Radio Afterglows and Host Galaxies of Gamma-Ray Bursts
Long-Biao Li, Zhi-Bin Zhang, Yong-Feng Huang, Xue-Feng Wu, Si-Wei Kong, Di Li, Heon-Young Chang, Chul-Sung Choi
aa r X i v : . [ a s t r o - ph . H E ] M a y Mon. Not. R. Astron. Soc. , 1–10 (2014) Printed 14 August 2018 (MN L A TEX style file v2.2)
Radio Afterglows and Host Galaxies of Gamma-Ray Bursts
Long-Biao Li , Zhi-Bin Zhang , ⋆ , Yong-Feng Huang † , Xue-Feng Wu ,Si-Wei Kong , Di Li , , Heon-Young Chang and Chul-Sung Choi Guizhou University, Department of Physics, College of Sciences, Guiyang 550025, China Department of Astronomy, Nanjing University, Nanjing 210093, China Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210008, China Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China National Astronomical Observatories of China, Chinese Academy of Sciences, 20A Datun Road, Beijing 100020, China Key Laboratory of Radio Astronomy, Chinese Academy of Sciences, China Department of Astronomy and Atmospheric Sciences, Kyungpook National University, 1370 Sankyuk-dong, Buk-gu,Daegu 702-701, Republic of Korea Korea Astronomy and Space Science Institute, 36-1 Hwaam, Yusong, Daejon 305-348, Republic of Korea Department of Physics and Astronomy, University of Nevada, Las Vegas, NV 89012, USA
14 August 2018
ABSTRACT
Considering the contribution of the emission from the host galaxies of gamma-raybursts (GRBs) to the radio afterglows, we investigate the effect of host galaxies onobservations statistically. For the three types of events, e.g. low-luminosity, standardand high-luminosity GRBs, it is found that a tight correlation exists between theratio of the radio flux (RRF) of host galaxy to the total radio peak emission and theobservational frequency. Especially, toward lower frequencies, the contribution fromthe host increases significantly. The correlation can be used to get a useful estimatefor the radio brightness of those host galaxies which only have very limited radioafterglow data. Using this prediction, we re-considered the theoretical radio afterglowlight curves for four kinds of events, i.e. high-luminosity, low-luminosity, standard andfailed GRBs, taking into account the contribution from the host galaxies and aimingat exploring the detectability of these events by the Five-hundred-meter ApertureSpherical radio Telescope (FAST). Lying at a typical redshift of z = 1, most of theevents can be detected by FAST easily. For the less fierce low-luminosity GRBs, theirradio afterglows are not strong enough to exceed the sensitivity limit of FAST at suchdistances. However, since a large number of low luminosity bursts actually happenvery near to us, it is expected that FAST will still be able to detect many of them. Key words: gamma–ray burst: general – methods: numerical – methods: statistical
Gamma-ray bursts (GRBs) are bright flashes of gamma-raysthat happen randomly in the sky. They were serendipitouslydiscovered in 1967 by the Vela satellites (Klebesadel et al.1973), but were poorly understood until Feb 28, 1997 whenthe first afterglow was detected (Groot et al. 1998). Sub-sequently, the counterpart of GRB 970508 was detected asthe first afterglow in radio bands (Frail et al. 1997). Thedetection of afterglows makes it possible for us to obtainbroadband observational data, to identify the host galax-ies, and to determine the redshifts of GRBs. The so called ⋆ E-mail: [email protected] † E-mail: [email protected] fireball-shock model was developed to explain the main fea-tures of GRBs and their afterglows (Rees & M´esz´aros 1994;Piran 1999; Zhang 2007), the latter are generally believed toarise from the interaction of the fireball with the surround-ing interstellar medium (ISM) (M´esz´aros & Rees 1997; Prian2000; M´esz´aros 2002).According to the fireball-shock model (e.g. M´esz´aros2002; Piran 2004; Zhang & M´esz´aros 2004; Zhang 2014), theoutflow of a GRB interacts with the ISM to form an externalshock. The shock accelerates electrons. At the same time, afraction of the shock energy is transferred to the magneticfield. The afterglow emission arises from synchrotron radi-ation of these accelerated electrons due to their interactionwith the magnetic filed. Within this framework, the mainfeatures of GRB afterglows can be well explained. c (cid:13) Table 1.
Observational properties of GRBs and their host galaxies in radio bands.
GRB E iso a T
90 a z a Frequency Host Flux Density Peak Flux Density b Peak Time References c RRF(10 erg) (s) (GHz) ( µ Jy) ( µ Jy) (days) (%)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)Low-Luminosity GRBs020903 n ±
66 294 ±
91 36 .
73 28,9 14 . ± . ±
34 782 ±
28 36 . ± . . ± . ±
30 1058 ±
19 23 .
