# A promising formation channel for symbiotic X-ray binaries: cases of IGR
J17329-2731 and 4U 1700+24

Iminhaji Ablimit1,2
1Key Laboratory for Optical Astronomy, National Astronomical Observatories,
Chinese Academy of Sciences, Beijing 100012, China
2Department of Astronomy, Kyoto University, Kitashirakawa-Oiwake-cho, Sakyo-
ku, Kyoto 606-8502, Japan E-mail<EMAIL_ADDRESS>
###### Abstract

Recent observations demonstrate that the symbiotic X-ray binary (SyXB) IGR
J17329-2731 contains a highly magnetized neutron star (NS) which accretes
matter through the wind from its giant star companion, and suggest that 4U
1700+24 may also have a highly magnetized NS. Accretion-induced collapse (AIC)
from oxygen-neon-magnesium white dwarf (ONeMg WD) + red giant (RG) star
binaries is one promising channel to form these SyXBs, while other long
standing formation channels have difficulties to produce these SyXBs. By
considering non-magnetic and magnetic ONeMg WDs, I investigate the evolution
of ONeMg WD + RG binaries with the mesa stellar evolution code for producing
SyXBs with non-magnetic or magnetized NSs. In the pre-AIC evolution with
magnetic confinement, the mass accumulation efficiency of the accreting WD is
increased at low mass transfer rate compared to the non-magnetic case. The
newborn NSs formed via AIC of highly magnetized WDs could inherit the large
magnetic field through conservation of magnetic flux, and the systems could
have a long age compatible with that of the red giant companions. These young
and highly magnetized NSs could accrete matters from the stellar wind of the
giant companions to that shine as those observed SyXBs, and could preserve
their high magnetic field during this time. The mesa calculation results show
that the initial parameter (initial RG mass and orbital period) space for the
AIC with magnetic confinement to form SyXBs with highly magnetized NSs shifts
to be lower and narrower compared to that of the no magnetic confinement case.

###### keywords:

binaries: close - stars: evolution - stars: magnetic fields - white dwarfs -
binaries: symbiotic.

††pagerange: A promising formation channel for symbiotic X-ray binaries: cases
of IGR J17329-2731 and 4U 1700+24–Data Availability

##  1 Introduction

Symbiotic X-ray binaries (SyXBs) are identified as a rare class of low-mass
X-ray binaries (LMXBs), and mainly composed of red giant (RG) stars and
neutron stars (NSs) (e.g., Masetti et al. 2006; Kaplan et al. 2007; Corbet et
al. 2008; Nespoli et al. 2008; Bozzo et al. 2011, 2012, 2013). Recently, two
known SyXBs have been reported and suggested to have highly magnetized NSs:
(1) IGR J17329-2731. Bozzo et al. (2018) analyzed the data collected quasi-
simultaneously with XMM-Newton and NuSTAR, and they estimated, according to
the observational results, that this SyXB contains an NS with a magnetic field
strength of $\sim 2.4\times 10^{12}$ Gauss. (2) 4U 1700$+$24\. Bozzo et al.
(2022) performed broad-band X-ray and soft $\gamma$-ray spectroscopy of 4U
1700$+$24, and found the presence of cyclotron scattering features. Based on
their observational data analysis, they suggest that 4U 1700$+$24 may have a
highly magnetized NS with a magnetic field strength of $\sim 1.4\times
10^{12}$ Gauss. There is a very interesting and unclear question: how are
these highly magnetized NS SyXBs as members of LMXBs are formed?

NSs in LMXBs can be formed via core-collapse supernovae during binary stellar
evolution, in this case NSs usually have low-mass main-sequence (MS) donors
($<1M_{\odot}$). In addition, LMXBs were suggested as the product of the
common envelope phase caused by unstable mass transfer from high-mass (donor’s
mass is larger than 10 $M_{\odot}$) X-ray binaries (van den Heuvel 1994).
Then, it was suggested that LMXBs may have evolved from intermediate-mass
(donor’s mass is between 3 and 7 $M_{\odot}$) X-ray binaries (King & Ritter
1999 ; Podsiadlowski & Rappaport 2000 ; Kolb et al. 2000). In these formation
routes, the donor stars fill their Roche-lobe due to nuclear evolution or
orbital angular momentum loss, and an accretion disk may formed nearby around
the NSs. Spin periods of the NSs can be accelerated, and the magnetic field of
the NSs may experience decay by the mass accretion. The magnetic field decay
induced by the mass accretion is not well studied in detail, however, it is
generally believed that the amount of field decay is closely related to the
accreted mass (Taam & van den Heuvel 1986, Shibazaki et al. 1989). As one
class of LMXBs, the formation of SyXBs with strongly magnetized NSs through
the formation routes introduced above is difficult, because it is hard to
explain the presence of a highly magnetized NS in SyXB, and the main reasons
are as follows: 1\. In those formation routes, the mass transfer and accretion
phase cannot be avoided when becoming an X-ray binary. If a highly magnetized
NS was born in LMXBs, the low mass MS star in this LMXB at least needs 9-10
Gyr to become an RG and fill its Roche-lobe to start the mass transfer. Then,
the NS at least accretes some amount of the transferred mass from the donor
and emits high energy particles, and this accretion could decay the magnetic
field of the NS. 2\. If the newborn NS in LMXBs has both a strong magnetic
field and very high rotational velocity, the propeller effect may take place,
and the NS ejects all matters transferred from the donor. Thus, no accretion
occurs in the case of the propeller effect, and the NS may retain its strong
magnetic field until the donor evolves to an RG. However, there is also no
high energy emissions in this case, and the binary cannot be an X-ray binary.

