Gamma-ray Burst Prompt Emission Spectrum and $E_{\text{p}}$ Evolution Patterns in the ICMART Model

Gamma-ray bursts (GRBs) are the most luminous explosions in the universe. Their bursty emission in the hard-X-ray/soft-$\gamma$-ray band is usually called the “prompt emission” (Zhang, 2018). The temporal structure of the prompt emission exhibits diverse morphologies (Fishman & Meegan, 1995), which can vary from a single smooth pulse to extremely complex light curves with many pulses overlapping within a short duration. Gao et al. (2012) proposed that the prompt emission light curves are typically the superposition of an underlying slow component and a more rapid fast component, where the fast component tends to be more significant in high energies and becomes less significant at lower frequencies (Vetere et al., 2006).

The photon number spectrum of prompt emission, both for time-resolved spectrum and time-integrated spectrum, can usually be fitted with a broken power law known as the Band function (Band et al., 1993)

where $A$ is the normalization factor, $E_{0}$ is the break energy in the spectrum, $\alpha$ and $\beta$ are the low-energy and high-energy photon spectral indices with $\alpha$ ranging at $(-2,0)$ and $\beta$ ranging at $(-4,-1)$ (Preece et al., 2000). The peak energy of the $E^{2}N(E)$ spectrum ($E_{p}=(2+\alpha)E_{0}$) distributes within several orders of magnitude but clusters around $200-300\text{keV}$ (Preece et al., 2000; Goldstein et al., 2013). For bright bursts, two types of evolution patterns between $E_{\text{p}}$ and flux are usually observed, i.e. the hard-to-soft pattern (Norris et al., 1986) where $E_{\text{p}}$ decreases throughout the pulse and the intensity-tracking pattern (Kargatis et al., 1994; Bhat et al., 1994; Golenetskii et al., 1983) where $E_{\text{p}}$ tracks the radiation intensity.

After decades of investigations, the origin of GRB prompt emission is still under debated. The main obstacle in front of theorists is the composition of GRB outflows. For matter-dominated scenario, the most representative model is the so-called the internal shock model (Rees & Meszaros, 1994). In this model, a relativistic unsteady outflow is generated from the central engine, which can be represented as a succession of shells ejected with random velocity, mass and width. Mechanical collisions between the faster late shells and the slower early shells could convert a certain fraction of the relative kinetic energy into internal energy by the resulting internal shocks. The internal energy thus generated is then radiated via synchrotron or inverse Compton scattering and giving rise to the prompt emission. In this scenario, the non-thermal spectrum is naturally expected as long as the collision radius is beyond the photosphere radius, and the temporal variability is attributed to the erratic activity of the central engine while the angular spreading time of shells at different radius causes the variable timescales (Rees & Meszaros, 1994; Kobayashi et al., 1997). This model is difficult to account for superposed slow and fast variability components, unless the central engine itself carries these two variability components in the time history of jet launching, which has no proper explanation in physics (Hascoët et al., 2012). For the evolution patterns between $E_{\text{p}}$ and flux, the internal shock model is easy to explain intensity-tracking pattern, but difficult to interpret the hard to soft pattern. In addition, the internal shock model also faces other challenges (Zhang & Yan, 2011, for a review), including the efficiency problem (Panaitescu et al., 1999; Kumar, 1999), fast cooling problem (Ghisellini et al., 2000; Kumar & McMahon, 2008), the electron number excess problem (Daigne & Mochkovitch, 1998; Shen & Zhang, 2009), the missing bright photosphere problem (Zhang & Pe’Er, 2009; Daigne & Mochkovitch, 2002) and so on.

