Evidence of High-latitude Emission in the Prompt Phase of GRBs: How Far from the Central Engine are the GRBs Produced?

One of the difficulties in nailing down the physical mechanism of gamma-ray bursts (GRBs) comes from the fact that there has been no clear observational evidence on how far from the central engine the prompt gamma rays of GRBs are emitted. Here we present a simple study addressing this question by making use of the “high-latitude emission” (HLE). We show that our detailed numerical modeling exhibits a clear signature of HLE in the decaying phase of “broad pulses” of GRBs. We show that the HLE can emerge as a prominent spectral break in F ν spectra and dominate the peak of ν F ν spectra even while the “line-of-sight emission” (LoSE) is still ongoing. This finding provides a new view of HLE emergence since it has been believed so far that the HLE can show up and dominate the spectra only after the LoSE is turned off. We remark, however, that this “HLE break” can be hidden in some broad pulses, depending on the proximity between the peak energies of the LoSE and the HLE. Therefore, this new picture of HLE emergence explains both the detection and nondetection of HLE signature in observations of broad pulses. Also, we present three examples of Fermi Gamma-ray Burst Monitor GRBs with broad pulses that exhibit the HLE signature. We show that their gamma-ray-emitting region should be located at ∼1016 cm from the central engine, which places a constraint on the GRB models.


INTRODUCTION
The gamma-ray bursts (GRBs) are believed to invoke highly relativistic jets with bulk Lorentz factors of a few hundreds (Kumar & Zhang 2015).For such a highly relativistic jet, the relativistic beaming and boosting of radiation plays an important role and gives rise to interesting results especially when combined with a spherical geometry of the emitting surface.The photons emitted from a jet location with high latitude, called the "highlatitude emission" (HLE), take longer to reach a distant observer and are boosted with a smaller Doppler factor than the photons traveling along the line of sight, called the "line-of-sight emission" (LoSE).These two aspects of HLE are known as the "curvature effect" of a relativistic spherical jet.It is known that the HLE satisfies a simple relation (Kumar & Panaitescu 2000;Dermer 2004;Uhm & Zhang 2015), α = 2 + β, between the temporal index α and the spectral index β in the convention of F ν obs ∝ t − α obs ν − β obs if the emitter remains a constant Lorentz factor.Here, F ν obs is the observed spectral energy flux, t obs the observer time, and ν obs the observed frequency.This relation was generalized later for relativistic jets that undergo bulk acceleration (α > 2 + β) or bulk deceleration (α < 2 + β) (Uhm & Zhang 2015).
The curvature effect of HLE is commonly invoked to account for the steep decay of X-ray and γ-ray flares seen in the afterglow phase of GRBs (Liang et al. 2006;Uhm & Zhang 2016a;Jia et al. 2016;Ajello et al. 2019) as well as during the steep-decay phase of X-ray emission following the prompt emission (Zhang et al. 2006(Zhang et al. , 2009)).As for the prompt phase of GRBs, several studies (Ryde & Petrosian 2002;Kocevski et al. 2003;Genet & Granot 2009;Shenoy et al. 2013) have investigated the role that the curvature effect has on temporal and spectral properties of individual pulses, but an unambiguous identification of HLE could not be achieved.
The prompt phase of GRBs contains an important observational feature called the "broad pulses".Observationally, the broad pulses exhibit two distinct patterns of peak evolution; i.e., the peak-energy (E p ) of νF ν spectra shows a "hard-to-soft" or a "flux-tracking" pattern across the pulses (Ford et al. 1995;Norris et al. 1996;Golenetskii et al. 1983;Lu et al. 2012).In addition, the light curves of broad pulses in different energy bands exhibit a sequential pattern in their peak time, known as the "spectral lags" (Norris et al. 1996(Norris et al. , 2000;;Kocevski & Liang 2003); softer emission lags behind harder emission ("positive" type) in most cases, whereas harder emission can lag behind softer emission ("negative" type) in some cases.The curvature effect of HLE was traditionally suggested as a plausible explanation for the positive type of spectral lags (Shen et al. 2005), but a detailed study (Uhm & Zhang 2016b) showed that the HLE cannot give rise to any spectral lags if the spectral shape is softer than F ν obs ∝ ν 2 obs .The HLE may produce some spectral lags for a spectral shape harder than this ν 2 obs , but the resulting spectral lags are essentially invisible due to the significant flux-level difference between the light curves (Uhm & Zhang 2016b).
The complex and intriguing characteristics of broad pulses carry crucial clues to unveil the nature of GRBs.For instance, a series of numerical studies (Uhm & Zhang 2016b;Uhm et al. 2018) showed that all those features of broad pulses can be successfully reproduced within a single physical picture that invokes a bulk acceleration of the emitting region and that keeps the LoSE ongoing across the production of broad pulses.Also, Li & Zhang (2021) found evidence of jet acceleration in an effort of searching for the curvature effect.
Here, we present a simple study that identifies a clear signature of HLE in the decaying phase of broad pulses and provide a new understanding on the HLE emergence.We also present three examples of Fermi-GBM (Meegan et al. 2009) GRBs that exhibit the HLE signature in their broad pulses.

