Periodic repeating fast radio bursts: interaction between a magnetized neutron star and its planet in an eccentric orbit
Abudushataer Kuerban, Yong-Feng Huang, Jin-Jun Geng, Bing Li, Fan Xu, Xu Wang
DDraft version February 9, 2021
Typeset using L A TEX preprint style in AASTeX62
Periodic repeating fast radio bursts: interaction between a magnetized neutron star and its planetin an eccentric orbit
Abudushataer Kuerban, Yong-Feng Huang,
1, 2
Jin-Jun Geng, Bing Li,
4, 5
Fan Xu, andXu Wang School of Astronomy and Space Science, Nanjing University, Nanjing 210023, People’s Republic of China Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing210023, People’s Republic of China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, People’s Republic of China Key Laboratory of Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing100049, People’s Republic of China Particle Astrophysics Division, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049,People’s Republic of China
ABSTRACTFast radio bursts (FRBs) are mysterious transient phenomena. The study of repeatingFRBs may provide useful information about their nature due to their re-detectability.The two most famous repeating sources are FRBs 121102 and 180916, with a period of157 days and 16.35 days, respectively. Previous studies suggest that the periodicity ofFRBs is likely associated with neutron star (NS) binary systems. Here we propose anew model that the periodic repeating FRBs are due to the interaction of an NS withits planet in a highly elliptical orbit. The periastron of the planet is very close to theNS so that it would be partially disrupted by tidal force every time it passes throughthe periastron. Fragments generated in the process will fall toward the NS and finallycollide with the compact star to give birth to observed FRBs. The model can naturallyexplain the repeatability of FRBs with a period ranging from a few days to severalhundred days, but it generally requires that the eccentricity of the planet orbit shouldbe large enough. Taking FRBs 121102 and 180916 as examples, it is shown that themain features of the observed repeating behaviors can be satisfactorily accounted for.
Keywords:
Radio transient sources, Neutron stars, Exoplanets, Tidal disruption INTRODUCTIONThe first discovery of fast radio bursts (FRBs) by Lorimer et al. (2007) and consequent reportof five similar sources by Keane et al. (2012) and Thornton et al. (2013) opened a new window inastronomy. Since then FRBs become a hot research topic. The isotropic energy released by FRBsis in the range of 10 − erg, and the duration is typical several milliseconds. The observeddispersion measure is ∼ − − (Petroff et al. 2019), which strongly hints that FRBs are Corresponding author: Yong-Feng [email protected] a r X i v : . [ a s t r o - ph . H E ] F e b Abudushataer Kuerban et al. of cosmological origin. According to the observed repeatability, these enigmatic events may comefrom two kinds of progenitors, i.e. repeating sources and non-repeating sources.Many models (see Platts et al. (2019) for a recent review) are proposed to interpret the propertiesof FRBs. However, the underlying physics – the progenitor as well as emission mechanism remainsunclear (Katz 2018; Petroff et al. 2019; Platts et al. 2019; Cordes & Chatterjee 2019; Zhang 2020).Repeating FRBs, in particular periodic repeating FRBs, may provide valuable information about thenature of this mysterious phenomenon.Here we will mainly focus on the periodic repeating activities of FRBs. The most famous periodicrepeating sources are FRB 121102 and FRB 180916. FRB 121102 has a period of 157 days (Rajwadeet al. 2020), and FRB 180916 has a period of 16.35 days (Chime/Frb Collaboration et al. 2020). Twokinds of models, single star model and binary model, have been proposed to interpret the periodicrepeatability of these FRBs. The single star models are mainly concerned with the precession ofneutron stars (Levin et al. 2020; Yang & Zou 2020; Sob’yanin 2020; Zanazzi & Lai 2020) while thebinary models associated FRBs with the interaction between the two objects in neutron star (NS)binary systems (Mottez & Zarka 2014; Dai et al. 2016; Zhang 2017, 2018; Lyutikov et al. 2020; Ioka& Zhang 2020; Dai 2020; Gu et al. 2020; Mottez et al. 2020; Geng et al. 2020; Decoene et al. 2021).Usually, the precession period of NS is unlikely to be so long as 16.35 days (Chime/Frb Collaborationet al. 2020). Additionally, the fixed emission region of FRBs in the precession models is not addressedproperly yet (Xiao et al. 2021). Various observational facts imply that binary models are more