2012 • 19 Pages • 2.19 MB • English

Posted April 14, 2020 • Uploaded
by gorczany.mattie

PREVIEW PDF

Page 1

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 doi:10.1088/0004-637X/761/2/147 ⃝C 2012. The American Astronomical Society. All rights reserved. Printed in the U.S.A. DYNAMICS AND AFTERGLOW LIGHT CURVES OF GAMMA-RAY BURST BLAST WAVES WITH A LONG-LIVED REVERSE SHOCK 1 1 2 2 2 3 Z. Lucas Uhm , Bing Zhang , Romain Hascoe¨t , Fre´de´ric Daigne , Robert Mochkovitch , and Il H. Park 1 Department of Physics and Astronomy, University of Nevada–Las Vegas, 4505 South Maryland Parkway, Las Vegas, NV 89154, USA; [email protected] 2 Institut d’Astrophysique de Paris, UMR 7095 Universite´ Pierre et Marie Curie-CNRS, 98 bis Boulevard Arago, F-75014 Paris, France 3 Department of Physics, Sungkyunkwan University, Suwon 440-746, Republic of Korea Received 2012 August 11; accepted 2012 October 29; published 2012 December 4 ABSTRACT We perform a detailed study on the dynamics of a relativistic blast wave with the presence of a long-lived reverse shock (RS). Although a short-lived RS has been widely considered, the RS is believed to be long-lived as a consequence of a stratiﬁcation expected on the ejecta Lorentz factors. The existence of a long-lived RS causes the forward shock (FS) dynamics to deviate from a self-similar Blandford–McKee solution. Employing the “mechanical model” that correctly incorporates the energy conservation, we present an accurate solution for both the FS and RS dynamics. We conduct a sophisticated calculation of the afterglow emission. Adopting a Lagrangian description of the blast wave, we keep track of an adiabatic evolution of numerous shells between the FS and RS. An evolution of the electron spectrum is also followed individually for every shell. We then ﬁnd the FS and RS light curves by integrating over the entire FS and RS shocked regions, respectively. Exploring a total of 20 different ejecta stratiﬁcations, we explain in detail how a stratiﬁed ejecta affects its blast wave dynamics and afterglow light curves. We show that, while the FS light curves are not sensitive to the ejecta stratiﬁcations, the RS light curves exhibit much richer features, including steep declines, plateaus, bumps, re-brightenings, and a variety of temporal decay indices. These distinctive RS features may be observable if the RS has higher values of the microphysics parameters than the FS. We discuss possible applications of our results in understanding the gamma-ray burst afterglow data. Key words: gamma-ray burst: general – radiation mechanisms: non-thermal – shock waves Online-only material: color ﬁgures was studied analytically by assuming a constant ratio of two 1. INTRODUCTION pressures at the FS and RS (Rees & Me´sza´ros 1998; Sari & The central engine of a gamma-ray burst (GRB) ejects a Me´sza´ros 2000). relativistic outﬂow (called an ejecta) with high Lorentz factors. The structure or stratiﬁcation of the ejecta and ambient As the ejecta interacts with a surrounding ambient medium, a medium could be in fact even more general. There is no reason relativistic blast wave develops. The blast wave consists of two why it should take only a constant or power-law proﬁle. Uhm shock waves: the forward shock (FS) wave sweeping up the (2011, hereafter U11) presented a semi-analytic formulation for ambient medium and the reverse shock (RS) wave propagating this class of general problems where the ejecta and ambient through the ejecta. The shocked ambient medium is separated medium can have an arbitrary radial stratiﬁcation. U11 takes from the shocked ejecta by a contact discontinuity, and a into account a radial spread-out and spherical expansion of such compressed hot gas between the FS and RS is called a “blast.” a stratiﬁed ejecta and ﬁnds which shell of this evolved ejecta Without an extended activity of the central engine, the RS gets passed by the RS at a certain time and radius. U11 then is expected to be short-lived if the ejecta is assumed to have a ﬁnds the dynamics of the blast wave with a long-lived RS, constant Lorentz factor Γej. The RS vanishes as it crosses the by employing two different methods: (1) an equality of pressure end of the ejecta. The blast wave then enters a self-similar stage across the blast wave (mentioned above) and (2) the “mechanical where the FS dynamics is described by the solution of Blandford model” (Beloborodov & Uhm 2006). U11 shows that the two & McKee (1976, hereafter BM76). This FS emission has been methods yield signiﬁcantly different dynamical evolutions and believed to be the main source of the long-lasting, broadband demonstrates that the method (1) does not satisfy the energy afterglows (Me´sza´ros & Rees 1997; Sari et al. 1998). The short- conservation law for an adiabatic blast wave while the method lived RS emission would then be important only brieﬂy in the (2) does. The mechanical model does not assume either an early afterglow phase. Thus, it was proposed to explain a brief equality of pressure across the blast wave or a constant ratio optical ﬂash detected in some GRBs (Me´sza´ros & Rees 1997, of two pressures at the FS and RS. It shows that the ratio of 1999; Sari & Piran 1999a, 1999b). The dynamical evolution of two pressures should in fact evolve in time as the blast wave such a short-lived RS with a constant Γej was studied analytically propagates. (Sari & Piran 1995; Kobayashi 2000), under the assumption of Besides these theoretical considerations, recent early after- an equality of pressure across the blast wave. glow observations led by Swift revealed a perplexing picture However, a general view on the structure of the ejecta should regarding the origin of GRB afterglows. In contrast of a sim- include the possibility that the ejecta emerges with a range of the ple power-law decay feature as expected from the standard Lorentz factors. The shells with lower Lorentz factors gradually afterglow theory, the X-ray data show more complicated fea- “catch up” with the blast wave as it decelerates to a comparable tures including initial rapid declines, plateaus, and ﬂares (e.g., Lorentz factor. Therefore, the RS wave is believed to be long- Tagliaferri et al. 2005; Burrows et al. 2005; Nousek et al. 2006; lived in general. An example with a long-lived RS, where a O’Brien et al. 2006; Chincarini et al. 2007) that reveal rich power-law ejecta interacts with a power-law ambient medium, physics in the early afterglow phase (Zhang et al. 2006; Zhang 1

