# Wide binary pulsars from electron-capture supernovae

Simon Stevenson1,2, Reinhold Willcox2,3, Alejandro Vigna-Gómez4 and Floor
Broekgaarden5
1 Centre for Astrophysics and Supercomputing, Swinburne University of
Technology, John St, Hawthorn, Victoria- 3122, Australia
2 The ARC Centre of Excellence for Gravitational Wave Discovery, OzGrav
3 School of Physics and Astronomy, Monash University, Clayton, Vic. 3800,
Australia
4 DARK, Niels Bohr Institute, University of Copenhagen, Jagtvej 128, 2200,
Copenhagen, Denmark
5 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Neutron stars receive velocity kicks at birth in supernovae. Those formed in
electron-capture supernovae from super asymptotic giant branch stars—the
lowest mass stars to end their lives in supernovae—may receive significantly
lower kicks than typical neutron stars. Given that many massive stars are
members of wide binaries, this suggests the existence of a population of low-
mass ($1.25<M_{\mathrm{psr}}/$M${}_{\odot}<1.3$), wide
($P_{\mathrm{orb}}\gtrsim 10^{4}$ day), eccentric ($e\sim 0.7$), unrecycled
($P_{\mathrm{spin}}\sim 1$ s) binary pulsars. The formation rate of such
binaries is sensitive to the mass range of (effectively) single stars leading
to electron capture supernovae, the amount of mass lost prior to the
supernova, and the magnitude of any natal kick imparted on the neutron star.
We estimate that one such binary pulsar should be observable in the Milky Way
for every 10,000 isolated pulsars, assuming that the width of the mass range
of single stars leading to electron-capture supernovae is $\lesssim 0.2$ M⊙,
and that neutron stars formed in electron-capture supernovae receive typical
kicks less than 10 km s-1. We have searched the catalog of observed binary
pulsars, but find no convincing candidates that could be formed through this
channel, consistent with this low predicted rate. Future observations with the
Square Kilometre Array may detect this rare sub-class of binary pulsar and
provide strong constraints on the properties of electron-capture supernovae
and their progenitors.

###### keywords:

pulsars: general - transients: supernovae - supernovae: general - stars:
neutron

††pubyear: 2021††pagerange: Wide binary pulsars from electron-capture
supernovae–Wide binary pulsars from electron-capture supernovae

##  1 Introduction

Stars with initial masses $\lesssim 8$ M⊙ end their lives as white dwarfs,
whilst more massive stars undergo core-collapse supernovae and form neutron
stars or black holes (Woosley et al., 2002; Doherty et al., 2017). Super
asymptotic giant branch (SAGB; Garcia-Berro & Iben, 1994; Doherty et al.,
2017) stars111Recently O’Grady et al. (2020) reported the observation of
$\sim$ 10 SAGB star candidates in the Magellanic Clouds. on the boundary
between these two regimes are thought to result in electron-capture supernovae
(ECSNe; Miyaji et al., 1980; Nomoto, 1984, 1987) leading to the formation of a
neutron star. The mass range of single stars expected to undergo ECSNe is
narrow and roughly in the range 8–10 M⊙, although the exact mass range is very
uncertain and depends on the details of the models (e.g., Poelarends et al.,
2008; Doherty et al., 2015, 2017; Jones et al., 2016; Leung et al., 2020). The
mass range for ECSNe may be wider in interacting binaries (e.g., Podsiadlowski
et al., 2004; Doherty et al., 2017; Poelarends et al., 2017; Siess &
Lebreuilly, 2018), which may result in the majority of ECSNe occurring in
binary systems. Whether ECSNe occur in single stars at all is an open question
(e.g., Willcox et al., 2021).

ECSNe are expected to have low luminosities compared to other classes of
supernovae, and be observed as type IIp or IIn supernovae (e.g., Tominaga et
al., 2013; Moriya et al., 2014; Hiramatsu et al., 2021). No observed
supernovae have been conclusively associated with an ECSN, but SN2018zd may be
the best candidate to date (Hiramatsu et al., 2021). Other candidates include
the supernova that formed the Crab nebula/pulsar (e.g., Nomoto et al., 1982;
Smith, 2013; Moriya et al., 2014), though see Gessner & Janka (2018).

Numerical simulations succeed in realising ECSNe from first principles (e.g.,
Kitaura et al., 2006; Dessart et al., 2006). Recently, Gessner & Janka (2018)
performed hydrodynamical simulations of ECSNe and found that the remnant
neutron stars receive a kick of only a few km/s at most. The small kicks arise
from the rapid explosion, which does not allow time for substantial
asymmetries to develop. Kicks this small are much lower than the typical
velocities of a few hundred km/s inferred for neutron stars from the proper
motions of isolated Galactic pulsars (e.g., Lyne & Lorimer, 1994; Hobbs et
al., 2005; Verbunt et al., 2017). For this reason, ECSNe play an important
role in the formation of double neutron star binaries (e.g., Vigna-Gómez et
al., 2018; Giacobbo & Mapelli, 2019), the retention of neutron stars in
globular clusters (Pfahl et al., 2002a), the population of wide neutron star
high-mass X-ray binaries (Pfahl et al., 2002b; Podsiadlowski et al., 2004;
Knigge et al., 2011) and the mass distribution of neutron stars in binaries
(Schwab et al., 2010; Ozel et al., 2012).

Given that many massive stars are members of wide, non-interacting binaries
(e.g., Moe & Di Stefano, 2017), a population of neutron stars receiving (very)
low kicks during ECSNe would lead to the prediction of wide, eccentric,
unrecycled binary pulsars, which would not be predicted if ECSNe lead to large
kicks (and thus disrupt wide binaries). Observations of wide binary pulsars
(or the lack of these systems) could therefore be a powerful test of whether
ECSNe can occur in effectively single stars. This could provide insights into
ECSNe and their progenitors.

