# The effect of aspherical stellar wind of giant stars on the symbiotic
channel of type Ia supernovae

Chengyuan Wu1, Dongdong Liu2,3,4, Xiaofeng Wang1, Bo Wang2,3,4
1Physics Department and Tsinghua Center for Astrophysics (THCA), Tsinghua
University, Beijing, 100084, China
2Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216, China
3Key Laboratory for the Structure and Evolution of Celestial Objects, Yunnan
Observatories, CAS, Kunming 650216, China
4University of Chinese Academy of Sciences, Beijing 100049, China
E-mail:wuchengyuan<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

The progenitor systems accounting for explosions of type Ia supernovae (SNe
Ia) is still under debate. Symbiotic channel is one of the possible progenitor
scenarios, in which the WDs in these systems increase in mass through wind
accretion from their red giant companions. The mass-loss processes of the
giants in the symbiotic systems could produce amount of circumstellar medium
(CSM), and the detections of the interaction signals between SN ejecta and CSM
can be used as an ideal way to distinguish the different progenitor models.
However, the density distribution and geometric structure of the CSM around
the symbiotic systems remain highly uncertain. By assuming that the tidal
torque from binary interaction can increase the mass-loss rate of the red
giant and cause the stellar wind concentrate towards the equatorial plane, we
provide a simplified method to estimate the density and the degree of
deviation from spherical symmetry of the CSM. Based on the calculations of the
binary evolutions of symbiotic systems using stellar evolution code MESA, we
obtained the parameter space for producing SNe Ia. We found that SNe Ia could
originate from symbiotic systems with massive carbon-oxygen white dwarfs (CO
WDs), while the half-opening angle of the stellar wind from red giant towards
the WD varies with the binary evolution, resulting in the formation of
surrounding CSM with different geometric structures. The corresponding
properties of ejecta-CSM interactions may be examined by the
spectropolarimetry observations in the future, from which one may find
additional relationship between circumstellar environment of SNe Ia and their
progenitor systems.

###### keywords:

binaries: symbiotic – stars: evolution – supernovae: general – white dwarfs

††pagerange: The effect of aspherical stellar wind of giant stars on the
symbiotic channel of type Ia supernovae–References††pubyear: 2019

##  1 Introduction

Type Ia supernovae (SNe Ia) are successfully used as standardizable candles in
measuring the cosmic distance (e.g. Riess et al. 1998; Perlmutter et al.
1999), and also have great contribution to the dynamical and chemical
evolution of galaxies (e.g. Greggio & Renzini 1983; Matteucci & Greggio 1986).
It has long been believed that SNe Ia are derived from thermonuclear
explosions of carbon-oxygen white dwarfs (CO WDs; Hoyle & Fowler 1960),
however, the progenitor systems are still difficult to be identified (e.g.
Wang & Han 2012; Maoz, Mannucci & Nelemans 2014; Livio & Mazzali 2018; Wang
2018; Jha, Maguire & Sullivan 2019; Ruiz-Lapuente 2019; Soker 2019a; Ruiter
2020). So far, at least two most popular progenitor models are widely
discussed: (1) The single-degenerate (SD) model. In this model, a CO WD
increases its mass by accreting material from its companion, and explodes as
SN Ia when its mass approaches Chandrasekar-mass limit (${M}_{\rm Ch}$).
Typically, the non-degenerate companion of the WD can be a main-sequence star
(MS), sub-giant, red giant (RG), AGB star or a helium star (e.g. Whelan & Iben
1973; Nomoto 1982; Li & van den Heuvel 1997; Langer et al. 2000; Han &
Podsiadlowski 2004; Wang et al. 2009). (2) The double-degenerate (DD) model.
In this model, supernova is originated from the coalescence of double CO WDs
(e.g. Iben & Tutukov 1984; Webbink 1984), although the final fates of double
WD merger may be somehow more complicated than what we thought (e.g. Nomoto &
Iben 1985; Schwab, Quataert & Kasen 2016; Wu, Wang & Liu 2019).

In the SD scenario, CO WD+RGB/AGB systems are thought to be progenitors of SNe
Ia, in which CO WDs are able to increase in mass to reach ${M}_{\rm Ch}$ by
accreting hydrogen-rich (H-rich) material from the stellar wind of their
evolved giant companions (e.g. Boffin & Jorissen 1988; Chen, Han & Tout 2011;
Iłkiewicz et al. 2019). This kind of wide binary systems, consisting of a
cool, evolved giant and a hot, luminous WD or neutron star, are known as
symbiotic stars (SySts). The evolved giants in symbiotic stars could be a
normal red giant for S-type, or a Mira variable embedded in an optically thick
dust shell for D-type (e.g. Milołajewska 2012). Up to now, hundreds of
symbiotic stars including several symbiotic recurrent nova (SyRN) systems such
as RS Oph, T CrB and V407 Cyg are confirmed. The WDs in the SyRN systems are
usually very massive or even close to ${M}_{\rm Ch}$, making the SyRNe to be
promising progenitor candidates for SNe Ia.

Observationally, an important way to discriminate different progenitor models
of SNe Ia is to detect the signal from the interaction between explosive
ejecta of the supernova and the circumstellar medium (CSM). Generally, SNe
that stem from the SD scenario should produce much denser CSM than the
counterpart in the DD scenario. This is because in the SD scenario, binary
mass-transfer process is indispensable before supernova explosion. The
material ejected by the WD wind and the mass donor during the binary evolution
may form the CSM of a certain density around the supernova. However, for the
DD model, mass-transfer process stopped long before the occurrence of the
supernova explosion due to the pretty long time-scale of the gravitational
wave radiation, which means that there is relatively low CSM around the
supernova (probably equals to the density of the interstellar medium, i.e.
${10}^{-24}\,\rm{g}\,\rm{cm}^{-3}$). Interaction of the SN ejecta with the CSM
is able to produce electromagnetic radiation from X-ray to radio bands,
depending on different radiation mechanism such as inverse Compton scattering
and synchrotron emission (e.g. Chevalier 1982). Such signals have been
detected in some cases. For instance, the first prototypical object for which
circumstellar hydrogen has been detected was SN 2002ic, followed by SN 2005gj
and SN 2008J (e.g. Hamuy et al. 2003; Aldering et al. 2006; Taddia et al.
2012). These SNe Ia show clear evidences for dense CSM, which indicates that
their progenitors may be the SySts. SN 2006X was the first SN Ia which was
found to show variable circumstellar Na I absorption lines in the spectra, and
its progenitor is deduced to be a WD+RG system (e.g. Patat et al. 2007). The
first case with both circumstellar interaction and time-varying narrow
absorption lines was PTF 11kx, and in this system, multiple shells of CSM are
detected, which can be described by a symbiotic nova progenitor, similar to RS
Oph (e.g. Dilday et al. 2012). However, some other literatures claim that the
CSM mass in PTF 11kx is larger than that predicted from symbiotic channel,
which indicated that this SN may result from core-degenerate scenario (e.g.
Soker et al. 2013). More recently, SN 2015cp and SN 2018fhw are also reported
to show H$\alpha$ emission lines likely resulting from ejecta-CSM interaction,
which can be attributed to the SD progenitor origins (e.g. Graham et al. 2019;
Vallely et al. 2019; Kollmeier et al. 2019). But it needs to be mentioned here
that the inferred hydrogen mass in SN 2018fhw may significantly less than what
theoretical predicted in SD scenario, which implies that alternative
explanations for the origination of hydrogen in SN 2018fhw may needed such as
the SN Ia ejecta unbinds a gas giant planet (e.g. Soker 2019b).

