Constraining the Beaming of Gamma- Ray Bursts with Radio Surveys

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Citation Perna, Rosalba, and Abraham Loeb. 1998. “Constraining the Beaming of Gamma-Ray Bursts with Radio Surveys.” The Astrophysical Journal 509 (2): L85–88. https:// doi.org/10.1086/311784.

Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:41393210

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CONSTRAINING THE BEAMING OF GAMMA-RAY BURSTS WITH RADIO SURVEYS Rosalba Perna and Abraham Loeb Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Received 1998 October 6; accepted 1998 October 22; published 1998 November 10

ABSTRACT The degree of beaming in gamma-ray bursts (GRBs) is currently unknown. The uncertainty in the g-ray±beaming ∝ 2 ∝ Ϫ2 angle, vb, leaves the total energy release (vbb ) and the event rate per galaxy (v ) unknown to within orders of magnitude. Since the delayed radio emission of GRB sources originates from a mildly relativistic shock and receives only weak relativistic beaming, the rate of radio-selected transients with no GRB counterparts can be Ϫ2 ∼ used to set an upper limit onvb . We ®nd that a VLA survey with a sensitivity of 0.1 mJy at 10 GHz could տ # 4 ⅙ Ϫ2 identify 2 10 (vb /10 ) radio afterglows across the sky if each source is sampled at least twice over a period of 1 month or longer. From the total number of տ0.1 mJy sources observed at 8.44 GHz and the fraction տ ⅙ of fading sources at 1.44 GHz, we get the crude limitvb 6 . Subject heading: gamma rays: bursts

1. INTRODUCTION ros, Rees, & Wijers 1998). The search for long-wavelength transients with no GRB counterparts can set important con- Gamma-ray burst (GRB) sources were discovered histori- straints on vb and hence on these models. cally in g-rays because they are rare and hence require con- In this Letter, we show that a 0.1 mJy radio survey at the tinuous monitoring of large areas of the sky. The necessary VLA1 (analogous to the FIRST survey; Helfand et al. 1996) all-sky monitoring program was ®rst made feasible in the g-ray can provide strong constraints on vb. The surveyed sources need regime, hence the name GRBs. By now, the prompt g-ray to be sampled at least twice over a timescale on the order of emission is known to be followed in most cases by delayed a month or longer, since short-term variability may also be emission in the X-ray (Costa et al. 1997), optical (van Paradijs caused by scintillations of steady pointlike sources. In our cal- et al. 1997; Fruchter et al. 1998a, 1998b), and radio (Frail 1998; culations, we assume that the GRB rate is proportional to the Frail et al. 1998) bands. Given the recent discovery of after- cosmic star formation rate and neglect the small fraction glows, it is now timely to explore the feasibility of searching (Շ10%) of GRBs that might be related to local radio super- for GRB sources at longer wavelengths than traditionally at- novae (Bloom et al. 1998). We model the time-dependent lu- tempted. The importance of complementary searches is high- minosity of the radio afterglows based on observational data lighted by the possibility that some sources might be g-ray and allow