches of field tend to dissipate quicker
(e.g., Aguilera et al. 2008).
The conclusion of this section is that because it is unclear
which (instantaneous)value of M
applies for the period when
Weng et al. (2022)timed the radio pulses (see Section 2),
several scenarios are theoretically possible. If the system is a
magnetar (i.e., B
p
10
14
G), lower values of M
(i.e., at
apastron)in conjunction with a quadrupole-like braking index
Figure 2. “Weak-field”P(t)tracks for LSI, with assumed accretion rates of =-
M
10 2
and -M
1
03Edd
for t5 kyr in the left and right panels, respectively, though with
negligible propeller torques at earlier times. The blue (red)curve, whose width accounts for a fiducial uncertainty in the maximum period derivative, viz.
-
P
3
.2 10 4.
4
10
, has ξ=0.09(0.04).Wefix the (intrinsic)braking index at n=3 and inclination angle at α=0. The polar field strength, as shown in the panel
legends, is chosen such that the torques produce the requisite
P
when P=0.269 s. The figure insets display the period derivative; the rapid rise occurs because strong
propeller torques are turned on.
6
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
n∼5 can explain ages t5 kyr and the maximum
P
values
(see Figure 3). For a nonmagnetar, a larger M
(i.e., at
periastron)combined with fields more typical of HMXBs could
easily explain the observations with B
p
10
13
Gif3n5
and 0.05 ξ0.1 (see Figures 1and 2).
4. TeV Gamma Rays
As confirmed by measurements taken with the MAGIC and
VERITAS gamma-ray telescopes (Albert et al. 2006; Acciari
et al. 2009), LSI is one of the few known gamma-ray binaries
emitting ∼TeV radiation. In this section, we consider a model
of particle acceleration near the magnetospheric boundary to
estimate the minimum polar field strength of LSI required to
produce ultrahigh-energy radiation.
4.1. Particle Acceleration
As put forth by Bednarek (2009a,2009b)and others,
electrons can be accelerated in the turbulent transition region
near the boundary of the Alfvén surface through Fermi
processes. The associated power for an electron of energy E
can be parameterized as
zz==W cE R ceB,11
acc Lar ()
where R
Lar
is the Larmor radius, eis the elementary charge, and
ζ=10
−1
ζ
−1
is the dimensionless “acceleration parameter,”
which effectively sets a cutoff energy for the emitted spectrum;
in the case of relativistic shocks, Khangulyan et al. (2007)
suggested that 10
−4
ζ10
−2
, though this parameter is
theoretically uncertain. We proceed by matching Equation (11)
with the energy losses from synchrotron processes, which apply
to the highest-energy particles, viz.
srg»Wc43, 12
A
syn T 2()
where σ
T
is the Thomson cross section, and
r
p»B8
AA
2is the
Alfvén energy density in the transition region. From the
definition of the magnetospheric radius (Equation (2)),wefind
x
»BGM M
BR .13
A
p
37 67
3573
()
() ()
Maintaining all of the various scalings, we therefore obtain
gzx»´ -
--
BMM R3.9 10 . 14
max 6
1
12 32 14
514
10
37
1.4
314
6
15 14
()
In general, the relativistic electron energy, E=γm
e
c
2
for
mass m
e
, acquires an ∼TeV value for γ
TEV
∼2×10
6
.
Demanding that
g
g
max TEV, we obtain the inequality
⎛
⎝
⎜⎞
⎠
⎟⎛
⎝⎞
⎠
zx
g
´
-
BMM
R
0.15 210 .15
14 1.4
314
10
37
6
15 14
1
12 32
14 5
TEV
6
14 5
()
Using the propeller inversion (Equation (6)), the above
expression becomes
⎛
⎝⎞
⎠
zx g
´
--
-
--
BLMR1.4 210,16
14 33
2
1.4
1
6
3
1
37TEV
6
6
()
which scales strongly with ξ. The details of the requisite B
value depend on when the gamma rays are observed relative to
the orbital cycle.