Page 2

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. 2007). More puzzlingly, some GRBs show clear chromatic be- haviors of the X-ray and optical afterglows (e.g., Panaitescu et al. 2006; Liang et al. 2007, 2008). It is now evident that the FS alone cannot interpret the broadband afterglow data for the majority of GRBs. Uhm & Beloborodov (2007) and Genet et al. (2007) inde- pendently showed that the RS-dominated afterglow ﬂux could reproduce some observed afterglow features, given the assump- tion that the FS emission is suppressed. In this paper, we study in great detail the dynamics and afterglow light curves of GRB central source blast waves with a long-lived RS. The purpose is to investigate how different ejecta stratiﬁcations affect their blast wave dy- namics and afterglow light curves. We explore various types of the ejecta stratiﬁcation and unveil that there exists a whole new class of the blast wave dynamics with a rapid and strong evo- lution of the RS strength. In order to ﬁnd an accurate solution for both the FS and RS dynamics, we make use of U11 with the Figure 1. Illustrative diagram of a spherical blast wave. A forward shock mechanical model. As explained above, this allows for the blast (FS) wave sweeps up the surrounding ambient medium, and a reverse shock wave with a long-lived RS to satisfy the energy conservation, (RS) wave propagates through the ejecta. The shocked ambient medium is separated from the shocked ejecta by a contact discontinuity (CD). A Lagrangian by introducing a pressure gradient across the blast wave region. description is employed to track an adiabatic evolution of all shells on the blast We perform a sophisticated calculation of afterglow emission, between the FS and RS. invoking a Lagrangian description of the blast wave. In the (A color version of this ﬁgure is available in the online journal.) widely used analytical afterglow model (e.g., Sari et al. 1998), it is assumed that the entire shocked material forms a single zone with same energy density and magnetic ﬁeld. The electron to be long-lived with a stratiﬁcation on the ejecta shells. As the energy distribution is solved only in the energy space, with blast wave with a long-lived RS is not in the self-similar stage no consideration of spatial distribution within the shocked of BM76, its deceleration deviates from the solution of BM76. region. Beloborodov (2005, hereafter B05) described a more In order to ﬁnd such dynamics of a blast wave with a sophisticated Lagrangian method, in which the postshock region long-lived RS, we make use of the semi-analytic formulation is resolved into subshells using a Lagrangian mass coordinate. presented in U11. The formulation has three input functions B05 studies an evolution of the magnetic ﬁeld and power- ρ1(r), Lej(τ ), and Γej(τ ), which deﬁne the initial setup of the law spectrum of electrons for each subshell as the blast wave blast wave (see Figure 1). The ambient medium density ρ1(r) is propagates. However, the postshock material in B05 is not allowed to take an arbitrary radial proﬁle, where r is the radius resolved in radius and all the subshells are located at the same measured from the central engine. The ejecta is completely radius. Also, the pressure and energy density in B05 are assumed speciﬁed by two other functions, i.e., its kinetic luminosity to be constant throughout the postshock material. Improving on Lej(τ ) and Lorentz factor Γej(τ ). Here, τ indicates an ejection B05, here we have a spatial resolution into the blast region, time of the ejecta shells. allowing for our Lagrangian shells to have their own radius. We When two functions Lej(τ ) and Γej(τ ) are known, a continuity further introduce a pressure proﬁle that smoothly varies over equation, which governs a radial spread-out and spherical the blast. This is because the pressure at the FS differs from expansion of a stratiﬁed ejecta, can be solved. This then yields the pressure at the RS, as discussed above. As the blast wave an analytic solution for the ejecta density ρej of any τ -shell at propagates, we keep track of an evolution of the pressure, energy radius r (U11, Section 3.1), density, and adiabatic index of every shell on the blast. We also [ ] −1 keep track of an evolution of the magnetic ﬁeld and power-law ′ Lej(τ ) r Γej spectra of electrons of all shells. ρ ej(τ, r) = 1 − ( ) . (1) Finally, in order to calculate synchrotron radiation from a 4πr2vej Γe2jc2 c Γ2 − 1 3/2 ej spherical shell on the blast, we analytically ﬁnd an observed spectral ﬂux for a distant observer, taking into account the 2 1/2 Here, c is the speed of light, and vej(τ ) = c (1 − 1/Γ ej) is the effects of the shell’s radial velocity and spherical curvature. velocity of τ -shell. Equation (1) is exact for a non-increasing We integrate this ﬂux over the entire blast to ﬁnd the sum of ′ proﬁle of Γej(τ ); Γ ej(τ ) ≡ dΓej/dτ ⩽ 0. emissions from all the shells between the FS and RS. We then self-consistently ﬁnd the path of the RS wave through In Section 2, we brieﬂy summarize how we ﬁnd the dynamics this evolved ejecta (U11, Section 3.3). When the RS wave is of a blast wave with a long-lived RS. In Section 3, we describe located at radius rr (t) at time t, we numerically determine which in detail our method of calculating afterglow light curves. τ -shell gets shocked by the RS, and name it as τr (t)-shell; the Numerical examples are presented in Section 4, which exhibit subscript r in rr and τr refers to the RS. In other words, the RS various features on the blast wave dynamics and afterglow light wave sweeps up the τr (t)-shell at time t, which has the Lorentz curves. Our results are summarized in Section 5 (Discussion) factor Γej(τr ) ≡ Γej(RS) and the density ρej(τr , rr ) ≡ ρej(RS). and Section 6 (Conclusion). The formulation ﬁnds a dynamical evolution of the blast wave using the “mechanical model” (Beloborodov & Uhm 2006; 2. DYNAMICS OF A BLAST WAVE U11, Section 4). The mechanical model was developed for a WITH A LONG-LIVED RS relativistic blast wave by applying the conservation laws of A schematic diagram of a spherical blast wave is shown in energy–momentum tensor and mass ﬂux on the blast between Figure 1. As mentioned in Section 1, the RS wave is expected the FS and RS. Speciﬁcally, we numerically solve a set of 2