The CSM properties such as the density distribution and the value of half-
opening angle ($\alpha$) can effect the observational features significantly
(e.g. Moriya et al. 2013; Nagao, Maeda & Ouchi 2020). Previous studies usually
assumed a spherical stellar wind that is lost by the giant donor of the WD
(e.g. Chomiuk et al. 2012; Meng & Han 2016; Lundqvist et al. 2020), however,
observations indicated that the giants in SySts could have aspherical stellar
wind. For example, (1) the outburst of RS Oph in 2006 suggested an asymmetric
shock wave which may result from the interaction between the nova ejecta and
an equatorial enhancement in the red giant wind (e.g. O’Brien et al. 2006);
(2) Spectropolarimetry observations of SN 2002ic suggested that its
circumstellar environment may be dense and clumpy with disc-like geometry
(e.g. Wang et al. 2004); (3) The multiple components of CSM presented in PTF
11kx implied a nonuniform material structure concentrated in the orbital plane
(e.g. Dilday et al. 2012); (4) Many SNe Ia with high photospheric velocities
(HVs) show significant excess flux in blue band at 40-100 days after their
maximum light, which could be explained by the light echoes from the
asymmetric shell or disc-like (model dependent) CS dust (e.g. Wang et al.
2019). Besides, mass-loss rates and wind velocities of the red giants remain
highly uncertainties. Typically, the red giants in binary systems may have
higher mass-loss rate than those in the field as the tidal interaction cannot
be ignored. Comparing with field giants, Zamanov et al. (2008) found that the
rotational velocities of giants in SySts are faster (i.e. by 1.5-4 times) and
the mass-loss rates are higher (i.e. by 3-30 times), which could be attributed
to the tidal interaction. In addition, metallicity can also alter the mass-
loss rates of the giants in SySts and hence affect the birthrates of SNe Ia
expected from the SySt channel. For example, (1) SNe Ia-CSM samples are almost
from the late-type spiral galaxies, which implies that these objects may
relate to young stellar population (e.g. Silverman et al. 2013). (2) The blue
excess resulted from light echoes mentioned above is only observed in HV SNe
Ia that may relate to metal-rich environment (e.g. Wang et al. 2013; Pan et
al. 2015; Pan 2020; Wang et al. 2019).

The wind-accretion process and the expected parameter space of SNe Ia in SySts
still remain uncertain. In present work, we plan to examine the effect of
aspherical wind of giant that resulted from the tidal torque from the WD on
the SySt channel leading to SNe Ia. According to our assumption, the WD in
SySt can accrete material from wind more effectively due to higher density of
the wind surrounding the WD. We calculated the parameter space for producing
SNe Ia in both solar and metal-poor environments based on our revised wind
prescription. Our article is organized as follows: In Sect. 2, we provide our
basic assumptions and methods for the numerical simulations. The results of
our simulations are shown in Sect. 3. Finally, we present the discussion in
Sect. 4 and brief summary in Sect. 5.

##  2 Numerical Methods

In order to obtain the evolutionary properties of the binaries in the
assumptions of aspherical wind and the predicted parameter space for producing
SNe Ia, we employ the stellar evolution code MESA (version: 10398; Paxton et
al. 2011, 2013, 2015, 2018) to calculate the evolutions of WD+FGB/AGB binaries
with solar metallicity (with Z=0.02) and in the metal-poor environment (with Z
= $1\times 10^{-4}$). The CO WDs are treated as point masses, with which
masses of WDs (${M}_{\rm WD}$) are between $1.0{M}_{\odot}$ and
$1.1{M}_{\odot}$ (the very massive CO WD evolved from single star evolution).
During the evolution, WDs can accrete portion of wind material, leading to the
mass growth. We assume that WDs explode as SNe Ia when their mass grow to
$1.378{M}_{\odot}$. In our simulations, the masses of MS companions ${M}_{\rm
com}$ vary from 1.5 to 4.0 ${M}_{\odot}$ with mass intervals of
$\Delta\,{M}_{\rm com}=0.1$, and the initial orbital periods (${P}^{\rm i}$)
are in the range of ${\rm log}({P}^{\rm i}/{\rm day})=1.0-4.0$ with period
intervals of $\Delta\,{\rm log}({P}/{\rm day})=0.1$.

###  2.1 Mass loss of the giants

Owing to the tidal force exerted on the companion stars from the WDs, stellar
wind from the giant may deviate from spherical symmetry, leading to the
formation of aspherical CSM around the binary systems. Zahn (1975)
investigated the dynamical tide in close binaries, and found that the torque
in tidal friction is related to $(R/{R}_{\rm L})^{6}$, where ${R}_{\rm L}$ is
the Roche Lobe radius of the mass donor.

Based on the tidal interaction in the close binaries, Tout & Eggleton (1988)
proposed tidal enhanced wind model, in which the mass-loss rate of the giant
can be expressed as

$\dot{M}=\dot{M}_{\rm wind}\cdot\\{1+B\times\,{\rm min}[(\frac{R}{{R}_{\rm
L}})^{6},\frac{1}{{2}^{6}}]\\},$ (1)

where $\dot{M}_{\rm wind}$ is the wind mass-loss rate of the RG in single star
system and B is a free parameter to control the effect of the tidal
enhancement. However, they did not consider the effect of wind material
concentrating towards the orbital plane. Here, we assumed that the tidal force
could cause (1) the increase in mass-loss rate of the donor, (2) the stellar
wind of the donor to be concentrated towards the equatorial disc. The
assumption of enhancement in mass-loss rate of the donor is same as Tout &
Eggleton (1988). The deflection angle ($\theta$) is defined as
($\frac{\pi}{2}-{\alpha}$), where ${\alpha}$ is the half-opening angle of the
conical stream from the giant to the WD, and we assumed that ${\rm sin}\theta$
may obey three different function forms as follows in our calculations.

(1) linear function:

${\rm sin}\theta={a}_{1}\times(\frac{R}{{R}_{\rm L}})^{6}+{b}_{1};$ (2)

(2) binomial function:

${\rm sin}\theta={a}_{2}\times(\frac{R}{{R}_{\rm
L}})^{12}+{b}_{2}\times(\frac{R}{{R}_{\rm L}})^{6};$ (3)

(3) exponential function:

${\rm sin}\theta={a}_{3}\times{\rm exp}\,[{b}_{3}\times(\frac{R}{{R}_{\rm
L}})^{6}].$ (4)

The free parameters of tidal effect can be constrained from the observations.
The RS Oph is a symbiotic system which involves a $1.33{M}_{\odot}$ WD, and a
$0.8{M}_{\odot}$ RG companion. The orbital period and separation of this
system are around 453.6 days and 1.48AU, respectively (e.g. Brandi et al.
2009; Booth, Mohamed & Podsiadlowski 2016). Mohamed, Booth & Podsiadlowski
(2010) did 3D hydrodynamic simulations of the RS Oph system, by assuming that
the radius of RG is approximated to 0.8 times of its the Roche-lobe radius,
and found that the half-opening angle of the wind material to the WD is about
$30^{\circ}$. By adopting their results, we assumed that ${\rm
sin}\theta=\frac{\sqrt{3}}{2}$ when the radius of a RG expands to $0.8{R}_{\rm
L}$. Besides, ${\rm sin}\theta$ should comply with additional boundary
conditions, i.e. (1) when $R<<{R}_{\rm L}$, ${\rm sin}\theta$=0; (2) when
${R}\sim{R}_{\rm L}$, ${\rm sin}\theta\approx 1$. Substituting the boundary
conditions into the three different function forms of ${\rm sin}\theta$, we
could constrain the corresponding parameters, i.e.

for the linear function:

${\rm sin}\theta=3.3\times(\frac{R}{{R}_{\rm L}})^{6};$ (5)

for the binomial function:

${\rm sin}\theta=-3.137\times(\frac{R}{{R}_{\rm
L}})^{12}+4.137\times(\frac{R}{{R}_{\rm L}})^{6};$ (6)

for the exponential function:

${\rm sin}\theta=39.3\times{\rm exp}\,[-(\frac{R}{{R}_{\rm L}})^{-6}].$ (7)

Figure 1: The varition of wind deflection angle ($\theta$) with ${R}/{R}_{\rm
L}$. Solid carves represent three different functional relationships between
($\theta$) and $(\frac{R}{{R}_{\rm L}})^{6}$, whereas dotted line represents
the upper limit of deflection angle.

