###### Abstract

At the moment, there are two neutron star X-ray binaries with massive red
supergiants as donors. De et al. (2023) proposed that the system SWIFT
J0850.8-4219 contains a neutron star at the propeller stage. We study this
possibility by applying various models of propeller spin-down. We demonstrate
that the duration of the propeller stage is very sensitive to the regime of
rotational losses. Only in the case of a relatively slow propeller model
proposed by Davies and Pringle (1981), the duration of the propeller is long
enough to provide a significant probability to observe the system at this
stage. Future determination of the system parameters (orbital and spin
periods, magnetic field of the compact object, etc.) will allow putting strong
constraints on the propeller behavior.

###### keywords:

neutron stars; X-ray binaries; accretion

1 1 0 https://doi.org/ Probing the Propeller Regime with Symbiotic X-ray
Binaries Probing the Propeller Regime with SyXBs Marina D. Afonina 1,2 and
Sergei B. Popov 3* Marina D. Afonina and Sergei B. Popov Afonina M.D.; Popov,
S.B. Correspondence<EMAIL_ADDRESS>Tel.: +39 040 2240 397

##  1 Introduction

Thousands of neutron stars (NSs) are known as sources of various natures
Nättilä and Kajava (2022); Popov (2023); Ascenzi et al. (2024). Observational
appearance of an NS depends on its parameters (rotation, mass, magnetic field,
etc.) and its interaction with the surrounding medium. Some regimes of the
interaction of an NS with external matter are not well studied. In particular,
this is true for the propeller regime proposed in 1970 by Shvartsman
Shvartsman (1970). At this stage, a rapidly rotating magnetosphere prevents
the accretion of the gravitationally captured plasma onto the NS surface.

Already in the early 1970s basic properties of the propeller stage and its
importance for the evolution of X-ray binaries were well understood Pringle
and Rees (1972); Davidson and Ostriker (1973); Shakura (1975); Illarionov and
Sunyaev (1975). However, the direct proof of the existence of this phase of NS
evolution was absent. The propeller phase of an NS evolution is an elusive one
as the energy release can be rather low and the duration of this stage can be
relatively short. Still, it is expected that some X-ray binaries can contain
NSs at this stage (see e.g., Campana et al. (2018) and references therein).
Probably, in several cases of millisecond pulsars Papitto and Torres (2015),
standard X-ray pulsars Tsygankov et al. (2016), accreting non-pulsating NSs
Zhang et al. (1998); Chen et al. (2006), and even ultra-luminous X-ray sources
Lutovinov et al. (2017); Vasilopoulos et al. (2021) it is possible to detect
transitions from accretion to the propeller stage (and back) by detection of
rapid changes in luminosity and spectral properties. Identification of an NS
in the propeller regime in a well-studied binary can provide important clues
for a better understanding of the interaction between the NS magnetic field
and ionized matter.

At the moment, mainly due to X-ray observations, many hundreds of NSs are
identified in interacting binary systems with different types of companions,
see Postnov and Yungelson (2014) for a review. For understanding the propeller
regime, so-called high-mass X-ray binaries (HMXBs) are especially interesting.
There are hundreds of X-ray binaries of this type Fortin et al. (2023);
Neumann et al. (2023). For our purposes, systems where the compact object
captures matter from a slow stellar wind of its companion, are the most
important. For some combinations of NS parameters (spin period and magnetic
field), parameters of the binary (semi-major axis, eccentricity), and the
donor (stellar mass, mass loss rate, and velocity of the stellar wind) the
rotating magnetosphere can expel the captured matter not allowing stable
accretion. Among various sub-classes of HMXBs, so-called symbiotic X-ray
binaries (SyXBs) can be one of the best candidates to host a propeller.

SyXBs are studied in many papers, see e.g. introductory part in Yungelson et
al. (2019) for a brief review. They consist of an NS and a donor at the red
giant branch (RGB) or the asymptotic giant branch. Mostly, donors are low-mass
late-type giants. Systems are not very numerous as they have a short lifetime
determined by the duration of the evolution of the donor. Yungelson et al.
Yungelson et al. (2019) estimated that there are less than 40-50 SyXBs in the
Galaxy. Still, there might be a few systems with massive donors, i.e. with red
supergiants.

Up to recent times, there was just one known X-ray binary with an accreting NS
and a red supergiant companion. This is the system 4U 1954+31 Hinkle et al.
(2020). The mass of the donor is estimated to be $7-15\,M_{\odot}$. The NS in
this system has a peculiarly long spin period — about five hours. Such a long
spin period can be explained, for example, assuming that the NS has a
magnetar-scale magnetic field Bozzo et al. (2022).

Recently, the second SyXB with a supergiant — SWIFT J0850.8-4219, — was
identified De et al. (2024). In this system, the donor is a red supergiant
with $T_{\mathrm{eff}}=3820\pm 100$ K and mass $\sim 10-20\,M_{\odot}$. The
spin period of the NS as well as the orbital period are not known. The semi-
major axis is at least $\gtrsim 300\,R_{\odot}$ as there is no Roche lobe
overflow in this system (the lower limit depends on the size of the donor and
probably it is a few times larger than the given estimate). The X-ray
luminosity is $(4\pm 1)\times 10^{35}$ erg s-1 and the spectrum $N(E)\propto
E^{-\Gamma}$ is rather hard with the photon index $\Gamma<1$. These properties
led the authors to the conclusion that the NS can be at the propeller stage.
Then a small fraction of the captured matter (not more than a few percent)
still can accrete onto the surface of the NS producing the observed emission.

If the hypothesis proposed by De et al. De et al. (2024) is correct then the
system SWIFT J0850.8-4219 provides a unique opportunity to probe the physics
of the propeller regime. In this paper, we analyze evolution under the
conditions measured for SWIFT J0850.8-4219 by applying several models of
propeller spin-down.

In the next section, we present the basics of the model we use. Section 3
contains a detailed description of various models of the propeller regime
proposed by different authors. Then in Sec. 4, results of the modeling are
described. In Sec. 5, we discuss some of our assumptions and present
calculations for alternative assumptions. In the final section, we present our
conclusions.

##  2 Model

In this section, we describe the basics of the model we use, except for the
part related to various approaches to specifying an NS spin evolution at the
propeller stage, which is presented in a separate section. At first, we
describe the stellar wind model and the calculations of the accretion rate,
$\dot{M}$. Then we demonstrate how the spin evolution of the NS is calculated
in our model.

###  2.1 Stellar wind

Stellar wind parameters are extremely important for our modeling.
Unfortunately, there are many unsolved problems related to the wind properties
at different phases of the evolution of massive stars, see e.g., Puls et al.
(2008) for a detailed review.

We need to specify stellar parameters throughout the evolution of the donor:
from the Main sequence to the supergiant stage. To reach this goal, we utilize
PARSEC evolutionary tracks Chen et al. (2015). We use the track for the single
mass value $M_{*}=14\,M_{\odot}$ and solar metallicity. For now, we assume
that the mass of the NS progenitor at the Zero Age Main sequence is $\sim
30\,M_{\odot}$ which is important to estimate the age of the NS. The PARSEC
track for the $30\,M_{\odot}$ ends at $6.3$ Myr with the onset of helium
burning. We assume that the NS is born at 7 Myr after the system formation.
Our results are not sensitive to the exact choice of this number in reasonable
limits defined by a possible progenitor mass. Everywhere below we will use the
time elapsed from the birth of the NS, which is calculated as
$t=\text{age}(14\,M_{\odot})-7$ Myr. Here the age corresponds to the moment of
time for which we take stellar parameters from the evolutionary track for the
$14\,M_{\odot}$ star.

