We use a simple pulse model to investigate the origin of the time lagluminosity relation (LLR) discovered by Norris et al.(ApJ, 534 (2000) 248). We show that, at least for single pulse bursts which satisfy both the 

hardness-intensity and the hardness-fluence correlations, the LLR can be simply obtained as a consequence of the Amati relation

Hafizi, M.

Mochkovitch, R.

English

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DOI 10.1393/ncc/i2005-10049-9IL NUOVO CIMENTO Vol. 28 C, N. 3 Maggio-Giugno 2005The time lag-luminosity relation: A consequenceof the Amati relation?(∗)R. Mochkovitch(1) and M. Hafizi(2)(1) Institut d’Astrophysique de Paris - 98 bis boulevard Arago, 75014 Paris, France(2) Faculty of Natural Sciences, Tirana University - Tirana, Albania(ricevuto il 23 Maggio 2005; pubblicato online il 9 Settembre 2005)Summary. — We use a simple pulse model to investigate the origin of the time lag-luminosity relation (LLR) discovered by Norris et al.(ApJ, 534 (2000) 248). We showthat, at least for single pulse bursts which satisfy both the hardness-intensity andthe hardness-fluence correlations, the LLR can be simply obtained as a consequenceof the Amati relation.PACS 98.70.Rz – γ-ray sources; γ-ray bursts.PACS 95.30.Gv – Radiation mechanisms.PACS 01.30.Cc – Conference proceedings.1. – IntroductionThe problem of the distance to gamma-ray bursts remained unsolved until the dis-coveries of the afterglows by Beppo-SAX. With now ∼ 40 measured redshifts, it hasbeen possible to calibrate relations linking absolute burst outputs (luminosity or totalradiated energy) to quantities directly available from the observations in gamma-rays.A Cepheid-like relation between variability and luminosity was for example proposed byReichart et al. [1]. More recently, Atteia [2] used the Amati relation [3] to introduce“pseudo-redshifts” which could for example be useful to rapidly identify high-z GRBsfrom their gamma-ray properties alone. Here we concentrate on the time lag-luminosityrelation discovered by Norris et al. [4]. The lags were computed by Norris et al.by cross-correlating burst profiles in BATSE bands 1 (20–50 keV) and 3 (100–300 keV). Theyfound that high-luminosity GRBs exhibit small lags and proposed a power law relationL = 1.3× 1053 (∆t13/0.01 s)−1.14 erg s−1(∗) Paper presented at the “4th Workshop on Gamma-Ray Burst in the Afterglow Era”, Rome,October 18-22, 2004.c© Societa` Italiana di Fisica 315316 R. MOCHKOVITCH and M. HAFIZIbetween the lag ∆t13 and the luminosity. The origin of the LLR was then investigatedby Kocevski and Liang [5] who found that the observed lags were a consequence of theburst spectral evolution. In this contribution we go a step further and show that theLLR must indeed be satisfied, at least by single pulse GRBs which follow the hardness-intensity (HIC [6]), the hardness-fluence (HFC [7]) and the Amati correlations.2. – The pulse modelWe suppose that the number of photons produced during the pulse per unit time andunit energy can be written N (E, t) = A(t)B [E/Ep(t)] where Ep(t) is the instantaneousvalue of the peak energy and B(x) the spectrum shape, assumed to be two smoothlyconnected power laws. The function B depends on x = E/Ep only if both the lowand high energy slopes α and β of the spectrum do not vary with time during pulseevolution. We make this rather crude approximation which allows a simple estimate ofthe bolometric photon rateNb(t) =∫ ∞0N (E, t)dE = A(t)Ep(t)B0,where B0 =∫∞0B(x)dx. We also suppose that Nb(t) and Ep(t) satisfy the HIC and theHFC after a time t0 during pulse decayEp(t) ∝ Nb(t)δ and Ep(t) ∝ e−aΦN(t)where ΦN(t) =∫ t0Nb(t′)dt′ is the photon fluence the photon fluence and a an exponentialdecay constant. Ryde and Svensson [8] have shown that this implies, for t > t0, thatNb(t) and Ep(t) behave asNb(t) =N01 + t−t0τ=N01 +Q(T − 1) , Ep(t) =E0(1 + t−t0τ)δ = E0(1 +Q(T − 1))δ ,where Q = t0/τ and T = t/t0. When t < t0 we adopt a parabolic form for Nb(t)Nb(t) = N0[(2 +Q)T − (1 +Q)T 2] ,i.e. Nb(t) and its time derivative are continuous at t = t0 (T = 1). Also, Nb(t) = 0 att = 0 and is maximum at Tmax = (2 + Q)/2(1 + Q). Since the maximum of hardnessgenerally precedes the maximum of count rate we tried for Ep(t < t0) different parabolicshapes having their maximum before Tmax but it appeared that a simple linear extensionof EpEp(t) = E0 [1 + δ Q(1− T )]gave essentially the same results. Expressions for Nb(t) and Ep(t) being known it ispossible to obtain the pulse shape Nij(t) in any band [Ei, Ej ]Nij(t) = Nb(t)Bij(Ep)B0 ,where Bij(Ep) =∫ xjxiB(x)dx with xi,j = Ei,j/Ep.THE TIME LAG-LUMINOSITY RELATION: A CONSEQUENCE OF THE AMATI RELATION? 