Similarity to earthquakes again: periodic radio pulses of the magnetar SGR 1935+2154 are accompanied by aftershocks like fast radio bursts

It was recently discovered that the time correlations of repeating fast radio bursts (FRBs) are similar to earthquake aftershocks. Motivated by the association between FRBs and magnetars, here we report correlation function analyses in the time-energy space for the 563 periodic radio pulses and the 579 X-ray short bursts from the magnetar SGR 1935+2154, which is known to have generated FRBs. Although radio pulses are concentrated near the fixed phase of the rotational cycle, we find that when multiple pulses occur within a single cycle, their correlation properties (aftershock production probability, aftershock rate decaying in power of time, and more) are similar to those of extragalactic FRBs and earthquakes. A possible interpretation is that the radio pulses are produced by rupture of the neutron star crust, and the first pulse within one cycle is triggered by external force periodically exerted on the crust. The periodic external force may be from the interaction of the magnetosphere with material ejected in an outburst. For X-ray bursts, we found no significant correlation signal, though correlation on the same time scale as radio pulses may be hidden due to the long event duration. The aftershock similarity between the periodic radio pulsation and FRBs is surprising, given that the two are energetically very different, and therefore the energy sources would be different. This suggests that the essence of FRB-like phenomena is starquakes, regardless of the energy source, and it is important to search for FRB-like bursts from neutron stars with various properties or environments.


INTRODUCTION
Fast radio bursts (FRBs) are extragalactic transient objects detected in radio waves with millisecond durations, and their source objects and emission mechanisms are largely still a mystery, though many theoretical models have been proposed (see Cordes & Chatterjee 2019;Platts et al. 2019;Zhang 2020;Petroff et al. 2022 for reviews).Some FRBs are known to produce recurring bursts, and these are likely to originate in neutron stars.In particular, magnetars (highly magnetized neutron stars, see Kaspi & Beloborodov 2017;Esposito et al. 2020 for reviews) have been considered a promising source of FRBs because of their abundant magnetic energy and the bursts of Xrays and gamma-rays that they occasionally induce.In fact, on April 28, 2020, two extremely bright radio bursts (FRB 20200428) similar to extragalactic FRBs were detected from the Galactic magnetar SGR 1935+2154, establishing that at least some FRBs are generated by magnetars (CHIME/FRB Collaboration et al. 2020;Bochenek et al. 2020).
More than several thousand FRB events have already been detected from several extragalactic FRB repeaters, and detailed statistical studies are possible.An interesting fact already established is that ★ E-mail: totani@astron.s.u-tokyo.ac.jp the burst wait-time distribution is bimodal.Although the long-side peak of the bimodal distribution can be explained by events occurring randomly by a Poisson process, the origin of the shorter peak has not been established (Wang & Yu 2017;Oppermann et al. 2018;Wang et al. 2018;Zhang et al. 2018Zhang et al. , 2021Zhang et al. , 2022;;Zhang et al. 2023;Li et al. 2019Li et al. , 2021;;Gourdji et al. 2019;Wadiasingh & Timokhin 2019;Oostrum et al. 2020;Tabor & Loeb 2020;Aggarwal et al. 2021;Cruces et al. 2020;Hewitt et al. 2022;Xu et al. 2022;Du et al. 2023;Jahns et al. 2022;Sang & Lin 2023;Wang et al. 2023b).In the previous study, we analyzed the two-point correlation function in the two-dimensional space of occurrence time and energy of repeating FRBs, and showed that the statistical characteristics of FRBs are remarkably similar to those of earthquakes but different from solar flares (Totani & Tsuzuki 2023, hereafter TT23).The similarities between FRBs and earthquakes can be listed as follows: (1) the probability of a single event followed by correlated aftershocks is about 10-50%, (2) the aftershock rate follows the Omori-Utsu law (Omori 1895;Utsu 1957Utsu , 1961)), which decays as a power of event time interval Δ, (3) the power law extends toward shorter Δ to the typical durations of FRBs and earthquakes, (4) even if the average event rate changes significantly due to fluctuations in activity, the aftershock rate (or probability) is universal and stable among different FRB sources, and (5) there is little correlation between the aftershock energies.These results suggest that FRBs are caused by ruptures of solid crusts at neutron star surfaces.
Then it is an interesting question whether a similar time correlation can be found in magnetar bursting phenomena.Thousands of short bursts from magnetars have been recorded in X-rays with a duration of about 0.01-10 seconds, which are one of the prominent features of magnetars (Kaspi & Beloborodov 2017;Esposito et al. 2020).SGR 1935+2154 has been one of the most active magnetars this decade, exhibiting several X-ray outbursts.These outbursts are characterized by a significant increase in persistent X-ray flux, typically by a factor of 10-1000, and can last from months to years.During its April 2020 outburst, NICER recorded hundreds of short X-ray bursts just half a day before FRB 20200428 (Younes et al. 2020).In October 2022, a similar pattern of bursting activity was observed, followed a few hours later by the detection of a bright radio burst (Giri et al. 2023;Maan et al. 2022;Hu et al. 2024).
Radio pulsations are also detected from several magnetars.These radio pulsations are a transient emission and appear in association with an X-ray outburst.The recently reported radio pulsation detected from SGR 1935+2154, which is the sixth member of the radio magnetar population, about half a year after FRB 20200428 is particularly interesting because it exhibits features (narrow-band emissions and frequency drifts) similar to FRBs, though much fainter (Zhu et al. 2023;Wang et al. 2023a).
Therefore, here we perform the same two-point correlation function analysis as TT23 on the periodic radio pulses (Zhu et al. 2023;Wang et al. 2023a) and X-ray short bursts of SGR 1935+2154 in 2020 and 2022 (Younes et al. 2020;Hu et al. 2024) to investigate the nature of time-energy correlation about bursting phenomena in magnetars.The magnetar SGR 1935+2154 rotates with a period of  = 3.24 s and has a spin-down rate of  = 1.43 × 10 −11 ss −1 .The important physical quantities derived from these are the surface dipole magnetic field strength of 2.2 × 10 14 G, the characteristic age of about 3.6 kyr, and the spin-down luminosity of 1.7 × 10 34 erg s −1 (Israel et al. 2016).Although the distance to SGR 1935+2154 is highly uncertain (Israel et al. 2016;Surnis et al. 2016;Zhou et al. 2020), here we adopt 6.6 kpc following Zhou et al. (2020); Zhu et al. (2023).