4.2. Orbital Scenarios
It has been argued by Torres et al. (2012)that the
(nonpulsational)radio emissions indicate that the highest-
energy emission occurs only at apastron, with lower, ∼GeV-
band emissions at periastron. If one assumes, therefore, that
high-energy particles are only emitted at apastron, where M
is
expected to achieve its local minimum, relatively weak fields
could permit ∼TeV emission for large enough ξ. If we take
ξ0.3 and fix the stellar parameters to their canonical values,
Equation (16)becomes
zx~´ --
BL610 0.3 G. 17
p,min 13 3 7 33
2
() ()
The above assumes an X-ray luminosity rate that is on the
low end, as relevant for the source being at apastron, together
with an extreme accelerator, ζ∼1. Significantly larger values
of ~-
MM10 2Edd
, as considered in Section 3, require polar
field strengths that approach the virial limit unless ξ?0.3.
Figure 3. “Strong-field”spin period evolutions: P(t)tracks for LSI permitting large polar field strengths, B
p
10
14
G, as shown in the panel legends. Larger braking
indices allow for
P
to be relatively small at early times t5 kyr (see insets, which display
P
as in Figure 2). Propeller torques may activate at this fiducial switching
time for an accretion rate
--
MM
1
010
6Edd 5
(left panel)or the surface field may decay to a dipole-like configuration with 2.5 n3(right panel)to account
for a large
P
shortly after t=5 kyr. We fixα=0.
7
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
Note that such values of ξare larger than those considered in
Section 3by a factor of 2, hinting again that the maximum
P
is overinflated unless the instantaneous L
X
was much lower
when the source pulsed in the radio.
For context, we consider how this analysis matches up with
other high-energy systems. Radio timing of the TeV binaries
hosting the millisecond pulsars PSR J2032+4127 and PSR
B1259–63, assuming standard dipole braking, gives B
p
≈
3.4 ×10
12
and 6.6 ×10
11
G, respectively (Johnston et al.
1994; Camilo et al. 2009). For the former source, however,
X-ray emissions are relatively weak, L
X
≈1.1 ×10
31
erg s
−1
(Camilo et al. 2009), so that even the propeller-minimum
Equation (17)imposes the modest constraint ´
B
8
p,min
zx
--
10 0.3 G
10 7 5 7
()to explain TeV outbursts. The X-ray
luminosity of the latter is more like that of LSI, with L
X
∼
10
33
erg s
−1
at apastron (Kawachi et al. 2004), though if ξ∼1,
the required minimum is z~´ -
B
1.4 10
p,min 11 3 G. The key
difference for LSI is that we anticipate small values of ξ,if
indeed --
P10 ss
11 1
.
A key conclusion is that there is an inverse relationship
between ξ, controlling the magnetospheric radius, and the polar
field strength required to explain ∼TeV emissions, at least
within the context of the Bednarek (2009a,2009b)model. This
is the reverse of that imposed by the radio
P
constraints
considered in Section 3;ifξis too small, synchrotron losses
prevent the system from emitting high-energy particles, while if
it is too large, the system cannot spin-down fast enough.
Overall, therefore, the analysis of this section suggests that
polar field strengths of order 10
13
G, at minimum, are required
for LSI. If relativistic shocks dominate the acceleration process
at the magnetospheric radius, however (i.e., if ζ=1;
Khangulyan et al. 2007), then magnetar-level fields appear to
be necessary.
5. Radio Pulsar Activation
It is generally put forth that pair production in charge-starved
magnetospheric “gaps”is a necessary action in the powering of
radio pulsations from neutron stars, magnetar or otherwise
(Ruderman & Sutherland 1975; though see Melrose et al.
2021). Depending on the rotation rate, the magnetic field, and
the structure and location of these gaps, a variety of possible
“death lines”arise, defining an overall “death valley”(Chen &
Ruderman 1993; Hibschman & Arons 2001; Szary et al. 2015).
This provides a basis for investigating the radio switch-on
of LSI.
5.1. Death Valley
Although many models exist, a promising location for these
gaps is above the polar cap(s), as the open field line bundle
ensures that material is continually expelled from there. The
maximum potential drop,
D
V
max,is
pnD»VBRc218
dmax 22 32
()
for dipole component B
d
. This must exceed the voltage drop
ΔV
pair
required for pair production; in a curvature-radiation
scenario, we have (Ruderman & Sutherland 1975)
⎜⎟
⎛
⎝⎞
⎠
D»
eV
mc mcR
H
R
B
B2
1
15 ,19
eecc
ppair
2
3
QED
()
where R
c
is the curvature radius, =»´
B
mc e 4.4
e
QED 23
1013 G denotes the Schwinger field, ÿis the reduced Planck
constant, and Hdenotes a characteristic gap width.