In our modeling, we use the following stellar parameters taken from the track:
mass $M_{*}(\mathrm{age})$, radius $R_{*}(\mathrm{age})$, mass loss rate
$\dot{M}_{\mathrm{w}}(\mathrm{age})$, and effective temperature
$T(\mathrm{age})$.

At the beginning of the red supergiant (RSG) stage (i.e., the star has an
inert helium core and an expanded hydrogen envelope), the second component
reaches the maximum radius of $740\,R_{\odot}$, then decreases to
$500\,R_{\odot}$ and starts to increase again at the latest stages of
evolution. If the donor star fills its Roche lobe then a stream of gas flows
through the inner Lagrange point. This is not the case for the system SWIFT
J0850.8-4219. Therefore, we can find the minimum separation $a$ of a circular
orbit using the value $740\,R_{\odot}$ for the maximum radius of the donor.
This value is not exceeded until the second component starts to expand after
the exhaustion of helium in the core (we do not consider further evolution of
the donor). From Eggleton (1983) we get the ratio of the effective radius of a
Roche lobe $R_{\text{L}}$ and the semi-major axis $a$:

$\frac{R_{\text{L}}}{a}=\frac{0.49q^{2/3}}{0.6q^{2/3}+\ln(1+q^{1/3})},$ (1)

where $q$ is the mass ratio. For the second component $q=M_{*}/M=10$, where
$M=1.4\,M_{\odot}$ is the NS mass. If $R_{\text{L}}=740\,R_{\odot}$ then the
tightest possible orbit is $a=1280\,R_{\odot}$.

The track is terminated when $R_{*}$ reaches $740\,R_{\odot}$ again. Thus,
during the evolution of the binary, the second component never fills its Roche
lobe.

With the mass, radius, and temperature known, we calculate the wind velocity
following the prescription from Vink et al. (2000). The evolution of the mass
loss rate $\dot{M}_{\text{w}}$ and the wind velocity $v_{\text{w}}$ near
$r=a=1280\,R_{\odot}$ are shown in Figure 1.

Figure 1: Donor mass loss rate $\dot{M}_{\text{w}}$ (blue solid curve) and
wind velocity $v_{\text{w}}(r)$ (black dashed curve) at $r=1280\,R_{\odot}$
over the time elapsed since the birth of the NS. Vertical lines indicate bi-
stability jumps. The second bi-stability jump determines the change in donor
evolution from the Main sequence to the red supergiant stage.

We now present a detailed description of our $v_{\text{w}}$ calculations. The
dependence of the wind velocity $v_{\text{w}}$ on the distance from the center
of the donor $r$ is given by a beta-type velocity law:

$v_{\text{w}}(r)=v_{\infty}\left(1-\frac{R_{*}}{r}\right)^{\beta},$ (2)

where $v_{\infty}$ is the terminal velocity.

The terminal velocity can be expressed in terms of the escape velocity near
the stellar surface $v_{\text{esc}}=\sqrt{2GM_{*}/R_{*}}$. The ratio
$v_{\infty}/v_{\text{esc}}$ depends on the evolutionary stage of the donor. We
can distinguish three stages. Each subsequent stage begins with a sharp jump
in $\dot{M}_{\text{w}}$. The authors of the paper Vink et al. (1999) suggest
that the jump in mass loss is closely related to the bi-stability jump, which
is the observed decrease in the ratio $v_{\infty}/v_{\text{esc}}$.

The first bi-stability jump corresponds to a decrease in the effective
temperature of the donor, $T$, down to 23,000 K during the evolution on the
Main sequence. In this case, Fe IV recombines to Fe III in the inner part of
the wind. This makes the wind acceleration more effective, increases
$\dot{M}_{\text{w}}$, but leads to a decrease in $v_{\infty}/v_{\text{esc}}$
from 2.6 to 1.3. According to the assumption from Vink et al. (2000), the
second bi-stability jump is also explained by the recombination of Fe III to
Fe II. This takes place at $T\sim$10,000 K and reduces
$v_{\infty}/v_{\text{esc}}$ from 1.3 to 0.7. For the $14\,M_{\odot}$ donor we
expect this transition to occur during the sharp drop in effective temperature
on the way to the RSG phase.

The parameter $\beta$ is about unity for O- and B-stars Vink et al. (2000).
So, we adopt $\beta=1$ for the first two stages. Now we determine the value of
$\beta$ value at the RSG stage. The SWIFT J0850.8-4219 counterpart is a K3-K5
type RSG. The dependence of the ratio $v_{\text{w}}/v_{\infty}$ for K4 type
RSGs is shown in Bennett (2010). For now, we are interested in $r<3R_{*}$,
where $\beta\approx 2\div 3$. We take $\beta=2$ for the RSG stage.

For the NS in the binary with the semi-major axis $a$, we calculate the rate
of matter capturing, $\dot{M}$, which is the NS accretion rate if accretion is
possible. According to Bondi Bondi (1952), for an NS surrounded by a stellar
wind with the density $\rho_{\text{w}}$ the accretion rate is
$\dot{M}\propto(GM)^{2}\rho_{\text{w}}v^{-3}$, where $v$ is the NS velocity
relative to the wind. In De et al. (2024) the authors use the following
equation:

$\dot{M}\approx 2.5\pi(GM)^{2}\rho_{\text{w}}(a)v^{-3}(a).$ (3)

The coefficient 2.5 is in correspondence with numerical modeling and
analytical estimates for moderate Mach numbers that are expected in the case
of SWIFT J0850.8-421, see Foglizzo and Ruffert (1997); Prust et al. (2024).

From the continuity equation $\dot{M}_{\text{w}}=4\pi
a^{2}\rho_{\text{w}}(a)v_{\text{w}}(a)$ we get $\rho_{\text{w}}(a)$. Here the
relative velocity $v\approx\sqrt{v_{\text{w}}^{2}+v_{\text{K}}^{2}}$ includes
the wind velocity $v_{\text{w}}$ and the sum of the Keplerian velocities of
both stars $v_{\text{K}}=\sqrt{{G(M_{*}+M)}/{a}}$. Here we neglect the sound
velocity as it is expected to be lower than the sum of the wind and orbital
velocities. Finally, we obtain

$\dot{M}=\frac{5}{8}\frac{(GM)^{2}}{a^{2}v_{\text{w}}v^{3}}\dot{M}_{\text{w}}.$
(4)

Using $\dot{M}_{\text{w}}$ and calculating $v_{\text{w}}$ from the properties
of the donor star, we can find $\dot{M}(t)$.

###  2.2 Spin evolution of NSs

In our description of the spin evolution, we basically follow the standard
approach presented e.g., in the book Lipunov (1992). A more recent description
can be found in the review Abolmasov et al. (2024).

The evolutionary status of an NS depends on the interplay between the NS spin
period, $P$, its magnetic field $B$, and the parameters of the surrounding
medium. It is convenient to describe the latter in terms of the mass accretion
rate, $\dot{M}$, even if there is no accretion onto the surface.

The initial spin period in our main calculations is assumed to be $P_{0}=100$
ms. This is in correspondence with usual estimates of typical NS parameters,
see e.g. Popov and Turolla (2012); Igoshev et al. (2022). We perform
calculations for different values of the magnetic field. However, as we focus
on the properties of SWIFT J0850.8-4219, we are mainly interested in the
values $\gtrsim 10^{12}$ G (see the field estimate in De et al. (2024)).