Fig. 1 shows the variation of ${\rm sin}\theta$ with $\frac{R}{{R}_{\rm L}}$
in three different functional forms. Note that the maximum value of ${\rm
sin}\theta$ cannot greater than 1, hence, we artificially limit the maximum
value of ${\rm sin}\theta=0.99$, and assumed that the stellar wind is
concentrated in the equatorial disc if ${\rm sin}\theta>0.99$. From the
figure, we can see that when ${R}<0.8{R}_{\rm L}$, $\theta$ in linear and
binomial forms have the similar increase rate, whereas $\theta$ changes
rapidly in the exponential form when ${R}>0.7{R}_{\rm L}$. In the current
work, we assumed that ${\rm sin}\theta$ is satisfied with binomial form, and
discuss the uncertainties in Sect. 4.

###  2.2 Mass accretion of WDs

When the companion enters into the giant branch, portion of the stellar wind
material could be accreted by the WD. Here, we assumed three different
accretion modes of the WD as follows.

(1) when the mass donor is in RGB phase (mass fraction of hydrogen in the
centre ${H}_{\rm centre}$ is less than ${10}^{-6}$, and carbon in the centre
(${C}_{\rm centre}$) is greater than ${10}^{-6}$), the accretion rate of WD is
in the Bondi-Hoyle accretion form (e.g. Bondi & Hoyle 1944; Boffin & Jorissen
1988) with an extra enhancement in the density of the wind material, which is

$\dot{M}_{\rm acc,BH}=-\frac{1}{\sqrt{1-{e}^{2}}}\,(\frac{G{M}_{\rm
WD}}{{v}_{\rm w}^{2}})^{2}\,\frac{{\alpha}_{\rm acc}\dot{M}_{\rm
loss}}{2{a}^{2}(1+\frac{{v}_{\rm orb}^{2}}{{v}_{\rm
w}^{2}})^{1.5}}\cdot\frac{\pi}{2\alpha},$ (8)

where ${\alpha}_{\rm acc}$ is accretion efficiency parameter which is set to
be $1.5$ (a default value in MESA), ${v}_{\rm orb}=\sqrt{\frac{G({M}_{\rm
com}+{M}_{\rm WD})}{a}}$ is the orbital velocity, ${v}_{\rm
w}=\frac{1}{2}\sqrt{\frac{G{M}_{\rm com}}{{R}_{\rm com}}}$ is the wind
velocity, and $\alpha$ is the half-opening angle of the conical stream from
the giant to the WD.

(2) Recent hydrodynamical simulations suggested that if the wind acceleration
radius of AGB star is larger than the Roche-lobe radius of itself in binary
system, the material in the wind acceleration zone could be accreted by the WD
through the inner Lagrangian point, which cause the formation of wind Roche-
lobe overflow mechanism (WRLOF; see Mohamed & Podsiadlowski 2007). WRLOF may
occur if the velocity of AGB wind is close to the orbital velocity of the WD.
Here, we assume that when the mass donor evolves to AGB phase (Both central
carbon and hydrogen are less than ${10}^{-6}$), and if
$\frac{1}{3}<\frac{{v}_{\rm orb}}{{v}_{\rm w}}<3$, the WD accretes material
from the wind of the donor in WRLOF mode with an extra enhancement in the
density of the wind material, in which the accretion rate can be expressed as

$\dot{M}_{\rm acc,WRLOF}=\frac{25}{9}{q}^{2}[-0.284(\frac{{R}_{\rm
d}}{{R}_{\rm L}})^{2}+0.918(\frac{{R}_{\rm d}}{{R}_{\rm
L}})-0.234]\cdot\frac{\pi}{2\alpha}\cdot\dot{M}_{\rm loss},$ (9)

where, $q=\frac{{M}_{\rm WD}}{{M}_{\rm giant}}$ is the mass ratio of WD to
companion, ${R}_{\rm d}=\frac{{R}_{\rm giant}}{2}\times(\frac{{T}_{\rm
d}}{{T}_{\rm eff,giant}})^{-2.5}$ is the dust radius of the AGB star, and
${T}_{\rm eff,giant}$ is the temperature of the giant, ${T}_{\rm d}=1500K$ is
the temperature of the carbon-rich dust (see Abate et al. 2013).

(3) When ${\rm sin}\theta\geq 0.99$, we assumed that the stellar wind is in an
equatorial disc form, in which the accretion rate of WD is related to the
crossing area of its gravitational radius in the orbital plane. The accretion
rate is approximately equals to

$\dot{M}_{\rm acc,DISC}=\frac{{R}_{\rm G}{v}_{\rm orb}}{{\pi}a{v}_{\rm
w}}\dot{M}_{\rm loss},$ (10)

where ${R}_{\rm G}=\frac{2G{M}_{\rm WD}}{{v}_{\rm w}^{2}+{v}_{\rm orb}^{2}}$
is the gravitational radius of the WD (e.g. Lv et al. 2009).

Note that the WD cannot accrete all of the wind matter ejected by the
companion, therefore, we artificially enforce the mass-accretion rate of the
WD from those three accretion forms mentioned above as $\dot{M}_{\rm
acc,BH}\leq 0.5$, $\dot{M}_{\rm acc,WRLOF}\leq 0.8$ and $\dot{M}_{\rm
acc,DISC}\leq 0.9$ (e.g. Iłkiewicz et al. 2019; Lv et al. 2009). In addition,
if the red giant fills its Roche-lobe during the evolution, dynamical unstable
mass transfer may occur since the mass-transfer rate increases dramatically,
leading to the formation of common envelope. In order to avoid the complicate
calculations, we artificially stop the simulations if mass-transfer rates from
Roche-lobe overflow are greater than ${10}^{-3}\,{M}_{\odot}\,\rm{yr}^{-1}$,
and assume that the corresponding systems are undergoing dynamically unstable
mass transfer.

###  2.3 Mass-loss rate estimation

We assumed that the wind mass-loss rate of the giant relates to its mass,
radius and luminosity. When the companion evolves to RGB phase, we adopt wind-
loss rate formula from Reimers (1975), which is

$\dot{M}_{\rm wind}=4\times{10}^{-13}{\eta}_{\rm R}\frac{LR}{M},$ (11)

where, ${\eta}_{\rm R}=0.5$ is a typical value in MESA. Here, the wind mass-
loss rate is enhanced by tidal force, thus, the real mass-loss rate in RGB
phase is

$\dot{M}=-4\times{10}^{-13}\,{\eta}_{\rm R}\,\frac{RL}{M}\,\\{1+B\times\,{\rm
min}[(\frac{R}{{R}_{\rm L}})^{6},\frac{1}{{2}^{6}}]\\},$ (12)

where R, L and M are radius, luminous and mass of the donor in solar units,
respectively. For the SySt RS Oph, the radius of the RG is about $\sim
59.1{R}_{\odot}$ which is obtained from the estimation of the average radius
for the corresponding spectral type of the RG (e.g. Zamanov et al. 2007). The
interval of recurrent nova eruption of RS Oph is about $20\,\rm{yrs}$,
corresponding to a mass-accretion rate of about $\dot{M}_{\rm
acc}=5\times{10}^{-8}\,{M}_{\odot}\,\mbox{yr}^{-1}$ (see Fig. 4 of Hillman et
al. 2016). To estimate the tidally enhanced parameter, we evolved binaries
with systematic parameters similar to RS Oph. The mass-loss rate of a
$0.8{M}_{\odot}$ star is about $\sim
7.6\times{10}^{-9}\,{M}_{\odot}\,\rm{yr}^{-1}$ when its radius expands to the
desired value without considering the effect of tidal enhancement, and the
Roche-lobe radius of the RG companion in RS Oph system is approximated as
$107{R}_{\odot}$ based on the estimation of Roche-lobe effective radius from
Eggleton (1983). Thus, the wind-accreting efficiency of the WD is about $\sim
30\%$ depending on the assumptions from Boffin & Jorissen (1988). Therefore,
we can estimate the tidally enhanced parameter ${B}$ is about a few thousand,
and we adopted ${B}=4000$ in our simulations. We adopted wind mass-loss rate
formula from Blöcker (1995) when the companion evolves to AGB phase. The mass-
loss rate can be expressed as

$\dot{M}_{\rm Blocker}=4.83\times{10}^{-9}{\eta}_{\rm
B}{M}^{-2.1}{L}^{2.7}\times{\dot{M}_{\rm Reimers}},$ (13)

where, the wind parameter ${\eta}_{\rm B}$ is set to be 0.1, which is a
typical value recommended in MESA.