In order to solve these problems, people proposed that GRB outflows should be Poynting flux dominated instead of matter-dominated. Significant magnetic dissipation may happen to power GRB prompt emission through different mechanisms, including MHD-condition-broken scenarios (Usov, 1994; Zhang & Mészáros, 2002), radiation-dragged dissipation model (Mészáros & Rees, 1997), the slow dissipation model (Thompson, 1994; Drenkhahn & Spruit, 2002; Giannios, 2008; Beniamini & Giannios, 2017), current-driven instabilities (Lyutikov & Blandford, 2003), and forced magnetic reconnection (Zhang & Yan, 2011; McKinney & Uzdensky, 2012; Lazarian et al., 2019). In this work, we focus on one representative model, i.e., the Internal-Collision-induced MAgnetic Reconnection and Turbulence (ICMART) model (Zhang & Yan, 2011). This model invokes a central engine powered, magnetically dominated outflow with magnetization factor $\sigma>1$. Similar to the internal shock model, mechanical collisions between mini-shells would distort the ordered magnetic field and trigger fast magnetic seeds, which would induce relativistic magnetohydrodynamics (MHD) turbulence in the interaction regions¹¹1It is worth noticing that rather than the mechanical collisions, the kink instability may be the cause of the magnetic turbulence but the physics processes they involve are similar (Lazarian et al., 2019).. The turbulence further distorts the magnetic lines, resulting in a reconnection cascade and thus a significant release of the stored magnetic field energy (an ICMART event). The particles, either accelerated directly in the reconnection region or accelerated randomly in the turbulence region, would radiate synchrotron radiation photons to power the observed GRB prompt emission.

A global simulation of an ICMART event has been presented in Deng et al. (2015), which shows significant energy dissipation when two highly magnetized blobs collide. Mini-jet-like reconnection events are seen from the simulation, even though local simulations are needed to display mini-jets in much smaller scales. On the other hand, Monte Carlo simulations have been performed to simulate the prompt emission light curves within the framework of the ICMART model (Zhang & Zhang, 2014). They show that the ICMART model can produce highly variable light curves with both fast and slow components, where the fast component is caused by many local Doppler-boosted mini-jets due to turbulent magnetic reconnection and the slow component is due to the runaway growth and subsequent depletion of these mini-jets. However, since these simulations were focusing on the temporal structure, they assumed that the radiation intensity arising from each reconnection event had the same spectral form (e.g., Band function with $\alpha=-1,~{}\beta=-3$ and $E_{p}=300\text{keV}$ in the observer frame), which made them ineffective in testing prompt emission spectrum properties.

In this work, we will further revise the previous simulation, by considering the evolution of the magnetic field with respect to radius and using a fast-cooling synchrotron spectrum to calculate the energy spectrum for each reconnection event, in order to judge whether the ICMART model could generate the Band spectrum and interpret the observed $E_{\text{p}}$ evolution patterns.

Within the ICMART scenario, a highly magnetized central engine would eject an unsteady outflow with variable Lorentz factors and luminosities but a nearly constant degree of magnetization. For simplicity, the outflow can be represented as a succession of discrete shells with variable Lorentz factors. Mechanical collisions between mini-shells would distort the magnetic field configurations and induce MHD turbulence, which will further distort the magnetic field and eventually trigger an ICMART event. Our simulation starts with tracking a shell with mass $M_{\text{bulk}}$, Lorentz factor $\Gamma_{0}$ and magnetized factor $\sigma_{0}=E_{\text{m, 0}}/E_{\text{k, 0}}$ ($E_{\text{m, 0}}$ marks the initial magnetic energy and $E_{\text{k, 0}}$ marks the initial kinetic energy), when the shell expands to radius $R_{0}$ and the reconnection cascade has been triggered.

Numerical simulations have shown that the reconnection-driven magnetized turbulence could self-generate additional reconnections (Takamoto et al., 2015; Kowal et al., 2017; Takamoto, 2018), inferring that the number of magnetic reconnections would increase exponentially. Here we assume that each reconnection event ejects a bipolar outflow and triggers two more reconnection events²²2Lacking detailed numerical simulations for a reconnection/turbulence cascade, the specific index of magnetic reconnection growth is still unknown. Here we assume each reconnection would trigger two more new reconnection events. The uncertainty brought by this assumption will not affect our analysis results about spectrum and $E_{p}$ evolution (because these results are already the superposition effect of sufficient magnetic reconnection events), but might affect the time when our simulated light curve would reach the peak. (Zhang & Zhang, 2014). Assuming the generation number of magnetic reconnections within the $1/\Gamma_{0}$ cone is $n_{\text{g}}$, the total number of magnetic reconnections $N_{\text{cone}}$ could be estimated as $N_{\text{cone}}\approx\sum_{n=1}^{n_{\text{g}}}2^{n}$. The duration for each reconnection in the lab frame could be estimated as

