Fast radio burst repeaters produced via Kozai-Lidov feeding of neutron stars in binary systems

Neutron stars are likely surrounded by gas, debris and asteroid belts. Kozai-Lidov 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), e.g., by asteroid-induced magnetic wake fields in the wind of the compact object. A few percent of binary neutron star systems in the Universe, such as neutron star-main sequence star, neutron star-white dwarf, double neutron star, and neutron star-black hole systems, can account for the observed non-repeating FRB rates. More remarkably, we find that wide and close companion orbits lead to non-repeating and repeating sources, respectively, and compute a ratio between repeating and non-repeating 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 to a few 10s of years, as their asteroid belts become depleted; 2) some non-repeaters could occasionally repeat, if we hit the short period tail of the FRB period distribution; 3) series of sub-Jansky level short radio bursts could be observed as electromagnetic counterparts of the mergers of binary neutron star systems.


Introduction
The origin of Fast Radio Bursts (FRBs), these brief, coherent and numerous radio pulses, has not yet been identified.Today, radio-astronomy 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 sub-Jansky up to more than 400 Jy, with steep energy spectra (James 2019).Consequently, the isotropic energy equivalent of a 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 appears to repeat, i.e. 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 non repeaters.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, Amiri et al. 2020).A large fraction of the repeate-mail: decoene@iap.fring FRBs have been discovered by CHIME, operating at around 400 MHz (CHIME/FRB Collaboration et al. 2019a; CHIME/FRB Collaboration 2019; Sch 2019; CHIME/FRB Collaboration et al. 2019b;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 (necessarily limited in 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, neither is there an accepted explanation for the two observed populations of repeaters and non-repeaters.A vast number of emission models exists, from exotic alien signals to cosmic strings, and can be found in Platts et al. (2018).The recent detection of two intense radio bursts, coincident with X-ray bursts and localised at the position of SGR1935+2154, points towards the magnetar hypothesis as a source of FRBs (Mereghetti et al. 2020;The CHIME/FRB Collaboration et al. 2020).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, beside 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 non-repeating 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 et al. 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 non-repeater FRBs.
Such scenarios require however both a large number of progenitors, and an efficient infall mechanism into the neutron-star Roche lobe.The Kozai-Lidov gravitational effects applied to the numerous binary neutron star systems naturally provide such a framework.
In the following, we study the effect of the Kozai-Lidov 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 Kozai-Lidov 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 Kozai-Lidov 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 comet-like materials (Stephan et al. 2017).
The first discovery of earth-mass exoplanets was indeed around a millisecond pulsar (Wolszczan & Frail 1992a).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 et al. 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 will 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 Kozai-Lidov time-scales for our binary system (Section 3) and discuss the implications in terms of FRB rates, taking into account the binary population rates (Section 4).We simulate the Kozai-Lidov effect on a mock solar-like asteroid belt in Section 5. Finally, we discuss the broader applications of this calculation in Section 6.

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 et al. 2014;Yu & Huang 2016;Mottez et al. 2013a,b).No asteroid belt has yet been observed at distances larger than 1 A.U., 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 A.U. 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 summarised in three steps: first the relativistic and magnetised 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 ∆f is the spectral bandwidth of the emission, 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 radio-waves 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 wind-like 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, magnetar-like 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 (The CHIME/FRB Collaboration et al. 2020), also coincident with X-ray 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.

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 pre-supernova mass, the asymmetry of the explosion, a possible impulsive "kick" velocity impinged on the neutron star at birth, etc.: a parameter-space explored with sophisticated numerical simulations (e.g., Lorimer 2008;Toonen et al. 2014 and references therein).
We focus here on neutron star-white dwarf (NSWD), neutron star-main sequence star (NSMS), neutron starneutron star (DNS) and neutron star-black 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 (i.e.non-recycled 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 non-detection 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 modelled 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 (non-recycled) 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 will 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 will mostly concentrate on the binaries mentioned above in this paper.However, we will 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 semi-major axis a c 1 A.U. are usually associated with recycled pulsars.
In our model, the FRB emission will happen in the first 10 4 yrs of the birth of the pulsar, and for close binaries, even within the first 10 yrs (see Section 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 (Faucher-Giguère & 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 Equation ( 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.