In this paper we develop a simple model for the formation of low-mass
($M_{\mathrm{psr}}\sim 1.3$ M⊙), wide ($P_{\mathrm{orb}}>10^{4}$ day),
eccentric ($e>0.5$), unrecycled ($P_{\mathrm{spin}}\sim 1$ s) binary pulsars
in the Galactic field formed by ECSNe. We describe our model in Section 2.1,
and present results of a Fiducial model in Section 2.2. We test the
sensitivity of our predictions to model uncertainties in Section 2.3. We
estimate the number of wide binary pulsars observable in the Milky Way in
Section 2.4, finding that roughly one wide binary pulsar should be observable
for every 10,000 isolated pulsars, assuming that the mass range of
(effectively) single stars that lead to ECSNe is $<0.2$ M⊙ (Willcox et al.,
2021). In Section 3 we search for candidate wide binary pulsars in the
observed pulsar population. We find no candidates that are well explained by
our model, consistent with our estimated rates; the observation of these
systems is hampered by the lack of sensitivity of pulsar observations to
orbital periods much longer than the observing baseline of a few decades. We
argue that a few (2–3) apparently isolated observed pulsars could actually be
in wide binaries. We conclude in Section 4.

##  2 Electron capture supernovae in a population of massive, wide binary
stars

###  2.1 Method and fiducial assumptions

We model a population of wide, massive, non-interacting binary stars at solar
metallicity, appropriate for pulsars formed recently in the Milky Way. A large
fraction of intermediate/massive stars are known to be part of binaries (e.g.,
Sana et al., 2012; Moe & Di Stefano, 2017). We are interested in the evolution
of single $\sim 8$ M⊙ SAGB stars at solar metallicity, that may end their
lives in ECSNe and produce observable pulsars in the Milky Way. Given the
uncertainties in the masses of SAGB stars which undergo ECSNe (Doherty et al.,
2017), we assume222We have checked that the exact mass assumed does not impact
our conclusions. that all primaries (initially the most massive star in the
binary) have a mass of 8 M⊙, as appropriate for solar metallicity SAGB stars.
We do not model binaries with primary stars with initial masses greater than 8
M⊙, as we assume that these will have already evolved and undergone core-
collapse supernovae, receiving high kicks of order a few hundred km/s,
resulting in the disruption of the binary (see e.g., Vigna-Gómez et al., 2018;
Renzo et al., 2019).

We sample the remaining properties of each binary from statistical
distributions based on observations of massive, wide binaries. We denote our
default set of assumptions our Fiducial model. We examine the predictions of
this model in Section 2.2, and consider some alternate assumptions in Section
2.3.

We determine the mass of the secondary star by drawing the mass ratio of the
binary $q=m_{2}/m_{1}$ from a power-law distribution with a slope of $-2$ for
mass ratios $q>0.1$, favouring typical mass ratios of around $q=0.2$, based on
observations of massive, wide binaries (Abt et al., 1990; Sana et al., 2014;
Aldoretta et al., 2015; Moe & Di Stefano, 2017).

The binary orbital periods are drawn from a distribution which is flat in the
log between $10^{4}$ days (assumed to be the minimum orbital period of non-
interacting binaries) and $10^{7}$ days (Öpik, 1924; Abt, 1983; Sana et al.,
2012; Moe & Di Stefano, 2017). Wide binaries have low binding energies, and
may be disrupted by flyby encounters with passing stars in the Galactic field
(e.g., Yabushita, 1966; Weinberg et al., 1987). As discussed by Igoshev &
Perets (2019), the typical lifetimes (before being disrupted this way) of the
massive, wide binaries we consider here are longer than the $\sim 50$–$200$
Myr the binary needs to survive to produce an observable binary pulsar (see
Section 2.2). Observations suggest that most massive wide binaries are
eccentric (e.g., Moe & Di Stefano, 2017). In our Fiducial model we assume that
the initial eccentricities of wide binaries are drawn from a uniform
distribution between 0 and 1. We assume that the supernova occurs at a random
time in the binary orbit.

The pre-supernova masses of SAGB stars are uncertain (Doherty et al., 2017).
For our Fiducial model, we assume that SAGB stars lose no mass during their
pre-supernova evolution (this is the No mass loss model from Section 2.3).
This is approximately what occurs for stars in this mass range using the
default mass-loss rate prescriptions in the binary population synthesis code
COMPAS (Stevenson et al., 2017; Vigna-Gómez et al., 2018) that we use for
estimating formation rates in Section 2.4. This implies that SAGB stars eject
a large amount of mass during the supernova, which translates to a large mass-
loss kick (Blaauw, 1961). Detailed SAGB models suggest that the envelope of
these stars may be lost prior to the supernova through a brief phase with high
wind mass-loss rates and thermal pulses (Doherty et al., 2017). We therefore
consider alternate assumptions regarding the pre-supernova mass loss of SAGB
stars in Section 2.3.

We assume that all ECSNe give rise to neutron star natal kicks of the same
magnitude. In our Fiducial model, we assume that the magnitude of this kick is
10 km/s, and we examine this in more detail in Section 2.3. In reality, the
natal kick will likely depend in detail on the properties of the progenitors.
However, to our knowledge, no such detailed prescription for ECSN kicks exists
at present. Our simple model is intended to allow us to easily understand the
impact that natal kicks have on the binary333We expect that our results would
be qualitatively similar if a narrow distribution of kick velocities was used.
We make the standard assumption that the distribution of kick directions is
isotropic (e.g., Tauris et al., 2017). We determine the post-supernova orbital
parameters following Kalogera (1996).

ECSNe are formed from the lowest mass stars to go supernova, with core masses
around the Chandrasekhar mass. The gravitational mass of the remnant neutron
star is less than its baryonic mass by around 10 %, with the exact mass
difference depending on the unknown neutron star equation of state. We assume
that the baryonic mass of all neutron stars formed through ECSNe is 1.36 M⊙
(e.g., Takahashi et al., 2013; Gessner & Janka, 2018); for our assumed
relation between the baryonic and remnant masses (Timmes et al., 1996), this
results in gravitational masses of 1.26 M⊙.

###  2.2 Fiducial model predictions

To elucidate the main predictions of this model, we first consider the results
of the Fiducial model described above. We examine the robustness of these
results to various assumptions in Section 2.3.

