A Unified Model for Repeating and Non-repeating Fast Radio Bursts

Since the first discovery a decade ago by Lorimer et al. (2007), fast radio bursts (FRBs) gave rise to a plethora of models. Some of those are catastrophic in origin, like mergers of two neutron stars (Totani 2013; Wang et al. 2016), mergers of two white dwarfs (Kashiyama et al. 2013), mergers of two black holes when at least one is charged (Zhang 2016), collapses of supramassive neutron stars into black holes (Falcke & Rezzolla 2014), etc., while some others are non-catastrophic like the magnetospheric activity of neutron stars (Popov & Postnov 2013; Pen & Connor 2015; Katz 2016), collisions between neutron stars and asteroids (Geng & Huang 2015; Dai et al. 2016), flares from stars in the Galaxy (Loeb et al. 2014), etc.

The repeating nature of FRB121102 (Scholz et al. 2016; Spitler et al. 2016) has ruled out the catastrophic models, unless this event is of an origin distinct from all other FRBs. Localization of this event to within a dwarf galaxy at a redshift of z = 0.19273(8) has excluded models involving Galactic origin as well (Chatterjee et al. 2017; Marcote et al. 2017; Tendulkar et al. 2017).

The fall of an asteroid of a mass of a few 10¹⁸ gm can cause an FRB (Dai et al. 2016 and references therein). When an asteroid comes close to a magnetized neutron star, a very large electric field is induced parallel to the magnetic field of the neutron star. This induced electric field detaches electrons from the surface of the deformed asteroid and accelerates these electrons to ultra-relativistic energies. Curvature radiation from these ultra-relativistic electrons moving along the magnetic field lines produces FRBs. This model has the potential to explain a diverse population of FRBs depending on the nature of the impact. If there is an asteroid belt around a neutron star, then a chance fall of an asteroid from that belt onto the neutron star would lead to a single burst. Geng & Huang (2015) argued that one neutron star is likely to face only one such asteroid impact, i.e., a non-repeating burst from a particular source. Later, Dai et al. (2016) demonstrated that a neutron star traveling through an asteroid belt would face several asteroid impacts resulting in a series of bursts. Although Dai et al. (2016) mainly concentrated on the phenomenon of isolated neutron stars moving in a galaxy and passing through asteroid belts around other stars, they also mentioned that if the neutron star comes too close to the asteroid-hosting star, then it might get captured by the other star, forming a binary. In this case, the neutron star would cross the asteroid belt repetitively leading to repeating series of bursts and this might be the case for FRB121102. In the present Letter, I extend this model. Note that it is not essential that the neutron star binary would form only via the capture process, it is also possible that the asteroid belt was created during the evolution of the binary. The neutron star would cross (twice in each orbital revolution) the asteroid belt around its stellar companion if the radius of the asteroid belt is smaller than the apastron distance of the neutron star, as well as the orbit of the neutron star being eccentric (at least mildly), as no crossing is possible in the case of two concentric circular orbits. When both of the above conditions are satisfied, there would be more than one plunge of asteroids onto the neutron star giving rise to a series of FRBs when the neutron star is inside the asteroid belt followed by a quiescent period (due to the absence of such plunges) when the neutron star is out of the belt, and the whole process would repeat due to the orbital motion of the neutron star. I elaborate this special situation in Section 2.

Sections 3 and 4 demonstrate the application of the model for the case of FRB121102 including possible system parameters that would agree with observations. Finally, in Section 5, I generalize the model and discuss how non-repeating FRBs can also be explained within this model.