Asteroid size
The radio emission depends crucially on the radius R ast and orbital distance a ast of the asteroid.One can infer from Equation (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 power-law (MPC-SAO-IAU 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.

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 A.U. 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 so-called plunge factor is taken into account (Ali-Haïmoud 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 Jansky-level bursts.
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 non-sphericity, 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); Mottez et al. (2020) can work only if asteroids actually fall close enough to the central object.We propose here that this can happen via the Kozai-Lidov effect.We set our fiducial asteroid belt distance to a ast = 1 A.U. 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 (Amiri et al. 2020).Indeed for favorable configurations, the alignment between the asteroid periodical motion and the observer's line-of-sight 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.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 will form after the supernova phases, in that case the system will already be 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.

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 time-scale for the asteroid eccentricity to shift from a mJy ∼ 0.1 AU to the Roche Limit in the emission zone (due to Kozai-Lidov effects) would be a fraction of the Kozai-Lidov 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 Kozai-Lidov 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 P ast = a 2 ast GM NS /4π, 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 Kozai-Lidov timescale, given by Eq. ( 12) (see Section 3), considering an outer body of mass 10 M with a semi-major axis of 10 A.U., 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 ω ∼ 10 −4 rad/s.Consequently, the emission beam wanders over an area proportional to the time of observation t obs and the Keplerian orbital period of the asteroid n ast = GM NS /a 3 ast ∼ 1.6 days(M NS /1.4 M ) a ast /10 −2 U.A. , 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 /1.4M ) 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 sub-emission components over short time-scales.Such events could explain the observations of FRB 121102, from which ∼ 90 bursts were detected during a five hour period (half falling within 30 minutes, Zhang et al. 2018).

Kozai-Lidov mechanisms
In the framework of asteroids orbiting a central pulsar and surrounded by an outer massive body (see Figure 1 for a sketch), we expect modifications of the orbits to occur through the exchanges of orbital momentum between the two two-body systems: (1) pulsar-asteroid (the inner binary), and (2) (pulsar-asteroid)-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.
As can be seen in Figure 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.

