A numerical jet model for the prompt emission of gamma-ray bursts
Ruben Farinelli, Rupal Basak, Lorenzo Amati, Cristiano Guidorzi, Filippo Frontera
aa r X i v : . [ a s t r o - ph . H E ] J a n MNRAS , 1– ?? (2020) Preprint 11 January 2021 Compiled using MNRAS L A TEX style file v3.0
A numerical jet model for the prompt emission of gamma–ray bursts
Ruben Farinelli, ⋆ Rupal Basak, , Lorenzo Amati, Cristiano Guidorzi, , , Filippo Frontera INAF – Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via P. Gobetti 101, I-40129 Bologna, Italy Dipartimento di Fisica e Scienze della Terra, Universit`a di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy INFN – Sezione di Ferrara, Via Saragat 1, I-44122 Ferrara, Italy
Accepted . Received 2020 ; in original form
ABSTRACT
Gamma-ray bursts (GRBs) are known to be highly collimated events, and are mostly de-tectable when they are seen on-axis or very nearly on-axis. However, GRBs can be seen fromo ff -axis angles, and the recent detection of a short GRB associated to a gravitational waveevent has conclusively shown such a scenario. The observer viewing angle plays an importantrole in the observable spectral shape and the energetic of such events. We present a numeri-cal model which is based on the single-pulse approximation with emission from a top-hat jetand has been developed to investigate the e ff ects of the observer viewing angle. We assume aconical jet parametrized by a radius R jet , half-opening angle θ jet , a comoving-frame emissivitylaw and an observer viewing angle θ obs , and then study the e ff ects for the conditions θ obs < θ jet and θ obs > θ jet . We present results considering a smoothly broken power-law emissivity lawin jet comoving frame, albeit the model implementation easily allows to consider other emis-sivity laws. We find that the relation E ip ∝ E . (Amati relation) is naturally obtained frompure relativistic kinematic when Γ > ∼
10 and θ obs < θ jet ; on the contrary, when θ obs > θ jet itresults E ip ∝ E . . Using data from literature for a class of well-know sub-energetic GRBs,we show that their position in the E ip − E iso plane is consistent with event observed o ff -axis.The presented model is developed as a module to be integrated in spectral fitting softwarepackage XSPEC and can be used by the scientific community. Key words: gamma-ray burst: general – radiation mechanisms: non-thermal – methods: numerical – soft-ware: simulations
Gamma-ray bursts (GRB) remain one of the most debated tran-sient phenomena even after decades of research. It is fairly estab-lished that a GRB is a highly collimated event powered by a rel-ativistic jet (Sari et al. 1999; Aloy et al. 2000; Zhang et al. 2003),launched during the catastrophic death of a massive star (Woosley1993) or a coalescence event between two neutron stars or a neutronstar - black hole pair (Eichler et al. 1989; Li & Paczy´nski 1998;Abbott et al. 2017a). The radiation process of the prompt emissionphase remains at the center of the debate ever since their discov-ery. From theoretical considerations, synchrotron emission in anordered or random magnetic field seeems to be a good descriptionof the broadband spectra (Meszaros & Rees 1993; Meszaros et al.1994; Katz 1994; Ghisellini & Lazzati 1999). But, based on thespectral index below the peak energy and the e ffi ciency of gamma-ray photon production, the models involving synchrotron pro-cess face some criticism (Crider et al. 1997; Preece et al. 1998; ⋆ E-mail: [email protected]
Kaneko et al. 2006, though see Burgess et al. 2019). Several al-ternative models have been proposed like synchrotron self-Compton (Dermer et al. 2000; Nakar et al. 2009), inverse-Comptonscattering (Lazzati et al. 2000; Barniol Duran et al. 2012), dif-ferent flavours of photospheric models (Beloborodov 2011;Lundman et al. 2013; B´egu´e & Pe’er 2015; Ahlgren et al. 2019), aswell as hybrid models in which di ff erent processes can also evolvein terms of their dominance (Zhang et al. 2018). Usually the promptGRB spectra are fitted with phenomenological models, such as cut-o ff powerlaw or the widely used Band function (Band et al. 1993).An additional photospheric component has been sometimes de-tected and modeled with a blackbody (Ryde et al. 2010). In addi-tion, there have been some recent developments to employ morephysically motivated models for the spectral fitting (Burgess et al.2019). The first physical model ( grbcomp also released for theXSPEC package) applied to the GRB spectral analysis has beendeveloped by Titarchuk et al. (2012) and later tested on a sample oftime-resolved spectra by Frontera et al. (2013).The spectral models, whether empirical or driven by a phys-ical scenario, mostly have an implicit assumption that the radi- © R. Farinelli et al. ation received by the observer comes directly on-axis from theGRB jet (Proga et al. 2003). Due to the relativistic e ff ects the jetis highly collimated and the radiation is indeed beamed within avery narrow cone, which justifies the assumption. However, it ishighly probable to see a GRB from an o ff -axis angle, not neces-sarily outside the cone, and in general it can potentially alter theobservables from an axisymmetric case (e.g., Yamazaki et al. 2003;Kathirgamaraju et al. 2018). The observer viewing angle ( θ obs ) thusbecomes an important parameter in the analysis of the observedlight curves and spectra.Naturally, the relatively nearby GRBs are the most interestingcases of highly o ff -axis GRBs. For instance, Salafia et al. (2016)calculated that a sizable fraction, anywhere between 10% to 80%of nearby GRBs (redshift, z < .