Figure 1:  Left: Orbital periods ($P_{\mathrm{orb}}$) and eccentricities of
wide binaries after an ECSN in our Fiducial model (blue points; see Section
2.1 for details). The black dot-dashed line shows the eccentricities above
which a binary would interact at periastron for a giant ($m_{2}=4$ M⊙,
$1\,000$ R⊙) companion as a function of orbital period. The black points show
the orbital periods and eccentricities of known binary pulsars (Manchester et
al., 2005). Recycled pulsars are shown in red. Uncertainties for the observed
systems are typically much smaller than the size of the points. Right:
Normalised probability distribution of the eccentricities of wide binary
pulsars. The solid black line shows the distribution after removing those
binaries that would interact at periastron, as discussed in Section 2.2.

We show the orbital periods and eccentricities of wide binary pulsars
following the ECSN in Figure 1. We see that essentially all eccentricities are
possible, although higher eccentricities are preferred; the median
eccentricity in this model is around 0.7. Around 10% of binaries that remain
bound are in high enough eccentricity orbits after the supernova that their
periapsis separation would be equal to the radius of a low-mass giant
companion ($R=1000$ R⊙), potentially leading to mass transfer or a binary
merger once the companion evolves. All other binaries do not interact, and
therefore we expect the pulsar to be unrecycled and have typical properties of
unrecycled pulsars, with a spin period $P_{\mathrm{spin}}\sim 0.1$–$1$ s and a
spin down rate of $10^{-17}<\dot{P}<10^{-13}$ (e.g., Boyles et al., 2011),
unless ECSNe preferentially produce pulsars with particular rotational
characteristics or luminosities. With the exception of the binaries with the
highest eccentricities (which may result in mergers in any case), these
binaries are sufficiently wide that gravitational radiation and tidal effects
are not expected to significantly affect the binary orbit after the supernova
on timescales when the pulsar is observable444The typical lifetime of an
unrecycled pulsar is only 10–100 Myr (e.g., Lynch et al., 2012; Chattopadhyay
et al., 2021)., and thus we neglect modelling these. We therefore expect that
the properties of observable systems should be very similar to their post-
supernova orbital properties. We find that, because of the low kicks
associated with ECSNe, these wide binary pulsars have low typical systemic
velocities of $\lesssim 10$ km/s.

The lifetime of an 8 M⊙ star is $\sim 40$ Myr (Pols et al., 1998; Hurley et
al., 2000). Assuming a pulsar lifetime of 10 Myr (100 Myr; see above), this in
turn implies that only companions more massive than 7 M⊙ (4 M⊙) can have
evolved to massive ($M>1$ M⊙) oxygen-neon white dwarfs whilst a pulsar is
still visible. Lower mass stars will still be unevolved main-sequence stars.
In our Fiducial model, we find that only 2 % (10 %) of these systems should
have massive oxygen-neon white dwarf companions with $M_{\mathrm{WD}}\sim 1$
M⊙, whilst the rest should have low-mass $M<7$ M⊙ (4 M⊙) main-sequence
companions. The median companion mass ($\sim 2.8$ M⊙) arises from a
competition between the initial distribution of binary mass ratios (which
favours low-mass companions) and the supernova dynamics (binaries with a more
massive companion are more likely to survive the supernova).

###  2.3 Variations from fiducial assumptions

Figure 2: Fraction of wide binaries ($P_{\mathrm{orb}}>10^{4}$ day, $m_{1}=8$
M⊙) which remain bound after an ECSN (orange) and their average post-supernova
eccentricity (blue), as a function of the kick velocity imparted to neutron
stars in ECSNe. The different line styles show different assumptions about the
amount of mass SAGB stars lose prior to the supernova (see Section 2.3). The
No mass loss model is shown with a dashed line, the Intermediate mass loss
model is shown with a dot-dash line whilst the Complete mass loss model is
shown with the dotted line. This figure shows results from the Uniform
eccentricity model.

Our model (described in Section 2.1) necessarily makes a number of simplifying
assumptions. One of the key uncertainties is the pre-supernova (envelope) mass
of SAGB star progenitors of ECSNe (Poelarends et al., 2008). Large pre-
supernova masses will lead to large amounts of mass ejected during the
supernova, leading to large mass loss kicks (Blaauw, 1961). In order to
investigate the impact, we use three simple models. The name of each model
refers to the amount of mass loss the SAGB stars experience prior to the ECSN.
In the No mass loss model we assume that the stars do not lose any mass
through stellar winds prior to the supernova explosion. This is approximately
what currently happens in binary population synthesis codes like COMPAS
(Stevenson et al., 2017; Stevenson et al., 2019; Vigna-Gómez et al., 2018).
Here, we do not intend this model to be realistic, but to demonstrate the
extreme case. At the other extreme is the Complete mass loss model where we
assume that the pre-supernova mass is equal to the baryonic mass of the
remnant neutron star, 1.36 M⊙ (Vigna-Gómez et al., 2018). This model is
motivated by detailed SAGB models that show that these stars have high mass-
loss rates of order $10^{-4}$ M⊙ yr-1 during the final stages of their lives
(see Doherty et al., 2017, for a review). However, we again expect that this
model is too extreme. In between these two extremes, in the Intermediate mass
loss model we assume that the star experiences some mass loss and then ejects
$M_{\mathrm{ej}}=5$ M⊙ of mass during the supernova, based on models for the
expected ejecta mass for ECSNe, along with that inferred from the Crab nebula
(Fesen et al., 1997; Tominaga et al., 2013; Hiramatsu et al., 2021). For
models that include pre-supernova mass loss we assume Jean’s mode mass-loss,
which leads to widening of the binary by a factor of a few. We neglect wind
mass transfer, which may be important even in wide binaries due to the slow
wind speeds of SAGB stars (e.g., Saladino et al., 2018; Höfner & Olofsson,
2018). We show the fraction of wide binaries that remain bound after the ECSN,
and the average eccentricity of the remaining bound binaries for each of these
models in Figure 2.