NSs also can form from WD binaries: (1) Merger-induced collapse of two WDs in
a binary (Nomoto & Iben 1985). (2) Core-merger-induced collapse during the
common envelope evolution of the WD binary (Ablimit et al. 2022). (3)
Accretion induced collapse (AIC) of the WD in a binary (e.g., Nomoto et al.
1979; Taam & van den Heuvel 1986). AIC of a highly magnetized WD in a binary
may be the promising solution for the formation puzzle of SyXBs with strongly
magnetized NSs. AIC from oxygen/neon/magnesium (ONeMg) composition WD binaries
has been studied as one of the comprehensive formation pathways for some
peculiar NS systems including millisecond pulsars (e.g., Chanmugam & Brecher
1987; Michel 1987; Kulkarni & Narayan 1988; Bhattacharya & van den Heuvel
1991; Yoon & Langer 2005; Hurley et al. 2010; Sabach & Soker 2014; Tauris et
al. 2013; Freire & Tauris 2014; Ablimit & Li 2015; Wang & Liu 2020; Perets
2022), redbacks/black widows (e.g., Ablimit 2019), and magnetars (Ablimit
2022). If the WD in the AIC process is highly magnetized, the high magnetic
flux could be conserved and even could be amplified during the AIC process
(e.g., Duncan & Thompson 1992).

The magnetic WDs have been observed in cataclysmic variables (CVs; e.g.,
Ferrario et al. 2015), symbiotic binaries and super-soft X-ray sources
(Kahabka 1995; Sokoloski & Bildsten 1999; Osborne et al. 2001). However, it is
worth noting that the AIC process (including conservation of magnetic flux)
and evolution of highly magnetized WD binaries as the progenitors of AIC or
type Ia supernovae (SNe Ia) are not well-known from the aspect of both
observation and theory. In addition, the origin of WDs’ magnetic fields also
remains unclear (see Mukhopadhyay & Bhattacharya [2022] for a recent review on
magnetized compact stars). Magnetic WDs could be the fossil remnants of their
progenitors (e.g., Angel et al. 1981), they would be formed from the field
amplification during the huliem-burning phase (Levy & Rose 1974), or they
could be generated from the binary interactions (Tout et al 2004; Nordhaus et
al. 2011; Garcia-Berro et al. 2012). It has been recently suggested that
single WDs’ magnetic fields can be generated by a dynamo of crystallization in
a process analogous to the Earth (Isern et al. 2017; Belloni et al. 2021;
Schreiber et al. 2021; Camisassa et al. 2022; Ginzburg et al. 2022).

Although polars (CVs containing highly magnetized WDs) known to date have
companion stars with relatively lower masses which cannot be progenitors of
AIC, very recent and interesting observational results demonstrated that
highly magnetized WDs like polars can have a two-pole accretion mode compared
to non-magnetic or weakly magnetized WDs (Schwope et al. 2021). Ablimit (2019;
see also Ablimit & Maeda 2019a,b) and Ablimit (2022) explored the role of
magnetism in hydrogen-rich or helium rich material accreting WD binaries as
progenitors of AIC (or SNe Ia), and they found that the initial donor mass -
orbital period space for producing AIC (or SNe Ia) can be different if
magnetic confinement is achieved.

The main goal of the work is that I study the detailed pre-AIC evolution of
highly magnetized, intermediate and non-magnetic ONeMg WDs with RG companions
as a promising formation channel for SyXBs with highly magnetized NSs or with
non-magnetic NSs. By employing the mesa stellar evolution code, I simulate
evolution of the ONeMg WD $+$ RG binaries for cases with and without magnetic
confinement. The details for the one dimensional mesa calculations of binary
evolution including magnetic confinement model are described in Section 2. In
Section 3, I give and discuss our results. The conclusion is presented in
Section 4.

##  2 The evolution of WD $+$ red giant star binaries (pre-AIC) without and
with the magnetic confinement

With the star and binary packages of version 15140 of mesa stellar evolution
code (Paxton et al. 2011, 2015), numerical simulations for the detailed
evolution of WD $+$ RG binaries are performed in this work. First of all, in
order to simulate the RG star models I employ the star package with a typical
Pop I composition with hydrogen abundance $X=0.70$, He abundance $Y=0.28$ and
metallicity $Z=0.02$. In the the stellar evolution, $initial\\_zfracs=6$ and
$kappa\\_file\\_prefix=$ ‘a09’ are set to call the opacity tables which are
built using the more recently available solar composition, and the Henyey
theory of convection with $mixing\\_length\\_alpha=1.8$ is used. The stellar
evolution is begun from a pre-main sequence model, and the final model is
saved to start the next step. The evolution is continued by loading the
previously save model, and stops it when the central helium fraction $\geq
0.98$ (it has a helium core and RG radius at this stage). RG star models with
masses ($M_{\rm RG}$) ranging from 0.5 to 2.0 $M_{\odot}$ are built by using a
mass interval of 0.1 $M_{\odot}$.

