Intensity Distribution Function and Statistical Properties of Fast Radio Bursts
aa r X i v : . [ a s t r o - ph . H E ] S e p Research in Astron. Astrophys.
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Vol. X No. XX , 000–000 R esearchin A stronomyand A strophysics Intensity Distribution Function and Statistical Properties of FastRadio Bursts
Long-Biao Li , , Yong-Feng Huang , ∗ , Zhi-Bin Zhang , Di Li , , Bing Li , , School of Astronomy and Space Science, Nanjing University, Nanjing 210046, China [email protected] Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry ofEducation, Nanjing 210046, China Department of Physics, College of Sciences, Guizhou University, Guiyang 550025, China National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China Key Laboratory for Radio Astronomy, Chinese Academy of Sciences, China Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China
Received XXXX ; Accepted XXXX
Abstract
Fast Radio Bursts (FRBs) are intense radio flashes from the sky that are char-acterized by millisecond durations and Jansky-level flux densities. We carried out a statis-tical analysis on FRBs discovered. Their mean dispersion measure, after subtracting thecontribution from the interstellar medium of our Galaxy, is found to be ∼
660 pc cm − ,supporting their being from cosmological origin. Their energy released in radio bandspans about two orders of magnitude, with a mean value of ∼ ergs. More inter-estingly, although the FRB study is still in a very early phase, the published collectionof FRBs enables us to derive a useful intensity distribution function. For the 16 non-repeating FRBs detected by Parkes telescope and the Green Bank Telescope, the intensitydistribution can be described as dN/dF obs = (4 . ± . × F − . ± . sky − day − ,where F obs is the observed radio fluence in units of Jy ms. Here the power-law indexis significantly flatter than the expected value of 2.5 for standard candles distributed ho-mogeneously in a flat Euclidean space. Based on this intensity distribution function, theFive-hundred-meter Aperture Spherical radio Telescope (FAST) will be able to detectabout 5 FRBs for every 1000 hours of observation time. Key words: pulsars: general – stars: neutron – radio continuum: general – intergalacticmedium – methods: statistical
Fast Radio Bursts (FRBs) are intense radio flashes that seem to occur randomly on the sky. They arecharacterized by their high brightness ( ≥ ), but with very short durations ( ∼ ms ). Until March2016, 16 non-repeating bursts and one repeat source have been discovered as unexpected outcome ofreprocessing pulsar and radio transient surveys (Lorimer et al. 2007; Keane et al. 2012; Thornton et al.2013; Burke-Spolaor & Bannister 2014; Spitler et al. 2014; Champion et al. 2016; Masui et al. 2015;Petroff et al. 2015a; Ravi, Shannon & Jameson 2015; Keane et al. 2016; Scholz et al. 2016; Spitler etal. 2016). Except for the possible counterpart and host galaxy of FRB 150418 identified in Keane et al.2016 (but see Vedantham et al. 2016 and Williams & Berger 2016 for different opinions), most previousefforts trying to search for the counterparts of other FRBs have led to a negative result (e.g. Petroff et al. Li, Huang, Zhang, Li & Li ∆ t ∝ ν − , and the pulse width is found to scale as W obs ∝ ν − (Lorimer et al. 2007; Thornton etal. 2013). Both characteristics are consistent with the expectations for radio pulses propagating througha cold, ionized plasma. These facts strengthen the view that FRBs are of astrophysical origin. Thedispersion measure, defined as the line-of-sight integral of the free electron number density, is a use-ful indication of distance. An outstanding feature of FRBs is that their dispersion measures are verylarge and exceed the contribution from the electrons in our Galaxy by a factor of 10 — 20 in mostcases. Lorimer et al. (2007) and Thornton et al. (2013) suggested that the large DM is