1) are detectable from an o ff -axis angle outside the jet cone by the Neil Gehrels Swift Obser-vatory (Gehrels et al. 2004).The recent discovery of a short GRBassociated to a gravitational wave signal (GW 170817) has con-clusively shown that the GRB was seen from an o ff -axis angle(Abbott et al. 2017a,b; Margutti et al. 2017).Another example is GRB 150101B ( z = .
13) which may havebeen seen from an o ff -axis angle of 13 ◦ (Troja et al. 2018) whilehaving a half-opening angle close to 9 ◦ (Fong et al. 2016). Thenearby low-luminosity long GRB sample is also interesting as be-ing faint and nearby there is a good probability that they may fall inthe category of GRBs seen o ff -axis (Fynbo et al. 2004). However,several studies based on radio observation as well as the luminosityfunction and local rate have suggested that they might be intrinsi-cally fainter rather than being seen o ff -axis, and can even belongto a di ff erent population compared to the cosmological high lu-minosity GRB class (Soderberg et al. 2004, 2006; Pian et al. 2006;Liang et al. 2007). On the other hand, hydrodynamic simulationsof GRB jets show that the jet break signature of the o ff -axis GRBscan be delayed by several weeks and may remain hidden in the data(van Eerten et al. 2010). The lack of jet break signature in Swift sample is indeed consistent, and thus provides room for the alter-native scenario.Some of the low-luminosity GRBs are found to be outliers ofthe Amati relation (Amati et al. 2002; Amati 2006), hereafter AR,which is a relation between the intrinsic peak energy of the EF ( E )spectrum ( E ip ) and the isotropically-equivalent total emitted energy( E iso ). They are GRB 980425, GRB 031203, GRB 080517, GRB100316D, GRB 171205A (Campana et al. 2006; Ghisellini et al.2006; Amati et al. 2007; Liang et al. 2007; Starling et al. 2011;Heussa ff et al. 2013; Stanway et al. 2015; D’Elia et al. 2018).Several studies have been performed to investigate the ob-servational e ff ects of an o ff -axis jet (e.g., Yamazaki et al. 2003;Guidorzi et al. 2009; Kathirgamaraju et al. 2018), albeit the prin-cipally focused on the light curve shape and temporal variability.The main goal of this work is to provide to the scientific com-munity the first relativistic jet model for the X-ray spectral fit-ting package XSPEC (Arnaud 1996). The intrinsic complexity ofthe jet physics (M´esz´aros & Rees 2001; Ramirez-Ruiz et al. 2002;Zhang et al. 2003, 2004; Morsony et al. 2007; Mizuta et al. 2011)and the need to achieve a trade-o ff between computational speedand model accuracy leads to introduce unavoidable simplificationsin code development. We have built a model based on the so-called Single Pulse Approximation (SPA), where a top-hat relativis-tic jet instantaneously emits a flash of radiation in the star frame(Yamazaki et al. 2003).The jet can be viewed at any angle and we obtain the to-tal spectra by integrating over the di ff erent areas on the emit-ting surface. We allow to consider di ff erent emissivity laws in the comoving-frame. This is thus not a radiative-transfer model, but itparametrises the geometry of the emission in terms of jet radius andopening angle as well as observer’s viewing angle.In addition to a simple top-hat jet, various theoretical ar-guments and hydrodynamic simulations, as well as observations,have proposed GRB jets with angular structure (Berger et al. 2003;Basak & Rao 2015; Margutti et al. 2017; Beniamini & Nakar2019; Salafia et al. 2020), i.e. non-constant Γ -factor. We then alsoconsider the possibility of a structured jet and present some results,albeit in the release for the XSPEC package we implemented thecase of a constant Γ -factor, to avoid having a too-high number offree-parameters.The paper is structured as follows: in Section 2 we presentthe mathematical formulation of the model. In Section 3 we reportresults as a function of the input parameters and compare the case ofconstant and variable Γ factor. In Section 4 we show the importantconsequences at observational level between on-axis and o ff -axiscases in terms of the well-know E ip − L iso or E ip − E iso relations.The Discussion and Conclusions are presented in Sections 5 and 6,respectively. We present here the mathematical details on which the jet geome-try and emission are based. The most important parameters for theproblem formulation are shown in Fig. 1. Let us first consider a ref-erence system
XYZ where the jet axis is aligned with the Z-axis.Applying a rotation of an angle θ obs around the X-axis, the top-hatcartesian coordinates in a system xyz where the z-axis is directedtowards the observer are x = R jet sin θ cos φ, y = R jet (cos θ sin θ obs + cos θ obs sin θ sin φ ) , z = R jet (sin θ cos θ obs − sin θ obs sin θ sin φ ) , (1)where θ ∈ [0 , θ jet ] and φ ∈ [0 , π ], with θ jet defined as the jethalf-opening angle. The distance of a point P on the top-hat surfaceto the observer located at position O with coordinates (0,0, d ) is r = q d − R jet cos θ obs cos θ + R jet d sin θ obs sin θ sin φ + R . (2)The cosine of the angle between the radial velocity vector andthe line from P to the observer ( PO ) iscos α = − R jet + d cos θ obs cos θ − d sin θ obs sin θ sin φ q d + R − dR jet cos θ obs cos θ + d R jet sin θ obs sin θ sin φ , (3)which becomes for r << d cos α ≈ cos θ obs cos θ − sin θ obs sin θ sin φ. (4)The cosine of the angle formed by PO and the z -axis iscos ω = r (cid:16) d − R jet cos θ obs cos θ + r sin θ obs sin θ sin φ (cid:17) , (5)and is ≈ MNRAS , 1– ?? (2020) et model of gamma-ray bursts ! ! ! " ! ! ! " Figure 1.
