The elementary spike produced by a pure e + e pair-electromagnetic pulse from a Black Hole: The PEM Pulse

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1 A&A 368, (21) DOI: 1.151/4-6361:2556 ESO 21 Atronomy & Atrophyi The elementary pike produed by a pure e + e pair-eletromagneti pule from a Blak Hole: The PEM Pule C. L. Biano 1, R. Ruffini 1, and S.-S. Xue 1 I.C.R.A.-International Center for Relativiti Atrophyi and Phyi Department, Univerity of Rome La Sapienza, 185 Rome, Italy Reeived 21 July 2 / Aepted 12 Deember 2 Abtrat. In the framework of the model that ue blak hole endowed with eletromagneti truture (EMBH) a the energy oure, we tudy how an elementary pike appear to the detetor. We onider the implet poible ae of a pule produed by a pure e + e pair-eletro-magneti plama, the PEM pule, in the abene of any baryoni matter. The reulting time profile how a Fat-Rie-Exponential-Deay hape, followed by a power-law tail. Thi i obtained without any peial fitting proedure, but only by fixing the energeti of the proe taking plaeinagivenembhofeletedma,varyingintherangefrom1to1 3 M and onidering the relativiti effet to be expeted in an eletron-poitron plama gradually reahing tranpareny. Speial attention i given to the ontribution from all regime with Lorentz γ fator varying from γ =1toγ =1 4 in a few hundred of the PEM pule travel time. Although the main goal of thi paper i to obtain the elementary pike intenity a a funtion of the arrival time, and it oberved duration, ome qualitative onideration are alo preented regarding the expeted petrum and on it departure from the thermal one. The reult of thi paper will be omparable, when data will beome available, with a ubfamily of partiularly hort GRB not followed by any afterglow. They an alo be propedeutial to the tudy of longer burt in preene of baryoni matter urrently oberved in GRB. Key word. blak hole phyi gamma ray: burt gamma ray: theory gamma ray: obervation 1. Introdution It i by now lear that the Gamma-Ray Burt phenomena, here generally alled GRB, far from repreenting a ingle phyial event, are quite omplex and are ompoite of very different phae, orreponding to ditintively different phyial proee. Any data analyi proedure hould be done taking into due aount thee different epoh and avoiding time averaging proedure on different epoh. In reent year muh attention ha been devoted to analyzing GRB afterglow, originating from the interation of a relativiti expanding fireball with the urrounding baryoni material, leading to important orrepondene between obervational data and phenomenologial model (ee e.g. Vietri 1998; Sari & Ein 2 and Djorgovki et al. 2). Le progre ha been made in undertanding the bai mehanim whih produe the fireball and in explaining the truture of the burt itelf. In the exiting literature, it ha been alternatively aumed that GRB originate: a) by the mooth tranition to tranpareny of an optially thik eletron-poitron plama, whoe origin i not diued (ee e.g. Goodman 1986); Send offprint requet to: R. Ruffini, ruffini@ira.it b) from external hok proee in the olliion of an expanding fireball, whoe origin i not diued, with baryoni matter at ret (ee e.g. Mézáro & Ree 1992); ) from the interation with eah other of different part of the fireball due to different expanion veloitie, by internal hok proee (ee e.g. Pazyńki & Xu 1994); d) from a mixed internal-external enario, in whih both effet are preent (ee e.g. Piran 1999); e) from a Compton-drag proe, aued by the interation of a relativiti fireball, whoe origin i not diued, with a very dene oft photon bath (ee e.g. Ghiellini et al. 2). Important a they are, thee emi-empirial approahe mi the ompletene of a omprehenive quantitative model. Some of the above proee ould indeed be preent in the different epoh of the GRB. In order to evaluate their relative magnitude and give a quantitative etimate of the aoiated relativiti effet i neeary to have a detailed model, with the orreponding equation of motion and the time evolution of the ytem. For thi reaon we onider the model whih aume the energy oure of GRB to be the proe of vauum polarization around a blak hole endowed with eletromagneti truture (EMBH) (ee Damour & Ruffini 1975 and Ruffini 1998). For uh a ytem the fundamental energeti apet have been larified by the introdution Artile publihed by EDP Siene and available at or

2 378 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule of the Dyadophere of an EMBH (Preparata et al. 1998b) a well a the equation of motion for the ytem have been integrated and are ummarized in the next etion. We proeed in four eparate tep: In thi paper, we make a firt tep. We onider the implet poible ae: the one orreponding to a pule produed by a pure e + e pair-eletro-magneti plama, the PEM pule, in the abene of any baryoni matter. The equation of motion of thi ytem have already been integrated (Ruffini et al. 1999). We here examine how the PEM pule gradually emit radiation during it evolution and the GRB imply our during the mooth approah to the tranpareny ondition, when the mean free path of the photon in the plama, L γ in the omoving frame and λ in the laboratory frame, given by 1 L γ = σ γ,e n e ±, λ = L γ γ, (1) approahe the thikne of the PEM pule itelf. Here σ γ,e i the photon and eletron Compton attering roetion, whoe Thomon limit i πα2 m m 2 2 and n e ± i the proper number denity of e + e pair. Thu the GRB i eentially determined by the eletron-poitron pair annihilation and the expanion and ooling of the PEM pule. It ha been lear ine the lai work of Ree on expanding radio oure that relativiti effet are entral to the undertanding of the atrophyi of extragalati oure (Ree 1966). The typial Lorentz γ fator onidered in that tudy wa of the order of γ 5 and eentially ontant over the time of obervation of the atrophyial oure. In the ae of GRB uh relativiti effet are alo important, but there are three major differene: 1. The relativiti effet are muh more extreme. It wa pointed out (ee Ruffini et al. 1999) that, in the ae of the abene of baryoni matter, a Lorentz γ fator of up to 1 4 an be reahed. Thi reult ha alo been extended to the ae of the expanion of the PEM pule in the preene of baryoni matter (ee Ruffini et al. 2) where even larger value of the Lorentz γ fator an be reahed; 2. The tranition from γ =1toγ 1 4 our in a few hundred of the harateriti travel time in the PEM pule (i.e. a few eond). It i therefore impoible, a hown in Fig. 2, in Fig. 8 and in Set. 4 7 of the preent paper, even a a rough approximation, to onider the Lorentz γ fator to be ontant during the