I use the binary package of mesa to simulate ONeMg WD $+$ RG binary evolution
with the initial orbital periods ($P_{\rm orb,i}$) of 0.5 - 2000 days. With RG
star models introduced above, and WDs are simulated as a point mass with
initial masses of 1.2 $M_{\odot}$. Because giant stars have lower surface
gravities and extended atmospheres, it is relatively complicated when
estimating the mass-transfer rate via the Roche-lobe overflow (RLOF). In this
work, the computational method in Kolb and Ritter (1990) is adopted for
estimating the RLOF mass transfer rate for RG stars, which should generate
reasonable results for stars at different evolutionary stages,

${\dot{M}_{\rm RL}}=-\dot{M}_{0}-2{\pi}F(q_{2})\frac{R^{3}_{\rm RL}}{GM_{\rm
RG}}\\\ \times{\int^{P_{\rm RL}}_{P_{\rm
ph}}{{\Gamma_{1}}^{1/2}{(\frac{2}{\Gamma_{1}+1})}^{(\frac{\Gamma_{1}+1}{2\Gamma_{1}-2})}{(\frac{\kappa_{\rm
B}T}{m_{\rm p}\mu})}}\,\mathrm{d}P},$ (1)

where $\dot{M_{\rm RL}}$ is the RLOF mass transfer rate, $\Gamma_{1}$ is the
first adiabatic exponent, and $P_{\rm ph}$ and $P_{\rm RL}$ are the pressures
at the photosphere and at the radius when the radius of the donor is equal to
its RLOF radius ($R_{\rm RL}$), respectively. $T$ is the temperature of the
donor, $\kappa_{\rm B}$ is Boltzmann constant and $\mu_{\rm ph}$ is the mean
molecular weight. $\dot{M}_{0}$ is

$\dot{M}_{0}=\frac{2\pi}{\rm exp(1/2)}F(q_{2})\frac{R^{3}_{\rm RL}}{GM_{\rm
RG}}{(\frac{\kappa_{\rm B}T_{\rm eff}}{m_{\rm p}\mu_{\rm ph}})^{3/2}}\rho_{\rm
ph},$ (2)

where $m_{\rm p}$ is the proton mass, and $T_{\rm eff}$ is the effective
temperature of the donor. $\mu_{\rm ph}$ and $\rho_{\rm ph}$ are the mean
molecular weight and density at its photosphere. For the fitting function
$F({q_{2}})$ ($q_{2}=M_{\rm accretor}/M_{\rm donor}$) and other physical
parameters in the evolutions are same as typical ones given in the
instrumental papers of mesa (e.g., Paxton et al. 2015). Stars in the giant
phase have relatively higher wind mass loss compared to stars at the early
stages; the prescription in Reimers (1975) is used for calculating wind mass
loss rate,

$\dot{M}_{\rm{wind}}=4\times 10^{-13}\frac{L_{\rm RG}R_{\rm RG}}{M_{\rm
RG}}{\eta_{\rm R}},$ (3)

where $L_{\rm RG}$ and $R_{\rm RG}$ are the luminosity and radius of the RG
star, and $\eta_{\rm R}=0.5$ is a typical value in the MESA code. The Reimers
scheme and Blocker scheme in mesa are set for the wind mass loss in the giant
phase by considering possible switch between RG and AGB branches (Paxton et
al. 2015).

In the accretion phase of the WD binary, the crucial role of mass transfer
rate is that it decides whether the material transferred from the donor onto
the WD can stably burn to let the WD grow in mass. When the mass transfer rate
(${\dot{M}_{\rm RL}}$) is suitable to realize stable hydrogen and helium
burning on the WD, the mass growth rate of the WD is,

$\dot{M}_{\rm{WD}}=\eta_{\rm H}\eta_{\rm He}{\dot{M}_{\rm RL}}.$ (4)

