# Excess close burst pairs in FRB 121102

Excess close burst pairs in FRB 121102 Abstract The repeating FRB 121102 emitted a pair of apparently discrete bursts separated by 37 ms and another pair, 131 d later, separated by 34 ms, during observations that detected bursts at a mean rate of ∼2 × 10−4 s−1. While FRB 121102 is known to produce multipeaked bursts, here I assume that these ‘burst pairs’ are truly separate bursts and not multicomponent single bursts, and consider the implications of that assumption. Their statistics are then non-Poissonian. Assuming that the emission comes from a narrow range of rotational phase, then the measured burst intervals constrain any possible periodic modulation underlying the highly episodic emission. If more such short intervals are measured a period may be determined or periodicity may be excluded. The excess of burst intervals much shorter than their mean recurrence time may be explained if FRB emit steady but narrow beams that execute a random walk in direction, perhaps indicating origin in a black hole's accretion disc. radio continuum: transients 1 INTRODUCTION The repeating FRB 121102 is unique. It is associated with a persistent radio source, possibly interpretable as a pulsar wind nebula, and therefore indicating a young neutron star. Its properties have been described by Scholz et al. (2016), Spitler et al. (2016), Chatterjee et al. (2017), Hardy et al. (2017), and Law et al. (2017). Its repetitions permitted accurate localization and identification with a star-forming dwarf galaxy at a redshift z = 0.193 (Bassa et al. 2017; Chatterjee et al. 2017; Tendulkar et al. 2017). Its redshift resolved all doubt that [if it is representative of fast radio bursts (FRBs), aside from its repetition rate] FRBs are at ‘cosmological’ distances, implying high emitted instantaneous power, despite a small duty cycle. Repetition demonstrated that FRBs, unlike gamma ray bursts (GRBs), are not the product of catastrophic events. Observations of repetitions over several years exacerbated the stringent requirements on the energy available, and test theoretical models, such as extreme pulsar pulses in which the bursts are powered by rotational energy and soft gamma repeater (GRBs) outbursts in which they are powered by magnetostatic energy (Katz 2016a,b). Comparatively, frequent and closely spaced bursts offer the opportunity to obtain information, such as the distribution of intervals between bursts, unavailable from FRBs not observed to repeat, and thereby to constrain the parameters and mechanism of the source. For example, if it is strictly periodic, its period can be constrained and, if a few more short intervals are measured, determined. Alternatively, the statistics of burst intervals can test models in which FRB emit narrow beams that wander in direction, such as those involving accretion discs around black holes. Scholz et al. (2017) at the Green Bank Telescope and Hardy et al. (2017) at Effelsberg have observed pairs of radio bursts from the repeating FRB 121102 separated by 37.3 ± 0.3 and 34.1 ± 0.3 ms, respectively (error estimates from Hardy et al. 2017, assumed the same for Scholz et al. 2017, and propagated as root sum of squares). If these are, in fact, separate bursts emitted within a narrow range of rotational phase, rather than substructure within long bursts, their observation has implications for any possible burst periodicity and for the mechanisms that make bursts. The greatest burst FWHM shown by Scholz et al. (2017) is about 3 ms while the greatest FWHM indicated by the Gaussian fits of Hardy et al. (2017) is about 5 ms. The signals shown in these latter two papers, with high S/N, have no indication of any power in the ∼30 ms between the two closely spaced peaks. The only structure of any FRB 121102 burst longer than 3–5 ms is one burst (#10) of Scholz et al. (2016) with either two components separated by about 10 ms or an FWHM of about 10 ms; the structure is frequency-dependent and the noise level is significant. There is no evidence that peaks ∼35 ms apart are substructure of a single broad burst. I therefore assume that the reported bursts are in fact separate bursts and consider the implications of that assumption. Using the mean rate of burst detections of ∼2 × 10−4 s−1 in Scholz et al. (2017) and Hardy et al. (2017), if Poissonian statistics applied only a fraction ≲10−5 of bursts would be found in such close pairs. The detection of two such pairs in observations comprising (together) 25 bursts is thus extraordinarily unlikely unless bursts are correlated on very short time-scales. Many possible models are consistent with correlation; for example, Cordes et al. (2017) have suggested multipath propagation resulting from plasma lensing. No single model is specifically indicated, but any successful model must admit such short-time correlations. If the burst source has an underlying periodicity with extensive nulling, like nulling pulsars (Gajjar, Joshi & Kramer 2012) and rotating radio transients (RRATs) (McLaughlin et al. 2006; McLaughlin 2009), then these observations constrain possible periods. If there is a single rotational phase of emission the period must be an integer fraction of all the observed pair intervals. The observation of more than one such interval requires also that the period be an integer fraction of the differences between each of the pair intervals, a strong constraint. The observation of multiple pair intervals could either unambiguously determine a millisecond period or exclude the possibility of such underlying periodicity. 