emiion proe of the GRB; 3. The tranpareny ondition given by Eq. (1) i reahed gradually by the effet of the ooling and the expanion of the plama and the orreponding dereae in the number of the eletron-poitron pair. Conequently, in priniple, all the tage from γ = 1 to γ 1 4 ontribute to the final burt truture. The main point of the preent paper i to larify the interplay of relativiti effet at work in thi extreme ae, in order to obtain the elementary pike intenity a a funtion of the arrival time, and it oberved duration. In partiular we here take into aount, till for impliity in the ae of a pherial geometry, the effet due to: the varying thikne of the emitting region; the energy flux, eentially modulated by a time varying reening fator (ee Appendie A and B); the time variation of the γ fator and onequene for the oberved arrival time. We examine expliitly, for impliity, the ae of Reiner- Nordtröm EMBH with Q M =.1 andm =1M, M = 1 2 M and M =1 3 M. Some onideration on the expeted petrum and it departure from the thermal one are alo preented in Set. 8 and will be further examined in forthoming publiation. All the reult of the treatment preented in thi paper an be obervationally relevant for a very peial la of hort GRB without any afterglow. They an alo be of qualitative interet for the firt pike and early feature of a more omplex long burt. It i important to tre that in the ae of the elementary pike here onidered the entire energy of the Dyadophere i emitted in the burt. Thi i not the ae for the long burt, the large majority of the urrently oberved GRB, where muh of uh energy i tranferred to the kineti energy of the baryoni omponent. In a forthoming paper (Biano, Ruffini, Xue, in preparation), we onider the emiion from a PEM pule interating with baryoni matter, before the ondition of tranpareny be reahed, and the orreponding onequene for the obervable effet on the intenity, petrum and time truture of the burt. The equation of motion for uh a pair-eletromagneti-baryoni pule (PEMB Pule) have been integrated and the relative intenity to be expeted for the GRB veru the kineti energy left in the aelerated baryoni material (Ruffini et al. 2) ha alo been given. In a final paper (Biano, Chardonnet, Frahetti, Ruffini, Xue, in preparation) we analyze the interation of the aelerated baryoni material (ABM Pule) with the intertellar medium. A the tranpareny ondition i reahed all the baryoni matter i left having aquired an enormou Lorentz γ fator reahed at the time of deoupling, typially in the range The interation of thee very high energy baryon with the urrounding intertellar medium give rie to the afterglow. We finally onider (Chardonnet & Ruffini, in preparation), within the above model, the prodution of very high energy omi ray by the eletrotati aeleration proe of the remnant EMBH. 2. Main aumption of the EMBH model The mot general blak hole expeted from gravitational ollape i an EMBH haraterized by a Kerr-Newmann geometry endowed with axial ymmetry and eletri and magneti field (Ruffini & Wheeler 1971). That indeed

3 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 379 an EMBH with ma maller than M, an give rie to vauum polarization and to the reation of e + e pair-eletromagneti plama outide the horizon via the Heienberg-Euler-Shwinger proe wa learly demontrated (ee Damour & Ruffini 1975). It wa there hown how thee proee an approah reveribility, in the ene of Chritodoulou and Ruffini (Chritodoulou & Ruffini 1971) and that thi phenomenon would lead to a mot natural model for GRB. The diovery of the afterglow of GRB by the Beppo- Sax atellite and the onequent lear determination of the ditane and energeti of the GRB ha motivated u to return to thi field. In order to give an etimate of the effiieny of the quantum eletrodynamial proe in the gravitational field of an EMBH we have onidered the idealized ae of a pherially ymmetri Reiner-Nordtröm geometry, negleting all the effet aoiated with rotation, to be later examined in a muh more omplex treatment. We have then omputed the phyial parameter, the patial extenion, the total energy, and the petrum of the relativiti plama of eletron-poitron pair reated by the vauum polarization proe and introdued the onept of the Dyadophere (Preparata et al. 1998a,b). The omputation of the relativiti hydrodynami of uh an e + and e pair and eletromagneti radiation plama in the field of an EMBH, till in the implified ae of pherial ymmetry, have been arried out both with imple emi-analytial and numerial approah arried out in Rome and their validation by the full hydrodynamial numerial omputation at Livermore (ee Ruffini et al. 1999). The main reult how the formation of a harp lab of e + and e pair and eletromagneti radiation: the PEM pule. Suh a PEM pule keep a ontant width in the laboratory frame and in a few hundred of harateriti roing time of the Dyadophere reahe extreme relativiti ondition with harateriti Lorentz γ fator of The pair denity at the beginning i o high that the plama i optially thik, and only very few photon an eape from the plama. With the expanion, ooling and annihilation of eletron-poitron pair, the ondition of tranpareny i gradually reahed (ee Eq. (1)). The detail of the temporal development of the burt emitted a the tranpareny ondition i reahed, a well ome indiation on the expeted petra, are the ubjet of the preent paper. For impliity, we have negleted the feedbak of the PEM pule on the adiabatiity ondition prior to reahing the final moment of tranpareny. The ae in whih ome baryoni matter i engulfed by the PEM pule, prior to reahing the ondition of tranpareny, ha been the ubjet of a ueive work (Ruffini et al. 2). Our implified model, a well a the validation by the full numerial ode at Livermore, have hown how the lab approximation i till valid in thi more general ae for a large range of the mae of the baryoni matter (a pair-eletromagneti-baryon pule, PEMB pule). Mot important, the addition of baryoni matter lead, through the eletroni omponent, to an inreae in the opaity of the PEMB pule. The tranpareny ondition i now reahed at later time, leading to an ever inreaing tranfer of the total energy of the PEMB pule to the kineti energy of the baryoni omponent. The larger the amount of baryoni matter, the maller i the energy of the PEMB pule releaed in the GRB, for fixed value of the ma and harge of the EMBH. All thee treatment refer to the idealized ae of an already formed EMBH, it i lear, however, that in reality the Dyadophere will be formed during the proe of gravitational ollape itelf, prior to the formation of the EMBH, and uh a proe may have ome ditint detailed obervational ignature. The general energeti feature of GRB here preented have been obtained from the idealized model whih onider a a tarting point an already formed blak hole (Preparata et al. 1998b). In order to obtain the fine detail oberved in the time truture of GRB, a well a their detailed petral evolution, there i no alternative but to tudy the gradual formation of the Dyadophere, a the horizon of the EMBH i approahed and formed. In preparation of thi more ompliated analyi and the one orreponding to axially ymmetri onfiguration, we have obtained ome preliminary reult onerning the relevant phyi of a harged ollaping hell in general relativity (Klippert & Ruffini, in preparation). The general energeti apet here onidered will not be modified but the time ontant will be longer due to general relativiti effet (Cherubini, Jantzen, Ruffini, in preparation). 