###  2.4 Mass growth of WDs

The WDs in our simulations are treated as point masses. In order to limit the
mass growth of WDs, we adopted the prescription provided by Hachisu et al.
(1999). In their prescriptions, the critical mass accretion rate of WD relates
to the mass fraction of hydrogen (${\chi}_{\rm H}$) and WD mass (${M}_{\rm
WD}$), i.e.

$\dot{M}_{\rm cr,H}=5.3\times{10}^{-7}\frac{(1.7-{\chi}_{\rm H})}{{\chi}_{\rm
H}}({M}_{\rm WD}/{M}_{\odot}-0.4){M}_{\odot}\,\mbox{yr}^{-1}.$ (14)

If the mass accretion rate is higher than $\dot{M}_{\rm cr,H}$, the WD
accumulates hydrogen as a rate of $\dot{M}_{\rm cr,H}$, the unburned material
is blown away by the optically thick wind (e.g. Kato & Hachisu 1994; Hachisu,
Kato & Nomoto 1996). If $\dot{M}_{\rm acc}$ is lower than $\dot{M}_{\rm cr,H}$
but higher than $\frac{1}{8}\dot{M}_{\rm cr,H}$, the WD can accumulate all the
accreted material. If $\dot{M}_{\rm acc}$ is lower than
$\frac{1}{8}\dot{M}_{\rm cr,H}$, strong H-shell flash is able to occur on the
surface of WD, and in this case, the WD cannot accumulate any material.
According to the assumptions above, the mass-accumulation efficiency of
hydrogen ${\eta}_{\rm H}$ can be expressed as follows:

${\eta}_{\rm H}=\left\\{\begin{array}[]{rcl}\frac{\dot{M}_{\rm
cr,H}}{\dot{M}_{\rm acc}},&&{\dot{M}_{\rm acc}>\dot{M}_{\rm cr,H}},\\\
1,&&{\frac{1}{8}\dot{M}_{\rm cr,H}\leq{\dot{M}_{\rm acc}}\leq{\dot{M}_{\rm
cr,H}}},\\\ 0,&&{{\dot{M}_{\rm acc}}\leq{\frac{1}{8}\dot{M}_{\rm
cr,H}}}.\end{array}\right.$ (15)

Hence, the mass-growth rate of helium layer can be expressed as

$\dot{M}_{\rm He}={\eta}_{\rm H}\dot{M}_{\rm acc}.$ (16)

Similarly, the mass-accumulation efficiency of He layer can be expressed as

${\eta}_{\rm He}=\left\\{\begin{array}[]{rcl}\frac{\dot{M}_{\rm
cr,He}}{\dot{M}_{\rm He}},&&{\dot{M}_{\rm He}>\dot{M}_{\rm cr,He}},\\\
1,&&{\dot{M}_{\rm st,He}\leq\dot{M}_{\rm He}\leq\dot{M}_{\rm cr,He}},\\\
{\eta}_{\rm He}^{{}^{\prime}},&&{\dot{M}_{\rm low,He}<\dot{M}_{\rm
He}<{\dot{M}_{\rm st,He}}},\\\ 0,&&{{\dot{M}_{\rm He}}\leq{\dot{M}_{\rm
low,He}}}.\end{array}\right.$ (17)

where, $\dot{M}_{\rm low,He}$ and $\dot{M}_{\rm cr,He}$ are the lowest and
highest critical transfer rate, $\dot{M}_{\rm st,He}$ is the minimum accretion
rate of stable He-shell burning, ${\eta}_{\rm He}^{{}^{\prime}}$ is the mass-
accumulation efficiency of helium in weak He-shell flashes. If $\dot{M}_{\rm
He}$ is lower than $\dot{M}_{\rm low,He}$, a helium detonation may cause a
supernova explosion, which is out of our consideration, and we assumed the
mass-accumulation rate of He layer to be zero. If $\dot{M}_{\rm He}$ is higher
than $\dot{M}_{\rm cr,He}$, the WD accumulates helium layer as a rate of
$\dot{M}_{\rm cr,He}$, the rest of material is blown away by the optically
thick wind. If $\dot{M}_{\rm He}$ is in the range of $\dot{M}_{\rm st,He}$ to
$\dot{M}_{\rm cr,He}$, the He-shell burning is stable. If $\dot{M}_{\rm He}$
is in the range of $\dot{M}_{\rm low,He}$ to $\dot{M}_{\rm st,He}$, the mass-
increase rate of CO core can be expressed as:

$\dot{M}_{\rm CO}={\eta}_{\rm He}\dot{M}_{\rm He}={\eta}_{\rm H}{\eta}_{\rm
He}\dot{M}_{\rm acc}.$ (18)

The critical transfer rates are

$\dot{M}_{\rm low,He}=4.0\times{10}^{-8}\,{M}_{\odot}\,\mbox{yr}^{-1}$ (19)

and

$\dot{M}_{\rm cr,He}=7.2\times{10}^{-6}\,({M}_{\rm
WD}/{M}_{\odot}-0.6)\,{M}_{\odot}\,\mbox{yr}^{-1},$ (20)

respectively, based on the previous simulations of mass-accretion WD
evolutions (e.g. Woosley, Taam & Weaver 1986; Nomoto 1982). ${\eta}_{\rm
He}^{{}^{\prime}}$ relates to the WD mass and $\dot{M}_{\rm He}$, and we
adopted the linearly interpolated from a grid computed by Kato & Hachisu
(2004).

For the low metallicity (Z=0.0001), we adopted the results of Chen et al.
(2019), in which they provided comprehensive models of nova at metal-rich and
metal-poor environment, for the upper and lower limits of steady hydrogen
burning, i.e.

${\rm log}(\dot{M}_{\rm lower})=-12.21+13.32\times{M}_{\rm
WD}-11.90\times{M}_{\rm WD}^{2}+3.82\times{M}_{\rm WD}^{3},$ (21) ${\rm
log}(\dot{M}_{\rm upper})=-9.81+7.25\times{M}_{\rm WD}-5.17\times{M}_{\rm
WD}^{2}+1.32\times{M}_{\rm WD}^{3}.$ (22)

In the metal-poor environment, we have not changed the polynomial fitting of
${\eta}_{\rm He}$ since the mass-accumulation rate of helium may be slightly
influenced by the metallicity (e.g. Wu et al. 2017).

###  2.5 Orbital angular-momentum losses

During the binary evolution, mass-loss from both donor and WD are needed to be
considered. The wind material which is lost from the surface of binary stars
can take away the specific angular momentum of mass donor and CO WD. The
angular-momentum-loss due to mass-loss from the donor is

$\dot{J}_{\rm ML}({\rm donor})=-\dot{M}_{\rm d}(\frac{{a}{M}_{\rm
WD}}{{M}_{\rm WD}+{M}_{\rm d}})^{2}\frac{2\pi}{{P}_{\rm orb}},$ (23)

and

$\dot{J}_{\rm ML}({\rm WD})=-(1-\eta)\dot{M}_{\rm WD}(\frac{{a}{M}_{\rm
d}}{{M}_{\rm WD}+{M}_{\rm d}})^{2}\frac{2\pi}{{P}_{\rm orb}}$ (24)

for the angular-momentum-loss from the vicinity of the CO WD, in which $\eta$
is the mass-accumulation rate of the CO WD.