Page 3

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. coupled differential Equations (78)– (80) and (92) of U11, which makes use of the FS and RS jump conditions (U11, Section 3.2). The FS and RS jump conditions in U11 are derived adopting photon a realistic equation of state (EOS) with a variable index κ, emitted at time t ( ) 1 1 2 p = κ (e − ρc ), κ = 1 + , (2) 3 γ¯ observer’s line of sight where p, e, and ρ are pressure, energy density, and mass density of the shocked gas, respectively, and γ¯ is the mean Lorentz factor of gas particles. The quantity κ smoothly varies between 2/3 (for a non-relativistic gas) and 1/3 (for an ultrarelativistic gas) as the gas temperature varies. This EOS differs from the exact EOS of a relativistic ideal gas (e.g., Synge 1957) by less than 5% Figure 2. Schematic diagram of a spherical shell at radius r at time t expanding 2 (U11, Section 2.2). Since e = γ¯ ρc , the EOS in Equation (2) with a Lorentz factor Γ. A photon emitted in the direction of the observer at time is the same as t by an electron at a polar angle θ is received by the observer at an observer time tobs as given in Equation (20). A thin ring between θ and θ + δθ is considered ( ) obs p 1 e ρc2 in order to derive an analytical expression for observed spectral ﬂux δFνobs ; see Equation (38). = − , (3) 2 2 ρc 3 ρc e (A color version of this ﬁgure is available in the online journal.) which was previously introduced by Mathews (1971) consid- Section 3.4.1) ering a relativistic “monoenergetic” gas, and later adopted by 5/2( 2 )5/2 p κ − κ Meliani et al. (2004) and Mignone & McKinney (2007) in their 3 = C = const. where pm(κ) ≡ ( ) . numerical simulations. p m(κ) κ − 1 4 3 (4) 3. LIGHT CURVES FROM A BLAST WAVE WITH A LONG-LIVED RS The function pm(κ) is monotonically decreasing in its valid range, (1/3) < κ < (2/3). Numerically solving for the dynamics of a blast wave with Let us consider a shell δmi either in the ambient medium a long-lived RS, we discretize the ambient medium and ejecta or in the unshocked ejecta; an index i is added to specify this into spherical mass shells δm. At every calculation step, a pair shell. When the shell δmi is shocked by the FS (or the RS), of shells is impulsively heated by shock fronts; one shell by the the jump conditions of the FS (or the RS) determine its initial FS and the other by the RS. We follow these shells subsequently thermodynamic quantities: pressure pi , energy density ei , mass 0 0 and use them as our Lagrangian shells for the blast. Thus, the i i density ρ , and quantity κ (U11, Section 3.2). Substituting the 0 0 blast is viewed as being made of many different hot shells that i i initial values p and κ into Equation (4), we determine the 0 0 pile up from the FS and RS, as depicted in Figure 1 with dotted i i constant C of the shell δm , (red) curves. i i Assuming an adiabatic blast wave, we keep track of an adi- p p i 0 = C = ( ) . (5) abatic evolution of these shells; an evolution of the thermody- i i pm(κ ) pm κ 0 namic quantities of shocked gas (pressure, energy density, adi- i i i abatic index, etc.) is followed individually for every shell. This Thus, if we know p or κ of the shell δm at later times, we can in turn yields an evolution of the magnetic ﬁeld for the shell subsequently follow an adiabatic evolution of the shell. (Section 3.1). We also keep track of an evolution of the power- Solving the mechanical model, at every calculation step, say law spectrum of electrons; a radiative and adiabatic cooling of at time t, we know r f (t), rr (t), pf (t), pr (t), and P (t), which are the spectrum is followed for every shell (Section 3.2). radii of the FS and RS, pressures at the FS and RS, and integrated Zooming in on the blast shown in Figure 1, we focus on pressure over the blast, respectively. Thus, an instantaneous a single spherical shell of radius r expanding with a Lorentz pressure proﬁle for the blast (r r < r < rf ) may be approximated factor Γ (see Figure 2). While taking into account the effects by a quadratic function p(r), which (1) matches two boundary of the shell’s Doppler boosting and spherical curvature on the values (i.e., p(r f (t)) = pf (t) and p(rr (t)) = pr(t)) and ∫ synchrotron photons emitted along an observer’s line of sight, rf (2) satisﬁes the integrated pressure P (t) = p(r) dr (U11, rr we analytically ﬁnd an observed spectral ﬂux in terms of an Section 5). observed frequency νobs and observer time tobs (Section 3.3). As i i As we also know the radius r of the shell δm at time t, we now the blast is made of many Lagrangian shells, this spectral ﬂux is i i i i have the pressure p of the shell δm at time t: p (t) = p(r (t)). summed over the blast; a sum of emissions from all the shells in i Equation (5) then allows us to numerically ﬁnd the quantity κ the shocked ambient medium (or the shocked ejecta) is denoted i of the shell δm at time t. All other thermodynamic quantities of by “FS emission” (or “RS emission”). the shell can be found accordingly. For instance, Equation (2) i gives the thermal energy density e of the shell as th 3.1. Adiabatic Evolution of the Blast i p (t) i i i 2 e (t) ≡ e (t) − ρ (t) c = . (6) Here we follow the adiabatic evolution of the shocked gas on th i κ (t) a shell, in order to ﬁnd the evolution of the magnetic ﬁeld for the i shell. An adiabatic process of a relativistic gas whose EOS is The electrons in the shell δm emit synchrotron radiation in a i speciﬁed by Equation (2) or Equation (3) is described by (U11, magnetic ﬁeld B . The ﬁeld is unknown and parameterized by 3