dominated by thecontribution from the ionized intergalactic medium (IGM). FRBs thus seem to occur at cosmologicaldistances. With their large DMs, FRBs may be a powerful probe for studying the IGM and the spatialdistribution of free electrons.The millisecond duration of FRBs suggests that their sources should be compact, and the highradio brightness requires a coherent emission mechanism (Katz 2014a; Luan & Goldreich 2014). SinceFRBs’ redshifts are estimated to be in the range of z ∼ . — . (Thornton et al. 2013; Championet al. 2016), the emitted energy at radio wavelengths can be as high as ∼ — ergs. Althoughthe physical nature of FRBs is still unclear, some possible mechanisms have been proposed, such asdouble neutron star mergers (Totani 2013), interaction of planetary companions with the magnetic fieldsof pulsars (Mottez & Zarka 2014), collapses of hypermassive neutron stars into black holes (Falcke &Rezzolla 2014; Ravi & Lasky 2014; Zhang 2014), magnetar giant flares (Kulkarni et al. 2014; Lyubarsky2014; Pen & Connor 2015), supergiant pulses from pulsars (Cordes & Wasserman 2016), collisions ofasteroids with neutron stars (Geng & Huang 2015; Dai et al. 2016) or the inspiral of double neutronstars (Wang et al. 2016). Keane et al. (2016) suggested that there may actually be more than one classof FRB progenitors.New FRB detections are being made and much more are expected in the near future. Thornton et al.(2013) have argued that if FRBs happen in the sky isotropically, their actual event rate could be as highas ∼ sky − day − . Hassall, Keane & Fender (2013) discussed the prospects of detecting FRBs withthe next-generation radio telescopes and suggested that the Square Kilometre Array (SKA) could detectabout one FRB per hour. Based on the redshifts estimated from the measured DMs, Bera et al. (2016)studied the FRB population and predicted that the upcoming Ooty Wide Field Array can detect FRBs ata rate of ∼ . — per day, depending on the power-law index of the assumed distribution function,which could vary from -5 to 5. Note that their predicted detection rate is in a very wide range, whichmainly stems from the uncertainty of the FRB luminosity function and their spectral index.The luminosities depends strongly on the measured redshifts. However, the redshifts of FRBs arenot directly measured, but are derived from their DMs. The reliability of these redshifts still needs tobe clarified (Katz 2014b; Luan & Goldreich, 2014; Pen & Connor 2015). In this study, we examine thestatistical properties of published FRBs, and use the directly measured fluences of FRBs to derive anintensity distribution function. Our distribution function is independent of the redshift measurements.We then use the intensity distribution function to estimate the detection rate of FRBs by the ChineseFive-hundred-meter Aperture Spherical radio Telescope (FAST), the opening ceremony of which isslated for the 25th of September, 2016. Our article is organized as follows. In Section 2, we describe thesample of 16 non-repeating FRBs and present the statistical analyses of their parameters. In Section 3,we derive the intensity distribution function of FRBs. In Section 4, the observational prospects of FRBswith FAST are addressed. Our conclusions and discussion are presented in Section 5. http://rac.ncra.tifr.res.in/ Note that the usage of the word ”fluence” here is different from its common definition and dimension. We follow earlier FRBpapers in this study. ntensity Distribution Function and Statistical Properties of FRBs 3
Table 1
Key parameters of the 16 non-repeating FRBs. Observational data are mainly takenfrom http://astronomy.swin.edu.au/pulsar/frbcat/ (Petroff et al. 2016).
FRB W obs S peak F obs DM a DM Galaxy a DM Excess b z c D L c E c (ms) (Jy) (Jy ms) ( pc cm − ) ( pc cm − ) ( pc cm − ) (Gpc) ( ergs)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)010125 10.60 +2 . − . +0 . − . +2 . − . . ± .
