We discuss the optical and radio early afterglow emission of the reverse
shock that crosses a baryonic ejecta as it interacts with the external
interstellar medium (ISM). We show that the peak of the optical flash divides
the light curve of the reverse shock into two distinctive phases. The emission
after the peak depends weakly on the initial conditions of the ejecta and
therefore it can be used as an identifiable signature of a reverse shock
emission. On the other hand, the emission before the optical peak is highly
sensitive to the initial conditions and therefore can be used to investigate
the initial hydrodynamic profile of the ejecta. In particular, if the prompt
$\gamma$-ray emission results from internal shocks, the early reverse shock
emission should resemble a smoothed version of the prompt $\gamma$-ray light
curve.Comment: Submitted to Il Nuovo Cimento, proceedings of the 4th Workshop
  Gamma-Ray Bursts in the Afterglow Era, Rome, 18-22 October 200

Nakar, Ehud

Piran, Tsvi

English

arXiv

We discuss the optical and radio early afterglow emission of the reverse shock that crosses a baryonic ejecta as it interacts with the external interstellar medium (ISM). We show that the peak of the optical flash divided the reverse shock light curve into two distinctive phases. The emission after the peak depends weakly on the initial conditions of the ejecta and therefore it can be used as an identifiable

signature of a reverse shock emission. On the other hand, the emission before the optical peak is highly sensitive to the initial conditions and therefore can be used to investigate the initial hydrodynamic profile of the ejecta. In particular, if the prompt γ-ray emission results from internal shocks, the early reverse shock emission should resemble a smoothed version of the prompt γ-ray light curve

Nakar, E.

Piran, T.

