Issue 
A&A
Volume 645, January 2021



Article Number  A122  
Number of page(s)  20  
Section  Astrophysical processes  
DOI  https://doi.org/10.1051/00046361/202038975  
Published online  22 January 2021 
Fast radio burst repeaters produced via KozaiLidov feeding of neutron stars in binary systems
^{1}
Sorbonne Université, CNRS, UMR 7095, Institut d’Astrophysique de Paris, 98bis boulevard Arago, 75014 Paris, France
email: decoene@iap.fr
^{2}
Department of Physics and Astronomy, The Johns Hopkins University, Homewood Campus, Baltimore, MD 21218, USA
^{3}
Department of Physics, University of Oxford, Keble Road, Oxford OX1 3RH, UK
Received:
20
July
2020
Accepted:
26
November
2020
Neutron stars are likely surrounded by gas, debris, and asteroid belts. KozaiLidov perturbations, induced by a distant, but gravitationally bound companion, can trigger the infall of such orbiting bodies onto a central compact object. These effects could lead to the emission of fast radio bursts (FRBs), for example by asteroidinduced magnetic wake fields in the wind of the compact object. A few percent of binary neutron star systems in the Universe, such as neutron starmain sequence star, neutron starwhite dwarf, double neutron star, and neutron starblack hole systems, can account for the observed nonrepeating FRB rates. More remarkably, we find that wide and close companion orbits lead to nonrepeating and repeating sources, respectively, and they allow for one to compute a ratio between repeating and nonrepeating sources of a few percent, which is in close agreement with the observations. Three major predictions can be made from our scenario, which can be tested in the coming years: (1) most repeaters should stop repeating after a period between 10 years to a few decades, as their asteroid belts become depleted; (2) some nonrepeaters could occasionally repeat, if we hit the short period tail of the FRB period distribution; and (3) series of subJansky level short radio bursts could be observed as electromagnetic counterparts of the mergers of binary neutron star systems.
Key words: stars: neutron / binaries: general / radiation: dynamics / radiation mechanisms: nonthermal / turbulence / submillimeter: general
© V. Decoene et al. 2021
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The origin of fast radio bursts (FRBs), these brief, coherent, and numerous radio pulses, has not been identified yet. Today, radioastronomy surveys from all over the world have detected more than 700 FRBs, among which 137 have been officially reported (Petroff et al. 2016).
The large inferred dispersion measures (DM) point towards these being mostly at cosmological distances. The extragalactic origin is further confirmed by the isotropic distribution of FRBs over the sky. So far, FRBs have been detected with fluences ranging from subJansky up to more than 400 Jy, with steep energy spectra (James 2019). Consequently, the isotropic energy equivalent of an FRB is more than ten billion times higher than galactic pulsar emissions, with, in addition, spectra that are radically different from most of the known radio sources.
A fraction of FRBs appear to repeat, that is with multiple bursts spaced over a few seconds to months, observed at the same location. This implies that FRBs could belong to two distinct populations: repeaters and nonrepeaters. Among the hundred of events published so far, about 22 appear to repeat, mostly with no apparent periodicity, even though one has been reported to be periodic (CHIME/FRB Collaboration 2020a). A large fraction of the repeating FRBs have been discovered by CHIME, operating at around 400 MHz (CHIME/FRB Collaboration 2019a,b; Scholz 2019; Andersen et al. 2019; Fonseca et al. 2020). The absence of real differences in their spectra, however, suggests that the two populations may originate from the same sources.
The event rate, extrapolated from current observations that are necessarily limited in the observation time and field of view, suggests that FRBs occurs at an extraordinarily high rate of thousands per day, implying that the objects at the origin of these emissions must be numerous in the Universe (Petroff et al. 2019).
From a theoretical perspective, no consensual emission mechanism has been found, nor is there an accepted explanation for the two observed populations of repeaters and nonrepeaters. A vast number of emission models exist, from exotic alien signals to cosmic strings, and they can be found in Platts et al. (2019). The recent detection of two intense radio bursts, coincident with Xray bursts and localized at the position of SGR1935+2154, points towards the magnetar hypothesis as a source of FRBs (Mereghetti et al. 2020; CHIME/FRB Collaboration 2020b). This might, however, apply to a subset of the population only, since the equivalent luminosity of the radio bursts from SGR1935+2154 seems to be 40 times dimmer than the dimmest FRB.
Although the number of FRB detections is growing fast, the observational constraints remain limited. The key observables at this stage, besides the energy budget and time variability, are the rates of bursts and of repeating events. These numbers are challenging to reconcile with the existing source models in the literature.
In this paper, we propose a global scenario which could explain the rates of both repeating and nonrepeating events with a population of neutron stars in binary systems. Several studies have shown that the infall of bodies onto a compact object should lead to observable electromagnetic signals. In particular, via the Alfvén wing emission mechanism presented in Mottez & Zarka (2014), this emission could be the source of FRBs. Other authors have proposed that FRBs result from the impact of asteroids and comets on central compact objects (Geng & Huang 2015; Dai 2016; Smallwood et al. 2019). Interestingly, the above models could naturally lead to repeating signals, as long as small bodies, such as asteroids, pass by the star at a rate corresponding to the observations. Furthermore, it provides a natural explanation to the dichotomy between repeater and nonrepeater FRBs.
Such scenarios require however both a large number of progenitors, and an efficient infall mechanism into the neutronstar Roche lobe. The KozaiLidov gravitational effects applied to the numerous binary neutron star systems naturally provide such a framework.
In the following, we study the effect of the KozaiLidov mechanism on a triple system consisting of a central neutron star, a binary companion, and sizeable bodies orbiting nearby, such as an asteroid belt around the neutron star. Bodies perturbed by gravitational effects leave their orbits and fall onto the central object (Naoz 2016). For instance in the Solar System, the KozaiLidov mechanism is responsible for the Kirkwood gap in the asteroid belt, under the influence of Jupiter (Delgrande & Soanes 1943). Furthermore many astrophysical systems have been found to be consistent with the implication of KozaiLidov perturbations: such as the formation of hot Jupiters systems via the planetplanet interactions (Naoz et al. 2011), the formation of close compact binaries via mass loss channels induced by secular effects (Shappee & Thompson 2013; Michaely & Perets 2014), and the pollution of white dwarf atmospheres due to the infall of asteroid and cometlike materials (Stephan et al. 2017).
The first discovery of earthmass exoplanets was indeed around a millisecond pulsar (Wolszczan & Frail 1992). The existence of asteroid belts around millisecond pulsars has been invoked to explain various timing variations and other observational features (Cordes & Shannon 2008; Shannon et al. 2013; Brook 2014; Yu & Huang 2016; Mottez et al. 2013a).
This study is strongly related to the one presented in Mottez et al. (2020), where the authors discuss the possible FRB emission from the interaction between an asteroid belt and a pulsar. This is why we often refer to their work regarding the radio emission mechanism. However, our work focuses on the orbital dynamic of the asteroids inside the belt. In this perspective, we first present the FRB emission model and the parameter sets required for the signal to be observed. We then compute the KozaiLidov timescales for our binary system (Sect. 3) and discuss the implications in terms of FRB rates, taking into account the binary population rates (Sect. 4). We simulate the KozaiLidov effect on a mock solarlike asteroid belt in Sect. 5. Finally, we discuss the broader applications of this calculation in Sect. 6.
2. FRB emission from asteroids orbiting a pulsar
Asteroid belts close to neutron stars have been previously proposed to explain observational timing and radio features (Cordes & Shannon 2008; Shannon et al. 2013; Brook 2014; Yu & Huang 2016; Mottez et al. 2013a,b). No asteroid belt has yet been observed at distances larger than 1 AU, but this is likely due to observational bias. Asteroid belts could be the remains of planetary objects destroyed by the supernova that led to the formation of the neutron star, or result from the supernova fallback itself (Menou et al. 2001; Shannon et al. 2008). The aggregation of the debris to form a planet depends mostly on external conditions (Morbidelli & Raymond 2016). In particular, the presence of Jupiter prevents the formation of planets in the Solar asteroid belt. The perturbations produced by an outer black hole at ≳few AU with a mass of 10 M_{⊙} would be several orders of magnitude more intense than the influence of Jupiter on the Solar system belt. Therefore it is likely that no planet would form inside this asteroid belt.
Mottez & Zarka (2014) presented the extension of the Alfvén wing theory (see e.g., Saur et al. 2004) to relativistic winds induced by a pulsar and interacting with a companion body (e.g., planet, comet, asteroid, etc.). The emission mechanism can be summarized in three steps: first the relativistic and magnetized wind enters in direct contact with the orbiting body, creating a magnetic coupling. This direct contact induces a current sheet called an Alfvén wing, extending from the body far into space. Finally, the interaction of the outflow plasma crossing the Alfvén wing results in radio emission through coherent mechanisms such as the cyclotron maser instability.
For an asteroid of radius R_{ast} orbiting at distance a_{ast} from a pulsar located at distance D from the observer, the average flux density of radio waves inside the cone of emission of opening angle 1/γ, with γ the Lorentz factor of the wind, reads (Mottez & Zarka 2014; Mottez & Heyvaerts 2020; Mottez et al. 2020):
here Δf is the spectral bandwidth of the emission, ϵ_{w} the wind power conversion efficiency, and R_{⋆}, P_{⋆}, B_{⋆} the pulsar radius, rotation period and dipole magnetic field strength. A_{cone} = 4π/Ω_{A} ≥ 1 is an anisotropy factor, with Ω_{A} the solid angle in which the radiowaves are emitted in the source frame. For an isotropic emission, A_{cone} = 1 and if, the instability triggering the radio emissions is the cyclotron maser instability, A_{cone} ∼ 100 (Mottez et al. 2020).
One should note that, in Mottez & Heyvaerts (2020) a revised version of the Alfvén wing mechanism is presented, where the magnetic flux Ψ of the wind is evaluated where the field lines are windlike and not estimated at the surface of the neutron star as previously done in Mottez & Zarka (2014). Although the physics of the process remains identical to the previous version of the study, the intensity of the radio emission is scaled down. In the present study, we use the revised version of the mechanism.
It is interesting to note that in this radio emission mechanism model, magnetarlike objects with a strong magnetic field could power FRB emission of hundreds of Janskies as observed in the ASKAP survey. Such phenomena are also suggested by the recently observed double radio bursts from the magnetar SGR1935+2154 (CHIME/FRB Collaboration 2020b), also coincident with Xray bursts (Mereghetti et al. 2020).
In light of this emission equation, we discuss below the parameters required for the pulsar and the asteroids in order to produce an observable FRB.
2.1. Pulsar parameters
Neutron stars are frequently formed in binary star systems, but the subsequent evolution of these systems leads to diverse final configurations, depending on the presupernova mass, the asymmetry of the explosion, a possible impulsive ‘kick’ velocity impinged on the neutron star at birth, etc.: a parameterspace explored with sophisticated numerical simulations (e.g., Lorimer 2008; Toonen et al. 2014 and references therein).
We focus here on neutron starwhite dwarf (NSWD), neutron starmain sequence star (NSMS), neutron starneutron star (DNS) and neutron starblack hole (NSBH) binaries, which are found to be common outcomes of the evolution of binary systems containing neutron stars (Portegies Zwart & Verbunt 1996; Nelemans et al. 2001).
In a majority of NSBH systems, the neutron star is born with normal pulsar characteristics (e.g., nonrecycled pulsars with large magnetic fields and mild spin periods). Various evolutionary studies show indeed that it is difficult to form recycled pulsars in these systems and their low inferred rates are compatible with their nondetection in radio so far (Sipior et al. 2004; Pfahl et al. 2005; Shao & Li 2018; Kruckow et al. 2018).
The case that, in the majority of NSWD systems, the white dwarf is formed first has also been studied numerically (Toonen et al. 2018) and supported observationally (e.g., Portegies Zwart & Yungelson 1999; Kaspi et al. 2000; Manchester et al. 2000) Hence these systems contain normal (nonrecycled) neutron stars in eccentric orbits.
Finally, observations confirm the natural scenario in which double neutron star systems contain at least one normal pulsar (Tauris et al. 2017), which serve in our framework as the central object.