Hillman et al. (2015, 2016) studied the efficiency of hydrogen burning
($\eta_{\rm H}$) in a detailed way by considering the WD’s properties, and I
adopted their prescription. Zero and non-zero values of $\eta_{\rm H}$ can be
derived in this physical process: $\eta_{\rm H}=0.0$ (no mass growth) when
${\dot{M}_{\rm RL}}<\sim 5\times 10^{-8}\,\rm{M_{\odot}\,yr^{-1}}$ and nova
bursts occurs, $0.0<\eta_{\rm H}\leq 1.0$ and WD can grow in mass if $\sim
5\times 10^{-8}\,\rm{M_{\odot}\,yr^{-1}}\leq{\dot{M}_{\rm
RL}}<10^{-4}\,\rm{M_{\odot}\,yr^{-1}}$ (see Hillman et al. (2016) for more
details), and the binary enters the CE phase ($\eta_{\rm H}=0.0$) if
${\dot{M}_{\rm RL}}>10^{-4}\,\rm{M_{\odot}\,yr^{-1}}$. Some of transferred
mass can burn stably and some of them can be ejected from the WD if the mass
transfer rate exceeds $\sim 10^{-6}\,M_{\odot}\,{\rm yr^{-1}}$ (see also
Nomoto et al. 2007; Wolf et al. 2013). The calculation methods of Kato &
Hachisu (2004) for the mass accumulation efficiency of helium ($\eta_{\rm
He}$) are utilized in this work. These mass retention efficiencies are very
important, and the prescriptions used in this work are widely recognized in
the field. The different episodes for carbon burning cannot significantly
affect the growth of the WD (Brooks et al. 2017), thus the carbon burning
model does not have influence on our results. Some transferred matter may be
lost from the system as the optically thick wind when $0.0<\eta_{\rm H}<1.0$,
and it takes the angular momentum away with it. The angular momentum evolution
caused by all mass loss and other possible mechanisms (i.e. gravitational
wave, magnetic braking) are included in the calculations (see Paxton et al.
2015).

Compared to the spherically symmetric accretion, a strongly magnetized WD
could have the stream/confined accretion and unique emissions (e.g., Fabian et
al. 1977; Livio 1983; King & Shaviv 1984; Hameury et al. 1986; King 1993;
Wickramasinghe & Ferrario 2000; Wickramasinghe 2014; ; Ferrario et al. 2015;
Ablimit 2019). The accretion disc around the WD may be fully or partially
disrupted by the WD’s strong dipolar magnetic field (e.g., Cropper 1990; Frank
et al. 2002), and matter transferred via the RLOF of the donor star follows
magnetic field lines and falls down onto the magnetic poles of the WD as an
accretion column (see schematic figure of Ablimit (2019)). In this case, the
motion of the accreting matter could be dominated by the strong magnetic field
of the WD, because the magnetic pressure ($P_{\rm B}$) increases rapidly as
the accreted material approaches the WD’s surface, and the magnetic pressure
could exceed the ram pressure ($P_{\rm ram}$) of the accreting matter at the
magnetospheric radius ($R_{\rm M};$Frank et al. 2002),

$P_{\rm{B}}=\frac{(BR^{3}_{\rm WD})^{2}}{8\pi R^{6}_{\rm M}}\geq
P_{\rm{ram}}=\frac{(2GM_{\rm WD})^{1/2}{\dot{M}_{\rm RL}}}{4\pi R^{5/2}_{\rm
M}}.$ (5)

Thus, a minimal condition to have the magnetic confinement in the accretion
process of the hydrogen-rich material (Livio 1983) is that the magnetic field
strength ($B$) of the WD satisfies,

$B\geq 9.3\times 10^{7}(\frac{R_{\rm WD}}{5\times 10^{8}\,\rm
cm})(\frac{P_{\rm b}}{5\times 10^{19}\,\rm{dyne\,cm^{-2}}})^{7/10}\\\
(\frac{M_{\rm WD}}{\rm{M_{\odot}}})^{-1/2}(\frac{\dot{M}_{\rm
RL}}{10^{-10}\,\rm{M_{\odot}\,yr^{-1}}})^{-1/2}\,\rm{Gauss},$ (6)

where the pressure at the base of accreted matter is taken as $P_{\rm
b}=5\times 10^{19}\,\rm{dyne\,cm^{-2}}$ (Livio 1983); the formula in Nauenberg
(1972) is adopted for the mass ($M_{\rm WD}$) – radius ($R_{\rm{WD}}$)
relation of the WD.

Once the magnetic field strength of WD meets the minimal condition, the nova
burst can be suppressed and the magnetic confinement can be realized in the
accretion phase of a sufficiently magnetized WD even at a low mass transfer
rate. Then, the highly magnetized WD can have the isotropic pole-mass transfer
rate ($\dot{M}_{\rm{p}}$), which is (see also Ablimit & Maeda 2019a),

$\dot{M}_{\rm{p}}=\frac{S}{\Delta\rm{S}}{\dot{M}_{\rm RL}},$ (7)

where the surface area of a WD is $S=4\pi R^{2}_{\rm WD}$, and the area of the
polar caps (on which material is accreted) can be estimated as $\Delta{S}=2\pi
R^{2}_{\rm WD}\,\rm{sin^{2}{\beta}}$ ($\beta$ is the half-angle of the polar
cap; Frank et al. 2002). If the relation between two angles ($\beta$ and
$\theta$ which is the angle between the rotation axis and the magnetic field
axis; see Frank et al. (2002) for more details) is used, then the ratio is
${S}/{\Delta{S}}={{2R_{\rm m}}}/({R_{\rm WD}\,\rm{cos^{2}{\theta}})}$, where I
take $\theta=0$ (see also Ablimit & Maeda (2019a), Ablimit (2019) and Ablimit
(2022) for more details) in the calculations. ${R_{\rm m}}$ is the
magnetospheric radius which is related with Alfven radius ${R_{\rm A}}$ (Lamb
et al. 1973; Norton & Watson 1989; Frank et al. 2002),

