**5. Discussions and Conclusions**

Currently alleviating the tension between the expectation of synchrotron and observations in the GRB prompt regime is a more and more important issue. Two classes of model have been proposed to explain the low-energy photon index of GRB prompt emission, Comptonized quasi-thermal emission from the photosphere within a relativistic outflow and synchrotron and/or SSC emission in the optically thin region. These models can experience difficulties. For Comptonized quasi-thermal emission, the most significant effect to obtain *α* ∼ −1 is the equal arrival time effect in this model, which is relevant to the end time of central engine activity, but may not be applicable during the prompt emission phase when a continuous wind is ejected from the central engine [45]. Synchrotron slow cooling in internal shocks may not provide a high radiative efficiency, and a dominant SSC component usually predicts an even more dominant 2nd-order SSC component, which significantly exceeds the total energy budget of GRBs [46,47]. Thus, some evolutional parameters, such as the magnetic field, the fraction of the accelerated electrons, and the energy equipartition factors, were suggested to explain the low-energy index.

In this paper, we have considered a straightforward model, that is, the fast cooling synchrotron radiation in internal shocks. We obtain the magnetic field evolutional form in a practical shell–shell collision, *B* 0 ∝ constant before *δt* and *B* 0 ∝ *t* <sup>−</sup><sup>1</sup> after this time, and recalculate the electron distribution for this evolutional magnetic field. When *t* < *δt*, the magnetic field is nearly constant, and the fraction of cooling electrons in the invariable magnetic field is high enough so that *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −2 for *γ* 0 *<sup>e</sup>* < *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* is expected. However, when *t δt* but *t* < *tcrs*, the fraction of cooling electrons in the evolutional magnetic field is higher than in the invariable magnetic field, so that *dN<sup>e</sup> dγ* 0 *e* will be gradually proportional to *γ* 0 *e* 0 . 2*γ* 2 *cδt* and 2*γ* 2 *ct* indicate roughly the collision radius and the propagation distance of a relativistic outflow after the collision takes place but before the shock crossing time *tcrs*, respectively. In other words, if the propagation distance of the outflow is smaller than the collision radius before the shock crossing time, the magnetic field can be treated as a constant and it is not necessary to consider the evolution of the magnetic field when calculating the electron cooling. However, if the propagating distance of the outflow is larger than the collision radius before the shock crossing time, we have to consider the evolution of the magnetic field and can obtain a different electron distribution. Actually, the outflow may undergo the first case and then the second case, so we can obtain a reasonable range of the low-energy photon index *α*, from −3/2 to −2/3 theoretically. Since *dNe dγ* 0 *e* proportional to *γ* 0 *e* 0 is a gradual process below *Ep*, it is usually difficult to get *α* to be exactly equal to −2/3, but this index is only slightly smaller than −2/3. Moreover, we also consider a rising electron injection rate, which may exacerbate this situation, inducing *α* to be closer to −2/3.

Ref. [22] considered a decaying magnetic field varying with the distance from the central engine to explore the range of a low energy photon index in the GRB prompt regime. They discussed the radiation spectra of a cloud of plasma in a decaying magnetic field with an arbitrary decaying index *b* for a simplified model, which is called a "toy box model". Different from their work, we adopt a more physical case for the internal shock by considering the collision of two shells and inducing the decaying form of the magnetic field. As a result, a time-dependent magnetic field is derived (as shown in Figure 1). In fact, a time-dependent magnetic field could be translated to a distance-dependent form due to

the propagation of relativistic outflow. For the evolutional magnetic field form obtained from the practical internal shock, we study the influence on the spectral index. In addition to the detailed treatment of shell–shell collision, the kinetic luminosity and the energy equipartition parameters, *e<sup>B</sup>* and *e<sup>e</sup>* are taken into account to obtain the radiation spectra, comparing them with the actual GRB spectra for GRB 080916c and GRB 080825c.

In our model, in order to obtain the high prompt emission luminosity, we assume that *γ*<sup>4</sup> *γ*1. This assumption is reasonable. This is because estimates based on four methods by Ref. [48] show that the mean observed value of the bulk Lorentz factors of GRB outflows is a few hundred, corresponding to *γ*<sup>2</sup> = *γ*<sup>3</sup> ' *γ*<sup>1</sup> ∼ 100 in our model. Furthermore, within the framework of the collapsar model, a prior relativistic jet-like shell (e.g., shell A) first has to propagate through the envelope of a massive star and clean up almost all of the baryons along the propagation direction of this shell, leaving behind a clean passage for a posterior jet-like shell (e.g., shell B). This, therefore, leads to a reasonable possibility that the Lorentz factor of shell B is much greater than that of shell A.

Usually, we have *<sup>γ</sup>*<sup>1</sup> <sup>∼</sup> 100, so *<sup>γ</sup>*<sup>4</sup> <sup>∼</sup> <sup>10</sup><sup>4</sup> <sup>∼</sup> *<sup>γ</sup>* 2 1 is a universal relationship to obtain the high prompt emission luminosity. Ref. [37] also mentioned that, when *γ*<sup>4</sup> ∼ *γ* 2 1 , the highest luminosity from internal shocks is expected. In fact, this assumption is not a special case. When collisions among a series of shells with different Lorentz factors occur, the highest luminosity from one collision will cover the others. In other words, we always see the brightest. According to Equation (13), if deeming that *γ*4*γ* −2 1 does not vary significantly among bursts, we can easily obtain the so-called "Yonetoku Relation", *E<sup>p</sup>* ∝ *L* 1/2 *iso* [49], and the "Amati Relation", *E<sup>p</sup>* ∝ *E* 1/2 *iso* [50]. However, this model is also confronted with some issues, for example, the spectrum is somewhat broad near *E<sup>p</sup>* in contrast to the observed data or the Band function [51], which can be seen in Figure 6.

It is important that we should beware of the empirical Band function. The thermal components and more spectral structures are found in the prompt regimes of some GRBs, which deviate from the so-called Band function [52,53]. The thermal emission generated by the photosphere is a natural prediction of the generic fireball scenario. The relative strength of thermal emission and non-thermal emission may depend on the various environments [54,55]. Ref. [53] also claimed that the GRB spectra below the peak energy may present an extra break energy around a few keV, inducing a consistent spectral shape with expectation from the classical synchrotron radiation. More spectral structures of GRBs may make the simple Band function become invalid, and result in an incorrect low-energy spectral index if one forcibly fits them using a Band function. Although our model can present a consistent low-energy spectral index with observations in a certain range, due to the complexities of GRB prompt spectra, more detailed studies are needed.

**Author Contributions:** All authors have contributed equally to the preparation of this manuscript. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the National Key Research and Development Program of China (grant No. 2017YFA0402600), the National SKA Program of China (grant No. 2020SKA0120300), the National Natural Science Foundation of China under grants No. 11833003, 12003007 and the Fundamental Research Funds for the Central Universities (No. 2020kfyXJJS039).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** We thank the anonymous referees for the helpful suggestions and comments.

**Conflicts of Interest:** The author declares no conflict of interest.
