Gamma-ray Bursts Physics Research Articles

GRB 190114C: Fireball Energy Budget and Radiative Efficiency Revisited

The jet composition of gamma-ray bursts (GRBs), as well as how efficiently the jet converts its energy to radiation, are long-standing problems in GRB physics. Here, we reported a comprehensive temporal and spectral analysis of the TeV-emitting bright GRB 190114C. Its high fluence (∼4.4 × 10−4 erg cm−2) allows us to conduct the time-resolved spectral analysis in great detail and study their variations down to a very short timescale (∼0.1 s) while preserving a high significance. Its prompt emission consists of three well-separated pulses. The first two main pulses (P 1 and P 2) exhibit independently strong thermal components, starting from the third pulse (P 3) and extending to the entire afterglow, the spectra are all nonthermal, and the synchrotron plus Compton upscattering model well interprets the observation. By combining the thermal (P 1 and P 2) and the nonthermal (P 3) observations based on two different scenarios (global and pulse properties) and following the method described in Zhang et al., we measure the fireball parameters and GRB radiative efficiency with little uncertainties for this GRB. A relevantly high GRB radiative efficiency is obtained based on both the global and pulse properties, suggesting that if GRBs are powered by fireballs, the efficiency can sometimes be high. More interestingly, though the observed parameters are individually different (e.g., the amount of mass loading M), the radiative efficiency obtained from P 1 (η γ = 36.0% ± 6.5%) and P 2 (η γ = 41.1% ± 1.9%) is roughly the same, which implies that the central engine of the same GRB has some common properties.

Constraints on short gamma-ray burst physics and their host galaxies from systematic radio follow-up campaigns

ABSTRACT Short gamma-ray bursts (GRBs) are explosive transients caused by binary mergers of compact objects containing at least one neutron star. Multiwavelength afterglow observations provide constraints on the physical parameters of the jet, its surrounding medium, and the microphysics of the enhanced magnetic fields and accelerated electrons in the blast wave at the front of the jet. The synchrotron radio emission can be tracked for much longer than in other spectral regimes, and it can pin down the evolution of the spectral peak. We present the results of a systematic observing campaign of eight short GRBs with the MeerKAT radio telescope. Additionally, we present observations of four of these short GRBs using the ATCA radio telescope and two of these short GRBs with the e-MERLIN radio telescope. Using these results we report one possible detection of a short GRB afterglow from GRB 230217A and deep upper limits for the rest of our short GRB observations. We use these observations to place constraints on some of the physical parameters, in particular those related to electron acceleration, the circumburst density, and gamma-ray energy efficiency. We discuss how deeper observations with new and upgraded telescopes should be able to determine if the gamma-ray efficiency differs between long and short GRBs. We also report detections of the likely host galaxies for four of the eight GRBs and upper limits for another GRB, increasing the number of detected host galaxies in the radio with implications for the star formation rate in these galaxies.

Insights into the physics of gamma-ray bursts from the high-energy extension of their prompt emission spectra

The study of the high-energy part (MeV-GeV) of the spectrum of gamma-ray bursts (GRBs) can play a crucial role in investigating the physics of prompt emission, but it is often hampered by low statistics and the paucity of GeV observations. In this work, we analyze the prompt emission spectra of the 22 brightest GRBs which have been simultaneously observed by Fermi/GBM and Fermi/LAT, spanning six orders of magnitude in energy. The high-energy photon spectra can be modeled with a power-law N(E)∝E−β possibly featuring an exponential cutoff. We find that, with the inclusion of the LAT data, the spectral index β is softer than what is typically inferred from the analysis of Fermi/GBM data alone. Under the assumption that the emission is synchrotron, we derived a median value of the index p ∼ 2.79 of the power-law energy distribution of accelerated particles (N(γ)∝γ−p). In nine out of 22 GRB spectra, we find a significant presence of an exponential cutoff at high energy, ranging between 14 and 298 MeV. By interpreting the observed cutoff as a sign of pair-production opacity, we estimate the jet bulk Lorentz factor Γ, finding values in the range 130–330. These values are consistent with those inferred from the afterglow light curve onset time. Finally, by combining the information from the high-energy prompt emission spectrum with the afterglow light curve, we exploited a promising method to derive the distance R from the central engine where the prompt emission occurs. The distances (R > 1013 − 15 cm) inferred for the only two GRBs in our sample that are suitable for the application of this method, which have only lower limits on their cutoff energies, suggest large emitting regions, although they are still compatible with the standard model. Larger samples of GRBs with measured cutoff energies and afterglow deceleration time will allow for more informative values to be derived. These results highlight the importance of including high-energy data, when available, in the study of prompt spectra and their role in addressing the current challenges of the GRB standard model.