We do not include in our calculations magnetic field decay (see Igoshev et al.
(2021) for a review). This is justified by three considerations. On the one
hand, the NS in SWIFT J0850.8-4219 cannot be older than $\sim 10^{7}$ yrs as
it has a massive companion. Thus, we can safely ignore a long-term field
evolution due to Ohmic dissipation in the crust. On the other hand, the NS
might not be very young (e.g., due to the absence of a supernova remnant).
Then, in the case of standard pulsar-like fields, we can ignore a possible
early episode of field decay suggested in Igoshev and Popov (2014). Also, we
can neglect the early rapid evolution of a magnetar-scale field. It is
expected that after a few e-folding times such evolution saturates when the
compact object reaches the so-called Hall attractor stage Gourgouliatos and
Cumming (2014). If such early rapid episodes of field evolution took place in
the life of the NS in SWIFT J0850.8-4219, then on the scale of our model it
just modifies the assumption about the initial spin period. Finally, we can
neglect field decay due to accretion, see e.g. Konar and Bhattacharya (1997),
as in the case of SWIFT J0850.8-4219 we are dealing with a young system with a
massive donor not overfilling its Roche lobe, so the total amount of accreted
matter (as well the duration of accretion) cannot be sufficiently high to
influence the magnetic field significantly.

The NS is assumed to have constant mass $M=1.4\,M_{\odot}$ and the moment of
inertia $I=10^{45}$ g cm2.

We distinguish four main evolutionary stages: ejector, propeller, accretor,
and georotator (see Lipunov (1992); Abolmasov et al. (2024)). They can be
characterized by ratios between some characteristic radii. They are the light
cylinder radius $R_{\mathrm{l}}$, gravitational capture radius
$R_{\mathrm{G}}$, corotation radius $R_{\mathrm{co}}$, magnetospheric radius
$R_{\mathrm{m}}$, and Shvartsman radius $R_{\mathrm{Sh}}$.

The light cylinder radius is:

$R_{\text{l}}=c/\omega,$ (5)

where $c$ is the velocity of light and $\omega=2\pi/P$ is the spin frequency.

The corotation radius is:

$R_{\text{co}}=(GM/\omega^{2})^{1/3}.$ (6)

Recently, Lyutikov Lyutikov (2023) proposed a modification to the standard
approach to calculate the centrifugal barrier. We discuss this possibility in
Sec. 5.

The gravitational (aka Bondi) radius:

$R_{\text{G}}=\frac{2GM}{v^{2}}.$ (7)

Here $G$ is the Newton constant, and $v$ is the velocity relative to the
medium.

Shvartsman radius:

$R_{\text{Sh}}=\left(\frac{2\mu^{2}(GM)^{2}\omega^{4}}{\dot{M}v^{5}c^{4}}\right)^{1/2}.$
(8)

Here $\mu=BR^{3}$ is the magnetic moment, $R=10$ km is the NS radius.

Alfvén radius:

$R_{A}=\left(\frac{\mu^{2}}{2\dot{M}\sqrt{2GM}}\right)^{2/7}.$ (9)

At the ejector stage, the external matter (the stellar wind from the second
component) is stopped at the Shvartsman radius $R_{\text{Sh}}$, which is
greater than both the light cylinder radius $R_{\text{l}}$, and the
gravitational capture radius $R_{\text{G}}$. As the NS spins down, the
characteristic radius, at which the outer matter is stopped, $R_{\text{Sh}}$,
decreases until either it reaches the light cylinder radius
$R_{\text{Sh}}=R_{\text{l}}$, or the external matter becomes gravitationally
captured when $R_{\text{Sh}}=R_{\text{G}}$. At the subsequent evolutionary
phase — propeller, — external matter penetrates inside the light cylinder.
Thus, the relativistic particle wind is terminated. However, the matter cannot
accrete, yet, as it is stopped by a rapidly rotating magnetosphere at
$R_{\text{m}}$. Finally, if the magnetospheric radius $R_{\text{m}}$ is
smaller than the corotation radius $R_{\text{co}}$, then accretion onto the NS
surface is allowed.

At all stages, the spin period of the NS changes due to external torques. The
Euler equation for the spin period evolution can be written as:

$\dot{P}=\frac{P^{2}}{2\pi I}K.$ (10)

Here $K$ is the external (decelerating or accelerating) moment.

The decelerating moment for the ejector stage is defined as:

$K_{\text{E}}=2\frac{\mu^{2}}{R_{\text{l}}^{3}}.$ (11)

In general, we assume that the ejector-propeller transition corresponds to the
equality $R_{\text{Sh}}=R_{\text{G}}$. However, for high values of $v$,
$R_{\text{G}}\propto v^{-2}$ can be less than $R_{\text{l}}$. So the
transition condition is $R_{\text{Sh}}=R_{\text{l}}$. Then for the critical
period we have:

$P_{\text{EP}}=\left\\{\begin{array}[]{ll}\displaystyle\frac{2\pi}{c}\left(\frac{2\mu^{2}}{4\dot{M}v}\right)^{1/4},&R_{\text{G}}>R_{\text{l}}\\\
\displaystyle\frac{2\pi}{c}\left(\frac{2\mu^{2}(GM)^{2}}{\dot{M}v^{5}}\right)^{1/6},&R_{\text{G}}\leq
R_{\text{l}}.\\\ \end{array}\right.$ (12)

After this transition the external matter can interact with the NS
magnetosphere. The magnetospheric radius $R_{\mathrm{m}}$ is always kept
smaller than $R_{\mathrm{l}}$, since the existence of a magnetosphere outside
of the light cylinder is not possible.

At the accretion stage, in addition to the decelerating moment
($K_{\mathrm{sd}})$, an accelerating one ($K_{\mathrm{su}}$) appears. Thus, we
have: $K_{\text{A}}=K_{\text{sd}}-K_{\text{su}}$. The accelerating term is
different for the cases of spherical and disc accretion. For the spin-down and
spin-up torques, we can write:

$K_{\text{sd}}=k_{\text{t}}\frac{\mu^{2}}{R_{\text{co}}^{3}},$ (13)

where $k_{\text{t}}$ is a dimensionless coefficient $\sim 1$, and

$K_{\text{su}}=\left\\{\begin{array}[]{ll}\displaystyle\dot{M}\sqrt{GMR_{\text{A}}},&\text{disc}\\\
\displaystyle\dot{M}\eta\Omega R_{\text{G}}^{2},&\text{no disc.}\\\
\end{array}\right.$ (14)

Here $\Omega=\sqrt{{G(M_{*}+M)}/{a^{3}}}$, and we assume $\eta={1}/{4}$.

An accretion disc is formed if the specific angular momentum of the accreting
matter is larger than the Keplerian momentum at the magnetospheric radius.
This corresponds to the condition:

$\sqrt{GMR_{\text{m}}}\leq\eta\Omega R_{\text{G}}^{2}.$ (15)

At the stage of accretion, spin-up and spin-down might balance each other and
so an equilibrium is reached. The equilibrium period is obtained from the
condition $K_{\text{su}}=K_{\text{sd}}$. The radius of the magnetosphere does
not depend on the period at the accretion stage, so $P_{\text{eq}}$ can be
written in the form:

$P_{\text{eq}}=2\pi\mu\sqrt{\frac{k_{\text{t}}}{GMK_{\text{su}}}}.$ (16)

To specify the exact value of $k_{\text{t}}$ we proceed in the following way.
In our case, accretion proceeds with a disc formation. The inner radius of the
disc is $R_{\mathrm{d}}=fR_{\text{A}}$, $f\sim 0.5-1.0$, e.g. Chashkina et al.
(2017). One can define the so-called fastness parameter:
$\varpi_{\text{crit}}=(R_{\mathrm{d}}/R_{\text{co}})^{3/2}$. Numerical
calculations, e.g. Wang (1995), demonstrate that $\varpi_{\text{crit}}$ is
such that the ratio $R_{\text{d}}/R_{\text{co}}$ is from $\approx 0.9$ up to
$\lesssim 1$. We use the mean value 0.96. Then, from

$k_{\text{t}}\frac{\mu^{2}}{R_{\text{co}}^{3}}=\dot{M}\sqrt{GMR_{\text{A}}}$
(17)

for $f=1$ we obtain $k_{\text{t}}=(1/0.96)^{3}/(2\sqrt{2})\approx 0.4$.

