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The Luminosity - E_p Relation within Gamma--Ray Bursts and Implications for Fireball Models

a r X i v :a s t r o -p h /0403397v 1 17 M a r 2004

Draft version February 2,2008

Preprint typeset using L A T E X style emulateapj v.04/03/99

THE LUMINOSITY–E P RELATION WITHIN GAMMA–RAY BURSTS AND IMPLICATIONS FOR

FIREBALL MODELS

E.W.Liang 1,2,3,Z.G.Dai 1,and X.

F.Wu 1

1Astronomy

Department,Nanjing University,Nanjing 210093,P.R.China;Email:ewliang@https://www.wendangku.net/doc/3b3999899.html, 2Physics Department,Guangxi University,Nanning 530004,P.R.China

3National Astronomical Observatories/Yunnan Observatory,Chinese Academy of Sciences,Kunming 650011,

P.R.China

Draft version February 2,2008

ABSTRACT

Using a sample of 2408time–resolved spectra for 91BATSE GRBs presented by Preece et al.,we

show that the relation between the isotropic–equivalent luminosity (L iso )and the peak energy (E ′

p )

of the νF νspectrum in the cosmological rest frame,L iso ∝E ′

2p ,holds within these bursts,and also holds among these GRBs,assuming that the burst rate as a function of redshift is proportional to the star formation rate.The possible implications of this relation for the ?reball models are discussed by

de?ning a parameter ω≡(L iso /1052erg s ?1)0.5/(E ′

p /200keV).It is found that ωis narrowly clustered in 0.1?1.We constrain some parameters for both the internal shock and external shock models from the requirement of ω～0.1?1,assuming that these model parameters are uncorrelated.The distributions of the parameters suggest that if the prompt gamma–rays are produced from kinetic–energy–dominated internal shocks,they may be radiated from a region around R ～1012?1013cm (or Lorentz factor ～130?410)with a combined internal shock parameter ζi ～0.1?1during the prompt gamma–ray phase,which are consistent with the standard internal shock model;if the prompt gamma–rays of these GRBs are radiated from magnetic–dissipation–dominated external shocks,the narrow cluster of ωrequires σ～1?470,Γ～216?511,E ～1051?1054ergs,n ～0.5?470cm ?3,and ζe ～0.36?3.6,where σis the ratio of the cold-to-hot luminosity components,Γthe bulk Lorentz factor of the ?reball,E the total energy release in gamma–ray band,n the medium number density,and ζe a combined external shock parameter,which are also in a good agreement with the ?ttings to the afterglow data.These results indicate that both the kinetic–energy–dominated internal shock model and the magnetic–

dissipation–dominated external shock model can well interpret the L iso ∝E ′

2p relation and the value of ω.

Subject headings:gamma rays:bursts—gamma rays:observations—methods:statistical

1.INTRODUCTION

Gamma–ray bursts (GRBs)are now believed to be pro-duced by jets powered by central engines with a stan-dard energy reservoir at cosmological distances (see se-ries reviews by Fishman &Meegan 1995;Piran 1999;van Paradijs et al.2000;Cheng &Lu 2001;M′e sz′a ros 2002;Zhang &M′e sz′a ros 2003).

The most impressive features of GRBs are the great di-versities of their light curves and spectral behaviors,and extremely large luminosities.These spectra are well ?t-ted by the Band function (Band et al.1993).However,the radiation mechanism at work during the prompt phase remains poorly understood.Although the spectral be-havior and the luminosity are dramatically di?erent from burst to burst,the isotropic–equivalent luminosity,L iso ,(or isotropic–equivalent energy radiated by the source,

E iso ),and E ′

p ,the peak energies of νF νspectrum in the rest frame among GRBs,obey an empirical relation of

