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Analytic Light Curves of Gamma-Ray Burst Afterglows: Homogeneous versus Wind External Media

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We compare the afterglow emission at different observing frequencies, for each type of external medium. It is found that observations at sub-millimeter frequencies during the first day provide the best way of discriminating between the two models. By
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    a  r   X   i  v  :  a  s   t  r  o  -  p   h   /   0   0   0   3   2   4   6  v   2   7   J  u  n   2   0   0   0 submitted to T HE  A STROPHYSICAL  J OURNAL , 23 February, 2000 ANALYTIC LIGHT-CURVES OF GAMMA-RAY BURST AFTERGLOWS:HOMOGENEOUS VERSUS WIND EXTERNAL MEDIA A. P ANAITESCU Dept of Astrophysical Sciences, Princeton University, Princeton, NJ 08544 and P. K UMAR School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540 ABSTRACTAssuming an adiabatic evolution of a Gamma-Ray Burst (GRB) remnant interacting with an external medium,we calculatetheinjection,cooling,andabsorptionbreakfrequencies,andtheafterglowfluxforplausibleorderingsof the break and observing frequencies. The analytical calculations are restricted to a relativistic remnant and, inthe case of collimated ejecta, to the phase where there is an insignificant lateral expansion. Results are given forboth a homogeneous external medium and for a wind ejected by the GRB progenitor.We compare the afterglow emission at different observing frequencies, for each type of external medium. It isfound that observations at sub-millimeter frequencies during the first day provide the best way of discriminatingbetween the two models. By taking into account the effect of inverse Compton (IC) scatterings on the electroncooling, a new possible time-dependence of the cooling break is identified. The signature of the up-scatteringlosses could be seen in the optical synchrotronemission from a GRB remnant interacting with a pre-ejected wind,as a temporary mild flattening of the afterglow decay. The up-scattered radiation itself should be detected in thesoft X-ray emission from GRB remnants running into denser external media, starting few hours after the mainevent. Subject headings:  gamma-rays: bursts - methods: analytical - radiation mechanisms: non-thermal 1.  INTRODUCTION One of the most important issues regarding Gamma-RayBursts (GRBs) is the nature of the object that releases the rela-tivistic ejecta generating the high energy emission of the mainevent and the lower frequency emission during the ensuing af-terglow. Some insight about the GRB progenitor can be ob-tained from the properties of the circum-burst medium, whichcan be inferred from the features of the afterglow emission. If the ejecta is expelled during the merging of two compact ob- jects (M´esz´aros & Rees 1997b), it is expected that the mediumsurrounding the GRB source is homogeneous. However, if acollapsing massive star (Woosley 1993, Paczy´nski 1998) is thesrcin of the relativistic fireball, the circum-burstmediumis thewind ejected by the star prior to its collapse, whose density de-creases outwards. The two models differ in the dependence onradius of the particle density of the circum-burstmediumwhichthe GRB remnant interacts with, and in the value of this den-sity at the deceleration length-scale. The former modifies therate of decline of the afterglow, while the latter determines theoverall afterglow brightness. Therefore it is possible to corre-late afterglow emission features to a specific type of externalmedium.Significant work in this direction has been done by manyresearchers. The two afterglows that exhibited breaks consis-tent with the effects arising from strong collimation of ejecta –GRB 990123 (Kulkarni et al .  