Radio pulses
The radio pulses from SGR 1935+2154 have a short pulse width of about 1 ms and are concentrated at the specific phase of the rotation cycle (Zhu et al. 2023).We use for our analysis the 563 pulses detected by the Five-hundred-meter Aperture Radio Telescope (FAST) during the total observation time of 17 hours from 9 October to 7 November 2020, as reported in Wang et al. (2023a, hereafter referred to as the R20 data set, see also Table 1).These pulses were detected at 464 rotational cycles among the 1.9 × 10 4 cycles during the total observation time, and hence the probability of detecting a radio pulse per cycle is about 2.5%.The radio pulses are narrow-band and showing frequency-drifts, which are rarely observed in normal pulsars, but rather commonly seen in extragalactic FRBs, although the average fluence of SGR 1935+2154 radio pulsations are 7-8 orders of magnitude lower than that of FRB 20200428.We use the solar system barycentric time  of the pulse peak arrival and emitted energy  of pulses from the table given in the original paper (Wang et al. 2023a).The distance to SGR 1935+2154 assumed in Wang et al. (2023a) is the same as that we adopt in this work (6.6 kpc), but it should be noted that our correlation function analysis is not affected by the distance uncertainty because we look only the logarithmic energy difference between a pair of two events.Observations were made over 14 runs on 13 days, and the start and end times for each run are taken from the table in Wang et al. (2023a).
To examine the relationship between activity levels and the nature of the correlations, we divided this sample into three sub-samples.The pulse detection rate was particularly high during the first three days ) of the entire observation period by Wang et al. (2023a, see their Fig. 4), and we call this the high-rate sample.During the following 17 days, activity was low and the pulse detection rate was less than 1/10 of the high-rate sample.We call this the low-rate sample.Activity increased again during the last two days, with a pulse detection rate about half that of the high-rate sample.We call this the mid-rate sample.These sub-samples are also summarized in Table 1.