The evolutionary path of NSMS suggests that main sequence stars should be companions to normal radio pulsars, and their (scarce) observations support this scenario (Lorimer 2008 and references therein).
Two neutron star systems containing a planet companion have been observed (e.g., Lorimer 2008 and references therein). Data and studies on these objects are scarce, hence we mostly concentrate on the binaries mentioned above in this paper. However, we also discuss the possible contribution from these planetary systems. The two planets have been detected around millisecond pulsars, but it is impossible as yet to infer population characteristics, and normal pulsars are more numerous than millisecond pulsars and statistically likely to host planets.
For all these binary systems, it appears to be justified to assume that the neutron star presents the characteristics of a normal pulsar. We note however that the systems in close orbit, with companion semimajor axis a_{c} ≪ 1 AU are usually associated with recycled pulsars.
In our model, the FRB emission happens in the first ≲10^{4} yr of the birth of the pulsar, and for close binaries, even within the first 10 yr (see Sect. 4). The relevant pulsar parameters are hence those at birth. It is commonly accepted that the dipole magnetic field strength of the pulsar experiences little decay, with an average initial value of 10^{12.65} G (FaucherGiguère & Kaspi 2006). Recent simulations show that the initial spin period could be as low as 20 ms (Johnston & Karastergiou 2017) and typically below P_{⋆} < 150 ms (Gullón et al. 2014).
The numerical values of Eq. (1) demonstrates that such fiducial normal pulsar parameters suffice to produce observable radio emission at the Jansky level, provided that the asteroid presents specific characteristics, which we detail in the next Section.
We notice also that recycled pulsars, that have low magnetic fields of B ≲ 10^{9} G and P ∼ few ms, are not powerful enough to produce FRB emissions at the Jansky level, except for extremely large asteroids.
2.2. Asteroid size
The radio emission crucially depends on the radius R_{ast} and orbital distance a_{ast} of the asteroid. One can infer from Eq. (1) that large asteroids with radius R_{ast} ≳ 3 km are favored to power observable FRBs. From simple fragmentation arguments, it can be shown that the asteroid size distribution roughly follows a powerlaw (MPCSAOIAU 2019)
Larger, less numerous asteroids could produce intense bursts, at a lower rate. Conversely, mJy emission, detectable with current instruments, could be produced by smaller (3 − 10 km), more numerous asteroids.
2.3. Asteroid belt distance
Equation (1) shows that short distances from the central neutron star are required for the body to be immersed in strong magnetic fields. Although mJy emission can be produced at a distance a_{mJy} ∼ 0.1 AU from the neutron star, shorter orbital distances are required to power more intense bursts.
The shortest possible distance corresponds to the Roche limit. The Roche limit for an asteroid falling onto a neutron star is computed to be
with R_{ast} the asteroid radius, M_{ast} its mass, ρ_{ast} its density, and M_{NS} the central compact object mass.
Asteroids could penetrate deeper than the Roche lobe if the socalled plunge factor is taken into account (AliHaïmoud et al. 2016), allowing for shorter a_{ast} to be reached at maximum eccentricities. This would enable smaller (R_{ast} ∼ 3 − 10 km) – more numerous (N_{ast} ∼ 10^{4 − 5}) – asteroids to emit Janskylevel bursts.
We note that even at these close distances, small objects like asteroids are in general not evaporated via induction heating by the winds of the central neutron star (Kotera et al. 2016). Their size is indeed shorter than the typical wind electromagnetic wavelength, in the framework of the Mie theory. The effects of nonsphericity, as is the case for asteroids, are ≲30% on light absorption coefficients (Mishchenko et al. 1999).
The required short orbital distances imply that, unless most asteroid belts are already created in this emission zone delimited by d_{Roche} and a_{mJy}, the process of Mottez & Zarka (2014) and Mottez et al. (2020) can only work if asteroids actually fall close enough to the central object. We propose here that this can happen via the KozaiLidov effect. We set our fiducial asteroid belt distance to a_{ast} = 1 AU in the following.
We note that observations of pulsars show that there might be asteroid belts at ∼R_{⊙} (Cordes & Shannon 2008; Mottez et al. 2013a,b): these do not need to undergo infall in order to produce FRBs, as they are already deep into the strong wind region to produce strong Alfvén wing emissions. The signals from such belts could present some periodicity due to the regular orbits as observed for FRB180916, which presents a ∼16.35 days periodicity (CHIME/FRB Collaboration 2020a). Indeed for favorable configurations, the alignment between the asteroid periodical motion and the observer’s lineofsight could result in a periodical observation of bursts. However, turbulence effects in these inner wind regions along the observer’s line of sight may play a role in modifying such periodicities, an effect that we do not address here. We note also that Jones (2008) shows that infrared emission limits the inner radius of an asteroid belt to a factor that is two or three times larger than ∼R_{⊙}.
Finally, regarding the orbital modifications of the asteroids due to the supernova phase, two cases can be distinguished: close systems where most probably the asteroid belt or debris belt forms after the supernova phases, in that case the system is already relaxed in some way. For wide systems the large distance of the companion should not affect small bodies highly bound to the central neutron star, except for secular effects.
2.4. Reconciling the emission beaming with the observed FRB rate
FRB emission would be observed when the radio beam of the Alfvén wings crosses the observer’s line of sight. This probability is diminished by the narrow emission beam (of opening angle 1/γ ∼ 10^{−6} − 10^{−5}) produced by the Alfvén wave mechanism of Mottez & Zarka (2014), but compensated by the large number of orbits achieved by the asteroids before reaching the Roche limit. The timescale for the asteroid eccentricity to shift from a_{mJy} ∼ 0.1 AU to the Roche Limit in the emission zone (due to KozaiLidov effects) would be a fraction of the KozaiLidov timescale, which is a secular effect, hence happening on times much larger than the orbital time period of the asteroids. Therefore the number of Keplerian orbits performed in the emission zone before reaching the Roche limit is large. A rough estimate of the number of orbits achieved by the asteroid in the emission zone can be obtain by comparing the KozaiLidov timescale to the orbital periods of the asteroid at the beginning of the emission zone and at the end (the Roche limit). Considering Keplerian orbits, the orbital period can be derived from Kepler’s third law , where P_{ast} is the orbital period of the asteroid, a_{ast} its semimajor axis, G the universal constant of gravitation and M_{NS} the mass of the orbited pulsar. For an asteroid position at the beginning of the emission zone a_{ast, mJy} ∼ 0.1 AU, and orbiting a pulsar of mass M_{NS} ∼ 1.4 M_{⊙}, this period is about P_{ast, mJy} ∼ 6 × 10^{3} days, while at the end of the emission zone (Roche limit) it is much shorter, about P_{ast, Roche} ∼ 5.5 h. Therefore the comparison of t_{KL} the KozaiLidov timescale, given by Eq. (12) (see Sect. 3), considering an outer body of mass 10 M_{⊙} with a semimajor axis of 10 AU, with the orbital periods of the asteroid gives the number of orbits achieved N ∼ t_{KL}/P_{ast} ∼ 10^{5} − 10^{7}. The large number of orbits can thus compensate for the strong beaming and lead to more than one emission burst per asteroid, as we assume in the rest of our discussion. Other asteroids can also enter the emission zone, leading to repetitions of bursts.
In addition, turbulence effects, wind fluctuations and asteroid proper motions will also randomly affect the beam position and orientation. From Mottez et al. (2020) the authors, derive a conservative value of the emission source velocity, due to the wind intrinsic oscillations, of about v_{s} ∼ 0.01c ≪ v_{wind}, equivalent to an angular velocity of about rad s^{−1}. Consequently, the emission beam wanders over an area proportional to the time of observation t_{obs} and the Keplerian orbital period of the asteroid , assuming the orbital motion is in the same plane as the observer line of sight for simplification. This area can be described with an opening angle α_{w} = n_{ast}t_{obs} ∼ 10^{−1} rad (M_{NS}/1.4 M_{⊙})(a_{ast}/10^{−2} U.A.)(t_{obs}/1 h)≫α_{beam} ∼ γ^{−1} and defines the probable detection region. During the observation time t_{obs}, multiple bursts can be observed if the beam crosses the observer’s line of sight several times. Another consequence of the beam wandering motion is the burst duration, which result from the sweep time of the beam across the observer’s line of sight, given by
Finally, the number of bursts observed and their durations depend on the position of the asteroids when the emission is produced, but also on the pulsar characteristics, which make possible configurations as diverse as the observed FRB burst durations and repetitions.
Our final picture corresponds to an emission zone filled with asteroids whose Alfven wings randomly cross the observer’s line of sight during the large number of orbits achieved to reach the Roche limit, where the asteroid disruption occurs. During the disruption, complex tidalinduced fragmentation could happen, especially for large asteroids, leading to a multitude of subemission components over short timescales. Such events could explain the observations of FRB 121102, from which ∼90bursts were detected during a five hour period (half falling within 30 min, Zhang 2018).
3. KozaiLidov mechanisms
In the framework of asteroids orbiting a central pulsar and surrounded by an outer massive body (see Fig. 1 for a sketch), we expect modifications of the orbits to occur through the exchanges of orbital momentum between the two twobody systems: (1) pulsarasteroid (the inner binary), and (2) (pulsarasteroid)outer body (the outer binary). These exchanges can translate into an increase of the eccentricity of the inner binary and therefore lead to configurations where the two bodies of the inner binary move very close to one another, when reaching the periapsis of their orbits, leading in some cases to a crossing of the Roche limit.
Fig. 1. Framework for the KozaiLidov perturbation calculations in our binary neutron star+ asteroid triple system. The neutron star is surrounded by the asteroid belt, and the binary companion orbits at a larger distance. All objects are represented by their distance to the neutron star (for instance a_{ast} and a_{c}) and their inclination (for instance i_{ast} and i_{c}) with respect to the invariant plane. 
As can be seen in Fig. 1, the subscripts 1, 2, 3 refer to the central object, the outer body and the “massless” body orbiting the central object respectively. We specify the notations in some numerical estimates and in the next sections by denoting these objects with the subscripts NS, c, ast, corresponding to the central neutron star, its binary companion and the orbiting asteroids.
3.1. Secular perturbations in threebody systems
The motion of the outer body, also referred to as the perturbing body, induces gravitational perturbations which happen on secular timescales, that is on timescales much longer than the typical orbital timescales. In the specific case of a hierarchical three body system, where the semimajor axis of the inner binary is much smaller than the semimajor axis of the outer binary a_{1}/a_{2} ≪ 1, this system is stable. Furthermore in the test particle approximation, where one of the bodies is considered “massless” (m → 0), only the motion of this “massless” body is affected by the secular dynamics. For large mass ratios within the inner binary, the inner binary orbit can flip from a prograde motion to a retrograde motion by rolling over its semimajor axis. During one of these flips, the orbit passes trough an inclination of 90° which leads to a large eccentricity excitation.
The threebody dynamics is usually decomposed into the dynamics of two twobody systems, plus a perturbation effect between these twobody systems. In terms of Hamiltonian, one can write
where ℋ is the total Hamiltonian of the threebody system, G is the gravitational constant, m refers to the mass, a to the semimajor axis, and the subscripts 1, 2, 3 to one the body or one of the two twobody systems (1 or 2). Finally, ℋ_{pert} represent the perturbation term between the two twobody systems and can be decomposed over Legendre polynomials
With r_{1} and r_{2} the distances between the two bodies of the inner binary and outer binary respectively, P_{j} Legendre polynomials, Φ angle between r_{1} and r_{2}, and ℳ_{j} = m_{0}m_{1}m_{2} a mass term.
It is possible to rewrite this series only for the two main terms
where ℋ_{quad} and ℋ_{oct} represent the quadrupolar and octupolar orders of the perturbation and ϵ is given by
Depending on the configuration of the threebody system, the value of ϵ indicates which order dominates the dynamic (either quadrupolar or octupolar). Furthermore stable systems are expected for values of epsilon ϵ ∼ 0.1 or if the eccentricity is null a_{1}/a_{2} ∼ 0.1. Figure 2 shows the evolution of the ϵ parameter depending on the inner binary (pulsarasteroid in our case) configuration and the outer binary (perturbing body) configuration. The shadowed region represent the configurations where the outer body is closer than the inner binary (between the pulsar and the asteroid in our case), which is not possible. The domain where the octupolar is fully dominant is delimited by the two solid black lines. One can see that this region corresponds to configurations where the outer body has an eccentric orbit and is not too far from the inner binary.