A SIMPLE PHYSICAL MODEL
Following the previous works (Uhm & Zhang 2016b;Uhm et al. 2018), we adopt a simple physical picture where a thin, relativistic spherical shell expands in space radially.The radiating electrons are distributed uniformly in the shell and emit synchrotron photons (Rybicki & Lightman 1979) isotropically in the comoving frame.Then we take fully into account the curvature effect to compute the HLE (Uhm & Zhang 2015).We assume a "Band" function shape (Band et al. 1993) for the emission spectrum in the co-moving frame since the observed gamma-ray spectra are traditionally fit to this function and since it is a good representation of synchrotron radiation (Uhm & Zhang 2014;Zhang et al. 2016).The strength of magnetic field B(r) in the emitting region globally decreases as the radius r from the central engine increases, which is expected for a spherical jet traveling in space.We note that this was the essential physical element to explain the low-energy photon index of the Band spectra for the majority of GRBs (Uhm & Zhang 2014;Geng et al. 2018).Moreover, the emitting region itself undergoes rapid bulk acceleration (Uhm & Zhang 2016a,b) during which the prompt gamma-rays are produced; i.e., the bulk Lorentz factor Γ(r) of the region has an increasing profile in radius r.Also, the characteristic Lorentz factor γ ch (r) of electrons in the co-moving frame is allowed to evolve with radius r.
We present three numerical models of broad pulses: and [w].The three models have different γ ch (r) profile as described in Appendix A. Other than γ ch profile, we keep all other model parameters the same for the three models, for simplicity.We assume a Band-function shape with typical low-and high-energy photon spectral index α B = −0.8 and β B = −2.3,respectively, for the emission spectrum in the co-moving frame.The number of radiating electrons is assumed to increase at a constant injection rate R inj = 10 47 s −1 .The bulk Lorentz factor of the jet takes a power-law profile in radius r, Γ(r) = Γ 0 (r/r 0 ) s , with Γ 0 = 250, r 0 = 10 15 cm, and s = 0.35, as used in Uhm & Zhang (2016b).We turn on the emission of spherical jet at radius r on = 10 14 cm and turn off its emission at radius r off = 3 × 10 16 cm.For the given profile of Γ(r), this turning-off happens at about t obs = 4.0 sec.We stress that the LoSE remains ongoing until this turn-off time.The magnetic field strength B(r) in the co-moving frame also takes a power-law profile, B(r) = B 0 (r/r 0 ) −b , with B 0 = 30 G and b = 1.5 (Uhm & Zhang 2016b).We calculate the luminosity distance to GRB for a flat ΛCDM universe with parameters Ω m = 0.31, Ω Λ = 0.69, and H 0 = 68 km/s/Mpc (Planck Collaboration et al. 2016) and take a typical value of redshift z = 1.
Figure 1 shows a modeling result of the three models [u], [v], and [w].The top panels show the light curves at 100 keV, 300 keV, and 1 MeV, which exhibit both the positive and negative types of spectral lags.The top panels also show the temporal evolution of E p curves exhibiting both the hard-to-soft and the flux-tracking patterns across the pulses1 .The middle panels show the time-dependent spectra at 1 sec, 2 sec, and 3 sec (solid lines).In the co-moving frame, we inject a Bandfunction spectrum with fixed α B and β B .However, the resulting spectra in the observer frame deviate significantly from this single Band function.Hence, in order to understand this deviation, we repeat the same calculations without considering the curvature effect; the resulting spectra are shown in dotted lines in the middle panel for model [u].Comparing the solid and dotted lines, one can clearly see that the curvature effect causes the deviation and that the HLE emerges as a prominent additional spectral break in F ν spectra during the decaying phase of the broad pulses2 .We again stress that the jet emission is not turned off until about 4 sec in our models and, therefore, the LoSE is still active and dominates the peak of F ν spectra as it should.The bottom panels show the νF ν spectra directly calculated from the solid lines in the middle panels, in which it is clear that the "HLE break" (ν HLE ) in F ν spectra now becomes the peak energy (E p ) in νF ν spectra in the decaying phase of these broad pulses.
If the peak of νF ν spectra is dominated by the HLE, there should exist a simple scaling relation expected from the HLE theory (Dermer 2004;Uhm & Zhang 2015).Here, F ν,Ep is the spectral energy flux F ν measured at the peak energy E p .
In Figure 2, we plot F ν,Ep against E p across the broad pulses of three numerical models [u], [v], and [w].An open circle in each model marks the first point in the beginning of the pulse.One can clearly see that the model curves closely follow Equation (1) (indicated by the dotted line) in the decaying phase of broad pulses, ascertaining that the peak of νF ν spectra indeed originates from the HLE.This is the clear signature of HLE, produced in our numerical models of broad pulses.

