# X-ray flares from dense shells formed in gamma-ray burst explosions

R. Hascoet, A.M. Beloborodov, F. Daigne, R. Mochkovitch

► To cite this version:

R. Hascoet, A.M. Beloborodov, F. Daigne, R. Mochkovitch. X-ray flares from dense shells formed in gamma-ray burst explosions. Monthly Notices of the Royal Astronomical Society, 2017, 472 (1), pp.L94-L98. 10.1093/mnrasl/slx143. hal-01704358

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# X-ray flares from dense shells formed in gamma-ray burst explosions

R. Hascoët,<fnref n="1" /> A. M. Beloborodov,<fnref n="1" /> F. Daigne<fnref n="2" /> and R. Mochkovitch<fnref n="2" />

Accepted 2017 September 5. Received 2017 September 1; in original form 2017 February 8

## ABSTRACT

Bright X-ray flares are routinely detected by the Swift satellite during the early afterglow of gamma-ray bursts, when the explosion ejecta drives a blast wave into the external medium. We suggest that the flares are produced as the reverse shock propagates into the tail of the ejecta. The ejecta is expected to contain a few dense shells formed at an earlier stage of the explosion. We show an example of how such dense shells form and describe how the reverse shock interacts with them. A new reflected shock is generated in this interaction, which produces a short-lived X-ray flare. The model provides a natural explanation for the main observed features of the X-ray flares – the fast rise, the steep power-law decline and the characteristic peak duration $\Delta t/t \simeq 0.1–0.3$.

**Key words:** radiation mechanisms: non-thermal – shock waves – gamma-ray burst: general.

## 1 INTRODUCTION

About 30 per cent of gamma-ray bursts (GRBs) show X-ray flares during their early afterglow (e.g. Burrows et al. 2005a,b; Falcone et al. 2007; Chincarini et al. 2010; Yi et al. 2016). The X-ray flares are characterized by a fast rise of luminosity (typically by a factor of ~10) to a sharp peak followed by a power-law decay (Chincarini et al. 2007). The ratio of the characteristic temporal width of the peak $\Delta t_f$ to its time of occurrence since the beginning of the GRB is typically $\Delta t_f/t_f \sim 0.1–0.3$. At the end of the flare, the X-ray flux resumes the underlying smooth decay of the early afterglow, with no apparent flux increment left. At other wavelengths (optical or gamma-rays), the flares often remain undetected (Troja et al. 2015; see however Li et al. 2012; Peng et al. 2014).

The fast rise, steep decay and relatively short duration of the X-ray flares motivated several authors to invoke late activity of the central engine and associate the flares with the jet produced by this activity – a scaled-down version of the prompt GRB emission (e.g. Burrows et al. 2005b; Fan & Wei 2005; Romano et al. 2006; Zhang et al. 2006; Lazzati & Perna 2007; Yu & Dai 2009; Bernardini et al. 2011; Margutti et al. 2011; Troja et al. 2015; Jia, Uhm & Zhang 2016; Uhm & Zhang 2016). This possibility requires a mechanism that keeps the central engine active for about one day, as some flares are observed as late as a day after the GRB explosion. It also requires the engine to be variable in a special way on a time-scale $\Delta t_f \sim (0.1–0.3) t_f$, in contrast to the much faster, multipeak variability observed during the prompt GRB. Conversely, Beniamini & Kumar (2016) explain flares by the photospheric emission from a slow jet ejected together with the ultrarelativistic ejecta, which does not require a late activity of the source. In this Letter, we propose an alternative

possibility, where flares are produced by the long-lived RS when it crosses the tail of the GRB ejecta.

Our model does not require a late activity of the central engine; it requires that the Lorentz factor in the end of the GRB explosion is significantly reduced, from $\Gamma \sim 100$ to $\Gamma \sim 10$, which leads to the formation of an extended tail during the propagation of the ejecta (e.g. Genet et al. 2007; Uhm & Beloborodov 2007). As described in Section 2, dense shells are expected to form in the tail of the expanding ejecta before the passage of the RS. In Section 3, we estimate the bolometric light curve of emission produced by the encounter of the RS with the dense shell, and find that it resembles the observed X-ray flares. Comparison of the model with observations and future possible ways of its development are discussed in Section 5.