Distributions of Energy, Luminosity, Duration, and Waiting Times of Gamma-Ray Burst Pulses with Known Redshift Detected by Fermi/GBM

Discovered more than 50 years ago, gamma-ray burst (GRB) prompt emission remains the most puzzling aspect of GRB physics. Its complex and irregular nature should reveal how newborn GRB engines release their energy. In this respect, the possibility that GRB engines could operate as self-organized critical (SOC) systems has been put forward. Here, we present the energy, luminosity, waiting time, and duration distributions of individual pulses of GRBs with known redshift detected by the Fermi Gamma-ray Burst Monitor. This is the first study of this kind in which selection effects are accounted for. The compatibility of our results with the framework of SOC theory is discussed. We found evidence for an intrinsic break in the power-law models that describe the energy and the luminosity distributions.

Off-axis Afterglow Closure Relations and Fermi-LAT Detected Gamma-Ray Bursts

Gamma-ray bursts (GRBs) are one of the most promising transient events for studying multiwavelength observations in extreme conditions. Observation of GeV photons from bursts would provide crucial information on GRB physics, including the off-axis emission. The Second Gamma-ray Burst Catalog (2FLGC) has been announced by the Fermi Large Area Telescope (Fermi-LAT) Collaboration. This catalog includes 29 bursts with photon energy higher than 10 GeV. While the synchrotron forward-shock model has well explained the afterglow data of GRBs, photon energies greater than 10 GeV are very difficult to interpret within this framework. To study the spectral and temporal indices of those bursts described in 2FLGC, Fraija et al. () proposed the closure relations (CRs) of the synchrotron self-Compton (SSC) emission of GRBs emitted from an on-axis jet, which decelerates in stellar-wind and constant-density medium. In this paper, we extend the CRs of the SSC afterglow from an on-axis scenario to an off-axis scenario, including the synchrotron afterglow radiation that seems to be off-axis. In order to investigate the spectral and temporal index evolution of those bursts reported in 2FLGC, we consider hydrodynamical evolution with energy injection in the adiabatic and radiative regimes for an electron distribution with a spectral index of 1 < p < 2 and 2 < p. The results show that the most likely scenario for synchrotron emission corresponds to the stellar wind whether or not there is energy injection and that the most likely scenario for SSC emission corresponds to the constant density when there is no energy injection and to the stellar wind when there is energy injection.

Prospects for Detecting γ-Ray Bursts at Very High Energies with the HADAR Experiment

Recent ground-based observations of TeV photons have significantly deepened our understanding of the nature of gamma-ray bursts (GRBs). However, many fundamental problems remain unsolved concerning the physical mechanisms behind GRBs, necessitating the need for sufficient statistical data. The High Altitude Detection of Astronomical Radiation (HADAR) experiment utilizes a wide-angle water Cherenkov telescope, presenting a novel approach to measure the spectra and variability of GRBs from 10 GeV to 10 TeV energy ranges with unprecedented photon statistics and thereby break new ground in elucidating the physics of GRBs, which is still poorly understood. In this study, a time-dependent numerical modeling technique is utilized to simulate extensive light curves and spectral energy distributions of synthetic GRB afterglow emissions. By harnessing the remarkable capabilities of HADAR, we evaluate its potential in detecting GRB afterglow emissions at energies >10 GeV. Through our analysis, we unveil the prospect of detecting an estimated 5.8 GRBs annually, facilitating a systematic investigation into their reliance on model parameters. Future HADAR observations would offer valuable insights into the magnetic field and the environmental conditions surrounding GRBs.