where $L^{\prime}$ is the size of the magnetic reconnection and $v^{\prime}_{\text{in}}$ is the inflow velocity of the magnetic field line. Hereafter, parameters denoted with (^′) are in the rest frame of the jet bulk. Note that this duration timescale is about $T_{\text{r, obs}}\sim 0.1\text{s}$ in the observer frame and therefore the reconnections cascade lasts for $n_{\text{g}}T_{\text{r, obs}}$ in the observer frame.

For each reconnection event, the magnetized factor of the reconnection area drops from $\sigma_{0}$ to $1$. Thus the dissipated magnetic energy approximately equals to $(\sigma_{0}-1)/\sigma_{0}$ times the magnetic energy within the reconnection area. We assume that half of the dissipated magnetic energy is used to boost the kinetic energy of the jet and the other half is initially distributed to electrons (Drenkhahn & Spruit, 2002), and then gets converted to photons through synchrotron radiation. Part of the photons that beams toward the observer is recorded as the prompt emission of GRB.

In order to simulate the observed GRB spectrum, three rest frames need to be invoked: $1)$ the rest frame of the jet bulk; $2)$ the rest frame of the mini-jet; $3)$ the rest frame of the observer (here we ignore the cosmological expansion effect). Hereafter, parameters denoted with (^′′) are in the rest frame of the mini-jet. The quantities within these three frames are connected through two Doppler factors, i.e.,

where $\gamma\approx\sqrt{1+\sigma}$ is the relative Lorentz factor of mini-jet with respect to the jet bulk (Zhang & Zhang, 2014), $\beta_{\rm bulk}$ and $\beta$ are the corresponding dimensionless velocities with respect to $\Gamma$ and $\gamma$, $\theta$ is the latitude of the mini-jet, i.e., the angle between the line of sight and the direction of the bulk at the location of the mini-jet and $\phi$ is the angle between the direction of the mini-jet and the direction of the bulk in the bulk comoving frame. In our simulation, we trace each mini-jet to calculate its radiation intensity and photon energy distribution in the direction of the observer, and stack the simultaneously arriving photons from all mini-jets to obtain the overall light curve and the corresponding time-resolved spectrum.

In the rest frame of the mini-jet, synchrotron radiation power at frequency $\nu^{\prime\prime}$ is given by (Rybicki & Lightman, 1979)

where $q_{e}$ is electron charge, $\nu_{\rm cr}^{\prime\prime}=3\gamma_{\rm e}^{2}q_{e}B^{\prime\prime}_{\text{e}}/(4\pi m_{\rm e}c)$ is the characteristic frequency of an electron with Lorentz factor $\gamma_{e}$, $B^{\prime\prime}_{\text{e}}$ is the comoving magnetic field strength in the emission region and $F(x)=x\int_{x}^{\infty}K_{5/3}(\xi)d\xi$ with $K_{5/3}(x)$ is the modified Bessel Function of five thirds order. Lacking full numerical simulations of magnetic turbulence and reconnection, the comoving magnetic field strength $B^{\prime\prime}_{\text{e}}$ in the mini-jet is rather difficult to make a direct estimation. Here we introduce a free parameter $k$ to connect $B^{\prime\prime}_{\text{e}}$ with the bulk magnetic field strength as $B^{\prime\prime}_{\text{e}}=\sqrt{k}B^{\prime}/\gamma$. The value of $k$ is justified by limiting $E_{p}$ in the simulation results to be consistent with the observational data. We assume $B^{\prime\prime}_{\text{e}}$ decays with the radius of the bulk as

where b would be much larger than 1, due to the rapid consumption of magnetic energy in the reconnection process. Considering the magnetic flux conservation, the bulk magnetic field strength $B^{\prime}$ decreases as the jet expanding. The radial part of the magnetic field decreases as $B^{\prime}_{r}\propto R^{-2}$ while the transverse part decreases as $B^{\prime}_{t}\propto R^{-1}$. At a large radius one has a transverse-dominated magnetic field

where $B^{\prime}_{0}$ is the initial magnetic strength of the bulk.