## 2 DENSE SHELLS AFTER INTERNAL SHOCKS

The observed variability of GRB emission (Fishman & Meegan 1995; Beloborodov, Stern & Svensson 2000; Guidorzi et al. 2012) suggests that the Lorentz factor of the relativistic ejecta $\Gamma$ fluctuates in a broad range of time-scales, from a few milliseconds to several minutes. In the expanding outflow, regions where the radial gradient of Lorentz factor is negative (i.e. where $\Gamma$ is decreasing outwards) are progressively compressed. If the ejecta is not magnetically dominated, internal shocks eventually form and propagate. The internal shocks can impact the prompt $\gamma$-ray emission mainly in two ways: (i) in the subphotospheric (optically thick) region, shock heating can offset the adiabatic cooling of radiation and change its spectrum from thermal to the observed Band-type shape through the Comptonization process and additional synchrotron emission (Mészáros & Rees 2000; Pe'er, Mészáros & Rees 2006; Beloborodov 2010; Vurm, Beloborodov & Poutanen 2011);

<footnote n="1">¹Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120th Street, New York, NY 10027, USA</footnote>

<footnote n="2">²UPMC-CNRS, UMR7095, Institut d'Astrophysique de Paris, F-75014 Paris, France</footnote>

<footnote>* E-mail: daigne@iap.fr</footnote>
<figure><img src="image_3.png" /><figcaption>Figure 1. Hydrodynamic simulation of a spherical outflow undergoing internal shocks during its expansion. The flow is injected cold with a constant kinetic power during $t_w = 10$ s at a radius $R_0 = 0.1$ ct$_w$. The figure shows the Lorentz factor profile as a function of normalized Lagrangian mass coordinate (the front edge of the outflow being at $m=0$) at three moments of time in the fixed (lab) frame attached to the central source: the initial (injected) state (dashed line), during the internal shock phase at $t_{lab} = 800$ s (dotted line) and after internal shocks ended at $t_{lab} = 10^5$ s (solid line). At $t_{lab} = 10^5$ s, well after the internal shock phase, the outflow has cooled adiabatically and entered a ballistic phase. As a result of internal shocks, two large portions of the ejecta have a uniform Lorentz factor with $\delta\Gamma/\Gamma \lesssim 0.01$. We have checked that adding a fast variability component in the outflow (leading to several generations of internal shocks) does not significantly change the final Lorentz factor distribution.</figcaption></figure>

(ii) outside the photosphere, shocks can continue to produce synchrotron emission in the gamma-ray band if they efficiently accelerate particles (Rees & Meszaros 1994; Kobayashi, Piran & Sari 1997; Daigne & Mochkovitch 1998).

As described in the various studies cited above, internal shocks will develop in a relativistic outflow which is initially inhomogeneous. These shocks can amplify the density variations, especially if they are radiatively efficient. More importantly, the outflow tends to develop a few massive dense shells long after the shocks. At observer time $t_0$ when the internal shock phase ends, the fluctuations of Lorentz factor $\Gamma$ have been damped and $\Gamma$ acquires a monotonic radial profile. Its slope $d\Gamma/dr > 0$ is significantly inhomogeneous, as illustrated below by a hydrodynamical simulation. A remarkable feature of the post-internal-shock flow is the presence of a few ‘plateaus’ with $\delta\Gamma/\Gamma \ll 1$ in the profile of $\Gamma(r)$ or $\Gamma(m)$ where $m$ is the Lagrangian mass coordinate in the flow (see Fig. 1). These plateaus contain a large mass and the subsequent ballistic expansion of the outflow at observer times $t \gg t_0$ generates a large contrast in density, as can be seen from the following estimate.