2 THE PERIOD Spitler et al. (2016) and Hardy et al. (2017) pointed out that the discovery of repeating pulses offers an opportunity to determine if they are periodic, and their period if so, even if (as in RRAT) pulses are observed very infrequently. This method becomes less effective if the periods are very much shorter than the intervals between observed pulses, and may fail entirely if the period varies so that the observed phase cannot be maintained between successive observations. This is likely if the FRBs are produced by a very fast and high-field neutron star that rapidly spins down, if there are orbital Doppler shifts or glitches, or if the emission region is in a surrounding nebula and moves. Phase may be lost if the intervals between observations $$t_{\text{int}} \gtrsim \sqrt{1/{\dot{\nu }}} \sim \sqrt{T/\nu }$$, where T is the spin-down time. For hypothetical T ∼ 100 y and ν ∼ 500 s−1, phase may be maintained over intervals of 1–2 h, comparable to the intervals observed (Cordes et al. 2017), provided irregular timing noise is small. Phase may be maintained over longer intervals if the spin-down rate follows a simple function like a power law that can be fitted. Suppose a strictly periodic underlying phenomenon in which an overwhelming majority of possible pulses (more than 99.9999 per cent in the recent observations) are nulled below the detection limit. This (with less extreme nulling) is the generally accepted model of RRAT. It would also describe giant pulsar pulses were there sufficiently high thresholds of detectability, and many pulsars show more or less frequent nulling. Pairs of pulses must be separated by an integer multiple of the period, so that the period   $$P = {\Delta T_i \over n_i}$$ (1)for some positive integer ni, where ΔTi is the interval between two bursts in the ith pair. In addition, for all pairs (i, j)   $$P = {\Delta T_i - \Delta T_j \over k_{i,j}}$$ (2)for some positive integer ki, j. These resemble Diophantine equations, but are complicated by the fact that the ΔTi have errors of measurement. Measurement uncertainty limits the efficacy of period determination. A single pair of bursts with an interval tint ∼ 5000 s, timed to an accuracy δt ∼ 0.3 ms, is consistent with   $$N_\nu \sim t_{\text{int}} \delta t\, \nu ^2 \sim 4 \times 10^5$$ (3)distinct possible spin periods of frequency $$\nu = {\cal O} (500$$/s). As Spitler et al. (2016) and Hardy et al. (2017) pointed out, the extant data are not sufficient to determine a fast spin period, if there be one, or to exclude its existence. Each additional independently measured interval of length ∼5000 s prunes the set of possible periods by a factor ∼νδt ∼ 0.1, so that many intervals with tint ∼ 5000 s must be measured to determine a unique fast period. If it is assumed that periodicity is exact, with bursts centred at the same phase (when they occur at all), then for the close pairs observed by Scholz et al. (2017) and Hardy et al. (2017), equation (2) is the strongest constraint. It then implies that P must be an integer fraction of 3.2 ± 0.4 ms, the difference between the two short measured intervals (propagating errors as root sum of squares). The integer is unlikely to be more than about five because a neutron star has a (somewhat uncertain) minimum rotational period of about 0.6 ms (Lasota, Haensel & Abramowicz 1996). Periods permitted by the extant data are 3.2 ± 0.4, 1.6 ± 0.2, 1.07 ± 0.13, 0.81 ± 0.10, 0.65 ± 0.08, 0.54 ± 0.07 ms etc. Such short periods are also required to meet the energetic requirements of pulsar models (Katz 2016a) provided they are not narrowly beamed and their instantaneous radiated power does not exceed the spin-down power (see however, Katz 2017a,b for speculative alternatives). Scholz et al. (2017) and Hardy et al. (2017)observed some bursts with fitted pulse widths as much as 5 ms, greater than these inferred periods. This is an argument against the preceding short inferred periods, and may point to much longer periods with the 3.2 ms interval representing variation in the phase of outburst. Another argument against very short periods may be the observation that FRB 121102 has been active for 5 yr, setting an upper bound on its magnetic moment in order that its spin-down time be at least that long; however, longer periods imply less available rotational energy. Collimated emission (Katz 2017a) is a possible resolution of this conundrum. Alternatively, these bursts might be the result of emission detected around an entire rotation, with its peak corresponding to the rotational phase at which the emission is strongest. These uncertainties in the allowed periods correspond to phase lags of many cycles over a ∼5000 s interval, so that the longer intervals, however accurately measured, cannot be used to select a valid, or exclude an invalid, short period directly from those permitted by the 34.1 and 37.3 ms intervals. It may be possible to ‘ladder up’ through a series of measured intervals ΔTn, n = 1, 2, … satisfying   $${\Delta T_{n+1} \over \Delta T_n} \lesssim {P \over 2 \delta t},$$ (4)where the Tn can be differences between measured intervals. For periods ∼1 ms this ratio is not large. Suitable intervals have not yet been measured. 