80 28,9 18 . ± . n ± h ±
60 65 . ± . . ± . ±
60 828 ±
28 58 . ± . . ± . ±
52 724 ±
19 48 ± . . ± . n . ± . ±
36 48 . ± . . ± . ±
51 373 ±
36 49 ± . . ± . n . ± . ±
58 4 . . ± . ±
32 245 ±
50 3 . ± . . ± . . ± . ±
83 2 ± . . ± . n ±
53 1218 ±
12 47 . ± . . ± . ±
32 381 ±
19 179 . ± . . ± . ±
44 780 ±
13 57 . ± . . ± . ±
19 958 ±
11 37 . ± . . ± . ± . h ±
81 25 . ± . . ± . . ± . h ±
30 9 . ± . . ± . . ± . h ±
30 10 ± . . ± . ± . ±
34 8 . ± . . ± . ±
33 240 ±
53 4 . . ± . ± h ±
24 14 . ± . . ± . ± h ± † .
56 4,2 22 . ± . ±
52 897 ±
39 27 ± . ± . ± h ±
22 18 . ± . . ± . ±
25 161 ±
20 31 . ± . . ± . . ± . ±
22 27 .
01 17,9 19 . ± . ±
16 162 ±
13 13 . ± . . ± . . ±
23 291 ±
21 12 . ± . . ± . ±
28 470 ±
26 32 . ± . . ± . . ± . ±
23 18 . ± . . ± . n ±
36 60 ± † .
85 11,11 20 . ± . n ±
109 2232 ±
30 78 . ± . . ± . ± . ±
33 32 . ± . . ± . ± . ±
28 17 . ± . . ± . n ±
27 156 ±
34 7 . . ± . ± . ±
20 140 . ± . . ± . ±
49 1028 ±
16 84 . ± . ± . ± . ±
114 12 . ± . . ± . ±
47 341 ±
41 4 . ± . . ± . ±
64 686 ±
26 16 . ± . . ± . ± . ±
21 10 . ± . . ± . ±
27 496 ±
24 13 ± . ± . ±
40 331 ±
30 4 . . ± . ±
22 57 ± † . . ± . ± . ±
31 7 . ± . . ± . ± . ± † .
54 32,32 28 . ± . ±
37 171 ±
14 91 . ± . . ± . ±
11 332 ±
11 33 . ± . . ± . . ± . ± † .
65 21,21 3 . ± . ± . ± † .
70 31,31 5 . ± . n
110 60 0.706 1.43 54 . ± . ±
49 8 . ± . . ± . . ± . ±
24 7 . ± . . ± . ± . ± † .
49 13,13 2 . ± . ± h ± † .
59 4,25 19 . ± . n
880 500 1.059 4.86 10 ±
25 71 ±
23 11 . . ± . . ± . ±
33 3 . ± . . ± . ± . ±
31 16 . ± . . ± . ± h ±
24 12 . ± . . ± . ± h ±
47 2 . . ± . . ± . ± † .
35 17,17 11 . ± . ±
42 349 ± † .
15 17,17 6 . ± . . ± . ± † .
24 17,17 9 . ± . ±
18 171 ±
23 6 . ± . ± . ±
49 377 ±
53 14 . ± . . ± . ±
51 256 ± † .
15 17,6 8 . ± . ±
30 634 ± † .
93 6,6 12 . ± . ±
27 76 ±
14 35 . ± . ± . ±
23 268 ±
32 5 . ± . . ± . c (cid:13)000
32 5 . ± . . ± . c (cid:13)000 , 1–10 adio Afterglows and Host Galaxies of GRBs Table 1 (continued).
Observational properties of 50 GRBs and their host galaxies in radio bands.
GRB E iso a T
90 a z a Frequency Host Flux Density Peak Flux Density b Peak Time References c RRF(10 erg) (s) (GHz) ( µ Jy) ( µ Jy) (days) (%)(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)High Luminosity GRBs070125 955 60 1.548 4.86 133 ±
21 308 ±
78 27 . . ± . ±
18 1028 ±
16 84 . ± . ± . ±
52 224 ±
54 3 . . ± . ±
45 616 ±
57 6 . ± . . ± . ±
38 243 ±
13 15 . ± . ± . ±
16 84 ±
16 14 . ± . . ± . ±
16 524 ±
19 8 ± . . ± . ±
27 330 ±
47 17 . ± . . ± . ±
28 205 ±
23 12 . ± . . ± . ±
61 222 ±
33 7 . . ± . ± . ±
68 4 . ± . . ± . ±
24 480 ±
69 7 ± . . ± . ±
48 473 ±
28 6 . ± . . ± . ±
42 518 ± † .
69 17,17 12 . ± . ±
69 623 ± † .
01 17,17 7 . ± . ± . ±
24 14 . ± . . ± . Notes. a Refer to Chandra & Frail (2012). b Observed radio peak flux density. c References are in the following order: host radio flux density, radio peak flux density.Abbreviations for the references are as follows: (1) Berger et al. (2000), (2) Berger et al. (2001a), (3) Berger, Kullarni & Frail (2001b), (4)Berger et al. (2003a), (5) Berger et al. (2003c), (6) Cenko et al. (2006), (7) Cenko et al. (2011), (8) Chandra et al. (2008), (9) Chandra & Frail(2012), (10) Djorgovski et al. (2001), (11) Fox et al. (2003), (12) Frail et al. (1999), (13) Frail et al. (2000a), (14) Frail et al. (2000b), (15)Frail, Waxman & Kulkarni (2000), (16) Frail et al. (2006), (17) Frail & Chandra (2014, private communication), (18) Galama et al. (2003),(19) Harrison et al. (2001), (20) Jakobsson et al. (2005), (21) Kulkarni et al. (1999), (22) Micha lowski et al. (2012), (23) Moin et al. (2013),(24) Perley & Perley (2013), (25) Piro et al. (2002), (26) Price et al. (2002), (27) Soderberg et al. (2004a), (28) Soderberg et al. (2004b), (29)Soderberg et al. (2006), (30) Soderberg et al. (2007), (31) Taylor et al. (2000), (32) Yost et al. (2002). h Host flux densities have been reported in Berger, Kullarni & Frail (2001b), Berger et al. (2003a), Micha lowski et al. (2012) and Perley & Perley(2013). n SN/GRB, i.e. SN-associated GRBs. † These values are the maximum observed flux densities, which are taken as the observed peak flux densities.