The magnitude of kicks that neutron stars formed in ECSNe receive is another
key uncertainty (Gessner & Janka, 2018). In Figure 2 we vary the kicks from
0.5–120 km/s. In all cases we assume that all ECSNe lead to the same kick
magnitude. We find that for kicks less than 5 km/s (Gessner & Janka, 2018) at
least 20 % of wide binaries would be expected to survive the supernova. If
ECSNe impart no kicks, around 30% of binaries remain bound in the No Mass Loss
model, whilst this fraction is close to 100 % for the Complete mass loss
model. Kicks $\gtrsim 10$ km/s start to disrupt a large fraction ($>90$ %) of
wide binaries in all models, with essentially all binaries disrupted by kicks
$>100$ km/s (where the fraction remaining bound is $<10^{-4}$). To summarise,
this suggests that, given our current understanding of kicks from ECSNe
(Gessner & Janka, 2018), a significant fraction of wide binaries should
survive, and a sizable population of wide binary pulsars should exist.

We also show in Figure 2 how the median eccentricity of the wide binaries that
remain bound varies with the kick velocity. All models lead to an average
post-supernova eccentricity of $e\sim 0.7$ regardless of the mass-loss model
or the magnitude of natal kicks. We also find that our models with higher
kicks than our Fiducial model result in slightly higher typical companion
masses (since only stars with comparable mass companions survive the
supernova).

Massive, wide binaries are observed to generally have eccentric orbits (Malkov
et al., 2012; Moe & Di Stefano, 2017). In addition to our Fiducial model where
the initial eccentricities of binaries are drawn from an uniform distribution,
we have also tested the impact of assuming that binaries initially have
circular orbits, or eccentricities drawn from a thermal distribution
($p(e)\propto e$). Eccentric binaries survive symmetric supernovae (i.e. with
no natal kicks) more often than a circular binary of the same mass and orbital
period due to the lower orbital velocity at, and greater time spent near,
apoapsis. The fraction of binaries which survive a symmetric supernova
increases from around 20% for the circular model to 50% in the thermal model.
The initial eccentricity distribution is unimportant for natal kicks $\gtrsim
10$ km/s.

Overall, with our 3 different assumptions regarding pre-supernova mass loss
from SAGB stars, and 3 eccentricity distributions for massive binaries, we
have simulated 9 different models with different assumptions. For each model
we used 12 different values for the kick velocity of neutron stars in the
range 0.5–120 km/s, resulting in a total of 108 models.

###  2.4 Estimate of the formation rate of wide binary pulsars

ECSNe are rare events, constituting a few percent of the core-collapse
supernova rate (Poelarends et al., 2008; Doherty et al., 2017; Jones et al.,
2019; Hiramatsu et al., 2021). The rate of ECSNe in wide binaries is sensitive
to the mass range of (effectively) single stars that lead to ECSNe. Willcox et
al. (2021) recently showed that the width of this mass range cannot be greater
than 0.2 M⊙ in order to not overproduce low velocity pulsars compared to
observations (e.g., Verbunt et al., 2017; Igoshev, 2020)555This also agrees
with recent constraints on the width of the ECSN mass range from s-process
element abundances in ultra faint dwarf galaxies (Hirai et al., 2019; Tarumi
et al., 2021). Using the population synthesis models from Willcox et al.
(2021), we have calculated the formation rate of wide binaries in which the
primary star is expected to undergo an ECSN i.e., the fraction of all binaries
that are born in the parameter space described in Section 2.1. We compare this
to the number of isolated pulsars (which we assume have similar radio
lifetimes, luminosities, beaming fractions etc) produced in the same model. We
find that one of these wide binaries is formed for every 100 isolated pulsars
formed in our population synthesis models. A significant fraction of wide
binaries are disrupted by the supernova in all of our models (cf. Figure 2),
with typically $\sim 10$% surviving the supernova. We therefore find one wide
binary that remains bound following the ECSN (hereafter wide binary pulsars)
is formed for every $\sim 1000$ isolated pulsars, leading to the possibility
that $\sim 3$ of the $\sim 3000$ observed isolated pulsars may have wide
binary companions. Wide binary pulsars may be misclassified as isolated
pulsars due to their long orbital periods compared to the typical durations
that pulsars have been observed (decades at most). Here we assume that
binaries with orbital periods greater than 100 years would be misclassified as
isolated pulsars (Willcox et al., 2021). In our models, we find that $\sim$
1–10% of these wide binary pulsars have orbital periods shorter than 100 yr,
with larger kicks generating a higher fraction. Putting this all together, we
estimate that one wide binary pulsar formed through an ECSN will be observed
for every 10,000 isolated pulsars. This is consistent with the current lack of
observations of such systems (see Section 3), but raises the possibility of
observing these systems with future pulsar surveys such as the Square
Kilometre Array (SKA) that will expand the number of known pulsars to
$>10,000$ (e.g., Keane et al., 2015). Given that ECSNe make up only a small
fraction of all supernovae, it is also worth considering whether a small
fraction of neutron stars formed from more massive stars in more common core-
collapse supernovae can also produce wide binary pulsars. According to our
models, for typical kicks greater than 100 km/s (Hobbs et al., 2005; Verbunt
et al., 2017), less than 1 in $10^{4}$ binaries survive the supernova (cf.
Figure 2). This indicates that the dominant formation channel for wide binary
pulsars will be through neutron stars formed with low kicks in ECSNe. We
emphasise that the rate estimates above are sensitive to several uncertain
model assumptions (cf. Section 2.3).

