Is the Rapid Decay Phase from High Latitude Emission?
aa r X i v : . [ a s t r o - ph . H E ] J a n 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
GRBsis the tail of the prompt emission. The most popular model to explain the SDP is Hight Latitude Emission (HLE). Manymodels for the prompt emission give rise to HLE, like the popular internal shocks (IS) model, but some models do not, suchas sporadic magnetic reconnection events. Knowing if the SDP is consistent with HLE would thus help distinguish betweendifferent prompt emission models. In order to test this, we model the prompt emission (and its tail) as the sum of independentpulses (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 theobserved spectrum is also a Band function, and naturally softens with time. The decay of the SDP is initially dominated bythe tail of the last pulse, but other pulses can dominate later. Modeling several overlapping pulses as a single broader pulsewould overestimates the SDP flux. One should thus be careful when testing the HLE.
Keywords:
Gamma-rays: bursts
PACS:
INTRODUCTION
Most gamma-ray bursts (GRBs) observed by the
Swift satellite show an early steep decay phase (SDP) in their X-raylight curve. It is usually a smooth spectral and temporal continuation of the GRB prompt emission, strongly suggestingthat it is the tail of the prompt emission [1]. It is generally explained by High Latitude Emission (HLE), where at latetimes the observer still receives photons from increasingly larger angles relative to the line of sight, due to the longerpath lenght caused by the curvature of the emitting region. These late photons have a smaller Doppler factor, whichresults in a steep decay of the flux and in a simple relation between the temporal and spectral indices a = + b , where F n ( T ) (cid:181) T − a n − b [2]. We test the consistency of HLE with the SDP by modeling the prompt emission as a sum of itsindividual pulses, including their tails. We calculate the flux for a single emission episode in the framework of internalshocks, and then combine several pulses to model the prompt emission. EMISSION OF A SINGLE PULSE
We consider an ultra-relativistic ( G ≫
1) thin (of width ≪ R / G ) spherical expanding shell emitting over a rangeof radii R ≤ R ≤ R f ≡ R + D R . The Lorentz factor of the emitting shell is assumed to scales as a power-law withradius, G = G ( R / R ) − m , where G ≡ G ( R ) . In order to calculate the flux received at any time T by the observerwe intergrate over the Equal Arrival Time Surface (EATS; [3]), which is the locus of points from which photonsthat are emitted at a radius R , angle q from the line of sight and lab frame time t reach the observer at the sameobserved time T . For a shell ejected at an observer time T e j , the first photon reaches the observer at a time T e j + T with T = ( + z ) R / [ ( m + ) c G ] . We also define T f ≡ T ( R f / R ) m + = T ( + D R / R ) m + , which is the last time atwhich 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, whichgenerally provides a good fit to the prompt GRB emission. The co-moving peak spectral luminosity is assumed toscale as a power-law with radius, L ′ n ′ p (cid:181) ( R / R ) a , where n ′ p ( R ) is the peak frequency of the emitted n F n spectrum.Since Internal shocks is the most popular model for the prompt emission, we consider it for the following. In thisframework, several simplifying assumptions can be made: the outflow is expected to be in the coasting phase ( m = n ′ p (cid:181) R d with d = −
1, and L ′ n ′ p (cid:181) ( R / R ) , i.e a =
1. Then, T = ( + z ) R / ( c G ) , T f = T ( + D R / R ) IGURE 1. Left:
Evolution of the shape of one normalized pulse with the normalized frequency n / n . Middle:
Evolution ofthe observed spectrum with time (corresponding to the values of ¯ T / ¯ T f written near each spectrum). The thin lines correspond tothe rising part of the pulse, the thick lines to the decaying part of the pulse. D R / R = n / n ( T ) = Right : Comparison of theevolution of the spectral (2 + b ; thin lines) and temporal ( a ; thick lines) slopes at fixed observed frequencies (for E ′ = . G = E , obs =