##  3 Numerical Results

###  3.1 Binary evolution calculations

Figure 2: A typical binary evolution example for producing SN Ia through
Bondi-Hoyle accretion mode in symbiotic channel with solar metallicity, in
which ${M}_{\rm WD}^{\rm i}=1.1{M}_{\odot}$, ${M}_{\rm com}^{\rm
i}=2.0{M}_{\odot}$ and ${\rm log}{P}^{\rm i}=2.6$, respectively. Panel (a):
solid black curve represents the evolutionary track of the companion in the HR
diagram, whereas the dashed line represents the evolutionary track of the
orbital period of the binary system. Panel (b): solid black curve represents
the evolution of the mass-loss rate of the companion star in logarithmic form
from the end of main-sequence stage to the SN explosion. Red and blue dash-
dotted curve show the evolution tracks of the masses of WD and companion;
panel (c): solid black curve and blue dash-dotted curve show the evolution of
Roche-Lobe, radius and the stellar radius of the companion, whereas the
magenta dash-dotted curve represents the change of half-opening angle of the
stellar wind material from the companion to the WD.

Figure 2-5 present typical examples of binary evolutions, in which the masses
of WDs could reach ${M}_{\rm Ch}$ ($1.378{M}_{\odot}$) through wind accretion.
In Fig. 2, the initial WD mass and orbital period are set to be ${M}_{\rm
WD}=1.1{M}_{\odot}$ and ${\rm log}({P}^{\rm i}/{\rm day})=2.6$, respectively.
The companion star is a $2.0{M}_{\odot}$ MS with solar metallicity (Z=0.02).
The wind mass-loss rate is quite small (on the order of magnitude of around
${10}^{-11}\,{M}_{\odot}\,\rm{yr}^{-1}$) during the main-sequence and
Hertzsprung gap phases. After about $1\times{10}^{9}$ years, the companion
enters into RGB phase and begins to expand gradually, resulting in the
increase of luminosity and mass-loss rate. According to the prescription
described in Sect 2, the WD could accretes portion of material in the stellar
wind. However, the WD cannot accumulate mass until the mass-accretion rate
increases to be greater than $1.0\times{10}^{-7}\,{M}_{\odot}\,\rm{yr}^{-1}$,
since all the accreted material is ejected by the strong hydrogen flash on the
surface of WD. Thus, the WD only increases in mass during the evolution of the
last $1.8\times{10}^{6}\,\rm{yr}$, which can be seen in panel (b) of Fig. 2.
The mass-loss from the vicinity of the WD carries away orbital angular
momentum, and the non-conservative mass-transfer process leads to the increase
of orbital period. As shown in panel (a) of Fig. 2, the binary system evolves
to the long orbital period during the RGB phase, and finally reached ${\rm
log}({P}^{\rm f}/{\rm day})=3.34$ when SN Ia explosion occurs. The evolution
of radius of the companion is shown in panel (c) of Fig. 2. As can be seen
from this figure that the radius of companion and its Roche-lobe are
$65.4{R}_{\odot}$ and $417.7{R}_{\odot}$, respectively. In this case, the
half-opening angle of the stellar wind from the companion to the WD is about
${87.2}^{\circ}$, which means that the WD is accreting material in the
spherical wind through Bondi-Hoyle accretion mode.

Figure 3: Binary evolution example for producing SN Ia through WRLOF accretion
mode in symbiotic channel with solar metallicity, in which the binary
comprises a $1.1{M}_{\odot}$ WD and a $2.0{M}_{\odot}$ MS with orbital period
of ${\rm log}{P}^{\rm i}=3.8$.

Fig. 3 shows an example of SN Ia arising from AGB channel, where the initial
companion mass ${M}_{\rm WD}=2.0{M}_{\odot}$ and initial orbital period is
${\rm log}({P}^{\rm i}/{\rm day})=3.8$, respectively. Due to the initially
wide separation, the wind mass-loss rate of the companion is lower than
${10}^{-7}\,{M}_{\odot}\,\rm{yr}^{-1}$ in the whole RGB phase since the effect
of tidal enhancement is relatively weak. However, when the companion evolves
into AGB phase, the radius and mass-loss rate increase rapidly since the star
has more effective mass-loss mechanism. In this case, the WD increases its
mass to ${M}_{\rm Ch}$ in $4.5\times{10}^{5}\,\rm{yr}$ through WRLOF wind
accretion mode.

Figure 4: Binary evolution example for producing SN Ia through equatorial disc
accretion mode in symbiotic channel with solar metallicity, in which the
binary comprises a $1.1{M}_{\odot}$ WD and a $2.0{M}_{\odot}$ MS with orbital
period of ${\rm log}{P}^{\rm i}=1.7$.

In Fig. 4, we present an example for producing SN Ia in which the wind
material is concentrated in the equatorial disc. The initial companion mass
and initial orbital period are ${M}_{\rm WD}=2.0{M}_{\odot}$ and ${\rm
log}({P}^{\rm i}/{\rm day})=1.7$, respectively. Similar to the first case, the
companion enters into RGB phase after about $7\times{10}^{9}\,\rm{yr}$ of
evolution and begins to increase in radius. Due to the relatively short
initial orbital period, a more effective tidal enhancement is exerted onto the
companion, resulting in the rapid increase of mass-loss rate. The evolution of
radius of the companion is shown in panel (c). As can be seen from this figure
that the companion nearly fills its Roche-Lobe at the final stage
(${R}/{R}_{\rm L}\sim 0.905$), and the half-opening angle of the stellar wind
from the companion to the WD is close to zero (${\rm sin}\alpha<0.01$), which
means that the stellar wind concentrates in the equatorial disc during this
stage (lasting for about $2.37\times{10}^{6}$ years).

Figure 5: Similar to Fig. 1, but for the evolution of different companion in
low metallicity environment, in which the mass and initial orbital period are
${M}_{\rm com}^{\rm i}=3.2{M}_{\odot}$ and ${\rm log}{P}^{\rm i}=3.5$,
respectively.

A typical example of binary evolution of low metallicity population is shown
in Fig. 5, where ${M}_{\rm WD}^{\rm i}=1.1{M}_{\odot}$, ${M}_{\rm com}^{\rm
i}=3.2{M}_{\odot}$, $Z=0.0001$ and ${\rm log}{P}^{\rm i}=3.5$. The companion
stars usually have smaller radius and low wind-mass-loss rate compared to
those with $Z=0.02$ during RGB stage due to the low opacity derived from low
metallicity. However, the mass-loss rate is high enough in AGB phase since the
AGB-wind is more sensitive to the luminosity. From this fiugre, we can see
that the radius of companion combined with mass-loss rate increase obviously
when the star enters into Thermally Pulsing AGB phase (TPAGB). In panel (c)
and (d), we plotted the evolution of dust radius and half-opening angle. The
dust radius ${R}_{\rm dust}$ exceeds ${R}_{\rm L}$ in the TPAGB phase, where
the WD can accrete material more efficiently through the assumption of WRLOF
comparing with BH. Due to the large amplitude in radial pulsation, half-
opening angle of the stellar wind can vaies from about $60^{\circ}-90^{\circ}$
in this stage.

###  3.2 Parameter space for producing SNe Ia

Figure 6: Regions of initial companion mass and orbital period which can
produce SNe Ia with ${M}_{\rm WD}^{\rm i}=1.1{M}_{\odot}$ and
$1.05{M}_{\odot}$ in solar metallicity and metal-poor environment. The open
circles denote that companion mass is not enough to increase the WDs for
reaching ${M}_{\rm Ch}$. Filled triangles, circles and squares represent that
the WDs explode as SNe Ia in the weak H-shell flash phase, stable hydrogen
burning phase and optical thick wind phase. Red and blue dots represent the
core of companion stars are helium core and carbon-oxygen core when WDs
explode as SNe Ia, in which the predicted ranges are enclosed by brown and
magenta curves. Open squares represent the systems that come across some
numerical problems when the mass donors are in TPAGB phase. The alternate
burning of hydrogen and helium shells makes the calculational time-scale
extremely short. For these systems, we just simply deduce that they may evolve
to SNe Ia if they are in the range surrounded by blue curve, otherwise they
may experience dynamical mass-transfer processes.