Page 4

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. a microphysics parameter ϵB, which is the ratio of ﬁeld energy For a fresh shell created at a shock front, γ¯p equals the shock 4 i 2 i density to thermal energy density : ϵB = ((1/8π)B )/e th. strength, i.e., the relative Lorentz factor of the preshock to the i i Thus, the magnetic ﬁeld B of the shell δm at time t is given by postshock medium (BM76; U11, Equation (14)). Thus, for the fresh shell at the FS, γ¯p equals Γ, the Lorentz factor of the blast, [ i ]1/2 i [ i ]1/2 p (t) as the ambient medium is at rest in the lab. frame. For the fresh B (t) = 8πϵB e th(t) = 8πϵB κi(t) . (7) shell at the RS, γ¯p equals γ43 given by [ ( )] Hereafter, we will omit the index i to simplify our notation. 2 2 1/2 γ43 = ΓΓej − (Γ − 1) Γ ej − 1 , (14) 3.2. Power-law Spectrum of Electrons where Γej is to be evaluated for the τr -shell; Γej = Γej(RS). We assume that a non-thermal electron spectrum is created in We use Equation (9) to ﬁnd the constant K and substitute it a fresh shell δm at a shock front (FS or RS); i.e., the electrons into the electron spectrum, are accelerated into a power-law distribution above a minimum [ ] −p Lorentz factor γm, p − 1 γe − 1 f (γe) = δN . (15) dN γm − 1 γm − 1 −p f (γe) ≡ = K (γe − 1) for γe ⩾ γm, (8) dγe This spectrum created at the shell δm evolves as the blast wave propagates. We track the adiabatic and radiative cooling of the where K is a constant, p is the slope of the spectrum, and γe is electron spectrum as follows. the Lorentz factor of the accelerated electrons in the ﬂuid frame. The total number of electrons within the spectrum is given by 1. Adiabatic cooling of γm: an adiabatic cooling of relativistic 1/4 ∫ electrons is described as γe ∝ p . Thus, the minimum Lorentz ∞ 1/4 K 1−p factor γm also evolves as γm ∝ p . Here, p is the pressure of δN = f (γe) dγe = (γm − 1) . (9) γm p − 1 the shell, not the slope of the electron spectrum. 2. Radiative and adiabatic cooling at high γe: the electrons The thermal energy of all electrons in the spectrum is found as at high γe in the spectrum experience a radiative and adiabatic ∫ ∞ cooling, which is described by the ﬁrst and the second term [ ] 2 δE = (γe − 1)mec f (γe) dγe below, respectively, γm = Kmec2 (γm − 1)2−p, p > 2, (10) γ˙e = − 61π mσeTc B2 (1 + Y ) γe2 + 41 p˙ γe. (16) p − 2 where me is the electron mass. Thus, we ﬁnd the mean thermal Here, σT is the Thomson cross section, and the dot indicates a ′ energy per electron, derivative with respect to t , the time measured in the comoving ﬂuid frame. The Compton parameter Y describes a relative δE p − 1 2 contribution of inverse Compton scattering to the cooling rate = (γm − 1) mec . (11) δN p − 2 of electrons. Equation (16) deﬁnes a cutoff Lorentz factor γc at the high For neutral plasma without pair-loading, if (1) all electrons 2 5 end of the spectrum. Dividing Equation (16) by −γe , we get passing through the shock become non-thermal, and (2) a 6 ( ) ( ) fraction ϵe of the shock energy goes to the electrons, then d 1 1 σT 2 1 1 d ln p the mean thermal energy per electron is alternatively given by = B (1 + Y ) − . (17) ′ ′ dt γc 6π mec 4 γc dt δE 2 = ϵe (γ¯p − 1) mpc . (12) The parameter Y can be evaluated as (Sari & Esin 2001) δN √ Here, mp is the proton mass and γ¯p is the mean Lorentz factor of ϵe 1 1 ϵe the protons in the postshock medium. Equations (11) and (12) (1 + Y )Y = η or Y = − + + η, (18) are combined to yield the lowest Lorentz factor γm as ϵB 2 4 ϵB p − 2 mp where γm = 1 + ϵe (γ¯p − 1). (13) p − 1 me { p−2 (γm/γc) for γm ⩽ γc, η = (19) 4 See Ioka et al. (2006) for a discussion on a possible time dependent 1 for γc < γm. evolution of the microphysics parameters. 5 This is a usual assumption in the afterglow literature, although it is not Note that p in Equation (17) is the pressure of the shell, and p necessarily true. See Genet et al. (2007) for examples of afterglow light curves in Equation (19) is the slope of the electron spectrum. When a in the long-lived RS scenario, obtained by assuming that only a small fraction ζ of the electrons become non-thermal. fresh shell δm is created at a shock front (FS or RS), γc = +∞ 6 The dominant fraction of shock energy goes to the protons, which dominate is adopted as its initial value. We then solve Equation (17) the pressure of the blast and evolve adiabatically. We may estimate the numerically and ﬁnd a subsequent evolution of 1/γc (and hence, pressure pe of the electrons prescribed by the fraction ϵe. For relativistic electrons, Equation (2) gives pe = (1/3) eth,e where eth,e is the thermal energy γc) for the shell δm. density of electrons. Equations (11) and (12) yield eth,e = n (p − 1)/(p − 2) γm mec2 = n ϵe (γ¯p − 1) mpc2 with the number 3.3. Curvature Effect and Light Curves 2 density n of electrons or protons. Thus, we get pe = (n/3) ϵe (γ¯p − 1) mpc . It Consider a spherical shell δm on the blast, which has radius can be compared to the proton pressure pp, for which we use Equation (19) of 2 2 U11; pp = (n/3) (γ¯ p − 1)/γ¯p mpc . This then gives the ratio r at time t expanding with a Lorentz factor Γ. An observer pe/pp = ϵe γ¯p/(γ¯p + 1). is located in the positive z-direction at a large (cosmological) 4