Fig. 2. Evolution of the octupolar efficiency parameter ϵ (Eq. (8)) as a function of the asteroid belt semimajor axis a_{ast} and the companion semimajor axis a_{c} (left), and the companion eccentricity e_{c} and ratio of asteroid belt semimajor axis a_{ast} and companion semimajor axis a_{c} (right). The solid black lines delimit the regions where the octupolar regime of the threebody dynamic is expected to be dominant ϵ = 0.1 and ϵ = 10^{−3}, and where a transition from octupolar regime to quadrupolar regime is expected to take place ϵ = 10^{−4}. 
When the outer body has a circular orbit, the dynamics is led by the quadrupolar term and results in the socalled classical KozaiLidov mechanism. In this mechanism, periodical exchanges of orbital momentum between the two twobody systems lead to a reduction of the inclination of the inner binary at a cost of an increase in eccentricity. These oscillations stem from the fact that in the test particle approximation (where one of the bodies of the inner binary has a mass close to zero), the zcomponent of the total orbital momentum, defined by the invariant plane, is conserved and can be rewritten as a function of the inclination and the eccentricity
where L_{z} is the zcomponent of the total orbital momentum of the threebody system, e_{2} is the eccentricity of the inner binary and i_{tot} is the total inclination of the system within the invariant plane. By conservation principles, it is straightforward to extract the maximal eccentricity reachable as a function of the initial inclination, assuming a total transfer of the inclination during the KozaiLidov effect. Therefore one can obtain
From the same argument, it is possible to derive the minimal initial inclination required to trigger classical KozaiLidov effects, which is 40 ° < i_{init, tot} < 140°.
In the case where the outer body has a non zero eccentricity (e_{2} ≠ 0), the octupolar term of the perturbation become nonnegligible. In particular in the regime where 10^{−3} ≲ ϵ ≲ 0.1, the octupolar term is dominant. In this regime, the KozaiLidov effects are called the eccentric KozaiLidov mechanism (EKM), where the classical KozaiLidov oscillations of the quadrupolar regime are modulated by a rotation of the orbits of the inner binary around its semimajor axis. This rotation leads to an increase of the inclination of upto i_{1} = 90° beyond which the orbit flips from a prograde motion to a retrograde motion. During the whole process, classical KozaiLidov oscillations continuously occur with intensity peaking when the orbit reaches a 90° inclination, triggering extreme eccentricities e_{1, max, EKL} → 1. The criteria for orbits to flip has been derived by Li et al. (2014)
where Ω_{1} and ω_{1} are the longitude of the ascending node and the argument of the periapsis of the inner binary respectively. Numerical results are consistent with this criteria (Li et al. 2014), and show once again how the ϵ parameter can be used to discriminate between the different dynamical regimes of the threebody system.
The EKM is characterized by longer timescales than the classical KozaiLidov effect, since it can be seen as the superposition of several classical KozaiLidov oscillations, but it leads to extremely high eccentricities. The intensity of the oscillations in the EKM depends on the value of ϵ and so on the dynamical regime of the threebody system. The EKM has been found to be possible for at least two distinct regimes: (i) Low eccentricityHigh inclination, and (ii) High eccentricityLow inclination. The first regime corresponds to the classical criteria on the initial inclination to trigger KozaiLidov oscillations (40 ° < i_{init, tot} < 140°), and more interestingly, the second regime corresponds to orbital configurations where the system can be almost coplanar but still trigger EKM thanks to the high eccentricity of the inner binary.
In the specific framework of KozaiLidov effects, the threebody dynamics can be described with three main regimes: the quadrupolar regime when the outer body has a circular trajectory, featured by classical KozaiLidov oscillations; the octupolar regime when 10^{−3} ≲ ϵ ≲ 0.1, enabling a richer dynamics with Eccentric KozaiLidov mechanisms and orbital flips; and finally, a combination of the previous two regimes where ϵ ≲ 10^{−3}, which depends on the specific configuration of the threebody system and is difficult to analyze in a general framework. In this study we consider the octupolar regime down to ϵ = 10^{−4}, where in fact a transition towards the quadrupolar regime operates. This choice is made for illustrative purposes, in order to map a larger parameter space (matching astrophysical objects) without falling into too much purely dynamical considerations. However in Appendix C, we provide a study focused on the quadrupolar regime, showing that the conclusions drawn from the octupolar regime also hold in this regime.
3.2. KozaiLidov timescales
As described before, the dynamics of each regime, quadrupolar or octupolar, is different and so are their characteristic timescales.
Interestingly in the quadrupolar regime (and the test particle approximation), the dynamics is fully integrable, meaning that the Hamiltonian equations of motion can be solved. In this perspective, Antognini (2015) derive the exact classical KozaiLidov period and study its behavior across the parameter space of the threebody dynamics. In particular, it is shown that this exact period only varies within a factor of a few from the standard (and wellknown) KozaiLidov timescale formula. It is worth noting that this is only true in general conditions, away from the boundary between the libration and rotation regime, where nonsecular effects are expected, as well as away from orbital resonances. This timescale is given by
In the octupolar regime, however, the dynamics is no longer integrable, as previous quantities are no longer integrals of motion, therefore the Hamiltonian equations of motion cannot any more be solved. Antognini (2015) shows that the exact period for the EKM can also be derived and this exact period can be well approximated with a KozaiLidov timescale in the EKM regime. This new time scale is given by
where t_{KL, i = 90°} represents the classical KozaiLidov timescale and we suppose that inclinations up to 90° can be reached thanks to the orbital flip mechanisms described earlier in Sect. 3.1. Furthermore the timescale of Eq. (13) describes a full EKM cycle, with two flips: from prograde to retrograde and back again.
The numerical values given in Eq. (14) correspond to a mildly close NSMS, NSWD or DNS case, with a_{c} the semimajor axis of the orbiting companion and M_{c} its mass. The estimate assumes a null eccentricity e_{c} = 0.
Figure 3 presents the EKM timescales for various threebody system configurations. Again, the shadowed region delimits the forbidden configurations. Generically, the timescale increases with the inner binary orbital width and with the distance of the outer binary, as expected from gravitational considerations: the farther the outer body, the lighter the gravitational perturbation on the inner binary, and similarly with the width of the inner binary.
Fig. 3. Evolution of the EKM timescale (given by Eq. (13)) for various configurations of threebody systems, as a function of the semimajor axis, for a companion eccentricity e_{c} = 0.1 (left), and outer binary eccentricity and semimajor axis, for an asteroid belt located at a_{ast} = 1 AU (right), for a low mass companion M_{c} = 1 M_{⊙} corresponding to DNS, NSMS and NSWD systems (top), and a high mass companion M_{c} = 10 M_{⊙} corresponding to NSMS and NSBH systems (bottom). The shadowed region on the left panels represents the forbidden configurations where the outer binary is closer than the inner binary. On the right panels, these forbidden configuration lies on the lefthand side of the vertical dashed line. The gray boxes on the right panels represent the parameters spaces where the different systems considered here are expected to lie. The solid lines show the limit where the timescale is shorter than one year (leading to transient events). On the left panels, the gray thin horizontal dashed lines and the arrows indicate the parameterspace in which the different systems would lie (Table 1). 
3.3. KozaiLidov relative time delays
In the specific case where the inner binary is in fact made of several small objects such as an asteroid belt or a comet cloud, orbiting a more massive central body, an additional interesting quantity is the relative time delay of the KozaiLidov timescales between closeby objects. The relative time delays between two small objects seperated by a distance Δa_{1} is given in the quadrupolar regime by
and in the octupolar regime by
Two objects separated by a distance Δa_{1} orbiting a central more massive object and perturbed by an outer body, undergo Eccentric KozaiLidov effects with a timescale difference given by Eqs. (15) and (16) depending on the dynamical regime.
Equation (16) can be also rewritten in a more compact way
This formula provides a more straightforward description of the relative time delay of the EKM.
Assuming that the initial distribution of a_{ast} in the asteroid belt follows a Normal distribution with mean ⟨a_{ast}⟩ and width σ_{a} = ε_{ast}⟨a_{ast}⟩, the mean distance between two consecutively falling asteroids can be estimated statistically as ⟨Δa_{ast}⟩≈σ_{a}/N_{ast, KL}, with N_{ast, KL} the number of asteroids undergoing KozaiLidov effects. In the octupolar regime, when the outer body has a non circular orbit, most asteroids undergo KozaiLidov effects and reach high eccentricities. Hence one can write N_{ast, KL} ∼ ϵ_{eff} N_{ast}, with N_{ast} the total number of asteroids in the belt and ϵ_{eff} ≳ 0.2 (see Sect. 5.3). The fraction of asteroids meeting the KozaiLidov criterion in the quadrupolar regime is calculated in Appendix C.2. One can then express the mean relative Eccentric KozaiLidov time delay as
where the numerical estimates are presented again for a mildly close NSMS or NSWD or DNS case.
Figure 4 describes the evolution of the relative time delays across the parameter space allowed for the threebody system. The time delay trend follows the EKM timescales as depicted in Fig. 3.
Fig. 4. Same as Fig. 3, but for the relative KozaiLidov time delay (given by Eq. (16)). The solid lines represent the limits where the relative delay equals 1 day (for sources producing dayrepeaters) and the dashed line is the limit where the relative delay equals 10 years (observational time beyond which sources cannot be observed as repeaters). 
4. FRB rates for close and wide NS binaries
In this section, we apply the formalism derived in the previous Sections to populations of neutron star binaries and derive corresponding FRB rates.
4.1. Neutron star binary system population characteristics
A neutron star is formed among a binary stellar system when the initially more massive star undergoes a supernova explosion. The companion can be a main sequence star, have already transformed into a white dwarf, or become a neutron star or a black hole following a second supernova explosion (Portegies Zwart & Verbunt 1996; Lorimer 2008). However, in most scenarios, the explosion or the kick experienced by the neutron star at birth disrupts the binary system (Hansen & Phinney 1997; Lu & Naoz 2019).
The majority of stellar binaries are initially wide (Kroupa 2008; Kroupa & PetrGotzens 2011), with orbital separation a_{c} ≳ few AU, and each object evolves mostly as single stars (Postnov & Yungelson 2014). Supernova kicks drastically reduce the rate of these wide binaries by disrupting them. Orbits with higher eccentricity are more likely to survive these kicks.
Numerical binary population synthesis indicate that systems containing main sequence stars could be of order ν_{NSMS} ∼ 5.8 × 10^{−5} yr^{−1} (Portegies Zwart & Verbunt 1996) in the Galaxy. These authors also show that wide NSMS systems represent about (6.7/65)% ∼ 0.1% of the total neutron star population (hence ∼1.7 × 10^{−5} yr^{−1}). The severe population cut by a factor of 65 compared to a produced number of wide binaries is due to the supernova kicks.
Observations concur, pointing to mildly close systems with a_{c} ∼ 1 − few AU, with mild to high eccentricities e_{c} ∼ 0.6 − 0.9 (e.g., PSR125963 and its 10 M_{⊙}mass Bestar companion, Johnston et al. 1992, J17403052 and its Btype star companion of mass M_{c} ∼ 11 M_{⊙}, Madsen et al. 2012; Hobbs et al. 2004, PSR J16384715 and its 4.5 M_{⊙}mass companion, Lyne et al. 2005, PSR J00457319 and its 4 M_{⊙}mass companion, Kaspi et al. 1994).
NSWD systems are naturally more numerous, as white dwarfs are common outcomes of main sequence stars, with simulated rates ∼4 times higher than for NSMS (Nelemans et al. 2001). Due to their formation channels, NSWD are frequently found in very close circular systems, in which case the neutron star is a recycled pulsar. The orbital semimajor axis distribution of NSWD binaries should however follow the same trend as NSMS systems, with 1/3 of wide binaries with high eccentricities.
For DNS, Nelemans et al. (2001) estimate a total Galactic population rate of ν_{DNS, all} ∼ 5.7 × 10^{−5} yr^{−1}, which includes binaries with recycled pulsars which are particularly close. Portegies Zwart & Van den Heuvel (1999) calculated numerically that wide systems with neutron stars that evolved mostly independently constitute again about a third of the total DNS population, with a rate of ν_{DNS, wide} ∼ 5.7 × 10^{−5} yr^{−1} (see also Kruckow et al. 2018).