X-ray bursts
We analyze two samples for NICER X-ray bursts (Table 1).The first sample (the X20 sample hereafter), reported by Younes et al. (2020), is bursts detected on April 28, 2020 (14 hrs before FRB 20200428).This period is the most intense burst active episode observed to date for SGR 1935+2154.We use 217 bursts detected in the first good time interval (GTI) whose duration is 1120 seconds for correlation analysis, omitting 6 bursts detected in later GTIs because of the small number.Energy flux  is not reported for 12 bursts in the 217 bursts of the first GTI, and hence these are dropped from the final sample.The total energy of a burst is estimated as  = 4 2   90 where  = 6.6 kpc is the distance and  90 is the burst duration estimated as the time interval during which 5-95% of the burst fluence is accumulated.
The second data set is 378 short X-ray bursts detected by NICER from October 12, 2022 to October 18, 2022 (Hu et al. 2024, the X22 sample hereafter).First, we identified burst candidates by the Bayesian-block analysis (Scargle et al. 2013).Subsequently, we calculate the chance probability that the total counts in each burst represent a Poisson random fluctuation, considering the count rate of approximately 3 seconds before and after the block.Bursts with a probability lower than 3 × 10 −7 , corresponding to a detection significance of 5, are classified as confirmed bursts.The total observation time during this period is 8.25 hr, and the mean event rate is about 10 times lower than the X20 sample.The total number of photons in  90 of a burst given in this data set is used as the proxy of the total energy in the correlation function analysis.As explained later in Section 3.1, the entire observation period is divided into sub-periods (each sub-period within one orbit of NICER, which is different from GTIs), and the number of pairs for correlation function calculation is counted only within each sub-period.Thus, if there is only one event in a sub-period, it is excluded from the correlation function analysis.Since 15 events are excluded for this reason, the final number of events in the X22 data set is 374.

Methods of correlation function calculations
The method for calculating the two-point correlation function is mostly the same as in TT23, and here we describe briefly focusing on the different points from the previous study.We compute the two-point correlation function  in the two-dimensional space of Δ  The mean event rates averaged over all sub-periods, with a weight by the number of events ( ev ).
and Δ lg , where Δ ≡  2 −  1 ( 2 >  1 ) is the time difference between a pair of two events (at time  1 and  2 with energies  1 and  2 ) and Δ lg  ≡ lg  2 − lg  1 is the logarithmic energy difference (lg ≡ log 10 ).The function  (Δ, Δ lg ) is defined as the excess of the number density of pairs compared with the uncorrelated case, and hence the number of pairs (  ) in a bin at (Δ, Δ lg ) can be written as where   is the expected pair number density in the uncorrelated case.
To estimate   we need to generate random and uncorrelated bursts by the Monte-Carlo method.Random data were generated by a Poisson process assuming a constant event rate and energy distribution, and the energy distribution was constructed empirically from the data sets, as in TT23.To reduce statistical error, the random data was generated with the sample size 100 times larger than that of the real data.We use the Landy-Szalay (LS) estimator (Landy & Szalay 1993), for the time correlation function  (Δ), because this estimator is known to have a small variance from the right value.Here,  and  are the number of pairs in the real and random samples (normalized by their different sample sizes), respectively, in a given bin of the Δ-Δlg  space, and   is the number of cross-pairs between the two samples.However, for the two-dimensional correlation function, the natural estimator / − 1 was used, because in bins with small , the LS estimator sometimes produces unphysical results of negative 1 + .
Poisson statistics about the number of pairs can most simply estimate the error in the correlation function, but it is known to underestimate if  is not zero.We also tried the jackknife method in our previous work, but jackknife errors were not much different from the Poisson errors, and the jackknife errors suffer from a large uncertainty because of the small sample sizes.Therefore, in this study, we will use the Poisson error and not consider the correlation of errors between different bins.This treatment is sufficient for this study, which does not require rigorous estimation of parameter errors.Poisson statistical errors for a small number of pairs (close to 1) were calculated according to Gehrels (1986).
Radio observations from the ground can be continuous for only a few hours at most during a day.Following TT23, the entire observation period is divided into sub-periods, with one sub-period consisting of data from the same observation date.We consider only pairs within each sub-period, and hence sub-periods in which only one event occurs are not used in our analysis because of no available event pairs.For random data generation, the event rate and energy distribution were held constant within each sub-period.If there is a gap in the middle of a day's observations, the correlation function was calculated by not generating random data events during the gap, using the information on the start and end times of each observing run.In the case of NICER X-ray data, a continuous observation is at most for 2000 s since the orbital period of the International Space Station is 90 minutes.Therefore, a sub-period is defined as a period of data taken within one orbit.There are many short interruptions of observations between GTIs within a single sub-period, and these were taken into account in the analysis by referring to the GTI information of NICER and by not generating random data during interruptions.