We evolved more than 1200 WD+MS systems in solar metallicity environment, and
more than 800 pairs of systems are in low metallicity environment, where the
initial WD masses are from $1.0{M}_{\odot}$ to $1.1{M}_{\odot}$. The companion
masses are in the range of $1.5-7.2{M}_{\odot}$ for solar metallicity and
$1.7-3.6{M}_{\odot}$ for low metallicity. The initial orbital periods ${\rm
log}({{P}^{\rm i}/{\rm day}})$ are from $1.0-4.0$. Fig. 6 present the initial
parameter space for producing SNe Ia in two metallicity environments under our
assumption. Regions surrounded by brown or blue curves represent that the WDs
can increase their masses to ${M}_{\rm Ch}$ when their companions evolve to
RGB or AGB phase, respectively. Binaries beyond the left and upper boundaries
would experience dynamically unstable mass-transfer and then may evolve to
common envelope phase to prevent the WD from increasing its mass, whereas
those beyond the right and lower boundaries tend to evolve to the double WD
systems since the companions fail to provide enough masses for WDs to accrete
effectively. Note that there exists possibility that binaries including a
massive companion can produce SNe Ia since these companions ignite central
helium non-degenerately and do not expand large enough to transfer sufficient
mass before they enter into AGB phase. The parameter space shrinks
dramatically with the decrease of initial mass of the CO WD, and vanished for
the systems with ${M}_{\rm WD}^{\rm i}=1.0{M}_{\odot}$ CO WDs in both solar
metallicity and metal-poor environment, which means that SNe Ia can only arise
in the SySts that consist of massive CO WDs under current assumptions.

Figure 7: Regions of initial (solid curves) and final (dashed curves)
companion masses and orbital periods for producing SNe Ia with ${M}_{\rm
WD}^{\rm i}=1.1{M}_{\odot}$ via wind-accretion channel in solar metallicity
(panel a) and metal-poor environment (panel b). Red and blue crosses represent
two symbiotic systems T CrB and RS Oph, respectively.

The final companion mass and orbital period distributions are shown in Fig. 7.
Comparing with the solar metallicity, the predicted parameter space for the
low metallicity environment is larger, since the opacity of the companion
decreases with decreasing metallicity, resulting in a smaller radius of
companion in the giant branch. This would cause the companion more easily to
avoid filling its Roche-lobe to trigger the dynamically unstable mass-
transfer. Particularly, ${M}_{\rm RG}$ and ${\rm log}{P}_{\rm orb}$ of two
well observed symbiotic stars, i.e. T CrB and RS Oph are located near the
final range, which means that they are two promising recurrent nova symbiotic
systems leading to SN Ia explosions.

###  3.3 Properties of circumstellar medium

The mass-loss rate and wind velocity are two critical parameters in estimating
the CSM properties. In Figs. 8 and 9, we present the mass-loss rates, wind
velocities and half-opening angles of the stellar wind from the companions to
WDs (i.e. conical streams from the giants to the WDs) in two metallicity
environments at the moment of SNe explosions. Since the stellar evolution has
a nuclear timescale, parameters at the time when SN explosion occurs can
approximately represent the CSM properties which are close to supernova.
Generally, the WD+AGB systems have larger mass-loss rate and lower wind
velocity than those from the WD+RGB channel due to higher luminosity and
larger radius of the AGB stars. The mass-loss rate for the companion in the
WD+AGB channel is mainly concentrated in the magnitude of
$10^{-5}\,{M}_{\odot}\,\rm{yr}^{-1}$, and some of the systems can have mass-
loss rate greater than $10^{-4}\,{M}_{\odot}\,\rm{yr}^{-1}$. For the WD+RGB
system, the mass-loss rate is in the range of
$3\times{10}^{-7}-3\times{10}^{-6}\,{M}_{\odot}\,\rm{yr}^{-1}$. The half-
opening angle can reflect the geometric shape of the CSM dust. The figures
show that the CSM usually has larger opening-angles if the WD experiences
optically thick wind phase, whereas those from stable H-shell burning or nova
ejection are in a wide range. In particular, some of the symbiotic recurrent
nova systems may have disc-like CSM, which are consistent with some observed
systems, like RS Oph and PTF 11kx. For the supernovae exploded in the WD+AGB
systems, the stellar wind typically has larger opening-angle since these
systems are in wide separations. Particularly, the half-opening angles of the
stellar wind for the WD+AGB systems in metal-poor environment are all larger
than ${75}^{\circ}$, meaning that the CSM dust formed in low metallicity
environment may be generally close to spherical symmetry geometry, which may
be verified by the spectropolarimetry observations.

Figure 8: Panel (a): wind velocities versus mass-loss rates of the SNe Ia
progenitors from the symbiotic systems in the solar metallicity at the
occurrence time of supernovae explosions. Open circles, filled circles and
filled squares indicate that the WDs in the corresponding systems are in the
H-shell flashes, stable H burning and optically thick wind phases,
respectively. Red and blue symbols represent that the supernovae explode in
the WD+RGB and WD+AGB systems, respectively. Panel (b): similar to panel (a),
but for the relationship between the half-opening angle of the stellar wind
and mass-loss rate. Figure 9: Similar to Fig. 8, but for the systems in the
metal-poor environment.

The density profile of the CSM is a key parameter in estimating the
interacting signal between the SN ejecta and CSM. In order to obtain the
variation of the CSM density, we assumed a wind-like CSM, in which the stellar
wind from the companion or ejected by the WD expands in a constant velocity,
and the mass-loss rate is adopted from the average mass-loss rate of the
progenitor in the final stage before SN Ia explosion. In this condition, the
density of CSM formed via stellar wind is inversely proportional to the square
of distance, i.e. ${\rho}_{\rm CSM}\propto{r}^{-2}$. For the systems with
accreting WDs, the optically thick wind or nova shell ejected by the WD is
able to interact with the CSM formed through the stellar wind from the
companion, making the density profile of the CSM more complicated. Here we
simply assumed that the interaction between WD ejection and stellar wind are
completely inelastic collision, which is satisfied with the conservation of
momentum, i.e. the interacted medium would have the same expanding velocity.
In Figs. 10 and 11, we estimated the possible density distributions of the CSM
under our assumption, which can provide initial physics for the interaction
model. The velocity of optically thick wind and nova ejecta are approximately
assumed to be $1000\,\rm{km}\,\rm{s}^{-1}$ (e.g. Hachisu, Kato & Nomoto 1999;
Yaron et al. 2005; Moriya et al. 2019), and the interaction between fast wind
and slow stellar wind can accelerate the CSM to a velocity of $\sim
100\,\rm{km}\,\rm{s}^{-1}$, which decreases the density of CSM in the systems
with high mass-loss rate. The CSM with higher density tend to appear in WD+AGB
systems since the predicted wind mass-loss rate in AGB phase is greater than
RGB phase; whereas for the WD+RGB channel, a dense CSM may appear in stable
H-shell burning phase or weak H-shell flash phase as the stellar wind is more
likely concentrated to the equatorial disk.

Figure 10: Density of CSM changing with distance at different phases in solar
metallicity. Panel (a): WD+RGB systems in which the WDs are experiencing
optically thick wind. Blue and red solid lines indicate the upper and lower
limits of the density of CSM in the regions within half-opening angle of these
systems, whereas the blue and red dash-dotted lines indicate the upper and
lower limits in the regions within the deflection angle. Panel (b): WD+RGB
systems in which the WDs are in stable H burning phase. Panel (c): WD+RGB
systems in which the WDs are in the weak H-shell flash phase (here, we simply
assume that the stellar wind filled up with the cavity rapidly during the
interval of the recurrent nova). Panel (d): WD+AGB systems in which the WDs
are in optically thick wind phase. Panel (e): WD+AGB systems in which the WDs
are in the weak H-shell flash phase. Figure 11: Similar to Fig. 10, but for
the systems in the metal-poor environment.

