arXiv subjects

En-Wei Liang

Publications and source records attributed to En-Wei Liang.

At least 145 records · Page 8Linked to original sources

arXiv2021

Evidence for gravitational lensing of GRB 200716C

Observationally, there is a small fraction of Gamma-ray bursts (GRBs) with prompt emission observed by Fermi/GBM that are composed of two pulses. Occasionally, the distance to a GRB may be lensed when a high mass astrophysical object resides in the path between the GRB source and observer. In this paper, we describe GRB 200716C, which has a two-pulse emission and duration of a few seconds. We present a Bayesian analysis identifying gravitational lensing in both temporal and spectral properties, and calculate the time delay ($Δt\sim 1.92$ s) and magnification ($γ\sim 1.5$) between those two pulses based on the temporal fits. One can roughly estimate the lens mass to be about $2.4\times 10^{5}~M_{\odot}$ in the rest frame. We also calculate the false alarm probability for this detection to be about 0.07\% with trial factors, and a present-day number density of about $808 \rm~Mpc^{-3}$ with an energy density $Ω\sim 1.4\times 10^{-3}$. If the first pulse of this GRB near the trigger time is indeed gravitationally echoed by a second pulse, GRB 200716C may be a short GRB candidate with extended emission.

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arXiv2021

Comparison of the Characteristics of Magnetars Born in Death of Massive Stars and Merger of Compact Objects With {\em Swift} Gamma-Ray Burst Data

Assuming that the shallow-decaying phase in the early X-ray lightcurves of gamma-ray bursts (GRBs) is attributed to the dipole radiations (DRs) of a newborn magnetar, we present a comparative analysis for the magnetars born in death of massive stars and merger of compact binaries with long and short GRB (lGRB and sGRB) data observed with the {\em Swift} mission. We show that the typical braking index ($n$) of the magnetars is $\sim 3$ in the sGRB sample, and it is $\sim 4$ for the magnetars in the lGRB sample. Selecting a sub-sample of the magnetars whose spin-down is dominated by DRs ($n\lesssim 3$) and adopting a universal radiation efficiency of $0.3$, we find that the typical magnetic field strength ($B_p$) is $10^{16}$ G {\em vs.} $10^{15}$ G and the typical initial period ($P_0$) is $\sim 20$ ms {\em vs.} $2$ ms for the magnetars in the sGRBs {\em vs.} lGRBs. They follow the same relation between $P_0$ and the isotropic GRB energy as $ P_0\propto E_{\rm jet}^{-0.4}$. We also extend our comparison analysis to superluminous supernovae (SLSNe) and stable pulsars. Our results show that a magnetar born in merger of compact stars tends to have a stronger $B_p$ and a longer $P_0$ by about one order of magnitude than that born in collapse of massive stars. Its spin-down is dominated by the magnetic DRs as old pulsars, being due to its strong magnetic field strength, whereas the early spin-down of magnetars born in massive star collapse is governed by both the DRs and gravitational wave (GW) emission. A magnetar with a faster rotation speed should power a more energetic jet, being independent of its formation approach.

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arXiv2021

GRB 101225A as Orphan Dipole Radiation of a Newborn Magnetar with Precession Rotation in an Off-Axis Gamma-Ray Burst

The unusual multiwavelength lightcurves of GRB 101225A are revisited by assuming that it is from an off-axis GRB powered by a newborn magnetar. We show that its optical afterglow lightcurve is fitted with the forward shock model by parameterizing its jet structure as a Gaussian function with a half opening angle of the jet core as $1.67^{\rm o}$. The derived initial Lorentz factor ($Γ_0$) is 120, and the viewing angle to the jet axis is $θ_v=3.7^{\rm o}$. Tentative QPO signatures of $P=488$ seconds and $P=250\sim 300$ seconds are found with a confidence level of 90\% by analysing its X-ray flares observed in the time interval of $[4900,\ 7500]$ seconds. Its global gamma-ray/X-ray lightcurve and the QPO signatures are represented with the magnetar dipole radiation (DR) model by considering the magnetar precession motion, assuming that the magnetar spindown is dominated by the GW emission. The bulk Lorentz factor of the DR ejecta is limited to 8, being much lower than $Γ_0$. Comparing GRB 101225A with the extremely off-axis GRB 170817A, we suspect that the nature of the two-component jet in GRB 170817A is a combination of a co-axial GRB jet and a DR ejecta. GRB 101225A would be among the brightest ones of the CDF-S XT2 like X-ray transient population driven by newborn magnetars. Discussion on detectability of its gravitational wave emission is also presented.

