arXiv subjects

Bing Zhang

Publications and source records attributed to Bing Zhang.

At least 415 records · Page 23Linked to original sources

arXiv2016

Constraints on binary neutron star merger product from short GRB observations

Binary neutron star mergers are strong gravitational wave (GW) sources and the leading candidates to interpret short duration gamma-ray bursts (SGRBs). Under the assumptions that SGRBs are produced by double neutron star mergers and that the X-ray plateau followed by a steep decay as observed in SGRB X-ray light curves marks the collapse of a supra-massive neutron star to a black hole (BH), we use the statistical observational properties of {\em Swift} SGRBs and the mass distribution of Galactic double neutron star systems to place constraints on the neutron star equation of state (EoS) and the properties of the post-merger product. We show that current observations already put following interesting constraints: 1) A neutron star EoS with a maximum mass close to a parameterization of $M_{\rm max} = 2.37\,M_\odot (1+1.58\times10^{-10} P^{-2.84})$ is favored; 2) The fractions for the several outcomes of NS-NS mergers are as follows: $\sim40\%$ prompt BHs, $\sim30\%$ supra-massive NSs that collapse to BHs in a range of delay time scales, and $\sim30\%$ stable NSs that never collapse; 3) The initial spin of the newly born supra-massive NSs should be near the breakup limit ($P_i\sim1 {\rm ms}$), which is consistent with the merger scenario; 4) The surface magnetic field of the merger products is typically $\sim 10^{15}$ G; 5) The ellipticity of the supra-massive NSs is $ε\sim (0.004 - 0.007)$, so that strong GW radiation is released post the merger; 6) Even though the initial spin energy of the merger product is similar, the final energy output of the merger product that goes into the electromagnetic channel varies in a wide range from several $10^{49}$ erg to several $10^{52}$ erg, since a good fraction of spin energy is either released in the form of GW or falls into the black hole as the supra-massive NS collapses.

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arXiv2016

Efficient Production of High-energy Nonthermal Particles during Magnetic Reconnection in a Magnetically-dominated Ion-Electron Plasma

Magnetic reconnection is a leading mechanism for dissipating magnetic energy and accelerating nonthermal particles in Poynting-flux dominated flows. In this letter, we investigate nonthermal particle acceleration during magnetic reconnection in a magnetically-dominated ion-electron plasma using fully kinetic simulations. For an ion-electron plasma with the total magnetization $σ_0=B^2/(4πn(m_i+m_e)c^2)$, the magnetization for each species is $σ_i \sim σ_0$ and $σ_e \sim (m_i/m_e) σ_0$, respectively. We have studied the magnetically dominated regime by varying $σ_{e} = 10^3 - 10^5$ with initial ion and electron temperatures $T_i = T_e = 5 - 20 m_ec^2$ and mass ratio $m_i/m_e = 1 - 1836$. The results demonstrate that reconnection quickly establishes power-law energy distributions for both electrons and ions within several ($2-3$) light-crossing times. For the cases with periodic boundary conditions, the power-law index is $1<s<2$ for both electrons and ions. The hard spectra limit the power-law energies for electrons and ions to be $γ_{be} \sim σ_e$ and $γ_{bi} \sim σ_i$, respectively. The main acceleration mechanism is a Fermi-like acceleration through the drift motions of charged particles. When comparing the spectra for electrons and ions in momentum space, the spectral indices $s_p$ are identical as predicted in Fermi acceleration. We also find that the bulk flow can carry a significant amount of energy during the simulations. We discuss the implication of this study in the context of Poynting-flux dominated jets and pulsar winds especially the applications for explaining the nonthermal high-energy emissions.

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arXiv2015

On the non-thermal kappa-distributed electrons in planetary nebulae and HII regions: the kappa index and its correlations with other nebular properties

Recently, a suspicion arose that the free electrons in planetary nebulae (PNe) and HII regions might have non-thermal energy distributions. In this scenario, a kappa index is introduced to characterize the electron energy distributions, with smaller kappa values indicating larger deviations from Maxwell-Boltzmann distributions. Assuming that this is the case, we determine the kappa values for a sample of PNe and HII regions by comparing the intensities of [OIII] collisionally excited lines and the hydrogen Balmer jump. We find the average kappa indices of PNe and HII regions to be 27 and 32, respectively. Correlations between the resultant kappa values and various physical properties of the nebulae are examined to explore the potential origin of non-thermal electrons in photoionized gaseous nebulae. However, no positive result is obtained. Thus the current analysis does not lend to support to the idea that kappa-distributed electrons are present in PNe and HII regions.

