Is the Rapid Decay Phase from High Latitude Emission?

📝 Abstract

There is good observationnal evidence that the Steep Decay Phase (SDP) that is observed in most Swift GRBs is the tail of the prompt emission. The most popular model to explain the SDP is Hight Latitude Emission (HLE). Many models for the prompt emission give rise to HLE, like the popular internal shocks (IS) model, but some models do not, such as sporadic magnetic reconnection events. Knowing if the SDP is consistent with HLE would thus help distinguish between different prompt emission models. In order to test this, we model the prompt emission (and its tail) as the sum of independent pulses (and their tails). A single pulse is modeled as emission arising from an ultra-relativistic thin spherical expanding shell. We obtain analytic expressions for the flux in the IS model with a Band function spectrum. We find that in this framework the observed spectrum is also a Band function, and naturally softens with time. The decay of the SDP is initially dominated by the tail of the last pulse, but other pulses can dominate later. Modeling several overlapping pulses as a single broader pulse would overestimates the SDP flux. One should thus be careful when testing the HLE.

💡 Analysis

There is good observationnal evidence that the Steep Decay Phase (SDP) that is observed in most Swift GRBs is the tail of the prompt emission. The most popular model to explain the SDP is Hight Latitude Emission (HLE). Many models for the prompt emission give rise to HLE, like the popular internal shocks (IS) model, but some models do not, such as sporadic magnetic reconnection events. Knowing if the SDP is consistent with HLE would thus help distinguish between different prompt emission models. In order to test this, we model the prompt emission (and its tail) as the sum of independent pulses (and their tails). A single pulse is modeled as emission arising from an ultra-relativistic thin spherical expanding shell. We obtain analytic expressions for the flux in the IS model with a Band function spectrum. We find that in this framework the observed spectrum is also a Band function, and naturally softens with time. The decay of the SDP is initially dominated by the tail of the last pulse, but other pulses can dominate later. Modeling several overlapping pulses as a single broader pulse would overestimates the SDP flux. One should thus be careful when testing the HLE.

📄 Content

arXiv:0901.0680v1 [astro-ph.HE] 6 Jan 2009 Is the Rapid Decay Phase from High Latitude Emission? F. Genet and J. Granot University of Hertfordshire Abstract. There is good observationnal evidence that the Steep Decay Phase (SDP) that is observed in most Swift GRBs is the tail of the prompt emission. The most popular model to explain the SDP is Hight Latitude Emission (HLE). Many models for the prompt emission give rise to HLE, like the popular internal shocks (IS) model, but some models do not, such as sporadic magnetic reconnection events. Knowing if the SDP is consistent with HLE would thus help distinguish between different prompt emission models. In order to test this, we model the prompt emission (and its tail) as the sum of independent pulses (and their tails). A single pulse is modeled as emission arising from an ultra-relativistic thin spherical expanding shell. We obtain analytic expressions for the flux in the IS model with a Band function spectrum. We find that in this framework the observed spectrum is also a Band function, and naturally softens with time. The decay of the SDP is initially dominated by the tail of the last pulse, but other pulses can dominate later. Modeling several overlapping pulses as a single broader pulse would overestimates the SDP flux. One should thus be careful when testing the HLE. Keywords: Gamma-rays: bursts PACS: 98.70.Rz INTRODUCTION Most gamma-ray bursts (GRBs) observed by the Swift satellite show an early steep decay phase (SDP) in their X-ray light curve. It is usually a smooth spectral and temporal continuation of the GRB prompt emission, strongly suggesting that it is the tail of the prompt emission [1]. It is generally explained by High Latitude Emission (HLE), where at late times the observer still receives photons from increasingly larger angles relative to the line of sight, due to the longer path lenght caused by the curvature of the emitting region. These late photons have a smaller Doppler factor, which results in a steep decay of the flux and in a simple relation between the temporal and spectral indices α = 2+β, where Fν(T) ∝T −αν−β [2]. We test the consistency of HLE with the SDP by modeling the prompt emission as a sum of its individual pulses, including their tails. We calculate the flux for a single emission episode in the framework of internal shocks, and then combine several pulses to model the prompt emission. EMISSION OF A SINGLE PULSE We consider an ultra-relativistic (Γ ≫1) thin (of width ≪R/Γ2) spherical expanding shell emitting over a range of radii R0 ≤R ≤R f ≡R0 + ∆R. The Lorentz factor of the emitting shell is assumed to scales as a power-law with radius, Γ2 = Γ2 0(R/R0)−m, where Γ0 ≡Γ(R0). In order to calculate the flux received at any time T by the observer we intergrate over the Equal Arrival Time Surface (EATS; [3]), which is the locus of points from which photons that are emitted at a radius R, angle θ from the line of sight and lab frame time t reach the observer at the same observed time T. For a shell ejected at an observer time Tej, the first photon reaches the observer at a time Tej + T0 with T0 = (1+z)R0/[2(m+1)cΓ2 0]. We also define Tf ≡T0(R f /R0)m+1 = T0(1+∆R/R0)m+1, which is the last time at which photons emitted from the line of sight reach the observer. We choose for the emission spectrum the phenomenological Band function (Band et al., 1993) spectrum, which generally provides a good fit to the prompt GRB emission. The co-moving peak spectral luminosity is assumed to scale as a power-law with radius, L′ ν′p ∝(R/R0)a, where ν′ p(R) is the peak frequency of the emitted νFν spectrum. Since Internal shocks is the most popular model for the prompt emission, we consider it for the following. In this framework, several simplifying assumptions can be made: the outflow is expected to be in the coasting phase (m = 0), and the electrons are expected to be in fast cooling regime. The emission mechanism is assumed to be synchrotron. This leads to ν′ p ∝Rd with d = −1, and L′ ν′p ∝(R/R0)1, i.e a = 1. Then, T0 = (1 + z)R0/(2cΓ2 0), Tf = T0(1 + ∆R/R0) FIGURE 1. Left: Evolution of the shape of one normalized pulse with the normalized frequency ν/ν0. Middle: Evolution of the observed spectrum with time (corresponding to the values of ¯T/ ¯Tf written near each spectrum). The thin lines correspond to the rising part of the pulse, the thick lines to the decaying part of the pulse. ∆R/R0 = 1. ν/ν0(T0) = 1. Right: Comparison of the evolution of the spectral (2 + β; thin lines) and temporal (α; thick lines) slopes at fixed observed frequencies (for E′0 = 0.5 keV and Γ0 = 300, so that E0,obs = 300 keV). and the luminosity is L′ ν′ = L′ 0  R R0 a S

ν′ ν′p ! , S(x) = e1+b1  xb1e−(1+b1)x x < xb, xb2xb1−b2 b e−(b1−b2) x > xb, (1) where S is the normalized Band function, x ≡ν′/ν′ p, with ν′ = (1 + z)ν/δ where ν is the observed frequency, xb = (b1 −b2)/(1 + b1), and b1 and b2 are the high and low energy slopes of the spectrum; z is the redshift of the so

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