R. Ruffini - ICRA, Rome Univ. & ICRANet

R. Ruffini
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R. Ruffini
ICRA, Rome Univ. & ICRANet

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High Energy Astrophysical Phenomena (35)
Solar and Stellar Astrophysics (11)
General Relativity and Quantum Cosmology (9)
Cosmology and Nongalactic Astrophysics (8)
Astrophysics of Galaxies (8)
Nuclear Theory (5)
High Energy Physics - Phenomenology (3)
Physics - Plasma Physics (2)
Physics - History of Physics (1)
Instrumentation and Methods for Astrophysics (1)

Publications Authored By R. Ruffini

Theoretical and observational evidences have been recently gained for a two-fold classification of short bursts: 1) short gamma-ray flashes (S-GRFs), with isotropic energy $E_{iso}<10^{52}$~erg and no black hole (BH) formation, and 2) the authentic short gamma-ray bursts (S-GRBs), with isotropic energy $E_{iso}>10^{52}$~erg evidencing a BH formation in the binary neutron star merging process. The signature for the BH formation consists in the on-set of the high energy ($0.1$--$100$~GeV) emission, coeval to the prompt emission, in all S-GRBs. Read More

The long lasting attempt to identify the nature of X-ray flares, observed by Swift after the prompt emission of some long gamma ray bursts (GRBs), is here addressed within the context of a special subclass of long GRBs with isotropic energy $E_{iso} > 10^{52}$~erg, recently indicated as binary-driven hypernovae (BdHNe). Such BdHNe have as progenitor a tight binary system composed of a carbon-oxygen core (CO$_\mathrm{core}$) and a neutron star (NS) undergoing an induced gravitational collapse (IGC) to a black hole (BH) triggered by the CO$_\mathrm{core}$ explosion as a supernova (SN). Only in the case of such BdHNe, when analysed in the rest frame of the sources, the time of occurrence, the temporal duration, the X-ray luminosity and the total energy of the flares correlate with the $E_{iso}$. Read More

We compute the binding energy and angular momentum of a test-particle at the last stable circular orbit (LSO) on the equatorial plane around a general relativistic, rotating neutron star (NS). We present simple, analytic, but accurate formulas for these quantities that fit the numerical results and which can be used in several astrophysical applications. We demonstrate the accuracy of these formulas for three different equations of state (EOS) based on nuclear relativistic mean-field theory models and argue that they should remain still valid for any NS EOS that satisfy current astrophysical constraints. Read More

We investigate the possibility that some SGRs/AXPs could be canonical rotation-powered pulsars using realistic NS structure parameters instead of fiducial values. We show that realistic NS parameters lowers the estimated value of the magnetic field and radiation efficiency, $L_X/\dot{E}_{\rm rot}$, with respect to estimates based on fiducial NS parameters. We show that nine SGRs/AXPs can be described as canonical pulsars driven by the NS rotational energy, for $L_X$ computed in the soft (2--10~keV) X-ray band. Read More

It has previously been discovered that there is a universal power law behavior exhibited by the late X-ray emission (LXRE) of a "golden sample" (GS) of six long energetic GRBs, when observed in the rest-frame of the source. This remarkable feature, independent of the different isotropic energy (E_iso) of each GRB, has been used to estimate the cosmological redshift of some long GRBs. This analysis is extended here to a new class of 161 long GRBs, all with E_iso > 10^52 erg. Read More

It has been shown that a self-gravitating system of massive keV fermions in thermodynamic equilibrium correctly describes the dark matter (DM) distribution in galactic halos and predicts a denser quantum core towards the center of the configuration. Such a quantum core, for a fermion mass in the range of $50$ keV $\lesssim m c^2 \lesssim 345$ keV, can be an alternative interpretation of the central compact object in Sgr A*. We present in this work the gravitational lensing properties of this novel DM model in Milky Way-like spiral galaxies. Read More

The generation of the circular polarization of Gamma Ray Burst (GRB) photons is discussed in this paper via their interactions with astroparticles in the presence or absence of background fields such as magnetic fields and non-commutative space time geometry. Solving quantum Boltzmann equation for GRB-photons as a photon ensemble, we discuss the generation of circular polarization (as Faraday conversion phase shift $\Delta \phi_{FC}$) of GRBs in the following cases: (i) intermediate interactions, i.e. Read More