Schematic view of the jet geometry for the on-axis ( top-figures ) and o ff -axis ( bottom-figures ) cases in a reference system xyz where the z -axis isaligned towards the observer. For computing the spectrum (see equation [13]) it is more convenient to define the polar and azimuthal angles θ and φ in areference system XYZ rotated by an angle θ obs around the x -axis and where the Z -axis is aligned with the jet axis. The point P represents the region on the jetsurface from where photons first reach the observer. a Lorentz transformation along with the cosmological correctionterm and is given byI jet ( k obs , E obs ) = I jet ( k jet , E jet )D (1 + z) , (6)where E jet = (1 + z ) E obs D f , (7)while D f = / Γ (1 − β cos α ) , (8)is the usual Doppler factor, Γ is the bulk Lorentz factor and β is thefluid’s velocity in units of the speed of light. We consider isotropicemissivity in the jet comoving frame so that I jet ( k jet , E jet ) = I jet ( E jet ).Let us now consider an element area on the jet-surface dA = R d u d φ , centered at coordinates [ u , φ ], where u = cos( θ ). Thee ff ective solid angle subtended by this element area is d Ω e ff i , j = (1 + z ) R cos α D d u d φ, (9)where we used the relation D L = (1 + z ) D A between the an-gular diameter distance and the luminosity distance, while the termcos α accounts for the e ff ective projected area.Combining equations (6) and (9) the flux received by the ob-server at O from the surface area dA is dF ( E obs ) = (1 + z ) R D I jet ( E jet ) D cos α d u d φ. (10) The jet is assumed to emit a single pulse in the stationary(redshift-corrected) rest-frame so that L ( t ∗ ) = δ ( t ∗ − t ∗ , ): because ofthe curvature e ff ect, the pulse signal duration in the observer framefor θ obs ≤ θ jet is t onp = (1 + z ) R jet c [1 − cos( θ obs + θ jet )] , (11)while for θ obs > θ jet it is t o ff p = (1 + z ) 2 R jet c sin θ obs sin θ jet . (12)In the remainder of the paper we will label as on-axis and o ff -axisthe events belonging to the first and second case, respectively. The flux from the whole top-hat surface averaged over the pulsetime t p is obtained as F ( E obs ) = (1 + z ) R D ( t p / s) Z π Z c I jet ( E jet ) D cos α d u d φ, (13)with u c = cos θ jet .The isotropic equivalent luminosity is computed with the relation L iso = π D (1 + z ) Z E / (1 + z ) E / (1 + z ) F [(1 + z ) E obs ] dE obs , (14) Note that some authors label as o ff -axis the events for which θ obs > ◦ ,no matter on the value of θ jet .MNRAS , 1– ?? (2020) R. Farinelli et al. and the fluence as E iso = ∆ t obs (1 + z ) L iso . (15)The code modularity allows to choose among di ff erent models forthe emissivity in the comoving frame; we consider first a smoothlybroken-powerlaw (SBPL) in the form I ( E ) = K EE ! − p + EE ! /δ ( p − p ) /δ , (16)with K in units of erg cm − s − keV − ster − and the parameter δ determines the smoothness of the transitions between the two PLregimes with index p and p , and the change of slope occurs inthe energy interval E − E such thatlog E E = log E E ∼ δ. (17)The second model is a cut-o ff powerlaw (CPL) defined as I ( E ) = K ( E / keV) − q e − E / E cut . (18)Finally, for a thermal component we use a simple blackbody(BB) function I ( E ) = K ( E / keV) e E / kT bb − . (19)The normalization K is allowed to be free in the case of BPLand CPL models, while in the BB case it is dictated by its thermo-dynamical limit value 5 × erg cm − s − keV − ster − . In this pa-per, all presented results have been obtained assuming a comoving-frame SBPL spectrum. Γ -factor In order to emphasize the importance of the observer viewing angle,we show in Figure 2 the observed surface brightness of the jet forfour di ff erent combination of Γ = ,
50 and θ obs = ◦ , ◦ , whilewe fix θ jet = ◦ . The values shown in the colour bars are not inabsolute units, but should be used to compare the relative brightnessbetween di ff erent cases. The relative brightness drops by severalorders of magnitude from a viewing angle inside the jet cone ( θ obs <θ jet , upper panels) to outside the jet cone ( θ obs > θ jet , lower panels).For the former case, i.e., θ jet < θ obs , the lateral spreading becomeshigher and the relative brightness becomes lower as Γ decreases(compare panel A with panel B). However, for θ jet > θ obs (see thelower panels), we note an opposite trend. Here, the brightness isrelatively higher by a factor of ∼ Γ (compare panel Cwith panel D). It is evident that we cannot always associate brighterGRBs with higher Γ . The observer needs to be within the jet conefor that to happen, otherwise a reverse variation will be seen.We then consider the observed spectra and first focus on thecases for di ff erent values of the Γ factor for both the on-axis ando ff -axis case. The results are shown in Figure 3. The simulationshave been performed assuming a jet with θ jet = ◦ and radius R jet = cm, and a SBPL emissivity law in the comoving framewith p = p = . E =
10 keV and δ = .