##  3 Search for candidate binary pulsars formed by electron-capture
supernovae

Despite the low expected observation rate of wide binary pulsars (Section
2.4), we have examined the pulsar catalogue to see if any known binary pulsars
are consistent with having formed in this way. We do not find any compelling
candidates, in agreement with our estimated formation rates. We show the
orbital periods and eccentricities of observed wide ($P_{\mathrm{orb}}>100$
day) binary pulsars in Figure 1, taken from version 1.65 of the ATNF catalogue
(Manchester et al., 2005), accessed using PSRQPY (Pitkin, 2018). The widest
observed binary pulsar PSR J1024-0719 (not shown in Figure 1 due to the large
uncertainties in its parameters) has an orbital period of 2 000–20 000 yr
(Bassa et al., 2016; Kaplan et al., 2016). PSR J1024-0719 is a 5 ms pulsar
(Bailes et al., 1997), and thus has likely been recycled; its formation likely
involves stellar dynamics (and hence it is not formed through the channel we
propose here), and it may originate from a triple or a globular cluster (Bassa
et al., 2016; Kaplan et al., 2016). Other than PSR J1024-0719, the upper limit
of orbital periods of known binary pulsars is around $10^{4}$ day (cf. Figure
1, see also Igoshev & Perets, 2019), with a preference for high
eccentricities.

Many of the known wide binary pulsars are consistent with formation channels
involving episodes of prior mass transfer, unlike the formation channel we
consider here. For example, PSR J0823+0159 (B0820+02; Manchester et al., 1980;
Hobbs et al., 2004; Xue et al., 2017) is a 0.8 s pulsar in a wide (¿1200 day)
binary with a low eccentricity ($e=0.01$) and a $0.6$ M⊙ white dwarf companion
(Kulkarni, 1986; Koester et al., 1992; Koester & Reimers, 2000). Tauris et al.
(2012) propose that PSR B0820+02 formed from a wide low-mass X-ray binary. In
addition, several candidates have massive B/Be star companions; these are not
compatible with our model since it predicts low-mass main-sequence companions.
One example of such a system is PSR J1740-3052 (Stairs et al., 2001), a young
570 ms pulsar in a 230 day, $e=0.57$ binary orbit with a massive ($M>11$ M⊙) B
star companion (Bassa et al., 2011). Similarly, PSR J0045-27319 is a young
binary pulsar with a 0.926 s spin period, in a 54 day orbit with an
eccentricity of $e=0.8$ whose companion is also a B star of $\sim 10$ M⊙
(Kaspi et al., 1994; Bell et al., 1995). PSR J1638-4725 (Lorimer et al., 2006)
is a binary pulsar with an orbital period of 1941 day and a high eccentricity
$e>0.95$. It has a companion mass of $\sim 8$ M⊙, which is likely again too
massive to be explained by our model. PSR B1259-63 (PSR J1302-6350, Johnston
et al., 1992, 1994) is a 47 ms pulsar with an orbital period of $1237$ days,
an eccentricity of $\sim 0.9$. PSR B1259-63 again has a massive ($M\sim 10$
M⊙) Be star companion, SS 2883. In this case both the spin period of the
pulsar (indicating recycling) and the companion mass make it incompatible with
our model. PSR J2032+4127 is a 143 ms pulsar in a wide (20–30 yr) binary with
a highly eccentric ($e>0.8$) orbit (Lyne et al., 2015). It too has a massive
Be star companion ($\sim 15$ M⊙). Another binary pulsar of interest is the
recently discovered PSR J1954+2529 (Parent et al., 2021), which is a 0.93 s
non-recycled pulsar in a wide (82.7 day), eccentric ($e=0.114$) orbit with a
low-mass companion. The relatively close orbit of this binary (compared to the
systems we have focused on in this paper) again suggests that this binary may
have experienced a phase of mass transfer prior to the formation of the
pulsar. Since the primary has already formed a neutron star in these systems,
it seems likely that they have undergone binary interactions in which some
mass from the primary was transferred to the secondary prior to the supernova,
particularly in the binaries hosting Be stars (e.g., Vinciguerra et al.,
2020).

There is also an observed population of wide, unrecycled (young) binary
pulsars known in globular clusters, which have been associated with ECSNe
(Lyne et al., 1996; Boyles et al., 2011; Lynch et al., 2012). However, the
formation of these binaries likely involved dynamical interactions, rather
than the isolated binary evolution channel we discuss here, so we do not
examine them further here.

In conclusion, we do not find any observed binary pulsars that are well
explained by our model. This is likely either due to radio pulsar observations
not being sensitive to orbital periods significantly greater than the
observing duration, leading to wide binary pulsars being misclassified as
isolated pulsars (Bassa et al., 2016; Kaplan et al., 2016), or because wide
binary pulsars do not exist, either because the mass range of single SAGB
stars leading to ECSNe is very narrow (cf. Willcox et al., 2021), or because
ECSNe lead to natal kicks $\gtrsim 10$ km/s (cf. Figure 2). These lead to the
observed pulsar population being too small to observe these rare wide binary
pulsars. Future radio observations with MeerKAT (Bailes et al., 2020) and the
SKA (Keane et al., 2015) will expand the known pulsar population, hopefully
observing these systems. It may also be possible to use GAIA to observe wide,
low-mass main-sequence star companions to young pulsars (e.g., Igoshev &
Perets, 2019; Antoniadis, 2021).

##  4 Summary and conclusion

Both theoretical and observational evidence point to neutron stars formed in
ECSNe receiving low kicks at birth (Pfahl et al., 2002b; Gessner & Janka,
2018). We argue that if ECSNe occur in a population of wide, non-interacting
binaries, low kicks predicts the existence of low-mass
($1.2<M_{\mathrm{psr}}/$M${}_{\odot}<1.3$), wide ($P_{\mathrm{orb}}>10^{4}$
day), eccentric ($e\sim 0.7$), unrecycled ($P_{\mathrm{spin}}\sim 1$ s) binary
pulsars in the Galactic field. These binary pulsars typically have low-mass
main-sequence companions (Section 2.2). Our model shows that at least 20 % of
wide binaries are expected to survive an ECSN (cf. Figure 2), if natal kicks
are $\lesssim 5$ km/s. The exact fraction depends on the pre-supernova masses
of SAGB stars, and the natal kicks imparted to neutron stars (cf. Figure 2,
Section 2.3).