DATA ANALYSIS
Fermi-GBM (Meegan et al. 2009) has accumulated invaluable observations for the prompt emission of GRBs.In search of the HLE signature above, we analyzed a sample of Fermi-GBM GRBs with broad pulses.We require a certain minimum on the observed fluence to select bright GRBs and then perform a Bayesian-block analysis and impose several criteria to collect relatively clean broad-pulses.We then perform a time-resolved spectral analysis on each GRB selected.Details on the analysis are presented in an accompanying paper (Tak et al. 2023, submitted).
In Figure 3, we present three examples: GRB 110301A, GRB 140329A, and GRB 160113A.The top panels show the light curves at three different energy bands, together with the temporal evolution of E p curves.The bottom panels show E p vs F ν,Ep obtained from the time-resolved spectral analysis.The dotted line indicates the theoretical HLE relation in Equation (1).As one can see, the F ν,Ep -E p points obtained from the time-resolved analysis of the three bursts are in good agreement with Equation (1) in the decaying phase of their broad pulse, implying that the HLE signature is indeed identified.The color gradient used in the bottom panels is in accordance with the color gradient encoded in E p points in the top panels, which helps locate where in the pulse the HLE signature starts to show up.

CONCLUSIONS AND DISCUSSION
In this paper, we showed that the HLE can imprint a clear spectral signature in prompt-emission gamma-ray spectra (i.e., additional spectral break ν HLE in F ν spectra and peak energy in νF ν spectra) in the existence of ongoing LoSE.This result provides a new view regarding the HLE, because it has been believed so far that the HLE can show up and dominate the spectra only after the LoSE is turned off.
We remark, on the other hand, that the HLE spectral break is not required to appear in all broad pulses.It is because the HLE break could be buried under the LoSE component when the peak energy of LoSE (at a given time) is not far below that of HLE (emitted at earlier times but belonging to the same equal-arrivaltime surface).Whether this condition is satisfied or not would depend on the physical parameters in the emitting region.Therefore, this new perspective on the HLE emergence provides flexibility to elucidate both detection and non-detection of HLE signatures in observations of broad pulses.
In this paper, we also showed that some Fermi-GBM broad pulses exhibit the HLE scaling relation between E p and F ν,Ep (Equation 1) in their decaying phase.
The HLE signature observed in some broad pulses leads to important implication regarding the emission radius of GRBs.The HLE emitted at radius r is received at an observer time given roughly by t obs ∼ r/(2c Γ 2 ) like in the case of LoSE, which yields r ∼ 2c Γ 2 t obs = (1.6 × 10 16 cm) Γ 300 where c is the speed of light.The duration of broad pulses in our examples is tens of seconds, and therefore the gamma-ray emitting region of those GRBs with HLE signature should be located at ∼ 10 16 cm from the central engine for a typical value of Γ = 300.This inference of the emission radius is robust (i.e., independent of details of our modeling) and sheds light on differentiating the GRB models.The estimated large emission radius ∼ 10 16 cm is consistent with the ICMART model (Zhang & Yan 2011), which invokes collision-induced magnetic dissipation as the origin of GRB prompt emission.In contrast, the photospheric emission models (Lazzati et al. 2013) and the internal shock models (Rees & Mészáros 1994) are disfavored 3 since the photospheric radii and the internal shock radii are typically at ∼ 10 11 − 10 12 cm and at ∼ 10 13 − 10 14 cm, respectively (Kumar & Zhang 2015).