3 CLOSE PAIRS If burst arrival times are described by Poissonian statistics with a mean rate τ−1, then the a priori likelihood that an interval between bursts is less than T is 1 − e−T/τ ≈ T/τ (if T/τ ≪ 1). For the close pairs observed by Scholz et al. (2017) and Hardy et al. (2017), τ ∼ 5000 s and T/τ ∼ 7 × 10−6. The a posteriori choice of the observed intervals as the criterion T introduces a bias that invalidates the quantitative applicability of the a priori likelihood, but it is still apparent that the process is far from Poissonian. The distribution of intervals may be consistent with a model in which a narrow beam executes a random walk in direction, whether the beam is emitted by a neutron star or a black hole accretion disc (Katz 2017a,c). In such models, the statistics of recurrence are non-Poissonian. If the beam once points to the observer, producing an observable burst, the interval before the next burst is likely to be much less than its mean for Poissonian statistics, the reciprocal of the mean burst rate. This may be estimated from the probability that a two-dimensional random walk in angle (the space of angular deviations is nearly Cartesian for small deviations) will return to a particular direction in the interval t to t + dt after its previous visit to that direction. That probability density, for small deviations, is ∝t−1 (because the dispersion σ ∝ t−1/2 in each of two orthogonal directions). Hence, the likelihood of a return within a time T  $$P(T) = {\int _{T_0}^T {\text{d}t \over t} \over \int _{T_0}^{T_{\text{max}}} {\text{d}t \over t}} = {\ln {T/T_0} \over \ln {(T_{\text{max}}/T_0)}},$$ (5)where T0 is a lower cut-off corresponding to the time required for the beam to wander its own width and Tmax an upper cut-off on the recurrence time (in this model, corresponding to the time for the beam to return to a direction to the observer following diffusion to the outer bounds of its angular range). Neither of these parameters is well known (the observed pulse widths set an upper bound to T0 but are broadened by scintillation, imperfect de-dispersion and instrumental response), but the dependence of P(T) on them is only logarithmic. For T0 = 1 ms (a plausible upper limit) and Tmax = τ ∼ 5000 s, P(50ms) ≈ 0.25. This result should not be taken quantitatively, but indicates that in a wandering beam model short recurrence times occur orders of magnitude (103–104× for our parameters) more frequently than would be indicated by Poissonian statistics. The data are shown in Fig. 1. Each subfigure shows the intervals reported in the indicated paper, binned in widths of $$\sqrt{10}$$ on a logarithmic scale. The solid lines show the predictions of Poissonian statistics with the mean burst rate observed during the period over which the recurrences were observed. This rate varies over times of hours, days, and longer [table 1 of Spitler et al. 2016 and table 2 of Scholz et al. 2016 indicate periods of greater and lesser apparent (Cordes et al. 2017) activity like those of SGR (Laros et al. 1987)] so the predictions are not quantitative, but confirm the conclusion that the existence of repetition intervals ∼35 ms is strong evidence against the Poissonian model even during periods of greater activity (when nearly all the observed bursts occur and intervals are measured). The dotted lines show the predictions of the random walk model (equation 5), with roll-offs allowing for the finite length of continuous observations (intervals longer than the time from a burst to the end of the observation cannot be observed). This model is statistically consistent with the observations of millisecond intervals, although there may be a significant deficiency of intervals between 0.1 s and hundreds of seconds that is not explained by the model. Figure 1. View largeDownload slide Distributions of burst intervals in bins $$\sqrt{10}$$ wide on a logarithmic scale, observed by Scholz et al. (2016) (Green Bank Telescope, 2 GHz), Spitler et al. (2016) (Arecibo, 1.4 GHz), Scholz et al. (2017) (Green Bank Telescope, 2 GHz with one interpolated Arecibo burst, 2017 January 12 data only), and Hardy et al. (2017) (Effelsberg, 1.4 GHz, data from 2017 January 16, 19, 25 only). The shorter intervals observed by Spitler et al. (2016) at Arecibo are consistent with a more sensitive telescope that can detect fainter bursts unobservable elsewhere. The solid curves are predictions for independent random events (Poissonian statistics). The likelihood of finding two intervals in the 103/2–102 ms range out of 21 Green Bank and Effelsberg intervals, assuming Poissonian statistics, is ≲10−5 (the Arecibo bursts (Spitler et al. 2016) are not included in this estimate because of their higher rate; there were no Arecibo intervals <10 s). The dotted curves are predictions for a narrow beam random walking in angle and are consistent with the two sub-second recurrences. They may not be consistent with the gap between the very short and the longer intervals, but the statistics are poor. Figure 1. View largeDownload