likelyfavored by the periodicity of FRBs. The binary interaction models can be further categorized intotwo main classes: wind-like models and accretion/collision-like models. The wind-like models include:the binary comb mechanism (Zhang 2017, 2018; Ioka & Zhang 2020), mild pulsars in tight O/B-starbinaries (Lyutikov et al. 2020), small bodies orbiting around a pulsar or a magnetar (Mottez & Zarka2014; Mottez et al. 2020), and Kozai-Lidov feeding of neutron stars in binary systems (Decoeneet al. 2021). The collision/accretion-like models include: collision between a magnetized NS and anasteroid belt (Dai et al. 2016; Smallwood et al. 2019; Dai 2020), and NS-white dwarf interactions(Gu et al. 2016, 2020). As suggested earlier by a few authors, collisions between small-bodies and NScan generate transient events such as gamma-ray bursts (Campana et al. 2011), glitch/anti-glitchesand X-ray bursts (Huang & Geng 2014; Yu & Huang 2016), and FRBs (Geng & Huang 2015).Tidal disruption of minor-planets/asteroids around white dwarfs (WDs) has also been extensivelystudied (Bear & Soker 2013; Vanderburg et al. 2015; Granvik et al. 2016). Recent simulations(Malamud & Perets 2020a,b) show that a planet in a highly eccentric orbit around WD could betidally disrupted by tidal force and materials in the inner side of orbit would be accreted by the WD.The accreted clumps of materials may be responsible for the pollution of WD atmosphere by heavyelements (Vanderburg et al. 2015; Malamud & Perets 2020a,b). Similar processes (disruption of aplanet and its in-fall to the host star) can also occur in NS-planet systems if the initial parametersof the planetary system fulfill the tidal disruption condition. In fact, GRB 101225A may occur inthis way (Campana et al. 2011). Much efforts have also been made to search for close-in exoplanetsaround pulsars (Geng & Huang 2015; Huang & Yu 2017; Kuerban et al. 2020).In this study, we propose a new model to explain the periodic repeating properties of FRB sources.We argue that when a planet is in a highly eccentric orbit around a neutron star, it would be partiallydisrupted every time it passes through the pericenter. The major fragments generated during thedisruption will finally fall onto the neutron star to produce a series of FRBs. This model can naturally epeating frb s from tidal disruption Figure 1.
Schematic illustration (not to scale) of a pulsar planet in a highly eccentric orbit. The centralstar is an NS and the planet is assumed to be a typical rocky object. r is the separation between the NSand the planet at phase θ . r p is the periastron distance of the orbit. 2 r td is a characteristic distance, atwhich partial tidal disruption will occur (see text for more details). explain the periodic behavior of repeating FRBs. The structure of our paper is as follows. In Section2, we present the basic framework of our model for repeating FRBs. In Section 3, the periodicity andactive window are described in view of the model, and are compared with observations. In Section4, we address the possible existence of pulsar planets in highly eccentric orbits. Finally, Section 5presents our conclusions and some brief discussion. MODELThe planet disruption interpretation for the pollution of WD atmosphere by heavy elements (Van-derburg et al. 2015; Granvik et al. 2016; Malamud & Perets 2020a,b) and the NS-asteroid collisionmodel for FRBs (Geng & Huang 2015; Dai et al. 2016) motivate us to investigate the periodic re-peating activities of FRBs in framework of the neutron star-planet interaction model. When a planetis in a highly elliptical orbit with the periastron distance being small enough, it might be partiallydisrupted every time it comes to the pericenter. This periodic partial disruption behavior will lead toregular collisions of the disrupted fragments with the host neutron star, producing periodic repeatingFRBs.Figure 1 illustrates the general picture of the NS-planet system in an eccentric orbit. We assumethat the central star is an NS with a mass of M = 1 . M (cid:12) , and the companion is a rocky planet witha mass of m , mean-density ¯ ρ , and an orbital period of P orb . The semi-major axis ( a ) and orbitalperiod are related by the Kepler’s third law as P a = 4 π G ( M + m ) . (1)The distance between the NS and planet at phase θ (the true anomaly, see Figure 1) in the eccentricorbit is r = a (1 − e )1 + e cos θ , (2) Abudushataer Kuerban et al. P orb (days)10.610.811.011.211.411.6 L o g r p ( c m ) e = e = . e = . e = . e = . e = . r td (100 g cm )2 r td (30 g cm )2 r td (10 g cm )2 r td (5 g cm )2 r td (3 g cm ) Figure 2.