In a new classification of merging binary neutron stars (NSs) we separate short gamma-ray bursts (GRBs) in two sub-classes. The ones with $E_{iso}\lesssim10^{52}$ erg coalesce to form a massive NS and are indicated as short gamma-ray flashes (S-GRFs). The hardest, with $E_{iso}\gtrsim10^{52}$ erg, coalesce to form a black hole (BH) and are indicated as genuine short-GRBs (S-GRBs). Read More

We have recently introduced a new model for the distribution of dark matter (DM) in galaxies, the Ruffini-Arg\"uelles-Rueda (RAR) model, based on a self-gravitating system of massive fermions at finite temperatures. The RAR model, for fermion masses above keV, successfully describes the DM halos in galaxies, and predicts the existence of a denser quantum core towards the center of each configuration. We demonstrate here, for the first time, that the introduction of a cutoff in the fermion phase-space distribution, necessary to account for the finite Galaxy size, defines a new solution with a compact quantum core which represents an alternative to the central black hole (BH) scenario for SgrA*. Read More

Following the induced gravitational collapse (IGC) paradigm of gamma-ray bursts (GRBs) associated with type Ib/c supernovae, we present numerical simulations of the explosion of a carbon-oxygen (CO) core in a binary system with a neutron-star (NS) companion. The supernova ejecta trigger a \emph{hypercritical} accretion process onto the NS thanks to a copious neutrino emission and the trapping of photons within the accretion flow. We show that temperatures 1--10~MeV develop near the NS surface, hence electron-positron annihilation into neutrinos becomes the main cooling channel leading to accretion rates $10^{-9}$--$10^{-1}~M_\odot$~s$^{-1}$ and neutrino luminosities $10^{43}$--$10^{52}$~erg~s$^{-1}$ (the shorter the orbital period the higher the accretion rate). Read More

In a recent letter we have outlined some issues on GW 150914, we hereby give additional details. We analyze the event GW 150914 announced by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) as the gravitational-wave emission of a black-hole binary merger. We show that the parameters of the coalescing system and of the newly formed Kerr black-hole can be extracted from basic results of the gravitational-wave emission during the inspiraling and merger phases without sophisticate numerical simulations. Read More

We analyze the event GW 150914 announced by the Advanced Laser Interferometer Gravitational-Wave Observatory (LIGO) as the gravitational-wave emission of a black-hole binary merger. We show that the parameters of the coalescing system and of the newly formed Kerr black-hole can be extracted from basic results of the gravitational-wave emission during the inspiraling and merger phases without sophisticated numerical simulations. Our strikingly accurate estimates are based on textbook formulas describing two different regimes: 1) the binary inspiraling analysis treated in Landau and Lifshitz textbook, and 2) the plunge of a particle into a black-hole, treated in the Rees-Ruffini-Wheeler textbook. Read More

We construct mass-radius relations of white dwarfs taking into account the effects of rotation and finite temperatures. We compare and contrast the theoretical mass-radius relations with observational data. Read More

We investigate isolated sub- and super-Chandrasekhar white dwarfs which lose angular momentum through magnetic dipole braking. We construct constant rest mass sequences by fulfilling all stability criteria of rotating configurations and show how the main structure of white dwarfs such as the central density, mean radius and angular velocity change with time. We explicitly demonstrate that all isolated white dwarfs regardless of their masses, by angular momentum loss, shrink and increase their central density. Read More

GRBs, traditionally classified as "long" and "short", have been often assumed, till recently, to originate from a single black hole (BH) with an ultrarelativistic jetted emission. There is evidence that both long and short bursts have as progenitors merging and/or accreting binaries, each composed by a different combination of carbon-oxygen cores (CO$_{\rm core}$), neutron stars (NSs), BHs and white dwarfs (WDs). Consequently, the traditional long bursts have been sub-classified as (I) X-ray flashes (XRFs), (II) binary-driven hypernovae (BdHNe), and (III) BH-supernovae (BH-SNe). Read More

There is mounting evidence for the binary nature of the progenitors of gamma-ray bursts (GRBs). For a long GRB, the induced gravitational collapse (IGC) paradigm proposes as progenitor, or "in-state", a tight binary system composed of a carbon-oxygen core (CO$_{core}$) undergoing a supernova (SN) explosion which triggers hypercritical accretion onto a neutron star (NS) companion. For a short GRB, a NS-NS merger is traditionally adopted as the progenitor. Read More