After its birth, the NS with $P_{0}=0.1$ s and $B\gtrsim 10^{9}$ G appears at
the ejector stage as the second star in the binary is on the Main sequence and
does not produce a strong wind. The NS then gradually spins down while the
rate of mass loss from the companion due to the stellar wind increases. The NS
can make transitions from ejector to propeller, and then to accretor. In the
next section, we present different propeller models used in our study, and
then in Sec. 4, we describe the whole modeled evolution.

##  3 Propeller stage

In our study, we use several propeller models to see if any of them provide a
significant probability of capturing a system like SWIFT J0850.8-4219 at this
stage.

At the propeller stage, the external matter is stopped at $R_{\text{m}}$ and
$R_{\text{m}}>R_{\text{co}}$. So, the accretion of large amounts of matter is
prevented by the centrifugal barrier. Around the magnetosphere of the NS, an
envelope of external material extends up to $R_{\text{G}}$. Various propeller
models are characterized by a density profile of this envelope which depends
on the energy release and transfer. Thus, pressure outside the magnetospheric
boundary can be different in different models. This leads to different
expressions for $R_{\text{m}}$. Additionally, as each propeller model
corresponds to a different regime of the interaction between the magnetosphere
and the envelope, it results in different spin-down torques $K$.

The basic properties — decelerating moment and magnetospheric radius, — of all
the models and the corresponding references are presented in Table 1. A
detailed review of the various propeller models can be found in Abolmasov et
al. (2024).

We apply five models of the propeller stage, denoted A, A1, B, C, and D.
Models A and A1 are based on the paper Shakura (1975). The difference between
the two models is related to different definitions of the magnetospheric
radius. In A1 we follow the original proposal by Shakura. In this case,
$R_{\mathrm{m}}$ depends on the spin period. In model A we use the constant
value $R_{\mathrm{m}}=R_{\text{A}}^{7/9}R_{\text{G}}^{2/9}$ as in all other
models (B, C, D). Thus, for A1 and other models, we have different values of
the critical period for the propeller-accretor transition.

The propeller-accretor transition period for models A, B, C, D (for
$R_{\text{m}}<R_{\text{G}}$):

$P_{\text{PA}}=\pi\left(\frac{\mu^{2}}{2\sqrt{2}GM\dot{M}v^{2}}\right)^{1/3}.$
(18)

For model A1 (for $R_{\text{m}}<R_{\text{G}}$):

$P_{\text{PA}}=\frac{2\pi}{(GM)^{5/7}}\left(\frac{\mu^{2}}{\sqrt{2}\dot{M}}\right)^{3/7}.$
(19)

We do not consider the subsonic propeller stage (see Davies and Pringle (1981)
about it), assuming that accretion starts since
$R_{\mathrm{m}}<R_{\mathrm{co}}$. Probably, soon after the accretion starts,
it can resemble the settling accretion regime Shakura et al. (2012) even for
relatively large inflow rates while the envelope around the magnetosphere is
cooling down.

At the propeller stage, it is necessary to consider separately the cases when
$R_{\text{m}}<R_{\text{G}}$ and $R_{\text{m}}>R_{\text{G}}$ because the
magnetospheric radius is calculated differently in these cases. We control
this carefully, because depending on the mass loss rate of the donor, one or
the other condition can be realised in a wide range of magnetic fields.
Therefore, in the Table 1 we provide formulae for both cases. However, we note
that for realistic parameters, there is no transition from the propeller to
the accretor stage for $R_{\text{m}}>R_{\text{G}}$, i.e. there are no
georotators. Therefore we do not provide the critical period for this case.

The decelerating moment is defined differently in the considered models, see
Table 1. In some cases, it depends on the additional parameter — free-fall
velocity at the magnetospheric radius:
$v_{\text{ff}}(R_{\text{m}})=\sqrt{2GM/R_{\text{m}}}$.

Among all the considered models, model A provides the most rapid spin-down as
the magnetospheric radius stays constant and the period grows exponentially.
The slowest evolution is the property of the model D. In this case, the spin-
down rate $\dot{P}$ can be even slower than at the stage of ejection.
Francischelli and Wijers Francischelli and Wijers (2002) provided some
argumentation against the fastest variants of the propeller spin-down.
Nevertheless, here we present results for various possibilities.

Table 1:  Propeller models and corresponding values of the decelerating torque $K$ and the magnetosphere radius $R_{\text{m}}$.  Model | Authors | $K=-I\dot{\omega}$ |  $R_{\text{m}}$, $R_{\text{m}}<R_{\text{G}}$ |  $R_{\text{m}}$, $R_{\text{m}}>R_{\text{G}}$
---|---|---|---|---
A1 | Shakura (1975) Shakura (1975) | $\dot{M}\omega R_{\text{m}}^{2}$ | $({\mu^{2}vR_{\text{G}}^{1/2}}/{(2\dot{M}\omega^{2})})^{2/13}$ | $({\mu^{2}vR_{\text{G}}^{2}}/{(2\dot{M}\omega^{2})})^{1/8}$
A | Shakura (1975) Shakura (1975) | $\dot{M}\omega R_{\text{m}}^{2}$ | $R_{\text{A}}^{7/9}R_{\text{G}}^{2/9}$ | $({\mu^{2}vR_{\text{G}}^{2}}/{(2\dot{M}v)})^{1/6}$
B | D&O (1973) Davidson and Ostriker (1973) | $\dot{M}\sqrt{2GMR_{\text{m}}}$
C | I&S (1975) Illarionov and Sunyaev (1975) | $\dot{M}v_{\text{ff}}^{2}/(2\omega)$
D | D&P (1981) Davies and Pringle (1981) | $\dot{M}v^{2}/(2\omega)$

##  4 Results

In this section we present the results of calculations of the evolution of an
NS in a binary system, taking into account the evolution of the donor.

###  4.1 Detailed spin evolution of a typical neutron star in a binary

We show the results for an NS with $P_{0}=100$ ms in a binary with the semi-
major axis $a=1280\,R_{\odot}$. The eccentricity is assumed to be zero
throughout the evolution. The results for the standard magnetic field
$B=10^{12}$ G are shown in Fig. 2 and for the higher value $B=4\times 10^{12}$
G in Fig. 3. The top panels show the evolution of $\dot{M}$ and the middle and
bottom panels show the absolute value of the period derivative $|\dot{P}|$ and
the period $P$ over the NS age $t$. The panels to the right are the zoomed
region where most of the transitions occur as the donor approaches the RGB.