Scientific Open-access Literature Archive and Repository

DOI 10.1393/ncc/i2005-10076-6IL NUOVO CIMENTO Vol. 28 C, N. 3 Maggio-Giugno 2005Early afterglow from a reverse shock as a tracer of the promptgamma-ray light curve(∗)E. Nakar(1) and T. Piran(1)(2)(1) Theoretical astrophysics, Caltech - Pasadena, CA 91125, USA(2) Racah Institute for Physics, The Hebrew University - Jerusalem 91904, Israel(ricevuto il 23 Maggio 2005; pubblicato online il 22 Settembre 2005)Summary. — We discuss the optical and radio early afterglow emission of the re-verse shock that crosses a baryonic ejecta as it interacts with the external interstellarmedium (ISM). We show that the peak of the optical flash divided the reverse shocklight curve into two distinctive phases. The emission after the peak depends weaklyon the initial conditions of the ejecta and therefore it can be used as an identifiablesignature of a reverse shock emission. On the other hand, the emission before theoptical peak is highly sensitive to the initial conditions and therefore can be usedto investigate the initial hydrodynamic profile of the ejecta. In particular, if theprompt γ-ray emission results from internal shocks, the early reverse shock emissionshould resemble a smoothed version of the prompt γ-ray light curve.PACS 98.70.Rz – γ-ray sources; γ-ray bursts.PACS 96.50.Fm – Shock waves.PACS 95.30.Lz – Hydrodynamics.PACS 01.30.Cc – Conference proceedings.1. – IntroductionAccording to the internal-external shocks model the prompt gamma-ray burst (GRB)is produced by internal shocks within a relativistic flow while the afterglow is producedby external shocks between this flow and the surrounding matter. The early afterglowappears during the transition from the prompt γ-ray emission to the afterglow. Duringthis transition the relativistic flow, ejected by the source, interacts directly with thecircum burst medium. This interaction can be used to pin down the nature of therelativistic flow (baryonic or Poynting flux). In a baryonic flow the reverse shock (RS)that propagates into the ejecta produces both optical and radio emission. A Poyntingflux flow is expected to produce only the higher energy forward shock emission. Whileother sources of early optical and radio emission may exist also in Poynting Flux, the RSemission has a very robust optical and radio observable signatures that are very unlikelyto be imitated by other phenomena [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 431432 E. NAKAR and T. PIRANFig. 1. – Top: An illustration of the prompt γ-ray emission (the flux is in arbitrary units)resulting from the collisions of four shells. The initial Lorentz factor and width of these fourshells is depicted in the inset. Bottom: The optical light curve of the reverse shock as simulatednumerically. The initial hydrodynamic profile of the simulation appears in the top panel inset.The total energy (isotropic equivalent) of the ejecta is 8 · 1051 erg and the external density is aconstant 10 cm−3. The energy equipartition parameters of the electrons and the magnetic fielde = 0.5 and B = 0.01, respectively. The electrons spectral index is p = 2.5 and the burst is ata redshift z = 1. Note that the time axis is logarithmic.In GRBs the reverse shock is mildly relativistic [2]. This implies that during onecrossing the shock dissipates almost all the bulk motion energy of the outflow to internalenergy. After a single crossing of the outflow a rarefaction wave is reflected and only theforward shock remains, forming the shocked external medium into a Blandford-Mckeeself-similar solution. The original ejecta expands and cools at the tail of the shockedexternal medium. The reverse shock emission can be divided into two distinctive phases,the RS crossing phase and the expanding-cooling phase. Observationally, the two areseparated by the peak of the optical flash. The evolution before the time of the opticalpeak, t0, is highly sensitive to the initial profile of the ejecta, namely to the density andthe Lorentz factor of the ejecta at the beginning of the RS crossing. Thus the light curvebefore and at t0 can be used as a diagnostic tool of these properties. On the other hand,the RS that crosses the shell erases, to a large extend, the initial shell profile. Moreover,the evolution during the expanding and cooling phase depends only weakly on this profile[3]. Therefore, the behavior after t0 depends weakly on the initial conditions and as suchit provides a very unique signature of an RS emission.EARLY AFTERGLOW FROM A REVERSE SHOCK ETC. 433Swift(1) is expected to provide detailed optical observation of the early afterglow. Insome cases the observation is expected to start before t0 and thus opens a unique windowinto the properties of the ejecta (in case that the ejecta is baryonic). Moreover, here weshow that, if the prompt emission results from internal shock, then the light curve ofthe early optical afterglow at t < t0 is expected to be correlated with the prompt γ-rayemission (see fig. 1).2. – The light curve at t > t0 — a test of the reverse shockAfter the reverse shock crosses the ejecta the shocked plasma cools and expand atthe tail of the shocked external medium. Kobayashi and Sari [3] explored this phasenumerically and they have shown that the evolution of the hydrodynamical parametersin the cooling ejecta depends weekly on the strength of the reverse shock (that in turndepends on the initial conditions of the ejecta). As a result the