$R_{\rm m}=\phi{R_{\rm A}}=\phi\,2.7\times 10^{10}{M^{-1/7}_{\rm
WD}}{\dot{M}^{-2/7}_{\rm acc}}{{\mu}^{4/7}}\,\rm{cm},$ (8)

where $\phi$ is a parameter ($\leq 1$) that takes into account the departure
from the spherically symmetric case, ${\mu}={B}{R^{3}_{\rm WD}}$ is the
magnetic moment of the WD in units of $10^{33}$ G $\rm{cm}^{3}$, $\dot{M}_{\rm
acc}$ is the mass-accretion rate (which is the actual mass transfer rate
$\dot{M}_{\rm RL}$ in this case) in units of $10^{16}$ g $\rm{s}^{-1}$ and
$M_{\rm WD}$ is the WD mass in solar units. $\dot{M}_{\rm{p}}$ is the rate to
calculate the hydrogen burning efficiency in the magnetic confinement model,
while $\dot{M}_{\rm RL}$ is one to derive the hydrogen burning efficiency in
the case of no magnetic confinement.

In this work, it is assumed that the ONeMg WD will collapse into an NS when it
grows in mass and reaches the Chandrasekhar limit mass of 1.38 $M_{\odot}$.
This limit mass to collapse may be higher for relatively fast rotating ONeMg
WDs but not all WDs rotate that fast (Yoon & Langer 2005; Freire & Tauris
2014; Ablimit & Li 2015), and it will not affect the main goal of this work.
It is also worth noting that the magnetic field generation for some WDs may
need the fast rotation (e.g., Ginzburg et al. 2022). The rotation of WDs is
not studied in the work, and the main focus is to examine the formation of
SyXBs with highly magnetized NSs with the assumption of magnetic flux
conservation during the AIC of highly magnetized WDs. In the binary evolution,
the fixed values for non-, intermediate- and high-magnetic field strengths of
the WDs are taken as 0, $1\times 10^{4}$ G and $2.5\times 10^{7}$ G (e.g.,
Livio 1983; Ferrario et al. 2015), respectively.

##  3 Results and discussion

###  3.1 Pre-AIC evolution

Figure 1 shows initial parameter spaces of WD$+$RG binary systems, the upper
panel is for systems having WDs with non- (0) or intermediate ($1\times
10^{4}$ G) magnetic field strengths (no magnetic confinement), and the lower
panel is for the highly magnetized ($2.5\times 10^{7}$ G) WD binaries
(magnetic confinement case). In the initial orbital period ($P_{\rm
orb,i}$)-secondary mass ($M_{\rm RG,i}$) planes, red star symbols mark the
binaries which lead to the AIC of ONeMg WDs, while green circles, black
squares and crosses represent binaries that cannot produce AIC due to
inefficient mass transfer or initiation of nova bursts. The WD binaries with
higher mass RG stars and too high mass transfer rate may undergo the CE
evolution, and this kind of system is not marked in the figure. The ranges of
initial donor mass and orbital period for the no magnetic confinement case are
$1.1-1.8\,M_{\odot}$ and $10-750$ days respectively. For the magnetic
confinement case, they are $0.7-1.8\,M_{\odot}$ and $5-120$ days. The initial
parameter space for the AIC shifts to lower regions compared to the no
magnetic confinement case, the one main reason is that the mass loss from the
system is generally higher for the higher values of $\dot{M}_{\rm p}$ (the WD
eject some of the transferred mass when $\dot{M}_{\rm
p}>10^{-6}\,M_{\odot}\,{\rm yr^{-1}}$). Compared with the previous work (e.g.,
Tauris et al. 2013; Wang et al. 2022), the progenitors of AIC in this work
have relatively shorter initial orbital periods and a wider range of initial
masses of the RG stars. The main reason causing the different results is that
the calculation method for RLOF mass transfer rate is different. Besides, I
considered both the wind mass loss and RLOF mass transfer in the evolution
while other works only considered one of them in their works. The RG in the
pre-AIC evolution can lose a certain amount of mass through the stellar wind
before the RLOF, and this mass loss could help the system have a relatively
stable RLOF mass transfer. In the pre-AIC evolution, it is also worth noting
that the wind mass loss rate before the RLOF is less than $3\times
10^{-8}\,\rm{M_{\odot}\,yr^{-1}}$ most of the time (besides only a small
fraction of the mass lost from the spherically (isotropic) stellar wind moves
toward the WD), thus the RLOF mass transfer rate dominates the mass accretion
phase in the WD binary evolution.