Page 5

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. 2 distance. See Figure 2 for a schematic diagram. An observer lab qeB γc ν = (0.15) . (27) c time tobs is set equal to zero when the observer detects the very mec Γ(1 − βμ) ﬁrst photon that was emitted at the explosion center at time t = 0. Also, consider a thin ring on the shell δm, which is The electron with γe = γνlab has the frequency νlab in the direction of the observer, speciﬁed by a polar angle θ with respect to the z-axis. Then photons emitted from this ring at time t will be detected by the 2 qeB γν lab observer at the observer time, νlab = (0.15) . (28) mec Γ(1 − βμ) [ ] r tobs = t − μ (1 + z), (20) The spherical shell emits photons continuously when it c expands, but we think of a series of “snapshots” of the shell. We where μ ≡ cos θ and z is the cosmological redshift of the burst. consider two consecutive snapshots (i.e., two calculation steps) The factor (1 + z) is introduced due to the expansion of the separated by a time interval δt , and then assume that the shell universe. accumulates its emission between two snapshots and emits all The spherical shell δm contains a total of δN electrons the accumulated energy instantaneously like a “ﬂash” when it uniformly distributed over the shell, retaining a non-thermal arrives at the second snapshot. In other words, the emission from spectrum of γm, γc, and slope p (see Section 3.2). Since the the shell is viewed as a series of ﬂashes. number of electrons on the thin ring (between θ and θ + δθ) is When the shell ﬂashes, the accumulated energy by the single given as (|δμ|/2) δN, the electron spectrum of the ring becomes electron during δt is emitted. The energy emitted into a solid angle δΩ in the direction of the observer is given in the lab. [ ] −p |δμ| δN p − 1 γe frame as ˜ f (γe) = , (21) ( ) 2 γm γm lab dE1 Le δΩ δt δE ≡ δΩ δt = . (29) 1 4 3 dΩ dt 4π Γ (1 − βμ) ˜ lab where we assumed γe ≫ 1 and γm ≫ 1. Here the tilde on f (γe) indicates the ring. Note that f˜(γe) is subject to the thickness of The next step is then to calculate the emission from the entire ˜ lab the ring, δθ or δμ. thin ring; let δE be the emission from the entire ring into We now focus on a single electron in the ring. On average, the the solid angle δΩ in the direction of the observer. With the electron is assumed to emit synchrotron photons isotropically7 deﬁnition of spectral energy δE˜ lab ≡ d(δE˜ lab)/dν, we consider ν at a single characteristic frequency νﬂuid in the ﬂuid frame, (e.g., B05) ∣ ∣ νﬂuid = (0.15)q meeBc γe2. (22) νlab(δE˜νlalab) = d d(δ(lEn˜ lνab)) ∣ = 2dd(δ(Eln˜ laγbe)) ∣ ν=νlab γe=γν lab ∣ Here, qe is the electric charge of the electron, and B is the ˜ lab ∣ magnetic ﬁeld of the shell. The synchrotron luminosity of the 1 d(δE ) ∣ = γν lab ∣ . (30) electron is given by 2 dγe ∣ γe=γν lab 1 Le = σT c B2 γ e2. (23) Here the second equality uses the relation δ(ln νlab) = 6π 2 δ(ln γ νlab ), which is veriﬁed from Equation (28); for the instan- taneous ﬂash of the shell, a variation in νlab results only from The photon frequency νlab in the lab. frame is different from δγν lab since the ring has a ﬁxed B, Γ, and μ instantaneously. νﬂuid due to radial expansion of the shell. Since the radial bulk Together with Equations (21) and (29), Equation (30) yields motion of the electron has the angle θ with the observer’s line of sight (see Figure 2), the frequency νlab in the direction of the ( ˜ lab ) 1 [ lab ˜ ]∣ observer is simply νlab δEνlab = 2 γνlab δE1 f (γe) γe=γνlab [ ] 1−p νﬂuid 1 p − 1 2 2 δΩ δt |δμ| δN γνlab νlab(θ) = . (24) = σ T c B γν lab 4 3 . Γ(1 − β cos θ) 24π 4π Γ (1 − βμ) γ m (31) The angular distribution of the energy of photons emitted by the electron in the direction of the observer is written in the lab. lab lab We remark that Equation (31) is valid only for ν m < νlab < νc frame as since the electron spectrum in Equation (21) is valid only for ( ) dE1 Le 1 γm < γνlab < γc. Dividing Equation (31) by νlab in Equation (28), = . (25) we ﬁnd the spectral energy emitted from the entire thin ring into 4 3 dΩ dt 4π Γ (1 − β cos θ) lab the solid angle δΩ in the direction of the observer, [ ] Here the subscript in E1 refers to the single electron. 2 1−p lab 5 p − 1 σT mec B δΩ δt |δμ| δN γνlab ˜ From Equations (22) and (24), we note that the Lorentz factors δE = . νlab 3 2 18π 4π qe Γ (1 − βμ) γm γm and γc of the electron spectrum correspond to the frequencies lab lab (32) ν and ν , respectively, m c 2 This energy is emitted instantaneously during the ﬂash, but the lab qeB γm ν = (0.15) , (26) thickness of the ring (between θ and θ + δθ) introduces a time m mec Γ(1 − βμ) interval δt˜lab along the observer’s line of sight, 7 r r See Beloborodov et al. (2011) for discussion of anisotropic emission in the δt˜lab = [cos θ − cos(θ + δθ)] = |δμ|. (33) ﬂuid frame. c c 5

