**3. Synchrotron Radiation with a Decaying Magnetic Field and a Variable Electron Injection Rate**

*3.1. Synchrotron Radiation with a Decaying Magnetic Field*

As usual, we assume that fractions *e<sup>B</sup>* and *e<sup>e</sup>* of the internal energy density in a GRB shock are converted into the energy densities of the magnetic field and electrons, respectively. Thus, using *B* 0 *<sup>i</sup>* = (8*πeBe* 0 *i* ) 1/2 for *i* = 2 or 3, we can calculate the strength of the magnetic field before the shock crossing time *tcrs* by

$$B\_2' = B\_3' = \left[\frac{\epsilon\_B L\_k}{2\gamma\_1^6 c^3 \left(t + \delta t\right)^2}\right]^{1/2},\tag{7}$$

and find that the change of the magnetic field before *δt* can be ignored (i.e., *B* 0 *i* ∝ constant), but after *δt* the magnetic field *B* 0 *i* decreases linearly with time *t* (i.e., *B* 0 *i* ∝ *t* −1 ). Actually, the evolution of the magnetic field is caused by the expansion of the shocked regions, which is presented in Figure 1. After the shock crossing time *tcrs* (here, *tcrs* is comparable with the peak time of the slow pulse in GRBs), the spreading of the hot materials into the vacuum cannot be ignored and the merged shell undergoes an adiabatic cooling. During this phase, the volume of the merged shell is assumed to expand as *V* 0 *i* ∝ *R s* , where *s* is a free parameter and its value is taken to be from 2 to 3. As a result, the particle number density would decrease as *n* 0 *i* ∝ *V* 0−<sup>1</sup> *<sup>i</sup>* ∝ *R* −*s* , the internal energy density as *e* 0 *i* ∝ *V* 0−4/3 *<sup>i</sup>* ∝ *R* <sup>−</sup>4*s*/3, and the magnetic field strength as *B* 0 *i* ∝ (*e* 0 *i* ) <sup>−</sup>1/2 ∝ *R* <sup>−</sup>2*s*/3 ∝ *t* <sup>−</sup>2*s*/3. Because no additional shock-accelerated electrons are injected after the shock crossing time *tcrs*, we only study the prompt emission before *tcrs* in the remaining part of this paper. What we want to point out is that the redefined time interval *δt* is not equal to the shock crossing time (*tcrs*), the

latter one is dependent on the thickness of the shells. In this paper, the two shells are must be thick enough so that *tcrs δt*.

**Figure 1.** The magnetic field as a function of time. The two blue vertical dotted line represents the redefined interval *δt* = 0.1 s and the shock crossing time *tcrs* = 3 s, respectively. After the shock crossing time, the merged shell expands adiabatically and *s* = 3 is assumed. The dynamics parameters *L<sup>k</sup>* = 10<sup>51</sup> erg s−<sup>1</sup> , *γ*<sup>1</sup> = 300, *γ*<sup>4</sup> = 30,000, *p* = 2.5, *e<sup>e</sup>* = 0.3, *e<sup>B</sup>* = 0.3, and *z* = 1 are taken from numerical calculations.

The electrons accelerated by the shocks are assumed to have a power-law energy distribution, *dNe*,*i*/*dγ* 0 *e*,*i* ∝ *γ* 0 *e* −*p* for *γ* 0 *<sup>e</sup>*,*<sup>i</sup>* ≥ *γ* 0 *e*,*m*,*i* , where *γ* 0 *e*,*m*,*i* is the minimum Lorentz factor of the accelerated electrons. The following electron cooling discussion is not based on the conventional synchrotron and SSC cooling, which always give us the electron distribution, *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>* and *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −*p*−1 for *γ* 0 *<sup>e</sup>* > *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* in the fast cooling case, *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −*p* for *γ* 0 *<sup>e</sup>* < *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* and *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −*p*−1 for *γ* 0 *<sup>e</sup>* > *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* in the slow cooling case. These electron distributions do not take into account the evolution of the magnetic field. Ref. [22] discussed the electron distribution affected by a decaying magnetic field based on *B* 0 ∝ *r* −*b* , where *r* is the fireball radius and *b* is the magnetic field decaying index. They considered the electron distribution of a group of plasma in a magnetic field with an arbitrary decaying index *b*, which is called a "toy box model". Here we consider a more physical process, internal shocks, which generate an evolutional magnetic field and a consistent spectrum with the observed Band spectral shape.