In short, we identified a clear signature of HLE in the prompt phase of GRBs both theoretically and observationally.Also, we presented a unique constraint on the validity of the competing GRB models.The characteristic Lorentz factor γ ch (r) of electrons evolves in radius r in our models.Considering the energy dissipation and particle acceleration process, which would depend, for instance, on the degree of magnetic energy dissipation, it is plausible to assume that the γ ch -profile evolves in radius as the emitting shell expands in space.The model [u] has a broken power-law profile with γ 0 ch = 10 5 , r 0 = 10 15 cm, g 1 = 1/2, and g 2 = 1.The model [v] also takes the same form of broken power-law but with γ 0 ch = 2 × 10 5 , r 0 = 2 × 10 15 cm, and g 1 = g 2 = 1.The model [w] has a profile made of four power-law segments with γ 0 ch = 10 5 , r 1 = 5 × 10 14 cm, r 2 = 10 15 cm, and r 3 = 4 × 10 15 cm.These three γ ch profiles are shown in Figure 4.
This work was supported by the National Research Foundation of Korea (NRF) grant, No. 2021M3F7A1084525, funded by the Korea government (MSIT).B.B.Z acknowledges the support by the National Key Research and Development Programs of China (2018YFA0404204, 2022YFF0711404, 2022SKA0130102), the National Natural Science Foundation of China (Grant Nos.11833003, U2038105, 12121003), the science research grants from the China Manned Space Project with NO.CMS-CSST-2021-B11, and the Program for Innovative Talents, Entrepreneur in Jiangsu.APPENDIX A. PROFILE OF CHARACTERISTIC LORENTZ FACTOR γ ch OF ELECTRONS

Figure 1 .
Figure1.Light curves and time-dependent spectra of our numerical models [u],[v], and [w].The top panels show the light curves at 100 keV, 300 keV, and 1 MeV, together with temporal evolution of Ep curves.The middle panels show time-dependent spectra at 1 sec, 2 sec, and 3 sec, with the curvature effect of HLE(Uhm & Zhang 2015) fully included (solid lines) or removed (dotted lines) for model[u].The νFν spectra in the bottom panels are derived from the corresponding solid lines in the middle panels.

Figure 2 .
Figure 2. The peak energy Ep vs the spectral flux Fν at Ep (i.e., Fν,E p ) across the broad pulses of our numerical models [u], [v], and [w].An open circle in each model marks the first point in the beginning of pulses.The dotted line indicates the relation Fν,E p ∝ E 2 p in Equation (1).

Figure 3 .
Figure 3. Results of our time-resolved spectral analysis performed on three example broad pulses in GRB 110301A, GRB 140329A, and GRB 160113A.The top panels show the light curves at three different energy bands, together with temporal evolution of Ep points.The bottom panels show the Ep vs Fν,E p points obtained from the analysis.The dotted line indicates the relation Fν,E p ∝ E 2 p in Equation (1).The color gradient in the bottom panels is in accordance with the color gradient encoded in Ep points in the top panels.

Figure 4 .
Figure 4. Profile of characteristic Lorentz factor γ ch of electrons in our numerical models [u], [v], and [w].