The periastron distance as a function of the orbital period for various eccentricities. Horizontallines corresponds to r p = 2 r td for different planet densities (marked in the brackets). The dash-dottedvertical line corresponds to an orbital period of 16.35 days, and the dashed vertical line is 157 days. where e is the eccentricity of the orbit. The characteristic tidal disruption radius of the planetdepends on its density as (Hills 1975) r td ≈ (cid:18) Mπ ¯ ρ (cid:19) / , (3)where ¯ ρ is the mean-density of the planet.Whether a planet will be tidally disrupted or not depends on its separation ( r ) with respect to theNS. If r is smaller than a critical value of 2 . r td , then it will begin to be partially disrupted (Liuet al. 2013). The separation between the planet and NS is different when the planet is at differentorbital phase. At periastron, it is r p = a (1 − e ) . (4)For a highly elliptical orbit on which the separation varies in a very wide range, the planet may betidally affected mainly near the periastron and it is relatively safe at other orbital phases. Here, wefocus on the disruption near the periastron. If the orbit is too compact (for example, r p ≤ r td ), thenthe disruption is violent and the planet will be completely destroyed. But when r td < r p < . r td ,then the planet will only be partially disrupted every time it passes by the periastron. Since the epeating frb s from tidal disruption r p = 2 r td for simplicity, which satisfies the condition for a partial dis-ruption. We can then calculate the relation between the periastron distance and the orbital period,which depends on the orbital eccentricity. The results are shown in Figure 2. For comparison, wehave also marked the partial tidal disruption distance (2 r td for the planet with a particular density)as horizontal lines. We see that a partial disruption would occur near the periastron for a wide rangeof orbital period. For example, for an orbit with e = 0 .
95, the partially disrupted planet will have anorbital period of ∼
20 days when its mean density is 30 g cm − . If the mean density is 10 g cm − , 5 gcm − or 3 g cm − , then the orbital period will be 30 days, 43 days and 60 days, correspondingly. Moregenerally, for a planet with the mean density ranging from 3 g cm − to 10 g cm − , partial disruptionwill occur for P orb ∼ e ∼ × g is needed (Geng & Huang 2015). Taking the density as ∼ − , the size of the asteroid will be ∼ ∼
100 km (Malamud &Perets 2020a,b).The above process of partial disruption happens periodically every time the survived main portionof the planet passes through the periastron. Consequently, the regular in-falling and collision canaccount for the periodic repeating FRBs.
Abudushataer Kuerban et al. PERIODICITY AND ACTIVE WINDOW OF FRBS3.1.
Periodicity
Observations indicate that FRB 180916 seems to have a repeating period of 16.35 days (Chime/FrbCollaboration et al. 2020), while FRB 121102 may have a period of 157 days (Rajwade et al. 2020).It hints that the period of repeating FRBs may vary in a relatively wide range. In our model,the period is mainly determined by the orbital motion of the planet. The observed periods thusexert some constraints on the parameters of our NS-planet systems. Here we show that the planetdisruption model can meet the observational requirements.As mentioned in Section 2, we take r p = 2 r td as the typical case for the partial disruption condition.It naturally leads to a relation of a (1 − e ) = 2 (cid:18) Mπ ¯ ρ (cid:19) / . (5)Combining Eq. (1) and Eq. (5), one can derive the relationship between various parameters of theNS-planet systems. In Figure 3, we have plot the relation between the eccentricity and orbital periodfor planets that satisfy the partial disruption condition. The calculations are conducted for planetswith a mean density of ¯ ρ = 3 g cm − , 5 g cm − , and 10 g cm − , respectively. We see that with theincrease of the period, the eccentricity should also increase. This is easy to understand. The key pointis that the periastron distance ( r p = 2 r td ) is almost fixed by the mean density in our framework. Atthe same time, to acquire a long orbital period, the semi-major axis should be large enough. As aresult, the eccentricity will have to be large. From Figure 3, we see that to get a period of ∼ e ∼ ∼
16 days, e ∼ P orb ≥
160 days, e ≥ .