For NSBH, the birth rate is estimated to 0.6 − 13 Myr^{−1} in the Galactic disk (Shao & Li 2018; Dominik et al. 2013; Lamberts et al. 2018). Recent simulations show that about 10% of the binary population could be wide binaries (Belczyński & Bulik 1999; Kruckow et al. 2018).
The orbits of close binary neutron star systems that have lowmass companions, such as lowmass white dwarfs (M_{c} ≲ 0.7 M_{⊙}) tend to be circular: e_{c} ∼ 10^{−5} − 10^{−2}. Close systems with highmass companions, such as neutron stars, some white dwarfs and main sequence stars (M_{c} ≲ 0.7 M_{⊙}) have more eccentric orbits e_{c} ∼ 10^{−2} − 0.9 (e.g., Lorimer 2008; Hobbs et al. 2004). NSBH systems have a wide range of eccentricities, that essentially span the full physically allowed range Kruckow et al. (2018). Wide systems have highly eccentric orbits.
We consider for our systems, masses of M_{NS} = 1.4 M_{⊙} (since this is the minimal mass required to produce a NS) for the central neutron star. The companion masses span over M_{c} = 0.01 − 10 M_{⊙} for white dwarfs to black holes. For illustration, we use M_{c} = 10 M_{⊙}, a typical value in NSBH (Kruckow et al. 2018) and NSMS systems. Estimates can easily be scaled for larger black hole masses, which would lead to higher asteroid infall rates.
Table 1 summarizes the typical parameter ranges discussed above for our binary populations.
Population characteristics (binary system Galactic birth rate and companion mass M_{c}) and orbital element distributions (eccentricity e_{c} and semimajor axis a_{c}) for neutron stars in binary systems: neutron starwhite dwarf (NSWD), double neutron star (DNS), and neutron starblack hole (NSBH).
We mentioned in Sect. 2.1 that systems with planet companions have also been observed (Lorimer 2008), but the rates and characteristics of these systems are not yet clear. A planet was detected in the triple Pulsar System PSR B162026, with a wide inferred semimajor axis of a_{c} ∼ 23 AU and moderate orbital eccentricity (Sigurdsson et al. 2003). Three planetary bodies were found orbiting at ∼AU distances around pulsar B1257+12 Wolszczan & Frail (1992). In both systems, the pulsar is recycled. More formation studies and observational data would be needed to derive population characteristics for neutron starplanet systems. We focus here on the other binaries mentioned above, that are more documented.
4.2. The octupolar regime dominates over most of the binary parameter space
Figure 2 shows the values of the octupolar efficiency term ϵ, depending on the companion orbital elements. Each type of companion (white dwarf, black hole or neutron star) covers a different region of the allowed parameter space.
Interestingly, one can see that most systems will be found in the region where ϵ = 0.1 − 10^{−4}, dominated by octupolar dynamics. Therefore, we concentrate in the following on the octupolar regime and derive our main estimates within these dynamics (the full derivation for the quadrupolar regime can be found in the Appendix C). For systems approaching ϵ = 10^{−4} (NSWD systems in particular), the quadrupolar dynamics will start to dominate. However, our results should be equally valid in this case. As we demonstrate in the Appendix C, the quadrupolar regime leads to a less efficient KozaiLidov mechanism, and hence to lower FRB rates per source. This is nevertheless compensated by a higher source population rate for NSWD (see Table 1).
We notice that the timescales of the octupolar regime and quadrupolar regime (see Eq. (16)) only differ by a factor . Therefore in principle, systems in the octupolar regime should be characterized with longer timescales than in the quadrupolar regime. From Fig. 3, we can see that for almost all systems, the absolute timescale of Eccentric KozaiLidov oscillations is longer than one year, except for very close or highly eccentric systems. Regarding the relative time delays of Eccentric KozaiLidov oscillations, Fig. 4 shows typical timescales below one day up to times longer than the age of the Universe.
As illustrated in Figs. 3 and 4, the EKM can occur over a large range of timecales. This flexibility makes this process a good candidate to explain the diversity of observed FRB rates.
4.3. Contributions of wide and close populations to FRBs and FRB repeaters
It appears from Eq. (16) and Figs. 3–4 that the main parameter governing the infall rate via KozaiLidov effect is the orbital separation between the neutron star and the black hole a_{c}. The distance at which the neutron star binary companion can be located spans several orders of magnitude, from a_{c} ∼ few 10^{−3} AU to 100 AU. From the previous Section, systems can be split into three populations: wide systems with a_{c} ≳ 10 AU, mildly close systems with a_{c} ∼ 0.3 − few AU, and close systems with a_{c} ≲ 0.3 AU.
The close systems are often associated to recycled pulsars, which are not magnetized enough to produce FRB emission at the Jansky level, except for extremely large asteroids (see Sect. 2.1 and Eq. (1)).
The timescale over which the KozaiLidov effects can take place, hence the lifetime of the system as an FRB source, is highly dependent on a_{c} (Eq. (13)). While wide binaries have t_{EKL} ≫ 10 yr (or t_{KL} ≫ 10 yr in quadrupolar regime) and can be viewed as longlived FRB sources, close and mildly close binaries have t_{EKL} < few 10s of yr and should be considered as shortlived FRB transients. For close binaries with a_{c} ≪ 1 AU, t_{EKL} ≪ 1 yr, leading to a “singleshot” transient, that will not be observed as repeating over a long timescale. Some mildly close binaries can live thousands of years, as can be seen in Fig. 3 for some NSBH systems.
Close, mildly close and wide binaries are expected to be observed as different types of FRB sources for our model. Indeed, for wide systems with a_{c} ≳ few AU, Δt_{KL} ≳ 10 yr, leading to nonrepeating sources. For close and mildly close binaries with a_{c} ≲ few AU, Δt_{KL} ≲ 10 yr, sources could be observed as repeating, with various emission frequencies. Close systems with a_{c} ≪ 1 AU will produce emissions with periods shorter than a day. As previously discussed, these systems are however likely to be too faint to produce the observed FRB signals.
As the gravitationalwave merger timescale is
the survival of both mildly close and wide binary systems over the age of the Universe is mostly guaranteed for a circular orbit (e_{c} = 0). For large eccentricity, the merger can however happen on a shorter timescale, down to ∼10^{4} years (Peters 1964). In any case, these timescales are longer than t_{EKL} and do not need to be considered here.
Therefore, in this scenario, mildly close binaries would produce day and monthrepeaters and wide binaries nonrepeaters. It is interesting to notice that in the current analysis (CHIME/FRB Collaboration 2019b; Fonseca et al. 2020), day to few day periods seem to be favored among repeaters. This could be consistent with the dichotomy between the signatures from mildly close and wide binaries.
This dichotomy is reflected in the calculation of the FRB rates from these two categories.
Mildly close binaries can be day/monthrepeater FRBs during t_{KL} < 10 yr. Their FRB rate density is hence directly linked to their birth rates as documented in Table 1. We calculate roughly the total density rates ṅ_{c} for each population using the Galactic birth rates estimated in the literature and assuming a local density of galaxies of n_{gal} = 0.02 Mpc^{−3}: . The rate density of dayrepeater FRB sources then reads
where ϵ_{rep} < 1 is a source efficiency factor, and ϵ_{mild − close} the fraction of mildly close systems among a population.
For wide binaries, the rate density of FRBs expected to be sourced by infalling asteroids can be estimated by convolving the mean infall rate 1/⟨Δt_{KL}⟩, the typical lifespan of the asteroid belt in its primordial configuration, t_{EKL} (or t_{KL}), and the rate density of wide binary systems ϵ_{wide}n_{c}, with ϵ_{wide} the fraction of wide systems among a population. It yields a rate density of apparently nonrepeating FRB events of
with ϵ_{wide} = 0.3 (Portegies Zwart & Verbunt 1996; Portegies Zwart & Van den Heuvel 1999; Kruckow et al. 2018), ϵ_{eff} ∼ 0.2 the KozaiLidov efficiency factor discussed in Sect. 5.3, and ϵ_{nrep} < 1 a similar source efficiency factor as in Eq. (20).
These calculations assume that these binaries undergo a flat source emissivity evolution, out to redshift z ∼ 1 (Postnov & Yungelson 2014). For a starformation type evolution, the number of sources would increase by a factor of ∼2.
In Table 1, we estimated the FRB rate densities produced by various binary populations, for mildly close and wide systems, leading to repeating (rep.) and apparently nonrepeating (nrep.) sources respectively. For DNS and NSWD, we have assumed that a fraction ϵ_{mildclose} = 1/3 of the whole population was in mildly close orbit. For NSMS and NSBH, we assumed that the majority of the population was in mildly close orbit ϵ_{mildclose} = 1. For NSWD, we assumed that ϵ_{wide} = 1/3, and used the rates provided in the literature for wide NSMS and DNS rates. For NSBH, we assumed ϵ_{wide} = 0.1. These fractions are discussed in Sect. 4.1.
The rate densities estimated from Eq. (21) can be directly compared to the cosmological FRB rate densities inferred from observations, of order ṅ_{FRB,obs} ∼ 2 × 10^{3} Gpc^{−3} yr^{−1}Petroff et al. (2019). Except for NSBH, for which the entire population would not suffice to produce the observed FRB rate densities, we notice that ṅ_{FRB,obs} ≪ ṅ_{FRB,nrep}. The inferred source efficiency can thus be of order ϵ_{nrep} ≲ 10% (NSWD: 2.0%, NSMS: 8.7%, DNS: 7.4%). This number leaves room for binary systems which do not fulfill the criteria to undergo KozaiLidov mechanisms, such as systems without asteroid belts, orbital inclinations, etc.
More than 700 FRBs have been observed as of today, (although only 137 have been published), among which 22 have been identified as repeaters (Andersen et al. 2019; Fonseca et al. 2020), yielding a possible ratio of ∼3%. Interestingly, this number match quite well the ratios estimated for our systems: ṅ_{FRB,rep}/ṅ_{FRB,nrep} ∼ 5.3% ϵ_{rep}/ϵ_{nonrep} for NSMS (for NSMS: 1.4% and for DNS: 1.4%). For a same population, one can assume that ϵ_{rep} = ϵ_{nonrep}. However, the formation of asteroid belts might differ for close and wide systems.
It is possible that all the neutronstar binaries mentioned above contribute to the FRB rates. If one assumes that their efficiencies ϵ_{nrep} and ϵ_{rep} are equal, the total rate density of apparently nonrepeating FRBs would be of ṅ_{FRB,nrep,all} ∼ 15 × 10^{4} ϵ_{nrep} Gpc^{−3} yr^{−1}, with a source efficiency that can be as low as ϵ_{nrep} = 1.4%. The repeater rate density would be of ṅ_{FRB,nrep,all} ∼ 3.1 × 10^{3}ϵ_{nrep} Gpc^{−3} yr^{−1}, which implies a repeater to nonrepeater ratio of ∼2%, compatible with the observed ratios. The scenario is surprisingly comfortable and consistent with the current observations.
5. Simulating numerically asteroid infall rates
In this section, we simulate numerically the FRB rates of close and wide binary systems with an asteroid belt undergoing KozaiLidov effects. We model a primordial asteroid belt (without any gaps such as the Kirkwood gaps of the Solar system), in analogy with the Solar asteroid belt.
5.1. Synthetic asteroid belt
We model the distribution of the orbital parameters of the current Solar belt using the data from the IAU Minor Planet Center (MPCSAOIAU 2019). A total number of 792041 asteroids of the Solar belt are inventoried in this database.
Numerous asteroids sensitive to the KozaiLidov effect are missing from the distribution of orbital elements of the current Solar asteroid belt, influenced by giant planets such as Jupiter. The Kirkwood gaps for instance, illustrate this effect. These features motivate the construction of a synthetic asteroid belt for our model, filling most of the gaps and mimicking the primordial population of the belt (see Fig. 5).
Fig. 5. Asteroid orbital parameter distributions inside the Solar asteroid belt (blue) and our reconstructed primordial model belt (orange). Top: asteroid semimajor axes distribution. Bottom: asteroid inclinations as a function of the semimajor axes. In both panels, the Kirkwood gaps are clearly visible in the Solar asteroid belt. 
The synthetic belts follow a Gaussian distribution fitting the general trend of the current Solar belt. We use for the semimajor axis a standard deviation of σ_{a} = 0.15⟨a_{ast}⟩, with the mean semimajor axis ⟨a_{ast}⟩ left as a free parameter. For the inclinations, we follow the Solar belt distribution with mean inclination ⟨i_{ast}⟩ = 0° and standard deviation σ_{i} = 30°. The initial eccentricities are not modeled since they are not relevant to our computation of the KozaiLidov effects. However Fig. 5 suggests that a fitting model similar to what is done for the distribution of semimajor axis could be easily achieved.
The number of these massive asteroids follows a power law distribution as a function of their size, as observed in the Solar System MPCSAOIAU (2019) (see Sect. 2.2. Their masses can be retrieved by assuming that they are roughly spherical, with a density ρ_{ast} = 2 g cm^{−2}.
This simple method allows us to construct a more generic asteroid belt, although it is restricted to our knowledge of the Solar system.
5.2. Simulations setup
Following the asteroid distribution computed in the previous section, we randomly draw a set of asteroid parameters (size R_{ast}, semimajor axis a_{ast}, inclination i_{ast}). We select the objects that meet the following three criteria