300 keV). and the luminosity is L ′ n ′ = L ′ (cid:18) RR (cid:19) a S n ′ n ′ p ! , S ( x ) = e + b (cid:26) x b e − ( + b ) x x < x b , x b x b − b b e − ( b − b ) x > x b , (1)where S is the normalized Band function, x ≡ n ′ / n ′ p , with n ′ = ( + z ) n / d where n is the observed frequency, x b = ( b − b ) / ( + b ) , and b and b are the high and low energy slopes of the spectrum; z is the redshift of thesource and d L the luminosity distance between the source and the observer. We define n = G n ′ / ( + z ) , where n ′ ≡ n ′ p ( R ) . One should note that most of the results derived in the following hold only in the model of internalshocks, and not in a more general case.The observed flux is then (in the framework of internal shocks): F n ( T ≥ T ej + T ) = F (cid:18) T − T ej T (cid:19) − "(cid:18) min ( T − T ej , T f ) T (cid:19) − S (cid:18) nn T − T ej T (cid:19) , (2)where F ≡ ( + z ) L / ( p d L ) . Figure 1 (left panel) shows the variation of a pulse shape with the normalized frequency n / n . Different shapes can be obtained, form spiky to rouder. For m = d = − n F n spectrum decreases with time as n p = n / ˜ T , and ˜ T = ( T − T ej ) / T = + ¯ T . This corresponds to a softeningof the spectrum with time (Fig. 1 right panel) which agrees with observations. On the same panel we compare theevolution of the instantaneous spectral slope b ≡ − d log F n / d log n with the temporal slope ˜ a ≡ − d log F n / d log ˜ T ,where ˜ T = ( T − T ej ) / T = + ¯ T : we can see that the HLE relation ˜ a = + b is valid as soon as ¯ T > ¯ T f . One should becareful that this is true only in the framework of internal shocks model, and with this definition of the temporal slope(for exemple, ¯ a ≡ − d log F n / d log ¯ T , which is another definition of the temporal slope, approaches 2 + b only at latetime). COMBINING PULSES TO OBTAIN THE PROMPT EMISSION
Within our model, the prompt emission is the sum of independent pulses, and the SDP is thus the sum of the tails ofthese pulses. For a prompt emission composed of several equal pulses, at late time the contribution of each pulse isequal, and the temporal slope just after the peak of a pulse increases with its ejection time T e j . When varying severalparameters among the different pulses, the late time flux ratio of the pulse tails is the ratio of their F peak T + b f . Justafter the peak of the last pulse, the SDP is dominated by the last pulse. This shows that several pulses can dominate IGURE 2. Left:
Exemple of a prompt emission consisting of three pulses with T ej = − s, s, s, T = s for all three pulses, D R / R = , , , and F peak / F = . , , . . Thin non-solid lines represent individual pulses, while the thick solid line shows thetotal prompt emission. Right:
Comparison between a fit with several pulses ( here) and a fit with one broad pulse. Thin non solidlines shows each individual pulse, the thin solid line shows the total prompt emission, and the thick solid line shows a possible fitwith one broad pulse. The normalized frequency is n / n = . . Both panels are in logarithmic scale. the SDP at different times, as one can see in the left panel of Fig. 2. Therefore, one should be careful to consider thiswhen studying the temporal and spectral behavior of the SDP.Figure 2 (right panel) shows what can happen if, because of noisy data or coarse time bins, a prompt emission (thinsolid line) which is actually composed by several pulses is fitted by one broad pulse (thick solid line): the fit wouldgive a tail with the same temporal slope at late time than the actual prompt tail, but with no higher temporal slopes justat the end of the prompt, the whole tail of this broad pulse being close to a power law. Moreover, this overestimatesthe flux of the SDP. It is important to keep this in mind when confronting such a model with actual data. CONCLUSION
We have outlined a model for the prompt emission and its tail. This model contains a restricted number of freeparameters, 10 per pulse: a , m , d , F , b , b , E ( T ) , T , T f and T e j . In the case for internal shocks, this can bereduced to 7: m = d = − a =
1; as in this framework D R ∼ R is expected, one can fix F f / T = + D R / R = N pulses, the total number of free parameters can be further reduced to 3 ( N + ) , insteadof 6 N , as we expect the Band function parameters ( b , b and E ( T ) ) to be similar for all pulses.The shape of a pulse can vary considerebly in our model, from very spiky to rounder, which qualitatively reproducesthe observed diversity. The observed spectrum is a pure Band function as the emitted one in the case of internal shocks,and our model naturally produces a softening of the spectrum, as is observed.When combining several pulses to model the prompt emission, the Steep Decay Phase is initailly dominated by thelast pulse, and is dominated at late times by the pulse with the largest F peak T + b f (essentially the widest pulse, exceptif there is a large difference of flux between the pulses), but can be dominated by other pulses in between.When fitting data, one should be careful not to consider several overlapping pulses as a single broad pulse, whichwould lead to an overestimate of the prompt tail flux and a misinterpretation of the steep decay phase. REFERENCES
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