3. The radiation flux from the PEM pule The frequeny ω and wave-vetor k of photon emitted from the PEM pule (ee Fig. 1) expreed in the laboratory frame are: k = ω ( in ϑu +oϑv), k = ω, (2) where ϑ i the angle (in the laboratory frame) between the radial expanion veloity and the diretion from the origin of the PEM pule to the oberver, v i a unit vetor along the radial expanion veloity of the PEM pule, and u i a unit vetor orthogonal to v oriented toward riing ϑ. We are auming here that k and R T are parallel, alo for photon emitted with ϑ.thiilearlyagood approximation, beaue the ditane R T orrepond to a redhift z 1, while the radiu of the emitting region i of the order of magnitude of a few light eond. Then the Lorentz boot along v to the omoving frame of the PEM pule yield the orreponding omoving quantitie: ω = γω (1 v ) o ϑ, ω = k, (3) ( k = k in ϑu + γ k o ϑ v ) v, (4) In the omoving frame photon radiating out of the PEM pule mut have (ee Eq. (4)): o ϑ v, (5)

4 38 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule beaue the omponent of the photon momentum in the omoving frame along the radial expanion veloity diretion mut be poitive in order to eape. The large amount of high-energy photon emiion i mainly due to eletron-poitron annihilation. Thu, a a preliminary onideration and approximation, we aume that photon are in equilibrium at the ame temperature T with eletron-poitron pair before and at deoupling. In the omoving frame of the photon eletron-poitron pair plama fluid, the blak-body petrum of photon that are in thermal with e + e -pair i given by dn γ d 3 = 1 1 k π 2 exp ( hω ), (6) kt 1 where n γ i the number-denity of photon and T i the temperature in the omoving frame. The non-thermal petrum due to multiple invere Compton attering may lightly modify thee aumption. Thee onideration relevant to the detail of the petrum (ee Set. 8) do not modify the temporal profile of the total deteted radiation flux, whih largely depend on the reening fator S(t) rather than on the radiative petrum. Note that due to the Liouville theorem on the invariane of the ditribution funtion (ee e.g. Ehler 1971), we an write: 1 exp ( 1 hω ) = ( ) (7) kt 1 hω exp kt lab 1 where ω and T are the photon energy and temperature in the omoving frame, while ω and T lab are the ame quantitie in the loal laboratory frame. We an now ue Eq. (3) for ω obtaining: T T lab = γ ( 1 v (8) o ϑ) Therefore we an ue the following formula for the photon petrum: dn γ d 3 k = 1 1 ( ) π 2 (9) hωγ(1 exp v o ϑ) 1 kt Integrating Eq. (9) over all photon momenta, we obtain the total radiation flux in the loal laboratory frame of the PEM pule. The number of photon (k k+dk) radiating per unit time from a mall urfae element d 2 Σ of the PEM pule to the detetor i given by (per unit area of detetor): dn γ d 3 k (ˆk v)d 2 Σ dω k A, dω k = A RT 2 (1) where ˆk = k k i direted toward the oberver, A i the area of detetor, R T the ditane to the detetor from the origin of the PEM pule and dω k i the olid angle ubtended by the detetor. From Eq. (5, 2), we have [ v ] ˆk v =oϑ, 1. (11) By uing Eq. (9), we integrate Eq. (1) over all photon energie and obtain the oberved infiniteimal energy flux in the ae of an optially thin PEM pule ( in erg m 2 ) : df T (t, ϑ) = 1 RT 2 a ( T (t) 4 ) γ (t) 1 v(t) o ϑ o ϑd 2 Σ. (12) Integrating only dφ [, 2π] in the urfae-element d 2 Σof the PEM pule, and adding the reening fator S (t), we finally obtain the infiniteimal photon energy flux direted toward the detetor and emitted at time t with an angle ϑ for an optially thik PEM pule: df (t, ϑ) =2πa ( T (t) 4 ) γ (t) 1 v(t) o ϑ r 2 (t) o ϑs (t) d o ϑ, (13) whereweomitthefatorr 2 T. df will be then meaured in term of erg, and we will till have to multiply the reult by R 2 T to obtain the oberved intenitie. Note that the radiation flux i axially ymmetri with repet to the axi direted toward the detetor. 4. The arrival time veru emiion time Due to the high value of the Lorentz γ fator ( ) for the bulk motion of the expanding PEM pule, the pherial wave emitted from it external urfae appear extremely ditorted to a ditant oberver (ee e.g. Ree 1966). The urfae emitting the photon deteted at an arrival time differene t a, meaured from the arrival of the firt photon, i not trivially the pherial urfae of the fireball at a fixed time differene t, meaured from the initial emiion, but photon emitted at different t and at different angle reah the detetor at the ame time t a. The relation between emiion time t and arrival time t a in the ae of a ontant γ for the expanding fireball, ha been found by Ree (for the definition of ϑ ee Fig. 1) (ee Ree 1966): t a = t (1 v ) o ϑ. (14) The external radiu r (t) of the fireball, due to the ontany of γ, i obviouly given by: r (t) =vt. (15) If, in Eq. (14), we fix the value of the arrival time t a = t a, uing Eq. (15) we an find the equation deribing the urfae emitting the photon deteted at arrival time t a: vt a r = 1 v (16) o ϑ,

