**1. Introduction**

The prompt radiation mechanism of gamma-ray bursts (GRBs) is still being debated, even though the prompt spectra can usually be fitted well by the Band function [1], which suggests a smoothly jointed broken power law with the low-energy photon index *α* ∼ −1, the high energy photon index *β* ∼ −2.2 and the peak energy *E<sup>p</sup>* ∼ 250 keV [2,3]. Currently, neither the possible one-temperature thermal emission from an ultrarelativistic fireball, nor the single synchrotron radiation from shock-accelerated electrons within this fireball, provide an explanation for such a low-energy photon index (see [4,5] for a review).

Generally, there are two mechanisms that explain the low energy photon index *α* ∼ −1 of the GRB prompt emission. The first mechanism is the Comptonized quasi-thermal emission from the photosphere of an ultrarelativistic outflow [6–20]. The second mechanism is synchrotron and/or synchrotron self-Compton (SSC) emission in the optically thin region. For fast-cooling synchrotron radiation in the internal shock model, possible solutions include invoking a small-scale rapidly decaying magnetic field [21], a decaying magnetic field with a power-law index in a relativistically-expanding outflow [22–24], a decaying magnetic field in a post-shock region [25], Klein–Nishina (KN) cooling [26,27], an adjustable synchrotron self-absorption frequency [28,29], or the acceleration process [30] and other evolutional model parameters [31]. Alternatively, slow cooling was introduced to understand the low-energy photon index [32]. In addition to the internal shock model, the other energy dissipation mechanisms, such as the ICMART model [33], were proposed to solve the low-energy spectral index issue. In some models (e.g., [34]), the observed prompt emission of GRBs is understood to be dominated by the SSC emission, while the synchrotron radiation is in much lower energy bands.

**Citation:** Wang, K.; Dai, Z.-G. The Low-Energy Spectral Index of Gamma-ray Burst Prompt Emission from Internal Shocks. *Galaxies* **2021**, *9*, 68. https://doi.org/10.3390/ galaxies9030068

Academic Editors: Elena Moretti and Francesco Longo

Received: 30 July 2021 Accepted: 13 September 2021 Published: 15 September 2021

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The fast cooling synchrotron radiation in the internal shock model is generally considered to be a straightforward and leading mechanism to explain the GRB prompt emission spectra, and the most important issue in this model is to explain the low-energy spectral index. An underlying assumption in the traditional synchrotron internal shock model is to calculate the electron cooling without considering the evolution of the magnetic field. In other words, the magnetic field is treated as a constant and its effect in the continuity equation of electrons is usually ignored (e.g., [35]). In the fast cooling case, the predicted low-energy photon index *α* ∼ −3/2 is much softer than observed. In this paper, we try to alleviate this problem. We calculate the magnetic field in the realistic internal shock model during a collision of two relativistic thick shells and obtain an evolutional form of the magnetic field, *B* 0 ∝ constant before the time *δt* that is nearly equal to the ejection time interval of the two shells, and *B* 0 ∝ *t* <sup>−</sup><sup>1</sup> after the time *δt*. We consider the cooling of electrons accelerated by internal shocks for this evolutional magnetic field, and find the resulting spectral index *α* ∼ −3/2 for *B* <sup>0</sup> ∝ constant and *α* ∼ −2/3 for *B* 0 ∝ *t* −1 , by adopting a cooling method similar to that in Ref. [22]. Actually, these two cases may coexist, and the outflow may undergo the first case and then the second case, so theoretically the actual index *α* will range from −3/2 to −2/3. Furthermore, below the peak energy *E<sup>p</sup>* there is a gradual process, so that *α* is only close to −1. In addition, we consider a rising electron injection rate, leading to a larger *α*, slightly smaller than −2/3.

This paper is organized as follows. We calculate the dynamics of a collision between two thick shells in Section 2. In Section 3, we investigate the electron cooling and its synchrotron radiation with an evolutional magnetic field and a rising electron injection rate. In the final section, discussions and conclusions are given.