##  4 Discussion

###  4.1 Uncertainties in the modeling

In our simulations, we assumed that the mass-loss rate of the giant donor can
be enhanced via the tidal interaction. However, the degree of the enhancement
is difficult to be measured, which somehow leaves large uncertainty. Chen, Han
& Tout (2011) calculated the binary evolutions under the assumption of
reinforced attrition from companion (e.g. Tout & Eggleton 1988). They found
that, if assuming a significantly great value of tidally enhanced parameter
(${B}\sim 10000$) which estimated from Z Her, the parameter space for
producing SNe Ia through wind-accretion symbiotic channel can extend to longer
orbital periods and their birthrate can increase to about
$6.9\times{10}^{-3}\,\rm{yr}^{-1}$. However, the enhanced effect of tidal
interaction may be overestimated in the symbiotic systems. In our simulations,
we assumed that both enhanced mass-loss rate and wind deflection angle are
related to $({R}/{R}_{\rm L})^{6}$, which means that material lost from red
giant is more concentrated towards the equatorial plane and hence increases
the mass-accretion rate of the WD. Owing to the difficulties in estimating the
relationship between the effect of tidal enhancement and $({R}/{R}_{\rm
L})^{6}$, we assumed three different forms of ${\rm sin}\theta$, and the
corresponding parameters can be estimated from the observational parameters of
RS Oph as mentioned in Sect. 2. Here, we calculate the binary evolutions of CO
WD+RGB channel with linear and exponential forms of ${\rm sin}\theta$, and the
parameter spaces as well as corresponding features are shown in Fig. 12.
Comparing with panel (a1) of Fig. 6, we can find that the parameter spaces for
producing SNe Ia via WD+RGB channel (red symbols within brown curves) under
the assumptions of linear and binomial functions have almost no difference,
whereas the parameter space under the assumption of exponential function
slightly shrinks, since the stellar wind has greater value of deflection angle
in linear and binomial functions when ${R}/{R}_{\rm L}<0.7$, resulting in a
higher density of stellar wind surrounding the WD. In this case, WD can
accrete wind material more efficiently and thus more easily to increase in
mass to ${M}_{\rm Ch}$. The distributions of SNe Ia systems in ${v}_{\rm
wind}$-${\rm log}\dot{M}$ panel and ${\theta}$-${\rm log}\dot{M}$ panel are
similar under different assumptions, which means that the influence of
different function forms on our results are negligible.

Another uncertainty in our simulations is the limitation of mass-accretion
rate of the WD. In wind-accretion process, the accretion rates of WDs are
calculated by approximated equations which related to the parameters of
binaries, and may overestimate the accretion rate of WDs in some cases during
the evolutions. Since the WDs in symbiotic systems cannot accrete all of the
wind material from their giant companions, we artificially set the upper limit
of mass-accretion fractions for different accretion modes, although the
proportion of angular momentum loss from RG or WD can influence the process of
binary evolution and thus the parameter space for producing SNe Ia. In our
calculations, we simply adopted the corresponding limitations based on
previous works (e.g. Iłkiewicz et al. 2019; Lv et al. 2009).

Figure 12: Parameter space for producing SNe Ia via symbiotic channel and the
predicted stellar wind properties at the moment when SNe explode under two
different function forms between ${\rm sin}\theta$ and $({R}/{R}_{\rm
L})^{6}$. Linear function: panel (a1)-(a3); Exponential function: panel
(b1)-(b3).

Previous studies inferred a relatively low rate of SNe Ia from the SySts. This
is because, (1) the Roche-lobe mass-transfer usually leads to irreversible
expansions of the giants. The extremely fast mass-transfer from the giant
donor to the WD may cause dynamically instable process, resulting in the
formation of common envelope, which could prevent the WD from evolving to SN
Ia in the traditional SD scenario (e.g. Li & van den Heuvel 1997; Wang, Li &
Han 2010); (2) The wind mass-transfer usually causes an increase of the
orbital separation between the WD and its giant companion, resulting in a low
efficiency with which the WD retains material (e.g. Kenyon et al. 1993;
Yungelson et al. 1995; Branch et al. 1995). We have not calculated the
birthrate of SNe Ia via symbiotic channel, since SNe Ia can hardly stem from
the binaries with initially less massive WDs under our assumptions, which
predicted a relatively low birthrate. Indeed, there are some strong
constraints on the symbiotic SD scenario from the observations which means
that the SNe from SySt are relatively rare although they are not completely
excluded. For instance, radio observations place strong constraints on the
presence of CSM, and the non-detections imply a low rate of thermonuclear SNe
which have symbiotic progenitors (e.g. Chomiuk et al. 2016; Lundqvist et al.
2020); The studies of supernova remnants in the Milky Way and LMC indicated
the absence of luminous red giants in SN Ia progenitor systems (e.g. Li et al.
2011; González Hernández et al. 2012; Edwards, Pagnotta & Schaefer 2012). On
the simulations, note that some other alternative paths for producing SNe Ia
via the symbiotic channel are proposed to increase the birthrate. Hachisu,
Kato & Nomoto (1999) suggested that the stellar wind from the WD may interact
with the RG and strip some of the mass from the RG surface, which can
stabilize the mass-transfer process and avoid the formation of common
envelope. Lv et al. (2009) estimated that the Galactic birthrate of SNe Ia via
symbiotic channel is between $1.03\times{10}^{-3}$ and
$2.27\times{10}^{-5}\,\rm{yr}^{-1}$ by considering a spherical + equatorial
disc concentrated stellar wind. By adopting an improved method to calculate
the mass-transfer rate, Liu et al. (2019) found that the parameter space for
producing SNe Ia via the semidetached symbiotic channel is significantly
enlarged. However, none of these models can be completely proved by the
observations so far, making the birthrate and circumstellar environment of
symbiotic systems still be an open question. The Roche-lobe related aspherical
symmetry stellar wind assumption proposed in the present work provide an
additional possibility of wind accretion, in which more complicated CSM
density distribution is predicted to be formed, linking the properties of
ejecta-CSM interactions with the evolutionary process of the progenitors,
which are hoped to be detected in the observations especially for the
spectropolarimetry observations in the future (e.g. Cikota et al. 2017; Yang
et al. 2018).

###  4.2 Comparing with observations

Observationally, some symbiotic novae consisting of massive WDs are thought to
be the progenitors of SNe Ia. Two typical examples are RS Oph and T CrB. The
symbiotic recurrent nova RS Oph consists of a $1.3-1.4{M}_{\odot}$ WD and a
$0.68-0.8{M}_{\odot}$ RG companion in an circular orbit with orbital period of
$\sim 454.1\pm 0.41\,\rm{d}$ and orbital inclination of about $50^{\circ}$
(e.g. Brandi et al. 2009; Booth, Mohamed & podsiadlowski 2016). The WD mass in
T CrB is about ${M}_{\rm WD}\sim 1.37\pm 0.13{M}_{\odot}$ and the mass of RG
${M}_{\rm RG}\sim 1.12\pm 0.23{M}_{\odot}$. The orbital period of T CrB
${P}_{\rm orb}\sim 226.57\,\rm{d}$ and orbital inclination ${i}\sim
67^{\circ}$ (e.g. Stanishev et al. 2004). Our calculations show that the
current companion masses and orbital periods of both systems are close to the
${M}_{\rm com}^{\rm f}-{P}_{\rm orb}^{\rm f}$ range for producing SNe Ia,
which indicate that RS Oph and T CrB are strong candidates in which the
massive WDs in both systems can grow in mass to ${M}_{\rm Ch}$ through the
wind-accretion process. Additionally, the RG in RS Oph may be close to filling
its Roche-lobe by measuring its rotation period (e.g. Zamanov et al. 2007).
According to our assumption, there exist symbiotic recurrent nova systems that
consist of an almost Roche-lobe filling RG with disc-like stellar wind can
produce SNe Ia, in which the companion mass and orbital period are close to
those of RS Oph. This implies that our model is consistent with the
observations.