Page 6

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. Hence, Equations (32) and (33) yield the spectral luminosity of the entire ring, ˜ lab δE lab νlab δL = νlab δt˜lab [ ] 3 1−p 5 p − 1 σT mec B δΩ δt δN γν lab = , 3 2 18π 4π qe r Γ (1 − βμ) γm (34) which shines into the solid angle δΩ in the direction of the lab observer. Note that we do not include a tilde for δL since the νlab thickness of the ring (namely, δμ) cancels out here. The burst is at a cosmological distance from the observer. For a ﬂat ΛCDM universe, the luminosity distance of a burst at redshift z is given by ∫ z ′ c (1 + z) dz DL = √ , (35) H0 0 Ω m(1 + z′)3 + ΩΛ −1 −1 Figure 3. Ejecta stratiﬁcations for examples 1a, 1b, and 1c. The ejecta Lorentz where H0 = 71 km s Mpc , Ωm = 0.27, andΩΛ = 0.73 (the factors Γej(τ ) are shown as a function of the ejection time τ . concordance model). The photons are redshifted while traveling (A color version of this ﬁgure is available in the online journal.) the cosmological distance, and the observed frequency νobs is Figure 4. Blast wave dynamics for examples 1a, 1b, and 1c. In panel (a), the curves denoted by Γ show the Lorentz factor of the blast wave as a function of the radius rr of the RS, and the curves by Γej(RS) show the Lorentz factor of the ejecta shell that gets shocked by the RS when the RS is located at radius rr. In panel (b), the curves denoted by pf show the pressure at the FS, and the curves by pr show the pressure at the RS. Panel (c) shows the relative Lorentz factor γ43 across the RS wave, as is given by Equation (14). Panel (d) shows the density nej(RS) = ρej(RS)/mp of the ejecta shell, which enters the RS at radius rr. (A color version of this ﬁgure is available in the online journal.) 6

Page 7

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. FS Figure 6. Same as in Figure 3, but for examples 2a, 2b, and 2c. (A color version of this ﬁgure is available in the online journal.) obs expression for δF νobs in terms of tobs and νobs, [ ] (1−p)/2 obs max νobs δF νobs (tobs, νobs) = δFνobs × obs , (38) ν m where 3 max 5(1 + z) p − 1 σT mec B δN δt δF νobs ≡ 2 [ ( )]2 , 18π 4πD L qe r 3 cβ tobs Γ 1 − t − r 1+z RS (39) 2 obs 0.15 qeB γm ν = [ ] . (40) m ( ) 1 + z mec cβ tobs Γ 1 − t − r 1+z Figure 5. Afterglow light curves for examples 1a, 1b, and 1c. Panel (a) shows the FS emissions in X-ray (1 keV) and R band as a function of the observer time When the spherical shell of radius r ﬂashes at time t, the observer tobs. Panel (b) shows the RS emissions in X-ray (1 keV) and R band. at cosmological distance receives its emission for a period of (A color version of this ﬁgure is available in the online journal.) the observer time t obs, r tobs r t − ⩽ ⩽ t + . (41) obtained by c 1 + z c νlab νobs = . (36) In other words, Equation (38) is to be evaluated only in 1 + z this period of tobs. For any tobs in this period, the observed obs The deﬁnition of the luminosity distance gives an observed spectral ﬂux δFν obs at the observed frequency νobs is expressed spectral ﬂux at νobs, analytically. Thus, we do not need to consider the ring any more. lab lab Equation (31) is valid for ν m < νlab < νc , and therefore lab obs obs obs (1 + z) δLνlab Equation (38) is valid only for νm < νobs < νc , where δF = νobs 2 2 D LδΩ obs 0.15 qeB γc ν = [ ] . (42) 3 c ( ) 5(1 + z) p − 1 σT mec B 1 + z mec Γ 1 − cβ t − tobs = r 1+z 2 18π 4πD L qe r [ ] (1−p)/2 In general, the observed spectral ﬂux at νobs can be obtained as δN δt νlab ⎧( ) × , (37) obs 1/3 obs obs Γ3(1 − βμ)2 ν mlab ⎪(νobs/νm ) for νobs < νm < νc , ⎨ obs (1−p)/2 obs obs obs max νobs/νm for νm < νobs < νc , bwyhe(νrelabE/qνu mlabt)i1o/n2s. (26) and (28) have been used to replace γνlab /γm δFνobs = δFνobs × ⎪⎩⎪(νobs/νcobs)1/3 for νobs < νcobs < νmobs, obs Finding μ = (c/r) [t − tobs/(1 + z)] from Equation (20) and 0 for νc < νobs. substituting it into Equation (37), we arrive at an analytical (43) 7