slide Distributions of burst intervals in bins $$\sqrt{10}$$ wide on a logarithmic scale, observed by Scholz et al. (2016) (Green Bank Telescope, 2 GHz), Spitler et al. (2016) (Arecibo, 1.4 GHz), Scholz et al. (2017) (Green Bank Telescope, 2 GHz with one interpolated Arecibo burst, 2017 January 12 data only), and Hardy et al. (2017) (Effelsberg, 1.4 GHz, data from 2017 January 16, 19, 25 only). The shorter intervals observed by Spitler et al. (2016) at Arecibo are consistent with a more sensitive telescope that can detect fainter bursts unobservable elsewhere. The solid curves are predictions for independent random events (Poissonian statistics). The likelihood of finding two intervals in the 103/2–102 ms range out of 21 Green Bank and Effelsberg intervals, assuming Poissonian statistics, is ≲10−5 (the Arecibo bursts (Spitler et al. 2016) are not included in this estimate because of their higher rate; there were no Arecibo intervals <10 s). The dotted curves are predictions for a narrow beam random walking in angle and are consistent with the two sub-second recurrences. They may not be consistent with the gap between the very short and the longer intervals, but the statistics are poor. 4 DISCUSSION Many FRB models are based on neutron stars. If FRB are analogous to giant pulsar pulses (Keane et al. 2012; Connor, Sievers & Pen 2016; Cordes & Wasserman 2016; Lyutikov, Burzawa & Popov 2016) the radiated power is drawn from its rotational energy. Bursts of FRB 121102 have observed flux densities in the range 0.1–0.8 Jy (Hardy et al. 2017), though the brighter bursts may be intensified by scintillation or lensing (Cordes et al. 2017). A flux density of 0.1 Jy (a conservative lower bound) implies, assuming isotropic emission, a power of 1.5 × 1041 ergs s−1. If spin-down power is converted to coherent GHz radiation with efficiency ε the maximum spin period would be $$120 \epsilon ^{1/4} B_{15}^{1/2}$$ ms. Unless B15 ≳ 10ε−1/2, requiring an unprecedentedly high ε [the frequency of giant pulses of the Crab pulsar sharply decreases for ε ≳ 10−5 (Karuppusamy, Stappers & van Straten 2010)] and B orders of magnitude greater than in any known magnetar, it is not possible to explain the ≈35 ms burst intervals as multiple phase windows within a single longer rotation period of a rotation-powered FRB. In SGR-like models (Popov & Postnov 2010, 2013; Kulkarni et al. 2014; Lyubarsky 2014; Pen & Connor 2015; Katz 2016c), the radiated power is drawn from a neutron star's magnetostatic energy and the preceding constraints do not apply. Hardy et al. (2017) suggested that the short intervals may represent multiple rotational phases of a single (slower) rotation. Such models are disfavoured by the absence of a FRB in a fortuitous radio observation during the extraordinary 2004 outburst of SGR 1806-20; its Galactic distance of ∼15 kpc would imply a brightness about 110 dB greater than that of FRBs at cosmological distances, more than compensating for a 70 dB sidelobe suppression at 35° from the beam, yet none was detected (Tendulkar, Kaspi & Patel 2016). This argument depends on the unverified assumption that FRBs emit roughly isotropically; if strongly beamed, the argument is vitiated. The remarkable discoveries by Scholz et al. (2017) and Hardy et al. (2017) of closely spaced pairs of bursts from the repeater FRB 121102 offer important clues to FRB mechanisms. If there is an underlying periodic clock, such as neutron star rotation, with period short enough that the pairs cannot be substructure of single bursts then these pairs strongly constrain possible periods. Because every pair of closely spaced bursts provides an additional independent ΔTi, the number of constraints provided by equation (2) grows quadratically with the number of pairs observed. The discovery of one or two more pairs with separations comparable to those recently observed could either conclusively demonstrate the existence of a periodicity and determine its period (and hence demonstrate the validity of pulsar-like models) or demonstrate the absence of periodicity (and hence disprove such models in which the emission region is stable in phase). The observation of close pairs with highly non-Poissonian statistics requires explanation. Even in a periodic model, the observation of close pairs requires that, apart from the periodicity, the source's activity be correlated in time. Many astronomical objects, including SGR (Laros et al. 1987), RRAT (McLaughlin 2009), and nulling pulsars (Gajjar, Joshi & Kramer 2012), have time-clustered non-Poissonian statistics. While these clearly distinguish periods of activity from less active periods, their statistics do not resemble the statistics of the close pairs of FRB 121102. Unlike the other episodically active objects (and phenomena outside astronomy such as earthquake swarms), a general period of enhanced activity is not sufficient to explain the close pairs of FRB 121102. As discussed in Section 3, the random walk model makes a specific prediction, equation (5), for the recurrence statistics. 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# Excess close burst pairs in FRB 121102

, Volume 476 (2) – May 1, 2018
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© 2018 The Author(s) Published by Oxford University Press on behalf of the Royal Astronomical Society