The rate of infalls of asteroids onto a neutron star passing through an asteroid belt, i.e., the rate of FRBs can be written as (Dai et al. 2016): ${ \mathcal R }\sim 1.25\times {10}^{10}$ R_ns,6 (M_ns/1.4 M_⊙) ${v}_{\mathrm{ns},7}^{-1}{N}_{a}$ ${({\eta }_{1}/0.2)}^{-1}\,{({\eta }_{2}/0.2)}^{-1}$ ${({R}_{\mathrm{belt}}/2\mathrm{au})}^{-3}\,{\mathrm{hr}}^{-1}$, where R_ns,6 is the radius of the neutron star in the unit of 10⁶ cm, v_ns,7 is the speed of the neutron star in the unit of 10⁷ cm s⁻¹, N_a is the total number of asteroids in the belt, η₁R_belt and η₂R_belt are the thickness and width of the belt, R_belt is the inner radius of the belt, and η₁ and η₂ are two fractional numbers. R_ns,6 = 1 is the standard value for the radius of neutron stars. Dai et al. (2016) used v_ns,7 = 2 as the speed of isolated neutron stars moving in their host galaxies. One needs to replace this value by the orbital speed of the neutron star in order to estimate the rate of asteroid infalls during the passage of the neutron star through an asteroid belt around its binary stellar companion. The orbital speed of the neutron star can be written as

$$\begin{eqnarray}&&{v}_{{\rm{b}},\mathrm{ns}}(f)=\sqrt{\displaystyle \frac{G({M}_{\mathrm{ns}}+{M}_{\mathrm{com}})}{{a}_{\mathrm{ns}}(1-{e}^{2})}}{[1+2e\cos f+{e}^{2}]}^{1/2},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, f is the true anomaly of the neutron star, a_ns and e are the semimajor axis and the eccentricity of the orbit of the neutron star, respectively. M_ns and M_com are the masses of the neutron star and the companion. a_ns is related P_b as ${a}_{\mathrm{ns}}$ $=\,\tfrac{{M}_{\mathrm{com}}}{({M}_{\mathrm{ns}}+{M}_{\mathrm{com}})}$ ${\left[{\left(\tfrac{{P}_{{\rm{b}}}}{2\pi }\right)}^{2}G({M}_{\mathrm{ns}}+{M}_{\mathrm{com}})\right]}^{1/3}$. Thus, to estimate ${ \mathcal R }$, one needs to know various parameters like a_ns (or P_b), e, f, M_ns, and M_com. The standard value of M_ns is 1.4 M_⊙.

Moreover, the path length of the neutron star within the asteroid belt can be estimated if the location of the belt in the orbit can be determined. This path length is the arc-length of the orbit inside the belt:

$$\begin{eqnarray}&&s={\int }_{{f}_{1}}^{{f}_{2}}\sqrt{{r}_{\mathrm{ns}}^{2}+{\left(\displaystyle \frac{{{dr}}_{\mathrm{ns}}}{{df}}\right)}^{2}}\,{df},\end{eqnarray} \tag{ 2 }$$

where f₁ and f₂ are true anomalies of the neutron star when it enters and exits the belt, and r_ns is the magnitude of the radius vector of the neutron star in its orbit, defined as

$$\begin{eqnarray}&&{r}_{\mathrm{ns}}(f)=\displaystyle \frac{{a}_{\mathrm{ns}}(1-{e}^{2})}{(1+e\cos f)}.\end{eqnarray} \tag{ 3 }$$

In the next section, I fit reported detections and non-detections of bursts from the direction of FRB121102 to extract a value of P_b, obtain realistic values of ${ \mathcal R }$ for a wide range of other parameters, and estimate values of the path length.

Epochs of detections and non-detections of bursts from the direction of FRB121102 are noticeable in the compilation of the published results in Table 1. In the present model, the epochs of non-detections (2015 December 9–2016 February 1 as mentioned in Scholz et al. 2016 and 2016 May 1–27 as mentioned in Chatterjee et al. 2017) can be explained as the neutron star being in the orbital phases out of the asteroid belt, while the epochs of detections are the times when the neutron star was inside the belt. The neutron star would pass through one of the turning points of the orbit between two such epochs of detections, i.e., it would exit the belt, pass through a turning point, and enter the belt again (at another location).