110 680.3 0.57 3.35 2.77010621 8.00 +4 . − . +0 . − . +5 . − . ±
523 223 0.19 0.93 0.12010724 20.00 +0 . − . +0 . − . ± +0 . − . +0 . − . +2 . − . . ± . +0 . − . +0 . − . +0 . − . . ± . +0 . − . . ± . +1 . − . +1 . − . +3 . − . . ± . +2 . − . +0 . − . +2 . − . . ± . +0 . − . +0 . − . +1 . − . . ± . +3 . − . +0 . − . +4 . − . . ± . +1 . − . +0 . − . +2 . − . . ± . +0 . − . +0 . − . +0 . − . . ± . +9 . − . +0 . − . +6 . − . ±
31 830 0.69 4.24 3.35131104 2.37 +0 . − . +0 . − . +2 . − . ± +3 . − . +0 . − . +2 . − . . ± . +0 . − . +0 . − . +1 . − . . ± . a DM and DM Galaxy are the total dispersion measure and the contribution from the local Galaxy, repectively. b The dispersion measure excess, which is defined as DM − DM Galaxy . c The redshifts are estimated from the corresponding DM excess. With these redshifts, the luminosity distances ( D L ) andthe emitted energies ( E ) can then be calculated. We extract the key parameters of 16 non-repeating FRBs detected by Parkes and GBT from the FRBCatalogue of Petroff et al. (2016) . The data are listed in Table 1. At the direction of FRB 121102,additional 16 repeating bursts have been detected (Spitler et al. 2016; Scholz et al. 2016), indicating thatall these events may be quite different from other non-repeating FRBs in nature. So we treat these 17repeating events separately in our following study.Column 1 of Table 1 provides the FRB names. The observed width or duration of the correspondingradio pulse ( W obs ) is presented in Column 2. Column 3 is the observed peak flux density ( S peak ) ofeach FRB. Column 4 tabulates the observed fluences ( F obs ) in units of Jy ms, which is calculated as F obs = S peak × W obs . Columns 5, 6, and 7 present the observed DMs of FRBs, the DM contributionsfrom the Galaxy ( DM Galaxy ), and the DM excesses ( DM Excess ), respectively. The DM excess is definedas DM Excess = DM − DM Galaxy . The estimated redshift ( z ) is presented in Column 8, assumingthat the density of electrons is a constant for the IGM. The corresponding luminosity distances ( D L )and the emitted energies ( E ) are presented in Columns 9 and 10, respectively. Note that there is noreliable estimate on the uncertainties of DM Galaxy , therefore, it is not included. Then the uncertaintiesof DM Excess , z , D L and E are also not available. http://astronomy.swin.edu.au/pulsar/frbcat/ Li, Huang, Zhang, Li & Li (a) N o . o f bu r s t s DM (pc cm -3 ) (b) N o . o f bu r s t s DM Excess (pc cm -3 ) (c) N o . o f bu r s t s log Energy (ergs) Fig. 1
Distributions of the DM (Panel a), the DM excess (Panel b), and the estimated energy(Panel c). The solid curve in each panel is the best-fit Gaussian function, with the fittingcorrelation coefficients being 0.90, 0.95 and 0.96 in Panels (a), (b) and (c), respectively. -1 -1 -1 -1 (a) S peak (Jy) D M E xc e ss ( p c c m - ) (b) F obs (Jy ms) (c) Energy (10 ergs) (d) S peak (Jy) W ob s ( m s ) (e) DM Excess (pc cm -3 ) (f) Energy (10 ergs) Fig. 2
Panels (a), (b), and (c) illustrate the observed peak flux density ( S peak ), the observedfluence ( F obs ), and the estimated energy ( E ) against the DM excess ( DM Excess ), respec-tively. Panels (d), (e) and (f) present S peak , DM Excess and E against the observed pulsewidth ( W obs ), respectively. The best fit line is shown in Panel (c) when the two parametersare clearly correlated. ntensity Distribution Function and Statistical Properties of FRBs 5 We first focus on the distribution of the observed DMs of 16 non-repeating FRBs in Table 1. Fig. 1illustrates the histogram of DMs (Panel a) and DM excesses (Panel b). Both DM and DM Excess roughlyfollow the normal distribution and can be well fitted with a Gaussian function. The Gaussian functionof DM peaks at ±
45 pc cm − , while DM Excess peaks at ±
60 pc cm − . The standard devia-tions of these two Gaussian fitting are comparable and of the magnitude of