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arXiv2021

The Study of Dust Formation of Four Type Ibn Supernovae

In this paper, we investigate the early-time optical$-$near-infrared (NIR) spectral energy distributions (SEDs) of four Type Ibn supernovae (SNe). We find that the SEDs of SN~2010al, LSQ13ddu, and SN~2015G can be well explained by the single-component blackbody model, while the SEDs of OGLE-2012-SN-006 cannot. We invoke the double-component model assuming that the SEDs were produced by the SN photosphere and the heated dust to fit the optical$-$NIR SEDs of the four SNe Ibn, finding that the derived temperatures of the dust associated with OGLE-2012-SN-006 favor the scenario that the dust consists of the graphite grains, and the mass and temperature of dust are $\sim$$0.5-2.0\times10^{-3}~M_\odot$ and $\sim$ $1200-1300$ K, respectively. Moreover, our fits for SN~2010al, LSQ13ddu, and SN~2015G show that the upper limits of the masses of the dust associated with the three SNe Ibn are respectively $1.45\times 10^{-5}~M_\odot$, $5.9\times 10^{-7}~M_\odot$, and $2.4\times 10^{-7}~M_\odot$. A further analysis demonstrates that the inferred radius of the dust shell surrounding OGLE-2012-SN-006 is significantly larger than that of the SN ejecta at early epochs, indicating that the NIR excesses of the SEDs of OGLE-2012-SN-006 were produced by a preexisting dust shell. Our study for the early-time SEDs of four SNe Ibn, together with the previous studies and the fact that some SNe showed the evidence of dust formation at the late-time SEDs, indicates that at least $\sim$1/3 of SNe Ibn show evidence for dust formation.

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arXiv2021

GRB 210121A: A Typical Fireball Burst Detected by Two Small Missions

The Chinese CubeSat Mission, Gamma Ray Integrated Detectors (GRID), recently detected its first gamma-ray burst, GRB 210121A, which was jointly observed by the Gravitational wave high-energy Electromagnetic Counterpart All-sky Monitor (GECAM). This burst is confirmed by several other missions, including \fermi and \textit{Insight}-HXMT. We combined multi-mission observational data and performed a comprehensive analysis of the burst's temporal and spectral properties. Our results show that the burst is relatively special in its high peak energy, thermal-like low energy indices, and large fluence. By putting it to the $E_{\rm p}$-$E_{\rmγ, iso}$ relation diagram with assumed distance, we found this burst can be constrained at the redshift range of [0.3,3.0]. The thermal spectral component is also confirmed by the direct fit of the physical models to the observed spectra. Interestingly, the physical photosphere model also constrained a redshift of $z\sim$ 0.3 for this burst, which help us to identify a host galaxy candidate at such a distance within the location error box. Assuming the host galaxy is real, we found the burst can be best explained by the photosphere emission of a typical fireball with an initial radius of $r_0\sim$ 3.2 $\times 10^7$ cm.

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arXiv2021

The Magnetar Model and the Ejecta--Circumstellar-Matter Interaction Model for 31 Luminous, Rapidly Evolving Optical Transients

We study 31 luminous, rapidly evolving optical transients (REOTs), and use the magnetar model and the ejecta--circumstellar-matter (CSM) interaction (CSI) model to fit their multiband light curves. We find that 28 events can be fitted by the magnetar model. and the derived masses ($\sim0.05$--5.58\,M$_\odot$) are consistent with that of stripped and ultrastripped core-collapse supernovae (CCSNe). On the other hand, the CSI model can fit 25 events; the derived ejecta masses are $\sim 3$--30\,M$_\odot$, consistent with that of SNe~Ibn. The results, together with the fact that many luminous REOTs have been confirmed to be luminous SNe~IIb/Ib/c or Ibn/IIn, suggest that at least a fraction of luminous REOTs spectroscopically unclassified might be rapidly evolving SNe~Ib/Ic or Ibn. Three events in the sample cannot be explained by the two models we use. We expect that future intense photometry, spectroscopic classifications, and systematic light-curve modeling of luminous REOTs will shed more light on their nature.