Consider a shell of width $\Delta_{sh}$ with the Lorentz factor variation across the shell $\delta\Gamma = (d\Gamma/dr)\Delta_{sh}$. The shell expands ballistically with a velocity difference of the order of $(\delta\Gamma/\Gamma^3)c$. Then, at observer time $t \gg t_0$ (with $dt = dr/2\Gamma^2c$), its width is given by

$$ \frac{\Delta_{sh}(t)}{\Delta_0} = 1 + 2 \frac{\delta\Gamma}{\Gamma} \frac{t}{t_0}, \qquad (1) $$

where $\Delta_0$ is the shell width at $t_0$. The relevant $\Delta_0$ at the end of the internal shock phase corresponds to the longest variability time-scale observed in the burst, $\Delta_0 \sim ct_0$, which is comparable to the burst duration, e.g. $t_0 \sim 1-10$ s. In contrast, the time $t$ given to ballistic expansion before the RS crosses the outflow in our model

is associated with the observed time of the X-ray flares, $t \sim t_f \sim 10^3-10^4$ s.

The large ratio $t/t_0$ implies that $\delta\Gamma$ should strongly affect the density structure of the outflow. The parts of the outflow with $\delta\Gamma \sim \Gamma$ will become much thicker than the parts with $\delta\Gamma \ll \Gamma$, which will form dense shells. Consider for example a region $\Delta_1$ with $|\delta\Gamma/\Gamma|_1 \sim 1$ and an adjacent region $\Delta_2$ with $|\delta\Gamma/\Gamma|_2 \ll 1$, i.e. a plateau resulting from the previous propagation of internal shocks. Assuming that the bulk Lorentz factor $\Gamma$ is similar in the two regions, the density contrast at time $t$ is roughly given by $\rho_2/\rho_1 \sim \Delta_1/\Delta_2$. From equation (1), this leads to

$$ \frac{\rho_2}{\rho_1} \sim 2 \frac{t}{t_0}. \qquad (2) $$

The plateaus appearing after the propagation of internal shocks (region 2) will therefore eventually form dense massive shells.

These plateaus evolving in dense massive shells were indeed observed and described in full hydrodynamical studies of internal shocks (Daigne & Mochkovitch 2000). In this work, we study again this effect in outflows with various initial $\Gamma(m)$ using a one-dimensional (spherically symmetric) hydrodynamic code with a second-order HLLC approximate Riemann solver (Mignone & Bodo 2005) on a moving mesh (Duffell & MacFadyen 2011). In agreement with Daigne & Mochkovitch (2000), we find that the plateaus typically have a small $\delta\Gamma/\Gamma$ of a few per cent, even in the case of adiabatic shocks (an example is shown in Fig. 1).

Similar (but exactly flat) plateaus are produced by simplified simulations in which the outflow is represented by a large number of discrete shells that interact by direct collisions (Daigne & Mochkovitch 1998). In this model, the ejecta is assumed to be cold and all pressure waves are neglected. This simplification is not crucial as it gives results similar to the detailed hydrodynamical simulations (cf. Daigne & Mochkovitch 2000). Therefore, we believe that the formation of massive dense shells in the outflow is a robust consequence of internal shocks and we study below the consequences during the afterglow phase. For this purpose, we extend the simplified ‘shell’ model to the external blast wave and the RS propagation through the outflow, following the procedure described in Genet et al. (2007) and Hascoët et al. (2011, 2012, 2014a).

### 3 FLARES: A SIMPLIFIED MODEL

We use this simplified ‘shell’ model to follow the RS propagation through the outflow and its interaction with the dense shells. An example is shown in Fig. 2. The central engine is assumed to produce an outflow of variable Lorentz factor distribution and duration $t_w = 10$ s. The injected kinetic power $\dot{E}_K = 10^{53}$ erg s$^{-1}$ is to be constant. The material ejected at the end has a low Lorentz factor, which will lead to the formation of a tail and a long-lived RS. Fig. 2 shows how the initial distribution of $\Gamma$ is changed after the internal shocks. Most of the outflow mass concentrates into three shells with Lorentz factors $\Gamma \approx 240, 120$ and $30$ carrying about 10, 20 and 40 per cent of the total mass. The last 30 per cent of the outflow have not been affected by internal shocks.