5 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 381 r + v u P k r(t) ϑ r d R T L R Oberver Fig. 1. Sheme to find the relation between the emiion time t in the loal laboratory frame and the arrival time t a. P i the point where a generi photon i emitted. L i the ditane of P from the oberver. R T itheditaneoftheblakholefromthe oberver. r d i the radiu of the dyadophere. r + i the radiu of the horizon. r (t) i the radiu of the external urfae of the def. fireball at time t. R i defined by R R T r d. ϑ i the angle between r (t) andr T. v i a unit vetor along the radial expanion veloity. u i a unit vetor orthogonal to v oriented toward riing ϑ. k i the momentum of the photon emitted toward the oberver. Note that we have aumed k R T,and thi i ertainly a very good approximation (ee text, Set. 3) whih deribe an ellipoid of eentriity v (ee Ree 1966). Thi i the approximation that ha been widely ued in the gamma ray burt literature: ee e.g. Sari & Piran (1997), Sumner & Fenimore (1997), Ramirez-Ruiz & Fenimore (1999), Fenimore et al. (1996), Granot et al. (1999), Fenimore (1999), Panaiteu & Mézáro (1998), Piran (1999) and referene therein. However, in our ae the veloity v i not ontant: in a few hundred of the PEM pule roing time 1 the γ fator of the bulk motion of the expanding PEM pule goe from to 1 4, and o we have to find the generalization of Eq. (14) for nonontant veloity. Thi an be done uing the geometry of Fig. 1. We et t = when the fireball tart to expand and the firt photon are emitted, o that r () = r d. Let a photon be emitted at time t from the point P.Itditane from the oberver i L. The time it take to arrive at the L detetor i, of oure,. So, it arrival time, meaured from the arrival of the firt photon a time R after it emiion at t =,i: t a = t + L R (17) where we have defined t a = when the firt photon emitted at t =andϑ = reahe the oberver. L i learly given by: L = RT 2 + r (t)2 2 R T r (t)oϑ (18) where at any given value of emiion time t, oϑ an aume any value between v(t) and 1 a noted above, where v (t) i the expanion peed of the fireball at time t 1 The roing time of the PEM pule i given by r d r +, and thi i 1 2, for a blak hole with M =1 3 M and Q =.1Q max (ee Eq. (5)). Now, r (t) i of the order of magnitude of ome light-eond, and R T orrepond to a redhift z 1. So we an expand the right hand ide of Eq. (18) in power of r(t) R T L R T ( 1 r (t) R T at firt order: ) o ϑ, (19) whih orrepond to negleting the lateral diplaement from the line of ight axi. Subtituting (19) into (17) yield: t a = t + r d r (t) o ϑ, (2) wherewehaveuedthefatthatr T = R + r d (ee Fig. 1). For r (t) we an ue the following expreion: r (t) = t v (t )dt + r d, (21) o that Eq. (2) an be written in the form: t t a = t v (t )dt + r d o ϑ + r d, (22) whih redue to Eq. (14) only if v i ontant and r d i negligible with repet to r (t). Alo in Eq. (22) we an fix t a = t a to obtain the equation deribing the urfae that emit the photon deteted at an arrival time t a. In thi ae, we no longer have ellipoid of ontant eentriity v. Sine the veloity i trongly varying from point to point, we have more ompliated urfae like the profile reported in Fig. 2 where at every point there will be a tangent ellipoid of ontant eentriity, but uh an ellipoid varie in eentriity from point to point. For a fixed time t of emiion in Eq. (22), the allowed angular interval v o ϑ 1 lead to a orreponding mearing of the arrival time t a over the interval r t a = γ 2 ( ), 1+ v (23) alled the angular time ale (ee e.g. Piran 1999). 5. The radiation flux with repet to arrival time for an infinitely thin fireball Equation (13), integrated with repet to ϑ, give u the value of F (t), the flux emitted from the PEM pule at time t. Intead we want to ompute F (t a ), the flux deteted at an arrival time t a. In priniple, we hould ubtitute Eq. (22) into Eq. (13) before the integration, but we have no analytial relation between v (t) andt, oanumerial integration mut be performed. We ued the following approah: 1. We fix the value of the emiion time t, andletoϑ aume a direte et of value between 1 and v(t) ; 2. For eah direte value of o ϑ, we ompute df (t, ϑ) and t a (t, ϑ);

6 382 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 1.2e+56 1e+9 1e+56 5e+8 8e+55 y (m) Flux (erg/) 6e+55 5e+8 4e+55 2e+55 1e+9 2e+11 4e+11 6e+11 8e+11 1e+12 x (m) Fig. 2. Equitemporal urfae, projeted into the plane of Fig. 1, with Carteian oordinate entered on the blak hole, for a PEM pule ariing from a blak hole with M =1 3 M and Q =.1Q max, negleting the fireball thikne effet (ee Fig. 3 and Set. 5), with t a going from.262 (the inner urfae) to.2662 (the outer one) with tep of The oberver i far away along the x axi, while y axi ale i expanded to ee the extremely elongated urfae, making the pherial urfae entered on the blak hole appear to be the vertial line given in the figure 3. Then, we hange the value of t and repeat, thu letting o ϑ vary with fixed t over it allowed range, and t vary from the beginning of the emiion to the deoupling time, and at eah tep, we ompute all the value of df (t, ϑ) andt a (t, ϑ); 4. Now we have a large number of orreponding value of df and t a, and we have to ompute F (t a )atheum of all the value of df orreponding to the ame t a. Unfortunately, we have omputed thee value numerially, and o they are not exat. Thu ome df that hould orrepond to the ame t a may orrepond to lightly different t a, and vie vera. So we define F (t a ) a the um of all the value of df orreponding to an arrival time between t a and t a + r, where the value of r i aigned by hand ; 5. Of oure, during thi integration we an ee whih value of t and ϑ give u the ame (in the meaning diued above) value of t a, and in thi way we obtain plot like the one in Fig. 2. Thi proe yield F (t a ), the o alled light urve of the GRB, and thi i hown in Fig. 3, whih reveal a very big problem with our omputation. In fat, we ee that the imulated time profile of the GRB eem to be ut after the peak, going to zero too fat, while experimentally long tail are oberved after the peak. Thi i beaue we have aumed that all photon are emitted from the external urfae of the expanding fireball, but they an alo be emitted from a ertain depth inide the urfae, and thi implie a time delay in the arrival time for photon emitted from different depth. Thi effet i almot negligible at the beginning, when the reening fator S (t) atifie S (t) 1, but thi ondition hange very rapidly Arrival Time (t a ) () Fig. 3. Time profile for a burt from a blak hole with M = 1 3 M and Q =.1Q max, negleting fireball thikne effet. The total energy releaed toward the detetor i E tot erg and the time duration (T 9) of the event i T The plot i made with r =1 6 (eetext) during the expanion, when S (t) 1. The time required by a photon to ro the entire PEM pule i about 1 2, when S (t) 1, in the ae reported in Fig. 3, while the duration of the peak we have obtained i about 1 4. Therefore the thikne effet annot be negleted and we mut hange our numerial integration heme. 6. The radiation flux with repet to arrival time for a thik fireball In the previou etion we omputed the emitted flux by onidering the uppreion fator λ D and auming that all the radiation i emitted from the external urfae of the expanding fireball. Now, we relax thi lat aumption, and modify the etimate of the arrival time given in Eq. (22) by inluding the delay due to the thikne of the emiion region in the fireball. We aume that, at a fixed emiion time t, only the region of the fireball between r (t) andr (t) λ (t) i ative in the photon emiion, o the new formula i: t t a = t v (t )dt + r d r 2 o ϑ + r d, (24) where r 2 λ (t). Thi interval, for a fixed value of t and for ϑ =, lead to a orreponding maximum mearing of the arrival time t a over the interval t a = λ, (25) alled the intrini time ale. Due to thi extra fator, the numerial integration i alo lightly different: 1. We pik the firt value of t and divide the emitting part of the fireball into N emitting ub-hell, eah of width λ(t) N and haraterized by different value of the depth r 2, and we aume that the flux emitted by eah ub-hell i df N. We pik the firt ub-hell at the