140 pc cm − . We see that DM Excess /DM ∼ , which supports FRBs’ cosmological origin. Panel (c) of Fig. 1 shows thatthe estimated radio energy approximatively follows a log-normal distribution. The log-normal peak isabout ergs, consistent with an earlier estimation by Huang & Geng (2016) when only 10 burstswere available.In Panels (a), (b) and (c) of Fig. 2, we plot the observed peak flux density, the observed fluence andthe estimated radio energy against the DM excess, respectively. Fig. (2a) shows that S peak does not haveany clear correlation with DM Excess , which is somewhat unexpected since a more distant source usuallytends to be dimmer. One possible reason is that S peak depends on the time and frequency resolution ofthe radio telescope, and another reason may be that the currently observed DM Excess values are stillin a relatively narrow range (the largest DM Excess is only ∼ times that of the smallest one, andthe estimated D L range is a factor of ∼ F obs does not correlatewith DM Excess . It indicates that the width of FRBs also does not depend on DM Excess , which will befurther shown in the following figures. In Fig. (2c), we see that the energies shows a strong correlationwith DM Excess , which is natural since the energy emitted has a square dependence on the distance. Infact, the best fit result of Fig. (2c) is E ∝ DM . ± . , with the correlation coefficient and P-value(rejection possibility) as 0.78 and . × − , respectively. It corresponds to a correlation between theenergy and the luminosity distance as E ∝ D . ± . . The power-law index here is roughly consistentwith the square relation within a still relatively large error box. Note that the correlation may also bepartly caused by the telescope selection effect, because weaker FRB events can be detected only atnearer distances, although they may also happen at far distances. In addition, the sample of FRBs is stilllimited. It is expected that more FRBs would be detected when radio telescopes with higher sensitivitiescome into operation in the future.Fig. (2d) demonstrates the trend for FRBs with brighter peaks ( S peak ) to have narrower width( W obs ). A similar tendency has been found for giant pulses from some pulsars (e.g. Popov & Stappers2007; Bhat, Tingay & Knight 2008; Popov et al. 2009; Cordes et al. 2016; Popov & Pshirkov2016). In the case of FRBs, this correlation can be explained by some possible models. For example,Geng & Huang (2015) argued that FRBs can be produced by the collisions of asteroids with neutronstars. In this scenario, when the asteroid collides with the neutron star with a very small impact pa-rameter, the collision process will finish very quickly and the brightness of the FRB should be high.On the other hand, if the asteroid collides with the neutron star with a slightly larger impact parame-ter, the collision process will be significantly prolonged and the peak flux of the induced FRB will becorrespondingly weaker. It can naturally account for the correlation of W obs and S peak as shown inFig. (2d). Finally, we see that neither E nor DM Excess correlate with W obs (Figs. (2e) and (2f)). Notethat while the observed pulse width is relatively clustered, emitted energies span two orders of magni-tude. In Panels (b) and (e), we mark the positions of FRBs 010621 and 010724. These two events seemto be quite different from others. We argue that they may form a distinct group, characterized by a lowDM and a large fluence. It suggests the existence of different FRB populations. More events detected inthe future will help to clarify such a possibility. An absolutely scaled luminosity function can help to reveal the nature of FRBs (Bera et al. 2016). Sincethe redshifts of FRBs have not been independently measured, the derived absolute luminosities and theemitted energies of FRBs are thus controversial (Katz 2014b; Luan & Goldreich 2014; Pen & Connor2015). On the other hand, the apparent intensity distribution function of astronomical objects can alsoprovide helpful information on their nature. A good example is the study of gamma-ray bursts (GRBs).Before 1997, when the redshifts of GRBs were still unavailable, a deviation from the − / power- Li, Huang, Zhang, Li & Li -1 F = 2.0 Jy ms (a) N F obs (Jy ms) (b) P o w e r- La w I nde x F (Jy ms)
Fig. 3