We followed the outflow to much larger radii where the RS eventually crosses all three shells. The dynamics of the RS depends on the external density. We have calculated two cases: a uniform external medium $\rho = \text{const}$ and a wind-like medium $\rho = Ar^{-2}$, where A is a constant which depends on the mass-loss rate of the GRB progenitor. Its typical range for Galactic Wolf-Rayet stars is $10^{11}-10^{12}$ g cm$^{-1}$ (Crowther 2007). Recent modelling of optical and
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Left: Lorentz factor distribution as a function of Lagrangian mass coordinate; the front edge of the outflow is at $m = 0$. The initial distribution $\Gamma(m)$ is shown by the dashed curve, and the final distribution (after internal shocks) is shown by the solid curve. Right: bolometric light curves (assuming a radiative efficiency $\epsilon_e = 0.1$) calculated for different ambient media (dotted lines). The wind density $\rho$ is parametrized by $A_{11}$ as follows: $\rho R^2 = 10^{11} A_{11} \text{ g cm}^{-1}$. In the uniform medium case, $n$ is the density in cm$^{-3}$. From top to bottom: wind medium with $A_{11} = 1$, uniform medium with $n = 1$ ($L_{\text{bol}}$ has been divided by $10^3$), and the case of a 'naked' burst (no ambient medium; $L_{\text{bol}}$ has been divided by $10^5$) shown for comparison. The solid lines show the same light curves for the case of an anisotropic emission with $A(\theta)$ given by equation (7) and $\theta_0 = 70^\circ$. In the case of the naked burst, the three early pulses (corresponding to the three pairs of internal shocks which initially propagate within the ejecta) are followed by high-latitude (off-axis) emission.</figcaption></figure>

GeV flashes in GRBs suggest that $A \sim 10^{11} \text{ g cm}^{-1}$ is typical for GRB progenitors (Hascoët et al. 2014b; Hascoët et al. 2015).

## 3.1 Bolometric light curve

The simulation gives the power dissipated in the RS and its bolometric luminosity with two assumptions: (i) that a constant fraction $\epsilon_e \sim 0.1$ of the dissipated energy is injected in shock-accelerated electrons and (ii) that these electrons promptly radiate their energy ('fast-cooling' regime). To find the bolometric light curve received by a distant observer, we transform the emission from the flow frame to the observer frame and take into account the spherical curvature of the emitting shells. The radiation received by the observer from the RS is shown in Fig. 2. In this figure, we also included the prompt radiation produced by the internal shocks, assuming the same $\epsilon_e = 0.1$.

When the RS goes through the dense shells, there is a sudden rise of dissipated power producing a flare in the light curve. The immediate rise of luminosity to the peak of the flare in Fig. 2 is an artefact of the simplified model, which does not resolve the propagation of the RS through the dense shell. An accurate hydrodynamical model would give a slower but still a steep rise; it is discussed and estimated in Section 4 below.

In our example with three dense shells, two flares are produced; the crossing of the first shell (at $\Gamma \approx 240$) corresponds to the initial rise of dissipated power from the RS. Using the approximate expression for the flare radius $R_f \sim \Gamma^2 ct_f$, one can estimate the observed time of the flare,

$$t_f = \begin{cases} \left( \frac{3E}{4\pi n m_p c^5 \Gamma^8} \right)^{1/3} & \approx 840 \left( \frac{E_{54}}{n} \right)^{1/3} \Gamma_2^{-8/3} \text{ s (uniform)}, \\ \frac{E}{4\pi A c^3 \Gamma^4} & \approx 300 \left( \frac{E_{54}}{A_{11}} \right) \Gamma_2^{-4} \text{ s (wind)}. \end{cases} \quad (3)$$

Here, $\Gamma$ is the Lorentz factor of the massive shell ($\Gamma_2 = \Gamma/100$), $E$ is the kinetic energy of the ejecta which has already crossed the RS

($E_{54} = E/10^{54}$ erg), $n$ is the density of the uniform medium in cm$^{-3}$ and $A_{11}$ is the density parameter of the wind medium in units of $10^{11} \text{ g cm}^{-1}$. One can see that the flare can be significantly delayed and this delay is sensitive to the Lorentz factor of the massive shell.

We also observe that the flares produced by this mechanism share two key features with the observed X-ray flares: (i) after the flare the light curve returns to the pre-flare decaying afterglow and (ii) the flare duration $\Delta t_f$ is proportional to its time of occurrence $t_f$.

The simulation shown in Fig. 2 assumes fast cooling at all times, which gives a constant radiative efficiency equal to $\epsilon_e$. The flux enhancement during the flare can be even higher if before and after the flare the emission is in the slow-cooling regime. Then the radiative efficiency can be significantly increased during the flare.