for a scatter in their peak luminosities. Our model faint for geometric or physical reasons. Such sources can only is described in detail in § 2. The number count of radio after- be found at long wavelengths, e.g., in radio surveys. Indeed, glows that it predicts for radio surveys is calculated in § 3. By if the g-ray emission is beamed, then there should be a pop- comparing this number with that of fading sources in existing ulation of radio afterglows that are g-ray faint (Rhoads 1997). radio surveys, we get a lower limit on vb. Finally, our main This follows from the fact that the bulk Lorentz factor of the conclusions are summarized in § 4. emitting material in GRB sources declines with time and is only of order unity when the late radio emission takes place (Waxman, Kulkarni, & Frail 1998), hence making the effect of relativistic beaming weak at that time. 2. STATISTICS OF RADIO GRB AFTERGLOWS The evolution of a jet resembles that of a spherical ®reball The isotropy of GRBs (Meegan et al. 1993; Briggs et al. (i.e., the jet behaves like a conical section of a spherical ®re- 1993) and the ¯attening of their number count distribution at ball), as long as the expansion Lorentz factor is larger than the faint ¯uxes suggest that most GRBs occur at cosmological 1 Ϫ1 inverse of the jet opening angle,G vb . The smooth power- distances. This hypothesis has been con®rmed by the detection law decline of optical afterglows, observed over a timescale of Fe ii and Mg ii absorption lines at a redshift of z ϭ 0.835 of days to months for GRB 970508 and GRB 970228 (Livio in the optical spectrum of GRB 970508 (Metzger et al. 1997), et al. 1997; Sokolov et al. 1998), suggests that these ®reballs the inference of a redshiftz ϭ 3.42 for the host galaxy of GRB behaved as if they were spherically symmetric on angular scales 971214 (Kulkarni et al. 1998), and the detection of emission ∼ 11∼ ranging from 1/G (1 day) 83to 1/G (1 month) (Rhoads and absorption lines atz ϭ 0.966 for the host of GRB 980703 1997, 1998; Waxman 1997a). However, there is still a missing (Djorgovski et al. 1998). ∼ 53 gap in afterglow observations on the time window of minutes The energy scale of cosmological GRBs, 10 fb ergs (Kul- to several hours following a GRB. During this time, the ®reball karni et al. 1998), corresponds to the rest mass energy of a decelerates fromG ∼ 100 toG ∼ 10 , and so there is uncertainty fraction of a solar mass and implies a link between their energy about the structure of the ®reball on angular scales of ∼0.01±0.1 budget and compact stars. Indeed, the most popular models for rad. It is still possible that the highly relativistic expansion with the origin of GRBs relate them to compact stellar remnants, G ∼ 10 2 is restricted only to a very small angular diameter, ∼ ϭ Њ vb 1/G 0.6. In fact, the popular GRB scenarios of binary 1 The VLA is a facility of the National Radio Astronomy Observatory, which coalescence of compact stars or failed supernovae favor strong is operated by Associated Universities, Inc., under contract with the National collimation over spherical expansion (Woosley 1998; MeÂszaÂ- Science Foundation. L85 L86 PERNA & LOEB Vol. 509