Intensity distribution functions of 16 non-repeating FRBs listed in Table 1 (Case I).Panel (a) shows an exemplar distribution function for a particular bin width, with the y -axisshowing the number of FRBs in each bin. Panel (b) illustrates the best-fitted a values fordifferent bin widths, in which the solid short horizontal line shows the average value of a fora preferable range of ∆ F when a is relatively stable.law in the peak flux distribution of GRBs was noted (Tavani 1998). It was explained as a hint for thecosmological origin of GRBs, which was later confirmed by direct redshift measurements.Here, we focus on the observed fluence ( F obs ) of FRBs, instead of S peak . Seriously affected by scat-ter and scintillation of IGM, the peak flux density can be relatively unstable. The combination of W obs and S peak , i.e. the observed fluence, can then more reliably indicate the fierceness of FRBs. Anotherreason is that due to the limited time resolving power of our radio receivers, a FRB should last longenough to be recorded so that the duration is also a key factor. In fact, a tentative cumulative distribu-tion vs. the fluence has been drawn by Katz (2016) based on a smaller data set consisting of 10 FRBs.Caleb et al. (2016) has also tried to compose a cumulative log N ( > F ) − log F correlation by using 9FRBs in the high latitude (Hilat) region of the Parkes survey.Although the energy distribution of FRBs spans about two orders of magnitude, it is still relativelyclustered, which indicates that FRBs can be considered as standard candles to some extent. We canthen use the brightness distribution of FRBs to hint their spatial distribution. We assume that the actualnumber density of FRBs occurring in the whole sky per day at a particular fluence F obs follows a power-law function, i.e. dN/dF obs = A F − a obs , where A is a constant coefficient in units of events sky − day − and a is the power-law index. Both A and a need to be determined from observations. We first onlyconsider the 16 non-repeating FRBs in Table 1 as the input data (Case I). We group the 16 FRBs intodifferent fluence bins with a bin width of ∆ F , count the number of FRBs in each bin and get the best-fitted power-law function for dN/dF obs . In Panel (a) of Fig. 3, we show the best-fitted result when thebin width is taken as ∆ F = 2 . Jy ms. The fitted power-law index is a = 0 . ± . , with the fittedcorrelation coefficient being 0.86 and the corresponding P-value being 0.001. Note that the error barsalong x -axis simply represent the bin size, and the y -axis error bars are the statistical errors, which arethe square root of the number in each bin.Obviously, since the total number of FRBs is still very limited, the choice of the bin width willseriously affect the fitting result. So, we have tried various bin width ranging from 0.3 Jy ms to 7.0Jy ms to study the effect. For these different bin widths, the derived power-law indices are illustrated inPanel (b) of Fig. 3. From this panel, we see that when the bin width is very small ( ∆ F ≤ . Jy ms), thefitted a value depends strongly on the bin width. The reason is that only two or three FRBs are groupedinto one bin generally, thus the fluctuation dominates the final result. Meanwhile, when the bin width istoo large ( ∆ F > . Jy ms), the error bar of the fitted a also becomes larger. In these cases, only two ntensity Distribution Function and Statistical Properties of FRBs 7 Table 2
The observable event rate of FRBs in the literature. F Limit R ( > F Limit ) Reference Derived coefficient A (Jy ms) ( sky − day − ) (10 sky − day − )3.0 Thornton et al. (2013) . ± . . × Spitler et al. (2014) . ± . . × Keane & Petroff (2015) . ± . . × Law et al. (2015) . ± . . × Rane et al. (2016) . ± . . × Champion et al. (2016) . ± . -1 (b) (a)F = 0.32 Jy ms N F obs (Jy ms) P o w e r- La w I nde x F (Jy ms)
Fig. 4
Intensity distribution functions of the 17 repeating FRBs from the source of FRB121102 (Case II). Symbols are the same as Fig. 3.or three bins are left with a non-zero number of FRBs so that the derived a becomes unreliable again.On the contrary, when ∆ F is in the range of 1.8 — 4.2 Jy ms, the best-fitted a keeps to be somehowconstant and the error box is also small. So we choose such a ∆ F range to derive the a parameter. Toreduce the effects of fluctuation as far as possible, we add up all the 12 a values derived for ∆ F rangingbetween . ≤ ∆ F ≤ . to get a mean value for a (designated as a for Case I), whichfinally gives a = 1 . ± . (see Fig. (3b)).Integrating the intensity distribution function, we can derive the FRB event rate above a particularfluence