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arXiv2021

Revisiting the analysis of axion-like particles with the Fermi-LAT gamma-ray observation of NGC1275

In this work, we re-analyze the Fermi-LAT observation of NGC 1275 to search for axion-like particle (ALP) effects and constrain ALP parameters. Instead of fitting the observed spectrum with ALP models, we adopt an alternative method for the analysis of this source which calculates the irregularity of the spectrum. With the newly used method, we find no spectral oscillation for the NGC 1275 and rule out couplings $g_{aγ}>3\times10^{-12}\,{\rm GeV^{-1}}$ around ALP mass of $m_a\sim$ 1 neV at 95\% confidence level, which is more stringent than the previous results. We also show that the constraints can be further improved by combining the observation of PKS 2155-304. We suggest that with more sources taken into account, we could obtain a much wider exclusion region.

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arXiv2021

Gamma-Ray Emission Produced by $r$-process Elements from Neutron Star Mergers

The observation of a radioactively powered kilonova AT~2017gfo associated with the gravitational wave-event GW170817 from binary neutron star merger proves that these events are ideal sites for the production of heavy $r$-process elements. The gamma-ray photons produced by the radioactive decay of heavy elements are unique probes for the detailed nuclide compositions. Basing on the detailed $r$-process nucleosynthesis calculations and considering radiative transport calculations for the gamma-rays in different shells, we study the gamma-ray emission in a merger ejecta on a timescale of a few days. It is found that the total gamma-ray energy generation rate evolution is roughly depicted as $\dot{E}\propto t^{-1.3}$. For the dynamical ejecta with a low electron fraction ($Y_{\rm e}\lesssim0.20$), the dominant contributors of gamma-ray energy are the nuclides around the second $r$-process peak ($A\sim130$), and the decay chain of $^{132}$Te ($t_{1/2}=3.21$~days) $\rightarrow$ $^{132}$I ($t_{1/2}=0.10$~days) $\rightarrow$ $^{132}$Xe produces gamma-ray lines at $228$ keV, $668$ keV, and $773$ keV. For the case of a wind ejecta with $Y_{\rm e}\gtrsim0.30$, the dominant contributors of gamma-ray energy are the nuclides around the first $r$-process peak ($A\sim80$), and the decay chain of $^{72}$Zn ($t_{1/2}=1.93$~days) $\rightarrow$ $^{72}$Ga ($t_{1/2}=0.59$~days) $\rightarrow$ $^{72}$Ge produces gamma-ray lines at $145$ keV, $834$ keV, $2202$ keV, and $2508$ keV. The peak fluxes of these lines are $10^{-9}\sim 10^{-7}$~ph~cm$^{-2}$ s$^{-1}$, which are marginally detectable with the next-generation MeV gamma-ray detector \emph{ETCC} if the source is at a distance of $40$~Mpc.

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arXiv2021

Late Afterglow Bump/Plateau around the Jet Break: Signature of a free-to-shocked wind Environment in Gamma-ray Burst

A number of gamma-ray bursts (GRBs) exhibit the late simultaneous bumps in their optical and Xray afterglows around the jet break. Its origin is unclear. Based on the following two facts, we suggest that this feature may sound a transition of circum-burst environment from a free-wind medium to a homogeneous medium. (I) The late bump followed by a steep decay is strongly reminiscent of the afterglows of GRB 170817A, which is attributed to an off-axis observed external-forward shock (eFS) propagating in an interstellar medium. (II) Observations seem to feature a long shallow decay before the late optical bump, which is different from the afterglow of GRB 170817A. In this paper, we study the emission of an eFS propagating in a free-to-shocked wind for on/off-axis observers, where the mass density in the shocked-wind is almost constant. The late simultaneous bumps/plateaux in the optical and X-ray afterglows are really found around the jet break for high-viewing-angle observers. Moreover, there is a long plateau or shallow decay before the late bump in the theoretical light-curves, which is formed during the eFS propagating in the free-wind. For low-viewing-angle observers, the above bumps appear only in the situation that the structured jet has a low characteristic angle and the deceleration radius of the on-axis jet flow is at around or beyond the free-wind boundary. As examples, the X-ray and optical afterglows of GRBs 120326A, 120404A, and 100814A are fitted. We find that an off-axis observed eFS in a free-to-shocked wind can well explain the afterglows in these bursts.