≈ such as neutron stars or black holes (see, e.g., Eichler et al. 980703 yield AFnm S 0.7 mJy. For the above cosmology, this ≈ # 31 Ϫ2 Ϫ1 Ϫ1 1989; Narayan, PaczynÂski, & Piran 1992; Usov 1992; Woosley implies AL nm S 1.5 10 h 70 ergs s Hz . 1993; MeÂszaÂros et al. 1998). Since these remnants form out of The calculation of the afterglow number counts should also massive stars not long after their birth, it is reasonable to assume incorporate the fact that not all GRBs have afterglows at radio that the GRB rate traces the star formation rate without a sig- frequencies. The fraction fradio of radio-loud GRBs is still un- ni®cant delay. This scenario was investigated by Wijers et al. certain, but the observation that four out of ∼15±17 well- (1997), who derived a best-®t constant of proportionality be- localized GRBs (namely, GRB 970508, GRB 980329, GRB tween the GRB occurrence rate, R(z), and the star formation 980519, and GRB 980703) were detected in the radio (D. Frail Ӎ rate per comoving volume, r(z), based on the requirement that 1998, private communication) implies that fradio 25%. Also, the former ®ts the observed number count distribution of GRBs. the afterglow radio ¯ux might have a break in its power-law The cosmic star formation rate as a function of redshift has evolution and decay more rapidly than predicted by equation been calibrated based on observations of the U- and B-band (1) after a particular time (e.g., during the nonrelativistic phase luminosity density evolution in the Hubble Deep Field (Madau of the shock hydrodynamics). We arti®cially introduce a free et al. 1996; Madau, Pozzetti, & Dickinson 1998). The ®t made parameter, tcutoff, after which we set the emitted radio ¯ux to by Wijers et al. (1997) yields a local GRB rate of R(z ϭ zero. Current data do not allow an empirical calibration of Ϫ9 Ϫ3 Ϫ1 ϭ ϭ # ע ϭ 0) (0.14 0.02) 10 Mpc yr in an Q 1, L 0, tcutoff, and so we leave it as a free parameter. We assume that ϭ Ϫ1 Ϫ1 տ and H0 70 km s Mpc cosmology. tcutoff 3 months since a simple power-law decline was ob- We model the frequency and time dependence of the after- served in GRB 970508 for over 3 months (Waxman et al. 1998). glow luminosity, L n(t), using the simplest unbeamed synchro- The late phase of the radio emission, several months after a tron model (see, e.g., Waxman 1997a, 1997b), with a luminosity GRB event, originates from semirelativistic material with per unit emitted frequency nÄ of G ∼ 1 and receives only weak relativistic beaming. However, the g-ray emission is emitted by material with G տ 10 2 and Ϫa K Ä could be con®ned to a cone with an angular diameter vb 1. ϭ n LnÄ(t) L nm , (1) In this case, the rate of radio afterglows would be enhanced [ ] Ϫ ⅙ Ϫ nm(t) 1 ϭ 2 ϭ # 2 2 by a factor fb 4p/p(vb /2) 5.25 10 (vb /10 ) , rela- tive to the unbeamed case. ϭ # 2 ϩ 1/2 Ϫ3/2 where nm(t) 8.8 10 (1 z) (t/month) GHz, t is time Typical radio surveys do not provide continuous monitoring at the source frame, and z is the source redshift. The observed of the sky, but rather they provide a sequence of ªsnapshotsº frequency n is related to the emitted frequency nÄ through of each resolution element on the sky. In observations where n ϭ nÄ/(1 ϩ z). The spectral index a is chosen to have the values the minimum detectable ¯ux of a source is F , the total number ϭ 1 ≤ ϭ 1 n a1 for n nm and a2 0.7 for n nm, so as to match the 3 of events across the sky that are brighter than Fn at observed temporal decay slope observed for GRB 970228 (Fruchter et frequency n is given by al. 1998b) and GRB 970508 (Galama et al. 1998). We consider a population of afterglow sources characterized ϱ by a total comoving rate per unit volume, R(z), and by a peak N(1 F ; n) ϭ f Ϫ1f ͵ dL P(L )N 0(1 F ; n, L ), ϭ ϩ 2 n b radio nm nm n nm ¯ux Fnm(z, L nm ) L nm(1 z)/4pDL (z), where DL(z) is the cos- 0 mology-dependent luminosity distance. For consistency with the numbers derived by Wijers et al. (1997), we assume Q ϭ (4) ϭ ϭ ϭ ϩ Ϫ ͱ ϩ 1, L 0, h 0.7, and DL(z) (2c/H0 )(1 z 1 z). Note that if the radio ¯ux of GRB sources is strictly proportional to where their g-ray ¯ux, then our number count results are independent z (L ) lim nm of the choice of cosmological parameters to zeroth order; this 0 dVc N (1 F ; n, L ) ϭ ͵ R(z)t∗(z, F , n, L ) dz. follows from the fact that Wijers et al. (1997) calibrate their n nm n nm 0 dz results based on GRB number count data. (5) We allow for a spread in the peak luminosity Lnm of the afterglows by using a lognormal probability distribution (and ϭ 3 ϩ Ϫ ͱ ϩ 2 ϩ Ϫ7/2 thus minimizing the number of free parameters), Here dVc 4(c/H0 ) (1 z 1 z) (1 z) dQ dz is the comoving volume element within a solid angle dQ and redshift 2 1 [ln (L ) Ϫ ln (L )] dL interval dz, and zlim(L nm ) is found from the algebraic relation ϭ Ϫ nm ? nm ϭ P(L nm )dL nm exp 2 . F (z , L ) F . The time t∗(z, F , n) represents the duration ͱ2 2 { 2j } L nm lim nm n n pj nm over which an event at a redshift z is brighter than the limiting (2) ¯ux Fn at the observed frequency n,