Table 1 shows that the mid-point of the last epoch of detection was at a separation of 295 days from that of the previous epoch of detection, which was again separated from the epoch of detection preceding it by 175 days. This fact suggests that the orbital period of the neutron star is around 470 days (295 + 175 days), and it takes around 295 days for the neutron star to travel from one mid-location in the belt to another through a turning point and around 175 days to return to the first location through the other turning point.

By changing the above intervals slightly (1 day each),¹ I can fit all the epochs shown in Table 1. I call the mid-point of the first epoch of detection position-1, which is MJD 56233, and can be considered a location well inside the belt. I get subsequent times for the neutron star to be inside the belt as: position-a = position-1+174 days = MJD 56407, position-b = position-a+294 days = MJD 56701, position-c = position-b+174 days = MJD 56875, position-2 = position-c+294 days = MJD 57169, position-3 = position-2+174 days = MJD 57343, position-4 = position-3+294 days = MJD 57637. These are the positions when the neutron star was inside the belt experiencing asteroid infalls that triggered series of FRBs. Note that these positions are very close to the mid-points of each epoch. This proximity supports the validity of the present model. None of these locations falls in the epochs of reported non-detections. This fit also allows me to assume P_b = 468 days.

Following the same logic, the next two epochs when the neutron star will pass the asteroid belt are position-x1 = position-4+174 days = MJD 57811 (2017 February 27) and position-x2 =position-x1+274 = MJD 58105 (2017 December 18). Bursts near those times are expected.

Currently, there is no reported data for position-a, position-b, and position-c, although the present model expects bursts during those epochs. It would be interesting if the observing team clarifies whether any sensitive radio telescope was pointed in this direction during those epochs. However, observed data lacking any noticeable burst would not rule out the present model. If the observed data were contaminated by RFI, then it would be difficult to identify bursts. The bursts can be even of very low luminosity. The luminosity of bursts in this model is given as $L\simeq 2.63\times {10}^{40}\,{m}_{18}^{8/9}$ if standard values for other parameters are used (Table 2 of Dai et al. 2016), where m₁₈ is the mass of the asteroids in the unit of 10¹⁸ gm. The luminosity would decrease by a factor of 7.7 if the mass of the asteroid is smaller by one order of magnitude and by a factor of 59.9 if the mass of the asteroid is smaller by two orders of magnitude. So it is possible to have a series of very faint bursts if all of the asteroids that fell onto the neutron star during a particular passage are of low mass. The last possibility is the true absence of bursts. This can be the case if the asteroid belt whose constituents are also orbiting around the companion of the neutron star has some local voids and the neutron star crossed the belt through those voids. At present, it is not possible to favor one scenario over the others—more published data motivated to test these possibilities are essential.

In the next section, I estimate ${ \mathcal R }$ and s for different test binaries by varying e and keeping P_b fixed at the value of 468 days. Neutron stars can exist in such wide binaries—two binary pulsars with similar values of P_b are PSR B1800-27 with P_b = 407 days, e = 0.0005, ${M}_{{\rm{c}},\mathrm{med}}=0.17\,{M}_{\odot }$ (Johnston et al. 1995) and PSR J0214+5222 with P_b = 512 days, e = 0.005, ${M}_{{\rm{c}},\mathrm{med}}=0.48\,{M}_{\odot }$ (Stovall et al. 2014). Note that, for such wide period binaries, the general relativistic effects (over a few orbits as discussed here) can be ignored.

Values of f at different positions are needed to calculate ${v}_{{\rm{b}},\mathrm{ns}}$ (hence ${ \mathcal R }$) and s. One can estimate f solving Kepler's equations if in addition to P_b and e, the time of the periastron passage is also known. I proceed with logical guesses for the periastron passage time as discussed below.