7 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 383 3e e e e+55 2e+55 1e+55 Flux (erg/) 1.5e+55 Flux (erg/) 8e+54 6e+54 1e+55 4e+54 5e+54 2e Arrival Time (t a ) () Arrival Time (t a ) () Fig. 4. Time profile for a burt from a blak hole with M =1 3 M and Q =.1Q max, onidering fireball thikne effet. The total energy releaed toward the detetor i E tot erg and the time duration (T 9) of the event i T Theplotimadewithr =1 4 (eetext) urfae with r 2 = and vary the value of o ϑ from v(t) to 1 with a large number of very mall finite tep. At eah tep we ompute df and t a. Next we pik the eond ub-hell, inreaing r 2 by one tep, and repeat the omputation of df and t a varying o ϑ, and o on for every ub-hell o that the intrini time ale of photon traveling within the emitting region i onidered; 2. Next we pik the eond value of t and divide again the emitting hell of the fireball of width λ (t) inton ub-hell of width λ(t) λ(t).notethat N i an inreaing funtion of t, while N i fixed, o thee ub-hell are not the ame a before. We repeat the omputation of df and t a varying o ϑ for eah ub-hell. We pik the third value of t andrepeat,andoonuntilthe deoupling time. In other word, we vary o ϑ at fixed r 2 and t, and vary r 2 at fixed t; 3. The lat tak to perform i the ame of in previou ae, aigning to the arrival time t a the total flux of all the df orreponding to an arrival time between t a and t a + r. The reult of thi new omputation are hown in Fig. 4, 5 and 6, repetively regarding a blak hole with ma equal to 1 3 M,1 2 M and 1 M with harge Q =.1Q max, and in Fig. 7 where the three different ae are hown together for omparion purpoe. The new feature here ome from the equitemporal urfae of the burt. In fat, due to the extra term r 2,the relation between t and ϑ at fixed t a i no longer monotoni. Thu it an happen that photon emitted with a ertain t, ϑ and r 2 arrive after the one emitted at a ubequent time t, at a larger angle ϑ, but from a lower depth r 2.The reult i that the point whih in Fig. 2 lie along urve are now pread on the plane, with no poibility of finding lean urfae, a an be een in Fig. 8. Fig. 5. Time profile for a burt from a blak hole with M =1 2 M and Q =.1Q max, onidering fireball thikne effet. The total energy releaed toward the detetor i E tot erg and the time duration (T 9) of the event i T Theplotimadewithr = (ee text) Flux (erg/) 5e e+54 4e e+54 3e e+54 2e e+54 1e+54 5e Arrival Time (t a ) () Fig. 6. Time profile for a burt from a blak hole with M = 1 M and Q =.1Q max, onidering fireball thikne effet. The total energy releaed toward the detetor i E tot erg and the time duration (T 9) of the event i T Theplotimadewithr = (ee text) In thi ae, the light urve are muh more imilar to the oberved one: we an ee that the firt part after the peak how an exponential behavior, followed by a tail with a power law dependene. So we uppoe that the eond part an be repreented by a funtion like: F =(p 1 t p2 a )+p 3, (26) with p 1, p 2 and p 3 free parameter, and the firt part by: F = ( p 4 e p5 ta) + p 6, (27) with p 4, p 5 and p 6 free parameter. We now apply a fit algorithm eparately on the two part of the tail to etimate the parameter. The reult are hown in Fig. 9, 1 and 11.

8 384 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 3e e+55 3e e+55 Computed Profile p 1 *t a p 2 +p 3 p 4 *e (p 5 *t a ) +p 6 2e+55 2e+55 Flux (erg/) 1.5e+55 Flux (erg/) 1.5e+55 1e+55 1e+55 5e+54 5e Arrival Time (t a ) () Arrival Time (t a ) () Fig. 7. The three peak of Fig. 4, 5 and 6 are hown on the ame plot, for viual omparion Fig. 9. Fit of the tail of the time profile of Fig. 4. The value obtained for the parameter are: y (m) 1e+9 5e+8 Parameter Value Meaure Unit 53 erg p 1 (1.91 ±.85) 1 (1+p 2 ) p ±.12 None p 3 54 ( 6.2 ±.82) 1 erg 58 erg p 4 (3.9 ±.68) 1 p ± erg p 6 (7.15 ±.14) 1 5e+8 1e+9 2e+11 4e+11 6e+11 8e+11 1e+12 x (m) Fig. 8. Equitemporal Surfae for a PEM pule from a blak hole with M =1 3 M and Q =.1Q max, onidering fireball thikne effet (ee Fig. 4 and Set. 6), with the ame ale a Fig. 2. Thi plot only how the point for t a =.3. The oberver i far away along the x axi and the blak hole i at the point (, ). The vertial line are a portion of pherial urfae entered on the blak hole. Note that the point doe not form definite line, but how the effet of the uperpoition of many urfae emitting independently It eem that there i a relation between the value of p 2 and p 5. In fat, dereaing the value of the blak hole ma, and o dereaing the energy of the burt, p 2 rie, making the power law dereae lower. In ontrat, a p 5 dereae the exponential part will be teeper. So the dereae in the energy releaed in the burt orrepond an L like hape, with a very teep exponential dereae, followed by an almot ontant power-law part. 7. The temporal truture and time duration Within the validity of our approximation, whih ha the very trong aumption of pherial ymmetry, we an now preent the main predited theoretial feature of the GRB whih hould be ompared with obervation: 1. The fat-rie, exponential-deay profile whih repreent a lear time trigger of the GRB ignal, haraterized by an exponential-deay profile with e βta (ee Fig. 9, 1 and 11). 2. The tail of the total radiation flux with a power law dependene given by t α a,whereα i loe to unity (ee Fig. 9, 1 and 11). From the temporal profile of the radiation flux a a funtion of the EMBH mae we find that the more maive ytem have an exponential behavior with a maller abolute value of the oeffiient of the exponent and, orrepondingly, a larger abolute value of the exponent of the power law. Thee effet an be ompared and ontrated with GRB obervation (ee Set. 6). The time duration of the GRB i uually obervationally haraterized by the o-alled T 9 riterion, that i the time interval tarting (ending) when the energy deteted i the 5% (95%) of the total emitted. Namely the T 9 i the time duration of the emiion of 9% of the energy. Thi riterion i very ueful in deribing oberved GRB, beaue, due to the bakground noie, it i very diffiult to find the exat tarting time and ending time of the emiion, while at 5% and 95% of the emiion the ignal i uually well above the noie. We an apply the ame