1999a) and GRB 990510 (Stanek et al .  1999, Harrison et al .  1999) – indicate that the externalgas was homogeneous (recent work by Kumar & Panaitescu2000 shows that jets interacting with winds cannot producesharp breaks in the afterglow light-curve). The optical emis-sion of three afterglows had a steeper than usual decline. GRB970228decayed as T  − 1 . 7 after the subtractionof an underlyingsupernova emission (Reichart 1999, Galama et al .  2000). Thelight-curveof GRB 980326fell off as T  − 2 . 1 (Groot et al . 1998)and an emission in excess of the early time extrapolation wasdetected  ∼  20  days after the main event, indicating a super-nova contribution (Bloom et al .  1999). A  T  − 2 decay was ob-served for the afterglow of GRB 980519 (Halpern et al .  1999).Such steep declines can be produced either by a fireball inter-acting with a pre-ejected wind (Chevalier & Li 1999) and anelectron index around 3, or by a narrow jet expanding later-ally in a homogeneous external medium and an electron indexslightly larger than 2. Chevalier & Li (1999) found that the ra-dio emission of the afterglowof GRB 980519is consistent withan external wind; however Frail et al .  (2000) point out that theinterstellar scintillation present in the radio data does not allowruling out the jet model. Nevertheless, the existence of super-novae associated with GRB 970228 and GRB 980326 pointstoward a massive star as the srcin of these bursts, implyinga pre-ejected wind as the external medium. From the analysisof the optical radio emission of the afterglow of GRB 970508,Chevalier & Li (2000) conclude that the surrounding mediumwas a wind. Frail, Waxman & Kulkarni (2000) argue that thesame radio afterglow can be explained by a homogeneous ex-ternal medium.In this work we investigate the differencesbetween the light-curves of afterglows arising for each type of external medium,with the aim of finding ways for distinguishing between thetwo models. This study is done within the usual framework of a relativistic remnant interacting with a cold external gas. Asthe fireball is decelerated, a shock front sweeps up the externalgas, accelerating relativistic electrons and generating a mag-netic field in the shocked gas. We ignore the emission fromelectrons accelerated by the reverse shock which propagatesthrough the ejecta at very early times. At optical wavelengthsthis emission is short lived, lasting up to few tens of seconds af-ter themainevent(Sari &Piran 1999),but it couldbeimportant1  2for the radio emission until few days (Kulkarni et al .  1999b).Analytical afterglow light-curves for spherical remnants in-teracting with homogeneous external media have been previ-ously published by Sari, Piran & Narayan (1998). Features of afterglowsfromspherical fireballs, such as peakflux, breakfre-quencies, andtime evolutionoffluxes at a fixedfrequency,havebeen studied by M´esz´aros & Rees (1997a), Waxman (1997b),Wijers & Galama (1999), and Dai & Lu (2000) for homoge-neous media, and by Chevalier & Li (2000) for pre-ejectedwinds. In this work we present and compare analytical andnumerical light-curves at various observing frequencies, cover-ing all cases of interest, for both types of external media, tak-ing into account the differential arrival-time delay and Dopplerboosting due to the spherical shape of the source. We take intoaccount first order IC scattering, calculate its effect on the elec-tron cooling and on the afterglow synchrotron emission, andstudy briefly the high-energy emission resulting from the up-scattering of synchrotron photons. The possible importance of IC scatterings for the early afterglow emission was pointed outby Waxman (1997a) and Wei & Lu (1998). 