As the interval (174 days) between position-1–position-a, position-b–position-c, and position-2–position-3 is shorter than that between position-a–position-b, position-c–position-2, and position-3–position-4 (294 days), it is natural to think that the neutron star passed through the periastron during the interval for the first group. However, the reverse is also possible if the asteroid belt cuts the orbit very close to the apastron. I call the first scenario "case-A" and the second scenario "case-B." In the next two subsections, I explore "case-A" and "case-B" successively.

4.1. Case-A

4.2. Case-B

4.3. Case-A and Case-B Together

Using Equation (1), I calculate v_ns(f_i) for each value of f = f_i (in intervals of 01) when the neutron star is inside the belt, and then compute the weighted average as ${v}_{\mathrm{ns},\mathrm{avg}}={\sum }_{i}{w}_{i}{v}_{\mathrm{ns}}({f}_{i})/{\sum }_{i}{w}_{i}$, where ${w}_{i}={(1-e)}^{2}/{(1+e\cos {f}_{i})}^{2}$ is a weight factor corresponding to the relative duration the neutron star spends at a particular value of f_i (Bagchi et al. 2013). This ${v}_{\mathrm{ns},\mathrm{avg}}$ is used while calculating ${ \mathcal R }$ for different cases. I use the standard value for the number density of asteroids in the belt, i.e., ${N}_{a}/({\eta }_{1}{\eta }_{2}{R}_{\mathrm{belt}}^{3})=1.5625\times {10}^{11}\,{\mathrm{au}}^{-3}$ (Dai et al. 2016). Resulting values of ${ \mathcal R }$ are consistent with the observed rate ∼3 hr⁻¹ for the wide range of parameters I chose (e, M_com, and the time of the periastron passage). This fact again supports the validity of the present model.

Table 1 shows that the longest burst period was around position-4, during MJD 57623–MJD 57651 (28 days), so the neutron star was inside the belt for at least this time span. I estimate arc-lengths around position-4 for different choices of M_com, both for case-A and case-B. Table 2 shows that the arc-length, i.e., the minimum extent of the belt along the path of the neutron star varies between 6.7 and 37 million km (0.04–0.25 au).

Table 2.  Calculation of Arc-lengths of the Orbit around Location-4, Average Orbital Speed of the Neutron Star in Those Arcs, and the Rate of Asteroid Plunges for Some Sample Cases

Periastron Passage	e	True Anomalies	Mcom	aR	ans	s	${v}_{{\rm{b}},\mathrm{ns},\mathrm{avg}}$	${ \mathcal R }$
		at MJD 57623 and MJD 57651
(MJD)		(f1, f2)	(M⊙)	(km)	(km)	(km)	(km s−1)	(hr−1)
			0.2	2.1 × 108	2.6 × 107	9.2 × 106	86.7	3.2
	0.5	(2276, 2447)	0.6	2.2 × 108	6.7 × 107	2.4 × 107	60.3	4.5
			1.0	2.4 × 108	9.8 × 107	3.5 × 107	54.4	5.0
56320 (case-A)
			0.2	2.1 × 108	2.6 × 107	9.7 × 106	90.8	3.0
	0.001	(2822, 3038)	0.6	2.2 × 108	6.7 × 107	2.5 × 107	63.1	4.3
			1.0	2.4 × 108	9.8 × 107	3.7 × 107	56.9	4.8
			0.2	2.1 × 108	2.6 × 107	6.7 × 106	63.3	4.3
	0.5	(1470, 1573)	0.6	2.2 × 108	6.7 × 107	1.7 × 107	44.0	6.2
			1.0	2.4 × 108	9.8 × 107	2.6 × 107	39.7	6.9
56554 (case-B)
			0.2	2.1 × 108	2.6 × 107	9.7 × 106	90.68	3.0
	0.001	(1024, 1239)	0.6	2.2 × 108	6.7 × 107	2.5 × 107	63.1	4.3
			1.0	2.4 × 108	9.8 × 107	3.7 × 107	56.9	4.8

Note. Canonical values for the mass and the radius of the neutron star have been used, i.e., M_ns = 1.4 M_⊙ and R_ns = 10 km.