A special treatment for radio pulsation data
As reported in Zhu et al. (2023), radio pulses are emitted only at a specific phase within the 3.24 s period.Therefore, as shown in Fig. 1, Δ of the DD pairs are distributed only at integer multiples of the period  in the region of Δ > 1 s, while pairs of Δ < 0.1 s appear because occasionally multiple pulses are emitted at intervals of 0.1 s or less within a single cycle.Therefore the random samples were generated separately for the two regimes, as follows (see Fig. 1 right panel).In Δ > 1 s, random events were assumed to occur randomly on grids separated by the period , allowing only one event on a grid.Then, in this region, the correlation analysis would examine the presence or absence of correlation compared to the hypothesis that the pulses occur randomly with a certain probability in a rotation cycle.On the other hand, for Δ < 1 s, the random events were assumed to occur randomly in a continuous and uniform time distribution, not on grids, at the observed event rate averaged over a sub-period.This allows us to verify whether correlated aftershocks of Δ < 1 s occur following a given radio pulse, as found in TT23 for extragalactic FRBs.

Radio pulses
Fig. 2 shows the two-dimensional correlation function of the R20 sample in the Δ-Δlg space.No clear signal is detected in the Δ > 1 s region, which means that radio pulses occur randomly with a certain probability in a given rotation cycle, and the occurrence of a pulse in one cycle does not affect the probability of the pulse occurring in subsequent cycles.On the other hand, a strong correlation signal is deteted in the Δ < 1 s region.The signal shows no dependence on the energy direction, and the one-dimensional time correlation function  (Δ) calculated by pair counts without binning in the energy direction is shown in Fig. 3. Similar to what was seen for extragalactic FRBs in TT23, we see a correlation signal that decays with a power-law and flattening on the shortest time scale comparable to typical pulse width.To make the distribution easier to see, we generated the random data with  ran = 10 times more than the actual data in the region of Δ < 1 s, while for Δ > 1 s we set  ran = 1.
We fit this signal with the function of the Omori-Utsu law for earthquakes, by minimizing  2 (Fig. 3).The region with the smallest Δ is close to the typical pulse width, and it is susceptible to sub-burst handling because multiple pulses may overlap and appear to be a single event.Therefore, following TT23, data with Δ smaller than the peak of  (Δ) (shown by open symbols in the figure) were removed from the fit.The best-fit values and their 1 errors of the model parameters are  = 2.5 +2.1 −1.3 × 10 3 ,  = 1.9 +0.8 −0.4 , and  = 3.2 +5.5 −2.1 msec.As expected, the flattening time scale  is close to the typical width of a single pulse, and the power-law correlation signal means that there is no characteristic time scale except for the duration of one event ().
In Fig. 4 we show (1 + ) and the aftershock rate   ≡   (1 + ), where   is the mean event rate of the analyzed data set.By definition of ,   ≡    is the occurrence rate of physically correlated aftershocks following a single event.In our analysis, a single data set is divided into sub-periods where the event rate is assumed to be constant, and event rates for different sub-periods may vary widely.In TT23,   was estimated by the mean event rate through all sub-periods weighted by the number of pairs,  pair, for the -th sub-period, as where  , is the event rate within the -th sub-period.However, here we use the weight by the number of events,  ev,i : because the latter is an appropriate weighting in the sense that   is also the average of those for each subperiod ( , ) weighted by  ev,i (see Appendix A).We applied eq. ( 5) to the TT23 calculation, and found that the TT23 results remained virtually unchanged, suggesting that the choice between the two averaging methods is not so critical in practice.Similar to the extragalactic FRBs analyzed in TT23, the correlated aftershock rate    is slightly lower than (Δ) −1 , indicating that the expected number of aftershocks that follow a single event (the branching ratio), is slightly lower than order unity.Using the best-fit values of , , and , we find  = 0.18.Then these results can be interpreted as follows.