2 The mean of this distribution equals AL S ϭ L exp (j /2). We Ϫ nm ? (1 ϩ z)n 2/3 normalize its value by matching the mean peak ¯ux, t∗(z, F , n, L ) ϭ min t , n nm cutoff [ ] ( nm0 ∫ϱ ∫ϱ ϩ 0 dL nmP(L nm ) 0 dz[R(z)/(1 z)]Fnm(z, L nm )dVc AF S ϭ ϱ ϱ , Ϫ Ϫ nm ∫ dL P(L ) ∫ dz[R(z)/(1 ϩ z)]dV F 2/3a2 F 2/3a1 0 nm nm 0 c # n Ϫ n , {[F (z, L )] [F (z, L )] } nm nm nm nm ) (3) (6) to its observed value in radio afterglows. The observed 8.44 { # 2 ϩ 1/2 3/2 GHz light curves of GRB 970508, GRB 980329, and GRB where nm0 8.8 10 (1 z) GHz month . No. 2, 1998 CONSTRAINING BEAMING OF GRBs L87

Fig. 1.ÐNumber of radio-loud afterglows that would be detected in an all- Fig. 2.ÐSame as in Fig. 1, but with a ®xed j r 0 and different values sky radio survey as a function of its ¯ux detection threshold Fn at 10 GHz. of tcutoff. The different lines correspond to different values of the width of the GRB luminosity function, j (see eq. [2]). The fraction of the radio-loud afterglows of GRB afterglows. This limit can be improved by considering ϭ is assumed to be fradio 25%, and the afterglow emission is truncated after a only those sources that are unresolved and, at the same time, ϭ time tcutoff 6 months in all sources. The normalization factor of the number variable over a timescale of a month or longer. Still tighter counts involves the opening angle of the -ray emission, , in units of 10Њ. g vb constraints can be obtained by considering only the subset of all unidenti®ed sources that, after a suf®ciently long time (տ6 months), fade away below the detection threshold. From a deep 3. CONSTRAINTS FROM RADIO SURVEYS VLA survey, Windhorst et al. (1993) derived a maximum like- The VLA FIRST survey monitored 1550Њ of sky with a lihood ®t to the source number counts at 8.44 GHz in the ¯ux sensitivity of 1 mJy at a frequency of 1.5 GHz (Helfand et al. range of 14.5 mJy±1.5 mJy. By comparing a Westerbork and 1996). The intervals between the observations of each source a VLA survey of the same ®eld at 1.4 GHz, Oort & Windhorst ranged between 3 minutes and 3 weeks. Unfortunately, the (1985) found that about 10% of the sources were variable on ∼ properties of this survey were not optimal for the purpose of a timescale of a year, and only 3% were detected in one survey ∼ identifying GRB afterglows. Here we consider a hypothetical but not in the other. Thus, only 3% of the sources in this survey that is optimized for this purpose. Such a survey would survey qualify as candidates for GRB afterglows. The sub- use a higher frequency in order to avoid the suppression of the millijansky sources were typically found to be unresolved. Bet- afterglow ¯ux by synchrotron self-absorption at the source (typ- ter angular resolution could, in principle, limit further the num- ically observed at Շ5[(1 ϩ z)/2]Ϫ1 GHz). Moreover, since ber of afterglow candidates by resolving other source short-term variability could also be caused by scintillations, the populations. surveyed sources should be monitored at least twice over an Figure 3 shows the ratio between the predicted number of 1 extended period on the order of a month or more. After a few GRB afterglows N( Fn ) and the total observed number of radio months, all the radio-loud afterglows would fade signi®cantly sources Nobs (Windhorst et al. 1993) at 8.44 GHz. We denote and thus be easily distinguishable from other variable source the fraction of variable sources that fade away after a year by populations. The variability level of ∼0.1±1 mJy sources can ffade and the fraction of pointlike (unresolved) sources by be determined accurately, given the high sensitivity of the VLA. fpoint. Based on the observed abundance of unresolved variable Figure 1 shows the number of radio afterglows that are ex- sub-millijansky sources at 1.4 GHz (Oort & Windhorst 1985), ϭ pected to be found in an all-sky survey at 10 GHz, as a function we set ffade fpoint 3%. The ratio between the predicted after- of its ¯ux threshold. The plotted number depends on a nor- glow counts and the observed radio counts peaks at around malization factor involving the -ray±beaming angle in units 0.4±0.6 mJy. From the requirement that this ratio be smaller g vb տ ⅙ of 10Њ. We assume f ϭ 25% and show results for different than unity, we get the lower limit vb 6 . This limit is based radio on the variability statistics of only ∼102 sources (Oort & values of j while keeping tcutoff ®xed at 6 months. Figure 2 shows the same for j r 0, but for different values of t . Windhorst 1985) and hence suffers from large statistical un- cutoff ∝ Ϫ1/2 Typically, a 0.1 mJy survey similar to FIRST but at 10 GHz certainties (min {vb} ffade ). Surveys with a larger coverage տ # 4 ⅙ Ϫ2 of the sky are necessary in order to ®rm up the statistical would identify 2 10 (vb /10 ) afterglows across the sky if the g-ray emission from GRBs is collimated to within an signi®cance of this limit. opening angle vb. 4. CONCLUSIONS Lower limits on the value of vb can already be placed by existing radio surveys. The total number of radio sources ob- We have found that a 0.1 mJy all-sky survey at 10 GHz ∼ # 4 ⅙ Ϫ2 served on the sky provides a robust upper limit on the number should identify 2 10 (vb /10 ) GRB afterglows and there- L88 PERNA & LOEB Vol. 509