We searched the catalogue of observed binary radio pulsars for systems with
characteristics matching those described above (Section 3). We did not find
any candidates which are well explained by this model. Using binary population
synthesis models (Willcox et al., 2021) we have estimated the formation rate
of these wide binary pulsars, finding that roughly one binary should be
observable for every 10,000 isolated pulsars. This is consistent with the lack
of detection of such a system in the current population of observed pulsars,
but raises the possibility of observing them in the future with the SKA. Our
low predicted observed rate is a result of a narrow mass range of single star
ECSN progenitors (Willcox et al., 2021), combined with the fact that pulsar
observations are not sensitive to orbital periods longer than a few $10^{4}$
day. The observation of a wide binary pulsar could provide evidence that ECSNe
can occur for effectively single stars.

Key uncertainties remain in the modelling of ECSNe. These uncertainties
include the mass range of progenitors which are expected to undergo ECSNe,
what their pre-supernova masses are (for single stars when their envelopes are
not stripped through binary interactions) and what magnitude kicks neutron
stars receive in ECSNe. If the mass range of (single) SAGB stars leading to
ECSNe is narrow, they will be inherently rare (or possibly even non existent),
whilst ECSNe kicks larger than predicted by recent models (Gessner & Janka,
2018) would disrupt most wide binaries (cf. Figure 2). We have assumed that
all ECSNe form neutron stars, whilst some may lead to thermonuclear explosions
or white dwarf formation (e.g., Jones et al., 2016; Jones et al., 2019; Tauris
& Janka, 2019; Leung et al., 2020). Our discussion has focused on ECSNe, but
the lowest mass iron core-collapse supernovae may also produce neutron stars
with low kicks (e.g., Müller et al., 2019; Stockinger et al., 2020).

In this paper we have focused on the formation of wide binary pulsars from
wide, non-interacting binaries, since these would provide a clean probe of
SAGB star evolution and ECSNe without any complications from binary
interactions. ECSNe are likely more common in close, interacting binaries
(e.g., Vigna-Gómez et al., 2018; Vinciguerra et al., 2020; Willcox et al.,
2021). Since a sizeable fraction of massive stars are in triples (Moe & Di
Stefano, 2017), these may also provide important contributions to the
formation of wide binary pulsars through distinct evolutionary channels (see
e.g., Bassa et al., 2016; Hamers & Thompson, 2019). Improved theoretical
modelling of SAGB stars will aid in solidifying the theoretical predictions
for populations of binary pulsars, whilst observing a large population of
binary pulsars will place ever greater observational constraints on the
evolution of these stars.

## Acknowledgements

We thank Ryan Shannon, Stefan Oslowski, Adam Deller, Eric Thrane, Ilya Mandel,
Debatri Chattopadhyay and Ryosuke Hirai for useful comments and discussions.
We also thank the referee for constructive queries and suggestions. The
authors are supported by the Australian Research Council Centre of Excellence
for Gravitational Wave Discovery (OzGrav), through project number CE170100004.
SS is supported by the Australian Research Council Discovery Early Career
Research Award project number DE220100241.

## Data availability

The results of this work are available from the authors at reasonable request.
This paper made use of results from the binary population synthesis code
COMPAS (v02.19.02) as presented in Willcox et al. (2021). The latest version
of COMPAS is publicly available at www.github.com/TeamCOMPAS/COMPAS.

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##  Appendix 0.A Decidability Proof for Simple Intuitionistic Grammar Logics

Recall that for a simple intuitionistic grammar logic
$\mathsf{IK_{m}}(\Upsigma,\mathcal{A})$ the set $\mathcal{A}$ of axioms is
defined as follows:

$\mathcal{A}\subseteq\\{(\langle y\rangle^{n}A\supset\langle x\rangle
A)\land([x]A\supset[y]^{n}A)\ |\ x,y\in\Upsigma,0\leq n\leq
1\\}\cup\\{\mathrm{D}_{x}\ |\ x\in\Upsigma\\}.$

For the remainder of the appendix, we assume that $\mathcal{A}$ is defined as
above. Furthermore, recall that for a simple intuitionistic grammar logic, the
propagation rules from Figure 9 are used in the refined labeled calculus
$\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$.

We will repeat definitions and lemmata due to Simpson to give the reader
intuition regarding the proof of decidability, but will omit various details
as they can be found in Simpson’s PhD thesis [47, Section 7.3].

We recursively define the modal depth $\mathrm{m_{d}}(A)$ of a formula $A$ as
follows: $\mathrm{m_{d}}(p)=\mathrm{m_{d}}(\bot)=0$, $\mathrm{m_{d}}(B\odot
C)=\max\\{\mathrm{m_{d}}(B),\mathrm{m_{d}}(C)\\}$ for
$\odot\in\\{\lor,\land,\supset\\}$, and $\mathrm{m_{d}}(\triangledown
B)=\mathrm{m_{d}}(B)+1$ for $\triangledown\in\\{\langle x\rangle,[x]\ |\
x\in\Upsigma\\}$. We also recursively define the set $\mathrm{S}(A)$ of
subformulae of a formula $A$ as follows: $\mathrm{S}(p)=\\{p\\}$,
$\mathrm{S}(\bot)=\bot$, $\mathrm{S}(B\odot C)=\\{B\odot
C\\}\cup\mathrm{S}(B)\cup\mathrm{S}(C)$ for
$\odot\in\\{\lor,\land,\supset\\}$, and $\mathrm{S}(\triangledown
B)=\\{\triangledown B\\}\cup\mathrm{S}(B)$ for $\triangledown\in\\{\langle
x\rangle,[x]\ |\ x\in\Upsigma\\}$. For a multiset $\Gamma$ of labeled
formulae, we define

$\mathrm{m_{d}}(\Gamma)=\max\\{\mathrm{m_{d}}(A)\ |\ w:A\in\Gamma\\}\text{ and
}\mathrm{S}(\Gamma)=\bigcup_{w:A\in\Gamma}\mathrm{S}(A).$