9 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule e e+55 Computed Profile p 1 *t a p 2 +p 3 p 4 *e (p 5 *t a ) +p 6 5e e+54 Computed Profile p 1 *t a p 2 +p 3 p 4 *e (p 5 *t a ) +p 6 4e+54 1e e+54 Flux (erg/) 8e+54 6e+54 Flux (erg/) 3e e+54 2e+54 4e e+54 1e+54 2e+54 5e Arrival Time (t a ) () Arrival Time (t a ) () Fig. 1. Fit of the tail of the time profile of Fig. 5. The value obtained for the parameter are: Fig. 11. Fit of the tail of the time profile of Fig. 6. The value obtained for the parameter are: Parameter Value Meaure Unit 52 erg p 1 (7.6 ± 1.8) 1 (1+p 2 ) p ±.46 None p 3 54 ( 3.16 ±.23) 1 erg 6 erg p 4 (1.8 ± 1.1) 1 p ± erg p 6 (5.15 ±.9) 1 Parameter Value Meaure Unit 52 erg p 1 (1.6 ±.75) 1 (1+p 2 ) p 2.9 ±.73 None p 3 54 ( 1.23 ±.15) 1 erg 56 erg p 4 (8.5 ± 2.5) 1 p 5 ( 25 ± 13) erg p 6 (1.332 ±.48) 1 proedure to our theoretially omputed burt, obtaining their T 9. The reult for the three different blak hole mae onidered are reported in the aption of Fig. 4, 5 and 6. Sine now the angular time ale orreponding to different emiion epoh are taken into aount a well a the intrini time ale of the emitting region, the T 9 omputed are ignifiantly larger (ee Fig. 12) than the one of Ruffini et al. (1999), omputed only on the bai of the angular time ale (ee Eq. (23)) at the lat tranparent point. We emphaize the fat that by breaking the aumption of pherial ymmetry that we have adopted and by introduing inhomogeneitie in the PEM pule, the omplex ub-truture of the oberved GRB an be eaily aommodated in our enario, uing the two different time ale (angular and intrini) of the emiion proe. An inhomogeneity in the PEM pule with time ale maller than the angular one (ee Eq. (23)), r δ< γ 2 ( ) 1+ v O ( 1 4), (28) i inviible in the light-urve. However, any inhomogeneitie with time ale δ within the range between the angular and the intrini time ale (ee Eq. (23) and (25)), r γ 2 ( ) 1+ v <δ< λ, (29) Time duration (T 9 ) () M BH /M Solar Fig. 12. The T 9 of the three imulated elementary burt of Fig. 4, 5 and 6, plotted a a funtion of the blak hole ma would give rie to omplex ub-truture in the light urve, up to a number given by D r ( γ 2 1+ v ) (3)

10 386 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule N(ε) 3e e+25 2e e+25 1e+25 5e+24 Computed photon petrum Blak body petrum ε (MeV) Fig. 13. Spetrum of the radiation oberved at t a =.3 from a blak hole with M =1 3 M and ξ =.1, together with a blak body petrum fitted on the peak 8. General onideration about the petrum Although the main topi of thi artile deal with the truture of the burt and it time truture a een from the oberver, we an add ome qualitative onideration about the petrum and epeially on it departure from a blak body one. If one take the aumption of Set. 3, it i eay to ompute the expeted petrum of the oberved radiation. The oberved number petrum N ɛ,per photon energy ɛ, per teradian, of photon emitted by a ingle hell i given by (in photon/ev) (ee Eq. (65) of Ruffini et al. 1999): N ɛ (v, T, R) dv u ɛ ɛ =( )4πR 2 dr ɛt vγ [ 1 exp[ γɛ(1 + v log )/T ] ] 1 exp[ γɛ(1 v )/T ], (31) whih ha a maximum at ɛ max = 1.39γT ev for γ 1. We an then um thi petrum over an equitemporal urfae of our PEM-pule to get the total petrum of the radiation oberved at a ertain arrival time, and thi i reported in Fig. 13, in the ae of a blak hole with M =1 3 M and ξ =.1, together with a blak body petrum fitted on the peak, for omparion. It i lear that thi petrum i already different from a blak body, both at low and high frequeny. Thee implified reult may be, however, affeted by two additional effet, whih are not very relevant for determining the equation of motion of the PEM pule and the time profile of the burt, but an indeed be of relevane for the definition of the petrum. The firt modifiation would affet ome of the onideration in Set. 6, where a radial dependene of the temperature of the radiation may be needed following a more detailed deription of the tranpareny ondition. Thi will lead to a further broadening of the petrum given in Fig. 13. The eond modifiation may be due to the onideration of multiple invere Compton attering of the photon, deribed by Kompaneet equation (Kompaneet 1956; ee alo Felten & Ree 1972). Sine the inreae of energy of a photon per attering i: E = 4kT E, (32) m e 2 the ondition for the ditortion beome y = kt m e 2 σ γ,en e dr = kt m e 2 max ( τ T,τT 2 ) 1 (33) with n e the number of the eletron, σ γ,e the Thomon ro etion and τ T the opaity due to Thomon attering τ T = n e σ γ,e D. (34) Then, the number of attering i ( ) 2 D N = = τ 2 λ T. (35) In the preent model, at emiion time, ay, 2 afterthe beginning of the expanion, the omptonization appear to be a ruial effet. Indeed, the optial depth for free-free aborption i: τ ff = k ff Σ (36) where k ff m p n e T 7/2 g 1 m 2 (37) and Σ i the urfae denity of the eletron. However, the medium i opaque with repet to the Compton attering (deribed by the Thomon ro etion): τ T (38) o that the ondition for omptonization (Eq. (33)) i fulfilled. Hene, the blak body petrum, whih exit in the omoving frame at the very initial tage of the expanion, quite oon will undergo modifiation due to the omptonization. The oberved petrum F ν, in thi ae, will depart even more from a thermal one, with blak body ditribution at low energie, and a plateau up to the Wien exponential utoff. In the Fig. 14 we give a qualitative form of the petrum. The variation of the petrum in the oure of the expanion of the fireball an be determined via the imultaneou olution of the hydrodynamial equation of motion and energy balane (given in Ruffini et al. 1999) together with the Kompaneet equation (Kompaneet 1956): n y = 1 x 2 x [x 4 ( n + n 2 + n x )], x = hν kt where n i the oupation number denity of photon: n (ν) = 3 8πhν 3 F ν (39) The loation of the utoff, whih depend on the number of olliion, will give independent information on the phyial parameter of the initial Dyadophere. Thi omputation i beyond the ope of the preent paper.