The electrons number density could be estimated with the continuity equation (Uhm & Zhang, 2014):

where $\dot{\gamma_{e}}(\gamma_{e})$ denotes the electron cooling rate and $Q(\gamma_{e},t^{\prime\prime})$ denotes the injected source function. Electrons would suffer both radiative and adiabatic cooling (Uhm & Zhang, 2014), so that

where $\sigma_{T}$ is the Thomson scattering cross section. The distribution of the accelerated electrons is usually assumed to be a power-law function,

where $Q_{0}$ is a normalization factor, $\gamma_{\text{e,m}}$ and $\gamma_{\text{e,M}}$ are the minimum and maximum injected electron Lorentz factor. $\gamma_{\text{e,M}}$ could be calculated as

One can use electron number and energy conservation law to solve $Q_{0}$ and $\gamma_{\text{e,m}}$ for a specific reconnection event, which should require

and

where $t_{\text{s}}$ and $t_{\text{e}}$ are the starting and ending time of the magnetic reconnection, $f_{\text{e}}$ is the fraction of accelerated electrons within the reconnection area, $n^{\prime}_{\text{e}}\propto R^{-2}$ is the number density of electrons in the bulk frame, and $\delta E^{\prime}_{\text{p, inj}}$ is the dissipated energy for a single magnetic reconnection, which could be estimated as

where $t^{\prime}_{\text{s}}$ is the beginning time. Considering both the expansion and reconnection effects, similar to the bulk magnetic field strength, here we assume a general evolution form for $L^{\prime}$,

where $L^{\prime}_{0}$ is the initial magnetic reconnection size.

As seen by the observer, each mini-jet radiation corresponds to a single pulse starting from $t^{\prime}_{\text{s}}/\mathcal{D}_{1}$ to $(t^{\prime}_{\text{s}}+T^{\prime}_{0})/\mathcal{D}_{1}$, with radiation power

As suggested by Zhang & Zhang (2014), here we use Gaussian shape to simulate each single pulse in the light curve. For a given time interval in the observer frame ($t_{1}$, $t_{2}$), the time resolved spectrum is given by

For one ICMART event, the peak time of the light curve corresponds to the total duration for the cascade process, which could be estimated as

We assume an abrupt cessation of the cascade process, so that the number of new mini-jets drops to 0. The light curve after $T_{p}$ is therefore contributed by the high-latitude emission from other mini-jets not along the line of sight due to the “curvature effect” delay.

In order to justify whether the ICMART model could generate the Band spectrum and the observed $E_{\text{p}}$ evolution patterns, we first run a series of simulation for one ICMART event with initial setup: $M_{\text{bulk}}\in\mathcal{U}(3.5\times 10^{-9},3.5\times 10^{-6})M_{\odot}$, $R_{0}=10^{14}\text{cm}$, $L^{\prime}_{0}=1.2\times 10^{11}\text{cm}$, $v^{\prime}_{\text{in}}=3\times 10^{9}\text{cm/s}$, $n_{\text{g}}\in\mathcal{U}(13,17)$, $\Gamma_{0}\in\mathcal{U}(100,300)$, $\sigma_{0}\in\mathcal{U}(5,50)$, $k\in\mathcal{U}(5\times 10^{-8},5\times 10^{-2})$, $p\in\mathcal{U}(2.3,2.8)$, $f_{e}\in\mathcal{U}(0.01,1)$, $b\in\mathcal{U}(1,50)$, $l\in\mathcal{U}(0,1)$, $\theta\in\mathcal{U}(0,2/\Gamma_{0})$ and $\phi\in\mathcal{U}(0,\frac{\pi}{2})$, where $\mathcal{U}$ refers to the Uniform distribution. In figs. 1, 2, 3, 4, 5 and 6 we plot the simulation results for selected situations that are relevant for illustrating the main conclusions, which can be summarized in the following subsections.