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arXiv2021

Highly Variable $γ$-Ray Emission of CTD 135 and Implications for its Compact Symmetric Structure

The $γ$-ray emission properties of CTD 135, a typical compact symmetric object (CSO), are investigated with $\sim$11-year Fermi/LAT observations. We show that it has bright and significantly variable GeV emission, with the $γ$-ray luminosity of $L_γ\sim10^{47}$ erg s$^{-1}$ and a variation index of TS$_{\rm var}=1002$. A quasi-periodic oscillation (QPO) with a periodicity of $\sim$460 days is detected in the global 95\% false-alarm level. These $γ$-ray emission features are similar to that of blazars. Its broadband spectral energy distribution (SED) can be attributed to the radiations of the relativistic electrons accelerated in the core region and the extended region. The SED modeling shows that the $γ$-rays are from the core region, which has a Doppler boosting factor of $δ\sim10.8$ and relativistically moves with a small viewing angle, being similar to blazar jets. On the base of the analysis results, we propose that the episodic activity of the central engine in CTD 135 results in a blazar-like jet and the bubble-like lobes as the Fermi bubbles in the Galaxy. The strong $γ$-ray emission with obvious variability is from the jet radiations and the symmetric radio structure is attributed to the bubbles. The jet radiation power and disk luminosity in units of Eddington luminosity of CTD 135 follow the same relation as other young radio galaxies, indicating that its jet radiation may be also driven by the Eddington ratio.

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arXiv2021

Detectability of "Merger-nova" emission from a long-lived magnetar in short gamma-ray bursts

One possible progenitor of short gamma-ray bursts (GRBs) is thought to be from a double neutron star (NS) merger, and the remnant of such a merger may be a supramassive NS, which is supported by rigid rotation and through its survival of hundreds of seconds before collapsing into a black hole (BH). If this is the case, an optical/infrared transient (namely merger-nova) is generated from the ejected materials and it is powered by radioactive decay from $r$-process, spin-down energy from a supramassive NS, as well as the magnetic wind from a newborn BH. In this paper, we systematically search for the signature of a supramassive NS central engine by analyzing the X-ray emission of short GRBs with internal plateau observed by {\em Swift}, and we find that five candidates of short GRBs have such feature with redshift measurement. Then, we calculate the possible merger-nova emission from those candidates for given the typical model parameters by considering the above three energy sources, and compare its brightness with the sensitivity of some optical telescopes. We find that the merger-nova emission of GRB 060801 in K-, r-, and U-bands with variations of $M_{\rm ej}$ ($10^{-4}-10^{-2} M_{\odot}$), $κ$ ($0.1-10 ~\rm cm^{2}~g^{-1}$), and $β$ ($0.1-0.3$) is very difficult to detect using the Vera C. Rubin, Panoramic Survey Telescope and Rapid Response System (Pan-STARRS), the Zwicky Transient Facility, and the Roman Space Telescope (Roman), except for the case of large ejecta mass $M_{\rm ej}=10^{-2} M_{\odot}$. However, we are very hopeful to detect the merger-nova emission of GRBs 090515, 100625A, and 101219A using more sensitive instruments, such as Vera C. Rubin, Pan-STARRS, and Roman. Moreover, the merger-nova emission of GRB 160821B is bright enough to detect in our calculations, and it is also consistent with current real observations of merger-nova emission.

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arXiv2021

Electron Spectrum for the Prompt Emission of Gamma-ray Bursts in the Synchrotron Radiation Scenario

Growing evidence indicates that the synchrotron radiation mechanism may be responsible for the prompt emission of gamma-ray bursts (GRBs). In the synchrotron radiation scenario, the electron energy spectrum of the prompt emission is diverse in theoretical works and has not been estimated from observations in a general way (i.e., without specifying a certain physical model for the electron spectrum). In this paper, we creatively propose a method to directly estimate the electron spectrum for the prompt emission, without specifying a certain physical model for the electron spectrum in the synchrotron radiation scenario. In this method, an empirical function (i.e., a four-order Bezier curve jointed with a linear function at high-energy) is applied to describe the electron spectrum in log-log coordinate. It is found that our empirical function can well mimic the electron spectra obtained in many numerical calculations or simulations. Then, our method can figure out the electron spectrum for the prompt emission without specifying a model. By employing our method on observations, taking GRB 180720B and GRB 160509A as examples, it is found that the obtained electron spectra are generally different from that in the standard fast-cooling scenario and even a broken power law. Moreover, the morphology of electron spectra in its low-energy regime varies with time in a burst and even in a pulse. Our proposed method provides a valuable way to confront the synchrotron radiation mechanism with observations.