We construct equilibrium configurations of uniformly rotating neutron stars for selected relativistic mean-field nuclear matter equations of state (EOS). We compute in particular the gravitational mass ($M$), equatorial ($R_{\rm eq}$) and polar ($R_{\rm pol}$) radii, eccentricity, angular momentum ($J$), moment of inertia ($I$) and quadrupole moment ($M_2$) of neutron stars stable against mass-shedding and secular axisymmetric instability. By constructing the constant frequency sequence $f=716$ Hz of the fastest observed pulsar, PSR J1748-2446ad, and constraining it to be within the stability region, we obtain a lower mass bound for the pulsar, $M_{\rm min}=[1. Read More

The induced gravitational collapse (IGC) paradigm explains a class of energetic, $E_{\rm iso}\gtrsim 10^{52}$~erg, long-duration gamma-ray bursts (GRBs) associated with Ic supernovae, recently named binary-driven hypernovae (BdHNe). The progenitor is a tight binary system formed of a carbon-oxygen (CO) core and a neutron star companion. The supernova ejecta of the exploding CO core triggers a hypercritical accretion process onto the neutron star, which reaches in a few seconds the critical mass, and gravitationally collapses to a black hole emitting a GRB. Read More

Binary-driven-hypernovae (BdHNe) within the induced gravitational collapse (IGC) paradigm have been introduced to explain energetic ($E_{\rm iso}\gtrsim 10^{52}$~erg), long gamma-ray bursts (GRBs) associated with type Ic supernovae (SNe). The progenitor is a tight binary composed of a carbon-oxygen (CO) core and a neutron star (NS) companion, a subclass of the newly proposed "ultra-stripped" binaries. The CO-NS short-period orbit causes the NS to accrete appriciable matter from the SN ejecta when the CO core collapses, ultimately causing it to collapse to a black hole (BH) and producing a GRB. Read More

This paper summarizes the limits on propagation of ultra high energy particles in the Universe, set up by their interactions with cosmic background of photons and neutrinos. By taking into account cosmic evolution of these backgrounds and considering appropriate interactions we derive the mean free path for ultra high energy photons, protons and neutrinos. For photons the relevant processes are the Breit-Wheeler process as well as the double pair production process. Read More

The properties of uniformly rotating white dwarfs (RWDs) are analyzed within the framework of Newton's gravity and general relativity. In both cases Hartle's formalism is applied to construct the internal and external solutions to the field equations. The white dwarf (WD) matter is described by the Chandrasekhar equation of state. Read More

We describe one of the so-called low magnetic field magnetars SGR 0418+5729, as a massive fast rotating highly magnetized white dwarf following Malheiro et. al. 2012. Read More

It has been shown previously that the DM in galactic halos can be explained by a self-gravitating system of massive keV fermions (`inos') in thermodynamic equilibrium, and predicted the existence of a denser quantum core of inos towards the center of galaxies. In this article we show that the inclusion of self-interactions among the inos, modeled within a relativistic mean-field-theory approach, allows the quantum core to become massive and compact enough to explain the dynamics of the S-cluster stars closest to the Milky Way's galactic center. The application of this model to other galaxies such as large elliptical harboring massive central dark objects of $\sim 10^9 M_\odot$ is also investigated. Read More

We consider applications of general relativistic uniformly-rotating white dwarfs to several astrophysical phenomena related to the spin-up and the spin-down epochs and to delayed type Ia supernova explosions of super-Chandrasekhar white dwarfs, where we estimate the "spinning down" lifetime due to magnetic-dipole braking. In addition, we describe the physical properties of Soft Gamma Repeaters and Anomalous X-Ray Pulsars as massive rapidly-rotating highly-magnetized white dwarfs. Particularly we consider one of the so-called low-magnetic-field magnetars SGR 0418+5729 as a massive rapidly-rotating highly-magnetized white dwarf and give bounds for the mass, radius, moment of inertia, and magnetic field by requiring the general relativistic uniformly-rotating configurations to be stable. Read More

We derive and solve by the spectral method the equations for a neutral system of ultra-relativistic electrons that are compressed to the radius of the nucleus and subject to a driving force. This driving force can be thought of as originating from a nuclear breathing mode, a possibility we discuss in detail. Read More