Figure 2: Evolution of $\dot{M}$, spin period $P$, and the absolute value of
its time derivative $\dot{P}$ for the NS with a constant magnetic field
$B=10^{12}$ G and initial spin period $P_{0}=100$ ms in a binary with the
semi-major axis $a=1280R_{\odot}$. For each propeller model, the NS makes a
transition from the ejector to the propeller and then to the accretor stage,
except for the model D where the accretor stage is not reached. Black solid
curves refer to all propeller models. Black dotted vertical lines and black-
filled circles show the transition to the propeller stage, which is the same
for all models. Panels on the right side, show the zoomed region near the
ejector-propeller transition. Colored vertical dashed lines and colored-filled
circles indicate the transition to the accretor stage for each model. The
dashed lines are not shown on the left panels, since there they would overlap
with the black dotted line. Colored curves for models A1, A, B, and C overlap
at the accretor stage and are shown in green on the left panels. The black
arrow in the left middle panel points to the sign change of $\dot{P}$ value at
$6.7$ Myr

.

In Fig. 2 we see that for all models (except model D) the propeller stage is
very brief. It is therefore highly improbable to detect one of just two known
systems with RSG donors at this evolutionary phase.

Figure 3: Evolution of $\dot{M}$, spin period $P$, and the absolute value of
its time derivative $\dot{P}$ for the NS with a constant magnetic field
$B=4\times 10^{12}$ G. Line styles, points, and colors are the same as in Fig.
2. The transition to the accretor stage is now shown in the left panels. Here
the grey vertical dashed line shows the transition to the accretor stage for
models A1, B, and C, while the red one is for propeller model A. The evolution
of an NS within model D does not lead to the accretor stage. In the middle
panels, the black arrow at $6.7$ Myr on the left and $6.679$ Myr on the right
show the change in sign of $\dot{P}$. After the NS period reaches
$P_{\text{eq}}$, the value of $\dot{P}$ fluctuates visibly.

The NS evolution at the ejector stage and the transition to the propeller
stage are independent of the propeller model and are common to all NSs with
the same $P_{0}$ and $B$. At this stage the NS is slowing down, so the period
derivative is positive. From Eq. 11, $\dot{P}\propto P^{-3}$, independent of
$\dot{M}$ and $v$, and decreases with time until the transition to the
propeller.

The time of the ejector-propeller transition is different for the two magnetic
field values. As $\dot{P}\propto B^{2}$, the NS with the higher field evolves
faster. On the other hand, $P_{\text{EP}}\propto B^{1/2}$. So, for a larger
field, it is necessary to reach a larger spin period to become a propeller.
Finally, under these external conditions
$P_{\text{EP}}\propto(\dot{M}v)^{-1/4}$ and decreases over time with the donor
evolution, because the quantity $\dot{M}v$ increases until the donor reaches
the RGB. It is difficult to predict the outcome of calculations with these
three dependencies. It is therefore necessary to calculate the evolution
numerically.

The NS with $B=4\times 10^{12}$ G reaches the propeller stage at $\approx 6$
Myr, before the donor becomes an RSG. So, this transition is mainly driven by
the spin evolution. For the lower field $B=10^{12}$ G the transition occurs
when the mass loss by the secondary component is drastically increased. At
this moment, the transition from the ejector is inevitable. With the peak of
$\dot{M}$ at $\sim 6.7$ Myr, the transition period $P_{\text{EP}}$ reaches its
minimum of $\sim 100\,B_{12}^{1/2}$ ms, where $B_{12}=B/10^{12}$ G. At the
same time, from the integration of the Euler equation (Eq. 10) with
$K_{\text{E}}$, the period reached by $6.7$ Myr is $\gtrsim 1\,B_{12}$ s,
which is sufficient for the transition.

After reaching the propeller stage, all NSs spin down at a rate that is highly
dependent on the propeller model. The difference between the models can be
seen well in the zoomed region of Fig. 2. The efficiency of this deceleration
decreases from model A to model D, while the spin-down rate of model A1 is
mostly of the same order as that of model B. The shape of the $\dot{P}(t)$
generally follows the shape of $\dot{M}(t)$, since the spin-down for each
propeller model depends strongly on $\dot{M}$, and among other parameters
($P$, $v$, and $B$), the accretion rate varies the most. Except for model D,
the propeller stage for $B=10^{12}$ G corresponds to the sharp increase of
$\dot{M}$ at $\sim 6.7$ Myr, well visible in the top left panels of Figs. 2
and 3, and therefore lasts just for less than 20,000 yrs for $B=10^{12}$ G.
The duration of the propeller stage of NSs with $B=4\times 10^{12}$ G is
higher, only because the ejector-propeller transition occurs earlier. For both
magnetic field values, the further transition to accretion corresponds to this
peak in $\dot{M}$ at $6.7$ Myr.

The spin period within the propeller model A grows at a very high rate. The
evolution of the NS with $4\times 10^{12}$ G shows that even for moderate
values of $\dot{M}\sim 10^{12}$ g s-1 this spin-down is effective enough to
bring the NS to the stage of accretion significantly earlier than $\dot{M}$
reaches its maximum $\sim 10^{18}$ g s-1. The NS with $B=10^{12}$ G also
starts to accrete before this maximum. As for models A1 and B, $\dot{P}$ is
approximately an order of magnitude smaller than for model A. In these cases,
for both magnetic field values, the propeller-accretor transition is not due
to spin-down, but due to an increase in $\dot{M}$ from $10^{16}$ to $10^{18}$
g s-1. The deceleration in model C is less effective, so to reach the
transition period $P_{\text{PA}}$ the NS must evolve an additional $\sim 2-3$
thousand years with high external pressure at the accretion rate $\dot{M}\sim
10^{18}$ g s-1, i.e. very close to the maximum value. Finally, the evolution
according to model D is too slow to produce an accretor for both magnetic
field values.

An NS can make a transition to the accretor stage either due to a spin-down
(models A and C) or due to a rapid increase in $\dot{M}$ (models A1 and B).
After the transition, the spin period may decrease or increase depending on
the relationship between $K_{\text{su}}$ and $K_{\text{sd}}$ and on the
accretion regime — spherical or disc. For NSs with $10^{12}$ G, the disc is
formed immediately after the onset of accretion. The transition period
$P_{\text{PA}}$ is larger than the equilibrium value $P_{\text{eq}}$ for disc
accretion (Eq. 16). That’s why, these NSs start to spin up. At the beginning
of the accretion, the value of the spin-up torque $K_{\text{su}}$ is
significantly larger than the decelerating one. So, the evolution of $\dot{P}$
depends strongly on the accelerating moment $K_{\text{su}}$, initially. Due to
the dependence of the accelerating moment on the accretion rate —
$K_{\text{su}}\propto\dot{M}^{6/7}$, — the $\dot{P}(t)$ curve for model A
resembles $\dot{M}(t)$ at the beginning of accretion, as can be seen in the
middle right panel of Fig. 2.

For NSs with the higher magnetic field value, the accretion stage starts in a
similar way, except for model A. Within this model, the transition occurs
before $\dot{M}$ has reached a value sufficient for the disc formation. So, in
model A, the accretion starts without a disc. For spherical accretion under
these conditions, the absolute value of $K_{\text{su}}$ is less than the
braking torque. Therefore, during the accretor stage, the NS spins down until
$\dot{M}$ is high enough that $K_{\text{su}}>K_{\text{sd}}$. At this moment,
the value of $\dot{P}$ changes its sign (black arrow in the middle right panel
of Fig. 3). Then the NS starts to accelerate as $\dot{M}$ increases with time.
After the age $6.686$ Myr, the accretion rate will be sufficient to form a
disc as in any other propeller model.