2.  SIMPLE DYNAMICS OF RELATIVISTIC REMNANTS Forthecalculationoftheafterglowemissionit isnecessarytoknow how the remnant Lorentz factor  Γ  evolves with observertime  T  , as all other quantities that appear in the expression of the spectral flux are functions of   Γ  and of the remnant radius r  and external medium density  n ( r ) . We shall assume that theremnant is adiabatic, i.e .  the energy carried away by the emit-ted photons is a negligible fraction of the total energy of theremnant. This assumption is correct if the energy density of theelectrons accelerated at the shock front is a fraction  ε e  ≪  1 of the total energy density in post-shock fluid or if most of theelectrons are adiabatic, i.e .  their radiative cooling timescale ex-ceeds that of the adiabatic losses due to the remnant expansion.Assuming that the internal energy of the ejecta is negligiblecompared to its rest-mass energy and that the ratio internal-to-rest mass energy in the energized external medium is  Γ  −  1 (i.e .  its “temperature” tracks that of the freshly shocked gas),conservation of energy leads to m ( r )Γ 2 + M  fb Γ − [ m ( r ) + M  fb Γ 0 ] = 0  (1)where M  fb  and  Γ 0  are the initial mass and Lorentz factor of thefireball (whose energy is  E   =  M  fb Γ 0 ) and m ( r ) = 4 π 3 − sm  p n ( r ) r 3 (2)is the mass of swept-up material ( m  p  being the proton’s mass).The external medium particle density is n ( r ) =  Ar − s ,  (3)with  s  = 0  for a homogeneous medium and  s  = 2  for a windejected by the GRB progenitor at a constant speed.Equation (2) is valid if the remnant is spherical, but canalso be used for collimated ejecta when the lateral spreading(Rhoads 1999) is insignificant if the quantity  E   above is de-fined as the energy the fireball would have if it were spherical.Throughout this work we shall assume that the remnant is a jetwith an initial half-angle larger than  > ∼  20 o , in which case thesideways expansion is negligible during the relativistic phase.The following analytical calculations of the afterglow emissioncan be extended to sideways expandingjets and non-relativisticremnantsbyfirstdetermining Γ( r ) . Thesetofcoupleddifferen-tial equations describing the evolution of the jet Lorentz factorand its opening can be solved analytically for  s  = 0  (Rhoads1999). The lack of a good approximation for the jet dynam-ics in the case of pre-ejected winds is the main motivation forrestricting the following analytical calculations to spherical orwide-angle remnants.The solution of equation (1) is Γ( r ) = 12   4 x 3 − s + 1 + (2 x 3 − s / Γ 0 ) 2 − 1  x s − 3 Γ 0  ,  (4)where  x  is the radial coordinate r  scaled to r 0  =  3 − s 4 πE m  p c 2 A Γ 20  1 / (3 − s ) ,  (5)the deceleration length-scale, at which  m ( r 0 ) =  E/ ( c 2 Γ 20 ) = M  fb / Γ 0 . The result given in equation (4) is also valid inthe non-relativistic regime. For  x  ≪  1 Γ  < ∼  Γ 0 , whilefor  1  ≪  x  ≪  x nr  we find  Γ =  x − (3 − s ) / 2 Γ 