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### Abstract

Abstract The repeating FRB 121102 emitted a pair of apparently discrete bursts separated by 37 ms and another pair, 131 d later, separated by 34 ms, during observations that detected bursts at a mean rate of ∼2 × 10−4 s−1. While FRB 121102 is known to produce multipeaked bursts, here I assume that these ‘burst pairs’ are truly separate bursts and not multicomponent single bursts, and consider the implications of that assumption. Their statistics are then non-Poissonian. Assuming that the emission comes from a narrow range of rotational phase, then the measured burst intervals constrain any possible periodic modulation underlying the highly episodic emission. If more such short intervals are measured a period may be determined or periodicity may be excluded. The excess of burst intervals much shorter than their mean recurrence time may be explained if FRB emit steady but narrow beams that execute a random walk in direction, perhaps indicating origin in a black hole's accretion disc. radio continuum: transients 1 INTRODUCTION The repeating FRB 121102 is unique. It is associated with a persistent radio source, possibly interpretable as a pulsar wind nebula, and therefore indicating a young neutron star. Its properties have been described by Scholz et al. (2016), Spitler et al. (2016), Chatterjee et al. (2017), Hardy et al. (2017), and Law et al. (2017). Its repetitions permitted accurate localization and identification with a star-forming dwarf galaxy at a redshift z = 0.193 (Bassa et al. 2017; Chatterjee et al. 2017; Tendulkar et al. 2017). Its redshift resolved all doubt that [if it is representative of fast radio bursts (FRBs), aside from its repetition rate] FRBs are at ‘cosmological’ distances, implying high emitted instantaneous power, despite a small duty cycle. Repetition demonstrated that FRBs, unlike gamma ray bursts (GRBs), are not the product of catastrophic events. Observations of repetitions over several years exacerbated the stringent requirements on the energy available, and test theoretical models, such as extreme pulsar pulses in which the bursts are powered by rotational energy and soft gamma repeater (GRBs) outbursts in which they are powered by magnetostatic energy (Katz 2016a,b). Comparatively, frequent and closely spaced bursts offer the opportunity to obtain information, such as the distribution of intervals between bursts, unavailable from FRBs not observed to repeat, and thereby to constrain the parameters and mechanism of the source. For example, if it is strictly periodic, its period can be constrained and, if a few more short intervals are measured, determined. Alternatively, the statistics of burst intervals can test models in which FRB emit narrow beams that wander in direction, such as those involving accretion discs around black holes. Scholz et al. (2017) at the Green Bank Telescope and Hardy et al. (2017) at Effelsberg have observed pairs of radio bursts from the repeating FRB 121102 separated by 37.3 ± 0.3 and 34.1 ± 0.3 ms, respectively (error estimates from Hardy et al. 2017, assumed the same for Scholz et al. 2017, and propagated as root sum of squares). If these are, in fact, separate bursts emitted within a narrow range of rotational phase, rather than substructure within long bursts, their observation has implications for any possible burst periodicity and for the mechanisms that make bursts. The greatest burst FWHM shown by Scholz et al. (2017) is about 3 ms while the greatest FWHM indicated by the Gaussian fits of Hardy et al. (2017) is about 5 ms. The signals shown in these latter two papers, with high S/N, have no indication of any power in the ∼30 ms between the two closely spaced peaks. The only structure of any FRB 121102 burst longer than 3–5 ms is one burst (#10) of Scholz et al. (2016) with either two components separated by about 10 ms or an FWHM of about 10 ms; the structure is frequency-dependent and the noise level is significant. There is no evidence that peaks ∼35 ms apart are substructure of a single broad burst. I therefore assume that the reported bursts are in fact separate bursts and consider the implications of that assumption. Using the mean rate of burst detections of ∼2 × 10−4 s−1 in Scholz et al. (2017) and Hardy et al. (2017), if Poissonian statistics applied only a fraction ≲10−5 of bursts would be found in such close pairs. The detection of two such pairs in observations comprising (together) 25 bursts is thus extraordinarily unlikely unless bursts are correlated on very short time-scales. Many possible models are consistent with correlation; for example, Cordes et al. (2017) have suggested multipath propagation resulting from plasma lensing. No single model is specifically indicated, but any successful model must admit such short-time correlations. If the burst source has an underlying periodicity with extensive nulling, like nulling pulsars (Gajjar, Joshi & Kramer 2012) and rotating radio transients (RRATs) (McLaughlin et al. 2006; McLaughlin 2009), then these observations constrain possible periods. If there is a single rotational phase of emission the period must be an integer fraction of all the observed pair intervals. The observation of more than one such interval requires also that the period be an integer fraction of the differences between each of the pair intervals, a strong constraint. The observation of multiple pair intervals could either unambiguously determine a millisecond period or exclude the possibility of such underlying periodicity. 