Minimum size R_{ast} > 50 km, large enough to trigger FRBlike emissions via the Alfvén mechanism (Eq. (1))

Allow the triggering of KozaiLidov oscillations (see Appendix C)

Can reach the Roche limit under the KozaiLidov effect (Eq. (C.5)).
The last two criteria are always met under the octupolar regime. Both quadrupolar and octupolar regimes are taken into account in this calculation, as well as the GR effects.
For the selected asteroids, we compute the KozaiLidov timescales needed to reach the maximal eccentricity and the relative time delays between two consecutive asteroid infalls. The distribution of these relative infall times can be directly compared to FRB rates. To avoid statistical fluctuations due to the MonteCarlo drawing, we average our results over 10^{4} simulations.
5.3. Asteroid infall rates for a Solarlike belt
Figure 6 shows the distribution of the relative time delays Δt_{KL} for asteroids falling onto the central neutron star, for a current (green) and primordial Solarlike belt, and for companion inclinations i_{c} = 5° (orange), i_{c} = 10° (green) and i_{c} = 45° (blue). The central neutron star has mass M_{NS} = 1.4 M_{⊙} and the outer companion M_{c} = 10 M_{⊙}. The initial number of asteroids is set to N_{ast} = 10^{2}, following the power law spectra observed in the Solar system belt for the most massive asteroids. We examine in Fig. 6 the effect of the companion inclination i_{c} on the relative timescales and efficiency of the KozaiLidov effect, in the case of a wide system with companion distance a_{c} = 10 AU and mean asteroid belt distance ⟨a_{ast}⟩ = 1 AU.
Fig. 6. Distribution of relative time differences Δt_{KL} of asteroids falling into the Roche lobe of the central compact object due to KozaiLidov oscillations (which can be directly interpreted as the FRB emission periods), for the current Solar asteroid belt (green) and the primordial belt, for an inclination of the outer companion plane i_{c} = 5° (orange), i_{c} = 10° (green) and i_{c} = 45° (blue), and initial asteroid number N_{ast} = 10^{2}. We consider a high mass M_{c} = 10 M_{⊙} wide system with a_{c} = 10 AU and ⟨a_{ast}⟩ = 1 AU. The number of asteroid infalls increase with the inclination of the system, as expected. Additionally, the relative time differences Δt_{KL} increase with lower inclinations, except for the Solar asteroid case. The present Solar asteroid belt has already undergone KozaiLidov effects during its lifetime, leading to a cleansing of its asteroids that is potentially sensitive to KozaiLidov oscillations, which explains its misleading behavior in this figure. 
For this wide system, the infall rates span from days to thousands of years, with a maximum around ⟨Δt_{KL}⟩∼10 − 100 years, depending on the inclination i_{2}. More interestingly the efficiency of the KozaiLidov process, the ratio of the number of falling asteroids over the number of drawn asteroids, is greater for inclined systems. More asteroids fall onto the central pulsar for a more inclined asteroid belt, which is consistent with the KozaiLidov process, since more asteroids will meet the KozaiLidov criterion on the inclination i_{ast} ≳ 40°. Nevertheless, these simulations show that even for low to mildly inclined systems, the efficiency remains around 20%.
The comparison between the current Solar belt (green) and the primordial belts (blue or orange) shows that the lack of Kirkwood gaps induces a drastic increase of short timescales in the asteroid infall rate, and depending on the inclination, a factor of a few to an order of magnitude more events in total (greater efficiency).
Figure 7 displays, similarly to Fig. 6, the distribution of the relative time delays Δt_{KL} for asteroids falling onto the central neutron star, for various neutron star systems. The left panel presents systems with low mass companions, such as DNS, NSMS and NSWD for close and wide systems. The right panel shows systems with high mass companions, namely NSBH and NSMS. Wide and close systems can be distinguished through their relative asteroid falling rates: wide systems induce higher rates than close systems, for both low and high mass companions. NSWD systems appears to be much less efficient than any other systems, this is due to the fact that the KozaiLidov timescale is a function of the mass of the companion (see Eqs. (15) and (16)). An opposite result can be seen for high mass NSBH systems, which are much more efficient and with shorter timescales than NSBH wide systems, as expected. Finally one can notice that most of the systems (except NSWD systems), present an efficiency above 50% in this KozaiLidov mechanisms.
Fig. 7. Same as Fig. 6, distribution of relative time differences Δt_{KL} of asteroids falling into the Roche lobe of the central compact object due to KozaiLidov oscillations, for the different systems considered here. Left: low mass companions. Right: high mass companions. The initial number of asteroids is N_{ast} = 10^{2} and the inclinations of the companions is i_{c} = 30°. Specific parameters for each configuration can be found in Table 2. 
5.4. Connection with FRB observations
The results of the simulations detailed in Sect. 5.3 show a high consistency with the analytical estimates computed in Sects. 3 and 4.3. These simulations demonstrate that the application of the KozaiLidov framework introduced in Sect. 3.1 to a multiplicity of small objects such as the ones found in the Solar asteroid belt, remains consistent with the conclusions drawn in Sect. 4.3. They validate that the various populations of binary pulsar systems, such as DNS, NSMS, NSWD and NSBH, can explain the dichotomy observed between repeating and nonrepeating FRBs.
The efficiency of the KozaiLidov process is illustrated on Fig. 7, where the number of asteroid infalls compared to the total number of asteroid simulated (N_{ast} = 100 in Fig. 7), corresponds to the efficiency of the KozaiLidov mechanism in driving asteroids down to the Roche limit (the factor ϵ_{eff} introduced in Sect. 3.3, see also Sect. 5.2). It is clear that for most pulsar binary systems this process is efficient with ratio largely above 50%, and even for NSWD systems which are the less efficient systems, this ratio is around 20%. Consequently the KozaiLidov process in pulsar binary systems is efficient in driving asteroids down to the Roche limit from a Solarlike asteroid belt in our model.
Another interesting result coming only from the simulations concerns the distribution tails displayed in Fig. 7. One can see that the Gaussian rates distributions (at a first approximation) possess extended distribution tails. This result implies that for some wide systems, with long time delays on average, events could occur with shorter time delays at some point in the process. This translates, in terms of FRB bursts, in the existence of some observed nonrepeating sources that can produce few repetition bursts once in a while. One should note that these repetitions would be highly irregular, and would not be sustained over time, as they are statistically rare.
This spread in the time delays is also valid for close systems, with short time delays, and associated with FRB repeaters. This numerical result is actually in agreement with observations since a fraction of FRB sources are found bursting with irregular short periods, ranging from days to monthtimescales. These bursts would correspond to the lefthand tail of distributions such as the one shown in Fig. 7 for close systems.
FRB121102 could be a good candidate for this tail scenario. Activity periods have been reported for hour scale periods, day scale periods and monthly periods (Table 2 in Scholz et al. 2016). Such an erratic behavior could well be explained as a tail of the asteroid falling rate distribution. This source also presents substructure in the signal, with fainter pulses arriving at shorter intervals (Zhang 2018). These could be explained by the fragmentation of asteroid during the disruption in the Roche lobe, as mentioned in Sect. 2.4 or simply the presence of asteroids clumping in the asteroid belt, as observed in the Solar system, which is explained by asteroid collisions leading to subgroups of asteroids closeby and with similar orbital paramaters, therefore leading to similar KozaiLidov time delays and so similar infall rates.
The close systems presented in Fig. 7 illustrate the possibility of having a population of shortlived repeaters, with dayscale periods. These sources will appear less numerous than the wide systems due to their short active timescale (see Fig. 3), which is consistent with the low percentage of repeaters observed so far.
Finally the existence of short transient sources is predicted with our model. From Fig. 3, it is possible to find sources with very short lifetimes, below one year. These sources would completely deplete their asteroid belt over very brief infall rates, resulting in a firework display of bursts. These close sources are associated with recycled pulsars, with magnetic fields that are too low to produce Jansky level bursts. These events should hence be difficult to observe because of their brevity and their low flux.
6. Conclusion and discussion
Fast radio bursts can be produced if asteroids pass close to the Roche limit of a compact object with an electromagnetic wind (Mottez & Zarka 2014; Mottez et al. 2020), or if they undergo collisions with this object (Dai 2016; Smallwood et al. 2019). The infall of asteroids from standard belts onto the central compact object can be triggered by KozaiLidov oscillations, in the presence of an outer black hole.
The asteroid dynamics described by our model is able to reproduce the overall observed ratio of repeating to nonrepeating FRBs and motivates an explanation to unify the two observed populations under one simple mechanism, already evidenced in the Solar system. FRBs could be comfortably produced by a population of neutron star binary systems, in particular by NSWD, NSMS and DNS binaries. NSBH systems are expected to have a lower contribution due to their lower population rates. We find that mildly close systems (companion semimajor axis a_{c} ∼ 0.3 − few AU) produce day/month scale repeaters that live < 10 yr, while wide systems (a_{c} ∼ few − 10s AU) are steady sources, which will be observed as nonrepeating.
We find that a comfortable fraction of a few percent (< 10%) of these binary systems in the Universe can account for the observed nonrepeating FRB rates. More remarkably, our wide/close orbit dichotomy model predicts a ratio between repeating and nonrepeating sources of a few percent, which is in good agreement with the observations.
Close systems with a_{c} ≪ 1 AU could also lead to beamed radio signals, but such systems being often associated with recycled pulsars with low magnetic fields, the FRB flux should be low. The signatures of such systems would be specific: a series of mJy level pulses arriving over seconds to hours, and that would never repeat again. SubJansky radio bursts arriving with short periods (≪ day) produced in a single shot could thus constitute an electromagnetic counterpart to NSWD, DNS and NSMS mergers. Such FRBs could also be a counterpart to NSBH mergers as was already predicted in Kotera & Silk (2016).
Simulations presented in Sect. 5 numerically validate the analytical conclusions drawn in Sect. 4. We find that our conclusions hold under more realistic conditions, for instance when taking into account a realistic distribution of asteroid parameters inside an asteroid belt. Finally, the simulations also show that the asteroid belt structure combined with the induced dynamics of specific pulsar systems can lead to a short timescale tail (or repetition tail) even for systems labeled as nonrepeaters.
Three major predictions can be made from our scenario, which can be tested in the coming years:

Most repeaters should stop repeating after t_{EKL} < few 10s of years, as their asteroid belts becomes depleted.

Some nonrepeaters could occasionally repeat, if we hit the short Δt_{EKL} tail of the FRB period distribution.