For both magnetic field values, in a few thousand years after the propeller-
accretor transition, all NSs reach the equilibrium regime of disc accretion.
This regime starts at $t\lesssim 6.70$ Myr for $B=10^{12}$ G and at $t\lesssim
6.69$ Myr for $B=4\times 10^{12}$ G. The equilibrium period depends on the
magnetic field and the accretion rate, so for every propeller model, the
curves tend to saturate at the same value $P_{\text{eq}}$. The spin period
obtained by integrating the Euler equation oscillates slightly near the
equilibrium value. To make our calculations more precise, we replace the
calculated curves $P(t)$ with $P_{\text{eq}}$ and $\dot{P}$ with
$\dot{P}_{\text{eq}}=-3P_{\text{eq}}/(7\dot{M})\,d\dot{M}/dt$ from the
beginning of this regime. Here $d\dot{M}/dt$ is calculated numerically, which
still leads to fluctuations in the calculated values of $\dot{P}$, because
$\dot{M}(t)$ is a tabulated curve. These fluctuations in $\dot{P}$ are visible
in the middle panels of Figs. 2 and 3 after the equilibrium is reached. The
spin period $P=P_{\text{eq}}$ varies only because of changes in $\dot{M}$. At
the onset of the equilibrium disc accretion after $\dot{M}$ peaks at $6.7$
Myr, the NSs start to spin down again. At $7.3$ Myr $P_{\text{eq}}$ reaches
its maximum value because $\dot{M}$ is at its minimum. Here $\dot{P}$ changes
its sign, which is shown as a V-shaped feature and indicated by a black arrow
in the left middle panels of Figs. 2 and 3. The subsequent spin-up phase
begins at $7.3$ Myr, when the accretion rate starts to increase again.

###  4.2 Evolutionary stages of NSs in a wide range of the magnetic field

Fig. 4 illustrates the evolution of NSs with various magnetic field values
within the propeller models C (left panel) and D (right panel). The other
parameters remain the same as in the previous calculations in Sec. 4.1:
$P_{0}=100$ ms, $a=1280\,R_{\odot}$. We are mainly interested in the
evolutionary stages of NSs when the donor star is an RSG, i.e. when the age of
the NS is $t\gtrsim 6.7$ Myr. Model C is a representative example of the
models in which the NS starts to accrete. Model D represents the case where
the NS with the standard magnetic field value is not entering the accretion
stage.

Figure 4: Evolutionary stages of NSs with $P_{0}=100$ ms, $a=1280\,R_{\odot}$
over the age $t$ of the NS and its magnetic field $B$. Propeller model C is
shown on the left panel and model D is shown on the right. Three evolutionary
stages are shown in color: ejector (blue), propeller (light blue), and
accretor (orange). White dashed lines correspond to the evolutionary phases of
the donor star, as in Figure 1.

All NSs start their evolution as ejectors because the stellar wind of the
donor is very weak when the NS is born. The ejector phase of evolution is
described in the same way for all NSs. The increase of the spin period is
insignificant for $B\lesssim 3\times 10^{12}$ G as the transition to the
propeller stage occurs due to the evolution of the donor. For extremely low
values of $B\lesssim 2\times 10^{9}$ G, the ejector stage ends at $5.8$ Myr,
while the donor is still in the Main sequence. The first jump in $\dot{M}$
immediately leads to $P_{\text{EP}}$ that is short enough to have $P(t)\gtrsim
P_{\text{EP}}$. NSs at the adjacent slope section of the graph with
$B\sim(2\times 10^{9}-3\times 10^{9})$ G show the same evolution with just one
difference — at the ejector-propeller transition $P_{\text{EP}}$ is slightly
higher, so it is necessary that $\dot{M}$ grows slightly more for the higher
$B$ to make $P_{\text{EP}}=P(t)$. For NSs with $B$ between $3\times 10^{9}$ G
and $3\times 10^{12}$ G the transition to the propeller stage occurs with the
second component approaching the RGB. For NSs in this range of $B$ the
transition is similar to the previously described transition of the NS from
Fig. 2 with $B=10^{12}$ G. For an NS with a higher magnetic field value
$B\gtrsim 3\times 10^{12}$ G the spin evolution starts to influence the onset
of the propeller regime. So, the NS with $3\times 10^{12}\text{~{}G}\leq B\leq
2\times 10^{13}$ G spins down enough that its spin period becomes
$P(t)=P_{\text{EP}}$ while $\dot{M}$ is only $\sim 10^{12}$ g s-1. The
evolution of NSs with higher magnetic field values is so fast that the ejector
stage ends even before the first jump in $\dot{M}$. So, in this case, the
evolution of the donor has almost no effect on the NS evolution.

The area of the propeller stage in Fig. 4 is different for the two models, C
and D. In model C, before $\dot{M}$ peaks at $6.7$ Myr, the spin-down at the
propeller stage is negligible. But after $\dot{M}$ reaches its maximum value
$\sim 10^{18}$ g s-1, the propeller spin-down is always sufficient to bring an
NS to the onset of accretion which starts at $t\approx 6.7$ Myr for all values
of $B$. Here we can see how strongly the propeller spin-down rate depends on
$\dot{M}$, and so $B$ is almost unimportant. While the propeller stage with
$\dot{M}\lesssim 10^{12}$ g s-1 takes $\sim 1$ million years or more (this can
be seen for $B\lesssim 3\times 10^{9}$ G and $B\gtrsim 10^{13}$ G), while for
$\dot{M}\sim 10^{18}$ g s-1 only a few thousand years is enough to reach the
accretor stage ($B\sim 10^{10}-10^{12}$ G). In the left panel of Fig. 4, for
values of $B$ near $10^{11}$ G, there is a non-monotonic dependence of the
time of the transition from propeller to accretor $t_{\text{PA}}$ while the
time of the ejector-propeller transition remains constant. This is because
within model C the spin-down rate at the propeller stage is $\dot{P}\propto
P^{3}B^{-4/9}$. At the very beginning of this regime $P=P_{\text{EP}}$, which
is longer for higher $B$. The interplay between the dependence of $\dot{P}$ on
$P_{\text{EP}}$ and of $\dot{P}$ on $B$ produces a maximum in $t_{\text{PA}}$.

Generally, the spin-down in model D is the least effective one. However, the
spin-down in model D can be more effective than in model C under some
conditions. The ratio of the braking torque in model D and model C is
$K_{\text{D}}/K_{\text{C}}\propto v^{2}/v_{\text{ff}}^{2}(R_{\text{m}})\propto
R_{\text{m}}/R_{\text{G}}$, and this ratio is $\sim 1-40$ at $t<6.7$ Myr for
$B\gtrsim 3\times 10^{13}$ G. Within these conditions
$K_{\text{D}}/K_{\text{C}}\gtrsim 1$, so the spin-down in model D is higher.
However, this factor does not make a significant difference, because the spin
period evolution with $\dot{M}\lesssim 10^{12}$ g s-1, which is $\dot{M}$
value before $6.7$ Myr peak, is very slow. So for both models C and D, at the
propeller stage, $P$ remains almost the same until the donor becomes an RSG.
After $6.7$ Myr $K_{\text{D}}/K_{\text{C}}\sim 10^{-4}-10^{-3}$. Therefore, in
contrast to model C, in model D it takes significantly longer for an NS to
begin to accrete.