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arXiv2015

IGR J12580+0134: The First Tidal Disruption Event with an Off-beam Relativistic Jet

Supermassive black holes (SMBHs) can capture and tidally disrupt stars or sub-stellar objects orbiting nearby. The detections of Sw J1644+57-like events suggest that at least some TDEs can launch a relativistic jet beaming towards Earth. A natural expectation would be the existence of TDEs with a relativistic jet beaming away from Earth. The nearby TDE candidate IGR J12580+0134 provides new insights into the jet phenomenon. Combining several constraints, we find that the event invokes a $8-40$ Jupiter mass object tidally disrupted by a $3 \times 10^5 - 1.8 \times 10^7 M_\sun$ SMBH. Recently, a bright radio transient was discovered by Irwin et al. in association with IGR J12580+0134. We perform detailed modeling of the event based on a numerical jet model previously developed for the radio emission of Sw J1644+57. We find that the radio data of IGR J12580+0134 can be interpreted within an external forward shock model in the Newtonian regime. Using Sw J1644+57 as a template and properly correcting for its luminosity, we argue that the observed X-ray flux in early times is too faint to allow an on-beam relativistic jet unless the Lorentz factor is very small. Rather, the X-ray emission is likely from the disk or corona near the black hole. From various constraints, we find that the data are consistent with an off-beam relativistic jet with a viewing angle $θ_{\rm obs} \gtrsim 30^{\rm o}$, and an initial Lorentz factor $Γ_j \gtrsim $ a few.This scenario can readily be tested in the upcoming VLBI observations.

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arXiv2015

Chemical Abundances of Planetary Nebulae in the Substructures of M31

We present deep spectroscopy of planetary nebulae (PNe) that are associated with the substructures of the Andromeda Galaxy (M31). The spectra were obtained with the OSIRIS spectrograph on the 10.4 m GTC. Seven targets were selected for the observations, three in the Northern Spur and four associated with the Giant Stream. The most distant target in our sample, with a rectified galactocentric distance >100 kpc, was the first PN discovered in the outer streams of M31. The [O III] 4363 auroral line was well detected in the spectra of all targets, enabling electron temperature determination. Ionic abundances are derived based on the [O III] temperatures, and elemental abundances of helium, nitrogen, oxygen, neon, sulfur, and argon are estimated. The relatively low N/O and He/H ratios as well as abundance ratios of alpha-elements indicate that our target PNe might belong to populations as old as ~2 Gyr. Our PN sample, including the current seven and the previous three observed by Fang et al., have rather homogeneous oxygen abundances. The study of abundances and the spatial and kinematical properties of our sample leads to the tempting conclusion that their progenitors might belong to the same stellar population, which hints at a possibility that the Northern Spur and the Giant Stream have the same origin. This may be explained by the stellar orbit proposed by Merrett et al. Judging from the position and kinematics, we emphasize that M32 might be responsible for the two substructures. Deep spectroscopy of PNe in M32 will help to assess this hypothesis.

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arXiv2015

Evidence for Two-Component Jet in Sw J1644+57

The continued observations of Sw J1644+57 in X-ray and radio bands accumulated a rich data set to study the relativistic jet launched in this tidal disruption event. We find that the re-brightening feature in the radio light curve can be naturally explained by the two-component jet model. The possible origin of this structured jet are the Blandford-Znajek and Blandford-Payne mechanisms. We also show that this two-component jet model can interpret the two kinds of quasi-periodic variations in the X-ray light curve: a 200 second quasi-periodic oscillation (QPO) and a 2.7-day quasi-periodic variation. The latter is interpreted by a precessing outer jet launched near the Bardeen-Petterson radius of a warped disk. The $\sim$ 200s QPO could be associated with a second, narrower jet sweeping the observer line-of-sight periodically, which is launched from a spinning black hole in the misaligned direction with respect to the black hole's angular momentum.