The Optical Two- and Three-dimensional Fundamental Plane Correlations for Nearly 180 Gamma-Ray Burst Afterglows with Swift/UVOT, RATIR, and the Subaru Telescope

Gamma-ray bursts (GRBs) are fascinating events due to their panchromatic nature. We study optical plateaus in GRB afterglows via an extended search into archival data. We comprehensively analyze all published GRBs with known redshifts and optical plateaus observed by many ground-based telescopes (e.g., Subaru Telescope, RATIR) around the world and several space-based observatories such as the Neil Gehrels Swift Observatory. We fit 500 optical light curves, showing the existence of the plateau in 179 cases. This sample is 75% larger than the previous one, and it is the largest compilation so far of optical plateaus. We discover the 3D fundamental plane relation at optical wavelengths using this sample. This correlation is between the rest-frame time at the end of the plateau emission, , its optical luminosity, L opt, and the peak in the optical prompt emission, L peak,opt, thus resembling the three-dimensional (3D) X-ray fundamental plane (the so-called 3D Dainotti relation). We correct our sample for redshift evolution and selection effects, discovering that this correlation is indeed intrinsic to GRB physics. We investigate the rest-frame end-time distributions in X-rays and optical (, ), and conclude that the plateau is achromatic only when selection biases are not considered. We also investigate if the 3D optical correlation may be a new discriminant between optical GRB classes and find that there is no significant separation between the classes compared to the Gold sample plane after correcting for evolution.

Gamma-Ray Bursts at TeV Energies: Theoretical Considerations

Gamma-ray bursts (GRBs) are the most luminous explosions in the Universe and are powered by ultra-relativistic jets. Their prompt γ-ray emission briefly outshines the rest of the γ-ray sky, making them detectable from cosmological distances. A burst is followed by, and sometimes partially overlaps with, a similarly energetic but very broadband and longer-lasting afterglow emission. While most GRBs are detected below a few MeV, over 100 have been detected at high (≳0.1 GeV) energies, and several have now been observed up to tens of GeV with the Fermi Large Area Telescope (LAT). A new electromagnetic window in the very-high-energy (VHE) domain (≳0.1 TeV) was recently opened with the detection of an afterglow emission in the (0.1–1)TeV energy band by ground-based imaging atmospheric Cherenkov telescopes. The emission mechanism for the VHE spectral component is not fully understood, and its detection offers important constraints for GRB physics. This review provides a brief overview of the different leptonic and hadronic mechanisms capable of producing a VHE emission in GRBs. The same mechanisms possibly give rise to the high-energy spectral component seen during the prompt emission of many Fermi-LAT GRBs. Possible origins of its delayed onset and long duration well into the afterglow phase, with implications for the emission region and relativistic collisionless shock physics, are discussed. Key results for using GRBs as ideal probes for constraining models of extra-galactic background light and intergalactic magnetic fields, as well as for testing Lorentz invariance violation, are presented.

The Detection of GRBs at VHE: A Challenge Lasting for More than Two Decades, What Is Next?

Unveiling the mystery of gamma-ray bursts (GRBs) has been the target of many multi-waveband observational and theoretical efforts during the last decades. The results collected by current and past space-based instruments have provided important insights into the mechanisms at the origin of their prompt and afterglow phases. On the other hand, many questions, such as the the origin of the multi-GeV signal observed in a large number of events, remained unanswered. Within this framework, the first firm detections of a very-high-energy (VHE, E≳100 GeV) emission component by MAGIC and H.E.S.S. collaborations represented an important, long-awaited result for the VHE astrophysics community. However, while such discoveries opened a new era in the study of GRBs, they also provided an unexpected complexity due to the differences between the phenomenology of the observed events. This revealed that we still have an incomplete comprehension of GRB physics. In the nearby future, observations by the Cherenkov Telescope Array Observatory (CTAO), with unprecedented sensitivity in the VHE band, will have a key role in the study of these enigmatic objects and their interactions with the surrounding environment. In this review we will cover the recent GRB history, highlighting the efforts of follow-up campaigns by the VHE community that led to the first VHE GRB detection, and outlining what we can expect from future facilities in the next decades.

Gamma: a new method for modelling relativistic hydrodynamics and non-thermal emission on a moving mesh

ABSTRACT In recent years, dynamical relativistic jet simulation techniques have progressed to a point where it is becoming possible to fully numerically resolve gamma-ray burst (GRB) blast-wave evolution across scales. However, the modelling of emission is currently lagging behind and limits our efforts to fully interpret the physics of GRBs. In this work we combine recent developments in moving-mesh relativistic dynamics with a local treatment of non-thermal emission in a new code: gamma. The code involves an arbitrary Lagrangian–Eulerian approach only in the dominant direction of fluid motion that avoids mesh entanglement and associated computational costs. Shock detection, particle injection, and local calculation of their evolution including radiative cooling are done at runtime. Even though gamma has been designed with GRB physics applications in mind, it is modular such that new solvers and geometries can be implemented easily with a wide range of potential applications. In this paper, we demonstrate the validity of our approach and compute accurate broad-band GRB afterglow radiation from early to late times. Our results show that the spectral cooling break shifts by a factor of ∼40 compared to existing methods. Its temporal behaviour also significantly changes from the previously calculated temporary steep increase after the jet break. Instead, we find that the cooling break does not shift with time between the relativistic and Newtonian asymptotes when computed from our local algorithm. gamma is publicly available at: https://github.com/eliotayache/gamma.