With more and more afterglows being detected, thestudy of GRBs has entered an era of full wavelengths (e.g.Gehrels & Razzaque 2013). However, according to Hancock,Gaensler & Murphy (2013),the detection rate of afterglowsis only ∼
30 % at radio wavelengths, although some authorshave recently compiled several larger datasets (de UgartePostigo et al. 2012; Chandra & Frail 2012; Ghirlanda et al.2013; Staley et al. 2013). Note that the search for radio af-terglows is already in great depth (Chandra & Frail 2011).Another exclusive property of radio afterglows is their de-tection at high redshifts (Frail et al. 2006; Chandra et al.2010). For example, there are about 350 GRBs with red-shifts measured, of which ∼
32% are at redshifts z > ∼
7% at z > ∼
2% at z >
6. The maximum redshift is9.4 for GRB 090429B, indicating that GRBs are potentiallypowerful probes of the early Universe. In fact, GRBs’ highluminosities make them potentially detectable up to veryhigh redshifts (Lamb & Reichart 2000). They may be ob-servable with the current VLA up to z ∼
30 at frequencieshigher than ∼ z = 5. They also argued that FAST can evendetect radio afterglows at ν = 2 .
50 GHz up to z = 15 oreven more. However, they did not consider the contributionfrom the host galaxies of GRBs. In fact, the observed ra-dio emission should include the afterglow component as wellas the contribution from its host galaxy. In order to modelthe observed radio afterglows and evaluate the detectabilityof FAST more realistically, it is necessary to consider thecontribution from the host galaxy.In this study, we have collected a large sample of radioafterglows. We examine the relative contribution from theafterglows and their hosts based on our sample. For thispurpose, we propose a useful method based on the flux ratioof the host to the afterglow to estimate radio flux density ata given frequency for different kinds of bursts. Taking intoaccount the contribution from the hosts, we then re-examinethe detectability of FAST for those weak radio afterglows atvery high redshifts. Our paper is organized as follows. Webriefly introduce the GRBs of our sample in Section 2. In c (cid:13) , 1–10 Section 3, we present our statistic results on the ratio of thehost flux to radio peak flux (RRF). In Section 4, we presentour theoretical radio afterglow light curves and evaluate thedetectability of FAST. Our conclusions and discussion aregiven in Section 5.
As of Dec 2013, the number of observed radio afterglowsis about 95, which is only ∼
30 % of all the GRBs withafterglows being observed. In practice, the observed radioemission should be composed of the GRB afterglow compo-nent and the contribution from the host galaxy. For GRB980703, Berger, Kullarni & Frail (2001b) found that there isa constant component among the observed radio emission,which was interpreted as the first example of host contribu-tion to the afterglow (see also Frail et al. 2003). They arguedthat radio and submillimetre observations of the GRB hostgalaxies will be very useful for studying the obscured starformation rate and the properties of starbursts at high red-shifts. Berger et al. (2003a) pointed out that about 20 % ofGRB host galaxies are ultra–luminous (
L > L ⊙ ) andthey can be utilized to probe a representative population ofstar–forming galaxies.For bursts with detected radio afterglows, we collect theobservational data of host galaxies together with their peakfluxes of radio afterglows at ν = 1 . ∼ . erg and 10 erg, respectively. 6 GRBswithout known redshifts are not included in the above threesub-samples, but are treated as an independent sub-samplelisted in Table 1.In Column 1 of Table 1, the GRB names are given. Theisotropic energies E iso , the T durations in the observer’sframe and the redshifts z are listed in Columns 2, 3 and 4,respectively. Column 5 is the observational frequency. Col-umn 6 tabulates the contribution from the host galaxies. Inthis column, 10 radio fluxes for 7 host galaxies have been di-rectly reported in the literature. Other host fluxes are basedon the assumption that the host flux density is constant atall stages of the afterglow so that the host flux dominatesat late times. We determine the host contribution only forthose GRBs which were monitored in radio bands for a longperiod and whose radio light curves obviously become flatat the final stages.Columns 7 and 8 tabulate the observed peak flux den-sities and the corresponding time of the peak. Most of thepeak flux densities and the peak time are taken from Chan-dra & Frail (2012) which provided a sample with manyevents. They are determined by fitting the observed lightcurves. Meanwhile, for 13 GRBs whose peak flux densitiesare marked with daggers, the peak flux densities are simplythe brightest measurements. This may cause the peak fluxdensity to be somewhat under-estimated, but not by enoughto affect our results. The references of the host fluxes andthe observed peak fluxes are listed in corresponding orderin Column 9. The data observed by Frail & Chandra have been kindly offered to us by them. It should be noted thatfor some GRBs, other researchers may give different reportsof host fluxes in the literatures. For example, Berger et al.