Following Chen & Ruderman (1993), we consider three
models. (I)Pure dipole. Here one simply takes =
R
RR
clc 12
()
,
=
H
RR R
lc 12
()
,andB
p
=B
d
. The death line,
D
V
max
D
Vpair,reads -=BP
4
log 7.5 log 49.
3
d.(II)Twisted dipole.
This case is similar to (I), though instead we assume that the
magnetospheric field lines are curved such that R
c
≈R
å
. The line
is given by -=BP
4
log 6.5 log 45.7
d.(III)Twisted multi-
poles. In this case, B
p
?B
d
and H≈R
å
.Thefield lines are so
curved that radiated photons cross other parts of the cap’sopen
field lines (see Figure 3 in Chen & Ruderman 1993). The line is
given by -»BP
4
log 6 log 43.8
d. It may be the case, though,
that the cap is polluted by various materials, either from
thermionic crustal emissions (Szary et al. 2015)or the accretion
flow, in which case local electric fields may always be screened.
So we also consider an outer-gap model, for which the death line
(IV)reads -=BP
5
log 12 log 7
2
d(see Equation (24)of Chen
& Ruderman 1993).
Figure 4shows death lines I–IV on a B–Pdiagram. The
dashed line with black stars illustrates the possible range of
field strengths for LSI. We also show for completeness the
(characteristic)location of known radio-loud magnetars (red
stars)and high-Bpulsars (black diamonds). It is clear that these
latter sources can easily accommodate the death valley
constraints, with the possible exception of the outer magneto-
sphere model (IV). Similarly, unless B
p
10
13
G for LSI, it,
too, easily survives the valley.
In a magnetar (i.e., strongly quantum electrodynamic)
environment, photons may split with nonnegligible probability,
quenching pair production (Baring & Harding 1998). Medin &
Lai (2010)numerically traced the development of pair cascades
in the magnetospheres of stars with B10
14
G, including
effects from photon splitting with realistic selection rules for
the polarization modes, one-photon pair production into low
Landau levels, and resonant inverse-Compton scattering from
geometrically diverse hot spots. Remarkably, they found that
untwisted dipole fields observe death lines similar to category
Figure 4. Death lines (I–IV), detailed in the main text, in B–Pspace. When a
source lies vertically above a given line, the model says that the star can
“switch on”as a radio pulsar. Known radio-loud magnetars (namely, SGR
J1745–2900, PSR J1622–4950, XTE J1810–197, 1E 1547.0–5408, SGR 1935
+2154, and Swift J1818.0–1607)are shown by red stars, and a few high-B
pulsars (PSR J1119–6127, PSR J1734–3333, and PSR J1718–3718)are shown
with black diamonds. The primary in LSI is shown as a black star, with the
dashed line delineating the possible B-field range.
8
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
(I)with »´
B
P10 s G
d,min 12 2
()(see their Equation (60)),
which is 10
11
G for LSI.
As such, conventional theory and numerical simulations
suggest that it is easy for LSI to switch on as a radio pulsar. So
why are pulsations only now observed? Aside from their
∼microjansky brightness—invisible prior to FAST—dynami-
cal phenomena related to the soft-gamma flares may be
responsible. Patches of crust with intense magnetic spots can
drift through a combination of plastic flow and the Hall effect,
suggesting that the emitting region may itself drift postquake
(Gourgouliatos 2022). Magnetospheric twist injections and
diffusive settling could also reconfigure the polar cap geometry
(Beloborodov 2009). Pollution of the gap region is also likely
to play a role in the nulling fraction (Torres et al. 2012).
Alternatively, the pulsar may be permanently “on”but have its
radio emissions absorbed by the thick wind of the companion.
Future observations will help to settle the issue. If the pulsar is
henceforth “off,”at least prior to the release of another soft-
gamma flare, this would point toward magnetospheric twist
injections and diffusive reconfiguration. If pulses are consis-
tently seen with orbitally modulated brightnesses, a nonmag-
netar wind scenario, like that favored for PSR B1259–63
(Johnston et al. 1994), may apply.