light curve during thisphase depends on the initial condition only by the relative values of the three reverseshock break frequencies at t0 (νra, the self-absorbtion frequency, νrm, the synchrotronfrequency and νrc the cooling frequency). Over a wide range of the parameter space(assuming an ISM(2)) these frequencies satisfy νradio < νrm(t0) < νra(t0) < νopt < νrc (t0),where νradio and νopt are the observed radio and optical frequencies, respectively [1].In this case the optical light curve decays at t > t0 as ∼ t−2 [4]. In contrast to theoptical emission, the radio continues to rise at t > t0 and it peaks at a later time, t∗,when νradio = νra. Over a wide range of initial conditions (when the shock is not ultrarelativistic) νradio < νrm(t0) < νra(t0) ≈ 1012−13Hz. In this case the radio emission isexpected to rise first as ∼ t0.5 until νradio = νrm and then as ∼ t1.25 until t∗. At t > t∗ it isexpected to decay, similarly to the optical emission, as ∼ t−2. This behavior implies that(1)F∗F0(t∗t0)(p−1)/2+1.3= C(νoptνradio)(p−1)/2∼ 1000,where F0[F∗] is the peak optical [radio] flux (at t0[t∗]). C is a factor of the order unitythat arises due to uncertainty in the hydrodynamics [3]. This entire evolution providesseveral independent tests to verify that we observe a reverse shock emission. It providesalso an independent measurement of νra(t0):(2) νra(t0) ≈t∗t0νradio.Detailed radio observations at t < t∗ that identify the break during the rising phase,when νrm = νradio would enable a determination of νrm(t0) as well.3. – Internal shocks and the reverse shock light curve at t < t0The reverse shock emission during the time that the shock crosses the ejecta dependsstrongly on the strength of the shock, and in particular on the value of n0Γ2/nejecta(1) http://swift.gsfc.nasa.gov/docs/swift/swiftsc.html(2) If the external medium is a typical wind from a massive star then the density is much higherand as a result νrc (t0) < νopt. In this case the light curve evolution at t > t0 is different.434 E. NAKAR and T. PIRANwhere n0 is the density of the ISM and Γ [nejecta] is the Lorentz factor [density] of theejecta. The density profile of the ejecta before the internal shocks is not smoothed by theshocks, on the contrary, a density contact discontinuity is produced whenever an internalshock occurs within the ejecta. Therefore at the end of the internal shocks the densityprofile of the ejecta reflects the initial shells that produced the internal shocks. Sincethe instantaneous emission of the RS depends strongly on the density of the ejecta itwould vary strongly whenever the RS crosses a density discontinuity that remains fromthe internal shocks. As a result of the angular smoothing [5, 6] the observed RS emissionis an average of the instantaneous emission over a range of radii. This smoothing limitsthe variability time scale of the RS to be of the order of the observed time since thebeginning of the prompt emission. Therefore, if the prompt γ-ray emission results frominternal shocks, the RS optical light curve at t < t0 is expected to resemble a smoothversion of the prompt γ-ray emission.In order to demonstrate this relation between the prompt γ-ray emission and thereverse shock light curve at t < t0, we have carried out a one-dimensional hydrodynamicnumerical simulation(3), starting before the internal shocks and ending long after thereverse shock crosses the ejecta. The initial conditions are four shells with Lorentzfactors that vary between 60 and 240 with constant energy (see the inset in the top panelof fig. 1). The shells collide at a radius of ∼ 2 × 1014 cm and produce a prompt γ-rayemission composed of two pulses with a duration of approximately a second each. Anillustration of the prompt γ-ray emission is depicted in the upper panel of fig. 1. Havingthe hydrodynamic evolution of the ejecta and the external medium we used a detailedsynchrotron radiation code (4) to calculate the RS emission. The resulting optical lightcurve is depicted in the bottom panel of fig 1. The similarity between the two light curvesis clear. Note also that in this specific case the optical reverse shock emission peaks longafter the prompt γ-ray emission ends. This delay is expected when the ejecta is thin. Inthis case the ejecta expands linearly with the radius between the internal shocks radius(∼ 1014 cm) and the external shock radius (∼ 1016 cm). Figure 1 shows also that afterthe reverse shock crosses the ejecta, t > t0 the optical light curve is insensitive to thecomplex initial profile of the ejecta and it decays as t−2.1.∗ ∗ ∗We thank R. Sari and S. Kobayashi for providing us with their relativistic hy-drodynamics code and J. Granot, R. Mochkovitch, F. Daigne and E. Rossi forhelpful discussions.REFERENCES[1] Nakar E. and Piran T., MNRAS, 353 (2004) 647.[2] Sari R. and Piran T., ApJ, 455 (1995) L143.[3] Kobayashi S. and Sari R., ApJ, 542 (2000) 819.[4] Sari R. and Piran T., ApJ, 520 (1999) 641.[5] Fenimore E. E., Madras C. D. and Nayakshin S., ApJ, 473 (1996) 998.[6] Sari R. and Piran T., ApJ, 485 (1997) 270.[7] Nakar E. and Granot J., in preperation (2005).(3) The hydrodynamics simulations were done using a one-dimensional Lagrangian code thatwas provided to us generously by Re’em Sari and Shiho Kobayashi [3].(4) The synchrotron radiation code is described in [7].

Early GRB afterglow from a reverse shock as a tracer of the prompt gamma-ray light curve

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