Series of subJansky level short radio bursts could be observed as electromagnetic counterparts of NSWD, DNS, NSMS and NSBH mergers.
The present study can be applied to other close binary systems, provided that the central object generates a magnetized wind. In particular, pulsar systems with planets could contribute to this scenario.
The recent observation of two intense radio bursts in coincidence with Xray flares (CHIME/FRB Collaboration 2020b; Mereghetti et al. 2020), expected to originate from the magnetar SGR1935+2154, has shown some similarities with FRB emissions. This observation, if attributed to an FRBlike signal, would be the first FRB event observed in our Galaxy but also the dimmest FRB ever observed, with 40 times less radiated energy. Our model is not incompatible with this observation, assuming that this magnetar is in a binary configuration (even with a very far away companion). Some dynamical configurations, resulting from KozaiLidov oscillations can result in the observation of a double radio burst: (i) the observation of two consecutive (and closeby) asteroids falling close to the Roche limit and radiating via the Alfvén wing mechanism, (ii) the observation of a single asteroid close to the Roche limit but observed twice thanks to the turbulence of the beam, crossing twice the lineofsight of the observer, (iii) the fortuitous observation of the disruption of an asteroid crossing the Roche limit and emitting multiple radio beams in random directions, and crossing twice the lineofsight of the observer. However the production of the coincident Xray flares remains more challenging.
One possibility relies in the accretion of tidally disrupted material from a single asteroid onto the magnetar.
A rough estimate can be made by assuming emission via disruption and Eddington accretion of an asteroid of size R_{ast} ∼ 100 km and mass M_{ast} ∼ 8 × 10^{24} g (for a density ρ_{ast} ∼ 2 g cm^{−3}) at the Roche limit d_{Roche} and falling onto the central neutron star of size R_{NS} ∼ 10 km and mass M_{NS} ∼ 1.4 M_{⊙}. The mass accretion rate can be estimated as Ṁ_{ast} = ϵ_{ast}M_{ast}/t_{fall}, where ϵ_{ast} ∼ 0.1 is the fraction of asteroid material accreted and h the infall time from the Roche limit down to the neutron star. The Eddington luminosity is hence given by L_{edd,ast} = ϵ_{edd}Ṁ_{ast}c^{2} ∼ 10^{36} erg s^{−1}, where ϵ_{edd} ∼ 10^{−5} represents the efficiency of the Eddington process (expected to be much less efficient than for black hole accretion). The corresponding isotropic equivalent energy is E_{iso, ast} ∼ 10^{39} erg, close to the value inferred from observations (E_{iso, obs} ∼ 1.4 × 10^{39} erg, Mereghetti et al. 2020). Finally, if the emission results from thermal processes, the effective blackbody temperature T_{eff, ast} can be obtained through the StefanBoltzman law, which enables the determination of the maximal photon energy E_{γ, ast} = hν ∼ k_{B}T_{eff} ∼ 204 keV also close to the observations (E_{gamma, obs} ∼ 20 − 200 keV, Mereghetti et al. 2020).
Alternatively the interactions of material from the plasma with the Alfvén wings could also lead to high energy photon emission, for instance following similar processes as suggested by Beloborodov (2013).
Acknowledgments
We thank the anonymous referee for the thoughtful comments which helped improve this paper. We also thank F. Antonini, A. BenoitLévy, G. Boué, F. Daigne, R. Duque, I. Dvorkin, A. Lamberts, and P. Zarka for very fruitful discussions. This work is supported by the APACHE grant (ANR16CE310001) of the French Agence Nationale de la Recherche. This research has made use of data provided by the IAU’s Minor Planet Center.
References
 AliHaïmoud, Y., Kovetz, E. D., & Silk, J. 2016, Phys. Rev. D, 93, 043508 [CrossRef] [Google Scholar]
 Andersen, B., Bandura, K., Bhardwaj, M., et al. 2019, ApJ, 885, L24 [NASA ADS] [CrossRef] [Google Scholar]
 Antognini, J. M. O. 2015, MNRAS, 452, 3610 [NASA ADS] [CrossRef] [Google Scholar]
 Belczyński, K., & Bulik, T. 1999, A&A, 346, 91 [NASA ADS] [Google Scholar]
 Beloborodov, A. M. 2013, ApJ, 762, 13 [NASA ADS] [CrossRef] [Google Scholar]
 Blaes, O., Lee, M. H., & Socrates, A. 2002, ApJ, 578, 775 [NASA ADS] [CrossRef] [Google Scholar]
 Brook, P. R., Karastergiou, A., Buchner, S., et al. 2014, ApJ, 780, L31 [NASA ADS] [CrossRef] [Google Scholar]
 CHIME/FRB Collaboration (Amiri, M., et al.) 2019a, Nature, 566, 230 [NASA ADS] [CrossRef] [Google Scholar]
 CHIME/FRB Collaboration (Amiri, M., et al.) 2019b, Nature, 566, 235 [NASA ADS] [CrossRef] [Google Scholar]
 CHIME/FRB Collaboration (Amiri, M., et al.) 2020a, Nature, 582, 351 [CrossRef] [Google Scholar]
 CHIME/FRB Collaboration (Andersen, B. C., et al.) 2020b, Nature, 587, 54 [CrossRef] [Google Scholar]
 Cordes, J. M., & Shannon, R. M. 2008, ApJ, 682, 1152 [NASA ADS] [CrossRef] [Google Scholar]
 Dai, Z. G., Wang, J. S., Wu, X. F., et al. 2016, ApJ, 829, 27 [NASA ADS] [CrossRef] [Google Scholar]
 Delgrande, J. J., & Soanes, S. V. 1943, JRASC, 37, 187 [Google Scholar]
 Dominik, M., Belczynski, K., Fryer, C., et al. 2013, ApJ, 779, 72 [NASA ADS] [CrossRef] [Google Scholar]
 Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298 [NASA ADS] [CrossRef] [Google Scholar]
 FaucherGiguère, C.A., & Kaspi, V. M. 2006, ApJ, 643, 332 [NASA ADS] [CrossRef] [Google Scholar]
 Fonseca, E., Andersen, B. C., Bhardwaj, M., et al. 2020, ApJ, 891, L6 [NASA ADS] [CrossRef] [Google Scholar]
 Geng, J. J., & Huang, Y. F. 2015, ApJ, 809, 24 [NASA ADS] [CrossRef] [Google Scholar]
 Gullón, M., Miralles, J. A., Viganò, D., & Pons, J. A. 2014, MNRAS, 443, 1891 [CrossRef] [Google Scholar]
 Hansen, B. M. S., & Phinney, E. S. 1997, MNRAS, 291, 569 [NASA ADS] [Google Scholar]
 Hobbs, G., Lyne, A. G., Kramer, M., Martin, C. E., & Jordan, C. 2004, MNRAS, 353, 1311 [NASA ADS] [CrossRef] [Google Scholar]
 Holman, M., Touma, J., & Tremaine, S. 1997, Nature, 386, 254 [NASA ADS] [CrossRef] [Google Scholar]
 James, C. W. 2019, MNRAS, 486, 5934 [NASA ADS] [CrossRef] [Google Scholar]
 Johnston, S., & Karastergiou, A. 2017, MNRAS, 467, 3493 [NASA ADS] [CrossRef] [Google Scholar]
 Johnston, S., Manchester, R. N., Lyne, A. G., et al. 1992, ApJ, 387, L37 [NASA ADS] [CrossRef] [Google Scholar]
 Jones, P. B. 2008, MNRAS, 386, 505 [NASA ADS] [CrossRef] [Google Scholar]
 Kaspi, V. M., Johnston, S., Bell, J. F., et al. 1994, ApJ, 423, L43 [NASA ADS] [CrossRef] [Google Scholar]
 Kaspi, V. M., Lyne, A. G., Manchester, R. N., et al. 2000, ApJ, 543, 321 [NASA ADS] [CrossRef] [Google Scholar]
 Katz, B., Dong, S., & Malhotra, R. 2011, Phys. Rev. Lett., 107, 181101 [NASA ADS] [CrossRef] [Google Scholar]
 Kotera, K., & Silk, J. 2016, ApJ, 823, L29 [NASA ADS] [CrossRef] [Google Scholar]
 Kotera, K., Mottez, F., Voisin, G., et al. 2016, A&A, 592, A52 [CrossRef] [EDP Sciences] [Google Scholar]
 Kroupa, P. 2008, Lect. Notes Phys., 181 [Google Scholar]
 Kroupa, P., & PetrGotzens, M. G. 2011, A&A, 529, A92 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Kruckow, M. U., Tauris, T. M., Langer, N., et al. 2018, MNRAS, 481, 1908 [NASA ADS] [CrossRef] [Google Scholar]
 Lamberts, A., GarrisonKimmel, S., Hopkins, P. F., et al. 2018, MNRAS, 480, 2704 [NASA ADS] [CrossRef] [Google Scholar]
 Li, G., Naoz, S., Kocsis, B., & Loeb, A. 2014, ApJ, 785, 116 [NASA ADS] [CrossRef] [Google Scholar]
 Lithwick, Y., & Naoz, S. 2011, ApJ, 742, 94 [NASA ADS] [CrossRef] [Google Scholar]
 Liu, B., Muñoz, D. J., & Lai, D. 2015, MNRAS, 447, 747 [NASA ADS] [CrossRef] [Google Scholar]
 Lorimer, D. R. 2008, Liv. Rev. Rel., 11, 8 [Google Scholar]
 Lu, C. X., & Naoz, S. 2019, MNRAS, 484, 1506 [Google Scholar]
 Lyne, A. G. 2005, in Binary Radio Pulsars, eds. F. A. Rasio, & I. H. Stairs, ASP Conf. Ser., 328, 37 [Google Scholar]
 Madsen, E. C., Stairs, I. H., Kramer, M., et al. 2012, MNRAS, 425, 2378 [NASA ADS] [CrossRef] [Google Scholar]
 Manchester, R. N., Lyne, A. G., Camilo, F., et al. 2000, in IAU Colloq. 177: Pulsar Astronomy  2000 and Beyond, eds. M. Kramer, N. Wex, R. Wielebinski, et al., ASP Conf. Ser., 202, 49 [Google Scholar]
 Menou, K., Perna, R., & Hernquist, L. 2001, ApJ, 559, 1032 [NASA ADS] [CrossRef] [Google Scholar]
 Mereghetti, S., Savchenko, V., Ferrigno, C., et al. 2020, ApJ, 898, L29 [CrossRef] [Google Scholar]
 Michaely, E., & Perets, H. B. 2014, ApJ, 794, 122 [NASA ADS] [CrossRef] [Google Scholar]
 Mishchenko, M. I., Travis, L. D., & Hovenier, J. W. 1999, J. Quant. Spectr. Rad. Transf., 63, 131 [CrossRef] [Google Scholar]
 Morbidelli, A., & Raymond, S. N. 2016, J. Geophys. Res.: Planets, 121, 1962 [NASA ADS] [CrossRef] [Google Scholar]
 Mottez, F., & Heyvaerts, J. 2020, A&A, 644, A145 [CrossRef] [EDP Sciences] [Google Scholar]
 Mottez, F., & Zarka, P. 2014, A&A, 569, A86 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mottez, F., Bonazzola, S., & Heyvaerts, J. 2013a, A&A, 555, A125 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mottez, F., Bonazzola, S., & Heyvaerts, J. 2013b, A&A, 555, A126 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Mottez, F., Zarka, P., & Voisin, G. 2020, A&A, 644, A145 [CrossRef] [EDP Sciences] [Google Scholar]
 MPCSAOIAU, 2019, Minor Planet Center Orbit Database [Google Scholar]
 Naoz, S. 2016, Ann. Rev. Astron. Astrophys., 54, 441 [NASA ADS] [CrossRef] [Google Scholar]
 Naoz, S., Farr, W. M., Lithwick, Y., Rasio, F. A., & Teyssandier, J. 2011, Nature, 473, 187 [NASA ADS] [CrossRef] [Google Scholar]
 Nelemans, G., Yungelson, L. R., & Portegies Zwart, S. F. 2001, A&A, 375, 890 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Peters, P. C. 1964, Phys. Rev., 136, B1224 [NASA ADS] [CrossRef] [Google Scholar]
 Petroff, E., Barr, E. D., Jameson, A., et al. 2016, PASA, 33, e045 [NASA ADS] [CrossRef] [Google Scholar]
 Petroff, E., Hessels, J. W. T., & Lorimer, D. R. 2019, A&ARv, 27, 4 [NASA ADS] [CrossRef] [Google Scholar]
 Pfahl, E., Podsiadlowski, P., & Rappaport, S. 2005, ApJ, 628, 343 [NASA ADS] [CrossRef] [Google Scholar]
 Platts, E., Weltman, A., Walters, A., et al. 2019, Phys. Rep., 821, 1 [NASA ADS] [CrossRef] [Google Scholar]
 Portegies Zwart, S. F., & Verbunt, F. 1996, A&A, 309, 179 [NASA ADS] [Google Scholar]
 Portegies Zwart, S. F., & Van den Heuvel, E. P. 1999, New Astron., 4, 355 [CrossRef] [Google Scholar]
 Portegies Zwart, S. F., & Yungelson, L. R. 1999, MNRAS, 309, 26 [NASA ADS] [CrossRef] [Google Scholar]
 Postnov, K. A., & Yungelson, L. R. 2014, Liv. Rev. Rel., 17, 3 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Saur, J., Neubauer, F. M., Connerney, J. E. P., Zarka, P., & Kivelson, M. G. 2004, Plasma Interaction of Io with its Plasma Torus (Fran Bagenal, University of Colorado, BoulderTimothy E. Dowling, University of Louisville, KentuckyWilliam B. McKinnon, Washington University, St Louis) [Google Scholar]
 Scholz, P. 2019, Early science from the CHIME/FRB Project [Google Scholar]
 Scholz, P., Spitler, L. G., Hessels, J. W. T., et al. 2016, ApJ, 833, 177 [NASA ADS] [CrossRef] [Google Scholar]
 Shannon, R., Cordes, J., Kramer, M., et al. 2008, A Search for Debris Disks around Variable Pulsars (Spitzer Proposal) [Google Scholar]
 Shannon, R. M., Cordes, J. M., Metcalfe, T. S., et al. 2013, ApJ, 766, 5 [NASA ADS] [CrossRef] [Google Scholar]
 Shao, Y., & Li, X.D. 2018, MNRAS, 477, L128 [CrossRef] [Google Scholar]
 Shappee, B. J., & Thompson, T. A. 2013, ApJ, 766, 64 [NASA ADS] [CrossRef] [Google Scholar]
 Sigurdsson, S., Richer, H. B., Hansen, B. M., Stairs, I. H., & Thorsett, S. E. 2003, Science, 301, 193 [NASA ADS] [CrossRef] [PubMed] [Google Scholar]
 Sipior, M. S., Zwart, S. P., & Nelemans, G. 2004, MNRAS, 354, L49 [NASA ADS] [CrossRef] [Google Scholar]
 Smallwood, J. L., Martin, R. G., & Zhang, B. 2019, MNRAS, 485, 1367 [CrossRef] [Google Scholar]
 Stephan, A. P., Naoz, S., & Zuckerman, B. 2017, ApJ, 844, L16 [CrossRef] [Google Scholar]
 Tauris, T. M., Kramer, M., Freire, P. C. C., et al. 2017, ApJ, 846, 170 [NASA ADS] [CrossRef] [Google Scholar]
 Toonen, S., Claeys, J. S. W., Mennekens, N., & Ruiter, A. J. 2014, A&A, 562, A14 [NASA ADS] [CrossRef] [EDP Sciences] [Google Scholar]
 Toonen, S., Perets, H. B., Igoshev, A. P., Michaely, E., & Zenati, Y. 2018, A&A, 619, A53 [CrossRef] [EDP Sciences] [Google Scholar]
 Wolszczan, A., & Frail, D. A. 1992, Nature, 355, 145 [NASA ADS] [CrossRef] [Google Scholar]
 Yu, Y.B., & Huang, Y.F. 2016, Res. Astron. Astrophys., 16, 75 [Google Scholar]
 Zhang, Y. G., et al. 2018, ApJ, 866, 149 [NASA ADS] [CrossRef] [Google Scholar]
Appendix A: Threebody dynamics
In this appendix, we shortly review the key ingredients to understand the threebody dynamics in the framework of the KozaiLidov mechanisms. We detail how to obtain the various KozaiLidov periods and timescales for the quadrupolar and octupolar regimes. The demonstration follows a standard approach, shown by Antognini (2015), Lithwick & Naoz (2011), Naoz (2016) and others.
We define the Delaunay variables in the framework of canonical angleaction variables
with . This set of variables preserves the canonical structure of the Hamiltonian description and allows to fully describe the threebody system.
A.1. The quadrupolar regime and the KozaiLidov oscillations
The quadrupolar term from Eq. (7) is a constant of motion since the energy is conserved at the quadrupole order. We can rewrite the quadrupolar term as follows
which gives a reduced Hamiltonian , also a constant of motion. Furthermore we have
from the AlKashi theorem applied to the angular momenta. In the test particle limit (where m_{1} → 0) Eq. (A.8) reduces to G_{tot} ≈ G_{2} + G_{1}cos(i_{tot}), and since G_{tot} and G_{2} are conserve quantities, it comes that j_{1}cos(i_{tot}) is therefore conserved. Usually the previous constant of motion is defined as
and called Kozai’s integral. Equation (A.9) represents the projection of the total orbital momentum along the zaxis: j_{z}, and allows to compute the transfer of inclination to eccentricities via
So the reduced Hamiltionian can be rewritten as
Finally the reduce Hamiltonian is a function of , hence with three degrees of freedom but since and Θ are constant of motion the system is fully integrable via the equations of motions.
From the canonical variable g_{1}, we can write the standard equation of motion
And since j_{1} is related to G_{1} via L_{1} (constant of motion), we have