Secular perturbations in three-body systems
The motion of the outer body, also referred to as the perturbing body, induces gravitational perturbations which happen on secular timescales, i.e., 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 semi-major 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 "mass-less" (m → 0), only the motion of this "mass-less" body is affected by the secular dynamics.For large mass ratios within the inner binary, the inner binary orbit can flip from a pro-grade motion to a retro-grade motion by rolling over its semi-major axis.During one of these flips, the orbit passes trough an inclination of 90°which leads to a large eccentricity excitation.
The three-body dynamics is usually decomposed into the dynamics of two two-body systems, plus a perturbation effect between these two-body systems.In terms of Hamiltonian, one can write where H is the total Hamiltonian of the three-body system, G is the gravitational constant, m refers to the mass, a to the semi-major axis, and the subscripts 1, 2, 3 to one the body or one of the two two-body systems (1 or 2).Finally, H pert represent the perturbation term between the two two-body 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 j 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 is given by (8) Depending on the configuration of the three-body system, the value of indicates which order dominates the dynamic (either quadrupolar or octupolar).Furthermore stable systems are expected for values of epsilon ∼ 0.1 or if the eccentricity is null a 1 /a 2 ∼ 0.1.Figure 2 shows the evolution of the parameter depending on the inner binary (pulsar-asteroid 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.
When the outer body has a circular orbit, the dynamics is led by the quadrupolar term and results in the so-called classical Kozai-Lidov mechanism.In this mechanism, periodical exchanges of orbital momentum between the two two-body 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 z-component 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 z-component of the total orbital momentum of the three-body 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 Kozai-Lidov effect.Therefore one can obtain M ast < l a t e x i t s h a 1 _ b a s e 6 4 = " r s M 4 G 0 9 j 0 N 6 L U h E p M e s g j i 6 < l a t e x i t s h a 1 _ b a s e 6 4 = " x q F q 0 I c p d K h T 3 d / M a 3 j L s t J p p X H e n Y 9 5 a 8 H J Z w 7 R H z i f P z P f l j 4 = < / l a t e x i t > asteroid belt < l a t e x i t s h a 1 _ b a s e 6 4 = " 2 i 5 l f l 9 t t 2 L H + K 1 < l a t e x i t s h a 1 _ b a s e 6 4 = " j g z 4 < l a t e x i t s h a 1 _ b a s e 6 4 = " Y D O 7 2 l + z q e a H n r r t z c u H l 8 9 D (1) < l a t e x i t s h a 1 _ b a s e 6 4 = " Y D O 7 2 l + z q e a H n r r t z c u H l 8 9 D  (1) < l a t e x i t s h a 1 _ b a s e 6 4 = " k y < l a t e x i t s h a 1 _ b a s e 6 4 = " k y < l a t e x i t s h a 1 _ b a s e 6 4 = " k y  8) as a function of the asteroid belt semi-major axis aast and the companion semi-major axis ac (left), and the companion eccentricity ec and ratio of asteroid belt semi-major axis aast and companion semi-major axis ac (right).The solid black lines delimit the regions where the octupolar regime of the three-body dynamic is expected to be dominant = 0.1 and = 10 −3 , and where a transition from octupolar regime to quadrupolar regime is expected to take place = 10 −4 .
From the same argument, it is possible to derive the minimal initial inclination required to trigger classical Kozai-Lidov 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 non-negligible.In particular in the regime where 10 −3 0.1, the octupolar term is dominant.In this regime, the Kozai-Lidov effects are called the Eccentric Kozai-Lidov mechanism (EKM), where the classical Kozai-Lidov oscillations of the quadrupolar regime are modulated by a rotation of the orbits of the inner binary around its semi-major axis.This rotation leads to an increase of the inclination of upto i 1 = 90°beyond which the orbit flips from a pro-grade motion to a retro-grade motion.During the whole process, classical Kozai-Lidov 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 pa-rameter can be used to discriminate between the different dynamical regimes of the three-body system.The EKM is characterised by longer timescales than the classical Kozai-Lidov effect, since it can be seen as the superposition of several classical Kozai-Lidov oscillations, but it leads to extremely high eccentricities.The intensity of the oscillations in the EKM depends on the value of and so on the dynamical regime of the three-body system.The EKM has been found to be possible for at least two distinct regimes: (i) Low eccentricity-High inclination, and (ii) High eccentricity-Low inclination.The first regime corresponds to the classical criteria on the initial inclination to trigger Kozai-Lidov oscillations (40°< i init,tot < 140°), and more interestingly, the second regime corresponds to orbital configurations where the system can be almost co-planar but still trigger EKM thanks to the high eccentricity of the inner binary.
In the specific framework of Kozai-Lidov effects, the three-body dynamics can be described with three main regimes: the quadrupolar regime when the outer body has a circular trajectory, featured by classical Kozai-Lidov oscillations; the octupolar regime when 10 −3 0.1, enabling a richer dynamics with Eccentric Kozai-Lidov mechanisms and orbital flips; and finally, a combination of the previous two regimes where 10 −3 , which depends on the specific configuration of the three-body system and is difficult to analyze in a general framework.In this study we will consider the octupolar regime down to = 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.

Kozai-Lidov 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 Kozai-Lidov period and study its behaviour across the parameter space of the three-body dynamics.In particular, it is shown that this exact period only varies within a factor of a few from the standard (and well-known) Kozai-Lidov 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 Kozai-Lidov timescale in the EKM regime.This new time scale is given by ∼ 4.8 yr −1/2 a ast 0.5 A.U.
where t KL,i=90°r epresents the classical Kozai-Lidov timescale and we suppose that inclinations up to 90°can be reached thanks to the orbital flip mechanisms described earlier in section 3.1.Furthermore the time-scale of Equation (13) describes a full EKM cycle, with two flips: from pro-grade to retro-grade and back again.
The numerical values given in Equation ( 14) correspond to a mildly close NSMS, NSWD or DNS case, with a c the semi-major axis of the orbiting companion and M c its mass.The estimate assumes a null eccentricity e c = 0.
Figure 3 presents the EKM time-scales 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.