## 3.2 Effect of anisotropic emission

In the simplest model, which assumes isotropic emission in the fluid frame, the characteristic temporal width of the flare is $\Delta t_f \sim \tau$ where

$$\tau = \frac{R}{2\Gamma^2 c^2} \sim t_f. \quad (4)$$

Then the ratio $\Delta t_f/t_f$ is close to unity, greater than the typical observed ratio $\Delta t_f/t_f \sim 0.1-0.3$.

Significant anisotropy is, however, expected and supported by the fast luminosity variations observed in GRB afterglows (Beloborodov et al. 2011). The effect of anisotropy on the flare light curve is demonstrated by the following simple model.

For the moment, let us picture the encounter of the RS with the dense shell as an instantaneous flash of energy $E_f$ at a radius $R$. The bolometric light curve received from such a flash is given by (Beloborodov et al. 2011),

$$L(t) = \frac{2E_f A(\theta)}{\tau} \left(1 + \frac{t - t_f}{\tau}\right)^{-3} \quad (t > t_f), \quad (5)$$
where

$$ \cos \theta = \frac{\tau - (t - t_f)}{\tau + (t - t_f)}, \qquad (6) $$

$\theta$ is the angle with respect to the radial direction in the fluid frame, and $A(\theta)$ describes the angular distribution of emission; for isotropic emission $A(\theta) = 1$. The magnetic field in GRB shocks is in the shock plane, and their synchrotron radiation is anisotropic even when the emitting electrons are isotropic; in this case $A(\theta) = (3/4)(1 + \cos^2\theta)$. Anisotropy of the electron distribution can further enhance the anisotropy of the emitted radiation, which significantly shortens the duration of the flare (see fig. 3 in Beloborodov et al. 2011), leading to a characteristic duration $\Delta t_f \ll t_f$, with the shorter duration corresponding to a steeper decay of luminosity after the peak. As an illustration, we consider an example where the anisotropy in the plasma rest frame is modelled by

$$ A(\theta) \propto \exp\left(\frac{|\cos \theta|}{\cos \theta_0}\right) \qquad (7) $$

This leads to an initial decay index

$$ \left. \frac{d \log L}{d \log t} \right|_{t=t_f} = - \left( 3 + \frac{2}{\cos \theta_0} \right) \frac{t_f}{\tau} \qquad (8) $$

We consider a moderate anisotropy by adopting $\theta_0 = 70^{\circ}$ (which leads to 90 per cent of the energy being beamed within $\theta_0$) and the resulting light curves resemble the observed X-ray flares as shown in the right-hand panel of Fig. 2.

## 4 SHOCK REFLECTION FROM THE DENSE SHELL

In the simplified model of Section 3, the RS crossing of the thin dense shell instantaneously dissipates part of the shell kinetic energy, and the observed light curve of the flash is determined by the curvature effect. In this section, we describe a more realistic picture of the RS interaction with the dense shell (see Kumar & Piran 2000 for a similar theoretical picture). We also estimate the time-scales for the rise and duration of the dissipation event.

Two main effects should be taken into account: (i) the RS is greatly slowed down as it enters the dense gas, and it takes a finite time (estimated below) for the RS to cross the shell; (ii) the RS by itself does not efficiently decelerate the shell. The shell continues to plow through the ejecta that was previously accumulated in the blast wave. This ploughing launches a new internal shock inside the blast wave (Fig. 3). Effectively, the RS splits in two when it enters the dense shell. One (slow) shock continues to propagate into the dense shell and the other (fast) shock is reflected and launched back into the blast wave material. The fast shock dissipates most of the kinetic energy of the massive shell measured in the rest-frame of the blast wave. Below we call it ‘internal forward shock’ (IFS). It is the IFS which emits the X-ray flare.