Detailed evolution of a WD + RG binary in a 100 day orbit is shown in Figure
2, and initial masses of the WD and RG star are 1.0 $M_{\odot}$ and $M_{\rm
WD,i}=1.2M_{\odot}$, respectively. The RG star loses mass through wind during
the evolution, and the RG star expands and overflows its Roche lobe while
burning the helium in its core. Without magnetic confinement in this example
(left panels of Figure 3), the accreted hydrogen-rich material burns unstably
on the WD surface and nova outbursts happen, thus the nova bursts prevent the
mass growth of the WD by blowing away the material transferred onto the WD. As
a result of this mass loss and the RG wind mass loss which affect the angular
momentum evolution of the system, the orbit of the binary is wider. The WD’s
mass stays still without magnetic confinement, and the orbital period evolves
from 100 to 920 days due to the angular momentum taken by these mass losses
away from the binary. Under the effect of WD’s strong magnetic field (right
panels of Figure 3), the transferred matter can be confined in the polar cap
regions, and the higher polar mass transfer rate (red line) alters the mass
accretion phase of the binary. The main difference of these two cases with the
same RLOF mass transfer from the donor is that the burning efficiency of
accreted hydrogen-rich material per unit area on the WD can be higher than
that in the spherical accretion (non-magnetic) case. Therefore, the WD is
gaining mass smoothly as the RG star is losing its mass under the the magnetic
confinement case, and it finally reaches the mass limit to explode as an SN Ia
(right panels of Figure 3). Correspondingly, the orbital period evolves in a
much slower way, because the system lost only a small amount of mass through
the RG stellar wind, and the binary with the magnetic confinement evolves
further until the SN Ia occurs, so that the radius of the RG star expands more
than that of the RG star in the no magnetic confinement case. Evolutionary
outcomes are very different in this example due to the magnetic field of the
WD.

In Figure 3, I show another example of WD $+$ RG binary evolution which the WD
($M_{\rm WD,i}=1.2M_{\odot}$) can reach $M_{\rm Ch}$ with and without the
magnetic confinement cases. The mass transfer in both cases proceeds on a
thermal timescale, the ratio of the mass of the RG to that of the WD is lower
than the critical mass ratio when the RG fills its Roche lobe, and the binary
systems undergo stable mass transfer until the WDs collapse. In this binary,
the wind mass loss rate of an RG star with an initial mass of 1.3 $M_{\odot}$
is higher than $10^{-10}\,\rm{M_{\odot}\,yr^{-1}}$ and up to
$10^{-8}\,\rm{M_{\odot}\,yr^{-1}}$; it takes more angular momentum away from
the binary, thus the wind mass loss widens the orbit significantly compared to
that of the previous example. With the higher mass donor (higher RLOF mass
transfer rate). The polar accretion rate is also much higher in the magnetic
confinement case (red line), and this higher polar accretion rate limits the
steady nuclear burning on the WD’s polar cap regions (this means the magnetic
field cannot efficiently/sufficiently confine all transferred material in the
polar cap regions). Thus, some of the transferred mass via RLOF would be lost
from the binary and taken away the angular momentum with it due to the higher
polar accretion rate (red line in right panels; the mass loss rate will be
higher if $\dot{M}_{\rm p}$ is higher [when it exceeds
$10^{-6}\,M_{\odot}\,{\rm yr^{-1}}$], and some of the transferred matter can
not burn stably on the WD and escape from the WD.), and the RG donor star
loses more mass in order to let the WD grow in mass to $M_{\rm Ch}$. This mass
loss also widen the binary orbit more compared to that of the no magnetic
confinement case (left panels), and the surviving companion star in the
magnetic confinement case is less massive than in the non-magnetic model. The
different mass accretion modes make the binary evolution run in different
ways, so that properties of surviving companions are also different.

###  3.2 Post-AIC evolution: Formation of SyXBs with highly magnetized NSs

When ${M}_{\rm WD}=M_{\rm Ch}$, I assume that the WD collapses to be an NS
with a gravitational mass of 1.25 $M_{\odot}$ (e.g., Schwab et al. 2010). The
orbital separation becomes wider due to the sudden mass-loss. According to the
prescription of angular momentum conservation, the relationship between the
orbital separations just before ($a_{0}$) and after ($a$) the collapse
(Verbunt et al. 1990) is calculated as,

$\frac{a}{a_{0}}=\frac{M_{\rm WD}+M_{2}}{M_{\rm NS}+M_{2}},$ (9)

where $M_{\rm NS}$ and $M_{2}$ are the NS and giant star masses. The newborn
NS through AIC would have a very small kick (Hurley et al.2010; Tauris et
al.2013), thus I do not include the effect of the kick. The processes of mass
accretion and mass loss for the NS binaries are also not well-known (these
processes are out of the main propose of this work; e.g., L$\ddot{\rm u}$ et
al. 2012), thus I just simply assumed that the NS accretes half of the
transferred matter, and the rest of the transferred matter is assumed to be
expelled. Corresponding to the accretion onto the NS ($\dot{M}$ is the
accretion rate), the apparent X-ray luminosity can be calculated as $L_{\rm
acc}=\eta\dot{M}c$ (Shakura & Sunyaev 1973), where $\eta$ is the radiative
efficiency of accretion onto the NS, and $c$ is the speed of light. The
typical luminosity of NS X-ray binaries is $\gtrsim 10^{32}$ erg $\rm s^{-1}$.
The angular momentum evolution ($\dot{J}$) in the post-AIC evolution is mainly
dominated by the angular momentum loss due to mass loss, and is also affected
by the gravitational wave radiation and magnetic braking, (and $\dot{J}_{\rm
ML}$, $\dot{J}_{\rm GR}$ and $\dot{J}_{\rm MB}$ ), which is
$\dot{J}=\dot{J}_{\rm GR}+\dot{J}_{\rm ML}$ \+ $\dot{J}_{\rm MB}$.

