**4. Application to the Actual GRB Spectra**

In order to compare with the actual GRB spectrum, we select the broad band spectrum of GRB 080916c in the interval "b" detected by Gamma-ray Burst Monitor (GBM) and the Large Area Telescope (LAT) aboard the Fermi satellite (see Ref. [43]), from 3.58 s to 7.68 s since the lightcurve during this period is presented as a single and pure pulse. Moreover, its low energy photon index is close to the typical value of the low energy photon index of the GRB, that is, *α* ∼ −1, harder than the expectation of synchrotron fast cooling (−1.5). The "b" spectrum of GRB 080916c can be well fitted by the Band function with the low energy photon index of *α* = −1.02 ± 0.02, the high energy photon spectral slope *β* = −2.21 ± 0.03

and the peak energy *E<sup>p</sup>* = 1170 ± 140 keV [43]. Since the observational data can be well fitted by this Band function with a very small error range, the Band function is precise enough to represent the actual GRB emission. We select some representative points (black points in Figure 5) in this Band function to present the tendency of the actual GRB emission. In addition, more black points around the peak energy in the figure are taken to present the gradual change in behavior there. In Figure 5, by using a time-averaged energy spectrum from *t* = 0 s to *t* = 3 s, the emission of GRB 080916c can be fitted well in our model with the proper parameters, which have been listed in Table 1.

**Figure 5.** The time-averaged spectrum to fit the interval "b" of GRB 080916c. The black points are selected from the Band function with the low-energy photon index *α* = −1.02 ± 0.02, the high-energy photon index *β* = −2.21 ± 0.03 and the peak energy *E<sup>p</sup>* = 1170 ± 140 keV provided in Ref. [43], which are precise enough to present the tendency of the actual GRB emission. This spectral duration is from 3.58 s to 7.68 s and we fit it by adopting the time-averaged spectrum from *t* = 0 s to *t* = 3 s (*tmax* ≤ *tcrs*). The fitting parameters are listed in Table 1.

We select the single pulse spectrum of GRB 080825c in the interval "a" detected by Fermi GBM and LAT (see Ref. [44]), from 0.0 s to 2.7 s, which has a harder photon index, *α* ∼ −0.76. The "a" spectrum of GRB 080825c can be well fitted by the Band function with the low energy photon index *α* = −0.76 ± 0.05, the high energy photon index *<sup>β</sup>* <sup>=</sup> <sup>−</sup>2.54+0.11 <sup>−</sup>0.17 and the peak energy *<sup>E</sup><sup>p</sup>* <sup>=</sup> <sup>291</sup>+<sup>25</sup> <sup>−</sup><sup>22</sup> keV [44]. Such a hard photon index could not be approached easily for a constant electron injection rate, that is, *q* = 0, so we consider a rising electron injection rate as suggested in Section 3.2. Some representative points (black points in Figure 6) in this Band function are selected to present the tendency of the actual GRB emission as the same as the treatment for the GRB 080916c. The observational spectrum can be reproduced well in Figure 6 phenomenally by using a time-averaged energy spectrum from *t* = 0 s to *t* = 3 s with an index of rising electron injection rate *q* = 2 and other reasonable parameters (all parameters are listed in Table 1).

**Figure 6.** The time-averaged spectrum to fit the interval "a" of GRB 080825c. The black points are selected from the Band function with the low-energy photon index *α* = −0.76 ± 0.05, the high-energy photon index *<sup>β</sup>* <sup>=</sup> <sup>−</sup>2.54+0.11 <sup>−</sup>0.17 and the peak energy *<sup>E</sup><sup>p</sup>* <sup>=</sup> <sup>291</sup>+<sup>25</sup> <sup>−</sup><sup>22</sup> keV provided in Ref. [44], which are precise enough to present the tendency of the actual GRB emission. This spectral duration is from 0.0 s to 2.7 s and we fit it by adopting the time-averaged spectrum from *t* = 0 s to *t* = 3 s (*tmax* ≤ *tcrs*). The fitting parameters are listed in Table 1.


**Table 1.** The parameters adopted to fit the spectra of GRB 080916c and GRB 080825c.

The main parameters to effect the final spectrum are listed in Table 1. The dependence of the break energy of the spectrum on the listed parameters could be found in Equation (13) and for the magnitude of peak flux the dependence could be derived roughly in Equation (16). During the model fitting, for simplification, the energy equipartition factors for electrons and the magnetic field, that is,*e<sup>e</sup>* and *eB*, and *γ*<sup>4</sup> are fixed, and then the Lorentz factor *γ*<sup>1</sup> and kinetic luminosity *L<sup>k</sup>* are adjusted to match the observational peak energy and the peak flux. The electron injection index *p* is determined by the observational high-energy photon index since the relation between them is *β* ∼ (−*p* − 2)/2, 1 suggested by the synchrotron radiation. The shock cross time is comparably adopted with the typical duration of the slow pulse of the GRB, namely, ∼3 s. Different values of *δt* could affect the evolutional form of the magnetic field (as shown in Figure 1) and adjust the weight of the cooling in a constant magnetic field and the cooling in a decaying magnetic field. In other words, a smaller *δt* could make it so that the electron synchroton cooling mainly takes place in a decaying magnetic field and leads the photon index to be harder, while for a larger *δt*, the electrons are mainly cooling in a constant magnetic field and generating a photon

index close to −1.5. As a result, for GRB 080916c with a photon index ∼−1, a relatively small *δt* = 0.1 s is adopted. A harder photon index ∼−0.76 for GRB 080825c and a rising electron injection rate with an index *q* = 2 are taken into account as suggested in Section 3.2. Therefore, a certain range of a low-energy photon index from ∼−3/2 to ∼−2/3 could be approached through the adjustment of *δt* and the index of the electron injection rate *q*. However, for a low-energy photon index harder than −2/3, this model would become invalid.