The velocity of RS propagation through the dense shell, $\beta_{RS} = v_{RS}/c \ll 1$, may be estimated using the approximate pressure balance between the RS and IFS. Let $\rho_2$ be the proper density of the shell just before it is reached by the RS. Assuming that the shell is cold and weakly magnetized, its pressure behind the RS jumps to

$$ p_{RS} = \frac{3}{4} \beta_{RS}^2 \rho_2 c^2. \qquad (9) $$

Now, let us estimate the pressure behind the IFS, $p_{IFS}$. The dense shell acts like a wall and the Lorentz factor downstream of the IFS

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Schematic illustration of the RS encounter with a thin massive shell. The RS becomes slow as it enters the dense material and a new 'IFS' is created, which propagates away from the dense shell into the blast wave material previously shocked by the RS. The figure also shows the FS in the external medium and the contact discontinuity (CD) that separates the ejecta from the external medium inside the blast wave.</figcaption></figure>

is almost equal to the shell Lorentz factor $\Gamma_{ej}$. The relative Lorentz factor of the upstream and downstream is

$$ \Gamma_{\text{rel}} = \frac{1}{2} \left( \frac{\Gamma_{\text{ej}}}{\Gamma_{\text{bw}}} + \frac{\Gamma_{\text{bw}}}{\Gamma_{\text{ej}}} \right), \qquad (10) $$

where $\Gamma_{bw}$ is the Lorentz factor of the blast-wave material upstream of the IFS, and we have used $\Gamma_{ej} \gg 1$ and $\Gamma_{bw} \gg 1$. The gas pressure may be written as $p \approx (\rho u/3)(1 + c^2/u)$ (Beloborodov & Uhm 2006), where $\rho$ is the density and $u$ is the energy per unit rest mass. The gas in the downstream of IFS has been shocked twice: first by the RS and then by the IFS, both times with the same relative Lorentz factor $\Gamma_{rel}$. Therefore, $u \approx \Gamma_{rel}^2 c^2$ and $\rho \approx (4\Gamma_{rel})^2 \rho_1$, where $\rho_1$ is the outflow proper density prior to the crossing by the RS and IFS. In the estimate for $\rho$, we used the compression factor of $4\Gamma_{rel}$ for a strong adiabatic shock. This gives

$$ p_{\text{IFS}} \approx \frac{16}{3} (\Gamma_{\text{rel}}^4 - 1) \rho_1 c^2. \qquad (11) $$

The approximate pressure balance $p_{RS} \sim p_{IFS}$ gives an estimate for $\beta_{RS}$,

$$ \beta_{\text{RS}} \approx \frac{8}{3} \left[ (\Gamma_{\text{rel}}^4 - 1) \frac{\rho_1}{\rho_2} \right]^{1/2} . \qquad (12) $$

Using the obtained $\beta_{RS}$, we estimate the time it takes the RS to cross the dense shell, $t_{cross}$. The shell thickness is comparable to

$$ \Delta_{\text{sh}} = \eta \left( \frac{\rho_1}{\rho_2} \right) \Delta_{\text{bw}}, \qquad (13) $$

where $\Delta_{bw} \lesssim R/\Gamma_{bw}^2 \approx ct$ is the blast wave thickness, and $\eta = m_{sh}/m_{bw} \lesssim 1$ is the ratio between the mass of the incoming dense shell, $m_{sh}$, and the outflow mass already accumulated in the blast wave, $m_{bw}$. Then, we find

$$ t_{\text{cross}} \approx \frac{\Delta_{\text{sh}}}{2c\beta_{\text{RS}}} \lesssim 0.2 \eta \left( \frac{\rho_1}{\rho_2} \right)^{1/2} (\Gamma_{\text{rel}}^4 - 1)^{-1/2} t. \qquad (14) $$

Typical expected $\rho_2/\rho_1 > 30$ implies a short $t_{cross} \lesssim (0.03-0.07)t$.

The shell Lorentz factor is reduced from $\Gamma_{ej}$ to $\Gamma_{bw}$ when it sweeps up the rest mass

$$ m_{\star} \sim \frac{m_{\text{sh}}}{\Gamma_{\text{rel}}^2}, \qquad (15) $$
where we took into account that the gas ahead of the shell has been heated by the RS, and so its inertial mass is increased by the factor of $Γ_{rel}$. Assuming a relativistic IFS speed ~c, this gives the following estimate for the observed duration of the shell deceleration:

$$t_{\text{diss}} \sim \frac{m_*}{m_{\text{bw}}} \frac{\Delta_{\text{bw}}}{c} \left( \frac{\Gamma_{\text{bw}}}{\Gamma_{\text{ej}}} \right)^2 \sim \frac{m_*}{m_{\text{bw}}} \left( \frac{\Gamma_{\text{bw}}}{\Gamma_{\text{ej}}} \right)^2 t. \quad (16)$$