For a multiset $\mathcal{R}$ of relational atoms, we define an _$\mathcal{R}$
-extension_ as follows: (1) $\mathcal{R}$ is an $\mathcal{R}$-extension, and
(2) if $\mathcal{R}^{\prime}$ is an $\mathcal{R}$-extension and
$u\not\in\mathrm{Lab}(\mathcal{R})$, then $\mathcal{R}^{\prime}\cup\\{wRu\\}$
is an $\mathcal{R}$-extension. If $\mathcal{R}^{\prime}$ is an
$\mathcal{R}$-extension, then a label $u\in\mathrm{Lab}(\mathcal{R}^{\prime})$
has depth $n\geq 0$ iff there exists a sequence
$w_{0}R_{x_{1}}w_{1},\ldots,w_{n-1}R_{x_{n}}w_{n}\in\mathcal{R}^{\prime}$ such
that $w_{0}\in\mathrm{Lab}(\mathcal{R})$,
$w_{i}\in\mathrm{Lab}(\mathcal{R}^{\prime})\setminus\mathrm{Lab}(\mathcal{R})$,
and $w_{n}=u$. The depth of a label is well-defined since every label has only
one such accessing sequence by the definition of an $\mathcal{R}$-extension.
We define an $\mathcal{R}$-extension $\mathcal{R}^{\prime}$ to be _bounded_
relative to a labeled sequent $\mathcal{R},\Gamma\vdash w:A$ iff the depth of
every label in $\mathcal{R}^{\prime}$ is less than or equal to
$\mathrm{m_{d}}(\Gamma,w:A)$, and we define the _bounded-restriction_ of
$\mathcal{R}^{\prime}$ relative to a labeled sequent $\mathcal{R},\Gamma\vdash
w:A$ to be the set of all relational atoms from $\mathcal{R}^{\prime}$ whose
labels have a depth less than or equal to $\mathrm{m_{d}}(\Gamma,w:A)$. A
labeled sequent $\mathcal{R}^{\prime},\Gamma^{\prime}\vdash
w^{\prime}:A^{\prime}$ is defined to be _bounded_ relative to a labeled
sequent $\mathcal{R},\Gamma\vdash w:A$ iff (1) $\mathcal{R}^{\prime}$ is a
bounded $\mathcal{R}$-extension, and (2) if
$u:B\in\Gamma^{\prime},w^{\prime}:A^{\prime}$, then
$B\in\mathrm{S}(\Gamma,w:A)$ and
$\mathrm{m_{d}}(B)\leq\mathrm{m_{d}}(\Gamma,w:A)-n$ where $n$ is the depth of
$u$ in $\mathcal{R}^{\prime}$.

We define a _pseudo-derivation_ of a labeled sequent $\mathcal{R},\Gamma\vdash
w:A$ in $\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$ to be a derivation in
$\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$ from any finite collection of labeled
sequents (which need not be instances of $(id)$ or $(\bot_{l})$), and we
define its _size_ to be the number of inferences it contains. When a pseudo-
derivation is derivable from from labeled sequents that are instances of
$(id)$ or $(\bot_{l})$, then it is a derivation. Moreover, we stipulate that a
pseudo-derivation $\mathcal{D}$ of a labeled sequent $\mathcal{R},\Gamma\vdash
w:A$ is _bounded_ iff every labeled sequent in $\mathcal{D}$ is bounded
relative to $\mathcal{R},\Gamma\vdash w:A$.

###### Lemma 16

Let $\mathcal{D}$ be a pseudo-derivation of $\mathcal{R},\Gamma\vdash w:A$
from the labeled sequents in $\\{\mathcal{R}_{i},\Gamma_{i}\vdash w_{i}:A_{i}\
|\ 1\leq i\leq n\\}$ and let $d=\mathrm{m_{d}}(\Gamma,w:A)$. Then, the
following holds:

  1. 1.

For each $1\leq i\leq n$, $\mathcal{R}_{i}$ is an $\mathcal{R}$-extension and
if $u:B\in\Gamma_{i}\cup\\{w_{i}:A_{i}\\}$, then $u$ has a depth $n\leq d$,
$B\in\mathrm{S}(\Gamma,w:A)$, and $\mathrm{m_{d}}(B)\leq d-n$.

  2. 2.

There exists a bounded pseudo-derivation of $\mathcal{R},\Gamma\vdash w:A$
from the labeled sequents in
$\\{\mathcal{R}_{i}^{\prime},\Gamma_{i}^{\prime}\vdash w_{i}:A_{i}\ |\ 1\leq
i\leq n\\}$ where each $\mathcal{R}_{i}^{\prime}$ is the bounded-restriction
of $\mathcal{R}_{i}$.

###### Proof

The result is shown by induction on the size of the given pseudo-derivation
$\mathcal{D}$. We only show the $(p_{\langle x\rangle}^{2})$ case of the
inductive step as all cases are argued in an almost identical fashion as in
[47, Lemma 7.3.5].

Suppose that $(p_{\langle x\rangle}^{2})$ was applied at the top of
$\mathcal{D}$ as shown below.

$\mathcal{R}^{\prime},vR_{\overline{y}}u,\Gamma^{\prime}\vdash v:B$
$(p_{\langle x\rangle}^{2})$
$\mathcal{R}^{\prime},vR_{\overline{y}}u,\Gamma^{\prime}\vdash u:\langle
x\rangle B$ $\cdots$ $\vdots$ $\mathcal{R},\Gamma\vdash w:A$

We know that $u$ occurs at a depth $n\leq d-1$ because $\mathrm{m_{d}}(\langle
x\rangle B)\leq d-1$ by the inductive hypothesis. This implies claim 1 since
$v$ occurs at a depth $n-1<n\leq d-1$, $B\in\mathrm{S}(\Gamma,w:A)$, and
$\mathrm{m_{d}}(B)\leq d-(n-1)\leq d-n$. For claim 2, we suppose that we have
a bounded pseudo-derivation of
$\mathcal{R}^{\prime\prime},\Gamma^{\prime}\vdash u:\langle x\rangle B$ as
shown below, where $\mathcal{R}^{\prime\prime}$ is the bounded-restriction of
$\mathcal{R}^{\prime}$. Observe that a single application of $(p_{\langle
x\rangle}^{2})$ gives the desired bounded pseudo-derivation.