Evidence of X-Ray Plateaus Driven by the Magnetar Spindown Winds in Gamma-Ray Burst Afterglows

Abstract The central engine of gamma-ray bursts (GRBs) remains an open and cutting-edge topic in the era of multimessenger astrophysics. X-ray plateaus appear in some GRB afterglows, which are widely considered to originate from the spindown of magnetars. According to the stable magnetar scenario of GRBs, an X-ray plateau and a decay phase ∼t −2 should appear in X-ray afterglows. Meanwhile, the “normal” X-ray afterglow is produced by the external shock from a GRB fireball. We analyze the Neil Gehrels Swift GRB data, then find three gold samples that have an X-ray plateau and a decay phase ∼t −2 superimposed on the jet-driven normal component. Based on these features of the lightcurves, we argue that the magnetars should be the central engines of these three GRBs. Future joint multimessenger observations might further test this possibility, which can then be beneficial to constrain GRB physics.

GRB Polarization: A Unique Probe of GRB Physics

Over half a century from the discovery of gamma-ray bursts (GRBs), the dominant radiation mechanism responsible for their bright and highly variable prompt emission remains poorly understood. Spectral information alone has proven insufficient for understanding the composition and main energy dissipation mechanism in GRB jets. High-sensitivity polarimetric observations from upcoming instruments in this decade may help answer such key questions in GRB physics. This article reviews the current status of prompt GRB polarization measurements and provides comprehensive predictions from theoretical models. A concise overview of the fundamental questions in prompt GRB physics is provided. Important developments in gamma-ray polarimetry including a critical overview of different past instruments are presented. Theoretical predictions for different radiation mechanisms and jet structures are confronted with time-integrated and time-resolved measurements. The current status and capabilities of upcoming instruments regarding the prompt emission are presented. The very complimentary information that can be obtained from polarimetry of X-ray flares as well as reverse-shock and early to late forward-shock (afterglow) emissions are highlighted. Finally, promising directions for overcoming the inherent difficulties in obtaining statistically significant prompt-GRB polarization measurements are discussed, along with prospects for improvements in the theoretical modeling, which may lead to significant advances in the field.

Searching for Fermi GRB optical counterparts with the prototype Gravitational-wave Optical Transient Observer (GOTO)

ABSTRACT The typical detection rate of ∼1 gamma-ray burst (GRB) per day by the Fermi Gamma-ray Burst Monitor (GBM) provides a valuable opportunity to further our understanding of GRB physics. However, the large uncertainty of the Fermi localization typically prevents rapid identification of multiwavelength counterparts. We report the follow-up of 93 Fermi GRBs with the Gravitational-wave Optical Transient Observer (GOTO) prototype on La Palma. We selected 53 events (based on favourable observing conditions) for detailed analysis, and to demonstrate our strategy of searching for optical counterparts. We apply a filtering process consisting of both automated and manual steps to 60 085 candidates initially, rejecting all but 29, arising from 15 events. With ≈3 GRB afterglows expected to be detectable with GOTO from our sample, most of the candidates are unlikely to be related to the GRBs. Since we did not have multiple observations for those candidates, we cannot confidently confirm the association between the transients and the GRBs. Our results show that GOTO can effectively search for GRB optical counterparts thanks to its large field of view of ≈40 deg2 and its depth of ≈20 mag. We also detail several methods to improve our overall performance for future follow-up programmes of Fermi GRBs.

IKI GRB-FuN: observations of GRBs with small-aperture telescopes.