limit in the whole sky per day as R ( > F Limit ) = A Z F max F Limit F − a obs dF obs , (1)where F Limit corresponds to the limiting sensitivity of a radio telescope, F max is the upper limit ofthe fluence of observed FRBs, which is set as 35 Jy ms in our calculations (the observed maximumfluence is ∼ Jy ms at present). A is an unknown coefficient. It still needs to be determined fromobservations. FRBs were mainly identified from the archival data of several radio surveys. Constraintson the actual event rates of FRBs above a certain fluence limit were also derived in these analyses andwere reported in the literature. We sum up these constraints in Table 2. According to Eq. (1), thesedata can be used to derive the coefficient A by using our best-fit a value. The results are also listedin the last column in Table 2. Combining all these different A values, we finally get a mean value as A = (4 . ± . × sky − day − . After getting the power-law index a and the constant coefficient Li, Huang, Zhang, Li & Li A , we now can write down the full apparent intensity distribution function as, dNdF obs = (4 . ± . × F − . ± . sky − day − , (2)where F obs is in units of Jy ms.In total, 17 repeating FRB events have been detected from the source of FRB 121102. We treat theseevents as a separate group (Case II) and also study their intensity distribution. Similar fitting processesas for Case I are applied to Case II, the final results are shown in Fig. 4. In Fig. (4a), we show theexemplar fitting result when taking the bin width as ∆ F = . ± . , with the correlation coefficient and P-value being 0.99 and 0.001, respectively. Similar toFig. (3b), Fig. (4b) shows the derived a values for different bin widths. The most probable mean value of a (designated as a for Case II) is calculated for ∆ F ranging between 0.2 Jy ms and 0.35 Jy ms, which is a = 1 . ± . . Note that although a and a do not differ from each other markedly, there are actuallysignificant differences between the overall characteristics of non-repeating FRBs and repeating FRBs.For example, the most obvious difference is that the repeating FRBs are generally weaker, indicatingthat they are mainly at the weak end of the fluence distribution. FAST (Nan et al. 2011; Li, Nan & Pan 2013), an ambitious Chinese mega-science project, is currentlybeing built in Guizhou province in southwestern China. With a geometrical diameter of 500 metersand an effective diameter of ∼ meters at any particular moment, it will be the largest single-dishradio telescope in the world when it comes into operation in September of 2016. FAST’s receivers willcover both low frequency (70-500 MHz) and middle frequency (0.5-3 GHz) bands. We here considerthe prospect of detecting FRBs with FAST by using the derived intensity distribution function. Ourcalculations are done at the L band (1400 MHz) of FAST, which is the central observational frequencyfor most detected FRBs.The sensitivity or the limiting flux density ( S limit ) of a radio telescope can be estimated as (Zhanget al. 2015), S limit ≃ (12 µ Jy)( 0 . × m / K A e /T sys )( S/N τ ) / ( 100MHz∆ ν ) / , (3)where T sys is the system temperature, ∆ τ is the integration time, ∆ ν is the bandwidth, S / N is thesignal-to-noise ratio which is usually taken as 10 for a credible detection of a FRB (Champion et al.2016), A e is the effective area and it equals to η A π ( d/ , with η A being the aperture efficiency and d being the illuminated diameter. FAST has a system temperature of T sys ∼ K at 1400 MHz. For otherparameters of FAST, we take d = 300 m , η A = 0 . (Zhang et al. 2015), ∆ ν = 300 MHz, ∆ τ = 3 ms (Law et al. 2015). The limiting fluence of FAST is then calculated to be F Limit = S limit × ∆ τ = 0 . Jy ms. Note that from Equations (1) and (2), the actual FRB event rate above the fluence limit of 0.03Jy ms is (3 . ± . × sky − day − .The beam size of a radio telescope is Ω ∼ πθ , where θ ∼ . λ/d ) is the half opening angle ofthe beam. For FAST, the beam size is Ω ∼ .
008 deg at 1400 MHz. FAST’s reciver has 19 beams in Lband, and the corresponding total instantaneous field-of-view (FoV) is .
15 deg . Considering Eq. (2)and all these parameters, we can get the daily detection rate of FRBs by FAST as, R FAST ∼ (3 . ± . × × .
15 deg day − = 0 . ± .