(2003b) and Soderberg et al. (2007) also reported flux den-sities for GRBs 020405 and 050416A, which are slightly dif-ferent from the data in Table 1. But these differences aregenerally not large enough to affect our results.To consider the contribution from the host galaxies, weintroduce a new parameter – the Ratio of the radio fluxes(RRF) of the host to the peak emission. The RRF is definedas RRF ≡ F host F o,peak = F host F host + F b,peak . (1)The observed peak flux in Table 1, F o,peak , is the sum of F host and F b,peak , where F host and F b,peak stand for thehost flux density and the peak flux density of the pure ra-dio afterglow component, respectively. The RRF values aregiven in Column 10 of Table 1. In this treatment, we alsoassume that the host flux density keeps constant in all thepost-burst stages, as Berger, Kullarni & Frail (2001b) oncedid. Using the RRF parameter, we can analyze the flux-fluxcorrelation between the host galaxies and the GRB after-glows. It is interesting to note that our RRF analysis is com-pletely independent of the distance (redshift) measurement. Considering the fact that most current GRB radio afterglowsare observed at 1.43, 4.86 and 8.46 GHz, we use all data inthese three bands except those without known redshifts inTable 1. In Figure 1, we plot the observed host fluxes andpeak fluxes versus frequency for different types of GRBs.The data points are relatively scattered. We then calculatethe mean values for the two fluxes at the three frequencies byassuming an equal weight for each data point in logarithmicscale. Using the mean values of the host flux densities andthe peak flux densities, we fit the correlations between thefluxes and the observational frequencies for low-luminosity,standard and high-luminosity GRBs, respectively. The re-sults are also plotted in Figure 1. Generally, the spectraof radio afterglows and host galaxies are usually consideredto be a power-law function of frequency, so we select thepower-law fit. In the three upper panels, the correlationsare F host ∝ ν +0 . ± . , F host ∝ ν − . ± . and F host ∝ ν +0 . ± . for low-luminosity, standard and high-luminosityGRBs, with the correlation coefficients and P-values (rejec-tion possibilities) being (0.47, 0.0003), (0.60, 0.004) and (-0.95, 0.003) correspondingly. The three lower panels showthat the observed peak fluxes are also correlated with thefrequency as F o,peak ∝ ν +0 . ± . , F o,peak ∝ ν +0 . ± . and F o,peak ∝ ν +0 . ± . for the three types of GRBs, withthe correlation coefficients and P-values being (0.80, 0.001),(0.61, 0.001) and (0.90, 0.002), respectively.Theoretically, the radio flux density of a GRB afterglowcan be simplified as a power-law function of time and fre-quency, say, F ∝ t α ν β (e.g. Sari, Piran & Narayan 1998; Wuet al. 2005; Paper I), where α and β are the temporal andspectral indices, respectively. Generally, the number densityof the surrounding medium is assumed to depend on the c (cid:13) , 1–10 adio Afterglows and Host Galaxies of GRBs F ho s t ( Jy ) Low luminosity GRBs F o , pea k ( Jy ) (GHz) High luminosity GRBsStandard GRBs
Figure 1.
Observed host flux densities (upper panels) and peak flux densities (lower panels) as a function of the radio frequencies forlow-luminosity (left), standard (middle), and high-luminosity GRBs (right), and our best fit to the observations. Filled circles with errorbars are the mean values of the host flux densities at the corresponding frequency. Filled triangles with error bars are the mean valuesof the observed peak flux densities. The solid line in each panel represents our best fit to the observations. For the observational dataand their references, please see Table 1 for details.
Table 2.
Best fit parameters for the linear RRF — ν correlation.GRB types Fitting Resultsa b correlation coefficient P-valueLow Luminosity GRBs 0 . ± . − . ± .
002 0.96 0.09Standard GRBs 0 . ± . − . ± .
003 0.95 0.02High Luminosity GRBs 0 . ± . − . ± .
002 0.98 0.06All GRBs 0 . ± . − . ± .