2. For the cases θ obs < θ jet , the spectral peak and flux are posi-tively correlated with the Γ . On the contrary, for observer’s viewingangle outside the cone ( θ obs > θ jet ), a reverse variation is noticed.It is also worth pointing the significant drop in the received fluxfor o ff -axis events, confirming that most of GRBs belonging to thisclass are likely below the threshold sensitivity of current instru-ments, unless either the Γ factor is low or they are neighbour (seeSection 4).The second parameter we investigated is the jet opening angle θ jet , and results are presented in Figure 4: for θ obs < θ jet and Γ suchthat θ jet >> / Γ , all the spectra are equivalent, with their normaliza-tion rescaled by the pulse duration t p (see equations [11] and [13]).Note that this renormalization unavoidably comes from the modeldefinition, but actually the bulk of the flux comes from a regionof angle θ ∼ / Γ << θ jet centered towards the observer’s view-ing angle, which is independent on θ jet . This can be seen noticingthat the observed peak energy E p remains unchanged, and confirmsthe well-know result that for θ jet > / Γ a radially symmetric jetis indistinguishable from an expanding sphere (e.g., Rhoads 1997;Salafia et al. 2016).Typical GRB Lorentz factors are Γ > ∼
100 (M´esz´aros 2006),and the e ff ect of observer’s viewing angle in the on-axis case wouldrequire very collimated jet opening angles θ jet < ∼ ◦ . This leads inturn to two consequences at the observational level; firstly, suchvery narrow jets (if present) have a very low probability of detec-tion, second, the dynamical range of observer’s viewing angle with θ obs ≤ θ jet would be so narrow to essentially suppress any detectablee ff ect.Nevertheless, for illustrative purposes we report in Figure 4 (top-left panel) the case of a moderate Lorentz factor Γ =
20 and valuesof θ jet < ∼ / Γ for a fully on-axis observer ( θ obs = ◦ ). Here, there isnot just a simple spectral rescaling due to t p but also a progressivedecreases of E p as θ jet increases.The situation changes when looking outside the jet cone no matterwhether θ jet is lower or higher than 1 / Γ : in Figure 4 (bottom panels)we report simulated spectra again for θ jet < / Γ and θ jet >> / Γ .In this case the trend is always the same, with E p increasing as θ jet does as well. This is understood because for o ff -axis events, thedominant contribution to the observed flux always comes from theregion around the bottom border of the jet (see also Fig. 2). Forfixed observer’s viewing angle, as θ jet increases the Doppler factorfrom this area increases, leading in turn o higher observed valuesof E pobs . We also claim that using only spectral fitting, the jet open-ing angle cannot be measured in the GRB prompt phase for on-axisevents apart for the unlikely case θ jet < ∼ ◦ (see top-left panel of Fig-ure 3), while in principle it could be done for o ff -axis events (ifdetectable) where the e ff ect of θ jet in the observed spectra is muchmore enhanced. Note that we do not present results for varying val-ues of R jet as the pulse-average flux simply rescales linearly with it(equation [13]). Concerning the parameters of the comoving-frameSBPL (equation [16]), apart for the low-energy and high-energyslopes (depending on p and p respectively) and the smoothnessof the transition at the peak energy E (depending on δ ), we pointout that the both the flux and E p linearly scales with E , for fixedvalues of θ obs and jet parameters. Γ -factor In a more complicated scenario, the GRB jet may have an angulardependence Γ ( θ ) of the bulk Lorentz factor (see references in Sec-tion 1). Numerical hydrodynamic simulations have been performede.g., by Zhang et al. (2003), who showed angular variation of den- MNRAS , 1– ?? (2020) et model of gamma-ray bursts Figure 2.
Top-surface brightness along the xy -plane of a jet with opening angle θ jet = ◦ and di ff erent values of the Γ factor and observer’s viewing angle. Top-left : (A)
Γ = , θ obs = ◦ , Top-right : (B)
Γ = , θ obs = ◦ , Bottom-left : (C)
Γ = , θ obs = ◦ , Bottom-right : (D)
Γ = , θ obs = ◦ . −1 Energy (keV)10 −13 −12 −11 −10 −9 −8 −7 −6 E F ( E ) e r g c m − s − Γ = 10Γ = 20Γ = 50Γ = 100Γ = 200Γ = 500Γ = 1000Θ obs = 5° 10 −1 Energy (keV)10 −24 −22 −20 −18 −16 −14 E F ( E ) e r g c m − s − Γ = 10Γ = 20Γ = 50Γ = 100Γ = 200Γ = 500Γ = 1000Θ obs = 25°
Figure 3.
Simulated spectra with Γ varied in the range 10 − θ jet = ◦ , R jet = cm and a SBPL emissivity law in the comoving framewith E =
10 keV, p = p = . δ = .