06 day − . (4)For a 1000 hours of observation time, this corresponds to ∼ ± detections. ntensity Distribution Function and Statistical Properties of FRBs 9 In this study, we analysed statistically the key parameters of the 16 non-repeating FRBs detected byParkes and GBT. The observed DM spans a range of 375 — 1629 pc cm − and peaks at ∼
723 pc cm − ,while the DM excess peaks at ∼
660 pc cm − and typically accounts for ∼ of the total DM. Theemitted radio energy spans about two orders of magnitude, with the mean energy being about ergs.While most of the parameters do not correlate with each other, a burst with stronger S peak tends to haveshorter duration. Meanwhile, a clear correlation between the radio energy released and the DM excesshas been found to be E ∝ DM . ± . (Fig. 2c), which may reflect the square dependence of theemitted energy on the distance. But note that the observational selection effect may also play a role inthe statistics. From these statistical analyses, we found that FRBs 010621 and 010724 are quite differentfrom others and they may form a distinct group.Using the observed fluence as an indication for the strength of FRBs and combining the con-straints on the event rate of FRBs from various observational surveys, we derived an apparent in-tensity distribution function for the 16 non-repeating FRBs as dN/dF obs = (4 . ± . × F − . ± . sky − day − . For FRB 121102 and its repeating bursts, the corresponding power-lawindex is derived to be a = 1 . ± . . Based on the intensity distribution function, we were able toestimate the detection rate of FRBs by China’s coming FAST telescope. With a sensitivity of 0.03 Jy msand a total instantaneous FoV of 0.15 deg (19 beams), FAST will be able to detect roughly 1 FRB forevery 10 days of observations, or about 5 events for every 1000 hours.A few authors have studied the cumulative distribution function of FRBs (Bera et al. 2016; Calebet al. 2016; Katz 2016; Wang & Yu 2016), which is usually assumed to be a power-law function of N >F obs ∝ F − α obs . For standard candles distributed homogeneously in a flat Euclidean space, the value of α should be 3/2 (Thornton et al. 2013). Oppermann, Connor & Pen (2016) has argued that the range of α may be . ≤ α ≤ . , Caleb et al. (2016) has derived α = 0 . for a small sample of 9 Hilat FRBs,while Wang & Yu (2016) considered α as . for FRB 121102 and its 16 repeating bursts. Note thatthe relation between the cumulative index α and our intensity distribution index a is α = a − . So,our derived index of a = 1 . ± . will correspond to α = 0 . ± . . It is significantly smallerthan Caleb et al.’s value. The difference could be caused by different sample capacity. We have 16 non-repeating FRBs in our sample. It strongly points toward a deficiency of the dimmest FRBs, which hasalso been indicated in earlier studies (Bera et al. 2016; Caleb et al. 2016; Lyutikov, Burzawa & Popov2016). There are a few factors that could lead to such a deviation. First, the total number of currentlyobserved FRBs is still very limited. It can result in a large fluctuation in the measured power-law index.In fact, Caleb et al. (2016) have estimated that at lease ∼ FRBs are needed to extract conclusiveinformation on the physical nature of FRBs. Second, FRBs are not ideal standard candles. But as longas the characteristics of FRBs does not evolve systematically with the distance, the index will not beaffected too much. A wider brightness range will only result in a larger error box for α , which canstill be reduced by increasing the FRB samples. Third, FRBs may not be homogeneous sources, theco-moving density or their brightness may evolve in space. For example, the co-moving FRB densitymay be smaller when the distance increases, or farther FRBs may not be as fierce as those nearer to us.Fourth, the space may not be a flat Euclidean space, such as for a curved Λ -CDM cosmology. In thiscase, the deviation of α from 3/2 can be used as a probe for the cosmology (Caleb et al. 2016). Finally, itshould also be noted that the apparent deficiency of the dimmest FRBs could also be due to the selectioneffect. Much weaker FRB events may actually have happened in the sky, but we were not able to recordthem or find them due to current technical limitations. To make clear which of the above factors has ledto the smaller α , more new FRB samples would be necessary in the future.As addressed above, the apparent intensity distribution function derived here is still a preliminaryresult. We need much more samples to determine the power-law index more accurately. With an enor-mous effective area for collecting radio emissions, the Chinese FAST telescope is very suitable for FRBobservations. It may be able to increase the FRB samples at a rate of ∼ events per year (assuming aneffective observation time of 2000 hours). More importantly, FAST can operate in a very wide frequency range and can hopefully provide detailed spectrum information for FRBs. It is expected to be a powerfultool in the field.At present, whether FRBs are beamed or not is still an unclear but important problem. If FRBs arehighly collimated, the actual energy released will be much smaller than the currently estimated energybase on an assumed solid angle of ∼ (Huang & Geng 2016). The emission mechanisms of FRBswill then involve complex jet effects (Borra 2013). Studying the jet effects of FRBs will be an importanttask and it will help us understand the explosion mechanisms of FRBs. When more FRBs are observedand localized, we may be able to get useful information on the beaming effects from direct observations. Acknowledgements
This study was supported by the China Ministry of Science and Technology un-der State Key Development Program for Basic Research (973 program, Grant Nos. 2014CB845800,2012CB821802), the National Natural Science Foundation of China (Grant Nos. 11473012, U1431126,11263002) and the Strategic Priority Research Program (Grant No. XDB09010302).
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