2 THE PERIOD Spitler et al. (2016) and Hardy et al. (2017) pointed out that the discovery of repeating pulses offers an opportunity to determine if they are periodic, and their period if so, even if (as in RRAT) pulses are observed very infrequently. This method becomes less effective if the periods are very much shorter than the intervals between observed pulses, and may fail entirely if the period varies so that the observed phase cannot be maintained between successive observations. This is likely if the FRBs are produced by a very fast and high-field neutron star that rapidly spins down, if there are orbital Doppler shifts or glitches, or if the emission region is in a surrounding nebula and moves. Phase may be lost if the intervals between observations $$t_{\text{int}} \gtrsim \sqrt{1/{\dot{\nu }}} \sim \sqrt{T/\nu }$$, where T is the spin-down time. For hypothetical T ∼ 100 y and ν ∼ 500 s−1, phase may be maintained over intervals of 1–2 h, comparable to the intervals observed (Cordes et al. 2017), provided irregular timing noise is small. Phase may be maintained over longer intervals if the spin-down rate follows a simple function like a power law that can be fitted. Suppose a strictly periodic underlying phenomenon in which an overwhelming majority of possible pulses (more than 99.9999 per cent in the recent observations) are nulled below the detection limit. This (with less extreme nulling) is the generally accepted model of RRAT. It would also describe giant pulsar pulses were there sufficiently high thresholds of detectability, and many pulsars show more or less frequent nulling. Pairs of pulses must be separated by an integer multiple of the period, so that the period   $$P = {\Delta T_i \over n_i}$$ (1)for some positive integer ni, where ΔTi is the interval between two bursts in the ith pair. In addition, for all pairs (i, j)   $$P = {\Delta T_i - \Delta T_j \over k_{i,j}}$$ (2)for some positive integer ki, j. These resemble Diophantine equations, but are complicated by the fact that the ΔTi have errors of measurement. Measurement uncertainty limits the efficacy of period determination. A single pair of bursts with an interval tint ∼ 5000 s, timed to an accuracy δt ∼ 0.3 ms, is consistent with   $$N_\nu \sim t_{\text{int}} \delta t\, \nu ^2 \sim 4 \times 10^5$$ (3)distinct possible spin periods of frequency $$\nu = {\cal O} (500$$/s). As Spitler et al. (2016) and Hardy et al. (2017) pointed out, the extant data are not sufficient to determine a fast spin period, if there be one, or to exclude its existence. Each additional independently measured interval of length ∼5000 s prunes the set of possible periods by a factor ∼νδt ∼ 0.1, so that many intervals with tint ∼ 5000 s must be measured to determine a unique fast period. If it is assumed that periodicity is exact, with bursts centred at the same phase (when they occur at all), then for the close pairs observed by Scholz et al. (2017) and Hardy et al. (2017), equation (2) is the strongest constraint. It then implies that P must be an integer fraction of 3.2 ± 0.4 ms, the difference between the two short measured intervals (propagating errors as root sum of squares). The integer is unlikely to be more than about five because a neutron star has a (somewhat uncertain) minimum rotational period of about 0.6 ms (Lasota, Haensel & Abramowicz 1996). Periods permitted by the extant data are 3.2 ± 0.4, 1.6 ± 0.2, 1.07 ± 0.13, 0.81 ± 0.10, 0.65 ± 0.08, 0.54 ± 0.07 ms etc. Such short periods are also required to meet the energetic requirements of pulsar models (Katz 2016a) provided they are not narrowly beamed and their instantaneous radiated power does not exceed the spin-down power (see however, Katz 2017a,b for speculative alternatives). Scholz et al. (2017) and Hardy et al. (2017)observed some bursts with fitted pulse widths as much as 5 ms, greater than these inferred periods. This is an argument against the preceding short inferred periods, and may point to much longer periods with the 3.2 ms interval representing variation in the phase of outburst. Another argument against very short periods may be the observation that FRB 121102 has been active for 5 yr, setting an upper bound on its magnetic moment in order that its spin-down time be at least that long; however, longer periods imply less available rotational energy. Collimated emission (Katz 2017a) is a possible resolution of this conundrum. Alternatively, these bursts might be the result of emission detected around an entire rotation, with its peak corresponding to the rotational phase at which the emission is strongest. These uncertainties in the allowed periods correspond to phase lags of many cycles over a ∼5000 s interval, so that the longer intervals, however accurately measured, cannot be used to select a valid, or exclude an invalid, short period directly from those permitted by the 34.1 and 37.3 ms intervals. It may be possible to ‘ladder up’ through a series of measured intervals ΔTn, n = 1, 2, … satisfying   $${\Delta T_{n+1} \over \Delta T_n} \lesssim {P \over 2 \delta t},$$ (4)where the Tn can be differences between measured intervals. For periods ∼1 ms this ratio is not large. Suitable intervals have not yet been measured. 