Gamma-ray bursts (GRBs) are the most energetic and mysterious events in the Universe, which are observed in all ranges of electromagnetic spectrum. Most valuable results about physics of GRB are obtained by optical observations. GRBs are initially detected in gamma-rays with poor localization accuracy, and an optical counterpart should be found. The faster the counterpart is found, the more it can give to physics. This first phase, as a rule, corresponds to an early afterglow. The next phases of the observations are multicolor photometry, polarimetry, spectroscopy, and few days later the search for a supernova or kilonova associated with the GRB, and finally, observations of the host galaxy. To manage the problem of fast optical observations, telescopes with a small aperture are suitable. They can have a large field of view, which is necessary to cover initial localizations of GRBs. The sensitivity of the telescope+detector may be sufficient to record statistically significant light curve with fine time resolution. We describe one of the networks of telescopes with a small aperture IKI-GRB FuN, and present the results of early optical observation of GRB sources, and discuss the design requirements of the optical observations for effective GRB research in the next decade.

AGILE and Konus-Wind Observations of GRB 190114C: The Remarkable Prompt and Early Afterglow Phases

Abstract GRB 190114C represents a breakthrough for the physics of gamma-ray bursts (GRBs), being the first GRB with delayed emission above 300 GeV, as reported by MAGIC. We present in this paper the sub-MeV/MeV data of the prompt and early afterglow emissions of GRB 190114C, as detected by AGILE and Konus-Wind, in the 20 keV–100 MeV energy range. The first stages of the burst exhibit multiple emission components, associated with an interesting spectral evolution. The first 2 s of the prompt emission can be described by a single “Band-like” spectral component. The successive 4 s show the presence of an additional high-energy spectral component, which quickly evolves into a “hard-flat” component of the νF ν spectrum, extending up to 10–100 MeV and likely produced by inverse Compton radiation, whose onset and evolution are clearly shown in our data. After this phase, the νF ν spectrum evolves into a “V shape,” showing the persistence and spectral hardening of the additional high-energy component in substantial agreement with Fermi and Swift results. We also analyze the first ∼200 s of the early afterglow that show a reflaring episode near T 0 + 15 s. We identify a new, so-far-unnoticed flux temporal break near T 0 + 100 s, which is detected in hard X-rays by both Konus-Wind and INTEGRAL/SPI-ACS. We find this break incompatible with the commonly assumed adiabatic evolution of a fireball in a constant-density medium. We interpret this break as a consequence of radiative evolution of the early afterglow from a fireball expanding in a wind-like circumburst medium.

To Power the X-Ray Plateaus of Gamma-Ray Bursts through Larger Amplitude Electromagnetic Waves

Abstract The origin of the gamma-ray burst (GRB) X-ray plateau, especially the internal plateau, is still unclear, but it could be related to a GRB’s magnetar central engine. It is generally believed that the spin-down power of the magnetar is injected into forward external shock; however, we propose here that most of the power will be dissipated behind the GRB jet through a larger amplitude electromagnetic wave. Based on this proposal, the relevant physical conditions and observational implications are analyzed and discussed, and various kinds of X-ray light curves could be reproduced. Although the chromatic multiband afterglow in the standard external afterglow fireball model is still a matter of debate, we can naturally explain this feature through this proposal, i.e., the electrons generating the X-ray plateau and emitting the optical afterglow are accelerated by different mechanisms. It is emphasized that both the GRB jet and the spin-down wind should have significant observational consequences in the magnetar scenario, and should be focused equally in GRB physics.

Nondetection of the Gamma-Ray Burst X-Ray Emission Line: The Down-Comptonization Effect

Abstract The detection of the gamma-ray burst (GRB) X-ray emission line is important for studying GRB physics and constraining the GRB redshift. Since the line-like feature in the GRB X-ray spectrum was first reported in 1999, several works on line searching have been published over the past two decades. Even though some observations on the X-ray line-like feature were performed, its significance remains controversial to date. In this paper, we utilize the down-Comptonization mechanism and present the time evolution of the Fe Kα line emitted near the GRB central engine. The line intensity decreases with the evolution time, and the time evolution depends on the electron density and the electron temperature. In addition, the initial line with a larger broadening decreases less over time. For instance, when the emission line penetrates material with an electron density above 1012 cm−3 at 1 keV, it generally becomes insignificant enough after 100 s for it not to be detected . The line-like profile deviates from the Gaussian form, and it finally changes to be similar to a blackbody shape at the time of the thermal equilibrium between the line photons and the surrounding material.