First, a radio pulse occurs, which is the mainshock event occurring near the specific phase in the rotation cycle for some reason.Then aftershocks following this mainshock occur with a probability , and their onset times from the mainshock follow the Omori-Utsu law, as in the cases of earthquakes and FRBs.
It is also interesting to see if there are "aftershocks of aftershocks," i.e., aftershocks of a similar nature following a single aftershock, which can be verified as follows.Of the 464 cycles in which the radio pulses were observed, 82 cycles had two or more pulses, and hence the probability that an aftershock occurs following the mainshock is 82/464 = 0.18 ± 0.02 (the error is by 1 Poisson statistics), which is consistent with the estimate of the branching ratio .Of the 82 cycles in which two or more pulses are found, three or more pulses are seen in 17 cycles, and hence the probability of a third pulse occurring after the second pulse is 17/82 = 0.21 ± 0.06, which is again consistent with the  value.There were no cases where four pulses or more occurred in a single cycle, but given that the expected value is 17 ×  = 3.06, it is consistent within about 2 sigma by the Poisson statistics.These results indeed indicate that aftershocks of aftershocks follow by similar laws, and are consistent with the concept of the epidemic-type aftershock sequence (ETAS) model (Ogata 1999;Saichev & Sornette 2006;de Arcangelis et al. 2016) which does not distinguish between mainshocks or aftershocks, but rather assumes that aftershocks occur after any event with a common rule.The ETAS model is known to reproduce earthquake data well, and is also consistent with extragalactic FRBs (TT23).
The two-dimensional correlation function (Fig. 2) shows that the signal is nearly constant in the energy direction, indicating a weak or no energy correlation among aftershocks, again similar to the trend seen for FRBs and earthquakes in TT23.Another common characteristic of FRBs and earthquakes found in TT23 was that correlated aftershock rate   remains relatively stable, even as source activity (i.e.,   ) largely changes.To test whether this holds also for the SGR 1935+2154 radio pulses, we show the time correlation functions for the three sub-samples of high, mid, and low rates in Fig. 4.Although   is not much different between the high-and mid-rate samples, that of the low-rate sample is relatively small, which is a different trend from that found by TT23.This may suggest that when activity is very low, the probability of aftershocks also decreases, but further research is needed because of the small statistics in the samples of this study.

X-ray bursts
The computed 2D correlation functions for the X20 and X22 data sets are shown in Figs. 5 and 6, respectively, and the 1D time correlation functions are shown in Figure 7.The values of the correlation function  are close to zero in most regions, and no strong correlation signals as seen in the radio pulses were detected.In both data sets,  is negative in the region Δ < 10 s, i.e., an anti-correlation signal is visible.This time scale is close to the typical duration of an  X-ray burst (Younes et al. 2020), and the anti-correlation is likely by an effect of multiple overlapping bursts that are not counted as two independent events.It should be noted that 65% of bursts in the X20 sample show multi-peaked structures (Younes et al. 2020).
On the other hand, non-zero  signals with significantly large signal-to-noise (S/N) ratios are visible on long-time scales of Δ > 100 s.However, this region has many pairs, and slight systematic errors can easily produce a false signal.In particular, if the event rate or energy distribution of the bursts changes over a long time, there is a risk of false correlation signals compared to the random sample, where they are assumed to be constant.In conclusion, no significant time/energy correlations were detected in the X-ray bursts, though the time scales of the correlated radio pulses cannot be examined because they are hidden by the long duration of X-ray burst.