In the comoving frame, the evolution of the Lorentz factor of an electron via synchrotron and SSC cooling and adiabatic cooling can be described by [22]

$$\frac{d}{dt'} \left(\frac{1}{\gamma\_{\varepsilon}}\right) = \frac{\sigma\_T (1 + \Upsilon\_i)}{6\pi m\_{\varepsilon}c} B\_i^{\prime 2} - \frac{1}{3} \left(\frac{1}{\gamma\_{\varepsilon}}\right) \frac{d \ln n\_i^{\prime}}{dt'},\tag{8}$$

where *Y<sup>i</sup>* ≈ [(4*ηiee*/*e<sup>B</sup>* + 1) 1/2 <sup>−</sup> <sup>1</sup>]/2 is the Compton parameter, which is defined by the ratio of the IC to synchrotron luminosity, with *η<sup>i</sup>* = min[1,(*γ* 0 *e*,*c*,*i* /*γ* 0 *e*,*m*,*i* ) 2−*p* ] [42]. *γ* 0 *e*,*c*,*i* is the cooling Lorentz factor and the comoving time *t* 0 = 2*γt*.

The minimum Lorentz factor of the accelerated electrons is *γ* 0 *<sup>e</sup>*,*m*,*<sup>i</sup>* = *mp me* ( *p*−2 *p*−1 )*ee*(*γ*rel − 1) (where *γ*rel = *γ*<sup>21</sup> or *γ*<sup>34</sup> in region 2 or 3), so it can be written as:

$$
\gamma\_{\varepsilon,m,3}^{\prime} \simeq 1.0 \times 10^4 g\_p \epsilon\_{\varepsilon,-1/2} \gamma\_{4,4.5} \gamma\_{1,2.5'}^{-1} \tag{9}
$$

$$
\gamma'\_{\varepsilon,m,2} \simeq \mathfrak{A} \mathfrak{g}\_p \mathfrak{e}\_{\varepsilon,-1/2\prime} \tag{10}
$$

where *g<sup>p</sup>* = 3(*p* − 2)/(*p* − 1). Moreover, the cooling Lorentz factor *γ* 0 *<sup>e</sup>*,*c*,*<sup>i</sup>* = 6*πmec*/(*yiσTB* 0 3 2 *γt*), can be written as

$$
\gamma'\_{\varepsilon,\varepsilon,3} = \gamma'\_{\varepsilon,\varepsilon,2} \simeq 3.4 \times 10^2 y\_{i,0}^{-1} \varepsilon\_{B,-1/2}^{-1} L\_{k,51}^{-1} \gamma\_{1,2.5}^5 \frac{(t+\delta t)\_{,0}^2}{t\_{,0}} \tag{11}
$$

where *y<sup>i</sup>* = 1 + *Y<sup>i</sup>* is the ratio of the total luminosity to synchrotron luminosity.