L iso ∝E ′

2p (Amati et al.2002;Yonetoku et al.2003;Sakamoto et al.2004;Lamb et al.2003a,b,c).This relation was revisited in standard synchrotron/inverse–Compton/synchro–Compton models (Zhang &M′e sz′a ros 2002).Recently,Sakamoto et al.(2004)and Lamb et al.(2003a,b,c)pointed out that HETE–2observations not only con?rm this correlation,but also extend it to the

population of X–ray ?ashes,which are thought to be a low energy extension of typical GRBs (Heise et al.2001,Kippen et al.2003).Based on this relation,Atteia (2003)also constructed a simple redshift indicator for GRBs.One may ask:whether or not this relation holds in any segment within a GRB?The answer remains un-known.If the answer is positive,combining the results mentioned above,one might suggest that this relation is a universal law during the prompt gamma–ray phase,and presents some constraints on ?reball models.In this Let-ter,we investigate this https://www.wendangku.net/doc/3b3999899.html,ing a sample of 2408time–resolved spectra for 91BATSE GRBs presented by Preece et al.(2000),we show that this relation holds within these bursts,and also holds among these GRBs,assuming that the burst rate as a function of redshift is proportional to the star formation rate.We suggest that both the kinetic–energy–dominated internal shock model and the magnetic–dissipation–dominated external shock model may well in-terpret this relation.

Throughout this work we adopt H 0=65km s ?1Mpc ?1,?m =0.3,and ?Λ=0.7.

2.THE L ISO ?E ′

P RELATION WITHIN A GRB

Within a GRB,the relation between L iso and E ′

p is equivalent to a relation between the observed ?ux (F )and peak energy,E p .Thus,we examine whether or not both

F and E p follow a relation of F∝E2p.The time–resolved spectral catalog presented by Preece et al.(2000)includes 156long,bright GRBs.Four spectral models were used in their spectral?ttings.Di?erent models might present di?erent?tting results.Among156GRBs,91GRBs were ?tted by the Band function(Band et al.1993).We only include these GRBs into our analysis.There are2408 time–resolved spectra for these GRBs.In our analysis,the values of E p are taken from this catalog.The data used for spectral?ttings were observed by di?erent BATSE de-tectors.Nominal energy coverage is25to1800keV,with some small variations between these detectors.The?uxes presented in the catalog are in an energy band correspond-ing to the detectors.Hence,the values of F in our analysis are not the?uxes presented in the catalog,but are the inte-grated?uxes in energy band30–10000keV(e.g.,Yonetoku et al.2003)derived from the model spectral parameters in the catalog.

We evaluate the relation of F∝E2p within a GRB by the linear correlation coe?cient of the two quantities,log F and log E2p.We calculate the linear correlation coe?-cients(r)and chance probabilities(p)for each burst with the Spearman rank correlation analysis method.The dis-tribution of r is presented in Figure1.We?nd that about 75%GRBs exhibit a strong correlation between the two quantities with r>0.5and p<0.0001.We illustrate16 cases in Figure2.These results show that this relation holds within these GRBs.

To examine whether or not this relation holds among these GRBs,we assume that the redshift distribution for these GRBs is same as that presented by Bloom(2003). Bloom(2003)assumed the burst rate as a function of red-shift is proportional to the star formation rate as a function of redshift,and presented the observed redshift distribu-tion incorporating with observational biases(SFR1model is used in this work,see Porciani&Madau2001).We derive a value of redshift for a given GRB from this distri-bution by a simple Monte Carlo simulation method.To do so,we?rst derive the accumulated probability distribution of the Bloom’s redshift distribution,P(z)(0

3.IMPLICATIONS FOR FIREBALL MODELS

The above results well suggest that the relation of L iso∝E′2p remains within a GRB.This implies that the relation is independent of the temporal evolution of a?reball.This might provide strong constraints on?reball models.We de?ne a quantity,ω,to discuss these possible constraints, which is given by

ω=

L1/2

iso,52

The parameters in Eqs.(2)and(3)seem to be un-correlated,although we do not know if this is really the case.We simply assume that they are uncorrelated, and constrain their distributions from the requirement of ω～0.1?1.We suggest that these parameters should be clustered in the same range as that ofω.Thus,we de-rive R～1012?1013cm andζi～0.1?1for the internal shock model,implying that most of the gamma–rays are radiated from a region around R～1012?1013cm with similar shock–acceleration and radiation mechanisms dur-ing the prompt gamma–ray phase.Since R?2Γ2cδt v～0.6×1013Γ22.5δt v,?3cm,where c is the speed of light,and t v,?3the variability timescale in units of10?3second,we obtainΓ～130?410.These parameters are consistent with the standard internal shock model.For the exter-nal shock model,we deriveσ～1?470,Γ～216?511,ζe～0.36?3.6,E～1051?1054ergs,and n～0.5?470 cm?3.The distributions of these parameters are in a good agreement with the?ttings to the afterglows(Panaitescu &Kumar2001).