We show the existence of two families of short GRBs, both originating from the merger of binary neutron stars (NSs): family-1 with $E_{iso}<10^{52}$ erg, leading to a massive NS as the merged core, and family-2 with $E_{iso}>10^{52}$ erg, leading to a black hole (BH). Following the identification of the prototype GRB 090227B, we present the details of a new example of family-2 short burst: GRB 140619B. From the spectral analysis of the early $\sim0. Read More

Globally neutral neutron stars, obtained from the solution of the called Einstein-Maxwell-Thomas-Fermi equations that account for all the fundamental interactions, have been recently introduced. These configurations have a more general character than the ones obtained with the traditional Tolman-Oppenheimer-Volkoff, which impose the condition of local charge neutrality. The resulting configurations have a less massive and thinner crust, leading to a new mass-radius relation. Read More

We have recently developed a neutron star model fulfilling global and not local charge neutrality, both in the static and in the uniformly rotating cases. The model is described by the coupled Einstein-Maxwell-Thomas- Fermi (EMTF) equations, in which all fundamental interactions are accounted for in the framework of general relativity and relativistic mean field theory. Uniform rotation is introduced following the Hartle's formalism. Read More

We investigate the distribution of dark matter in galaxies by solving the equations of equilibrium of a self-gravitating system of massive fermions (`inos') at selected temperatures and degeneracy parameters within general relativity. Our most general solutions show, as a function of the radius, a segregation of three physical regimes: 1) an inner core of almost constant density governed by degenerate quantum statistics; 2) an intermediate region with a sharply decreasing density distribution followed by an extended plateau, implying quantum corrections; 3) an asymptotic, $\rho\propto r^{-2}$ classical Boltzmann regime fulfilling, as an eigenvalue problem, a fixed value of the flat rotation curves. This eigenvalue problem determines, for each value of the central degeneracy parameter, the mass of the ino as well as the radius and mass of the inner quantum core. Read More

The induced gravitational collapse (IGC) paradigm has been successfully applied to the explanation of the concomitance of gamma-ray bursts (GRBs) with supernovae (SNe) Ic. The progenitor is a tight binary system composed by a carbon-oxygen (CO) core and a neutron star (NS) companion. The explosion of the SN leads to hypercritical accretion onto the NS companion which reaches the critical mass, hence inducing its gravitational collapse to a black hole (BH) with consequent emission of the GRB. Read More

The properties of the electric field of a two-body system consisting of a Reissner-Nordstr\"om black hole and a charged massive particle at rest have recently been analyzed in the framework of first order perturbation theory following the standard approach of Regge, Wheeler and Zerilli. In the present paper we complete this analysis by numerically constructing and discussing the lines of force of the "effective" electric field of the sole particle with the subtraction of the dominant contribution of the black hole. We also give the total field due to the black hole and the particle. Read More

Affiliations: 1http://www.iasfbo.inaf.it/~amati/game, 2http://www.iasfbo.inaf.it/~amati/game, 3http://www.iasfbo.inaf.it/~amati/game, 4http://www.iasfbo.inaf.it/~amati/game, 5http://www.iasfbo.inaf.it/~amati/game, 6http://www.iasfbo.inaf.it/~amati/game, 7http://www.iasfbo.inaf.it/~amati/game, 8http://www.iasfbo.inaf.it/~amati/game, 9http://www.iasfbo.inaf.it/~amati/game, 10http://www.iasfbo.inaf.it/~amati/game, 11http://www.iasfbo.inaf.it/~amati/game, 12http://www.iasfbo.inaf.it/~amati/game, 13http://www.iasfbo.inaf.it/~amati/game, 14http://www.iasfbo.inaf.it/~amati/game, 15http://www.iasfbo.inaf.it/~amati/game, 16http://www.iasfbo.inaf.it/~amati/game, 17http://www.iasfbo.inaf.it/~amati/game, 18http://www.iasfbo.inaf.it/~amati/game, 19http://www.iasfbo.inaf.it/~amati/game, 20http://www.iasfbo.inaf.it/~amati/game, 21http://www.iasfbo.inaf.it/~amati/game, 22http://www.iasfbo.inaf.it/~amati/game, 23http://www.iasfbo.inaf.it/~amati/game, 24http://www.iasfbo.inaf.it/~amati/game, 25http://www.iasfbo.inaf.it/~amati/game