The prompt emission of GRBs is still a mystery. The main uncertainty is the composition of the outflow, Poynting flux dominated or matter-dominated, which essentially determines the energy dissipation mechanism, particle acceleration mechanism, and radiation mechanism. Putting together all the observational evidence, including the properties of the light curve, the nature of the spectrum and the coordination between the spectrum and the light curve (e.g. the correlation between $E_{\text{p}}$ and flux), may help to solve the problem.

In this work, we have developed a numerical code to simulate the prompt emission light curves, time resolved spectrum and $E_{\text{p}}$ evolution patterns for GRBs produced by the Internal-Collision-induced MAgnetic Reconnection and Turbulence (ICMART) model. Our simulation results could be summarized as follows:

• •

  The ICMART model could produce highly variable light curves, which can be decomposed as the superposition of an underlying slow component and a more rapid fast component. Such result is consistent with the previous simulations (Zhang & Zhang, 2014) and the observation facts (Gao et al., 2012).

• •

  The ICMART model could produce a Band shape spectrum, whose parameters ($E_{\text{p}}$, $\alpha$, $\beta$) could distribute in the typical distribution from GRB observations, as long as the magnetic field and the electron acceleration process in the emission region are under appropriate conditions.

• •

  For one ICMART event, the spectral peak $E_{\text{p}}$ evolution is always hard-to-soft pattern. But if one individual broad pulse in the GRB light curve is composed of multiple ICMART events, the $E_{\text{p}}$ evolution related to the overall light curve could be disguised as the intense-tracking pattern. Therefore, mixed $E_{\text{p}}$-evolution patterns can coexist in the same burst, with a variety of combined patterns.

Our results show that the ICMART model can explain the main characteristics of GRB light curve and spectrum, making it a very competitive model to produce GRB prompt emission. However, to interpret GRB observations in great detail, the ICMART model still faces some difficulties, which are worth pointing out.

In principle, the total energy and magnetic field strength carried by the shell of ICMART event can vary randomly in a wide range. However, in order to make GRBs from ICMART events, it is required that for all magnetic reconnection process, there is a preferred range for the magnetic field and the electron acceleration process in the emission region. Detailed numerical simulations of magnetic reconnection and particle acceleration processes are needed to address whether the $B^{\prime\prime}_{\text{e}}$ and $\gamma_{\text{e,m}}$ range demanded by the model could be achieved. One possibility is that the magnetic reconnection process is so intense that there is an upper limit on the strength of $B^{\prime\prime}_{\text{e}}$ (e.g. smaller than $10^{4}$ G). It is worth noticing that for most GRBs, $B^{\prime\prime}_{\text{e}}$ can not reach this upper limit, but needs to be lower than 100 G and needs to decay very rapidly (down 1-2 orders of magnitude) during the radiation process, otherwise the lower energy index $\alpha$ for majority of GRBs would distribute around $-1.5$ (entering deep fast cooling regime), instead of around $-1$ as suggested by the observations.

It is possible that in some ICMART event, $B^{\prime\prime}_{\text{e}}$ is even lower than 10 G. For these cases, $E_{\text{p}}$ of the spectrum would shift into the soft X-ray band. This might be the physical origin for the phenomenon so called “X-ray flashes (XRFs)”, which are generally believed not to be a different population from GRBs, but rather the natural extension of GRBs to the softer, less luminous regime (Sakamoto et al., 2008). Accumulated sample of more XRFs in the future with sky survey detectors in the X-ray band (e.g. Einstein Probe; (Yuan et al., 2018)) would help to justify this hypothesis.