3 CLOSE PAIRS If burst arrival times are described by Poissonian statistics with a mean rate τ−1, then the a priori likelihood that an interval between bursts is less than T is 1 − e−T/τ ≈ T/τ (if T/τ ≪ 1). For the close pairs observed by Scholz et al. (2017) and Hardy et al. (2017), τ ∼ 5000 s and T/τ ∼ 7 × 10−6. The a posteriori choice of the observed intervals as the criterion T introduces a bias that invalidates the quantitative applicability of the a priori likelihood, but it is still apparent that the process is far from Poissonian. The distribution of intervals may be consistent with a model in which a narrow beam executes a random walk in direction, whether the beam is emitted by a neutron star or a black hole accretion disc (Katz 2017a,c). In such models, the statistics of recurrence are non-Poissonian. If the beam once points to the observer, producing an observable burst, the interval before the next burst is likely to be much less than its mean for Poissonian statistics, the reciprocal of the mean burst rate. This may be estimated from the probability that a two-dimensional random walk in angle (the space of angular deviations is nearly Cartesian for small deviations) will return to a particular direction in the interval t to t + dt after its previous visit to that direction. That probability density, for small deviations, is ∝t−1 (because the dispersion σ ∝ t−1/2 in each of two orthogonal directions). Hence, the likelihood of a return within a time T  $$P(T) = {\int _{T_0}^T {\text{d}t \over t} \over \int _{T_0}^{T_{\text{max}}} {\text{d}t \over t}} = {\ln {T/T_0} \over \ln {(T_{\text{max}}/T_0)}},$$ (5)where T0 is a lower cut-off corresponding to the time required for the beam to wander its own width and Tmax an upper cut-off on the recurrence time (in this model, corresponding to the time for the beam to return to a direction to the observer following diffusion to the outer bounds of its angular range). Neither of these parameters is well known (the observed pulse widths set an upper bound to T0 but are broadened by scintillation, imperfect de-dispersion and instrumental response), but the dependence of P(T) on them is only logarithmic. For T0 = 1 ms (a plausible upper limit) and Tmax = τ ∼ 5000 s, P(50ms) ≈ 0.25. This result should not be taken quantitatively, but indicates that in a wandering beam model short recurrence times occur orders of magnitude (103–104× for our parameters) more frequently than would be indicated by Poissonian statistics. The data are shown in Fig. 1. Each subfigure shows the intervals reported in the indicated paper, binned in widths of $$\sqrt{10}$$ on a logarithmic scale. The solid lines show the predictions of Poissonian statistics with the mean burst rate observed during the period over which the recurrences were observed. This rate varies over times of hours, days, and longer [table 1 of Spitler et al. 2016 and table 2 of Scholz et al. 2016 indicate periods of greater and lesser apparent (Cordes et al. 2017) activity like those of SGR (Laros et al. 1987)] so the predictions are not quantitative, but confirm the conclusion that the existence of repetition intervals ∼35 ms is strong evidence against the Poissonian model even during periods of greater activity (when nearly all the observed bursts occur and intervals are measured). The dotted lines show the predictions of the random walk model (equation 5), with roll-offs allowing for the finite length of continuous observations (intervals longer than the time from a burst to the end of the observation cannot be observed). This model is statistically consistent with the observations of millisecond intervals, although there may be a significant deficiency of intervals between 0.1 s and hundreds of seconds that is not explained by the model. Figure 1. View largeDownload slide Distributions of burst intervals in bins $$\sqrt{10}$$ wide on a logarithmic scale, observed by Scholz et al. (2016) (Green Bank Telescope, 2 GHz), Spitler et al. (2016) (Arecibo, 1.4 GHz), Scholz et al. (2017) (Green Bank Telescope, 2 GHz with one interpolated Arecibo burst, 2017 January 12 data only), and Hardy et al. (2017) (Effelsberg, 1.4 GHz, data from 2017 January 16, 19, 25 only). The shorter intervals observed by Spitler et al. (2016) at Arecibo are consistent with a more sensitive telescope that can detect fainter bursts unobservable elsewhere. The solid curves are predictions for independent random events (Poissonian statistics). The likelihood of finding two intervals in the 103/2–102 ms range out of 21 Green Bank and Effelsberg intervals, assuming Poissonian statistics, is ≲10−5 (the Arecibo bursts (Spitler et al. 2016) are not included in this estimate because of their higher rate; there were no Arecibo intervals <10 s). The dotted curves are predictions for a narrow beam random walking in angle and are consistent with the two sub-second recurrences. They may not be consistent with the gap between the very short and the longer intervals, but the statistics are