After $t_{diss}$ the IFS will continue to propagate in the blast wave material. However, its luminosity must decline as the relative Lorentz factor of the shock decreases after $t_{diss}$ (one can make an analogy with the standard afterglow decay after the transition to self-similar deceleration). The fact that $t_{diss} \ll t$ implies that the IFS luminosity declines on a time-scale much shorter than t, i.e. it implies a steep decline of dissipated power.

The IFS may be mildly relativistic, if the flares occur long after the formation of the ballistic outflow ($t \gg t_0$). If the blast wave dynamics between the flares is approximated as self-similar, one finds $Γ_{ej}/Γ_{bw} = \sqrt{2}$ and 2 in the case of the wind and uniform external medium, respectively.

The time $t_{cross}$ is associated with the rise of the energy injection event and $t_{diss}$ determines its duration. The above results give typical $t_{cross} \ll t_{diss} \ll t$. A fast rise is predicted, on a time-scale $t_{cross} \lesssim 5 \times 10^{-2} t$, in agreement with the shape of observed flares (Chincarini et al. 2007). For short $t_{diss}$ the observed duration of the flare will be determined by the curvature (and anisotropy) effects, as described in Section 3.2. Here, we note an additional effect which reduces the observed duration of the flare. While the pre-flare afterglow is emitted by gas moving with $Γ_{bw}$, the flare is emitted by gas moving with $Γ_{ej} > Γ_{bw}$ (the gas behind the IFS). Therefore, even without the anisotropy effect, the curvature time-scale for the flare is compressed by the factor of $Δt_f/t_f \sim (Γ_{bw}/Γ_{ej})^2$, which is of the order of 0.2–0.5 (see above). The temporal properties of the bolometric emission of the flares are then in good agreement with observations showing a fast rise $t_{rise} \ll t$ and a slower decay with $t_{decay} \simeq 0.1–0.3t$. If electrons accelerated in the flare radiate mainly in X-rays in fast cooling regime, the shape of the flare in X-rays should have the same properties.

## 5 CONCLUSIONS

If GRB outflows have tails where the Lorentz factor is variable and extends down to low values $Γ \lesssim 10$, a few dense shells will develop as a result of internal shocks. This Letter then suggests that the propagation of a long-lived RS through this structured ejecta can provide a new explanation for the X-ray flares. The dense shells give sudden energy injections to the GRB afterglow. We examined the dynamics of the injection event and showed that it generates a new internal shock behind the blast wave, which produces a short-lived flare. This mechanism provides a natural explanation for the fast rise of the X-ray flares and their observed duration $Δt_f/t_f \sim 0.1–0.3$.

The proposed scenario assumes two radiative properties of the shock: (i) a fraction of the shock energy is transferred to non-thermal accelerated electrons capable of emitting synchrotron X-rays and (ii) the electrons radiate X-rays in the fast cooling regime. Most of the post-shock electrons behind the IFS may have moderate energies and radiate in low-frequency bands. As long as the low-energy electrons are in the slow-cooling regime, most of the shock luminosity will be produced at higher frequencies (X-rays) by the more energetic electrons in the tail of a non-thermal distribution. In this case, the optical counterpart of the X-ray flares will be weak and occur on a much longer time-scale. Detailed modelling of hydrodynamics and radiative properties of the flares are deferred

to a future work. As described in Section 4, the flare is emitted by gas that has been shocked twice and has high density and pressure. This gas can also possess strong magnetic fields and can radiate more efficiently compared with the pre-flare afterglow. This will also help to satisfy the conditions (i) and (ii).

## ACKNOWLEDGEMENTS

It is a pleasure to thank H. van Eerten and A. I. MacFadyen for useful discussions on hydrodynamics simulations. This work has been partially supported by the Programme National Hautes Energie (PNHE) and the French Space Agency (CNES). AMB was supported by NSF grant AST-1412485, NASA grant NNX15AE26G, and a grant from the Simons Foundation (no.446228, Andrei Beloborodov).

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