In the right panel of Fig. 4, at later stages of the donor evolution (above
the second dashed line), the border between the propeller and the accretor
stage, $t_{\text{PA}}(B)$, is curved. I.e., in this case, the magnetic field
value influences the time of the propeller-accretor transition. The spin-down
within this model does not depend directly on $B$, but it depends on the spin
period $\dot{P}\propto P^{3}$. Since all NSs start as ejectors, at the
beginning of the propeller regime the spin-down is initially much higher for
higher values of $B$ as $P_{\text{EP}}\propto B^{1/2}$. The transition period
$P_{\text{PA}}\propto B^{2/3}$ also depends on $B$, but this dependence is
weaker than $\dot{P}(B)\propto B^{3/2}$ and does not change $t_{\text{PA}}(B)$
much. Thus, due to the lower initial value of the spin period at the onset of
the propeller stage, NSs with lower magnetic fields start to accrete later,
and NSs with $B\lesssim 10^{13}$ do not have enough time to reach the accretor
stage at all.

The shape of the accretor stage region in the right panel of Fig. 4 has a
feature at $B\approx 2\times 10^{13}$ G. Near this value $t_{\text{PA}}(B)$ is
non-monotonic. Mainly, this feature is related to the fact that the transition
ejector-propeller happens due to the increase of $\dot{M}$. This means that
for lower fields it occurs for shorter periods. Spin-down at the propeller
stage in model D is very ineffective. So, the transition propeller-accretor
again happens due to the increase in $\dot{M}$ and the spin-period is nearly
equal to the one at the ejector-propeller transition. As
$P_{\mathrm{PA}}\propto(B^{2}/\dot{M}v^{2})^{1/3}$, $P_{\mathrm{EP}}\propto
B^{2}$, and $\dot{M}v^{2}$ increases at the latest stages that we consider,
there is a complicated interplay between the parameters resulting in the non-
monotonic behavior visible in the Figure.

In Fig. 4 we clearly see that the evolution of the donor is the main driver of
the NS evolution for a wide range of magnetic fields (including standard
fields $B\sim 10^{10}$ few $\times 10^{12}$ G), except some boundary cases
discussed above. This justifies that we do not consider various values of the
age difference between the NS and the donor. The evolution of the donor
remains the same for any value of this parameter and influences the evolution
of the NS in almost the same way.

##  5 Discussion

###  5.1 Magnetar evolution

In our model, we neglect the magnetic field decay. In particular, we ignore
the possibility that the initial field could be high and then rapidly decay.
It is expected that after a few e-foldings, the field is saturated at some
value ($\sim 1/20$ of the initial field) that corresponds to the Hall
attractor stage Gourgouliatos and Cumming (2014). Effectively, on the scale of
our problem, rapid decay of the magnetar scale field with subsequent
saturation results just in a larger initial spin period $\sim 10$ s, i.e. a
typical magnetar value. In this subsection, we calculate how this could
influence our conclusions.

In Fig. 5 we present evolution of an NS with $P_{0}=10$ s and constant field
$B=4\times 10^{12}$ G. All notations are the same as in Figs. 2, 3.

In this case, the NS is born already at the propeller stage. I.e., this means
that if we have in mind a magnetar scale initial magnetic field and short spin
period then the transition ejector-propeller happens very quickly in the
history of the compact object due to the rapid spin-down. However, as soon as
the mass loss by the donor starts to increase, the NS (in the cases of models
A, A1, B, and C) becomes an accretor approximately as fast as in Fig. 3. I.e.,
the duration of the propeller stage relative to the accretor stage at high
$\dot{M}$ is very similar to the case of a standard (short) initial spin
period. In the case of model D, as in Fig. 3, the NS remains at the propeller
stage.

We conclude that large initial spin periods $P\sim 10$ s (mimicking magnetar
fields with rapid decay and Hall attractor) do not influence the main results
of our analysis.

Figure 5: The evolution of an NS with $P_{0}=10$ s, $B=4\times 10^{12}$ G in a
binary with $a=1280\,R_{\odot}$. The first evolutionary stage of the NS is the
propeller stage. Therefore there is no vertical dotted line for the ejector-
propeller transition. Otherwise, all curves, lines, and dots have the same
style as in Fig. 3. Black arrows indicate changes in the sign of $\dot{P}$ as
in Fig. 2 and Fig. 3. In the middle right panel, the curve for model A1 has an
additional sign change at $6.685$ Myr after the propeller-accretor transition,
similar to the model A ($6.676$ Myr), which is also indicated by a black
arrow.

###  5.2 Relative spin-down rates for ejectors and different propeller regimes

In Lipunov and Popov (1995) the authors claimed that for a very wide range of
realistic parameters, spin-down at the propeller stage is more efficient than
at the ejector stage. Figs. 2, 3 demonstrate that for our parameters this is
not the case for models C and D. Even for model B there is an interval when
$\dot{P}$ at the propeller is smaller than at the ejector stage.

We think that basically, this claim assumed that
$R_{\mathrm{m}}=R_{\mathrm{A}}\approx R_{\mathrm{co}}.$ This can be true if
the density profile in the envelope is close to the one for the free-falling
matter and if the NS is close to the propeller-accretor transition. However,
if, according to the suggestion by Davies and Pringle Davies and Pringle
(1981), $R_{\mathrm{m}}=R_{\mathrm{A}}^{7/9}R_{\mathrm{G}}^{2/9}\gg
R_{\mathrm{A}}$ then the propeller spin-down can be less effective than the
ejector spin-down rate, in correspondence with our results.

In our study, we also neglect the phase of the ‘very rapid rotator’, see
Davies and Pringle (1981). In this regime when $R_{\mathrm{m}}\approx
R_{\mathrm{l}}$, the spin-down rate, according to Davies and Pringle, can be
even lower. This phase is still a hypothetical one. Its existence is not
supported by studies of NS evolution in binaries. Finally, if evolution is
mainly driven by the increase of $\dot{M}$ (as it is in the case we analyze),
not by the spin-down, then this regime can be neglected.

###  5.3 Disc formation at the propeller stage

In our study, we account for the possibility of disc formation at the accretor
stage. However, a disc can be formed already at the propeller stage. We
neglect this possibility in all our models. Disc formation might result in a
reduction of the magnetospheric radius. Thus, on the one hand, our
deceleration moment can be lower than in the case of disc formation (see e.g.,
Alpar (2001) about magnetosphere-disc interaction at the propeller stage). On
the other hand, the critical period for the transition can be shorter than we
consider. Both effects work in one direction: they shorten the propeller
stage. I.e., our conclusions regarding models A, A1, B, and C are conservative
in this respect.

However, we have to note that in many cases, the beginning of accretion is
related not to a gradual spin-down, but to a rapid increase of $\dot{M}$ due
to enhanced mass loss by the donor. In such cases, we expect that the disc
appearance would not significantly modify our results even quantitatively.
Still, this question deserves a more detailed analysis, especially for model
D, which is beyond the scope of this paper as most probably it requires
detailed numerical modeling of the interaction between the magnetic field and
surrounding medium.

###  5.4 Wind accretion flow in a red supergiant binary

In this study, we assumed a usual simplified description of the wind
accretion, see e.g. Lipunov (1992); Edgar (2004). However, the realistic
situation can be much more sophisticated. This is especially true for the
systems with massive evolved donors, see El Mellah et al. (2020) and
references therein.

Wind parameters of massive evolved stars can demonstrate a very complicated
behavior including intense pulsations. The flow can significantly deviate from
the spherical symmetry and wind velocity can vary in a complicated way
depending on the distance from the donor, not following the $\beta$-law that
we apply.