0 . Here x nr  = (Γ 20 / 3) 1 / (3 − s ) marks the end of the relativistic regime: Γ( x nr ) = 2 . For the ease of analytical calculations we shallassume that the power-law behavior of   Γ  lasts from  x  = 1  to x  =  x nr .The Lorentz factor given in equation (4) represents a “dy-namical” average of the Lorentz factors at which different re-gions of the shocked remnant move (the Blandford – McKeesolution). The Lorentz factor of the shock front that propagatesinto the external gas  Γ sh  = √  2Γ , with  Γ  given by equation(4), matches that given in equation (69) of Blandford & McKee(1976) for the power-law regime  1  ≪  x  ≪  x nr  if   E   is multi-plied by  (17 − 4 s ) / (12 − 4 s ) . This correction factor ( 17 / 8  for s  = 0  and  9 / 4  for  s  = 2 ) will be used in the following results.The constant  A  in equation (3) is the number density  n ∗  of the external homogeneous medium for  s  = 0 , while for  s  = 2 A  = 14 π . M m  p v  = 3 . 0 × 10 35 A ∗  cm − 1 ,  (6)where . M   is the mass loss rate of the massive star that ejectedthe wind at constant speed  v , and  A  was scaled to A ∗  = . M  / 10 − 5 M ⊙ yr − 1 v/ 10 3 kms − 1  ,  (7)as in Chevalier & Li (2000) for a Wolf-Rayet star. For fur-ther calculations it is convenient to use equation (3) in the form n ( r ) =  n 0 ( r/r 0 ) − s where  n 0  =  n ∗  for  s  = 0 , while for  s  = 2 n 0  = 1 . 9 × 10 4 E  − 253  Γ 40 , 2 A 3 ∗  cm − 3 (8)is the wind particle density at the deceleration radius (eq .  [5]): ( s  = 2)  r 0  = 4 . 0 × 10 15 E  53 Γ − 20 , 2 A − 1 ∗  cm  .  (9)The usual notation  C  n  = 10 − n C   is used throughout this work.Note that, for the referencevaluesused here, the decelerationradius in the wind model is 1.5 orders of magnitude lower thanthat for a homogeneous external medium: ( s  = 0)  r 0  = 1 . 3 × 10 17 E  1 / 353  Γ − 2 / 30 , 2  n − 1 / 3 ∗ , 0  cm  ,  (10)  3and even smaller for higher initial fireball Lorentz factors orslower winds (i.e .  a larger parameter  A ∗ ). If GRBs are due tointernal shocks occurring in unstable relativistic fireballs (Rees& M´esz´aros 1994, Paczy´nski & Xu 1994, Piran 1999), then the external shock resulting from the interaction of the fireball withthe pre-ejected (non-relativistic) wind may occur before the in-ternal shocksareover. Inthis case successiveinternalcollisionsoccurwhenfasterpartsoftheejectacatchupwith thedecelerat-ing leading edge of the fireball, a scenario suggested within the s  = 0  modelby Fenimore& Ramirez-Ruiz(2000),butwhich ismore likely to happen if the external medium is the gas ejectedby a massive star. The GRB itself would then exhibit the erraticvariability characteristic of internal shocks until a time of theorder r 0 c Γ 20 ≃ 10 E  53 Γ − 40 , 2 A − 1 ∗  s ,  (11)(which has a strong dependence on  Γ 0 ), after which there maybe significant emission from internal shocks on the outermostpart of the fireball and from the external shock that plowsthrough the external gas. The former mechanism generatespulses of increasing duration as the fireball expands, while thelater leads to a continuous emission.The time  T   when the observer receives a photon emittedalong the line of sight toward the fireball center can be calcu-lated by integrating d T   = (1 − β  )d t  = 12d t Γ 2  ,  (12)where  β   is the shocked fluid speed and  t  =  r/c  is the timemeasured in the laboratory frame. Approximating the solu-tion given in equation (4) with  Γ = Γ 0  for  x <  1  and Γ =  x − (3 − s ) / 2 Γ 0  for  1  < x < x nr , one obtains T   =  T  0 ( x 4 − s + 3 − s )  , T  0  ≡  12(4 − s ) r 0 c Γ 20 .  (13)Note that  T   given in equation (13) is the earliest time a photonemitted by the remnant at time  t  can reach the observer. Pho-tons emitted by the fluid moving at an angle  θ  = 1 / Γ  off thecenter–observe axis arrive at  T   =  t (1 − cos θ ) =  t/ 2Γ 2 , whichis a factor  4 − s  larger than the time corresponding to  θ  = 0 : T   =  t/ 2(4 − s )Γ 2 .From equation (13)  r ( T  )  can be found and then substitutedin the expressions for  Γ( r )  and  n ( r )  to obtain these quantitiesas a function of the observer time. For the power-law phase theresults are ( s  = 0) Γ( T  ) = 6 . 3  E  1 / 853  n − 1 / 8 ∗ , 0  T  − 3 / 8 d  ,  (14) ( s  = 0)  r ( T  ) = 8 . 2 × 10 17 E  1 / 453  n − 1 / 4 ∗ , 0  T  1 / 4 d  cm  ,  (15) n  =  n ∗  (constant) for a homogeneous external medium and ( s  = 2) Γ( T  ) = 7 . 9  E  1 / 453  A − 1 / 4 ∗  T  − 1 / 4 d  ,  (16) ( s  = 2)  r ( T  ) = 6 . 4 × 10 17 E  1 / 253  A − 1 / 2 ∗  T  1 / 2 d  cm  ,  (17) ( s  = 2)  n ( T  ) = 0 . 73  E  − 153  A 2 ∗ T  − 1 d  cm − 3 ,  (18)for an external wind,  T  d  being the observer time measured indays. Note that, at least for the scaling values chosen here,  Γ , r , and  n  have about the same values at  T   = 1  day in both mod-els. Also note that the above quantities (and thus the afterglowemission) are independent of the fireball initial Lorentz factor Γ 0 . 3.  BREAK FREQUENCIES Within the synchrotron emission model there are three ex-pected breaks in the afterglow spectrum:  ( i )  an injection break,at the synchrotron frequency  ν  i  at which the bulk of the elec-trons injected by the shock front radiate,  ( ii )  a cooling break,at the synchrotron frequency  ν  c  of electrons whose radiativecooling time equals the expansion timescale, and  ( iii )  an ab-sorption break, at  ν  a  below which the synchrotron photons areabsorbedby electronsin free-freetransitions in a magneticfield(synchrotron self-absorption).The break frequencies can be calculated if the distribution of the injected electrons and the strength of the magnetic field areknown. The distribution of the injected electrons is assumedto be  N  i ( γ  )  ∝  γ  −  p starting from a minimum random Lorentzfactor given by γ  i  =  m  p m e ε e (Γ − 1)  ,  (19)where m e  is the electron mass. The energy carried by this elec-tron distribution is a fraction  p − 1  p − 2 ε e  of the total internal energy.The post-shock magnetic field strength in the co-moving frameis given by B 2 8 π  =  ε B m  p c 2 n ′ e (Γ − 1) = 4 ε B m  p c 2 n ( r )(Γ − 1)  Γ + 34   , (20)where  ε B  is the fractional energy carried by the magnetic fieldand  n ′ e  is the co-moving frame electron density behind theshock front. Equations (19) and (20) are based on that the in- ternal to rest-mass energy density ratio in the shocked fluid is Γ − 1 ; the derivation of the latter equation also used that the co-moving particle density is  4Γ + 3  times larger than that aheadof the shock.3.1.  Injection Break  Using the relativistic Doppler factor  2Γ  correspondingto themotion of the source toward the observer (i.e . θ  = 0 ), the syn-chrotron emission from a power-law distribution of electronspeaks at the observer frame frequency ν  i  = 3 x  p 2 πem e cγ  2 i B Γ = 8 . 4 × 10 6 x  p γ  2 i B Γ Hz  ,  (21)wherethe factor x  p  is calculatedin Wijers & Galama (1999)forvarious values of the electron index  p . We shall use  x  p  = 0 . 