97 is necessary. In general, Figure 3 demonstrates thatpartial disruption does can happen periodically under proper condition, and repeating FRBs withperiod ranging from ∼ ∼ Active window
Interaction time scale
In a binary-interaction-like model for FRBs, the active window of FRBs should generally be relatedto the binary interaction time scale (Du et al. 2021). In our framework, the active window is basicallydetermined by the duty cycle of the partial disruption process. In previous sections, for simplicity,we have assumed a critical condition of r p = 2 r td and assume that the partial disruption happensonly at the periastron. In reality, a partial disruption will occur as long as r td ≤ r p ≤ r td , andin this case the disruption should happen both shortly before and shortly after the passage of theperiastron, i.e. during the orbital phases of − θ c < θ < θ c , where θ is the true anomaly and θ c is thecritical angle when the separation is r c = 2 r td . r c and θ c are connected by the orbit equation as, r c = a (1 − e )1 + e cos θ c . (6)The phase-dependent orbital angular velocity of the planet in an elliptical orbit is expressed as(Sepinsky et al. 2007) ω = dθdt = 2 πP orb (1 + e cos θ ) (1 − e ) / . (7) epeating frb s from tidal disruption e P o r b ( d a y s ) = 3 g cm = 5 g cm = 10 g cm Figure 3.
Orbital period as a function of the eccentricity under the partial tidal disruption condition of r p = 2 r td . The calculation is conducted for three different densities. The two horizontal short lines representthe orbital periods of 16.35 days and 157 days respectively. From this equation, one can calculate the duty cycle (i.e., the duration of the partial disruption, orthe time interval between phase − θ c and θ c ) as (Du et al. 2021) (cid:52) P orb = P orb π (cid:112) (1 − e ) (cid:90) θ c − θ c e cos θ ) dθ. (8)3.2.2. Confronting observed repeating FRBs
Here we assume a realistic configuration of r td ≤ r p ≤ r td , and calculate the interaction timescaledetermined by − θ c < θ < θ c . For a planet with a certain density and orbital period, the criticalphase angle θ c and orbit eccentricity e are connected by r c = 2 r td ( note that in this case, r p < r td ),which leads to, a (1 − e )1 + e cos θ c = 2 (cid:18) Mπ ¯ ρ (cid:19) / . (9)As we know, FRB 121102 has a period of 157 days (Rajwade et al. 2020), and FRB 180916 has aperiod of 16.35 days (Chime/Frb Collaboration et al. 2020). Taking these two sources as examples,we have tried to derive the observational constraints on the parameters of the planetary systems. Abudushataer Kuerban et al.
The calculations can be easily done by combining Eqs. (1), (8), and (9). Again, we assume threetypical densities for the planet, ¯ ρ = 3 g cm − , 5 g cm − , and 10 g cm − .Panel (a) of Figure 4 shows the relation between e and θ c . For a fixed period and planet density, θ c increases monotonously with the increment of the eccentricity. A smaller planet density can help toslightly ease the requirement of the eccentricity, but generally a large e is necessary. Most strikingly,the orbital period has a marked influence on the eccentricity. To get a period of 157 days, e ≥ . P orb = 16 .