Kozai-Lidov 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 Kozai-Lidov time-scales between close-by objects.The relative time delays between two small objects seperated by a distance ∆a 1 is given in the quadrupolar regime by and in the octupolar regime by Two objects separated by a distance ∆a 1 orbiting a central more massive object and perturbed by an outer body, undergo Eccentric Kozai-Lidov effects with a time-scale difference given by Equation ( 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 ∆a ast ≈ σ a /N ast,KL , with N ast,KL the number of asteroids undergoing Kozai-Lidov effects.In the octupolar regime, when the outer body has a non circular orbit, most asteroids undergo Kozai-Lidov effects and reach high eccentricities.Hence one can write N ast,KL ∼ eff N ast , with N ast the total number of asteroids in the belt and eff 0.2 (see Section 5.3).The fraction of asteroids meeting the Kozai-Lidov criterion in the quadrupolar regime is calculated in Appendix C.2.One can then express the mean relative Eccentric Kozai-Lidov 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 three-body system.The time delay trend follows the EKM timescales as depicted in Figure 3.  13) for various configurations of three-body systems, as a function of the semi-major axis, for a companion eccentricity ec = 0.1 (left), and outer binary eccentricity and semi-major axis, for an asteroid belt located at aast = 1 A.U. (right), for a low mass companion Mc = 1 M corresponding to DNS, NSMS and NSWD systems (top), and a high mass companion Mc = 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 left-hand 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 parameter-space in which the different systems would lie (Table 1).

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.

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 & Petr-Gotzens 2011), with orbital separation a c few A.U., 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 A.U., with mild to high eccentricities e c ∼ 0.6 − 0.9 (e.g., PSR1259-63 and its 10 M -mass Bestar companion, Johnston et al. 1992 16).The solid lines represent the limits where the relative delay equals 1 day (for sources producing day-repeaters) and the dashed line is the limit where the relative delay equals 10 years (observational time beyond which sources cannot be observed as repeaters).2018) Kroupa (2008) Belczyński & Bulik (1999) Table 1.Population characteristics (binary system Galactic birth rate νc and companion mass Mc) and orbital element distributions (eccentricity ec and semi-major axis ac) for neutron stars in binary systems: neutron star-white dwarf (NSWD), double neutron star (DNS), and neutron star-black hole (NSBH).The FRB rate densities ṅFRB are estimated using Eqs.(20-21), see Section 4.3.mass companion, Lyne 2005, PSR J0045-7319 and its 4 Mmass 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).
The orbits of close binary neutron star systems that have low-mass companions, such as low-mass white dwarfs (M c 0.7 M ) tend to be circular: e c ∼ 10 −5 − 10 −2 .Close systems with high-mass 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.
We mentioned in Section 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 B1620-26, with a wide inferred semi-major axis of a c ∼ 23 A.U. and moderate orbital eccentricity (Sigurdsson et al. 2003).Three planetary bodies were found orbiting at ∼ A.U. distances around pulsar B1257+12 Wolszczan & Frail (1992b).In both systems, the pulsar is recycled.More formation studies and observational data would be needed to derive population characteristics for neutron star-planet systems.We will focus here on the other binaries mentioned above, that are more documented.

The octupolar regime dominates over most of the binary parameter space
Figure 2 shows the values of the octupolar efficiency term , 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 = 0.1 − 10 −4 , dominated by octupolar dynamics.Therefore, we will 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).For systems approaching = 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, the quadrupolar regime leads to a less efficient Kozai-Lidov 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 time-scales of the octupolar regime and quadrupolar regime (see Equation 16) only differ by a factor 128 √ 10/(15π √ ) ∼ 8.6 −1/2 .Therefore in principle, systems in the octupolar regime should be characterised with longer time-scales than in the quadrupolar regime.
From Figure 3, we can see that for almost all systems, the absolute time-scale of Eccentric Kozai-Lidov oscillations is longer than one year, except for very close or highly eccentric systems.Regarding the relative time delays of Eccentric Kozai-Lidov oscillations, Figure 4 shows typical time-scales 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 time-cales.This flexibility makes this process a good candidate to explain the diversity of observed FRB rates.