We describe the GRB and All-sky Monitor Experiment (GAME) mission submitted by a large international collaboration (Italy, Germany, Czech Repubblic, Slovenia, Brazil) in response to the 2012 ESA call for a small mission opportunity for a launch in 2017 and presently under further investigation for subsequent opportunities. The general scientific objective is to perform measurements of key importance for GRB science and to provide the wide astrophysical community of an advanced X-ray all-sky monitoring system. The proposed payload was based on silicon drift detectors (~1-50 keV), CdZnTe (CZT) detectors (~15-200 keV) and crystal scintillators in phoswich (NaI/CsI) configuration (~20 keV-20 MeV), three well established technologies, for a total weight of ~250 kg and a required power of ~240 W. Read More

Essay selected for Honorable mention 2014 by the Gravity Research Foundation. We study an isothermal system of semi-degenerate self-gravitating fermions in General Relativity (GR). The most general solutions present mass density profiles with a central degenerate compact core governed by quantum statistics followed by an extended plateau, and ending in a power law behaviour $r^{-2}$. Read More

We have performed our data analysis of the observations by Swift and Fermi satellites in order to probe the induced gravitational collapse (IGC) paradigm for GRBs associated with supernovae (SNe), in the "terra incognita" of GRB 130427A. We compare and contrast our data analysis with those in the literature. We have verified that the GRB 130427A conforms to the IGC paradigm by examining the power law behavior of the luminosity in the early $10^4$ s of the Swift-XRT observations. Read More

Context: The induced gravitational collapse (IGC) paradigm addresses the very energetic (10^{52}-10^{54}erg) long gamma-ray bursts (GRBs) associated to supernovae (SNe). Unlike the traditional "collapsar" model, an evolved FeCO core with a companion neutron star (NS) in a tight binary system is considered as the progenitor. This special class of sources, here named "binary driven hypernovae" (BdHNe), presents a composite sequence composed of four different episodes [. Read More

CONTEXT: The induced gravitational collapse (IGC) scenario has been introduced in order to explain the most energetic gamma ray bursts (GRBs), Eiso=10^{52}-10^{54}erg, associated with type Ib/c supernovae (SNe). It has led to the concept of binary-driven hypernovae (BdHNe) originating in a tight binary system composed by a FeCO core on the verge of a SN explosion and a companion neutron star (NS). Their evolution is characterized by a rapid sequence of events: [. Read More

One longstanding mystery in bio-evolution since Darwin's time is the origin of the Cambrian explosion that happened around 540 million years ago (Mya), where an extremely rapid increase of species occurred. Here we suggest that a nearby GRB event ~500 parsecs away, which should occur about once per 5 Gy, might have triggered the Cambrian explosion. Due to a relatively lower cross section and the conservation of photon number in Compton scattering, a substantial fraction of the GRB photons can reach the sea level and would induce DNA mutations in organisms protected by a shallow layer of water or soil, thus expediting the bio-diversification. Read More

It has at times been indicated that Landau introduced neutron stars in his classic paper of 1932. This is clearly impossible because the discovery of the neutron by Chadwick was submitted more than one month after Landau's work. Therefore, and according to his calculations, what Landau really did was to study white dwarfs, and the critical mass he obtained clearly matched the value derived by Stoner and later by Chandrasekhar. Read More

Based on the Thomas-Fermi solution for compressed electron gas around a giant nucleus, $Z\approx 10^6$, we study electric pulsations of electron number-density, pressure and electric fields, which could be caused by an external perturbations acting on the nucleus or the electrons themselves. We numerically obtain the eigen-frequencies and eigen-functions for stationary pulsation modes that fulfill the boundary-value problem established by electron-number and energy-momentum conservation, equation of state, laws of thermodynamics, and Maxwell's equations, as well as physical boundary conditions. We choose a proton number of $Z=10^6$ and assume the nucleons in $\beta$-equilibrium at nuclear density. Read More

We study an isothermal system of semi-degenerate self-gravitating fermions in general relativity. Such systems present mass density solutions with a central degenerate core, a plateau and a tail which follows a power law behaviour $r^{-2}$. The different solutions are governed by the free parameters of the model: the degeneracy and temperature parameters at the center, and the particle mass $m$. Read More