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arXiv2020

Very High Energy Emission and Cascade Radiation of Gamma-Ray Burst Afterglows: Homogeneous Versus Wind External Media

Recent detection of sub-TeV emission from gamma-ray bursts (GRBs) represents a breakthrough in the GRB study. The multi-wavelength data of the afterglows of GRB 190114C support the synchrotron self-Compton (SSC) origin for its sub-TeV emission. We present a comparative analysis on the SSC emission of GRB afterglows in the homogeneous and wind environment in the framework of the forward shock model. The $γγ$ absorption of very high-energy photons due to pair production within the source and the Klein-Nishina effect on the inverse-Compton scattering are considered. Generally a higher SSC flux is expected for a larger circum-burst density due to a larger Compton parameter, but meanwhile the internal $γγ$ absorption is more severer for sub-TeV emission. The flux ratio between the SSC component and the synchrotron component decreases more quickly with time in the wind medium case than that in the homogenous-density medium case. The light curves of the SSC emission are also different for the two types of media. We also calculate the cascade emission resulted from the absorbed high-energy photons. In the ISM environment with $n> 1\,\rm cm^{-3}$, the cascade synchrotron emission could be comparable to the synchrotron emission of the primary electrons in the optical band, which may flatten the optical afterglow light curve at early time ($t<1$ h). In the wind medium with $A_{\ast}> 0.1$, the cascade emission in the eV-GeV band is comparable or even larger than the emission of the primary electrons at early time.

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arXiv2020

Lorentz Invariance Violation Limits from the Spectral Lag Transition of GRB~190114C

The spectral lags of gamma-ray bursts (GRBs) have been viewed as the most promising probes of the possible violations of Lorentz invariance (LIV). However, these constraints usually depend on the assumption of the unknown intrinsic time lag in different energy bands and the use of a single highest-energy photon. A new approach to test the LIV effects has been proposed by directly fitting the spectral lag behavior of a GRB with a well-defined transition from positive lags to negative lags. This method simultaneously provides a reasonable formulation of the intrinsic time lag and robust lower limits on the quantum-gravity energy scales ($E_{\rm QG}$). In this work, we perform a global fitting to the spectral lag data of GRB~190114C by considering the possible LIV effects based on a Bayesian approach. We then derive limits on $E_{\rm QG}$ and the coefficients of the Standard Model Extension. The Bayes factors output in our analysis shows a very strong evidence for the spectral-lag transition in GRB~190114C. Our constraints on a variety of isotropic and anisotropic coefficients for LIV are somewhat weaker than existing bounds, but they can be viewed as comparatively robust and have the promise to complement existing LIV constraints. The observations of GRBs with higher-energy emissions and higher temporal resolutions will contribute to a better formulation of the intrinsic time lag and more rigorous LIV constraints in the dispersive photon sector.

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arXiv2020

$L_{\rm syn}-E_{\rm syn, p}-δ$ relation in Active Galactic Nucleus Jets and Implication for the physical origin of the $L_{\rm p}-E_{\rm p,z}-Γ_0$ relation of Gamma-Ray Bursts

High energy photon radiations of gamma-ray bursts (GRBs) and active galactic nuclei (AGNs) are dominated by their jet radiations. We examine wether the synchrotron radiations of jets in BL Lacs, flat spectrum radio quasars (FSRQs), and Narrow Line Seyfert 1 galaxies (NLS1s) follow the relation between the prompt gamma-ray emission and the initial Lorentz factor ($Γ_0$) of GRBs. It is showed that the AGNs sample does not agree with the $L_{\rm p}-E_{\rm p,z}-Γ_{0}$ of GRBs. In addition, we obtain a tight relation of $L_{\rm syn}\propto E^{0.45\pm0.15}_{\rm syn,p}δ^{3.50\pm0.25}$ for FSRQs and NLS1 galaxies, where $L_{\rm syn}$ is the luminosity at peak photon energy $E_{\rm syn, p}$ of the synchrotron radiations. This relation is different from the $L_{\rm p}-E_{\rm p,z}-Γ_{0}$ relation of GRBs. The dependence of $L_{\rm syn}$ to $δ$ is consistent with the expectation of the Doppler boosting effect for the FSRQs and NLS1 galaxies, but it is not for GBRs. We argue that $Γ_0$ may be a representative of the kinetic power of the radiating region and the tight $L_{\rm p}-E_{\rm p, z}-Γ_0$ relation is shaped by the radiation physics and the jet power together.