The origin of pulses/bursts from SGR 1935+2154
The most notable result of our analysis is that the radio pulses of SGR 1935+2154 are accompanied by aftershocks of a similar nature to those of extragalactic FRBs and earthquakes, satisfying four of the five similarity points between FRBs and earthquakes listed in Section 1.The only difference between FRBs and earthquakes is the value of  the Omori-Utsu index ( ∼ 2 for FRBs and ∼ 1 for earthquakes), and the value of SGR 1935+2154 is close to that of FRBs.In addition to the similarity between FRBs and SGR 1935+2154 radio pulses in the characteristics of the radio signals (narrow-band emissions and frequency drifts) already reported by Zhu et al. (2023); Wang et al. (2023a), the results of this work strongly suggest that these two are physically similar phenomena.
However, there is one significant difference between FRBs and SGR 1935+2154; the SGR 1935+2154 radio pulses are concentrated in a specific phase of the rotation cycle.This can be interpreted as the first event in a cycle being triggered with a probability of 2.5% per cycle by some external force synchronized with the rotation of the neutron star, and the following aftershocks within the same cycle being similar in nature to those of FRBs or earthquakes.The average single pulse luminosity is about 5 orders of magnitude smaller than the spin-down luminosity (4.3 × 10 34 erg/s), and the energy emitted as radio pulses during an observation time is 10 orders of magnitude smaller than the spin-down energy (Zhu et al. 2023).Therefore, the rotational energy of the star is sufficient to explain the radio pulse.
Given the similarity to earthquakes and the popular theoretical concept linking starquakes and magnetar activities, one promising hypothesis is that they are related to breakup or cracking of the neutron star crust.In this case, an interesting question is what is the periodic external force that triggers the first radio pulse in one rotation cycle.An isolated neutron star rotating in a vacuum is not expected to feel periodically varying external forces or torques.In binary systems, tidal forces from the companion star may be periodic external forces, but no signs of binarity have been reported for SGR 1935+2154 (Lyman et al. 2022;Chrimes et al. 2022).A noteworthy observation here is that radio pulses of magnetars are generally transient and are observed in association with an X-ray outburst (Kaspi & Beloborodov 2017;Esposito et al. 2020).During the active period including an outburst, there is likely to be mass ejection from the magnetar and hence relatively dense gas is thought to exist in the vicinity of the neutron star magnetosphere compared to low activity phases.Interestingly, NICER observations of SGR 1935+2154 (Younes et al. 2023) show that a large spin-down glitch occurred three days before FAST detected three moderately bright FRB-like radio bursts (Good & Chime/Frb Collaboration 2020; Pleunis & CHIME/FRB Collaboration 2020), and then the periodic radio pulses were detected within one day from the FRB-like bursts.This suggests that there was a decrease in the stellar angular momentum associated with a mass loss before the emergence of the periodic radio pulses.Then non-axisymmetric magnetic interactions between such surrounding material and the central star may produce torques that vary with the rotational period of the star.Whether this mechanism causes the first pulse within a cycle to occur at precisely a particular phase requires a more detailed study, and we encourage theoretical studies on this direction.
A mechanism commonly considered to explain a pulse with a short time width at a particular rotational phase is that the pulse is visible when the collimated radiation crosses the line of sight.The radio pulses from SGR 1935+2154 are detected from only 2.5% of rotation cycles.Such low duty cycles and multiple pulses within one cycle have been observed from other pulsars and are commonly explained by considering patchy beams.A potential problem with this model, however, is whether it can naturally account for the systematic aftershock law observed in the delay time distribution of radio pulses from SGR 1935+2154.Theoretical investigation about this issue is also an interesting future research direction.
We did not detect any significant time correlation signals for the X-ray bursts.However, the time scale of the correlations observed in radio pulses is less than 100 msec, which is shorter than the typical duration of X-ray bursts (∼ 0.1-10 s, Younes et al. 2020).Therefore X-ray bursts may have correlations similar to those of radio pulses, but they are not detected because of the superposition of radiation from successive bursts.On the other hand, X-ray bursts may be essentially a different phenomenon from radio pulses, given the following observed facts.The peak time of X-ray bursts occurs at random phases in the rotation cycle (Younes et al. 2020).The luminosity of X-ray bursts (10 37−39 erg/s) exceeds the spin-down luminosity, as does the average energy release rate estimated from the total energy of all bursts (4.8 × 10 40 erg) that occurred during the 1120 s observation time of the X20 sample (Younes et al. 2020).Therefore, X-ray bursts cannot be explained by rotational energy, but are most likely produced by magnetic energy of the magnetar, which is in contrast to radio pulses synchronized with rotation.However, much brighter FRB-like radio bursts observed from SGR 1935+2154 may have a common physical origin with X-ray bursts, as discussed next.