2. The two cases for θ obs = ◦ (on-axis) and θ obs = ◦ (o ff -axis) are reported.MNRAS , 1– ?? (2020) R. Farinelli et al. −1 Energy (keV)10 −11 −10 −9 −8 −7 E F ( E ) e r g c m − s − Θ obs = 0° Θ jet =0.5°Θ jet =1°Θ jet =1.5°Θ jet =2°Θ jet =2.5° −1 Energy (keV)10 −13 −12 −11 −10 −9 −8 −7 E F ( E ) e r g c m − s − Θ obs = 2° Θ jet =5°Θ jet =10°Θ jet =15°Θ jet =20°Θ jet =25° −1 Energy (keV)10 −14 −13 −12 −11 −10 E F ( E ) e r g c m − s − Θ obs = 5° Θ jet =0.5°Θ jet =1°Θ jet =1.5°Θ jet =2°Θ jet =2.5° −1 Energy (keV)10 −22 −21 −20 −19 −18 −17 −16 −15 E F ( E ) e r g c m − s − Θ obs = 30° Θ jet =5°Θ jet =10°Θ jet =15°Θ jet =20°Θ jet =25° Figure 4.
Simulated spectra for di ff erent values of the jet opening angle θ jet . Top panels : on-axis case with
Γ =
20 such that θ jet < / Γ ( left ) and Γ =
100 with θ jet >> / Γ ( right ). Bottom panels : o ff -axis case. The jet radius is 10 cm, while the parameters of the comoving-frame SBPL emissivity law are the same ofFigure 3. sity, flux and Γ . Later, Lundman et al. (2013) (hereafter L13) usedan empirical form to characterize the derived Γ -profile. In this sec-tion, we explore the e ff ects of a non-constant Γ on the observedproperties of our model. In particular, we adopt the analytical func-tion proposed by L13, with slight modification, where we explicitlywrite the dependence of Γ on θ : Γ = Γ min + Γ max − Γ min p ( θ/θ jet ) p + , (20)where Γ max is the maximum value of the Lorentz factor at thecenter of the jet. It is important to keep in mind that Γ min is not thevalue of Γ for θ = θ jet , but it represents an asymptotic value of Γ atthe outer layer, for instance in the presence of a sub-relativistic sur-rounding cocoon. To better show this e ff ect, in Figure 5, we reportthe behaviour Γ as a function of θ for the case Γ max = Γ min = ff erent values of the index p .To investigate the observational e ff ects of the Γ -stratification,we show in Fig. 6 the observed E p and flux for di ff erent valuesof the observer’s viewing angle, with θ jet = ◦ and for the caseof constant and variable Γ -factor, respectively. The quantities arenormalised to the value at θ obs = ◦ as we are interested to con-sider relative rather than absolute variations. For the first case, we Γ (cid:1)(cid:2)(cid:3) (cid:4)(cid:5) (cid:6) (cid:4) Γ (cid:1)(cid:2)(cid:3) (cid:4)(cid:5) (cid:6) (cid:7)(cid:8) (cid:4) (cid:9) (cid:10) (cid:11) (cid:7)(cid:8)(cid:7)(cid:9)(cid:8)(cid:7)(cid:10)(cid:8)(cid:7)(cid:11)(cid:8)(cid:4)(cid:8)(cid:8) θ Γ Figure 5.
Dependence of the Γ -factor as a function of θ (see equation 20)with θ jet = ◦ , Γ max = Γ min = ff erent values of p . choose Γ = Γ max = Γ min = . p = Γ -constant: for θ obs ≤ θ jet , E p and remains constant, while the pulse-average flux simply rescales MNRAS , 1– ?? (2020) et model of gamma-ray bursts -3 -2 -1 N o r m a li s ed E p Γ constant Γ variable10 -12 -10 -8 -6 -4 -2 N o r m a li s ed f l u x Θ obs Figure 6.
Dependence of the observed E p and flux as a function of theobserver viewing angle θ obs for a constant Lorentz factor Γ =
100 and for Γ obeying the law in equation (20) with Γ max = Γ min = . p = θ jet = ◦ , and the values are normalized to the case θ obs = ◦ . with t p , as defined in Equations (11) and (12). This is easily under-stood in view of the considerations made in previous section when θ jet > / Γ – the strong beaming e ff ect suppresses any dependenceof the spectral shape on θ obs . On the other hand, when θ obs > θ jet , thestrong boosting of the photons along the direction of motion, with-out any light-of-sight intercepting the jet top-hat, causes a strongdrop in the observed flux, which progressively decreases as θ obs in-creases. The main contribution comes here from the bottom part ofthe jet surface, as previously outlined (see Figure 2).The result changes slightly when considering a variable Γ -factor: when θ obs ≤ θ jet , both E p and the flux are mostly dictatedby the value Γ ( θ obs ) which is lower than Γ (0). At the observationallevel however, a structured jet viewed at an angle θ obs is indistin-guishable from a Γ -constant jet with Γ = Γ ( θ obs ). On the contrary,when θ obs > θ jet both E p and the flux are higher than the case of con-stant Γ , because at the jet border from which most of the emissioncomes the Γ -factor achieves its minimum, and the Doppler boostingout of the direction towards the observer is less pronounced.The situation may be of course di ff erent in the more compli-cated scenario of a structured jet as reported e.g. in Zhang et al.(2003), with a surrounding sub-relativistic cocoon, but this e ff ectwhich results from magneto-hydrodynamical situations cannot betaken into account by our model.In the version released for the XSPEC package we did not im-plement the dependence of Γ on the polar angle, to avoid havinga too-high number of degrees of freedoms (additional Γ min and p ,see equation 20), which actually overcomes the number of obser-vational spectral parameters. E p − E iso RELATION FOR ON-AXIS ANDOFF-AXIS EVENTS
The main parameter driving the observed spectral peak energiesand fluxes in relativistic outflows is the Γ -factor of the emittingmaterial (Dermer 2004). We investigated this e ff ect with a seriesof simulations at di ff erent Γ -values (not depending on θ ), assumingthe same SBPL spectrum in the comoving-frame with E = p = p = . δ = . R jet = cm. The results are presented in Figure 7.We obtain a clear dichotomy between the on-axis and o ff -axiscases; when θ obs ≤ θ jet both E ip and L iso increases as Γ does. Inparticular, for Γ > ∼