On the basis of a fermionic dark matter model we fit rotation curves of The HI Nearby Galaxy Survey THINGS sample and compare our 3-parametric model to other models widely used in the literature: 2-parametric Navarro--Frenk--White, pseudoisothermal sphere, Burkhert models, and 3-parametric Einasto model, suggested as the new "standard dark matter profile" model in the paper by Chemin et. al., AJ 142 (2011) 109. Read More

The problem of modeling the distribution of dark matter in galaxies in terms of equilibrium configurations of collisionless self-gravitating quantum particles is considered. We first summarize the pioneering model of a Newtonian self-gravitating Fermi gas in thermodynamic equilibrium developed by Ruffini and Stella (1983), which is shown to be the generalization of the King model for fermions. We further review the extension of the former model developed by Gao, Merafina and Ruffini (1990), done for any degree of fermion degeneracy at the center ($\theta_0$), within general relativity. Read More

We propose a unified model for dark matter haloes and central galactic objects as a self-gravitating system of semidegenerated fermions in thermal equilibrium. We consider spherical symmetry and then we solve the equations of gravitational equilibrium using the Fermi integrals in a dimensionless manner, obtaining the density profile and velocity curve. We also obtain scaling laws for the observables of the system and show that, for a wide range of our parameters, our model is consistent with the so called universality of the surface density of dark matter. Read More

The Feynman-Metropolis-Teller treatment of compressed atoms has been recently generalized to relativistic regimes and applied to the description of static and rotating white dwarfs in general relativity. We present here the extension of this treatment to the case of finite temperatures and construct the corresponding equation of state (EOS) of the system; applicable in a wide regime of densities that includes both white dwarfs and neutron star outer crusts. We construct the mass-radius relation of white dwarfs at finite temperatures obeying this new EOS and apply it to the analysis of ultra low-mass white dwarfs with $M\lesssim 0. Read More

Following the recently established "Binary-driven HyperNova" (BdHN) paradigm, we here interpret GRB 970828 in terms of the four episodes typical of such a model. The "Episode 1", up to 40 s after the trigger time t_0, with a time varying thermal emission and a total energy of E_{iso,1st} = 2.60 x 10^{53} erg, is interpreted as due to the onset of an hyper-critical accretion process onto a companion neutron star, triggered by the companion star, an FeCO core approaching a SN explosion. Read More

We review recent progress in our understanding of the nature of gamma ray bursts (GRBs) and in particular, of the relationship between short GRBs and long GRBs. The first example of a short GRB is described. The coincidental occurrence of a GRB with a supernova (SN) is explained within the induced gravitational collapse (IGC) paradigm, following the sequence: 1) an initial binary system consists of a compact carbon-oxygen (CO) core star and a neutron star (NS); 2) the CO core explodes as a SN, and part of the SN ejecta accretes onto the NS which reaches its critical mass and collapses to a black hole (BH) giving rise to a GRB; 3) a new NS is generated by the SN as a remnant. Read More

Optically thick energy dominated plasma created in the source of Gamma-Ray Bursts (GRBs) expands radially with acceleration and forms a shell with constant width measured in the laboratory frame. When strong Lorentz factor gradients are present within the shell it is supposed to spread at sufficiently large radii. There are two possible mechanisms of spreading: hydrodynamical and thermal ones. Read More

The luminosity function (LF) statistics applied to the BATSE GRBs (sources of GUSBAD catalog) is the theme approached in this work. The LF is a strong statistical tool to extract useful information from astrophysical samples, where the key point of this statistical analysis is in the detector sensitivity, where we have performed careful analysis. We applied the tool of the LF statistics to three GRB classes predicted by the Fireshell model. Read More

It has been proposed that the temporal coincidence of a gamma-ray burst (GRB) and a type Ib/c supernova (SN) can be explained with the concept of induced gravitational collapse (IGC), induced by the matter ejected from an SN Ib/c accreting onto a neutron star (NS). We found a standard luminosity light curve behavior in the late-time X-ray emission of this subclass of GRBs. We interpret this as the result of a common physical mechanism in this particular phase of the X-ray emission, possibly related to the creation of the NS from the SN process. Read More

We consider the formation of photon spectrum at the photosphere of ultrarelativistically expanding outflow. We use the Fokker-Planck approximation to the Boltzmann equation, and obtain the generalized Kompaneets equation which takes into account anisotropic distribution of photons developed near the photosphere. This equation is solved numerically for relativistic steady wind and the observed spectrum is found in agreement with previous studies. Read More