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arXiv2015

On the polarization properties of magnetar giant flare pulsating tails

Three giant flares have been detected so far from soft gamma-ray repeaters, each characterized by an initial short hard spike and a pulsating tail. The observed pulsating tails are characterized by a duration of $\sim100\,\rm{s}$, an isotropic energy of $\sim 10^{44}\,\rm{erg}$, and a pulse period of a few seconds. The pulsating tail emission likely originates from the residual energy after the intense energy release during the initial spike, which forms a trapped fireball composed of a photon-pair plasma in a closed field line region of the magnetars. Observationally the spectra of pulsating tails can be fitted by the superposition of a thermal component and a power-law component, with the thermal component dominating the emission in the early and late stages of the pulsating tail observations. In this paper, assuming that the trapped fireball is from a closed field line region in the magnetosphere, we calculate the atmosphere structure of the optically-thick trapped fireball and the polarization properties of the trapped fireball. By properly treating the photon propagation in a hot, highly magnetized, electron-positron pair plasma, we tally photons in two modes (O mode and E mode) at a certain observational angle through Monte Carlo simulations. Our results suggest that the polarization degree depends on the viewing angle with respect to the magnetic axis of the magnetar, and can be as high as $Π\simeq30\%$ in the $1-30\,\rm{keV}$ band, and $Π\simeq10\%$ in the $30-100\,\rm{keV}$ band, if the line of sight is perpendicular to the magnetic axis.

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arXiv2015

A Tight $L_{\rm p, iso}-E'_{\rm p}-Γ_0$ Correlation of Gamma-Ray Bursts

We select a sample of 34 gamma-ray bursts (GRBs) whose $Γ_0$ values are derived with the onset peaks observed in the afterglow lightcurves (except for GRB 060218 whose $Γ_0$ is estimated with its radio data), and investigate the correlations among $Γ_0$, the isotropic peak luminosity ($L_{\rm iso}$), and the peak energy ($E_{\rm p,z}$) of the $νf_ν$ spectrum in the cosmological rest frame. An analysis of pair correlations among these observables well confirms the results reported by the previous papers. More interestingly, a tight correlation among $L_{\rm iso}$, $E_{\rm p,z}$, and $Γ_0$ is found from a multiple regression analysis, which takes the form of $L_{\rm iso} \propto E_{\rm p,z}^{1.34\pm 0.14} Γ_0^{1.32\pm 0.19}$ or $E_{\rm p,z} \propto L_{\rm iso}^{0.55\pm 0.06}Γ_0^{-0.50\pm 0.17}$. Nine other GRBs whose $Γ_0$ are derived via the pair production opacity constraint also follow such a correlation. Excluding GRB 060218, the $L_{\rm iso}-E_{\rm p,z}-Γ_0$ correlation is valid, and it even holds in the jet co-moving frame. However, GRB 060218 deviates the $L^{'}_{\rm iso}-E^{'}_{\rm p}$ relation of typical GRBs in the jet co-moving frame with $3σ$. We argue that the $L_{\rm iso} - E_{\rm p, z} - Γ_0$ correlation may be more physical than the $L_{\rm iso} - E_{\rm p,z}$ correlation, since physically the relationship between the observed $L_{\rm iso}$ and $E_{\rm p,z}$ not only depends on radiation physics, but also depends on the bulk motion of the jet. We explore the possible origins of this correlation and discuss its physical implications for understanding GRB jet composition and radiation mechanism.

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arXiv2015

The extension of variability properties in gamma-ray bursts to blazars

Both gamma-ray bursts (GRBs) and blazars have relativistic jets pointing at a small angle from our line of sight. Several recent studies suggested that these two kinds of sources may share similar jet physics. In this work, we explore the variability properties for GRBs and blazars as a whole. We find that the correlation between minimum variability timescale (MTS) and Lorentz factor, $Γ$, as found only in GRBs by Sonbas et al. can be extended to blazars with a joint correlation of $\rm MTS\proptoΓ^{-4.7\pm0.3}$. The same applies to the $\rm MTS\propto \it L_γ^{\rm -1.0\pm0.1}$ correlation as found in GRBs, which can be well extended into blazars as well. These results provide further evidence that the jets in these two kinds of sources are similar despite of the very different mass scale of their central engines. Further investigations of the physical origin of these correlations are needed, which can shed light on the nature of the jet physics.