Figure 4 shows the properties of the binaries and the donor stars at the time
just after the AIC. From the AIC with highly magnetized WDs, newborn NSs in
these binaries are strongly magnetized (even magnetars) as discussed in the
above sections, and they can be symbiotic X-ray sources after the highly
magnetized NSs start mass accretion via stellar wind from the giant star. The
ranges of companion masses and orbital periods for the SyXBs with these newly
born highly magnetized NSs (dashed line area in Figure 4) are 5-652 days and
0.41-1.21 $M_{\odot}$, respectively. For the SyXBs formed with the non-
magnetic NSs (solid line area), the ranges of companion masses and orbital
periods are 21.3-2521.5 days and 0.59-1.23 $M_{\odot}$, respectively. There is
a partially overlapped region in Figure 4, I suggest that finding lower-mass
and relatively more luminous giant stars after the AIC may provide important
clue to understand the magnetic case of this symbiotic channel. Besides,
companions of newborn NSs after the AIC will evolve to He WDs, therefore
finding low mass He WDs - NS systems may also be crucial for understanding
this symbiotic progenitor scenario. Observational results for the binary and
donor properties of 4U 1700+24 and IGR J17329-2731 are not available yet, and
only the orbital period of 4U 1700+24 (404 days; Masetti et al. 2002) is
known. Our results from AIC under the non-magnetic and magnetic cases can
reproduce the orbital period of 4U 1700+24 (dotted line in Figure 4). I expect
that future observations will provide more parameters of these known SyXBs to
test the AIC models in this work.

###  3.3 Discussion

Compared to previous studies (e.g. Ivanova et al. 2008; Sutantyo & Li 2000),
our model in this work has following advantages to explain the existence of
highly magnetized NSs in old stellar populations like 4U 1700+24 and IGR
J17329-2731: 1\. The presence of a strong magnetic field and its crucial role
in the pre-AIC evolution is more likely to help for producing highly
magnetized NSs in SyXBs. The conservation of B-field might be uncertain, but
what would take away or decay that high magnetic flux during the AIC process?
The mass accretion in the pre-AIC or the mass loss during the AIC is not
enough to decrease so magnetic flux stays high. 2\. The timescale of the pre-
AIC evolution in this work is very long (9-10 Gyr), thus the highly magnetized
NSs can be born in old systems in wider orbits. 3\. The newborn NSs also can
be low rotators to avoid the propeller effect, so that the newly born and
highly magnetized NSs can shine as X-ray sources through the wind mass
accretion from the giant companions, and this does not need a long timescale
(no too much accretion) and cannot decay the magnetic field of NSs
significantly.

These X-ray binaries with highly magnetized NSs formed via AIC may evolve to
become ultra-luminous X-ray sources (e.g. Abdusalam et al. 2020) In addition,
the AIC of a highly magnetized WD also may form fast radio burst sources with
a young and highly magnetized NS such as FRB 20200120E (Kirsten et al. 2021).
It has also been studied that the AIC channel can form magnetars in binaries
(Ablimit 2022). Recent observational results for CXOU J171405.7-381031 and SGR
0755-2933 (Chrimes et al. 2022) show that these two candidates may have
stellar companions associated with magnetars, and one alternative evolutionary
pathway to form this kind of candidate may be the AIC channel.

The AIC channel suffers from two main disadvantages (e.g. Kitaura et al. 2006;
Dessart et al. 2006): (1) The lack of direct observational evidences for AIC.
(2) The uncertainties/unclearness about the spin rate and the magnetic field
strength of a newborn NS formed via AIC. The rotational velocities and the
magnetic activity depend on the binary evolutionary history (Soker 2002).
Tauris & van den Heuvel (2006) demonstrated that a Chandrasekhar mass WD with
an equatorial radius of about 3000 km and a magnetic field strength of
$10^{3}$ G could undergo AIC and directly produce an NS with B $10^{8}$ G by
assuming flux conservation. Thus, producing a strongly magnetized NS (at the
level of $10^{12}$ G) through the AIC of a highly magnetized WD (as used in
this work) is not a problem. Besides, Dessart et al. (2006) studied that only
a few 0.001 $M_{\odot}$ of material is ejected during the AIC, thus this small
amount of mass ejection indicates the higher possibility of conservation.
However, it is also the reason why the AIC is hard to be observed (see Dessart
et al. (2006) and Piro & Kulkarni (2013) for more discussion). The another
uncertainty is the accretion rate and magnetic field decay in the post-AIC
evolution. It is still not well-known that the newborn NS could accrete how
many percent of the transferred material, and how the accretion, magnetic
field and spin (propeller effect) will affect the evolution. Compared to other
formation pathways, to our current knowledge, the SyXBs with strongly
magnetized NSs is more likely formed via the AIC channel.