6. GW Observability
As explored in the previous sections, there are a number of
inferences that can be made about the nature of LSI from the
electromagnetic observations. Future pulse recordings from the
source will be useful in narrowing down the intrinsic
P
values
relative to L
X
, which seem to be the most important quantities
as concerns the magnetar interpretation. If it turns out that LSI
houses a subsecond magnetar, the system could be visible in
GWs. Having a precise spin period for the object now allows
for targeted, narrowband GW searches to be carried out, which
can enhance the signal-to-noise ratio by a factor of ∼T
obs for
observing time T
obs
(e.g., Watts et al. 2008).
6.1. Magnetic Deformations
Magnetic fields introduce mass–density asymmetries in a
neutron star because the Lorentz force is not spherically
symmetric (Chandrasekhar & Fermi 1953). The inclusion of a
toroidal field, which is necessary for the hydromagnetic
stability of the system and possibly for powering the
magnetar-like flares from the source, can also boost the
effective deformation (Cutler 2002; Mastrano et al. 2011;
Glampedakis & Gualtieri 2018). The resulting mass quadrupole
moment varies in time as the star spins, resulting in the
emission of GWs at a frequency of f
GW
=2ν. For LSI, this
implies that f
GW
=7.43 Hz, which, unfortunately, is difficult to
observe with second-generation interferometers because the
combination of seismic, gravitational gradient (sometimes
called “Newtonian”), and thermal noises severely limits
detector sensitivity at 10 Hz (Martynov et al. 2016). This
may not be a problem, however, for the future third-generation
Einstein Telescope (ET), which is designed to avoid these
issues by being underground and including cryogenic suspen-
sion technologies to reduce thermal vibrations (Punturo et al.
2010).
Introducing the ellipticity ò=(I
zz
−I
xx
)/I
å
, for the moment
of inertia tensor I
ij
, the characteristic GW amplitude reads (e.g.,
Glampedakis & Gualtieri 2018)
⎛
⎝⎞
⎠⎛
⎝⎞
⎠⎛
⎝⎞
⎠
pn
n
=
»´
-
-
hGI
cd
d
16
2.2 10 10 3.7 Hz
2.65 kpc .20
0
22
4
28
5
2
()
The ellipticity depends critically on the relative contributions
of the toroidal and poloidal field bulks. To quantify its
magnitude, we follow Mastrano et al. (2011)and introduce the
quantity 0 <Λ1, which denotes the ratio between the
poloidal and total energy; Λ=0.5, for example, indicates
equipartition between the internal toroidal and poloidal sectors.
Mastrano et al. (2011)estimated the ellipticity as
»´ - L
--
BRM6 10 1 0.389 . 21
p
615
2
6
4
1.4
2
()()
Multipoles, proton superconductivity in the core, or accreted
mountains can lead to increases in ò. Note the local minimum
in òat Λ=0.389, which occurs because the prolate distortion
induced by the toroidal field balances the poloidal, oblate
distortion, leading to a force-free configuration.
Figure 5shows the characteristic strain (Equation (20)) as a
function of the poloidal-to-total energy ratio Λthrough
Equation (21)for various field strengths B
p
(e.g., the orange
bands correspond to B
p
=7.2 ×10
13
G). In plotting
Equation (20)for a given B
p
, we consider both a canonical
star with M
å
=1.4 M
e
and R
å
=10 km (lower curves)and an
extreme one with M
å
=2M
e
and R
å
=14 km (upper curves).
Overlaid are relevant detection thresholds for aLIGO
6
and ET
7
for a few different observational times T
obs
. The figure
demonstrates that the source is relatively dim in GWs, even
for the extreme case B
p
=7.2 ×10
14
G, as predicted if
propeller torques are totally absent and the star spins down
due to pure dipole braking (see Section 3). If, however, the star
is relatively compact and contains a toroidal field that houses
95% of the total magnetic energy (i.e., Λ0.05), the system
should be visible to ET given an observation time of T
obs
6
months. A superconducting core could allow for GWs to be
Figure 5. Characteristic strains h
0
from Equation (20)as a function of the
poloidal-to-total magnetic energy ratio, Λ, using the dipolar Tolman-VII model
(Equation (21)) of Mastrano et al. (2011)for LSI. The colored bands represent
the prediction for different B
p
, with the lower curves having M
å
=1.4 M
e
and
R
å
=10 km and the upper curves having M
å
=2M
e
and R
å
=14 km.
