Gamma-ray burst afterglows have been observed for months or even years in a
few cases. It deserves noting that at such late stages, the remnants should
have entered the deep Newtonian phase, during which the majority of
shock-accelerated electrons will no longer be highly relativistic. To calculate
the afterglows, we must assume that the electrons obey a power-law distribution
according to their kinetic energy, not simply the Lorentz factor.Comment: Poster at the 4th workshop "Gamma-Ray Bursts in the Afterglow Era"
  (Rome, 2004), accepted for publication in the proceedings. 4 pages, with 3
  figures inserte

Cheng, K. S.

Huang, Y. F.

Lu, T.

English

arXiv

Gamma-ray burst afterglows have been observed for months or even years in a few cases. It is worth noting that at such late stages, the remnants should have entered the deep Newtonian phase, during which the majority of shock-accelerated electrons will no longer be highly relativistic. To calculate the afterglows, we must assume that the electrons obey a power law distribution according to their kinetic energy, not simply the Lorentz factor. © Società Italiana di Fisica.link_to_subscribed_fulltex

Lu, T

Huang, YF

Cheng, KS

HKU Scholars Hub

GRB afterglows: Deep Newtonian phase and its application

Gamma-ray burst afterglows have been observed for months or even years in a few cases. It is worth noting that at such late stages, the remnants should have entered the deep Newtonian phase, during which the majority of shockaccelerated electrons will no longer be highly relativistic. To calculate the afterglows, we must assume that the electrons obey a power law distribution according to their kinetic energy, not simply the Lorentz factor