From the electron injection rate based on Equations (5) and (6), one can obtain the injected electrons number between *t* 0 and *t* 0 + *dt*0 . Assuming the original electron injection distribution *dNe*,*i*/*dγ* 0 *e*,*i* ∝ *γ* 0 *e* −*p* for *γ* 0 *<sup>e</sup>*,*<sup>i</sup>* ≥ *γ* 0 *e*,*m*,*i* , the injected electrons number between *t* 0 and *t* 0 + *dt*0 and between *γ* 0 *<sup>e</sup>* and *γ* 0 *<sup>e</sup>* + *dγ* 0 *e* can be derived. So we cut the injected electrons into small pieces in the time space *t* 0 and the energy space *γ* 0 *e* . At the beginning, time *t* 0 = 0, a number of electrons *dN* will be injected into the shocked region in a time interval *dt*0 and will be cooled in the initial magnetic field, so one can obtain the change of electron Lorentz factor ∆*γ* 0 *<sup>e</sup>*,1 based on Equation (8) for the electrons between *γ* 0 *<sup>e</sup>* and *γ* 0 *<sup>e</sup>* + *dγ* 0 *e* . In the next time interval *dt*0 , these electrons with the Lorentz factor between *γ* 0 *<sup>e</sup>* + ∆*γ* 0 *<sup>e</sup>*,1 and *γ* 0 *<sup>e</sup>* + *dγ* 0 *<sup>e</sup>* + ∆*γ* 0 *<sup>e</sup>*,1 will be cooled in the instantaneous magnetic field based on the evolutional magnetic field in Equation (7), and one can obtain another ∆*γ* 0 *<sup>e</sup>*,2 (∆*γ* 0 *<sup>e</sup>*,2 6= ∆*γ* 0 *<sup>e</sup>*,1). At the same time, another group electrons are injected and cooled in this instantaneous magnetic field. These processes are continuous before 2*γtcrs*. The shocked electrons are injected as time and all the electrons are cooled in the instantaneous magnetic field. We sum all electrons at time *t* 0 in the energy space, obtain the electron distributions at time *t* 0 and present them in Figure 2 (*tobs* = *t* 0/2*γ*). As shown in Figure 2, when *t* < *δt*, the magnetic field does not change significantly (see Figure 1), the electron distribution in the fast cooling case, *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −2 for *γ* 0 *<sup>e</sup>* < *γ* 0 *<sup>e</sup>*,*m*, and *dN<sup>e</sup> dγ* 0 *e* ∝ *γ* 0 *e* −*p*−1 for *γ* 0 *<sup>e</sup>* > *γ* 0 *<sup>e</sup>*,*m*, are expected. However, when *t* > *δt*, the electron distribution below *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* would be flattened because of the decaying magnetic field. Due to the magnetic field decay, the electrons injected at later times would cool more slowly than the electrons injected at early times (here, all times are before *tcrs*). In other words, the cooling efficiency would become smaller due to the decaying magnetic field, which induces more electrons accumulating at . *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* than in the invariable magnetic field case. When *t δt* and *t* < *tcrs*, the electron spectral index for *γ* 0 *<sup>e</sup>* < *γ* 0 *<sup>e</sup>*,*<sup>m</sup>* is even flattened to zero.

**Figure 2.** The electron distribution in energy space after cooling time *t* in the evolutional magnetic field in Figure 1. The same *δt*, *tcrs*, and dynamics parameters as in Figure 1 are taken in numerical calculations.

In order to find these new results for *t δt* and *t* < *tcrs*, we can evaluate the continuity equation of electrons in energy space, *<sup>∂</sup> ∂t* <sup>0</sup>(*dNe*,*<sup>γ</sup>* 0 *e* /*dγ* 0 *e* ) + *<sup>∂</sup> ∂γ*0 *e* [*γ*˙0 *e* (*dNe*,*<sup>γ</sup>* 0 *e* /*dγ* 0 *e* )] = *Q*(*γ* 0 *e* , *t* 0 ), where *dNe*,*<sup>γ</sup>* 0 *e* /*dγ* 0 *e* is the instantaneous electron spectrum at time *t* 0 , and *Q*(*γ* 0 *e* , *t* 0 ) = *Q*0(*t* 0 )(*γ* 0 *<sup>e</sup>*/*γ* 0 *<sup>e</sup>*,*m*) −*p* is the electron injection distribution accelerated by shocks above the minimum injection Lorentz factor *γ* 0 *<sup>e</sup>*,*m*. By ignoring the inconsequential adiabatic cooling term, we can get *<sup>d</sup> dt*0( 1 *γ* 0 *e* ) ∝ (1 + *Yi*)*B* 0 *i* 2 ∝ *t* 0−<sup>2</sup> , where *Y<sup>i</sup>* is assumed to be a constant before the shock crossing time *tcrs* in the fast cooling case. Then, we can obtain *γ* 0 *<sup>e</sup>* ∝ *t* 0 , and thus *γ*˙ 0 *<sup>e</sup>* ∝ *γ* 0 *e* 2 *t* 0−<sup>2</sup> ∝ *γ* 0 *e* 0 . For *γ* 0 *<sup>e</sup>*,*<sup>c</sup>* < *γ* 0 *<sup>e</sup>* < *γ* 0 *<sup>e</sup>*,*m*, *Q*(*γ* 0 *e* , *t* 0 ) = 0, to obtain the final and quasi-steady electron spectral shape at the arbitrary time *t* 0 , by considering a quasi-steady-state system (*∂*/*∂t* = 0), we can easily find *dNe*,*<sup>γ</sup>* 0 *e* /*dγ* 0 *<sup>e</sup>* ∝ *γ* 0 *e* 0 below *γ* 0 *<sup>e</sup>*,*m*.