Some SNe Ia have been reported to show apparently evidence of interaction
between supernova ejecta and CSM. SN 2006X is a normal SN Ia which show clear
evolution of Na I D doublet lines (e.g. Patat et al. 2007). The mean velocity
of CSM is about $\sim 50\,\rm{km}\,\rm{s}^{-1}$, and the circumstellar
hydrogen is likely concentrated in a thin spherical shell with a radius of
$\sim{10}^{17}\,\rm{cm}$. The upper limit to the shell mass can be estimated
as $3\times{10}^{-4}\,{M}_{\odot}$ based on some conservative assumptions, and
the dust surrounding the supernova may have multiple shells (e.g. Patat et al.
2007; Wang et al. 2008). According to our calculations, the mass-loss rates of
RGs in symbiotic recurrent nova systems is around $\sim 3\times
10^{-7}-1\times{10}^{-6}\,{M}_{\odot}\,\rm{yr}^{-1}$ and the mass of CSM can
be approximately calculated by

${M}_{\rm CSM}\approx\frac{{R}_{\rm distance}\dot{M}_{\rm loss}}{{v}_{\rm
wind}}-\Delta{M}_{\rm WD},$ (25)

where ${R}_{\rm distance}$ is the distance between the CSM and SN Ia, and
$\Delta{M}_{\rm WD}$ is the increasing mass of the WD during the time when the
CSM shell expands to the corresponding distance. Assuming that the nova ejecta
pushed the CSM to the distance of about ${10}^{17}\,{\rm cm}$ and formed a
thin shell, we can estimate that the mass of CSM shell is about
$2-8\times{10}^{-4}\,{M}_{\odot}$, which can explain the observational
properties of SN 2006X. This implies that the progenitor system of SN 2006X
may consist of a $\sim 2.0{M}_{\odot}$ RG and a massive CO WD with an orbital
period of about $40-60\,{\rm d}$, which is similar to the symbiotic system RS
Oph. However, it should be mentioned that the earlier explanation of the time-
variable Na I absorption lines from SN 2006X may suffer from some
difficulties. Soker (2014) argued that in order to account for such
variations, large hydrogen density is needed, and such multiple shells must be
at a distance of larger than $0.1{\rm pc}$ from the explosion, which implied a
large shell mass that cannot be explained by the SD scenario.

SN 2002ic and PTF 11kx are two SNe Ia for which clear evidence of
circumstellar interaction are detected (e.g. Hamuy et al. 2003; Dilday et al.
2012). Chugai & Yungelson (2004) estimated the mass of CSM around SN 2002ic by
fitting the interacted bolometric light curve under the assumption of
spherically symmetric CS envelope, and found that the total mass of CSM within
$7\times{10}^{15}\,{\rm cm}$ is about $0.4{M}_{\odot}$. According to the
spectropolarimetry data, Wang et al. (2004) suggested that the SN 2002ic
exploded inside a dense, clumpy, disc-like circumstellar environment. PTF 11kx
also show the evidence of existing massive CSM. Soker et al. (2013) estimated
that the mass of CSM may greater than $0.2{M}_{\odot}$ at a distance of
$\sim{10}^{16}\rm{cm}$, which is larger than what most SD scenarios predict,
implying that the massive CSM may originate from common envelope ejection
(i.e. PTF 11kx is stem from the merger of a WD with the core of its giant
companion). Graham et al. (2017) later estimated that the total mass of CSM is
about $0.06{M}_{\odot}$ within ${10}^{16}\,{\rm cm}$ by assuming a solar
abundance and a spherical shell geometry for the CSM in the PTF 11kx system,
depending on the reassessed of column density of Ca II. From our results, we
can estimate that the CO WD+MS binary systems with initial companion mass of
about $2.4-2.8{M}_{\odot}$ and initial orbital period of about
$3000-6000\,{\rm d}$ can evolve into the AGB symbiotic sysytems with a mass-
loss rate of AGB star in the range of
$1-2\times{10}^{-4}\,{M}_{\odot}\,\rm{yr}^{-1}$, which can produce enough
dense CSM comparing with the observational properties of PTF 11kx inferred
from Graham et al. (2017). But for the circumstellar environment of SN 2002ic,
none of the symbiotic systems can explain the dense CSM with disc-like
geometry, unless the RG has a significantly high mass-loss rate (i.e.
${10}^{-3}-{10}^{-2}\,{M}_{\odot}\,\rm{yr}^{-1}$) and it almost fills its
Roche-lobe simultaneously. However, binaries in this case may easily evolve to
the common envelope phase through the dynamically unstable mass-transform, and
may finally form double WD systems or undergo merger process which depends on
whether they can survive from common envelope phase or not. Therefore, SN
2002ic may originate from some other scenarios such as the explosion of the
core of an AGB star (e.g. Livio & Riess 2003).

There exist evidences showing that metallicity has an influence on the
formation of SNe Ia through symbiotic channel. Silverman et al. (2013) found
that the host galaxies of SNe Ia-CSM are all late-type spiral galaxies like
Milky Way with solar metallicity or dwarf irregulars like the Large Magellanic
Cloud with subsolar metallicity. One exception is ASASSN-18tb, which exhibits
a fairly typical light curve and has a luminosity of an underluminous or
transitional SN Ia, while it is hosted by an early-type galaxy dominated by
old stellar populations (e.g. Kollmeier et al. 2019; Vallely et al. 2019).
Wang et al. (2013, 2019) suggested that SNe Ia with high photospheric
velocities tend to have metal-rich progenitor systems and originate from
symbiotic systems. Pan (2020) found that the SNe Ia with high photospheric
velocities are likely formed from a distinct population which favors massive
host environments, and argued that the metallicity is the only important
factor in forming high-velocity SNe Ia. Our results indicated that the
parameter space for producing SNe Ia through the symbiotic channel with both
rich and poor metallicity dramatically shrinks with the decrease of initial WD
mass. Combining with the previous results that the stars in metal-rich
environment generally produce less massive WDs (e.g. Umeda et al. 1999), we
can estimate that SNe Ia stem from symbiotic channel which can produce much
more dense CSM gas or dust in high metallicity environment have a relatively
lower birthrate than those in low metallicity environment.

Besides of the symbiotic systems, SNe Ia could also stem from the CO WD+He
star systems. Moriya et al. (2019) estimated the CSM properties in CO WD+He
systems, and found that this kind of system can also produce high enough
density of CSM when SNe explode through “optically thick wind” scenario whcih
can be revealed by radio observations. Li et al. (2020) investigated a sample
of 16 SNe Ia, and found that all HV samples that exist in metal-rich
environment have redshifted of the velocity of late-time [Fe II] features,
which can be explained by the He-detonation model (e.g. Townsley et al. 2019),
linking the HV SNe Ia with He star systems. Therefore, it is still early to
make conclusion about the origin of CSM around SNe Ia and the effect of
metallicity. Probably more SNe Ia samples of different subclasses observed in
the future could provide the opportunities to reveal the corresponding
questions.

##  5 Summary

By assuming that the tidal force from the WD can alter the wind mass-loss rate
of the giant and the degree that the stellar wind deviate from spherical
symmetry, we investigated the production of SNe Ia through the symbiotic
channel. We obtained the parameter space for producing SNe Ia in both solar
metallicity and metal-poor environment and analyzed the property of the binary
systems including the density distribution of CSM at the moment of supernova
explosions. Our results enable to reproduce some observational properties of
stellar wind in SySts and CSM around SNe Ia with aspherical symmetric geometry
structures, by using which one may make better constrains on the progenitors
when combining with the observations of CSM interaction.

## Acknowledgments

CYW is supported by the National Natural Science Foundation of China (NSFC
grants 12003013). DDL is supported by the National Natural Science Foundation
of China (No. 11903075), the Western Light Youth Project of Chinese Academy of
Sciences and Yunnan province (No. 202001Au070054). XFW is supported by the
National Natural Science Foundation of China (NSFC grants 11325313, 11633002
and 11761141001), the National Program on Key Research and Development Project
(grant no. 2016YFA0400803). BW is supported by the NSFC (Nos 11873085 and
11521303), the Chinese Academy of Sciences (No QYZDB-SSW-SYS001), and the
Yunnan Province (Nos 2018FB005, 2019FJ001 and 202001AS070029).

## DATA AVAILABILITY

Results will be shared on reasonable request to corresponding author.

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