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DOI 10.1393/ncc/i2005-10141-2IL NUOVO CIMENTO Vol. 28 C, N. 3 Maggio-Giugno 2005GRB afterglows: Deep Newtonian phase and its application(∗)Y. F. Huang(1), T. Lu(2) and K. S. Cheng(3)(∗∗)(1) Department of Astronomy, Nanjing University - Nanjing 210093, China(2) Purple Mountain Observatory, Chinese Academy of Sciences - Nanjing 210008, China(3) Department of Physics, the University of Hong Kong - Hong Kong, China(ricevuto il 23 Maggio 2005; pubblicato online il 14 Ottobre 2005)Summary. — Gamma-ray burst afterglows have been observed for months or evenyears in a few cases. It is worth noting that at such late stages, the remnantsshould have entered the deep Newtonian phase, during which the majority of shock-accelerated electrons will no longer be highly relativistic. To calculate the afterglows,we must assume that the electrons obey a power law distribution according to theirkinetic energy, not simply the Lorentz factor.PACS 95.30.Qd – Astrophysical magnetohydrodynamics and plasmas .PACS 97.60.Bw – Supernovae.PACS 97.60.Lf – Black holes.PACS 01.30.Cc – Conference proceedings.1. – IntroductionGamma-ray bursts (GRBs) have been recognized as the most relativistic phenom-ena in the Universe. In 1997, Wijers et al.once discussed GRB afterglows of the non-relativistic phase [1]. However, for quite a long period, many authors were obviouslybeclouded by the energetics of GRBs and emission in the non-relativistic phase was gen-erally omitted. In 1998, Huang et al.stressed the importance of the Newtonian phase forthe first time [2]. In fact, the Lorentz factor of GRB blastwave evolves as γ ≈ (200 –400)E1/851 n−1/80 t−3/8s in the ultra-relativistic phase. It is clear that the shock will no longerbe ultra-relativistic within tens of days. Today, the importance of non-relativistic phasehas been realized by more and more authors [3-12]. For example, in the famous caseof GRB 030329, the transition to the non-relativistic regime is believed to be detected,since its X-ray and radio afterglow light curves flattened achromatically at t ∼ 40–50day [11,12].Recently it was further noted that GRB afterglows may enter the deep Newto-nion phase typically in a few months [13], when the minimum Lorentz factor of shock-accelerated electrons (γe,min ∼ ξe(γ − 1)mp/me) becomes less than a few. At this stage,(∗) Paper presented at the “4th Workshop on Gamma-Ray Burst in the Afterglow Era”, Rome,October 18-22, 2004.(∗∗) e-mail: hyf@nju.edu.cn.c© Societa` Italiana di Fisica 415416 Y. F. HUANG, T. LU and K. S. CHENGFig. 1. – Evolution of the bulk Lorentz factor (γ) vs. radius for an isotropic fireball [15]. Thedashed line illustrates the famous Sedov solution for a Newtonian fireball. The solid line is plotaccording to Huang et al.’s generical dynamical model, which is correct in both the relativisticand the Newtonian phases [14]. The dash-dotted line shows the result of an older dynamicalmodel, which is not correct in the Newtonian phase.most electrons will cease to be ultra-relativistic and their distribution function needs tobe reconsidered [13].2. – ModelTo describe the deceleration of GRB ejecta, we use the refined generic dynamicalmodel proposed by Huang et al. [14], which is mainly characterized bydγdm= − γ2 − 1Mej + 	m+ 2(1− 	)γm.(1)Figure 1 shows clearly that this equation is applicable in both the ultra-relativistic phaseand the Newtonian phase. Detailed description concerning the overall dynamical evolu-tion of isotropic fireballs as well as collimated jets can be found in Huang et al. [14,16,17].Shock-accelerated electrons are usually assumed to distribute as dN ′e/dγe ∝ γ−pe .However, in the deep Newtonian phase, most electrons are non-relativistic. To calculateafterglows, the distribution function now needs to be revised as [13]dN ′e/dγe ∝ (γe − 1)−p.(2)Optical afterglows can then be calculated conveniently by integrating synchrotron emis-sion from those electrons with Lorentz factors above a critical value (γe,syn) [13].Detailed numerical results for isotropic fireballs and highly collimated conical or cylin-drical jets have been presented by Huang and Cheng [13]. Here we show some exemplarresults in fig 2. Figure 2a illustrates optical afterglows from isotropic fireballs. Notethat the deep Newtonian phase typically begins at about 107 s. The light curves steepenslightly after that. Our results are consistent with analytical solutions. On the contrary,afterglow light curves of conical jets (fig. 2b) flatten in the deep Newtonian phase, whichis also consistent with analytical solutions.GRB AFTERGLOWS: DEEP NEWTONIAN PHASE AND ITS APPLICATION 417Fig. 2. – R-band optical afterglows from isotropic fireballs (a, left panel) and conical jets (b, rightpanel) [13]. The black dot on each light curve indicates the