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arXiv2015

Extra-galactic high-energy transients: event rate densities and luminosity functions

Several types of extra-galactic high-energy transients have been discovered, which include high-luminosity and low-luminosity long-duration gamma-ray bursts (GRBs), short-duration GRBs, supernova shock breakouts (SBOs), and tidal disruption events (TDEs) without or with an associated relativistic jet. In this paper, we apply a unified method to systematically study the redshift-dependent event rate densities and the global luminosity functions (ignoring redshift evolution) of these transients. We introduce some empirical formulae for the redshift-dependent event rate densities for different types of transients, and derive the local specific event rate density, which also represents its global luminosity function. Long GRBs have a large enough sample to reveal features in the global luminosity function, which is best characterized as a triple power law. All the other transients are consistent with having a single power law luminosity function. The total event rate density depends on the minimum luminosity, and we obtain the following values in units of ${\rm Gpc^{-3}~yr^{-1}}$: $0.8^{+0.1}_{-0.1}$ for high-luminosity long GRBs above $ 10^{50}~{\rm erg~s^{-1}}$, $164^{+98}_{-65}$ for low-luminosity long GRBs above $5\times 10^{46}~{\rm erg~s^{-1}}$, $1.3^{+0.4}_{-0.3}$, $1.2^{+0.4}_{-0.3}$, and $3.3^{+1.0}_{-0.8}$ above $ 10^{50}~ {\rm erg~s^{-1}}$ for short GRBs with three different merger delay models (Gaussian, log-normal, and power law), $1.9^{+2.4}_{-1.2}\times 10^4$ above $ 10^{44}~{\rm erg~s^{-1}}$ for SBOs, $ 4.8^{+3.2}_{-2.1}\times10^2 $ for normal TDEs above $10^{44}~ {\rm erg~s^{-1}}$, and $0.03^{+0.04}_{-0.02}$ above $ 10^{48}~ {\rm erg~s^{-1}}$ for TDE jets as discovered by Swift. Intriguingly, the global luminosity functions of different kinds of transients, which cover over 12 orders of magnitude, are consistent with a single power law with an index of -1.6.

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arXiv2015

Can life survive Gamma-Ray Bursts in the high-redshift universe?

Nearby Gamma-Ray Bursts (GRBs) have been proposed as a possible cause of mass extinctions on Earth. Due to the higher event rate of GRBs at higher redshifts, it has been speculated that life as we know it may not survive above a certain redshift (e.g. $z>0.5$). We examine the duty cycle of lethal (life-threatening) GRBs in the solar neighborhood, in the Sloan Digital Sky Survey (SDSS) galaxies and GRB host galaxies, with the dependence of the long GRB rate on star formation and metallicity properly taken into account. We find that the number of lethal GRBs attacking Earth within the past 500 Myr ($\sim$ epoch of the Ordovician mass extinction) is $0.93$. The number of lethal GRBs hitting a certain planet increases with redshift, thanks to the increasing star formation rate and decreasing metallicity in high-$z$ galaxies. Taking 1 per 500 Myr as a conservative duty cycle for life to survive, as evidenced by our existence, we find that there are still a good fraction of SDSS galaxies beyond $z=0.5$ where the GRB rate at half-mass radius is lower than this value. We derive the fraction of such benign galaxies as a function of redshift through Monte Carlo simulations, and find that the fraction is $\sim 50\%$ at $z\sim 1.5$ and $\sim 10\%$ even at $z \sim 3$. The mass distribution of benign galaxies is dominated by Milky-Way-like ones, thanks to their commonness, relatively large mass, and low star formation rate. GRB host galaxies are among the most dangerous ones.

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arXiv2015

A morphological analysis of gamma-ray burst early optical afterglows

Within the framework of the external shock model of gamma-ray bursts (GRBs) afterglows, we perform a morphological analysis of the early optical lightcurves to directly constrain model parameters. We define four morphological types, i.e. the reverse shock dominated cases with/without the emergence of the forward shock peak (Type I/ Type II), and the forward shock dominated cases without/with $ν_m$ crossing the band (Type III/IV). We systematically investigate all the Swift GRBs that have optical detection earlier than 500 s and find 3/63 Type I bursts (4.8%), 12/63 Type II bursts (19.0%), 30/63 Type III bursts (47.6%), 8/63 Type IV bursts (12.7%) and 10/63 Type III/IV bursts (15.9%). We perform Monte Carlo simulations to constrain model parameters in order to reproduce the observations. We find that the favored value of the magnetic equipartition parameter in the forward shock ($ε_B^f$) ranges from $10^{-6}$ to $10^{-2}$, and the reverse-to-forward ratio of $ε_B$ ($R_B$) is about 100. The preferred electron equipartition parameter $ε_e^{r,f}$ value is 0.01, which is smaller than the commonly assumed value, e.g., 0.1. This could mitigate the so- called "efficiency problem" for the internal shock model, if $ε_e$ during the prompt emission phase (in the internal shocks) is large (say, $\sim 0.1$). The preferred $R_B$ value is in agreement with the results in previous works that indicates a moderately magnetized baryonic jet for GRBs.