Finally, cos(i_{tot}) can be solved thanks to Eq. (A.14) injected in Eq. (A.8), which provides a full set of integrable equations to describe the full dynamics of the threebody system at the quadrupole order.
Antognini (2015) has shown that thanks to the integrability of the quadrupole order, the exact KozaiLidov period can be derived.
over a full period. Hence from Eq. (A.14), Antognini (2015) shows that the exact period can be written as
where
a constant of motion found by Lidov in 1962, discriminating between libration regime (C_{KL} < 0) and rotation regime (C_{KL} > 0).
Finally, Antognini (2015) has shown that the exact period of Eq. (A.17) differs from a factor of a few from the timescale described in Eq. (12) (and obtained via t_{KL} ∼ L_{1}/(15C_{2})), in dynamical regimes far from any resonances (see Fig. 1 from Antognini 2015).
A.2. The octupolar regime and the Eccentric KozaiLidov Mechanism
Even though the threebody system is not integrable at the octupolar order, Antognini (2015) shows that nevertheless it is possible to derive the exact period of the Eccentric KozaiLidov mechanism.
In a first step Antognini (2015) follows the analysis of Katz et al. (2011), where the authors introduce the eccentricity vector
this vector is pointing towards the periapsis of the inner binary, and allows one to describe the motion of the periapsis with time. Since ℋ_{quad} is constant, from Eq. (A.18) it is possible to define another constant of motion
We note that at the octupolar order, Θ and C_{KL} are no longer constants of motion. However, it is possible to assume that Θ and C_{KL} remain approximately constant over timescales of single KL oscillations. This motivates the rescaling of the times for the analysis as τ = t/t_{KL, i = 90°}, hence averaging over KL cycles. Katz et al. (2011) show that the evolution of Ω_{e} and Θ are given by
where K(x) and E(x) are complete elliptic functions of the first kind with x(C_{KL}) = 3(1 − C_{KL})/(3 + 2C_{KL}).
Another constant of motion found by Katz et al. (2011) is
which connects the dynamics of C_{KL} with Ω_{e}, and where
From Eq. (A.20), we can compute the evolution of C_{KL} as
from which we can extract the period of the motion
Now, by combining Eqs. (A.26) with (A.25) and thanks to Eqs. (A.21) and (A.22), Antognini (2015) has shown that the exact EKM period can be obtained as
Finally, Antognini (2015) extracts the EKM timescale (see Eq. (13)) as a function of the correct ϵ dependency and shows that it matches the EKM period (see Fig. 5 from Antognini 2015).
Appendix B: General relativity effects
General relativity (GR) effects, such as the periapsis precession can prevent KozaiLidov oscillations by stopping the Kozai resonance (Holman et al. 1997). For the resonance to occur, the KozaiLidov timescale must be shorter than the GR precession timescale (Blaes et al. 2002)
where the GR precession timescale is given by
following Fabrycky & Tremaine (2007), Blaes et al. (2002). In the quadrupolar regime
for circular orbits and, m_{0} = 1.4 M_{⊙}, m_{1} = 10^{−11} M_{⊙} and m_{2} = 10 M_{⊙}. In the octupolar regime, combining Eq. (13), we found
for elliptical orbits e_{1} = e_{2} = 0.1 and, m_{0} = 1.4 M_{⊙}, m_{1} = 10^{−11} M_{⊙} and m_{2} = 10 M_{⊙}. For both regimes, the GR precession effects are subdominant.
In the quadrupolar regime, the GR precession effects can suppress high eccentricity excitation e_{1, max, KL} for the inner orbit. The ratio between the inner orbit GR precession timescale and the KozaiLidov timescale can be rewritten as (Naoz 2016; Liu et al. 2015)
where
The maximal eccentricity reachable e_{1, max, GR} taking into account GR effects satisfies the following equation
with . For ϵ_{GR} ≪ 1, this yields . Figure B.1 displays both maximal eccentricities computed in the classical and general relativistic frameworks. As demonstrated also in Liu et al. (2015), GR effects on the maximum eccentricity reached will be stronger for higher inclinations.
Fig. B.1. GR effects in the quadrupolar regime: maximal eccentricity reachable via KozaiLidov perturbations as a function of the inclination, for the classical computation (dotted line) and the general relativistic corrections (straight line). The GR corrections tends to reduce the maximal eccentricity reachable. 
Appendix C: Quadrupolar treatment
In this section, we focus on the quadrupolar regime, where we show that despite the dynamical differences between the two regimes, quadrupolar effects can also be efficient in driving asteroids close to the pulsar and produce FRBlike emissions on timescales compatible with the observed FRB rates. However unlike the octupolar regime, the maximal eccentricity reachable is bounded by the initial inclination of the asteroid (since no flip occurs, the eccentricity does not increase to i = 90°). This difference implies that not all asteroids triggering KozaiLidov oscillations are able to reach the Roche limit (closest position to the central pulsar, hence required to produce the strongest radio emissions). Therefore in the first part of this Appendix, we derive a criterion to discriminate between asteroids potentially able to produce radio emissions and those which are not. From this criterion, we then compute the fraction of asteroids inside a belt participating to the emissions. We also take into account GR corrections, due to precession effects, on the maximal eccentricity reachable. And finally, we present the results of the quadrupolar approach in this KozaiLidov induced FRB emissions.
C.1. Roche limit crossing criteria
Under the influence of KozaiLidov oscillations, the eccenctricity of the inner binary can reach values sufficiently high so that its periastron crosses the Roche limit. This limit represents the closest possible position for an object in orbit, beyond which it be disrupted by tidal forces. The periastron of the elliptical orbit of the inner binary is given by . If the periastron crosses the Roche limit (Eq. (3)), the following equation is verified:
where R_{1} and R_{2} are the inner binary object radius. KozaiLidov oscillations can drive the inner binary orbit down to the Roche limit if the following condition is fulfilled:
Namely, we require the maximal KozaiLidov eccentricity to be larger than the required eccentricity for the periapsis to cross the Roche limit. Any inner binary, with orbital parameters matching the above equation, will be disrupted by tidal forces on a secular timescale following the KozaiLidov timescales.
C.2. Fraction of asteroids crossing the Roche limit due to KL oscillations
Assuming that the initial distribution of a_{ast} in the belt follows a Normal distribution with mean ⟨a_{ast}⟩ and width σ_{a} = ε_{ast}⟨a_{ast}⟩, the mean distance between two consecutively falling asteroids can be estimated statistically as ⟨Δa_{ast}⟩≈σ_{a}/N_{ast, KL}, with N_{ast, KL} the number of asteroids meeting the KozaiLidov criterion. One can express N_{ast, KL} = f(i_{c}) N_{ast}, with N_{ast} the total number of asteroids in the belt and f(i_{c}) the fraction of asteroids meeting the KozaiLidov criterion.
The fraction f(i_{c}) of asteroids meeting the KozaiLidov criterion depends on the inclination i_{c} as
where is the normal distribution function of mean ⟨x⟩ and variance . The KozaiLidov maximum semimajor axis to reach the Roche limit reads
Figure C.1 presents the values of f(i_{c}) in the classical derivation (blue). However, we will see in the next paragraph that in our regime, General Relativity (GR) effects dominate and lead to lower f(i_{c}).
Fig. C.1. Fraction of asteroids reaching the Roche limit via the quadrupolar KozaiLidov effect in the classical calculation (blue) and in the general relativity case (orange), as a function of the outer body inclination i_{2}. Its mass is set to m_{1} = 10 M_{⊙}. The density of the asteroids are set to ρ_{ast} = 2 g cm^{−3} and their semimajor axes follow a Normal law with mean semimajor axis ⟨a_{ast}⟩ = 1 AU and standard deviation σ_{a} = 0.15 ⟨a_{ast}⟩. 
C.3. General Relativity corrections
As shown in Appendix B, the GR corrections tend to reduce the maximal eccentricity reachable and therefore translates to a lower fraction of asteroids capable of reaching the Roche limit.
The KozaiLidov oscillations in the GR regime can drive an asteroid to disruption inside the Roche limit if the following condition is fulfilled
This maximum semimajor axis replaces a_{ast, KL} in Eq. (C.3), leading to a reduction of f(i_{c}) by a factor ∼3, as can be seen in Fig. C.1. Numerically, including GR corrections, f(i_{c} = 45 ° ) ∼ 0.02, leading to N_{ast, KL} = f(i_{c} = 45 ° )(N_{ast}/1000)∼20.
In this calculation, we have neglected the tidal and rotation terms, which can also affect the maximum eccentricity reached by the body. These terms are negligible compared to the GR term in our model. We note that the quadrupole approximation leads to a good analytical estimate of the orbital evolution, even when the octupole effects are strong (Liu et al. 2015).
C.4. Results
Thanks to the framework detailed in Appendices C.2, B and C.1 we can compute the infall rates of asteroids in binary pulsar systems, in the quadrupolar regime of the KozaiLidov effect. The mean relative KozaiLidov time between two consecutive asteroid disruptions can then be estimated as
where we have assumed e_{c} = 0 for the numerical estimate. Here, we have used the parameter values of typical NSBH/NSMS systems. The value of ε_{ast} is chosen so as to fit the parameters of the Solar belt (see Sect. 5).
The factor N_{ast, KL} = f(i_{c})N_{ast} corresponds to the number of asteroids which experience the KozaiLidov effect. Higher i_{c} can boost the FRB rates estimated above by 1 − 2 orders of magnitude, due to a larger f(i_{c}). Larger N_{ast} and higher i_{c} would also shorten ⟨Δt_{KL}⟩, consequently increasing the event rates of Sect. 4.
Figure C.2 displays the evolution of the KozaiLidov time t_{KL} and relative delay Δt_{KL} as a function of the outer perturbing body semi major axis a_{c} (here a_{BH}). A clear distinction between close and wide systems can be made based on the typical delay, the first ones, present relative delay on dayscales while the other ones have relative delay of several tens of years.
Fig. C.2. Mean KozaiLidov time ⟨t_{KL}⟩ (Eq. (12)) and relative time delay ⟨Δt_{KL}⟩ (Eq. (C.6)) as a function semimajor axis a_{c} of the outer perturbing body semimajor axis. 
These analytical results are confirmed thanks to the simulations detailed in Sect. 5, and here used only in the quadrupolar regime. Figure C.3 shows the distribution of the relative time delays for asteroids falling onto the central neutron star, in the quadrupolar regime, for a current (green) and primordial Solarlike belt for companion inclinations i_{c} = 5° (orange) and i_{c} = 45° (blue). The central neutron star has mass M_{NS} = 1.4 M_{⊙} and the outer companion M_{c} = 10 M_{⊙}. The initial number of asteroids is set to N_{ast} = 10^{3}. We examine the case of a wide system with companion distance a_{c} = 10 AU and mean asteroid belt distance ⟨a_{ast}⟩ = 1 AU (left panel).
Fig. C.3. Same as Figs. 6 and 7, distribution of relative time differences Δt_{KL} of falling asteroids in the Roche lobe of the central compact object due to quadrupolar KozaiLidov oscillations, for the current Solar asteroid belt (green) and the primordial belt, for an inclination of the outer body hole plane i_{c} = 5° (orange) and i_{c} = 45° (blue), and initial asteroid number N_{ast} = 10^{3}. We consider a high mass M_{c} = 10 M_{⊙} wide system (left) with a_{c} = 100 AU and ⟨a_{ast}⟩ = 1 AU, and a close system (right) with a_{c} = 10 AU and ⟨a_{ast}⟩ = 1 AU. 
For the wide system, the infall rates span days to millions of years, with a maximum around ⟨Δt_{KL}⟩∼10s − 100s years, depending on the inclination i_{c}. For close systems, the rates are of order dayscales.
The comparison between the current Solar belt and the primordial belt shows that the lack of Kirkwood gaps induces a drastic increase of short timescales in the asteroid infall rate, and depending on the inclination, a factor of a few to an order of magnitude more events in total. Larger inclinations i_{c} lead to shorter timescales, and to higher event rates since the shifting timescales due to inclination i_{c} is dominant over the 1/f(i_{c}) effect. Systems with larger inclinations i_{c} and with higher rates (Δt_{KL} ∼ 10 yr) will thus dominate in the sky.
Furthermore, one can notice the tail distribution at large timescales for the Solar belt in the right panel of Fig. C.3. It results from the Kirkwood gaps, where groups of asteroids with lower inclinations can reach the Roche limit due to the closer position of the outer black hole 10 AU (rather than 100 AU in the left panel). However, their lower inclinations result in larger timescales.
Figure C.4 shows a consistent result with Fig. C.3, a low mass companion in a more close system induces short timescales. The reduction of the KozaiLidov effect due to the low mass companion is compensated with shorter distances in that system, ending up with a similar result than in the left panel of Fig. C.3 (high mass companion in close system). Finally the result obtained for the quadrupolar treatment are fully consistent with the octupolar approach, in the sense that the dichotomy between repeaters and nonrepeaters is explained thanks to the populatin of systems involved in the KozaiLidov mechanism: either close systems leading preferentially to repeater FRBs or wide systems leading preferentially to nonrepeater FRBs.
Fig. C.4. Same as Fig. C.3, but for a low mass M_{c} = 1 M_{⊙} close system a_{c} = 2 AU and ⟨a_{ast}⟩ = 0.2 AU The initial asteroid number is N_{ast} = 10^{3}. 
All Tables
Population characteristics (binary system Galactic birth rate and companion mass M_{c}) and orbital element distributions (eccentricity e_{c} and semimajor axis a_{c}) for neutron stars in binary systems: neutron starwhite dwarf (NSWD), double neutron star (DNS), and neutron starblack hole (NSBH).
All Figures
Fig. 1. Framework for the KozaiLidov perturbation calculations in our binary neutron star+ asteroid triple system. The neutron star is surrounded by the asteroid belt, and the binary companion orbits at a larger distance. All objects are represented by their distance to the neutron star (for instance a_{ast} and a_{c}) and their inclination (for instance i_{ast} and i_{c}) with respect to the invariant plane. 