20 we obtain E ip ∝ Γ and L iso ∝ Γ , which in turnleads to the relation E ip ∝ L / . On the contrary, when θ obs > θ jet the two observable parameters progressively increase for moderatevalues of Γ up to ∼
40, above which a decreasing powerlaw-likebehavior occurs with E ip ∝ Γ − and L iso ∝ Γ − – consequently, oneobtains E ip ∝ L / .As a next step, we moved to the E ip − E iso plane in order toreproduce the AR with a possibly qualitative representation of itsintrinsic dispersion, which is known to have a variance higher thanthat coming from statistical uncertainties (Amati et al. 2002; Amati2006). The data dispersion of the AR indicates that one or morephysical and / or geometrical parameters play a role in addition to Γ which is claimed to have the leading role in dictating the observedslope E ip ∝ E . .Further considerations to point out for on-axis events are thefollowing:- E ip values (Y-axis) are ∝ E Γ - E iso values (X-axis) are ∝ E K Γ R Moreover, all the parameters must be combined in order to repro-duce not only the observed E ip − E iso slope, but also the normaliza-tion which is of order of ∼
100 (Amati et al. 2009).For o ff -axis events, E ip and E iso have the same dependence on E , K and R jet , but with an inverse proportionality on Γ as shownabove.Based on the above considerations and constraints, we pro-ceeded in the following way: let us define G ( P ) = N ( P c , σ P ) as thenormal gaussian distribution of a given parameter P with P c and σ P its mean and standard deviation, respectively.Performing simulations, we assumed G ( Γ ) = N (100 , Γ -values from such a distribution poses con-straints about the allowed range of E -values when comparing nu-merical results to observed values of E ip which cluster, a-part froma couple of cases below 10 keV, in a range of values from few tenskeV to few MeV (Amati et al. 2009). With the assumed distributionof Γ , we found that a good choice for the comoving-frame break en-ergy is G ( E ) = N (5 , G ( p ) = N (0 , .
5) and G ( p ) = N (1 . , . BATSE and
Fermi / GBM sample).The value of the jet half-opening angle θ jet is instead drawnfrom a uniform distribution in the range 5 ◦ − ◦ , while the ob-server’s viewing angle is sampled from a uniform distribution overcos( θ obs ) with 0 ≤ θ obs ≤ θ jet for on-axis events, θ jet < θ obs <θ jet + ◦ in the other case. Finally, for the product of jet ra-dius and SBPL normalization in such a way that G ( R / K ) = MNRAS , 1– ?? (2020) R. Farinelli et al. N (150 , R and K are in units of 10 cm and 10 ergs cm − s − keV − ster − , respectively.It is worth pointing that the independent sampling of all pa-rameters implies a diagonal covariance matrix which leads in turnto a variance of the data dispersion higher than that expected if atleast some physical quantities are correlated. However, we are hereinterested in testing the E ip − E iso main trend rather than its intrinsicdispersion, and the random parameter sampling has been adoptedjust to simulate a qualitative representation of the data dispersion.The results are reported in Figure 8: as expected from the be-havior of E ip and L iso as a function of Γ (see Fig. 7), we obtaintwo di ff erent slopes for the E ip − E iso relation for the two cases. Foron-axis events the index is ∼ .
5, and the dominant contributionto the data variance around the best-fit straight here comes fromthe random sampling of the comoving-frame SBPL parameters aswell as the product R / K . The jet half-opening angle and the ob-server viewing angle play instead no role (see Figure 6) under thecondition θ jet > / Γ whichis always satisfied here. Similar resultshave been obtained also by considering a structured jet accordingto equation (20). In this case indeed, the net e ff ect is to put an eventobserved at given angle θ obs in the same location of the E ip − E iso plane of events with constant Γ viewed at any angle θ obs < θ jet butwith Γ = Γ ( θ obs ) or Γ = Γ ( θ jet ) for on-axis and o ff -axis case, respec-tively.For events with θ obs > θ jet , we have also added data takenfrom the literature for a few well-known outliers of the AR, namelyGRB980425 (z = / SN2017iuk(z = = = = E ip − E iso plane is more con-sistent with the theoretical one derived for o ff -axis sources, and forwhich the slope is E ip ∝ E . . This result strengthens the claimthat the outliers of the AR are likely not intrinsically sub-luminousGRBs, but simply o ff -axis events which could be detected becauseof their nearness (Ramirez-Ruiz et al. 2005; Ghisellini et al. 2006).The simple geometric argument has additionally the advantage ofavoiding to search for other unknown physical properties at the ori-gin of the observable quantities. Our simulations endorse the in-terpretation that the AR relation arises from the observation of on-axis, highly-relativistic jets and originates from relativistic kine-matics e ff ects of sources with given distribution of the Lorentz Γ -factor, the latter playing the role of leading parameter. On the otherhand, the AR observed dispersion is due to intrinsic dispersion ofGRB properties such as the comoving-frame spectral emissivityshape and the typical radius R jet where the bulk of observed ra-diation during the prompt phase is released. For o ff -axis events, acorrelation in the E ip − E iso plane is still expected, but with a dif-ferent slope ( ∼ .