Contributions of wide and close populations to FRBs and FRB repeaters
It appears from Equation ( 16) and Figs.3-4 that the main parameter governing the infall rate via Kozai-Lidov 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 A.U. to 100 A.U.. From the previous Section, systems can be split into three populations: wide systems with a c 10 A.U., mildly close systems with a c ∼ 0.3 − few A.U., and close systems with a c 0.3 A.U..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 Section 2.1 and Equation 1).
The time-scale over which the Kozai-Lidov effects can take place, hence the lifetime of the system as an FRB source, is highly dependent on a c (Equation 13).While wide binaries have t EKL 10 yrs (or t KL 10 yrs in quadrupolar regime) and can be viewed as long-lived FRB sources, close and mildly close binaries have t EKL < few 10s of yrs and should be considered as short-lived FRB transients.For close binaries with a c 1 A.U., t EKL 1 yr, leading to a "single-shot" transient, that will not be observed as repeating over a long time-scale.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 A.U., ∆t KL 10 yrs, leading to non-repeating sources.For close and mildly close binaries with a c few A.U., ∆t KL 10 yrs, sources could be observed as repeating, with various emission frequencies.Close systems with a c 1 A.U. 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 gravitational-wave merger time-scale 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 time-scale, down to ∼ 10 4 years (Peters 1964).In any case, these time-scales are longer than t EKL and do not need to be considered here.Therefore, in this scenario, mildly close binaries would produce day/month-repeaters and wide binaries non-repeaters.It is interesting to notice that in the current analysis (CHIME/FRB Collaboration et al. 2019b;Fonseca et al. 2020), day to few day periods seem to be favoured 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/month-repeater FRBs during t KL < 10 yrs.Their FRB rate density is hence directly linked to their birth rates as documented in Table 1.We calculate roughly the total density rates ṅc 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 : ṅc = νc n gal .The rate density of dayrepeater FRB sources then reads where rep < 1 is a source efficiency factor, and 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/ ∆t 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 wide n c , with wide the fraction of wide systems among a population.It yields a rate density of apparently non-repeating FRB events of with wide = 0.3 (Portegies Zwart & Verbunt 1996; Portegies Zwart & Van den Heuvel 1999; Kruckow et al. 2018), eff ∼ 0.2 the Kozai-Lidov efficiency factor discussed in Section 5.3, and nrep < 1 a similar source efficiency factor as in Equation ( 20).These calculations assume that these binaries undergo a flat source emissivity evolution, out to redshift z ∼ 1 (Postnov & Yungelson 2014).For a star-formation 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 non-repeating (nrep.)sources respectively.For DNS and NSWD, we have assumed that a fraction mild−close = 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 mild−close = 1.For NSWD, we assumed that wide = 1/3, and used the rates provided in the literature for wide NSMS and DNS rates.For NSBH, we assumed wide = 0.1.These fractions are discussed in Section 4.1.
The rate densities estimated from Equation ( 21) can be directly compared to the cosmological FRB rate densities inferred from observations, of order ṅFRB,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 ṅFRB,obs ṅFRB,nrep .The inferred source efficiency can thus be of order nrep 10 % (NSWD: 2.0%, NSMS: 8.7%, DNS: 7.4%).This number leaves room for binary systems which do not fulfil the criteria to undergo Kozai-Lidov mechanisms: e.g., 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: ṅFRB,rep / ṅFRB,nrep ∼ 5.3% rep / non−rep for NSMS (for NSMS: 1.4% and for DNS: 1.4%).For a same population, one can assume that rep = non−rep .However, the formation of asteroid belts might differ for close and wide systems.
It is possible that all the neutron-star binaries mentioned above contribute to the FRB rates.If one assumes that their efficiencies nrep and rep are equal, the total rate density of apparently non-repeating FRBs would be of ṅFRB,nrep,all ∼ 15 × 10 4 nrep Gpc −3 yr −1 , with a source efficiency that can be as low as nrep = 1.4%.The repeater rate density would be of ṅFRB,nrep,all ∼ 3.1 × 10 3 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.

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 Kozai-Lidov 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.

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 (MPC-SAO-IAU 2019).A total number of 792041 asteroids of the Solar belt are inventoried in this database.
Numerous asteroids sensitive to the Kozai-Lidov 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 Figure 5).The synthetic belts follow a gaussian distribution fitting the general trend of the current Solar belt.We use for the semi-major axis a standard deviation of σ a = 0.15 a ast , with the mean semi-major 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 Kozai-Lidov effects.However Figure 5 suggests that a fitting model similar to what is done for the distribution of semi-major 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 MPC-SAO-IAU (2019) (see Section 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. to Kozai-Lidov 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 ic = 5°(orange), ic = 10°(green) and ic = 45°( blue), and initial asteroid number Nast = 10 2 .We consider a high mass Mc = 10 M wide system with ac = 10 A.U. and aast = 1 A.U..The number of asteroid infalls increase with the inclination of the system, as expected.Additionally, the relative time differences ∆tKL increase with lower inclinations, except for the Solar asteroid case.The present Solar asteroid belt has already undergone Kozai-Lidov effects during its lifetime, leading to a cleansing of its asteroids that is potentially sensitive to Kozai-Lidov oscillations, which explains its misleading behaviour in this figure.Table 2. Configuration parameters for the Monte-Carlo simulations presented in Section 5 and results in Figure 7, for low/high mass systems and close/wide companion orbits.