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arXiv2015

GRB 080503 late afterglow re-brightening: signature of a magnetar powered merger-nova

GRB 080503 is a short gamma-ray burst (GRB) detected by \emph{Swift} and has been classified as a compact-star-merger-origin GRB. The soft extended emission and the simultaneous late re-brightening in both the X-ray and optical afterglow lightcurves raise interesting questions regarding its physical origin. We show that the broad-band data of GRB 080503 can be well explained within the framework of the double neutron star merger model, provided that the merger remnant is a rapidly-rotating massive neutron star with an extremely high magnetic field (i.e. a millisecond magnetar). We show that the late optical re-brightening is consistent with the emission from a magnetar-powered "merger-nova". This adds one more case to the growing sample of merger-novae associated with short GRBs. The soft extended emission and the late X-ray excess emission are well connected through a magnetar dipole spin-down luminosity evolution function, suggesting that direct magnetic dissipation is the mechanism to produce these X-rays. The X-ray emission initially leaks from a hole in the merger ejecta pierced by the short GRB jet. The hole subsequently closes after the magnetar spins down and the magnetic pressure drops below ram pressure. The X-ray photons are then trapped behind the mergernova ejecta until the ejecta becomes optically thin at a later time. This explains the essentially simultaneous re-brightening in both the optical and X-ray lightcurves. Within this model, future gravitational wave sources could be associated with a bright X-ray counterpart along with the mergernova, even if the short GRB jet beams away from Earth.

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arXiv2015

How Bad/Good Are the External Forward Shock Afterglow Models of Gamma-Ray Bursts?

The external forward shock (EFS) models have been the standard paradigm to interpret the broad-band afterglow data of gamma-ray bursts (GRBs). One prediction of the models is that some afterglow temporal breaks at different energy bands should be achromatic. Observations in the Swift era have revealed chromatic afterglow behaviors at least in some GRBs, casting doubts on the EFS origin of GRB afterglows. In this paper, we perform a systematic study to address the question: how bad/good are the external forward shock models? Our sample includes 85 GRBs well-monitored X-ray and optical lightcurves. Based on how well the data abide by the EFS models, we categorize them as: Gold sample: (Grade I and II) include 45/85 GRBs. They show evidence of, or are consistent with having, an achromatic break. The temporal/spectral behaviors in each afterglow segment are consistent with the predictions (closure relations) of the EFS models. Silver sample: (Grade III and IV) include 37/85 GRBs. They are also consistent with having an achromatic break, even though one or more afterglow segments do not comply with the closure relations. Bad sample: (Grade V), 3/85 shows direct evidence of chromatic behaviors, suggesting that the EFS models are inconsistent with the data. These are included in the Bad sample. We further perform statistical analyses of various observational properties ($α$, $β$, $t_b$ and model parameters (energy injection index q, p, $θ_j$, $η_γ$, etc) of the GRBs in the Gold Sample, and derive constraints on the magnetization parameter $ε_B$ in the EFS. Overall, we conclude that the simplest EFS models can account for the multi-wavelength afterglow data of at least half of the GRBs. When more advanced modeling (e.g., long-lasting reverse shock, structured jets) is invoked, up to $>90 \%$ of the afterglows may be interpreted within the framework of the ESF models.