002 0.98 0.07 shock radius as n ∝ R − k . Liang et al. (2013) and Yi, Wu &Dai (2013) argued that the k value should be in the range of0 . − . k ∼
1. Ac-cording to Wu et al. (2005), in the case of homogenous ISMwith k = 0, F b,peak ∝ ν . But in the case of a typical stellarwind environment with k = 2, F b,peak ∝ ν / , assuming theradio peak is caused when the synchrotron self-absorptionfrequency, ν a , crosses the observational band. That is to say,the range of the power-law index for the F b,peak − ν relationwould be 0 − /
3. Our fitting results are consistent with thisrange.In our study, we average the original RRF values of theindividual GRBs at a frequency to get a mean RRF valueby assuming an equal weight for each observed data point.These mean values are then used in our final fitting process.For each kind of GRBs, we originally adopt two fittingfunctions, the linear function and the power-law function,to investigate the correlation between RRF and the obser- vational frequency ν . The detailed fitting functions are RRF = g ( ν ) = a + bν, Linear Case ,Aν B , Power-law Case . (2)The power-law case is proved to be worse than the linearone and will have been neglected subsequently. In Figure2, we present our best fits in the linear case for the threedifferent types of GRBs and all the GRB sample as listedin Table 1. The derived fitting parameters ( a and b ), thecorrelation coefficients and the P-values are listed in Table2. The correlation coefficients are generally high, and the P-values are small, indicating that the correlation is real and isnot a phenomenon by chance. For the different sub-samplesof GRBs in Table 1, we also use the Jackknife resamplingmethod to test the linear model and find that the linearmodel is reasonable in a statistical sense. In Figure 3, weplot the best linear fit lines for all the subsamples. We can c (cid:13) , 1–10 Low-Luminosity GRBs(N=12) RR F Standard GRBs(N=31)
High-Luminosity GRBs(N=32)
All GRBs(N=82) (GHz)
Figure 2.
The best linear fits to RRF versus ν for the low-luminosity, standard, high-luminosity GRBs and all GRBs as listed in Table1. The observed frequencies ν are the three corresponding frequencies, in addition, one averaged data point at a frequency of 4.5 GHzis also included in the second panel. N in each panel is the number of original data points used to derive the mean observational values(solid circles). see clearly that the four lines differ from each other onlyslightly.In fact, according to the definition, RRF should bewithin the range of 0 – 1. An RRF near 0 indicates thatthe afterglow component dominates the observed flux, whilean RRF close to 1 indicates that the host flux is muchlarger than F b,peak , which generally happens in low fre-quency ranges.Using Eqs. (1) and (2), for the linear case, one can easilyderive F host = g ( ν ) F o,peak = ( a + bν ) F o,peak . (3)Eq. (3) can be used to calculate the host fluxes of thoseGRBs with radio afterglows being detected but no hostsbeing identified. We can also get F host = g ( ν )1 − g ( ν ) F b,peak = a + bν − ( a + bν ) F b,peak . (4)Eq. (4) can also potentially be used to predict the flux den-sity of the peak afterglow or the host at particular frequen-cies when no direct observational data are available.This RRF-predicted host fluxes can help us to subtract RR F (GHz) Total GRBs Low-Luminosity GRBs Standard GRBs High-Luminosity GRBs
Figure 3.
Comparison of the best linear fits of the RRF- ν correla-tion for different kinds of bursts. The dash-dotted, dashed, dottedand solid lines represent the fit to low-luminosity, standard andhigh-luminosity GRBs and all GRBs, respectively.c (cid:13)000
Comparison of the best linear fits of the RRF- ν correla-tion for different kinds of bursts. The dash-dotted, dashed, dottedand solid lines represent the fit to low-luminosity, standard andhigh-luminosity GRBs and all GRBs, respectively.c (cid:13)000 , 1–10 adio Afterglows and Host Galaxies of GRBs Table 3.
Typical physical parameter values used in theoretically modeling of GRB afterglows.Source E iso γ θ n p ξ B ξ e ǫ θ obs (erg) (rad) ( cm − ) (rad)Standard GRBs a ............... 1 . ×
300 0.10 1.0 2.5 1 . × − a .................... 1 . ×
30 0.10 1.0 2.5 1 . × − a ... 1 . ×
300 0.10 1.0 2.5 1 . × − a .... 1 . ×
300 0.10 1.0 2.5 1 . × − a Refer to Paper I. the potential radio background emission from a GRB sourceso that the pure afterglow component can be identified, espe-cially at lower frequencies. The RRF- ν correlation can alsoexplain why the low-frequency radio afterglows are usuallydifficult to be detected, in contrast with the high-frequencycase. According to the RRF- ν relations, radio fluxes aredominated by the host component at lower frequencies, asshown in Figures 1-3. According to Eqs. (2), (3) and (4), the host flux density de-pends on the observing frequency. Thus, we can use the RRFat a certain frequency to calculate the host flux density ifthe peak flux density of the radio afterglow is available. Herewe use the theoretical peak flux density calculated from thefireball-shock model to estimate the host flux density. Takinginto account the contribution from the host galaxy, we willthen study the detectability of radio afterglows by FAST. AsZhang et al. (2015) did in Paper I, we also calculate the radioafterglow light curves for four kinds of GRBs, i.e. standard,failed, high luminosity, and low luminosity GRBs.
Here, we adopt the generic dynamical equations for beamedGRB outflows (e.g. Huang et al. 1998, 1999a, 1999b, 2000a,2000b) to calculate the radio afterglows of GRBs. The for-mulae of the dynamical model were widely applied in manystudies such as the overall afterglow modeling (Cheng &Wang 2003; Wu et al. 2004; Kong et al. 2009), the rebright-ening at multi-wavelengths (Dai et al. 2005; Xu & Huang2010; Kong et al. 2010; Yu & Huang 2013; Hou et al. 2014),and the beaming effect (Huang, Dai & Lu 2000c; Wei & Lu2002; Huang et al. 2003), etc. These equations are valid inboth the ultra-relativistic and non-relativistic stages, thusthey are especially convenient for calculating radio after-glows which usually last for a very long period and mayinvolve the Newtonian regime. In our calculations, the ef-fects of electron cooling, lateral expansion and equal arrivaltime surfaces are included.The parameters assumed in our calculations are listedin Table 3, where E iso is the initial isotropic energy, γ is theinitial bulk Lorentz factor, θ is the initial half-opening angleof the jet, θ obs is the angle between the axis of the jet and theline of sight, n is the number density of surrounding ISM, p isthe electron distribution index, ξ e and ξ B are respectively theenergy fractions of electrons and magnetic field with respectto the total energy. The radiative efficiency ǫ equals 1 in Table 4.