Simulations set-up
Following the asteroid distribution computed in the previous section, we randomly draw a set of asteroid parameters (size R ast , semi-major axis a ast , inclination i ast ).We select the objects that meet the following 3 criteria  6, distribution of relative time differences ∆tKL of asteroids falling into the Roche lobe of the central compact object due to Kozai-Lidov oscillations, for the different systems considered here.Left: low mass companions.Right: high mass companions.The initial number of asteroids is Nast = 10 2 and the inclinations of the companions is ic = 30°.Specific parameters for each configuration can be found in Table 2. 3. can reach the Roche limit under the Kozai-Lidov effect (Equation 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 Kozai-Lidov 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 Monte-Carlo drawing, we average our results over 10 4 simulations.

Asteroid infall rates for a Solar-like belt
Figure 6 shows the distribution of the relative time delays ∆t KL for asteroids falling onto the central neutron star, for a current (green) and primordial Solar-like 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 Figure 6 the effect of the companion inclination i c on the relative timescales and efficiency of the Kozai-Lidov effect, in the case of a wide system with companion distance a c = 10 A.U. and mean asteroid belt distance a ast = 1 A.U..For this wide system, the infall rates span from days to thousands of years, with a maximum around ∆t KL ∼ 10 − 100 years, depending on the inclination i 2 .More interestingly the efficiency of the Kozai-Lidov 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 Kozai-Lidov process, since more asteroids will meet the Kozai-Lidov 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 time-scales 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 Figure 6, the distribution of the relative time delays ∆t 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 Kozai-Lidov time-scale 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 Kozai-Lidov mechanisms.

Connection with FRB observations
The results of the simulations detailed in Section 5.3 show a high consistency with the analytical estimates computed in Sections 3 and 4.3.These simulations demonstrate that the application of the Kozai-Lidov framework introduced in Section 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 Section4.3.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 non-repeating FRBs.
The efficiency of the Kozai-Lidov process is illustrated on Figure 7, where the number of asteroid infalls compared to the total number of asteroid simulated (N ast = 100 in Figure 7), corresponds to the efficiency of the Kozai-Lidov mechanism in driving asteroids down to the Roche limit (the factor eff introduced in Section 3.3, see also Section 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 Kozai-Lidov process in pulsar binary systems is efficient in driving asteroids down to the Roche limit from a Solar-like asteroid belt in our model.
Another interesting result coming only from the simulations concerns the distribution tails displayed in Figure 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 month-time-scales.These bursts would correspond to the left-hand tail of distributions such as the one shown in Figure 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 sub-structure in the signal, with fainter pulses arriving at shorter intervals (Zhang et al. 2018).These could be explained by the fragmentation of asteroid during the disruption in the Roche lobe, as mentioned in Section 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 sub-groups of asteroids close-by and with similar orbital paramaters, therefore leading to similar Kozai-Lidov time delays and so similar infall rates.
The close systems presented in Figure 7 illustrate the possibility of having a population of short-lived repeaters, with day-scale periods.These sources will appear less numerous than the wide systems due to their short active timescale (see Figure 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 Figure 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.