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arXiv2020

Afterglow Synchrotron Radiations follow the $L_{\rm p, iso}-E_{\rm p,z}-Γ_0$ relation of Gamma-Ray Bursts? Cases of GRBs 190114C, 130427A, and 180720B

Bimodal spectral energy distributions (SEDs) of gamma-ray burst (GRB) afterglow of GRBs 190114C, 130427A and 180720B confirm that they are originated from the synchrotron emission (Syn) and synchrotron self-Compton Scattering process (SSC) of electrons accelerated in the jets. The radiation mechanism and the physics of the observed spectrum-luminosity/energy relations of GRBs remain as open questions. By extracting the Syn component through fitting their early afterglow SEDs with the Syn+SSC model, we find that their luminosity ($L_{\rm syn}$), peak energy ($E_{\rm p,syn,z}$), and the Lorentz factor of the afterglow fireball ($Γ_t$) follow the $L_{\rm p, iso}-E_{\rm p,z}-Γ_{0}$ relation of prompt gamma-rays, where $L_{\rm p, iso}$ is the isotropic luminosity, $E_{\rm p, z}$ is the peak energy of the $νf_ν$ spectrum in the burst frame, and $Γ_0$ is the initial Lorentz factor of the fireball. To examine whether late afterglows is consistent with this relation, we calculate the synchrotron component at late afterglows. It is found that they also follow the same $L_{\rm p, iso}-E_{\rm p,z}-Γ_{0}$ relation, albeit they are not consistent with the $L_{\rm p, iso}-E_{\rm p,z}$ relation. Our results may imply that the $L_{\rm p, iso}-E_{\rm p,z}-Γ_{0}$ would be an universal feature of synchrotron radiations of electrons accelerated in GRB jets throughout the prompt and afterglow phases among GRBs. Its origin is not fully understood and possible explanations are briefly discussed.

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arXiv2020

Constraining the Jet Launching Time of GRB 170817A by Utilizing the Baryon Loading

The observed delay of GRB 170817A relative to GW170817 carries significant information about gamma-ray burst (GRB) physics and is subject to intense debate. In this letter, we put forward an approach to discuss the major source of this time delay. First of all, we use the structured jet model to fit the X-ray/optical/radio afterglows of GRB 170817A together with superluminal motion measured by the Very Long Baseline Interferometry. Our structured jet is modelled with angle-dependent energy and baryon loading. It is found that our model can well fit the afterglows of GRB 170817A. After that, the baryon loading in the jet is inferred based on our fitting results. By comparing the baryon loading to the mass outflow in different stages, we infer that the time lag of the jet launch relative to the merger is less than hundreds or tens of milliseconds. It suggests that the time delay of GRB 170817A relative to GW170817 is defined mostly by the spreading time of the jet propagating to its dissipation radius.

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arXiv2020

The electromagnetic and gravitational-wave radiations of X-ray transient CDF-S XT2

Binary neutron star (NS) mergers may result in remnants of supra-massive or even stable NS, which have been supported indirectly by observed X-ray plateau of some gamma-ray bursts (GRBs) afterglow. Recently, Xue et al. (2019) discovered a X-ray transient CDF-S XT2 that is powered by a magnetar from merger of double NS via X-ray plateau and following stepper phase. However, the decay slope after the plateau emission is a little bit larger than the theoretical value of spin-down in electromagnetic (EM) dominated by losing its rotation energy. In this paper, we assume that the feature of X-ray emission is caused by a supra-massive magnetar central engine for surviving thousands of seconds to collapse black hole. Within this scenario, we present the comparisons of the X-ray plateau luminosity, break time, and the parameters of magnetar between CDF-S XT2 and other short GRBs with internal plateau samples. By adopting the collapse time to constrain the equation of state (EOS), we find that three EOSs (GM1, DD2, and DDME2) are consistent with the observational data. On the other hand, if the most released rotation energy of magnetar is dominated by GW radiation, we also constrain the upper limit of ellipticity of NS for given EOS, and it is range in $[0.32-1.3]\times 10^{-3}$. Its GW signal can not be detected by aLIGO or even for more sensitive Einstein Telescope in the future.

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