Parameters of the 9 sets of FAST’s receivers.No. Bands a ν c a flux density limit a (GHz) (GHz) ( µJy )1 0.07 - 0.14 0.10 843.52 0.14 - 0.28 0.20 238.03 0.320 - 0.334 0.328 376.54 0.28 - 0.56 0.40 63.55 0.55 - 0.64 0.60 44.56 0.56 - 1.02 0.80 18.07 1.23 - 1.53 1.38 9.58 1.15 - 1.72 1.45 7.09 2.00 - 3.00 2.50 5.5 a These data are taken from Nan et al. (2011) andPaper I. The flux density limit is the 5 σ detectionlimit of FAST for a 30-minute integration time. the highly radiative case and equals 0 in the adiabatic case.The initial Lorentz factor and isotropic energy are evaluateddifferently for the four kinds of GRBs, while the followingcommon parameters are assumed to be universal, namely n = 1 cm − , p = 2 . , ξ e = 0 . , ξ B = 0 . , θ = 0 . θ obs =0. Meanwhile, we take ǫ = 0 since we are mainly consideringthe late-time afterglows, which should be in the adiabaticregime. In order to make a comparative analysis with Paper I, were-calculate the radio afterglow light curves for four typesof GRBs according to FAST’s nine passbands, assuming atypical redshift of z = 1 .
0. As in Paper I, we also adopt thelimiting flux densities under an integral time of 30 minutesas the limiting sensitivity for FAST. The difference is thatwe now include the contribution from the host, whose flux isestimated from the linear RRF- ν correlation. Our numericalresults are plot in Figure 4. We see that the host flux densi-ties estimated from Eqs. (2) and (4) are often significantlylarger than the afterglow component at early and late times,which makes the total light curves much flatter than that inPaper I which did not consider the host galaxy contribution.This would make the radio afterglows more difficult to beidentified due to the influence of the relatively strong radiobackground.According to Figure 4, standard GRBs could be ob-served by FAST at frequencies ν > .
80 GHz, though at ν = 0 .
80 GHz, the predicted peak flux density is 19.6 µ Jy,and the 5 σ detection limit of FAST for a 30-minute integra-tion time is 18.0 µ Jy. Note that high luminosity GRBs havethe brightest radio afterglows so that they can be detected c (cid:13) , 1–10 -2 -2 -2 Standard5 Detection Limit Low Luminosity High Luminosity Failed0.10GHz0.10GHz F l u x D en s i t y ( Jy ) Time Since Burst(s)
Figure 4.
Predicted radio light curves within FAST’s nine energy passbands for four kinds of GRBs lying at z = 1 . ν case. The thin solid, dash-dotted, dashed and thick solid lines correspond to failed, high-luminosity, low-luminosity and standardGRBs, respectively. The horizontal dotted lines represent the 5 σ limiting sensivities of FAST for a 30-minute integration time (Paper I).Note that the host flux densities predicted by the linear RRF- ν correlation have been added in these light curves. -2
5 Detection Limit z=0.5z=0.1z=0.01z=0.0085 F l u x D en s i t y ( Jy ) Time Since Burst(s) z=0.001
Figure 5.
Predicted radio light curves of low-luminosity GRBsat 1.45GHz for the linear RRF- ν case. The redshifts for thethick solid, dashed, dotted, dash-dotted and thin solid lines are z = 0 . , . . , . .
5, respectively. The horizontal dotted line represents 5 σ limit-ing sensivity of FAST for a 30-minute integration time (Paper I).Note that the host flux densities predicted by the linear RRF- ν correlation have been added in these light curves. easily in most bands with ν > .
40 GHz. At ν = 0 .
40 GHz,FAST can detect high-luminosity GRBs from ∼ . ∼ ν > .
38 GHz. The reason is thatthe initial bulk Lorentz factor of failed GRBs is about tentimes smaller than that of standard ones. A lower Lorentzfactor then leads to a lower flux density at early times aswell as a lower peak flux density. However, the late timeafterglow depends mainly on the intrinsic energy of the jet,so the light curves differ from each other only slightly at latestages, especially at low frequencies (Wu et al. 2004). Theradio emission can be detected from ∼ ∼
58 daysat ν = 1 .
38 GHz. At ν > .
45 GHz, FAST can potentiallydetect very early radio emission.Radio afterglows of low-luminosity bursts are the weak-est ones in each panel of Figure 4. For example, the strongestpeak flux is only 0.14 µ Jy at ν = 2 .