Page 8

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. Figure 7. Same as in Figure 4, but for examples 2a, 2b, and 2c. (A color version of this ﬁgure is available in the online journal.) Equation (43) needs to be summed over the shocked region as Note that two summations are not commutative since the FS and the blast is made of many different shells. RS waves create new shocked shells as time goes. We ﬁnd the Here we recover the index i to name our Lagrangian shells “FS emission” (or the “RS emission”) by taking the summation ∑ i i {δm }. Each δm is impulsively heated at some point by the FS i over all shells only in the shocked ambient medium (or {δm } shd or the RS and becomes a shocked shell on the blast. We denote only in the shocked ejecta). For a ﬁxed observed frequency ν obs, these shocked shells by {δm sihd}. As mentioned above, the shells Equation (44) gives light curves Fνo obs(tobs) at νobs. For a ﬁxed i i {δm } on the blast have their own individual radius r , number obs shd observer time tobs, Equation (44) yields ﬂux spectra F νobs (νobs) i i i of electrons δN , magnetic ﬁeld B , and the Lorentz factors γ m at tobs. i and γ . c j Let us now use another index j to specify the time t of each 4. NUMERICAL EXAMPLES calculation step (or ﬂash). Solving for the blast wave dynamics at a calculation step with time t j, we ﬁnd the Lorentz factor Γ(tj ) Now we present a total of 20 different numerical examples, of the blast and the radii {ri(t j )} of the shells {δmi }. We also which are named as follows; 1a, 1b, 1c, 2a, 2b, 2c, 3a, 3b, 3c, shd i j i j i j 4a, 4b, 4c, 4d, 5a, 5b, 6a, 6b, 6c, 6d, and 6e. For all 20 examples, evaluate the shells’ emission properties {B (t ), γ (t ), γ (t )} m c j we keep the followings to be the same: (1) a constant density at time t (Sections 3.1 and 3.2). −3 8 ij obs ρ1(r)/mp = 1 cm is assumed for the ambient medium. Then, Equation (43) gives the spectral ﬂux δFν obs of each i j i (2) The ejecta has a constant kinetic luminosity Lej(τ ) = L0 = shell δm at time t . We sum it over all shocked shells {δm } shd shd 53 −1 j obs ∑ ij obs 10 erg s for a duration of τb = 10 s, so that the total isotropic to ﬁnd a ﬂux δFν obs = {δmi } δFνobs of the entire blast at 54 shd energy of the burst is to be Eb = L0 τb = 10 erg. (3) The burst j time t . Lastly, this needs to be summed over all ﬂashes separated is assumed to be located at a redshift z = 1. (4) The emission j by time intervals {δt } to give a total spectral ﬂux, 8 Although our method described in this paper allows us to study other types ∑ ∑ −2 obs ij obs of environment, such as a stellar wind with ρ1(r) ∝ r , we will keep the F νobs (tobs, νobs) = δFνobs (tobs, νobs). (44) same density ρ1(r)/mp = 1 cm−3 in all 20 examples. This is to focus on the j i {δt } {δm shd} study of the ejecta stratiﬁcations. 8

Page 9

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. FS Figure 9. Same as in Figure 3, but for examples 3a, 3b, and 3c (A color version of this ﬁgure is available in the online journal.) the same duration τb = 10 s. Since we are mainly interested in the afterglow light curves, we ignore the initial variation of Γej and take a simple uniform proﬁle of high Lorentz factors early on. Thus, for all 20 examples, we assume a constant Lorentz factor Γej = 500 for the initial 3 s. From 3 to 10 s, the examples 9 have a decreasing proﬁle of Γej, exhibiting various types of the ejecta stratiﬁcations. Note that only a comparable amount, 70% of the burst energy, has been distributed over the shells RS with lower Lorentz factors in order to maintain a long-lived RS. Thus, the deceleration of the blast wave would deviate only mildly from the solution of BM76. The examples with the same number in their names share a similar shape of the ejecta stratiﬁcations, and we cate- Figure 8. Same as in Figure 5, but for examples 2a, 2b, and 2c. gorize 20 examples into seven different groups; (1a/1b/1c), (2a/2b/2c), (3a/3b/3c), (4a/4b/4c/4d), (1a/5a/5b), (A color version of this ﬁgure is available in the online journal.) (6a/6b/4d), and (6c/6d/6e). This is to provide an efﬁcient com- parison among examples. Note that we use 1a and 4d twice in −1 −2 parameters are ϵe = 10 , ϵB = 10 , and p = 2.3 for the RS the comparisons. −2 −4 light curves, and ϵe = 10 , ϵB = 10 , and p = 2.3 for the FS For each group of examples, we present three ﬁgures: (1) the light curves. 1st ﬁgure shows the ejecta stratiﬁcations (e.g., see Figure 3). The Note that we have adopted different microphysics parameters ejecta Lorentz factors Γej(τ ) are shown as a function of ejection for the FS and RS. This is because the FS and RS shocked time τ , in different line (color) types. (2) The 2nd ﬁgure shows regions originate from different sources and the strengths of the blast wave dynamics of the examples (e.g., see Figure 4). the two shocks can be signiﬁcantly different. Indeed, afterglow In panel (a), the curves denoted by Γ show the Lorentz factor modeling suggested that the RS can be more magnetized, and of the blast wave as a function of the radius rr of the RS, and ϵe of the two shocks can also be different (e.g., Fan et al. the curves by Γej(RS) show the Lorentz factor of the ejecta shell 2002; Zhang et al. 2003; Kumar & Panaitescu 2003). Since the that gets shocked by the RS when the RS is located at the radius GRB central engine is likely magnetized, a natural consequence rr. In panel (b), the curves denoted by pf show the pressure at would be to invoke a larger ϵB in the RS. The bright optical ﬂash the FS as a function of the radius rr, and the curves by pr show observed in GRB 990123 (Akerlof et al. 1999) is likely related the pressure at the RS. Since the ambient medium is assumed to such a case (Zhang et al. 2003). Our emission parameters to have a constant density ρ1, the pf curves resemble the Γ above are chosen such that the FS and RS ﬂuxes are comparable 2 curves; pf ∝ Γ ρ1. Panel (c) shows the relative Lorentz factor to each other. Of course, either the FS or the RS emission could γ43 across the RS wave, as is given by Equation (14). Panel (d) be further enhanced or suppressed by varying their parameters ϵB and/or ϵe. 9 It is natural to assume a decreasing proﬁle of Γej since the internal shocks The only difference among examples then goes on the proﬁle during the prompt emission phase tend to smoothen the Γej distribution and of the ejecta Lorentz factors Γej(τ ) as a function of ejection time lead to a decreasing Γej proﬁle (otherwise, additional internal shocks would τ ; i.e., the examples have a different ejecta stratiﬁcation within occur). 9