35 days, the duration of disruption is generally less than 2 days; while for the case of P orb = 157 days it is less than ∼
10 days. The durations are significantly less than the correspondingobserved active windows of the two FRBs. We note that the duration of disruption is not necessarilyequal to the active window of FRBs in our model. The active window in fact corresponds to theinterval of the collisions of the fragments with the NS. We will discuss this issue in more detail below.Panel (c) shows r p versus e . Note that our partial disruption model requires the periastron distanceto be in the range of r td ≤ r p ≤ r td . So, Panel (c) can help to determine the range of the eccentricity.For example, when the planet density is ¯ ρ = 3 g cm − , to obtain an orbital period of P orb = 16 . . ≤ e ≤ .
94; and for the orbital period of P orb = 157 days, e ∼ .
98 is needed.The repeating period of FRB 180916 is 16.35 days (Chime/Frb Collaboration et al. 2020). For thissource, a total number of 38 bursts have been observed from MJD 58377 to MJD 58883, i.e. in a timeinterval of 507 days. There are 33 duty cycles in this time interval, of which FRBs were detected in15 cycles. No events were reported in the other 18 cycles. The active window in a typical cycle isabout 5 days and the average burst number per cycle ranges in 0.3 — 1.2 (Chime/Frb Collaborationet al. 2020). From Panel (c) of Figure 4, we see that for a period of P orb = 16 .
35 days, to ensurethe partial disruption process ( r td ≤ r p ≤ r td ), the eccentricity should roughly be 0 . ≤ e ≤ . ∼ P orb = 157 days, to ensure the partial disruption process ( r td ≤ r p ≤ r td ),the eccentricity should roughly be e ∼ .
98. Consequently, the duration of disruption in each cycle is4 — 5 days (see Panel (b) ). This duration is also less than the active window of ∼
87 days for FRB121102. Again, we argue that the impact interval of the generated fragments can be significantlyprolonged due to gravitational perturbations that are necessary to lead clumps to fall onto the hostNS.In a typical duty circle, the observed FRB number is of the order of a few. It indicates that usuallyless than 10 major fragments are generated during the passage of the periastron. It is interesting tonote that the number of observed bursts is related with fluence as N ∝ F α +1 , where α = − . ± . epeating frb s from tidal disruption FORMATION OF HIGH ECCENTRIC PLANETARY SYSTEMSIn Section 3, we see that to account for the observed repeating FRB period ranging from tens ofdays to over one hundred days, a highly elliptical planet orbit with e ≥ . (Schneideret al. 2011)). Among them, more than 10 objects are pulsar planet candidates. Although theeccentricities of these pulsar planet candidates are generally very small, high eccentricity pulsarbinaries do have been discovered (see references in the databases – Pulsars in globular clusters andThe ATNF pulsar catalogue (Manchester et al. 2005)). Additionally, a few planets with a largeeccentricity orbiting around other types of stars have also been detected (see the EU database). Goodexamples for these include HD 20782 b ( e = 0 . ± . e = 0 . ± . e = 0 . ± . e = 0 . ± . • Formation from the capture of FFPs by NSFFPs are common in space (Smith & Bonnell 2001; Hurley & Shara 2002; van Elteren et al.2019; Johnson et al. 2020; Mr´oz et al. 2020). They may be formed from various dynamicalinteractions (see Figure 1 in Kremer et al. (2019)), such as ejection from dying multiple-starsystems (Veras & Tout 2012; Wang et al. 2015; van Elteren et al. 2019), planet-planet scattering(Hong et al. 2018; van Elteren et al. 2019), or encounter of a star with other planetary systems(Hurley & Shara 2002). In a cluster’s full lifetime, about 10% — 50% of primordial planetarysystems experience various dynamical encounters and many planets become FFPs. About 30%— 80% of them escape the cluster due to strong dynamical encounters and/or tidal interactions(Kremer et al. 2019) and travel freely in space. The velocity of these FFPs is typically in therange of 1 — 30 km s − (Smith & Bonnell 2001; Hurley & Shara 2002). FFPs may be capturedby other stars or planetary systems and form highly eccentric planetary systems (Parker &Quanz 2012; Wang et al. 2015; Li & Adams 2016; Goulinski & Ribak 2018; Hands et al. 2019;van Elteren et al. 2019). A simulation by Goulinski & Ribak (2018) shows that more than ∼ pfreire/GCpsr.html Abudushataer Kuerban et al. e c ( d e g r ee ) (a) P orb = 16.35 days, = 3 g cm P orb = 16.35 days, = 5 g cm P orb = 16.35 days, = 10 g cm P orb = 157 days, = 3 g cm P orb = 157 days, = 5 g cm P orb = 157 days, = 10 g cm e D u r a t i o n o f d i s r u p t i o n ( d a y s ) (b) e r p / ( c m ) r td (100 g cm )2 r td (30 g cm )2 r td (10 g cm )2 r td (5 g cm )2 r td (3 g cm )(c) Figure 4.