##  4 Conclusion

Recent observational results show that IGR J17329-2731, one of the known
SyXBs, contains a highly magnetized NS, and suggest that another SyXB 4U
1700+24 may also have a strongly magnetized NS. Most previous theoretical
formation scenarios fail to explain the presence of strongly magnetized NSs in
SyXBs. The AIC channel from highly magnetized ONeMg WDs – RG binaries is
probably the promising formation channel to produce SyXBs with strongly
magnetized NSs. However, the detailed evolution for this AIC channel has not
yet been studied. In this work, with the mesa stellar evolution code to
simulate large grids of stellar and binary evolution, I have investigated both
the pre- and post-AIC binary evolution to show that the AIC channel from WD–RG
binaries is the favorable way to form SyXBs with strongly magnetized NSs. In
the pre-AIC evolution of ONeMg WD - red giant star binaries, I consider cases
where the ONeMg WD is non-magnetic and cases where the magnetic field is
strong and forces accretion onto two small caps on the poles of the WD.
Performing stellar evolution simulations our findings are as follows:

  * •

If the ONeMg WD in the AIC channel is strongly magnetized, the magnetism has a
crucial role in the pre-AIC evolution. In the AIC evolution, the newborn NS
may inherit the large magnetic field through conservation of magnetic flux,
and the newly formed highly magnetized NS system has a long age like the RG
companion (9-11 Gyr). The wind accretion of the NS (without significant
magnetic field decay) from the old giant companion can shine as an SyXB which
is similar to the observations of IGR J17329-2731.

  * •

When the ONeMg WD that accretes mass from an RG star has a strong magnetic
field, lower mass RG stars can lead to AIC, because the confined accretion
onto a small area allows more efficient hydrogen burning to suppress nova
outbursts. As a result of the magnetic confinement, the WD mass can grow to
the Chandrasekhar mass limit and collapse as showed in this work or explode as
an SN Ia in the symbiotic channel of the single degenerate scenario (Ablimit
et al. 2023). Our results also show that, with the magnetic confinement the
initial parameter space of this symbiotic channel for SNe Ia shrinks toward
shorter orbital periods ($5-120$ days) and lower donor masses
($0.7-1.8\,M_{\odot}$) compared with that ($10-750$ days and
$1.1-1.8\,M_{\odot}$) in the no magnetic confinement case.

  * •

After the AIC evolved from the symbiotic channel with different cases, newborn
NSs can be normal NSs or peculiar NS objects as pulsars or magnetars. For
observations, searches for relatively lower-massive giant stars or He WDs as
companions of these NSs will be helpful to improve our understanding on this
scenario.

## Acknowledgements

I thank the referee for the useful comments that improved this work. This work
is supported by NSFs.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

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Figure 1: Initial parameter space of ONeMg WD $+$ RG star binaries which lead
to AIC in the initial orbital period ($P_{\rm orb,i}$)-initial secondary mass
($M_{\rm He,i}$) plane. The upper panel shows results under the magnetic
confinement case, and the lower panel is for the no magnetic confinement case
(non- or low magnetic WDs). The initial WD masses are 1.2 $M_{\odot}$ for both
cases. Red stars represents for binaries which can lead to AIC, while other
symbols show that these WD + RG binaries cannot produce AIC. The regions
encircled by solid red lines are for producing AIC from WD + RG systems. The
upper limits of the regions are based on the occurrence of a common envelope.
The lower limits (i.e. black cross) are determined by the initiation of nova
bursts. The left and right limits (i.e. green circles and black square) are
derived according to inefficient mass transfer.

Figure 2: Pre-AIC evolutions of ONeMg WD $+$ RG binaries: the left panels for
the no magnetic confinement case, and right panels are for the magnetic
confinement case (a highly magnetized WD with B$=2.5\times 10^{7}$ G. The
initial masses of the WD and RG donor star are 1.2 $M_{\odot}$ and 1.0
$M_{\odot}$ in this example, respectively. The initial orbital period is 100
days. Evolutions of mass transfer including the wind mass loss (polar mass
accretion for the magnetic confinement case), the WD/donor mass, orbital
period, and the radius evolutions of the donors with time are shown in the
figure. The wind mass loss is low ($<1\times 10^{-8}M_{\odot}/{\rm yr})$
almost in the whole evolution time, and RLOF mass transfer dominates the
evolution.

Figure 3: Another example for detailed evolution of ONeMg WD $+$ RG star
binaries using the MESA code: evolutions of mass transfer (polar mass
accretion for the magnetic case), wind mass loss, the WD/donor mass, orbital
period , and the radius evolutions of the donors with time are given. Right
and left panels show the non-magnetized (no magnetic confinement case) and
highly magnetized (B$=2.5\times 10^{7}$ G; with magnetic confinement case) WD
cases, respectively. The initial masses of the WD and RG donor are 1.2
$M_{\odot}$ and 1.3 $M_{\odot}$, and the initial orbital period is 40 days.
The wind mass loss is very low ($<10^{-11}M_{\odot}/{\rm yr})$ which only
affects the angular momentum evolution in a relatively significant way.

Figure 4: Companion giant star mass distributions versus Orbital periods for
SyXBs evolved from ONeMg WD $+$ RG binaries of Figure 1.