50 GHz. According to ourcalculations, FAST can hardly detect any radio afterglowsof low-luminosity GRBs at the redshift of z = 1. However,many low-luminosity GRBs are likely to happen very nearto us, then they can also be observed by FAST. A good ex-ample is the low-luminosity GRB 980425 ( z = 0 . c (cid:13) , 1–10 adio Afterglows and Host Galaxies of GRBs shown in Figure 5, radio afterglows of low-luminosity GRBscan be detected by FAST at a redshift of z < .
1. Fan, Pi-ran & Xu (2006) argued that such under-luminous GRBswith less kinetic energy might be very common. We expectthat much more low-luminosity GRBs would be detected byFAST in the future.
In this study, we investigate the connection between hostfluxes and peak afterglow fluxes in radio bands statistically.The observed GRBs are classified into three types, i.e. low-luminosity, standard and high-luminosity GRBs. It is foundthat there is an anti-correlation between the RRF and theobservational frequency. At a higher frequency, the corre-sponding RRF is smaller. This could be due to more sig-nificant self-absorption at longer wavelengths (Rybicki &Lightman 1979).Based on this correlation, the host flux den-sities at different radio frequencies can be estimated. ThisRRF prediction is especially helpful for those GRBs whoseradio afterglow data are very limited. Meanwhile, we alsore-considered the capability of detecting GRBs with FAST.FAST have 9 energy channels in its first phase, ranging from0.10 GHz to 2.50 GHz (Nan et al. 2011). It will mainly op-erate in a relatively low frequency range. Our results showthat at a typical redshift of z = 1 .
0, the radio emission ofstandard GRBs can be well detected by FAST at ν > . ∼ F is a power-law function of F ∝ ν (e.g. Sari, Piran & Narayan 1998; Wu et al. 2005) if the ob-serving frequency ν is below the synchrotron self-absorptionfrequency ν ssa . The dependence of the radio afterglow lightcurves on various parameters has been investigated by someauthors, such as Chandra & Frail (2012). They found thatthe radio afterglow is relative bright when the ISM densityis between n = 1 and 10 cm − . This may explain why someGRBs bright in X-ray/optical bands are dim in radio bands.Also, the radio brightness strongly depends on the intrinsicenergy of the outflow. So, there is evidently a close relation- ship between the detectability of the radio afterglow andthe intrinsic physical parameters of the GRB (Zhang et al.2015). ACKNOWLEDGMENTS
We thank the anonymous referee for valuable commentsand suggestions that lead to an overall improvement of thisstudy. We acknowledge D. A. Frail and P. Chandra forkindly offering their invaluable radio afterglow data to us.We are thankful to B. Zhang, B.B. Zhang and Y. Z. Fanfor helpful discussions. This work is partly supported by theNational Basic Research Program of China (973 Program,Grant No. 2014CB845800) and the National Natural Sci-ence Foundation of China (Grant Nos. 11263002, 11473012and 11322328). XFW and DL acknowledges support bythe Strategic Priority Research Program “The Emergenceof Cosmological Structures” (Grant No. XDB09000000) ofthe Chinese Academy of Sciences. SWK acknowledges sup-port by China Postdoctoral science foundation under grant2012M520382. HYC was supported by BK21 Plus of theNational Research Foundation of Korea and a National Re-search Foundation of Korea Grant funded by the Koreangovernment (NRF-2013K2A2A2000525).
REFERENCES