Discussion on the connection with FRBs
Although we have argued that both FRBs and the radio pulses of SGR 1935+2154 are likely related to the crustal breakup of neutron stars, there is a significant difference in energy between FRB-like bursts (∼ 10 34−35 erg for FRB 20200428, CHIME/FRB Collaboration et al. 2020;Bochenek et al. 2020) and radio pulses (∼ 10 26−28 erg, Zhu et al. 2023;Wang et al. 2023a).The peak radio luminosity of FRB 20200428 (∼ 3 × 10 36 erg/s, CHIME/FRB Collaboration et al. 2020) exceeds the spin-down luminosity, which is difficult to explain by spin-down unless rotational energy is somehow stored in a certain time and released all at once.Furthermore, the phases of the FRBlike bright radio bursts from SGR 1935+2154 observed so far do not match that of the periodic radio pulses (Zhu et al. 2023).These facts suggest that the FRB-like bursts of SGR 1935+2154 and even brighter extragalactic FRBs are driven by a completely different energy source than rotational energy, most likely magnetic energy.
These facts imply that even if the energy sources and scales are completely different, radio bursts with remarkably similar radiation properties and aftershock nature can occur.The similarity may be attributed to physical processes such as starquakes rather than energy sources.This then suggests that there may be yet other populations of FRB-like radio burst phenomena driven by an energy source that is neither magnetic nor rotational.For example, radio bursts may occur as tidal forces induce starquakes in close encounters involving a neutron star in globular clusters, which may be the case for the repeating FRB source in a globular cluster of the nearby galaxy M81 (Kirsten et al. 2022) .It is important to search for short-duration radio bursts from more magnetars and non-magnetar neutron star populations to investigate their diversity and commonality.