25) whose value is closer to 1 / ff -axis assumption (Dado & Dar2019).It is also worth noticing that for both on-axis and o ff -axissources, the assumption of a gaussian distribution of the Γ -factors(if E has a narrow distribution as well) leads to a clustering ofpoints in the top-right and bottom-left part of the E ip − E iso dia-gram, respectively. This is actually observed in the true data (e.g.,Amati et al. 2008) and we claim that this is not due to observationalbias e ff ects, but arises from the intrinsic properties of the GRB pop-ulation. E i p ( k e V ) -4 -3 -2 -1 L i s o ( e r g / s ) Γ E i p ( k e V ) -8 -7 -6 -5 -4 -3 -2 L i s o ( e r g / s ) Γ Figure 7.
Behaviour of E ip and L iso as a function of the Lorentz factor Γ ,for the two cases of jet viewed on-axis ( top panels ) and o ff -axis ( bottompanels ). Jet parameters are θ jet = ◦ and R jet = cm, while observerviewing angles are θ obs = ◦ and θ obs = ◦ . Overplotted to data are thebest-fit powerlaw functions for Γ > ∼
20. For the on-axis case the slopes are E ip ∝ Γ and L iso ∝ Γ , for the o ff -axis case E ip ∝ Γ − and L iso ∝ Γ − . Despite the huge amount of theoretical work done until nowto describe the spectral emission of the GRB prompt phase,the scientific community still uses phenomenological models forthe X-ray spectral fitting. As mentioned earlier, grbcomp wasthe first physical model released for the XSPEC package. Themodel is based on hydro-dynamical simulations performed byChardonnet et al. (2010) who investigated the SN formation due topair-instability in very massive stars ( M > ∼ M ⊙ ), a phenomenonwhich has received observational evidence (Gal-Yam & Leonard2009; Gal-Yam et al. 2009).The bulk of the emission in grbcomp is due to Comptonizationof blackbody-like seed photons ( kT bb ∼ few keV) by a Maxwellianpopulation of hot electrons ( kT e ∼
100 keV) moving outward thestellar surface at sub-relativistic speed. However, an association be-tween pair-instability SN and GRBs still lacks, and grbcomp , albeitsuccessful in fitting data and providing results consistent with sim-ulations, appears strongly dependent on the GRB progenitor class,besides the fact of working out of the relativistic paradigm.To overcome the phenomenological approach in the spec-
MNRAS , 1– ?? (2020) et model of gamma-ray bursts −1 E iso (10 erg)10 E i p ( k e V ) On-axis
K=114 ± 4, m=0.500 ± 0.009 10 −9 −7 −5 −3 −1 E iso (10 erg)10 E i p ( k e V ) Off-axis
K=191 ± 9, m=0.257 ± 0.004
Figure 8.
Simulated E ip − E iso relation for events observed on-axis ( left panel ) and o ff -axis ( right panel ). References for true GRBs data are reported inSection 4. For GRB031203 and GRB080517 the upper and lower limits on E ip respectively, are reported. The two continuous black lines correspond to the ± σ dispersion region of the observed E ip − E iso relation reported by Amati et al. (2008). tral analysis and work within the well-consolidated relativisticframework, we developed a numerical model which assumesemission from a top-hat jet using the single-pulse approxima-tion (Yamazaki et al. 2003). Note that if the relativistic outflow isviewed on-axis ( θ obs < θ jet ), the observer can see only the top-hatsurface and there is essentially no di ff erence between a cone-likegeometry extended over the radial distance from the center of thesystem, and a geometrically thin shell.We now briefly discuss the reliability of the best-fit parame-ters while using the model for X-ray spectral analysis. First let usdefine F CR ( E obs , ~ P , t ) as the observed single-pulse light curve dueto curvature radiation at some energy E obs , and for a given set ofparameters ~ P . It is possible to reformulate equation (13) as F CR ( E obs , ~ P ) = t spa Z t spa F CR ( E obs , ~ P , t ) dt , (21)where t spa is the single-pulse duration in the observer framedefined in equations (11) and (12). On the other hand, the time-average spectrum over a general observed interval t grb can be writ-ten as F ( E obs ) = t grb Z t grb F [ E obs , ~ P ( t ) , t ] dt , (22)where now F [ E obs , ~ P ( t ) , t ] is the true source light curve and ~ P ( t ) is the time-dependent array of parameters describing the spec-trum. The SPA model best-fit parameters are thus the ones whichminimize the di ff erence between the right-hand terms in equations(21) and (22). In this context, the array ~ P is a proxy of < ~ P ( t ) > ,where the latter quantity is to be intended as averaged over t grb .In a subsequent paper, we will present detailed time-dependent results in the framework of SPA for on-axis and o ff -axis events, together with mathematical tools for reproducing lightcurves as compliant as possible with observations.Despite the above described approximation, we outline howthe proposed model is able to naturally reproduce the observed E ip − E iso relation (AR) for on-axis events, at the same time pro-viding a straightforward explanation for the outliers in terms ofsimple viewing angle e ff ects. This results strengthen the idea thatthe main characteristics of the sources are caught from the observa- tional point of view, allowing to consider with good confidence thephysical and / or geometrical parameters inferred from the spectralfitting procedure.For pratical purposes, it is also important to point out that for θ obs < θ jet the observed peak energy E p and the flux are essentiallyindependent on θ obs (see Figure 6), and this allows to reduce thespace