The Three-parameter Correlations About the Optical Plateaus of Gamma-Ray Bursts

Abstract Well-sampled optical light curves of 50 gamma-ray bursts (GRBs) with plateau features are compiled from the literature. By empirical fitting, we obtained the parameters of the optical plateaus, such as the decay slopes (α 1 and α 2), the break times (T b), and the corresponding optical fluxes (F b) at the break times. The break time of optical plateaus ranges from tens of seconds to 106 s, with a typical value of about 104 s. We have calculated the break luminosity, and it mainly ranges from 1044 erg s−1 to 1047 erg s−1, which is generally two or three orders of magnitude less than the corresponding break luminosity of the X-ray afterglow plateaus. We reanalyzed the optical plateaus and also found that a significantly tighter correlation exists when we added the isotropic equivalent energy of GRBs E γ,iso into the L b,z–T b,z relation. The best-fit correlation is obtained to be L b , z ∝ T b , z − 0.9 E γ , iso 0.4 . We next explored the possible correlations among L b,z, T b,z and E p,i, and found there is also a tight correlation between them, which takes the form of L b , z ∝ T b , z − 0.9 E p , i 0.5 . We argue that these two tight L b,z–T b,z–E γ,iso and L b,z–T b,z–E p,i correlations are more physical, and it may be directly related to the radiation physics of GRBs. The tight correlations can possibly be used as standard candles.

Special gamma-ray bursts and special radiation components from gamma-ray bursts

Gamma-ray bursts (GRBs) were first detected in 1960s. Since then, GRBs have been observed extensively in gamma-rays. The afterglows of several hundred GRBs were also detected, providing us with abundant clues for understanding the physics of GRBs. However, X-ray observations of GRBs at the prompt burst phase are still quite lacking. These very early X-ray observations should be very important for probing the activities of GRB central engines. The Einstein Probe, a unique satellite that works mainly in soft X-rays (0.5–4 keV) with a large field of view (1.1 sr), will open up a new window for us to observe the prompt X-ray emission of GRBs. In this article, we review the current status of the studies on some special kinds of GRBs such as those rich in X-rays and those baring special emission components. Especially, we investigate possible contributions of EP to the studies of X-ray flashes (XRFs), low-luminosity GRBs, ultra-long GRBs, and the precursors of GRBs. It is found that: (1) EP is expected to detect about 810 GRB-like events, among which 95% are soft XRFs and only 1% are typical GRBs; (2) EP could detect 0.2–8 low-luminosity GRBs each year (the exact detection rate is strongly dependent on the exact spectral distribution of low luminosity GRBs, which is still largely uncertain currently); (3) EP would detect at least 20–200 ultra-long GRBs each year; (4) EP may be able to detect a lot of GRB precursors (at least 80 events per year), which are still poorly understood. We believe that the EP satellite will lead to encouraging discovers and will deepen our understanding of GRBs in various aspects such as their classifications, their progenitors, the triggering mechanisms and the beaming features, etc. We also note that the EP satellite has great advantages on its wide field of view and good sensitivity, over other similar world-wide instruments. We suggested that the project should be carried out as soon as possible.

Constraining external reverse shock physics of gamma-ray bursts from ROTSE-III limits

Assuming that the early optical emission is dominated by the external reverse shock (RS) in the standard model of gamma-ray bursts (GRBs), we intend to constrain RS models with the initial Lorentz factor $\Gamma_0$ of the outflows based on the ROTSE-III observations. We consider two cases of the RS behavior: the relativistic shock and the non-relativistic shock. For homogeneous interstellar medium (ISM) and wind circum-burst environment, the constraints can be achieved by the fact that the peak flux $F_{\rm \nu}$ at the RS crossing time should be lower than the observed upper limit $F_{\rm \nu,limit}$. We consider the different spectral regimes that the observed optical frequency $\nu_{\rm opt}$ may locate in, which are divided by the orders for the minimum synchrotron frequency $\nu_{\rm m}$ and the cooling frequency $\nu_{\rm c}$. {\bf Considering the homogeneous and wind environment around GRBs, we find that the relativistic RS case can be constrained by the (upper and lower) limits of $\Gamma_0$ in a large range from about hundreds to thousands for 36 GRBs reported by ROTSE-III. The constraints on the non-relativistic RS case are achieved with limits of $\Gamma_0$ for 26 bursts ranging from $\sim 30$ to $\sim 350$.} The lower limits of $\Gamma_0$ achieved for the relativistic RS model is disfavored based on the previously discovered correlation between the initial Lorentz factor $\Gamma_0$ and the isotropic gamma-ray energy $E_{\rm \gamma, iso}$ released in prompt phase.