11 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule 387 N(x) x x 2 x -1/3 exp(-x 2 ) Fig. 14. Qualitative plot of the expeted petra due to multiple invere Compton attering 9. Conluion In our model the mot general GRB i haraterized by ix different phae (ee e.g. Ruffini 2), eah one with ditint phyial proee whih an lead to peifi obervational feature: 1. The identifiation of the preuror. Sine the ma range of the EMBH in whih the vauum polarization proe an our varie from 3.2 M to M, it i partiularly important to identify the preuror of the ollaping ore. While in the ae of 1 M EMBH the undertanding of the proe of gravitational ollape ha been ahieved on the ground of binary X-ray oure (ee Giaoni & Ruffini 1978), in the ae of more maive ollaping ytem ( M ) variou enario an be onidered (ee e.g. Gurzadyan & Ruffini, in preparation). In addition to the identifiation of the preuror, it i important in our model to identify the magnetohydrodynamial ondition leading to the harge eparation proe in the ollaping ore (ee Ruffini 2); 2. The proe of gravitational ollape whih lead to the formation of the Dyadophere. A mentioned in Set. 2, the ae of pherial ymmetry i urrently been invetigated (Klippert 2). In nonpherial gravitational ollape, thi phae may lead to the obervation of gravitational wave and the aoiated gravitationally indued eletromagneti radiation and eletromagnetially indued gravitational radiation (ee Johnton et al and Johnton et al. 1974); 3. The aeleration of the ole e + e eletromagneti plama omponent (the PEM pule) to relativiti Lorentz γ fator greater than 1 2. The dynami of thi phae ha been treated in a previou paper (ee Ruffini et al. 1999). Thi phae may lat all the way to the reahing of the tranpareny ondition. The obervational onequene of thi phae are the topi of the preent paper; 4. The interation of the PEM pule, prior to reahing the tranpareny ondition, with ome baryoni matter. The poible aeleration to even larger value of the Lorentz γ fator ( ) of the plama ompoed of γ, e +,e and the eletron and nuleon of the baryoni matter (the PEMB pule) ha been treated in a previou paper (Ruffini et al. 2). The obervational apet of thi phae will be preented in a forthoming paper (Biano et al., in preparation) where we ompute the duration T 9 value predited by thi more general model a well a their temporal truture and intenity variation and ompare and ontrat thee reult with the oberved one; 5. A ubtantial part of the energy of the Dyadophere, in the preene of a large quantity of baryoni matter, will be arried away by the kineti energy of the baryon. The relative importane of the energy tranfer from the Dyadophere to the kineti energy of the baryoni omponent ha been etimated in a previou paper (Ruffini et al. 2). The obervational apet of thi phae, partiularly relevant to the oberved afterglow and poibly to neutrino, will be preented in a forthoming paper (Biano et al., in preparation); 6. The aeleration of ultra high energy omi ray by the remnant eletrodynami truture of the EMBH (ee Ruffini 1999). Thi problem i alo under urrent examination (ee Chardonnet & Ruffini, in preparation). We have ued the detailed model of the reation of photon and eletron-poitron pair around an EMBH by the proe of vauum polarization (ee Damour & Ruffini 1975; Ruffini 1998 and Preparata et al. 1998b) and their ubequent time evolution (ee Ruffini et al. 1999) in order to ompute the different relativiti effet ourring in the determination of the time profile of the oberved radiation flux of a GRB with repet to arrival time, a the PEM pule gradually reahe tranpareny. If thee theoretial predition will be upported by obervational evidene, they will offer an important tool and a trong onnetion between the obervation of GRB and the propertie of the entral engine whih upplie the energy. Thi i even more ompelling ine the implified model ha only two bai parameter, the ma M and harge to ma ratio Q M of the EMBH, giving the firt lear evidene, in an atrophyial ytem, of uing the extratable ma-energy of a blak hole (ee Chritodoulou & Ruffini 1971). In addition, we make preie predition of the truture of the burt and of their time variability whih an be obervationally verified. The onideration preented in thi paper refer only to the above mentioned phae 3 or to the initial evolution of phae 4. The reahing of tranpareny by a pure e + e and γ omponent an only our for very low baryoni matter denity around EMBH, ρ B 1 9 g/m 3. Thi very peial irumtane will define a peial la of GRB with the peifi ignature preented in the preent paper: hort and elementary time variability and abene of afterglow.