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arXiv2015

On the Curvature Effect of a Relativistic Spherical Shell

We consider a relativistic spherical shell and calculate its spectral flux as received by a distant observer. Using two different methods, we derive a simple analytical expression of the observed spectral flux and show that the well-known relation $\hat α= 2+\hat β$ (between temporal index $\hat α$ and spectral index $\hat β$) of the high-latitude emission is achieved naturally in our derivation but holds only when the shell moves with a constant Lorentz factor $Γ$. Presenting numerical models where the shell is under acceleration or deceleration, we show that the simple $\hat α= 2+\hat β$ relation is indeed deviated as long as $Γ$ is not constant. For the models under acceleration, we find that the light curves produced purely by the high-latitude emission decay initially much steeper than the constant $Γ$ case and gradually resume the $\hat α= 2+\hat β$ relation in about one and half orders of magnitude in observer time. For the models under deceleration, the trend is opposite. The light curves made purely by the high-latitude emission decay initially shallower than the constant $Γ$ case and gradually resume the relation $\hat α= 2+\hat β$ in a similar order of magnitude in observer time. We also show that how fast the Lorentz factor $Γ$ of the shell increases or decreases is the main ingredient determining the initial steepness or shallowness of the light curves.

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arXiv2015

The Millisecond Magnetar Central Engine in short GRBs

One favored progenitor model for short duration gamma-ray bursts (SGRBs) is the coalescence of two neutron stars (NS-NS). One possible outcome of such a merger would be a rapidly spinning, strongly magnetized neutron star (known as a millisecond magnetar). These magnetars may be "supra-massive", implying they would collapse to black holes after losing centrifugal support due to magnetic dipole spindown. By systematically analyzing the BAT-XRT light curves of all short GRBs detected by {\em swift}, we test how well the data are consistent with this central engine model of short GRBs. We find that the so-called "extended emission" observed with BAT in some short GRBs are fundamentally the same component as the "internal X-ray plateau" as observed in many short GRBs, which is defined as a plateau in the lightcurve followed by a very rapid drop. Based on how likely a short GRB hosts a magnetar, we characterize the entire {\em Swift} short GRB sample into three categories: the "internal plateau" sample, the "external plateau" sample, and the "no plateau" sample. The derived magnetar surface magnetic field $B_{\rm p}$ and the initial spin period $P_0$ fall into the reasonable range. No GRBs in the internal plateau sample have the total energy exceeding the maximum energy budget of a millisecond magnetar. Assuming that the rapid fall time at the end of the internal plateau is the collapse time of a supra-massive magnetar to a black hole, and applying the measured mass distribution of NS-NS systems in our Galaxy, we constrain the neutron star equation of state (EOS). The data suggest that the NS equation of state is close to the GM1 model, which has a maximum non-rotating NS mass $M_{\rm TOV} \sim 2.37 M_\odot$.

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arXiv2015

Synchrotron Origin of the Typical GRB Band Function - A Case Study of GRB 130606B

We perform a time-resolved spectral analysis of GRB 130606B within the framework of a fast-cooling synchrotron radiation model with magnetic field strength in the emission region decaying with time, as proposed by Uhm & Zhang. The data from all time intervals can be successfully fit by the model. The same data can be equally well fit by the empirical Band function with typical parameter values. Our results, which involve only minimal physical assumptions, offer one natural solution to the origin of the observed GRB spectra and imply that, at least some, if not all, Band-like GRB spectra with typical Band parameter values can indeed be explained by synchrotron radiation.

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arXiv2015

A Correlated Study of Optical and X-ray Afterglows of GRBs

We study an extensive sample of 87 GRBs for which there are well sampled and simultaneous optical and X-ray light-curves. We extract the cleanest possible signal of the afterglow component, and compare the temporal behaviors of the X-ray light-curve, observed by Swift XRT, and optical data, observed by UVOT and ground-based telescopes for each individual burst. Overall we find 62\% GRBs that are consistent with the standard afterglow model. When more advanced modeling is invoked, up to 91\% of the bursts in our sample may be consistent with the external shock model. A large fraction of these bursts are consistent with occurring in a constant interstellar density medium (ISM) (61\%) while only 39\% of them occur in a wind-like medium. Only 9 cases have afterglow light-curves that exactly match the standard fireball model prediction, having a single power law decay in both energy bands which are observed during their entire duration. In particular, for the bursts with chromatic behavior additional model assumptions must be made over limited segments of the light-curves in order for these bursts to fully agree with the external shock model. Interestingly, for 54\% of the X-ray and 40\% of the optical band observations the end of the shallow decay ($t^{\sim-0.5}$) period coincides with the jet break ($t^{\sim-p}$) time, causing an abrupt change in decay slope. The fraction of the burst that consistent with the external shock model is independent of the observational epochs in the rest frame of GRBs. Moreover, no cases can be explained by the cooling frequency crossing the X-ray or optical band.

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