Overlaid are the detection thresholds for aLIGO and ET for a variety of
different observation times T
obs
, as indicated by the labeled dashed lines.
6
Data from https://dcc.ligo.org/public/0149/T1800044/005.
7
Data from http://www.et-gw.eu/index.php/etsensitivities.
9
The Astrophysical Journal, 940:128 (11pp), 2022 December 1 Suvorov & Glampedakis
detected by the star given a weaker toroidal field strength and
fixed observation window (Cutler 2002; Lander 2013). These
considerations strengthen the case for the construction of ET,
as a detection of GWs from LSI would be able to significantly
constrain its compactness and magnetic field strength.
7. Discussion
We have provided a multifaceted analysis of LSI, with the
primary aim of deciphering the observational clues concerning
its polar magnetic field strength. Although the P
P
,timing data
(modulo the caveats noted in Section 2)point to LSI being a
“typical”magnetar with B
p
≈7×10
14
G, other aspects of the
system suggest otherwise. The size of its magnetospheric radius
and mass accretion rate (as estimated by the persistent X-ray
luminosity)imply that LSI probably experiences a strong
propeller torque, leading to spin-down. Attributing the bulk of
a given
P
to a propeller torque implies that the system’s true
polar field instead lies in the ballpark of 10
13
G(see
Figure 1), as is more typical for HMXBs. Other physical
aspects of LSI lead to weaker constraints; a field of B
p
10
13
G
places LSI well inside the radio-loud portion of the magnetar
“fundamental plane”of Rea et al. (2012)and allows for the
emission of ∼TeV radiation from ultrarelativistic particles in
relevant boundary layers (Bednarek 2009b). All together, our
results suggest that LSI could be identified as a “low-B”
magnetar or a “high-B”radio pulsar, the distinction between
the two categories being, to some extent, a matter of semantics
(Pons & Perna 2011).
Future observations will shed more light on the true nature of
LSI. On the electromagnetic side, one of the most critical
parameters is
P
, for which a long-term, timing-based solution is
unavailable. Even if a lower
P
is recorded in future, which
would imply a weaker polar field, this does not necessarily
exclude the magnetar hypothesis. The presence of multiple
short (∼0.3 s)flares from LSI points toward the existence of an
internal toroidal or buried multipolar poloidal field. Indeed,
simulations of magnetically induced crust breaking in neutron
stars, which are popular models for the progenitors of
magnetar-like bursts (e.g., Pons & Perna 2011), suggest that
local fields of order 10
14
G are necessary to induce a fracture
(Lander et al. 2015). Tighter constraints on a possible thermal
component to the X-ray luminosity would be useful in this
direction, as toroidal field decay is likely to provide a strong
heat reservoir (see Section 2.5). A strong toroidal field also
boosts the predicted ellipticity of the star, which provides a
more optimistic outlook for GW detection (see Figure 5).A
confirmed detection of a continuous GW signal from LSI
should be taken as strong evidence for the presence of an
intense, B>10
14
G, magnetic field independent of the
P
value.
To date, there have been no confirmed detections of
magnetars in binaries (though see Israel et al. 2017 and
Igoshev & Popov 2018 for some candidates). It was argued by
King & Lasota (2019)that such systems must be rare. Notably,
aside from the observations discussed in this paper, there are
formation considerations that we have not explored. For
example, it may be the case that a large stockpile of angular
momentum is required at birth to instigate substantial dynamo
or magnetorotational activity, which suggests that (aside from
merger possibilities)only rapidly rotating stars undergoing core
collapse could produce magnetars. Whether any would-be
companions can survive the supernova explosion in this case is
unclear. The magnetic field may have decayed significantly by
the time the magnetar can find a companion, unless the decay
process is stalled by plastic flows or strong crustal impurities,
which extend the ohmic timescale (Igoshev & Popov 2018).
Even so, further investigation into the theoretical and
observational existence of magnetars in binaries is needed.
A.G.S. thanks Stefan Osłowski for helpful discussions on
pulse folding and general timing considerations. K.G. acknowl-
edges support from research grant PID2020-1149GB-I00 of the
Spanish Ministerio de Ciencia e Innovación. The research
leading to these results has received funding from the European
Union’s Horizon 2020 Programme under the AHEAD2020
project (grant No. 871158). We are grateful to the anonymous
referee for providing helpful and specific feedback that
improved the quality of the manuscript.
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