$\mathcal{R}^{\prime\prime},vR_{\overline{y}}u,\Gamma^{\prime}\vdash u:\langle
x\rangle B$ $\cdots$ $\vdots$ $\mathcal{R},\Gamma\vdash w:A$

∎

We now define a pre-order on all labeled sequents
$\mathcal{R}^{\prime},\Gamma^{\prime}\vdash w^{\prime}:A^{\prime}$ which are
bounded relative to a given labeled sequent $\mathcal{R},\Gamma\vdash w:A$.
This will be used to define an equivalence relation over the set of such
sequents, partitioning the set into a finite number of equivalence classes,
and ultimately permitting us to restrict the number of proofs of
$\mathcal{R},\Gamma\vdash w:A$ considered during proof-search to a finite
number. For the remainder of the appendix, we fix the labeled sequent
$\Lambda=\mathcal{R},\Gamma\vdash w:A$ and only consider labeled sequents
bounded relative to this one.

Let $\Lambda_{0}=\mathcal{R}_{0},\Gamma_{0}\vdash w_{0}:A_{0}$ and
$\Lambda_{1}=\mathcal{R}_{1},\Gamma_{1}\vdash w_{1}:A_{1}$ be bounded relative
to $\mathcal{R},\Gamma\vdash w:A$. We define a _morphism_ from $\Lambda_{0}$
to $\Lambda_{1}$ to be a function $f$ such that (1) for all
$u\in\mathrm{Lab}(\Lambda)$, $f(u)=u$, (2) if $u:B\in\Gamma_{0}$, then
$f(u)\in\Gamma_{1}$, (3) $f(w_{0})=w_{1}$, and (4) if
$uR_{x}v\in\mathcal{R}_{0}$, then $f(u)R_{x}f(v)\in\mathcal{R}_{1}$. We then
define the pre-order $\preceq$ as follows: $\Lambda_{0}\preceq\Lambda_{1}$ iff
$A_{0}=A_{1}$ and there exists a morphism $f$ from $\Lambda_{0}$ to
$\Lambda_{1}$. The equivalence relation $\cong$ is then defined as:
$\Lambda_{0}\cong\Lambda_{1}$ iff $\Lambda_{0}\preceq\Lambda_{1}$ and
$\Lambda_{1}\preceq\Lambda_{0}$.

###### Lemma 17

The equivalence relation $\cong$ partitions the set of labeled sequents
bounded relative to $\Lambda$ into a finite number of classes.

###### Proof

The proof is similar to the proof of [47, Proposition 7.3.6], but with the
exception that one must generalize the arguments of Simpson to account for
multiple modalities. ∎

We now define an _irredundant_ pseudo-derivation $\mathcal{D}$ to be a bounded
pseudo-derivation relative to a labeled sequent $\mathcal{R},\Gamma\vdash w:A$
such that no two labeled sequents
$\Lambda_{0}=\mathcal{R}_{0},\Gamma_{0}\vdash w_{0}:A_{0}$ and
$\Lambda_{1}=\mathcal{R}_{1},\Gamma_{1}\vdash w_{1}:A_{1}$ occur in
$\mathcal{D}$, with the former above the latter, such that
$\Lambda_{0}\preceq\Lambda_{1}$.

###### Lemma 18

Let $\Lambda=\mathcal{R},\Gamma\vdash w:A$ with
$\Lambda_{i}=\mathcal{R}_{i},\Gamma_{i}\vdash w_{i}:A_{i}$ bounded relative to
$\Lambda$ for $i\in\\{1,2\\}$. Then,

  1. 1.

If $\Lambda_{0}\preceq\Lambda_{1}$ and $\Lambda_{0}$ has a bounded derivation
of size $n$ relative to $\Lambda$, then $\Lambda_{1}$ has a bounded derivation
of size $n$ relative to $\Lambda$.

  2. 2.

If $\Lambda$ is derivable in $\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$, then
$\Lambda$ has an irredundant derivation.

###### Proof

The proof of claim 1 is similar to [47, Lemma 7.3.7] and the proof of claim 2
is similar to [47, Proposition 7.3.8]. ∎

###### Theorem 0.A.1

Let $\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$ be a refined labeled calculus for
a simple intuitionistic grammar logic $\mathsf{IK_{m}}(\Upsigma,\mathcal{A})$.
Then, it is decidable to check if
$\models_{\mathcal{A}}^{\Upsigma}\mathcal{R},\Gamma\vdash w:A$ for any
arbitrary labeled sequent $\mathcal{R},\Gamma\vdash w:A$.

###### Proof

We decide the validity of a labeled sequent $\mathcal{R},\Gamma\vdash w:A$ by
searching for an irredundant proof of it in
$\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$. The proof-search algorithm is
similar to the one given in [47, Section 7.3.3]. First, we take the labeled
sequent $\mathcal{R},\Gamma\vdash w:A$ as input and check if it is an instance
of $(id)$ or $(\bot_{l})$. If so, then we know that
$\models_{\mathcal{A}}^{\Upsigma}\mathcal{R},\Gamma\vdash w:A$, and if not,
then we continue proof-search, searching for all irredundant derivations of
size $1$, and then of size $2$, and so forth by applying all relevant rules of
$\mathbf{L}_{\Upsigma}^{*}(\mathcal{A})$ bottom-up. Only a finite number of
bottom-up rule applications are possible at each stage modulo the choice of
fresh labels in the $(\langle x\rangle_{l})$, $([x]_{r})$, and $(d_{x})$
rules. However, this peculiarity can be overcome by fixing how fresh labels
are chosen during proof-search. Also, note that it is computable to check
irredundancy since the $\preceq$ relation is decidable on labeled sequents and
it is decidable to check if a labeled sequent is bounded relative to the input
$\mathcal{R},\Gamma\vdash w:A$. Last, since the equivalence relation $\cong$
partitions the set of all labeled sequent bounded relative to
$\mathcal{R},\Gamma\vdash w:A$ into a finite number of equivalence classes, we
know that eventually proof-search will terminate. ∎

###### Theorem 6.4

The validity problem for simple intuitionistic grammar logics is decidable.

###### Proof

Follows from Theorem 6.4 above. ∎