52 ,which is strictly correct only for  p  = 2 . 5 . With the aid of equa-tions (14), (16), (18), (19), and (20) one obtains: ( s  = 0)  ν  i  = 0 . 92 × 10 13 E  1 / 253  ε 2 e, − 1 ε 1 / 2 B, − 2 T  − 3 / 2 d  Hz  , (22) ( s  = 2)  ν  i  = 1 . 9 × 10 13 E  1 / 253  ε 2 e, − 1 ε 1 / 2 B, − 2 T  − 3 / 2 d  Hz  . (23)Note that  ν  i  has the same scalings with the model parametersfor  s  = 0  and  s  = 2 . The ratio of the two frequencies is   1772 .  43.2.  Cooling Break  The relativistic electrons cool radiatively through syn-chrotron emission and IC scatterings of the synchrotron pho-tons on a co-moving frame timescale t ′ rad ( γ  ) = t ′ sy ( γ  ) Y   + 1 = 6 πY   + 1 m e cσ e 1 γB 2  ,  (24)where  t ′ sy ( γ  )  is the co-moving frame synchrotron coolingtimescale of electrons of Lorentz factor  γ  ,  Y   is the Comptonparameter, and  σ e  the cross-section for electron scatterings 1 .Using equation (20), the Lorentz factor of the electrons thatcool radiatively on a timescale equal to the remnant age t ′ = 1 c    d r Γ = 25 − src Γ  (25)can be written as γ  c  = 3 π (5 − s ) Y   + 1 m e c 2 σ e Γ B 2 r  = 77(5 − s )( Y   + 1) ε B 1 n Γ r 18 ,  (26)with  n  in  cm − 3 .The observer-frame frequency ν  c  of the cooling break is ( s  = 0)  ν  c  = 3 . 7 × 10 14 E  − 1 / 253  n − 1 ∗ , 0 ( Y  +1) − 2 ε − 3 / 2 B, − 2 T  − 1 / 2 d  Hz  , (27) ( s  = 2)  ν  c  = 3 . 5 × 10 14 E  1 / 253  A − 2 ∗  ( Y   +1) − 2 ε − 3 / 2 B, − 2 T  1 / 2 d  Hz  . (28)Hereafter we shall use the terminology “radiative electrons”for the case where the  γ  i -electrons cool mostly through emis-sion of radiation (i.e . t ′ rad ( γ  i )  < t ′ and  γ  c  < γ  i ), and we shallreferto “adiabaticelectrons”ifthe γ  i -electronscoolmostlyadi-abatically (i.e . t ′ rad ( γ  i )  > t ′ and  γ  i  < γ  c ).3.2.1.  Compton Parameter and Electron Distribution For the calculation of the Compton parameter  Y  , we takeinto account only one up-scatteringof the synchrotronphotons.Multiple IC scatterings of the same photon have an importanteffect on the electron cooling only if the Y   parameter for singlescatterings is above unity. As shown in  § 3.2.2 and  § 3.2.3 thisoccurs if   1) ε B  < ∼ 10 − 2 ε e, − 1  and if   2)  T < T  y , where  T  y  is thetime when  Y   falls below unity.Using equations (19) and (20), it can be shown that IC scatterings of order higher than two are suppressed bythe Klein-Nishina effect. For adiabatic electrons, a secondIC scattering occurs in the Thomson regime if   3 a )  T  > ∼ 1  E  1 / 353  n − 1 / 9 ∗ , 0  ε 20 / 9 e, − 1 ε 2 / 9 B, − 4  d  for  s  = 0 , and if   3 b )  T  > ∼ 1  E  1 / 253  A − 1 / 4 ∗  ε 5 / 2 e, − 1 ε 1 / 4 B, − 4  d  for  s  = 2 . If a second IC scatteringis ignored in these cases then a higher energy component peak-ing around1 GeV is left out, otherwisethe synchrotronand firstIC emissions remain unaltered  if   the electrons are adiabatic.The effect of second order up-scatterings is more importantwhen electrons are radiative, i.e .  for  4)  T < T  r  with  T  r  calcu-lated in § 3.2.2, as in this case it reduces the intensity of the syn-chrotron and first stage IC components. With the aid of equa-tions (19) and (20) it can be shown that for  s  = 0  a secondup-scatteringoccurs in the Thomsonregime and at T >  10 − 2 d if   5 a )  n > ∼  25  E  − 7 / 1153  ε − 10 / 11 e, − 1  ε − 2 / 11 B, − 3  cm − 3 , while for  s  = 2 the second IC emission is not suppressed by the Klein-Nishinaeffect if   5 b )  T  > ∼ 0 . 