Conclusion, 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 et al. 2016;Smallwood et al. 2019).The infall of asteroids from standard belts onto the central compact object can be triggered by Kozai-Lidov 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 semi-major axis a c ∼ 0.3 − few A.U.) produce day/month scale repeaters that live < 10 yrs, while wide systems (a c ∼ few − 10s A.U.) are steady sources, which will be observed as non-repeating.
We find that a comfortable fraction of a few percent (< 10%) of these binary systems in the Universe can account for the observed non-repeating FRB rates.More remarkably, our wide/close orbit dichotomy model predicts a ratio between repeating and non-repeating sources of a few percent, which is in good agreement with the observations.
Close systems with a c 1 A.U. 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.Sub-Jansky 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 Section 5 numerically validate the analytical conclusions drawn in section 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 time-scale tail (or repetition tail) even for systems labeled as non-repeaters.
Three major predictions can be made from our scenario, which can be tested in the coming years: 1. Most repeaters should stop repeating after t EKL < few 10s of years, as their asteroid belts becomes depleted.2. Some non-repeaters could occasionally repeat, if we hit the short ∆t EKL tail of the FRB period distribution.3. Series of sub-Jansky 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 magnetised wind.In particular, pulsar systems with planets could contribute to this scenario.
The recent observation of two intense radio bursts in coincidence with X-ray flares (The CHIME/FRB Collaboration et al. 2020;Mereghetti et al. 2020), expected to originate from the magnetar SGR1935+2154, has shown some similarities with FRB emissions.This observation, if attributed to a FRB-like 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 Kozai-Lidov oscillations can result in the observation of a double radio burst: (i) the observation of two consecutive (and close-by) 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 line-of-sight 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 line-of-sight of the observer.However the production of the coincident X-ray 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 Ṁast = ast M ast /t fall , where ast ∼ 0.1 is the fraction of asteroid material accreted and t fall = 2d Roche /GM NS ∼ 1.5 h the infall time from the Roche limit down to the neutron star.The Eddington luminosity is hence given by L edd,ast = edd Ṁast c 2 ∼ 10 36 erg.s −1 , where 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 Stefan-Boltzman 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, e.g.following similar processes as suggested by Beloborodov (2013).12) and relative time delay ∆tKL (Equation C.6) as a function semi-major axis ac of the outer perturbing body semi-major axis.
These analytical results are confirmed thanks to the simulations detailed in Section 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 Solar-like 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 A.U. and mean asteroid belt distance a ast = 1 A.U. (left panel).
For the wide system, the infall rates span days to millions of years, with a maximum around ∆t KL ∼ 10s − 100s years, depending on the inclination i c .For close systems, the rates are of order day-scales.
The comparison between the current Solar belt and the primordial belt shows that the lack of Kirkwood gaps induces a drastic increase of short time-scales 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 time-scales, and to higher event rates since the shifting time-scales due to inclination i c is dominant over the 1/f (i c ) effect.Systems with larger inclinations i c and with higher rates (∆t KL ∼ 10 yr) will thus dominate in the sky.
Furthermore, one can notice the tail distribution at large time-scales for the Solar belt in the right panel of Figure 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 time-scales.in the sense that the dichotomy between repeaters and nonrepeaters is explained thanks to the populatin of systems involved in the Kozai-Lidov mechanism: either close systems leading preferentially to repeater FRBs or wide systems leading preferentially to non-repeater FRBs.
X e P O 2 9 e O / e x 7 y 1 4 O U z h / A H 3 u c P h 0 a Q O g = = < / l a t e x i t > M c 7 p a 9 u P 2 Z p h N I w Q b X u e G 5 i / I w q w 5 n A S b G b a k w o G 9 E B d i y V N E L t Z 7 N T J + T U K n 0 S x s q W N G S m / p 7 I a K T 1 O A p s Z 0 T N U C 9 6 U / E / r 5 O a 8 N r P u E x S g 5 L N F 4 W p I C Y m 0 7 9 J n y t k R o w t o U x x e y t h Q 6 o o M z a d o g 3 B W 3 x 5 m T T P q 9 5 l 1 b 2 / K N d u 8 j g K c A w n U A E P r q A G d 1 C H B j A Y w D O 8 w p s j n B f n 3 f m Y t 6 4 4 + c w R / I H z + Q N B D Y 0 e < / l a t e x i t > 7 p a 9 u P 2 Z p h N I w Q b X u e G 5 i / I w q w 5 n A S b G b a k w o G 9 E B d i y V N E L t Z 7 N T J + T U K n 0 S x s q W N G S m / p 7 I a K T 1 O A p s Z 0 T N U C 9 6 U / E / r 5 O a 8 N r P u E x S g 5 L N F 4 W p I C Y m 0 7 9 J n y t k R o w t o U x x e y t h Q 6 o o M z a d o g 3 B W 3 x 5 m T T P q 9 5 l 1 b 2 / K N d u 8 j g K c A w n U A E P r q A G d 1 C H B j A Y w D O 8 w p s j n B f n 3 f m Y t 6 4 4 + c w R / I H z + Q N B D Y 0 e < / l a t e x i t > (1) < l a t e x i t s h a 1 _ b a s e 6 4 = " Y D O 7 2 l + z q e a H n r r t z c u H l 8 9 D V K 8 = " > A A A B 6 n i c b V B N S 8 N A E J 3 4 W e t X 1 a O X x S L U S 0 l E 0 W P R i 8 e K 9 g P a U D b b S b t 0 s w m 7 7 p a 9 u P 2 Z p h N I w Q b X u e G 5 i / I w q w 5 n A S b G b a k w o G 9 E B d i y V N E L t Z 7 N T J + T U K n 0 S x s q W N G S m / p 7 I a K T 1 O A p s Z 0 T N U C 9 6 U / E / r 5 O a 8 N r P u E x S g 5 L N F 4 W p I C Y m 0 7 9 J n y t k R o w t o U x x e y t h Q 6 o o M z a d o g 3 B W 3 x 5 m T T P q 9 5 l 1 b 2 / K N d u 8 j g K c A w n U A E P r q A G d 1 C H B j A Y w D O 8 w p s j n B f n 3 f m Y t 6 4 4 + c w R / I H z + Q N B D Y 0 e < / l a t e x i t >