Berger E., 2014, ARA&A, 52, 43Berger E. et al., 2000, ApJ, 545, 56Berger E. et al., 2001a, ApJ, 556, 556Berger E., Kulkarni S.R., Frail D.A., 2001b, ApJ, 560, 652Berger E., Cowie L.L., Kulkarni S.R., Frail D.A., AusselH., Barger A.J., 2003a, ApJ, 588, 99Berger E., Soderberg A.M., Frail D.A., Kulkarni S.R.,2003b, ApJ, 587, L5Berger E. et al., 2003c, Nature, 426, 154Cenko S.B. et al., 2006, ApJ, 652, 490Cenko S.B. et al., 2011, ApJ, 732, 29Chandra P., Frail D.A., 2011, Bull. Astr. Soc. India, 39,451Chandra P., Frail D.A., 2012, ApJ, 746, 156Chandra P. et al., 2008, ApJ, 683, 924Chandra P. et al., 2010, ApJ, 712, L31Cheng K.S., Wang X.Y., 2003, ApJ, 593, L85Dai Z.G., Wu X.F., Wang X.Y., Huang Y.F., Zhang B.,2005, ApJ, 629, L81de Ugarte Postigo A. et al., 2012, A&A, 538, 44Djorgovski S.G., Frail D.A., Kulkarni S.R., Bloom J.S.,Odewahn S.C., Diercks A., 2001, ApJ, 562, 654Fan Y.Z., Piran T., Xu D., 2006, J. Cosmol. Astropart.Phys., 09, 013Fox D.W. et al., 2003, ApJ, 586, L5Frail D.A., Kulkarni S.R., Nicastro L., Feroci M., TaylorG.B., 1997, Nature, 389, 261Frail D.A. et al., 1999, ApJ, 525, L81Frail D.A. et al., 2000a, ApJ, 538, L129Frail D.A. et al., 2000b, ApJ, 534, 559Frail D.A., Waxman E., Kulkarni S.R., 2000, ApJ, 537, 191Frail D.A. et al., 2003, ApJ, 590, 992Frail D.A. et al., 2006, ApJ, 646, L99 c (cid:13) , 1–10 Galama T.J., Frail D.A., Sari R., Berger E., Taylor G.B.,Kulkarni S.R., 2003, ApJ, 585, 899Gehrels N., Razzaque S., 2013, Frontier of Physics, 8, 661Ghirlanda G. et al., 2013, MNRAS, 435, 2543Groot P.J. et al., 1998, ApJ, 493, L27Hancock P.J., Gaensler B.M., Murphy T., 2013, ApJ, 776,106Harrison F.A. et al., 2001, ApJ, 559, 123Huang Y.F., Dai Z.G., Lu T., 1998, A&A, 336, L69Huang Y.F., Dai Z.G., Lu T., 1999a, Chin. Phys. Lett., 16,775Huang Y.F., Dai Z.G., Lu T., 1999b, MNRAS, 309, 513Huang Y.F., Dai Z.G., Lu T., 2000a, MNRAS, 316, 943Huang Y.F., Gou L.J., Dai Z.G., Lu T., 2000b, ApJ, 543,90Huang Y.F., Dai Z.G., Lu T., 2000c, A&A, 355, L43Huang Y.F., Lu T., Dai Z.G., Cheng K.S., 2003, in Li X.D.,Trimble V., Wang. Z.R., eds, Proc. IAU Symp. 214, HighEnergy Processes and Phenomena in Astrophysics. As-tron. Soc. Pac., San Francisco, p. 321Hou S.J., Geng J.J., Wang K., Wu X.F., Huang Y.F., DaiZ.G., Lu J.F., 2014, ApJ, 785, 113Ioka K., M´esz´aros P., 2005, ApJ, 619, 684Jakobsson P. et al., 2005, ApJ, 629, 45Klebesadel R.W., Strong I.B., Olson R.A., 1973, ApJ, 182,L85Kulkarni S.R. et al., 1999, ApJ, 522, L97Kong S.W., Huang Y.F., Cheng K.S., Lu T., 2009, Sciencein China G: Physics and Astronomy, 52, 2047Kong S.W., Wong A.Y.L., Huang Y.F., Cheng K.S., 2010,MNRAS, 402, 409Lamb D.Q., Reichart D.E., 2000, ApJ, 536, 1Levesque E.M., 2014, PASP, 126, 1Liang E.W. et al., 2013, ApJ, 774, 13Mesler R.A., Whalen Daniel J., Smidt Joseph, Fryer ChrisL., Lloyd-Ronning N.M., Pihlstr¨om Y.M., 2014, ApJ, 787,91M´esz´aros P., 2002, ARA&A, 40, 137M´esz´aros P., Rees M.J., 1997, ApJ, 476, 232Micha lowski M.J. et al., 2012, ApJ, 755, 85Moin A. et al., 2013, ApJ, 779, 105Nan R.D. et al., 2011, International Journal of ModernPhysics D, 20, 989Perley D.A., Perley R.A., 2013, ApJ, 778, 172Piran T., 1999, Phys. Rep., 314, 575Piran T., 2000, Phys. Rep., 333, 529Piran T., 2004, Reviews of Modern Physics, 76, 1143Piro L. et al., 2002, ApJ, 577, 680Price P.A. et al., 2002, ApJ, 573, 85Rees M.J., M´esz´aros P., 1994, ApJ, 430, L93Rybicki G.B., Lightman A.P., 1979, Radiative Processes inAstrophysics (New York:Wiley)Sari R., Piran T., Narayan R., 1998, ApJ, 497, L17Staley T.D. et al., 2013, MNRAS, 428, 3114Soderberg A.M. et al., 2004a, Nature, 430, 648Soderberg A.M. et al., 2004b, ApJ, 606, 994Soderberg A.M. et al., 2006, Nature, 442, 1014Soderberg A.M. et al., 2007, ApJ, 661, 982Taylor G.B., Bloom J.S., Frail D.A., Kulkarni S.R., Djor-govski S.G., Jacoby B.A., 2000, ApJ, 537, L17Wei D.M., Lu, T., 2002, MNRAS, 332, 994 Wu X.F., Dai Z.G., Huang Y.F., Ma H.T., 2004, Chin. J.Astron. Astrophys., 4, 455Wu X.F., Dai Z.G., Huang Y.F., Lu T., 2005, ApJ, 619,968Xu M., Huang Y.F., 2010, A&A, 523, A5Yi S.X., Wu X.F., Dai Z.G., 2013, ApJ, 776, 120Yost S.A. et al., 2002, ApJ, 577, 155Yu Y.B., Huang Y.F., 2013, Res. Astron. Astrophys., 13,662Zhang B., M´esz´aros P., 2004, International Journal of Mod-ern Physics A, 19, 2385Zhang B., 2007, Chin. J. Astron. Astrophys., 7, 1Zhang B., 2014, International Journal of Modern PhysicsD, 23, 30002Zhang Z.B., Kong S.W., Huang Y.F., Li D., Li L.B., 2015,Res. Astron. Astrophys., 15, 237 (Paper I)Zhang Z.B. et al., 2015, ApJ, to be submitted. c (cid:13)000