In the text 
Fig. 2. Evolution of the octupolar efficiency parameter ϵ (Eq. (8)) as a function of the asteroid belt semimajor axis a_{ast} and the companion semimajor axis a_{c} (left), and the companion eccentricity e_{c} and ratio of asteroid belt semimajor axis a_{ast} and companion semimajor axis a_{c} (right). The solid black lines delimit the regions where the octupolar regime of the threebody dynamic is expected to be dominant ϵ = 0.1 and ϵ = 10^{−3}, and where a transition from octupolar regime to quadrupolar regime is expected to take place ϵ = 10^{−4}. 

In the text 
Fig. 3. Evolution of the EKM timescale (given by Eq. (13)) for various configurations of threebody systems, as a function of the semimajor axis, for a companion eccentricity e_{c} = 0.1 (left), and outer binary eccentricity and semimajor axis, for an asteroid belt located at a_{ast} = 1 AU (right), for a low mass companion M_{c} = 1 M_{⊙} corresponding to DNS, NSMS and NSWD systems (top), and a high mass companion M_{c} = 10 M_{⊙} corresponding to NSMS and NSBH systems (bottom). The shadowed region on the left panels represents the forbidden configurations where the outer binary is closer than the inner binary. On the right panels, these forbidden configuration lies on the lefthand side of the vertical dashed line. The gray boxes on the right panels represent the parameters spaces where the different systems considered here are expected to lie. The solid lines show the limit where the timescale is shorter than one year (leading to transient events). On the left panels, the gray thin horizontal dashed lines and the arrows indicate the parameterspace in which the different systems would lie (Table 1). 

In the text 
Fig. 4. Same as Fig. 3, but for the relative KozaiLidov time delay (given by Eq. (16)). The solid lines represent the limits where the relative delay equals 1 day (for sources producing dayrepeaters) and the dashed line is the limit where the relative delay equals 10 years (observational time beyond which sources cannot be observed as repeaters). 

In the text 
Fig. 5. Asteroid orbital parameter distributions inside the Solar asteroid belt (blue) and our reconstructed primordial model belt (orange). Top: asteroid semimajor axes distribution. Bottom: asteroid inclinations as a function of the semimajor axes. In both panels, the Kirkwood gaps are clearly visible in the Solar asteroid belt. 

In the text 
Fig. 6. Distribution of relative time differences Δt_{KL} of asteroids falling into the Roche lobe of the central compact object due to KozaiLidov oscillations (which can be directly interpreted as the FRB emission periods), for the current Solar asteroid belt (green) and the primordial belt, for an inclination of the outer companion plane i_{c} = 5° (orange), i_{c} = 10° (green) and i_{c} = 45° (blue), and initial asteroid number N_{ast} = 10^{2}. We consider a high mass M_{c} = 10 M_{⊙} wide system with a_{c} = 10 AU and ⟨a_{ast}⟩ = 1 AU. The number of asteroid infalls increase with the inclination of the system, as expected. Additionally, the relative time differences Δt_{KL} increase with lower inclinations, except for the Solar asteroid case. The present Solar asteroid belt has already undergone KozaiLidov effects during its lifetime, leading to a cleansing of its asteroids that is potentially sensitive to KozaiLidov oscillations, which explains its misleading behavior in this figure. 

In the text 
Fig. 7. Same as Fig. 6, distribution of relative time differences Δt_{KL} of asteroids falling into the Roche lobe of the central compact object due to KozaiLidov oscillations, for the different systems considered here. Left: low mass companions. Right: high mass companions. The initial number of asteroids is N_{ast} = 10^{2} and the inclinations of the companions is i_{c} = 30°. Specific parameters for each configuration can be found in Table 2. 

In the text 
Fig. B.1. GR effects in the quadrupolar regime: maximal eccentricity reachable via KozaiLidov perturbations as a function of the inclination, for the classical computation (dotted line) and the general relativistic corrections (straight line). The GR corrections tends to reduce the maximal eccentricity reachable. 

In the text 
Fig. C.1. Fraction of asteroids reaching the Roche limit via the quadrupolar KozaiLidov effect in the classical calculation (blue) and in the general relativity case (orange), as a function of the outer body inclination i_{2}. Its mass is set to m_{1} = 10 M_{⊙}. The density of the asteroids are set to ρ_{ast} = 2 g cm^{−3} and their semimajor axes follow a Normal law with mean semimajor axis ⟨a_{ast}⟩ = 1 AU and standard deviation σ_{a} = 0.15 ⟨a_{ast}⟩. 

In the text 
Fig. C.2. Mean KozaiLidov time ⟨t_{KL}⟩ (Eq. (12)) and relative time delay ⟨Δt_{KL}⟩ (Eq. (C.6)) as a function semimajor axis a_{c} of the outer perturbing body semimajor axis. 

In the text 
Fig. C.3. Same as Figs. 6 and 7, distribution of relative time differences Δt_{KL} of falling asteroids in the Roche lobe of the central compact object due to quadrupolar KozaiLidov oscillations, for the current Solar asteroid belt (green) and the primordial belt, for an inclination of the outer body hole plane i_{c} = 5° (orange) and i_{c} = 45° (blue), and initial asteroid number N_{ast} = 10^{3}. We consider a high mass M_{c} = 10 M_{⊙} wide system (left) with a_{c} = 100 AU and ⟨a_{ast}⟩ = 1 AU, and a close system (right) with a_{c} = 10 AU and ⟨a_{ast}⟩ = 1 AU. 

In the text 
Fig. C.4. Same as Fig. C.3, but for a low mass M_{c} = 1 M_{⊙} close system a_{c} = 2 AU and ⟨a_{ast}⟩ = 0.2 AU The initial asteroid number is N_{ast} = 10^{3}. 

In the text 
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.