Fig. 1 .Fig. 2 .
Fig.1.Framework for the Kozai-Lidov 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 aast and ac) and their inclination (for instance iast and ic) with respect to the invariant plane.

Fig. 3 .
Fig.3.Evolution of the EKM timescale (given by Equation13) for various configurations of three-body systems, as a function of the semi-major axis, for a companion eccentricity ec = 0.1 (left), and outer binary eccentricity and semi-major axis, for an asteroid belt located at aast = 1 A.U. (right), for a low mass companion Mc = 1 M corresponding to DNS, NSMS and NSWD systems (top), and a high mass companion Mc = 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 left-hand 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 parameter-space in which the different systems would lie (Table1).

Fig. 4 .
Fig. 4. Same as Figure 3 but for the relative Kozai-Lidov time delay (given by Equation16).The solid lines represent the limits where the relative delay equals 1 day (for sources producing day-repeaters) and the dashed line is the limit where the relative delay equals 10 years (observational time beyond which sources cannot be observed as repeaters).

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

Fig. 6 .
Fig.6.Distribution of relative time differences ∆tKL of asteroids falling into the Roche lobe of the central compact object due to Kozai-Lidov 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 ic = 5°(orange), ic = 10°(green) and ic = 45°( blue), and initial asteroid number Nast = 10 2 .We consider a high mass Mc = 10 M wide system with ac = 10 A.U. and aast = 1 A.U..The number of asteroid infalls increase with the inclination of the system, as expected.Additionally, the relative time differences ∆tKL increase with lower inclinations, except for the Solar asteroid case.The present Solar asteroid belt has already undergone Kozai-Lidov effects during its lifetime, leading to a cleansing of its asteroids that is potentially sensitive to Kozai-Lidov oscillations, which explains its misleading behaviour in this figure.

Fig. 7 .
Fig.7.Same as Figure6, distribution of relative time differences ∆tKL of asteroids falling into the Roche lobe of the central compact object due to Kozai-Lidov oscillations, for the different systems considered here.Left: low mass companions.Right: high mass companions.The initial number of asteroids is Nast = 10 2 and the inclinations of the companions is ic = 30°.Specific parameters for each configuration can be found in Table2.
Fig. C.3.Same as Figures 6 and 7, distribution of relative time differences ∆tKL of falling asteroids in the Roche lobe of the central compact object due to quadrupolar Kozai-Lidov oscillations, for the current Solar asteroid belt (green) and the primordial belt, for an inclination of the outer body hole plane ic = 5°(orange) and ic = 45°(blue), and initial asteroid number Nast = 10 3 .We consider a high mass Mc = 10 M wide system (left) with ac = 100 A.U. and aast = 1 A.U., and a close system (right) with ac = 10 A.U. and aast = 1 A.U.