05  E  − 1 / 353  A 11 / 6 ∗  ε 5 / 6 e, − 1 ε 2 / 3 B, − 3  d .Concluding,secondstageup-scatteringscanbeignorediftheset of conditions  1) , 2) , 3)  or  1) , 4) , 5)  are not simultaneouslysatisfied. For a single up-scattering, the Compton parameter is Y   = 43    N  e γ  2 d τ  e  = 43 τ  e    N  e γ  2  N  e ( γ  )d γ ,  (29)where  N  e ( γ  )  is the normalized electron distribution and  τ  e  isthe optical thickness to electron scattering, given by τ  e  = 14 πσ e m ( r ) m  p r 2  = 13 − sσ e nr .  (30)If the injected  γ  i -electrons cool faster than the timescale of theirinjection,then γ  c  givenbyequation(26)isthetypicalelec-tron Lorentz factor in the remnant, and the electron distributionin the shocked fluid can be approximated by  N  ( r ) e  ∝   γ  − 2 γ  c  < γ < γ  i γ  − (  p +1) γ  i  < γ  ,  (31)with  p >  2 . In the opposite case most electrons have a randomLorentz factor  γ  i , and the electron distribution is  N  ( a ) e  ∝   γ  −  p γ  i  < γ < γ  c γ  − (  p +1) γ  c  < γ  .  (32)3.2.2.  Radiative Electrons For  γ  c  ≪ γ  i  equations (29) and (31) lead to Y  r  = 43 γ  i γ  c τ  e  .  (33)Substituting  γ  c  with the aid of equation (26), one obtains Y  r ( Y  r  + 1) = 5 − s 8(3 − s ) n ′ e m e c 2 γ  i B 2 / 8 π  = 5 − s 8(3 − s ) ε e ε B ,  (34)wherewe usedequations(19)and (20). Thereforethe Compton parameter during the electron radiative phase is Y  r  = 12    5 − s 2(3 − s ) ε e ε B + 1 − 1   .  (35)Hence the electron cooling is dominated by IC scatterings(i.e . Y  r  >  1 ) for  ε B  <  ˜ ε B , where ˜ ε B  = 5 − s 16(3 − s ) ε e  .  (36)Note that  Y  r  is time-independent. Therefore the  ν  c  given byequations (27) and (28) decreases with time for  s  = 0  and in-creases in the  s  = 2  model. Thus, for observations made at afixed frequency,the electrons emitting at that frequencychangetheir cooling regime from adiabatic to radiative in the case of ahomogeneous external gas, and from radiative to adiabatic foran external wind (Chevalier & Li 2000). 1 The up-scattering of the  ν  i  synchrotron photons on the  γ  c - and  γ  i -electrons occurs in the Thomson regime for  T >  10 − 3 day, i.e .  the scattering cross-sectionin equation (24) is not reduced by the Klein-Nishina effect  5With theaidofequations(19),(20),and(24),it canbeshown that the electrons are radiative if  nr Γ 2 >  3(5 − s )32( Y  r  + 1)( m e /m  p ) 2 σ e ε e ε B = 4 . 2 × 10 16 ε e ε B 5 − sY  r  + 1 cm − 2 , (37)which, with the further use of equations (14) – (18), leads to the conclusion that the electrons are radiative until the observertime  T  r  given by ( s  = 0)  T  r  = 0 . 025  E  53 n ∗ , 0 ( Y  r  + 1) 2 ε 2 e, − 1 ε 2 B, − 2  day  , (38) ( s  = 2)  T  r  = 0 . 23  A ∗ ( Y  r  + 1) ε e, − 1 ε B, − 2  day  .  (39)3.2.3.  Adiabatic Electrons For  γ  i  ≪  γ  c  equations (29) and (32) give the Compton pa- rameter Y  a  = 43  τ  e ×   γ   p − 1 i  γ  3 −  pc  2  < p <  3 γ  2 i  3  < p .  (40) Case 1:  2  < p <  3 . By substituting equations (19) and (26) in equation (40), one obtains Y  a ( Y  a  + 1) 3 −  p = F   p ( T  ) ≡ c s (  p ) ε  p − 1 e  ε  p − 3 B  ( n Γ 2 r 18 )  p − 2 , (41)where  log c s (  p ) = (3 −  p )log(5 − s ) − log(3 − s )+1 . 4  p − 3 . 7 .The Compton parameter can be obtained by solving numeri-cally the above equation. For analytical purposes, one can ap-proximate Y  a  = F   p  for F   p  <  1 , in which case the IC losses areless important, and Y  a  = F  14 − p  p  for  F   p  >  1  .  (42)In the latter case the IC scatterings affect the electron cooling.Note that the quantity  n Γ 2 r  in equation (41) decreases withtime. Thus, for  ε B  <  ˜ ε B , the Compton parameter  Y  a  is aboveunity until a time  T  y  which can be determined by substitutingequations (14) – (18) in (42): ( s  = 0)  T  y  = 10 8 − 3 pp − 2 E  53 n ∗ , 0 ε 2 p − 1 p − 2 e, − 1  ε − 2 3 − pp − 2 B, − 3  day  ,  (43) ( s  = 2)  T  y  = 10 4 . 9 − 1 . 6 pp − 2 A ∗ ε p − 1 p − 2 e, − 1 ε − 3 − pp − 2 B, − 3  day  .  (44)Thus, for  ε B  <  ˜ ε B  and  T  r  < T < T  y , the Compton param-eter determines the evolution of the cooling break frequency(eqs .  [27] and [28]): ν  c ( s =0) = 10 15+ 2 . 5 p − 5 . 54 − p  E  − p 2 53  n − 2 ∗ , 0 ε − 2(  p − 1) e, − 1  ε − p 2 B, − 3 T  3 p − 82 d  14 − p Hz  , (45) ν  c ( s =2) = 10 15+ 2 . 2 p − 5 . 54 − p E  12 53  A − 4 ∗  ε − 2(  p − 1) e, − 1  ε − p 2 B, − 3 T  3 p − 42 d  14 − p Hz  . (46)Note that for a homogeneous medium ( s  = 0 ) and  8 / 3  < p < 3 , the cooling break frequency increases with time, unlike thedecreasing behavior it has for  T < T  r . Case 2:  p >  3 . This case is treated here for completeness,as there are no afterglows for which such a steep electron indexhas been found. Equations (19), (30), and (40) lead to Y  a  = 33 − sε 2 e n Γ 2 r 18  .  (47)For  ε B  <  ˜ ε B  the Compton parameter is above unity until ( s  = 0)  T  y  = 0 . 11  E  53 n ∗ , 0 ε 4 e, − 1  day  ,  (48) ( s  = 2)  T  y  = 0 . 87  A ∗ ε 2 e, − 1  day  .  (49)For  ε B  <  ˜ ε B  and  T  r  < T < T  y , the evolution of the coolingbreak frequency is ( s  = 0)  ν  c  = 1 . 1 × 10 17 E  − 3 / 253  n − 2 ∗ , 0 ε − 4 e, − 1 ε − 3 / 2 B, − 3 T  1 / 2 d  Hz  , (50) ( s  = 2)  ν  c  = 1 . 5 × 10 16 E  1 / 253  A − 4 ∗  ε − 4 e, − 1 ε − 3 / 2 B, − 3 T  5 / 2 d  Hz  . (51)Note that in this regime  ν  c  increases with time in both models.3.3.  Absorption Break  The synchrotron self-absorption frequency  ν  a  can be calcu-lated with the aid of equation (6.50) from Rybicki & Light-man (1979). With the notations  γ   p  = min( γ  i ,γ  c ) ,  ν   p  =min( ν  i ,ν  c ) , and  ν  0  = max( ν  i ,ν  c )  it can be shown that opticalthickness to synchrotron self-absorption can be approximatedby τ  ab ( ν  ) ≃ 5  e Σ Bγ  5  p ×   ( ν/ν   p ) − 5 / 3 ν < ν   p ( ν/ν   p ) − ( q +4) / 2 ν   p  < ν < ν  0 ,  (52)where  Σ = (3 − s ) − 1 nr  is the remnant electron column den-sity,  q   = 2  for radiative electrons ( γ  c  < γ  i ),  q   =  p  for adiabaticelectrons ( γ  i  < γ  c ).3.3.1.  Radiative Electrons Equations (20), (26), and (52) give the optical thickness at the cooling break frequency τ  c  = 53 − senrBγ  5 c = 1 . 1 × 10 − 3 (3 − s )(5 − s ) 5 ( Y  r +1) 5 ε 9 / 2 B  n 11 / 2 Γ 4 r 618  . (53)For  s  = 0  equations (14), (15), and (53) lead to ( s  = 0)  τ  c  = 0 . 11  E  253 n 7 / 2 ∗ , 0  ( Y  r  + 1) 5 ε 9 / 2 B  .  (54)For  τ  c  <  1  the optical thickness to synchrotron self-absorptionis unity at  ν  a  given by  ν  a  =  ν  c τ  3 / 5 c  : ( s  = 0)  ν  a  = 6 . 5 × 10 9 E  7 / 1053  n 11 / 10 ∗ , 0  ( Y  r +1) ε 6 / 5 B, − 1 T  − 1 / 2 d  Hz  . (55)For  s  = 2  equations (16) – (18), and (53) give ( s  = 2)  τ  c  = 0 . 44  E  − 3 / 253  A 7 ∗ ( Y  r  + 1) 5 ε 9 / 2 B  T  − 7 / 2 d  .  (56)For  T > T  a  we have  τ  c  <  1  and  ν  a  < ν  c : ( s  = 2)  ν  a  = 1 . 4 × 10 12 E  − 2 / 553  A 11 / 5 ∗  ( Y  r +1) ε 6 / 5 B, − 2 T  − 8 / 5 d, − 2  Hz  . (57)
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