Wide binary pulsars from electron-capture supernovae

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_⊙.

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.

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).

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.

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).