moment when γe,min = γe,syn ≡ 5,and open circles mark the time when the bulk Lorentz factor γ = 2, 5 and 10, respectively.3. – ApplicationIt has been shown clearly that GRB ejecta enters the deep Newtonian phase typicallyin about 3 months. Afterglows in the deep Newtonian phase are thus very important,especially in the following three cases. Case 1, late afterglows. Optical afterglows fromsome GRBs have been detected for more than six months. Radio afterglows are detectableeven three years later. The deep Newtonian phase is unavoidable when such observationsare to be accounted for. Case 2, GRBs with a dense medium. For some GRBs, thedensity of circum-burst medium may be as large as 103 cm−3, or even 106 cm−3 insome rare cases. The GRB ejecta will then decelerate very rapidly and may enter thedeep Newtonian phase in less than 20 days. Case 3, fireballs with relatively small initialLorentz factor. This includes failed GRBs and the two-component jet model of GRBs,which will be discussed in more details below.Failed GRBs [18], or dirty fireballs as named by Dermer et al. [19], are relativisticfireballs with initial Lorentz factor γ0  100 – 1000. They cannot produce normal GRBs,but may give birth to X-ray flashes and contribute to orphan afterglows. The simplediscovery of orphan afterglows then does not necessarily mean that GRBs are highlyFig. 3. – Effects of the viewing angle on the afterglow from a two-component jet (a, left panel),and its fit to the optical afterglow of XRF 030723 (b, right panel) [23].418 Y. F. HUANG, T. LU and K. S. CHENGcollimated [18], although this was once regarded as a hopeful method for measuring thebeaming angle of GRBs [20, 21]. To judge whether an orphan afterglow comes from afailed GRB or a jetted but off-axis GRB, Huang et al. suggested that the most importantthing is to monitor the orphan for a relatively long period [18]. Obviously, the calculationof afterglows in the deep Newtonian phase is necessary in such studies.Recently it was proposed that some GRB jets may have two components: a centralnarrow ultra-relativistic outflow and an outer, wider, mildly relativistic ejecta [22, 23].This two-component jet model can potentially give a unified description for GRBs andX-ray flashes [23]: if our line of sight is within the narrow component, a normal GRB willbe observed; On the contrary, if the line of sight is outside the narrow component butwithin the wide component, an X-ray flash will be witnessed. In both cases, long-lastingafterglows can be detected. In such a model, since the outer ejecta is midly relativisticat the beginning, radiation in the deep Newtonian phase will be inevitably involved incalculating its afterglows. Afterglow behaviors of two-component jets have been studieddetailedly by Huang et al. [23]. Here, as an example, we illustrates the effects of theviewing angle on the optical light curves in fig 3a. Figure 3b shows clearly that the two-component jet model can give a perfect explanation for the observed optical afterglowsfrom the X-ray flash XRF 030723.∗ ∗ ∗This research was supported by the National Natural Science Foundation of China(10003001, 10221001, 10233010 and 10473023), the FANEDD (Project No: 200125), theNational 973 Project, and a RGC grant of Hong Kong SAR.REFERENCES[1] Wijers R., Rees M. J. and Me´sza´ros P., MNRAS, 288 (1997) L51.[2] Huang Y. F., Dai Z. G. and Lu T., A&A, 336 (1998) L69.[3] Livio M. and Waxman E., ApJ, 538 (2000) 187.[4] Frail D., Waxman E. and Kulkarni S. R., ApJ, 537 (2000) 191.[5] Dermer C. D., Bo¨ttcher M. and Chiang J., ApJ, 537 (2000) 255.[6] Dermer C. D. and Humi M., ApJ, 556 (2001) 479.[7] Piro L. et al., ApJ, 558 (2001) 442.[8] in’t Zand J. J. M. et al., ApJ, 559 (2001) 710.[9] Panaitescu A. and Kumar P., MNRAS, 350 (2004) 213.[10] Zhang B. and Me´sza´ros P., Int. J. Mod. Phys. A, 19 (2004) 2385.[11] Tiengo A., Mereghetti S., Ghisellini G., Tavecchio F. and Ghirlanda G., A&A,423 (2004) 861.[12] Frail D. A. et al., ApJ, 619 (2005) 994.[13] Huang Y. F. and Cheng K. S., MNRAS, 341 (2003) 263.[14] Huang Y. F., Dai Z. G. and Lu T., MNRAS, 309 (1999) 513.[15] Huang Y. F., preprint astro-ph/0008177.[16] Huang Y. F., Gou L. J., Dai Z. G. and Lu T., ApJ, 543 (2000) 90.[17] Huang Y. F., Dai Z. G. and Lu T., MNRAS, 316 (2000) 943.[18] Huang Y. F., Dai Z. G. and Lu T., MNRAS, 332 (2002) 735.[19] Dermer C. D., Chiang J. and Bo¨ttcher M., ApJ, 513 (1999) 656.[20] Rhoads J. E., ApJ, 487 (1997) L1.[21] Granot J., Panaitescu A., Kumar P. and Woosley S. E., ApJ, 570 (2002) L61.[22] Berger E. et al., Nature, 426 (2003) 154.[23] Huang Y. F., Wu X. F., Dai Z. G., Ma H. T. and Lu T., ApJ, 605 (2004) 300.

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GRB afterglows: deep Newtonian phase and its application

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