CONCLUSION
In order to investigate the time and energy correlation properties of burst phenomena of SGR 1935+2154, the Galactic magnetar that is known to produce FRBs, we analyzed the two-point correlation functions in time and energy space for the periodic radio pulses (563 pulses) and X-ray short bursts (two data sets, each containing 205 and 374 events).
Radio pulses occur at a particular phase of the rotational cycle, but randomly with a probability of 2.5% of all the cycles.By examining time correlations for multiple pulses within a single cycle, we found that their aftershock properties are similar to those of FRBs and earthquakes in four of the five similarities reported in TT23.Namely, (1) each pulse is accompanied by an aftershock with a probability of about  ∼ 20%, (2) its distribution of the delay time Δ follows the Omori-Utsu law, ∝ (Δ + ) −  with  ∼ 2, (3) the flattening time scale  is close to the typical pulse width (1-10 msec), and (4) there is no strong correlation about energy of aftershock pulses.In this picture, there is no distinction between mainshocks and aftershocks, and aftershocks of aftershocks occur with the same statistical properties.This is confirmed by the ratio of the number of cases in which two and three pulses occurred in a single rotation cycle.This is consistent with the picture of the ETAS model, which is known to be in good agreement with the earthquake data.Regarding another similarity reported in TT23, namely that the aftershock rate is constant regardless of source activity, the SGR 1935+2154 radio pulses may differ at this point, since the aftershock rate is lower in the low-rate sample, during which the radio pulse detection rate was about 10 times lower than other sub-samples.However, the present sample lacks sufficient statistics to conclude this unequivocally, and further data and study are needed.
Thus, a natural interpretation of these results would be to assume that periodic radio pulses are due to seismic ruptures in the surface crust of the neutron star, though we cannot rule out the possibility that an entirely different physical process than starquakes could cause aftershocks of similar properties.In this scenario, the power-law time correlation function requires that the time scale of crustal ruptures is less than msec, and this should provide constraints on the physical properties of the neutron star crust (e.g., the propagation speed of seismic waves) and the size of rupture regions, which can be verified by future theoretical studies.Furthermore, the pulses occur around the specific phase within the 3.24-second period, suggesting that the initial pulse in a cycle is triggered by periodically varying external force on the crust, followed by accompanying aftershocks similar to FRBs and earthquakes.The energy emitted as periodic radio pulses is sufficiently small to be explained by the spin-down luminosity of the star.There is no sign of a binary system for SGR 1935+2154, and magnetar radio pulses are generally transient and appear in association with outbursts.Therefore we speculate that this periodic external force may be the interaction between the rotating magnetosphere and dense surrounding material ejected by an outburst, rather than tidal forces from the companion star.Another possibility is that the short-width pulses originate from beaming of radiation crossing the line of sight, and for some reason they are followed by pulses with a power-law distribution of delay times.
For X-ray short bursts, on the other hand, we could not find any significant correlated signals in the Δ ≳ 1 s region.Although we cannot rule out the possibility that the correlation of Δ ≲ 100 ms found in radio pulses are masked by the longer duration (∼ 1 s) of X-ray bursts, it is also possible that X-ray bursts are essentially a different phenomenon from periodic radio pulses.The phase of X-ray bursts is random relative to the rotation cycle, and both the luminosity of individual bursts and the burst energy generation rate averaged over an observation period exceed the spin-down luminosity.Thus, unlike periodic radio pulses, the energy source of X-ray bursts cannot be stellar rotation.
It is surprising that periodic radio pulses of SGR 1935+2154 exhibit aftershock properties similar to those of extragalactic FRBs, even though the radio pulses are ∼ 10 7 times fainter than FRB 20200428.Since the luminosity of the FRB-like bursts in SGR 1935+2154 exceeds the spin-down luminosity and the phases at which they occur are random relative to the rotational cycle, stellar rotation is unlikely to be the energy source for these bursts and the even brighter extragalactic FRBs.Then, the similarity in the nature of radio emissions and aftershocks between FRBs and the magnetar periodic radio pulses should be attributed not to energy sources but to physical processes such as crustal rupture.This hypothesis suggests that the common essence of the FRB phenomenon is not what the energy source is, but starquake processes in neutron stars.FRB-like events could then occur in neutron stars with various properties and environments, and it is important to search for FRB-like radio bursts of various energy scales from neutron stars of various populations including non-magnetars.It would also be interesting to investigate whether a series of radio pulses from other magnetars or ordinary part-time pulsars also show a similar time correlation to SGR 1953+2154.

Figure 1 .
Figure1.Distribution of DD (left) and RR (right) pairs of the SGR 1935+2154 radio pulses (the R20 data set).Since the method of generating RR pairs is different for Δ > 1 s and < 1 s, they are shown separately.To make the distribution easier to see, we generated the random data with  ran = 10 times more than the actual data in the region of Δ < 1 s, while for Δ > 1 s we set  ran = 1.

Figure 2 .Figure 3 .
Figure 2. The two-dimensional correlation function  in the Δ-Δ lg  space of the radio pulses of SGR 1935+2154 (the R20 sample).Bins with zero DD pairs are not colored.

Figure 4 .
Figure 4. Top: the time correlation functions 1 +  (Δ ) for the R20 sample and the three sub-samples of high, mid, and low rates.Bottom: the same as the top panel, but for the aftershock rate   (1 +  ) following an event, where   is the mean event rate in a sample.

Figure 5 .Figure 6 .
Figure 5.The correlation function  (left) and its S/N ratio in the 2D space of Δ and Δ lg  for the X20 sample of X-ray bursts from SGR 1935+2154.Bins with zero DD pairs are not colored.

Table 1 .
The pulse/burst samples of SGR 1935+2154 used in this study.