fore could place important constraints on the GRB beaming տ angle, vb. In fact, the total number of 0.1 mJy sources on the sky at 8.44 GHz (Windhorst et al. 1993) and the fraction of fading unresolved sources at 1.4 GHz (Oort & Windhorst 1985) տ ⅙ already yield the crude lower limit vb 6 . This result is only weakly sensitive to the width of the afterglow luminosity func- tion (see Fig. 1) and is mainly uncertain because of the limited size of the sample of sub-millijansky sources that had been monitored for long-term variability (Oort & Windhorst 1985). Nevertheless, the derived constraint is interesting, given the fact that the g-ray emission in GRB sources originates from material with a bulk Lorentz factor G տ 10 2 and could have ∼ Շ Њ been collimated to within an angle vb 1/G 0.6, which is narrower by more than an order of magnitude than our limit. In comparison, radio jets in quasars are collimated to within ∼15Њ and possess Lorentz factors G Շ 10 (Begelman, Bland- ford, & Rees 1984). A future VLA survey, optimized to search for radio after- glows by monitoring sub-millijansky source variability at ∼10 GHz over a timescale of several weeks to several months, could

improve the above upper limit on vb considerably. Alternatively, such a search might identify a new class of radio afterglow Fig. 3.ÐThe ratio between the predicted cumulative number of GRB af- events that fade away after a few months and have no GRB 1 terglows N( Fn) and the observed cumulative number of radio sources Nobs counterparts.

(Windhorst et al. 1993), as a function of the ¯ux threshold Fn at 8.44 GHz. The fraction of unresolved fading .40%ע The observed number is uncertain by ϭ sources in the observed population is assumed to be ffade fpoint 3%, based on the 1.4 GHz data (Oort & Windhorst 1985). Model parameters are chosen as in Fig. 1, with j r 0. The condition that the plotted ratio be less than unity We thank Dale Frail for useful discussions. This work was տ ⅙ at any ¯ux threshold sets a lower limit on the g-ray±beaming angle vb 6 . supported in part by the NASA grant NAG5-7039.

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