We selected the size of the binary system sufficiently large to avoid the
Roche lobe overflow. However, when the radius of the donor is about the Roche
lobe radius, a new regime of mass transfer can be realized Mohamed and
Podsiadlowski (2007). We do not account for this possibility and so
underestimate the accretion rate for smaller values of the orbital
separations.

In addition, the accretion flow in the vicinity of the compact object might
have a complicated topology, e.g. Huarte-Espinosa et al. (2013). Thus, the
exact amount of matter gravitationally captured by the NS, its angular
momentum, and its dependence on time can deviate from the values we assume in
our modeling. Conditions for disc formation also can be different from those
that we use. Precise modeling of a particular system might be based on
realistic 3D calculations for the actual parameters of the binary. Still, at
the moment, the properties of SWIFT J0850.8-4219 are not precisely known. So,
a more simplified analysis presented above is justified.

###  5.5 The corotation radius value and magnetic inclination

Usually, the corotation radius is taken in the form of eq. (6). However,
recently it was admitted Lyutikov (2023) that even in the case of an aligned
rotator without a disc formation, the equatorial value of the centrifugal
barrier is $0.87R_{\text{co}}$, where $R_{\text{co}}$ is defined by eq. (6).
We checked if this modification could influence our results. With a smaller
value of $R_{\mathrm{co}}$, we obtained slightly higher values of
$P_{\text{eq}}$ at the accretor stage, but in general, it did not change our
main results and conclusions.

The situation with the centrifugal barrier becomes more complicated if the
magnetic inclination and a realistic magnetosphere structure are taken into
account. In all our calculations we neglect that surfaces corresponding to the
critical radii ($R_{\mathrm{co}}$, $R_{\mathrm{m}}$, $R_{\mathrm{Sh}}$) are
not spheres. Also, we do not account for the magnetic inclination, i.e. for
the misalignment between spin and magnetic axis.

Accounting for these details might result in slight modifications of spin-
up/down torques, in the value of the equilibrium period, and in the accretion
luminosity as the fraction of matter that reaches the NS surface might be
different for various inclinations. Still, it is reasonable to assume that our
general conclusions might stay intact.

###  5.6 Luminosity at the propeller stage

One of the arguments in favor of the propeller interpretation of the system
SWIFT J0850.8-4219 is related to its low X-ray luminosity $L\approx 4\times
10^{35}$ erg s-1 De et al. (2024). At the accretor stage, assuming a realistic
binary separation and properties of the secondary component, this value is
expected to be much higher $L_{\text{A}}\sim 10^{37}$ erg s-1. In the
propeller regime, the matter flow is stopped by a rapidly rotating
magnetosphere. Thus, just a tiny fraction of matter can reach the NS surface.
We can determine the relative accretion efficiency as $\zeta=L/L_{\text{A}}$.
According to De et al. (2024) this parameter is $\zeta\sim 10^{-2}$.

Let us estimate the relative accretion efficiency expected in our model.
First, we consider model D. This is the only scenario that results in a long
propeller stage in a binary with the NS in the closest possible orbit
$a=1280\,R_{\odot}$. As can be seen in Figs. 2, 3, and 5 when the donor is an
RSG the value of the accretion rate is $\dot{M}\approx 2\times 10^{17}$ g s-1.
So, the accretion luminosity would be $L_{\text{A}}=GM\dot{M}/R\approx 4\times
10^{37}$ erg s-1. From this we obtain $\zeta\approx 10^{-2}$. This is
consistent with the expected accretion efficiency suggested in the original
paper De et al. (2024) and seems realistic.

If we consider a slightly more effective propeller spin-down, model C, we need
to increase the separation $a$ up to at least $3950\,R_{\odot}$ in order to
produce a long propeller stage (otherwise, the probability to detect one of
just two known systems at this stage is negligibly small). In this case, the
rate of matter capturing is lower $\dot{M}\approx 7\times 10^{15}$ g s-1, so
$L_{\text{A}}\approx 10^{36}$ erg s-1 and the efficiency is higher
$\zeta\approx 0.3$. This value seems to be too high for the propeller regime.
From this, we can conclude that the propeller model C does not describe the
system SWIFT J0850.8-4219 well even if the duration of the propeller stage is
made long. The remaining propeller models A1, A, and B require even higher
values of the semi-major axis $a$, which results in a higher efficiency
$\zeta$. Therefore, these propeller models appear less realistic than model D
in the context of this binary within our scenario of evolution.

###  5.7 The possibility of settling accretion in SWIFT J0850.8-4219

Reduced accretion luminosity can be explained also in the model of settling
accretion. This accretion regime has been proposed in Shakura et al. (2012)
and later considered in detail in the series of papers Shakura et al. (2013,
2014); Popov et al. (2015). This type of accretion can occur if the X-ray
luminosity is below the critical value $L_{\text{SA}}\sim 10^{36}$ erg s-1. It
is determined by the characteristic cooling time $t_{\text{cool}}$ and the
free fall time $t_{\text{ff}}$. Otherwise, if $L\gtrsim L_{\text{SA}}$ then
$t_{\text{cool}}\lesssim t_{\text{ff}}$, i.e. cooling due to the Compton and
radiative processes is effective. In this case, the usual Bondi accretion
proceeds.

Generally, the subsonic settling accretion is characterized by the existence
of a hot convective shell around the NS magnetosphere. Due to the interchange
instability, a small amount of matter enters the magnetosphere. This could
explain the low X-ray luminosity in SWIFT J0850.8-4219 in comparison with the
maximum value $L_{\text{A}}$. However, our modeling shows that for realistic
parameters the accretion stage starts when $\dot{M}$ is already too high and
in addition, an accretion disc is formed around the NS which prevents the
appearance of the settling accretion regime. This regime can be realized (and
lasts long enough) only if the separation between the two components is
substantially large $a>6000\,R_{\odot}$. The NS circular velocity in such an
orbit is only $21$ km s-1, while typical kick velocities are $100-1000$ km
s-1, see e.g., Igoshev et al. (2021) and references therein. So, after a
supernova explosion of the NS progenitor, very fine-tuning would be required
to maintain this binary. We therefore conclude that the settling accretion is
not a viable option for SWIFT J0850.8-4219.

##  6 Conclusions

In this paper, we applied different models of spin-down aiming to analyze the
hypothesis proposed in De et al. (2024) that the NS in SWIFT J0850.8-4219 is
at the propeller stage. We demonstrate that for all but one of the models
duration of this stage is too short to provide a significant probability of
detection of one of just two known systems as a propeller. Only for the model
of the supersonic propeller by Davies and Pringle Davies and Pringle (1981)
the NS remains at the propeller stage with a supergiant companion for the
parameters suggested for SWIFT J0850.8-4219. Measurements of the NS spin
period and its derivative are necessary to confirm if the NS is in the
propeller regime and to distinguish between different scenarios of the
propeller stage. In general, studies of SyXBs might be fruitful in
understanding the NS behavior at the propeller stage.

The authors contributed to this study equally. All authors have read and
agreed to the published version of the manuscript.

###### Acknowledgements.

We thank all the anonymous referees for their comments and suggestions that
helped us to improve the manuscript. MA acknowledges support from the Basis
Foundation grant 23-2-1-74-1. SP acknowledges support from the Simons
Foundation which made possible the visit to the ICTP. The code used for
modeling is accessible on request. Please, contact the first author. The
authors declare no conflicts of interest. Abbreviations The following
abbreviations are used in this manuscript:
HMXB | High-mass X-ray binary
---|---
NS | Neutron star
RGB | Reg giant branch
RSG | Red supergiant
SyXB | Symbiotic X-ray binary
no References yes

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