Dependence of various parameters on the eccentricity. Panel (a) shows the critical phase θ c (corresponds to r c = 2 r td ) as a function of eccentricity. Panel (b) illustrates the interaction timescale asa function of eccentricity. Panel (c) shows the periastron distance versus eccentricity. The calculations areconducted by taking the orbital period as 16.35 days (thick lines) or 157 days (thin lines), corresponding tothe repeating events of FRB 180916 and 121102, respectively. Three different densities, ¯ ρ = 3 g cm − (solidlines), 5 g cm − (dashed lines) and 10 g cm − (dotted lines), are assumed for the planet. Note that the linestyles in Panels (b) and (c) are the same as those in Panel (a). epeating frb s from tidal disruption e > .
85, and the masses of FFPs do notaffect the eccentricity significantly. • Formation from NS exchange/snatch a planetPulsars can obtain a kick velocity when they are born in the supernova explosion. If a planetsurvives in supernova, the newborn high-speed pulsar and the survived planet may form aneccentric planetary system by gravitational interaction. Additionally, when a pulsar move witha kick velocity of 100 — 400 km s − in space, it may pass by a planetary system. Duringthis process, the pulsar can also exchange/snatch a planet from other planetary systems viagravitational perturbations. Planetary systems formed in this way may also be eccentric. • Formation from the Kozai-Lidov effect in a multi-body systemThe Kozai-Lidov effect (Naoz 2016) can explain the dynamics of multi-body systems in whichone companion in an outer orbit can change (increase) the eccentricity of objects in innerorbits by gravitational perturbations. The timescale for forming a high eccentricity system isdetermined by the initial parameters. If the central star of such a multi-body system is an NSthen a highly eccentric NS-planet system may form.From the above descriptions, we see that there are many routes to form high eccentricity planetsaround neutron stars. The requirement of e ≥ . CONCLUSIONS AND DISCUSSIONIn this study, we aim to explain the periodic repeatability of FRBs by considering a NS-planetinteraction model. In our framework, a planet moves around its host NS in a highly eccentric orbit.The periastron of the planet satisfies a special condition as r td ≤ r p ≤ r td , so that the crust of theplanet will be partially disrupted every time it pass through the periaston. Fragments of the sizeof a few kilometers are produced in the process. They will fall toward the NS due to gravitationalperturbations and finally collide with the compact host star, producing the observed FRBs. Theperiod of repeating FRBs corresponds to the orbit period of the planet. To account for the observedperiod of ∼
10 — 100 days, an orbit eccentricity larger than ∼ . Abudushataer Kuerban et al. birth to coherent radio emissions and FRBs. Detailed procedure following the disruption thus stillneed further studies. Note that the disruption distance of rocky planets is ∼ cm (Mottez et al.2013a,b). At this distance, the evaporation takes a long time, ∼ yr (Kotera et al. 2016), and itdoes not affect our model significantly. ACKNOWLEDGMENTSThis work is supported by National SKA Program of China No. 2020SKA0120300, by the Na-tional Natural Science Foundation of China (Grant Nos. 11873030, 12041306, U1938201, U1838113,11903019, 11833003), and by the Strategic Priority Research Program of the Chinese Academy ofSciences (“multi-waveband Gravitational-Wave Universe”, Grant No. XDB23040000). This workwas also partially supported by the Strategic Priority Research Program of the Chinese Academy ofSciences under Grant No. XDA15360300. REFERENCES
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