Page 10

The Astrophysical Journal, 761:147 (19pp), 2012 December 20 Uhm et al. Figure 10. Same as in Figure 4, but for examples 3a, 3b, and 3c. (A color version of this ﬁgure is available in the online journal.) shows the density nej(RS) = ρej(RS)/mp of the ejecta shell, BM76 (denoted by a dot-dashed line). The examples 1b and 1c which enters the RS wave at radius rr. We omit panels (c) and have higher RS pressure pr than the example 1a, and therefore (d) for the last two groups of examples. (3) The 3rd ﬁgure shows their Γ curves deviate from BM76 even stronger than the case the afterglow light curves (e.g., see Figure 5). In the upper panel of 1a while their RS waves exist. However, once their RS waves −3/2 (a), we show the FS emissions in X-ray (1 keV) and R band as vanish, their Γ curves start to follow Γ ∝ r r as they should. a function of the observer time tobs. In the lower panel (b), we As a result, all three Γ curves completely agree with one another show the RS emissions in X-ray (1 keV) and R band. after all three RS waves disappear. This is expected because the Each group has a comparison point to help readers understand same amount of burst energy E b has been injected into the blast the effects of ejecta stratiﬁcation on the FS and RS dynamics waves which swept up the same amount of ambient medium out and further on the FS and RS afterglow light curves. to a certain radius. In other words, three Γ curves are shaped by three different plans or time schedules of “spending” the same 4.1. Group (1a/1b/1c) energy budget Eb. Once the budget is used up, the outcome or the Lorentz factor of the blast waves should be the same. Figure 3 shows the ejecta stratiﬁcations of examples 1a, Also notice that three pf curves exhibit the same behaviors as Γ 2 1b, and 1c. For a duration from 3 to 10 s, the ejecta Lorentz curves, accordingly, since pf ∝ Γ ρ1 and ρ1 = const. √ factors Γej(τ ) decrease exponentially from 500 to 5, 5 × 50, Three Γej(RS) curves stay close to the Γ curves since the and 50, respectively. Note that an exponential decrease implies ejecta shells catch up with the blast waves only when Γej ∼ Γ; (d/dτ )(ln Γej) = Γ e′ j/Γej = const. the resulting relative Lorentz factors γ43 are shown in panel (c). The blast wave dynamics of these three examples are shown A constant kinetic luminosity Lej(τ ) = L0 is assumed here, and −2 ′ in Figure 4. The Γej(RS), pr, γ43, and ρej(RS) curves vanish at thus Equation (1) yields ρej(τ, r) ∝ r when Γej(τ ) = 0. −2 an earlier time or radius for higher ending values of Γej (i.e., Hence, ρ ej(RS) = ρej(τr , rr ) ∝ rr while the RS waves examples 1b and 1c), as the RS waves cross the end of ejecta. sweep up the initial 3 s of ejecta shells with Γ ej = 500 (see The Γ curve of example 1a shows that its blast wave decelerates panel (d)). When the RS waves arrive at τ = 3 s, they encounter −3/2 ′ slightly slower than Γ ∝ rr , the self-similar solution of a discontinuity in the value of Γ ej(τ ), which results in a sudden 10

X-RAY, ULTRAVIOLET, AND OPTICAL FLARES IN GAMMA-RAY BURST LIGHT CURVES

2014 • 160 Pages • 1.86 MB

Advances in Gamma Ray Resonant Scattering and Absorption: Long-Lived Isomeric Nuclear States

2015 • 201 Pages • 4.68 MB

The Extragalactic Background Light and the Gamma-ray Opacity of the Universe

2012 • 95 Pages • 2.14 MB

Nonlinear Absorption of Light in Materials with Long-lived

2012 • 31 Pages • 797 KB

Nonlinear Absorption of Light in Materials with Long-lived Excited

2012 • 31 Pages • 797 KB

Demonstration of an Approach to Precisely Measure -ray Branching Ratios for Long-Lived Emitters

2017 • 198 Pages • 11.07 MB

Shock Waves: A Practical Guide to Living with a Loved One's PTSD

2010 • 143 Pages • 792 KB

Shock Waves: A Practical Guide to Living with a Loved One's PTSD

2010 • 136 Pages • 868 KB

How to Generate Supernova Light Curves: Photometry with the - MIT

2002 • 97 Pages • 10.71 MB

Long lived light scalars as probe of low scale seesaw models

2017 • 43 Pages • 9.49 MB

Generation and Characterization of Coherent Soft X-Ray Light with High Harmonic Generation

2016 • 101 Pages • 9.89 MB

Time resolved photoelectron spectroscopy with ultrafast soft x-ray light

2001 • 171 Pages • 2.74 MB

Long-lived charge carrier dynamics in polymer/quantum dot blends and organometal halide ...

2014 • 89 Pages • 3.62 MB

Searching for transits in the WTS with the difference imaging light curves

2014 • 187 Pages • 19.33 MB

Title Study of Microlensing Exoplanets with Combination of Light Curves and AO Images Author(s)

2012 • 126 Pages • 17.23 MB

Economical long-life light sources with plug-in bases

2003 • 82 Pages • 415 KB