12 388 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule The etimate of the T 9 and the hape of the burt may be an important tool for their identifiation. The petra feature an alo be an important tool for verifying the oniteny of the model. The theoretial tool developed in thi paper are applied in the forthoming publiation to the more general ae where baryoni matter i preent. In thi more general ae only a mall part of the energy of the Dyadophere will be radiated in the burt. A fration of the energy, inreaing with the amount of baryoni matter, will be tranferred to the kineti energy of the baryoni matter, leading to obervational onequene and to the afterglow epoh. Appendix A: Eletron-poitron pair annihilation and GRB The evolution of the PEM pule i ompletely deribed by the general relativiti hydrodynamial equation (ee Ruffini et al. 1999) and the rate equation for the numberdenitie of eletron and poitron Temperature (MeV) M BH =1 3 M Solar M BH =1 2 M Solar M BH =1M Solar (n e ±U µ ) ;µ =(n e ±U t ),t + 1 r 2 (r2 n e ±U r ),r = σ v [n e (T )n e +(T ) n e n e +], (A.1) where σ i the mean annihilation-reation ro-etion of eletron-poitron pair, v i the thermal veloity of eletron-poitron pair, n e +(T )andn e (T ) are the proper number-denitie of eletron and poitron, given by appropriate Fermi integral. We learly have n e ±(T )=n γ (T ), (A.2) where n γ (T ) i the number-denity of photon given by the Boe integral, integrated from 2m e to infinity. In the initial evolution tage of the PEM pule, the temperature T i larger than the energy-threhold.5 MeV of eletron-poitron pair reation (ee Fig. A.1). The eletron and poitron reated in dyadophere are in thermal equilibrium with photon, by the proe e + + e γ + γ. n e ± n e ±(T ) n γ (T ). The rate equation beome (A.3) (n e ±U µ ) ;µ =, (A.4) that i jut the onervation of the total number of eletron-poitron pair (ee Ruffini et al. 1999). A the temperature T (ee Fig. A.1) drop firt loe to and then below the threhold.5 MeV, the eletron and poitron that were in thermal equilibrium with photon annihilate to photon, n γ (T ) >n e+ e (A.5) when PEM pule approah tranpareny. The ratio between the number-denitie of photon n γ (T ) and pair n n e ±, defined a e ± n γ(t ), are indiated in Fig. A.2, for T 1 MeV, for eleted EMBH t (eond) Fig. A.1. The temperature in the omoving frame a a funtion of the loal laboratory time t for typial PEM pule Appendix B: The reening fator in the radiation flux At the beginning, the PEM pule i optially thik, o themeanfreepathλ of the photon in the plama (ee Eq. (1)) i more than ten order of magnitude lower than the thikne D of the fireball: for a PEM pule reated by an EMBH with M =1 3 M and Q =.1Q max, the initial numerial value are: D 1 9 m and λ 1 6 m. So pratially we have no radiation emiion. During the expanion, the plama denity and temperature both dereae, while λ rie, and alo the emitted flux rie, until λ D, when the plama i optially thin and all the remaining photon eape. We then aume that the blak-body radiation flux F BB (in erg/) emitted by our fireball i given by: F BB (t) def. S (t) F T BB (t) 4πaT 4 r 2 γ 2 S (t), erg (B.1) where a , T i the omoving frame m 3 MeV 4 plama temperature given in MeV, γ i the Lorentz γ fator of the bulk motion of the expanding plama, r i the radiu of the external urfae of the PEM pule and S (t),

13 C. L. Biano et al.: Elementary pike from an EMBH Blak Hole: The PEM pule e-6 Sreening Fator (S).1 1e-6 1e-8 1e-1 1e-12 1e-14 pair-photon ratio 1e-8 1e-1 1e-12 1e-16 M BH =1 3 M Solar, Q=.1Q max M BH =1 2 M Solar, Q=.1Q max M BH =1M Solar, Q=.1Q max 1e Time (eond) Fig. B.1. The reening fator S (t) (ee Eq. (B.2)) for three different EMBH with harge Q =.1Q max and ma M = 1 M, M =1 2 M and M =1 3 M repetively 1e-14 1e+56 1e+54 1e-16 1e+52 1e+5 1e-18 M BH =1 3 M Solar M BH =1 2 M Solar M BH =1M Solar 1e t (eond) Fig. A.2. The ratio between number-denitie of photon n n γ(t ) and n e ±, defined a e ± n γ (T ), a funtion of the loal laboratory time t for the three EMBH of Fig. B.1 i the reening fator, given by (ee Preparata et al. 1999): 2 λ (t) S (t) = 3 D (B.2) In Fig. B.1 are plotted the reening fator S (t) for different plama, originated around EMBH haraterized by harge Q M =.1 andbymam =1M, M =1 2 M and M =1 3 M repetively. Uing the quantitie obtained by the numerial imulation of the expanion of the PEM pule (ee Ruffini et al. 1999), we an now make a plot of the emitted flux veru emiion time t, again for eleted EMBH (ee Fig. B.2). A lear radiation flah orreponding to an inreae of the order of magnitude of 1 1 in the radiation flux i oberved a the reening fator S (t) approahe unity. Aknowledgement. It i a pleaure to thank V. Gurzadyan, J. Salmonon and J. Wilon for helpful diuion, a well a the anonymou referee for attrating our attention on the petral ditribution, omplementary to the reult of the preent work, a well a for the wording of the manuript. Flux (erg/) 1e+48 1e+46 1e+44 1e+42 1e+4 M BH =1 3 M Solar, Q=.1Q max 1e+38 M BH =1 2 M Solar, Q=.1Q max M BH =1M Solar, Q=.1Q max 1e Time (eond) Fig. B.2. The reened radiation fluxe F BB (t) (ee Eq. (B.1)) are hown for the ame three ae of Fig. B.1 Referene Chritodoulou, D., & Ruffini, R. 1971, Phy. Rev. D, 4, 3552 Damour, T., & Ruffini, R. 1975, Phy. Rev. Lett., 35, 463 Djorgovki, S. G., Frail, D., et al. 2, Proeeding of the Ninth Marel Gromann Meeting on General Relativity, in pre, Rome 2 Ehler, J. 1971, Proeeding of Coure 47 of the International Shool of Phyi: Enrio Fermi, ed. R. K. Sah (Aademi Pre, New York) Felten, J. E., & Ree, M. J. 1972, A&A, 17, 226 Fenimore, E. E. 1999, ApJ, 518, 375 Fenimore, E. E., Madra, C. D., & Nayakhin, S. 1996, ApJ, 473, 998 Frontera, F., et al. 2, ApJS, 127, 59 Ghiellini, G., Lazzati, D., Celotti, A., & Ree, M. J. 2, MNRAS, 316, L45 Giaoni, R., & Ruffini, R. (ed.), Phyi and Atrophyi of Neutron Star and Blak Hole (North Holland, Amterdam 1978) Goodman, J. 1986, ApJ, 38, L47 Granot, J., Piran, T., & Sari, R. 1999, ApJ, 513, 679