317Fig. 1. – Lag-luminosity relation obtained with our model compared to the Norris et al. [4] datapoints. The squares and circles respectively correspond to lags calculated from profiles goingdown to 0.1 and 0.5 of the peak intensity.3. – The lag-luminosity relation3.1. Method of calculation. – We obtain the time lag-luminosity relation by using theAmati relation coupled to our pulse model. For a given pulse shape (fixed by the valuesof Q, δ and t0) we choose E0, the value of Ep at t = t0 and cross-correlate the profilesin BATSE bands 1 and 3 to get the time lags. We then compute the spectrum for thewhole pulse and determine its global Ep. When Ep is known the Amati relationEiso = 1052[Ep200 keV]2.17erggives the isotropic radiated energy Eiso =∫∞0Lb(t)dt, where Lb(t) is the bolometricluminosity. The bolometric luminosity readsLb(t) ∝ Nb(t)Ep(t)and the value of Eiso from the Amati relation fixes the proportionality constant whichfinally allows to compute the peak luminosity Lmax of the pulse.3.2. Results. – We present in fig. 1 the lag-luminosity relation for Q = 0.5, δ = 0.75,t0 = 2 s and z = 1. These values lead to a pulse lasting 10–15 s and the adopted spectralparameter δ is typical of the results obtained by Ryde and Svensson [8] (0.4 < δ < 1).The agreement with the data points [4] is satisfactory (for each of the bursts the Norris etal. [4] is satisfactory (for each of the bursts the squares and circles respectively correspondto lags calculated from profiles going down to 0.1 and 0.5 of the peak intensity).318 R. MOCHKOVITCH and M. HAFIZIWe have then considered the effects of a variation of the pulse parameters δ andQ on the lag-luminosity relation. It appears that changes of the spectral evolution (δ)and pulse shape (Q) do not strongly affect the LLR. Moreover, these quantities can inprinciple be determined from the data if it is of sufficient quality.But this does not mean that estimating burst distances with the LLR is an easytask. The curve in fig. 1 was obtained assuming a fixed redshift z = 1 but in practicethe redshift is just the quantity to be determined. Variations of redshift affect the LLRthroughK-corrections on the pulse profiles in different energy bands and time dilations ofthe lags. A method to correct the observed lags from these cosmological effects has beenproposed by Norris [9]. The method allows to fix the redshift via an iterative procedurewhen the modeled peak flux agrees with the observed one. A preliminary analysis inthe context of our model however shows that this method remains partially degenerate,especially for large lags where it appears difficult to distinguish between weak bursts atsmall z and bright ones at large z.4. – ConclusionWe have presented a simple pulse model to show that the lag-luminosity relationdiscovered by Norris et al. [4] can be considered as a consequence of the HIC, HFC andAmati relation. We have studied the sensitivity of the LLR relative to the parametersfixing the pulse shape and spectral evolution. Concerning the use of the LLR as a distanceindicator we believe that it is not accurate enough to yield the redshift of any specificburst but could still be useful for the analysis of large samples.REFERENCES[1] Reichart D., Lamb D. and Fenimore E. et al., ApJ, 552 (2001) 57.[2] Atteia J. L., A&A, 407 (2003) L1.[3] Amati L., Frontera F. and Tavani M. et al., A&A, 390 (2002) 81.[4] Norris J., Marani G. and Bonnell J., ApJ, 534 (2000) 248.[5] Kocevski D. and Liang E., ApJ, 594 (2003) 385.[6] Golenetskii S. V., Mazets E. P. and Aptekar R. L. et al., Nature, 306 (1983) 451.[7] Liang E. and Kargatis V., Nature, 381 (1996) 49.[8] Ryde F. and Svensson R., ApJ, 529 (2000) L13.[9] Norris J., ApJ, 579 (2002) 386.

The time lag-luminosity relation: A consequence of the Amati relation?

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The time lag-luminosity relation: A consequence of the Amati relation?

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