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Now I demonstrate the orbital geometry and location of the neutron star in the orbit for sample cases in Figure 1. In the left panel, locations on the orbit where the neutron star crosses the asteroid belt for the periastron passage on MJD 56320 (blue squares, case-A) and on MJD 56554 (brown diamonds, case-B) are shown for e = 0.5. The right panel shows the variation in f with time for case-A. The green lines (curved) are for e = 0.5, while the purple lines (almost straight) are for e = 0.001. Filled squares denote the epochs (and locations in the orbit) when bursts were detected. Unfilled squares are the expected epochs of bursts in the past (position-a, position-b, and position-c; see Section 2). The asterisks ("∗") are the next two epochs (MJD 57811 and MJD 58105) when the neutron star will be inside the belt.

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Detection of bursts close to the predicted epochs, i.e., around 2017 February 17 and 2017 December 18 would be a stronger support of the present model. Ruling out via non-detection would better be done only after a number of successive failures, as I have already argued for the absence of bursts.

A future discovery of an FRB with only one active epoch of several bursts can be explained either by a very wide binary in which the neutron star has crossed the asteroid belt around its companion only once (after such burst searches have been initiated) or by the passage of an isolated neutron star through an asteroid belt around another star.

Non-repeating FRBs can occur under different circumstances, e.g., (i) the asteroid belt is around the neutron star itself and a chance fall of an asteroid from that belt onto the neutron star leads to a single burst, (ii) a neutron star grazing an asteroid belt around another star, or (iii) a binary neutron star grazing an asteroid belt around its companion (possible if the orbit of the neutron star and the asteroid belt are non-coplanar) and the orbit is so wide that only one such grazing event has occurred so far. Scenario (i) seems to be the most likely as scenarios (ii) and (iii) are very restrictive—the neutron star should not pass through the middle of the belt, it must graze the belt so that it would suffer only one impact during each grazing. Under scenario (i), the rate (10³–10⁴ sky⁻¹ day⁻¹) calculated by Geng & Huang (2015) holds valid. However, as the other two scenarios cannot be ruled out, we note that the first FRB (Lorimer burst) occurred on MJD 52146, which was almost 10 years ago (5631 days ago if I choose the current date as 2017 January 24), and it is possible for a neutron star to have an orbital period larger than 5631 days—as an example, the orbital period of PSR J2032+4127 is 8578 days and the eccentricity is 0.93 (Lyne et al. 2015).

Thus, a diverse variety of FRBs can be explained with this model without changing the basic physics behind generation of bursts, only by considering different configurations of the asteroid belt.

One of the non-repeating FRBs, FRB131104 has been recently associated with a bright gamma-ray transient (DeLaunay et al. 2016). Although soft gamma-ray emission is possible after the asteroid–neutron star impact (Dai et al. 2016), the gamma-ray flux² ${F}_{\gamma }={\dot{E}}_{{\rm{G}}}/4\pi {d}_{L}^{2}$ for d_L = 3.5 Gpc is too low (∼10⁻¹⁶ erg s⁻¹ cm⁻²) if the values of the parameters for the neutron star and the asteroids are as usual (Dai et al. 2016, see their Equation (3) for ${\dot{E}}_{{\rm{G}}}$). However, because of the uncertainties in both the claimed association and the estimation of d_L (mainly due to the uncertainties in the models of the dispersion measure in both the interstellar medium and the intergalactic medium), it is not yet possible to exclude the present model being the cause of this FRB.

This model will remain valid even in the case of a future detection of a low dispersion measure FRB, i.e., an FRB in the Galaxy, as a binary neutron star with an asteroid belt around the companion or a neutron star having an asteroid belt around itself can very well exist in the Galaxy.