parameter dimension by one degree of freedom by setting θ obs to any value from 0 to θ jet in the X-ray spectral fitting proce-dure for all GRBs obeying the AR, i.e. are seen on-axis. The latestcondition can be preliminarly tested by performing spectral fittingwith e.g., the usual Band function (Band et al. 1993) and checkingthe position of the GRB in the E ip − E iso plane. Moreover, under thesame on-axis condition, most of the contribution to the flux comesfrom a region of width θ ∼ / Γ centered along the direction to theobserver (see Figure 2), which is independent of the jet openingangle as long as θ jet > ∼ / Γ . This allows to further keep θ jet frozento reasonable values (let say 10 ◦ − ◦ ) during the fit, further low-ering the number of free parameters. Some degree of degeneracyis instead expected between the comoving-frame break energy E (see equation 16) and the Γ -factor as E p ∝ E Γ and L iso ∝ E Γ . Inthis case, one should try to leave free both parameters and evaluatethe magnitude of errors of the best-fit values or keep free one of thetwo quantities from other independent evaluations.Another point which deserves to be outlined is that thecomoving-frame emissivity and the jet Γ -factor are here treatedas potentially independent parameters. Actually, for the internalshock model particle acceleration as well as magnetic field val-ues of the emission zone depend on the hydrodynamical condi-tions of the shocks forming between colliding shells. These inturn depend on the relative shell velocities and densities (e.g.,Daigne & Mochkovitch 1998; Boˇsnjak et al. 2009). The spectralshape and normalization in the comoving frame are related to thefinal Γ -factor of each couple of merged shells. It is however verydi ffi cult to provide within the context of a model for spectral fittingsome analytical or numerical dependencies of the comoving-frameemissivity parameters on the Γ -factor. This would indeed requirea di ff erent approach to the problem, with a set of coupled radia-tive transfer and hydrodynamical simulations from which eventu-ally deriving explicit correlations to be tabulated and later importedinto a model which, we outline again, needs to achieve a trade- MNRAS , 1– ?? (2020) R. Farinelli et al. o ff between computational speed and complexity. At the observa-tional level, correlations between the jet Γ -factor and local emissiv-ity need to be derived downstream from the model best-fit parame-ters. The presented model, albeit focused on the GRBs, can be con-sidered general , having no limitations in the relativistic outflow Γ -values. A possible drawback is given by the fact that the emissionfrom the lateral walls of a jet is neglected; this assumption is ex-pected to essentially have little or no e ff ect for θ obs ≤ θ jet , while foro ff -axis events (if detectable) the fluxes computed from the modelshould provide a lower limit to the actual values, in particular forjets surrounded by a sub-relativistic cocoon (Kathirgamaraju et al.2018). This would require however calculations of the emissivityprofile across the jet radial direction, with definition of a Γ ( R )-law,which is outside the scope of the present work. Note however thatemission from the lateral walls may be important for a jet havingan appreciable radial extension, while for geometrical thin config-urations with ∆ R / R << ffi ciently good approximation. The main purpose of our work was to make available for theXSPEC package the first relativistic non-phenomenological modelfor fitting the spectra of GRBs during the prompt phase. We thusdeveloped a model for reproducing the observed spectra arisingfrom the emission of a top-hat relativistic jet or a geometricallythin shell using the single pulse approximation. Despite unavoid-able simplifications, necessary to have reasonable computationaltimes with the XSPEC package, we have shown that the modelreproduces the observed slope ∼ . E ip − L iso or E ip − E iso (AR) for on-axis events ( θ obs < θ jet ), and this e ff ect natu-rally arises from pure relativistic kinematic e ff ects, no-matter onthe emissivity law in the comoving frame, provided a peak en-ergy is of course present in the EF(E) spectrum. For o ff -axis events( θ obs > θ jet ) the slope is instead ∼ .
25. Many e ff orts have beenmade over years for explaining the physical origin of the AR (e.g.,Guida et al. 2008; Dermer & Menon 2009; Ghirlanda et al. 2012;Titarchuk et al. 2012; Vyas et al. 2020) as well as its outliers, andin this work the disentangling between two distinct classes of ob-served events, which depends on the observer viewing angle, hasbeen achieved with a thorough mathematical and numerical treat-ment.We outline that the observational testing of this theoretical predic-tion can be also a very important scientific goal for the next gener-ation of GRB observatories such as THESEUS (Amati et al. 2018),whose great enhanced sensitivity is expected to be able to catch alarge sample of weak o ff -axis with enough statistics to allow time-resolved spectral analysis. ACKNOWLEDGMENTS
This project has received funding from the European Union’sHorizon 2020 research and innovation program under the MarieSklodowska-Curie grant agreement n. 664931 (RB).
DATA AVAILABILITY
The source code of the model, labeled grbjet , is available at thewebsite addresshttps: // heasarc.gsfc.nasa.gov / xanadu / xspec / newmodels.html REFERENCES
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