The above results indicate that both the lowσinternal shock model and the highσexternal shock model can well interpret the L∝E′2p relation and the value ofω.

4.CONCLUSIONS AND DISCUSSION

Using a sample of2408time–resolved spectra for91 long,bright GRBs presented by Preece et al.(2000), we show that the L iso～E′2p relation holds within these BATSE bursts,and this relation also holds among these GRBs by assuming that the burst rate as a function of redshift is proportional to the star formation rate.

We discuss possible implications of this relationship for the?reball models by de?ning a parameterω≡(L iso/1052erg s?1)0.5/(E′p/200keV).It is found thatωis not in?uenced by the Doppler–boosting e?ect,and it is de-termined by the gamma–ray emission region and shock pa-rameters in the kinetic–energy–dominated internal shock model or determined by the parameters of both the shock and the environment in the magnetic–dissipation–dominated external shock model.We derive the distri-butions of some parameters for both the internal shock model and the external shock model from the requirement ofω～0.1?1.We suggest that if the prompt gamma–rays are produced from a kinetic–energy–dominated in-ternal shock,they may be radiated from a region around R～1012?1013cm(or Lorentz factor～130?410)with an internal shock parameterζi～0.1?1,which is consis-tent with the standard internal shock model;if the prompt gamma–rays of these GRBs are radiated from magnetic–dissipation–dominated external shocks,theω～0.1?1 requiresσ～1?470,Γ～216?511,ζe～0.36?3.6, E～1051?1054ergs,and n～0.5?470cm?3.Please note that the distributions for these model parameters for both the internal and external shock models are based on the assumption that they are uncorrelated.Although these parameters seem to be uncorrelated,we do not know if it is really the case.If these parameters are correlated dur-ing prompt gamma–ray phase,these distributions are not valid.

We would like to thank the referee,Dr.Don Lamb, for his valuable comments,which have enabled us to im-prove greatly the manuscript.This work is supported by the National Natural Science Foundation of China (grants10233010and10221001),the National973Project (NKBRSF G1*******),the Natural Science Foundation of Yunnan(2001A0025Q),and the Research Foundation of Guangxi University.

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0.00.10.20.30.40.50.60.70.80.9 1.0

N u m b e r

Fig.1.—Distribution of the linear coe?cients for log F –log E 2p

2.2 2.4 2.6 2.8

3.0 3.2

1.6 1.8

2.0 2.2 2.4

1.6 1.8

2.0

2.2

1.8

2.1 2.4 2.7

3.0

2.0 2.4 2.8

3.2

2.0 2.2 2.4 2.6 2.8 2.0 2.2 2.4 2.6 2.8

1.8

2.1 2.4 2.7

3.0

1.8

2.1 2.4 2.7

1.8

2.1 2.4 2.7

3.0

2.2 2.4 2.6 2.8

3.0 3.2

2.4 2.6 2.8

3.0 3.2

2.0 2.1 2.2 2.3 2.4

1.6 1.8

2.0 2.2

2.4 2.0 2.2 2.4

2.6 2.1 2.4 2.7

3.0

L o g F (e r g c m

-2

-1

)

Log E p

(keV)

Fig.2.—The observed ?ux as a function of E p for 16GRBs.

1001000

(e r g s -1

)

E p

(1+z) (keV)

Fig.3.—The L iso ,52as a function of E p(1+z)for 2408GRB spectra.The solid line is L iso ,52=10?5.1×[E p (1+z )]2.

0.010.1110

100

120

140

160

N u m b e r

Fig.4.—Distribution of ωfor 2408spectra.

-0.10-0.050.000.050.100.150.200.25

N u m b e r

Fig.5.—Distribution of k for 91GRBs.