poor. Figure 1. View largeDownload slide Distributions of burst intervals in bins $$\sqrt{10}$$ wide on a logarithmic scale, observed by Scholz et al. (2016) (Green Bank Telescope, 2 GHz), Spitler et al. (2016) (Arecibo, 1.4 GHz), Scholz et al. (2017) (Green Bank Telescope, 2 GHz with one interpolated Arecibo burst, 2017 January 12 data only), and Hardy et al. (2017) (Effelsberg, 1.4 GHz, data from 2017 January 16, 19, 25 only). The shorter intervals observed by Spitler et al. (2016) at Arecibo are consistent with a more sensitive telescope that can detect fainter bursts unobservable elsewhere. The solid curves are predictions for independent random events (Poissonian statistics). The likelihood of finding two intervals in the 103/2–102 ms range out of 21 Green Bank and Effelsberg intervals, assuming Poissonian statistics, is ≲10−5 (the Arecibo bursts (Spitler et al. 2016) are not included in this estimate because of their higher rate; there were no Arecibo intervals <10 s). The dotted curves are predictions for a narrow beam random walking in angle and are consistent with the two sub-second recurrences. They may not be consistent with the gap between the very short and the longer intervals, but the statistics are poor. 4 DISCUSSION Many FRB models are based on neutron stars. If FRB are analogous to giant pulsar pulses (Keane et al. 2012; Connor, Sievers & Pen 2016; Cordes & Wasserman 2016; Lyutikov, Burzawa & Popov 2016) the radiated power is drawn from its rotational energy. Bursts of FRB 121102 have observed flux densities in the range 0.1–0.8 Jy (Hardy et al. 2017), though the brighter bursts may be intensified by scintillation or lensing (Cordes et al. 2017). A flux density of 0.1 Jy (a conservative lower bound) implies, assuming isotropic emission, a power of 1.5 × 1041 ergs s−1. If spin-down power is converted to coherent GHz radiation with efficiency ε the maximum spin period would be $$120 \epsilon ^{1/4} B_{15}^{1/2}$$ ms. Unless B15 ≳ 10ε−1/2, requiring an unprecedentedly high ε [the frequency of giant pulses of the Crab pulsar sharply decreases for ε ≳ 10−5 (Karuppusamy, Stappers & van Straten 2010)] and B orders of magnitude greater than in any known magnetar, it is not possible to explain the ≈35 ms burst intervals as multiple phase windows within a single longer rotation period of a rotation-powered FRB. In SGR-like models (Popov & Postnov 2010, 2013; Kulkarni et al. 2014; Lyubarsky 2014; Pen & Connor 2015; Katz 2016c), the radiated power is drawn from a neutron star's magnetostatic energy and the preceding constraints do not apply. Hardy et al. (2017) suggested that the short intervals may represent multiple rotational phases of a single (slower) rotation. Such models are disfavoured by the absence of a FRB in a fortuitous radio observation during the extraordinary 2004 outburst of SGR 1806-20; its Galactic distance of ∼15 kpc would imply a brightness about 110 dB greater than that of FRBs at cosmological distances, more than compensating for a 70 dB sidelobe suppression at 35° from the beam, yet none was detected (Tendulkar, Kaspi & Patel 2016). This argument depends on the unverified assumption that FRBs emit roughly isotropically; if strongly beamed, the argument is vitiated. The remarkable discoveries by Scholz et al. (2017) and Hardy et al. (2017) of closely spaced pairs of bursts from the repeater FRB 121102 offer important clues to FRB mechanisms. If there is an underlying periodic clock, such as neutron star rotation, with period short enough that the pairs cannot be substructure of single bursts then these pairs strongly constrain possible periods. Because every pair of closely spaced bursts provides an additional independent ΔTi, the number of constraints provided by equation (2) grows quadratically with the number of pairs observed. The discovery of one or two more pairs with separations comparable to those recently observed could either conclusively demonstrate the existence of a periodicity and determine its period (and hence demonstrate the validity of pulsar-like models) or demonstrate the absence of periodicity (and hence disprove such models in which the emission region is stable in phase). The observation of close pairs with highly non-Poissonian statistics requires explanation. Even in a periodic model, the observation of close pairs requires that, apart from the periodicity, the source's activity be correlated in time. Many astronomical objects, including SGR (Laros et al. 1987), RRAT (McLaughlin 2009), and nulling pulsars (Gajjar, Joshi & Kramer 2012), have time-clustered non-Poissonian statistics. While these clearly distinguish periods of activity from less active periods, their statistics do not resemble the statistics of the close pairs of FRB 121102. Unlike the other episodically active objects (and phenomena outside astronomy such as earthquake swarms), a general period of enhanced activity is not sufficient to explain the close pairs of FRB 121102. As discussed in Section 3, the random walk model makes a specific prediction, equation (5), for the recurrence statistics. 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Monthly Notices of the Royal Astronomical SocietyOxford University Press

Published: May 1, 2018

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