Next, the four characteristic frequencies in regions 2 and 3 that can be calculated from *ν* = *qe* 2*πmec B* 0*γ* 0 *e* 2 *γ* are derived as [36]

$$h\nu\_{m,2} \simeq 2.1 \times 10^{-4} g\_p^2 \varepsilon\_{\varepsilon, -1/2}^2 \varepsilon\_{\text{B,-1}/2}^{1/2} L\_{k,51}^{1/2} \gamma\_{1,2.5}^{-2} (t + \delta t)\_{\text{\textquotedblleft 1}}^{-1} \text{ keV} \tag{12}$$

$$
\hbar \nu\_{\rm m,3} \simeq 26 g\_p^2 \varepsilon\_{\varepsilon, -1/2}^2 \varepsilon\_{\rm B, -1/2}^{1/2} L\_{k,51}^{1/2} \gamma\_{4,4.5}^2 \gamma\_{1,2.5}^{-4} (t + \delta t)\_{\beta}^{-1} \text{ keV} \tag{13}
$$

and

$$h\nu\_{\rm c,2} = h\nu\_{\rm c,3} \simeq 3.6 \times 10^{-2} y\_{,0}^{-2} \epsilon\_{B,-1/2}^{-3/2} L\_{k,51}^{-3/2} \gamma\_{1,2.5}^8 \frac{(t+\delta t)\_{,0}^3}{t\_{,0}^2} \text{ keV.} \tag{14}$$

Here, if *γ*<sup>1</sup> = 100 and *γ*<sup>4</sup> = 10,000, we obtain *hνm*,3 ' 186 keV at time *t* = 1 s, which is approximatively equal to the typical value of *E<sup>p</sup>* of the GRB prompt emission.

We also present the spectrum of region 3 in the top panel of Figure 3 based on the electron distribution shown in Figure 2. However, we do not present the spectrum of region 2 because, from NFS, (1) its photon peak frequency is much smaller than the typical GRB prompt emission *Ep*, (2) the radiation efficiency can not be high enough as a result of slow cooling, and (3) the flux of region 2 is much lower than that of region 3. The last reason can be evaluated from [35,37]

$$F\_{\nu, \text{max}, i} < \frac{N\_{\varepsilon, i}}{4\pi D\_L^2} \frac{m\_\varepsilon c^2 \sigma\_T}{\Im q\_\varepsilon} B\_i' \gamma\_\prime \tag{15}$$

where *D<sup>L</sup>* is the luminosity distance of the burst and *Ne*,*<sup>i</sup>* is the total number of injected electrons until the time *t*. Since a portion of the electrons have cooled to a much smaller value than *γbreak* (where the break Lorentz factor *γbreak* of an electron distribution, *γbreak* = *γe*,*<sup>m</sup>* for fast cooling, and *γbreak* = *γe*,*<sup>c</sup>* for slow cooling), the actual number of electrons near *γbreak* is less than *Ne*,*<sup>i</sup>* and thus the actual *Fν*,max,*<sup>i</sup>* is smaller than the right term of inequality Equation (15). So, we can obtain

$$\begin{split} \nu\_{m,3} F\_{\nu, \text{max},3} &< 1.5 \times 10^5 g\_p^2 \varepsilon\_{\varepsilon, -1/2}^2 \varepsilon\_{B, -1/2} L\_{k, 51}^2 \gamma\_{4,4} \gamma\_{1,2}^{-6} \\ &\times \frac{t\_{\text{ $\boldsymbol{\gamma}$ }}}{(t + \delta t)\_{\boldsymbol{\gamma}0}^2} D\_{\text{L,28}}^{-2} \text{ keV cm}^{-2} \text{s}^{-1}, \end{split} \tag{16}$$

and

$$\begin{aligned} \, \_{\mathrm{V},2}F\_{\mathrm{V},\mathrm{max},2} &< 8.7 \times 10^{-1} \, \_{\mathrm{V},0}^{-2} \epsilon\_{B-1/2}^{-1} \, \_{\mathrm{V},2}^{5} \\ &\times \, \frac{(t+\delta t)\_{\mathrm{0}}^{2}}{t\_{,0}} D\_{\mathrm{L},28}^{-2} \, \_{\mathrm{keV}} \mathrm{ev} \, \mathrm{cm}^{-2} \mathrm{s}^{-1} \, \_{\mathrm{\prime}} \end{aligned} \tag{17}$$

where *γ*<sup>1</sup> = 100 and *γ*<sup>4</sup> = 10,000 are taken.

From Figure 3, we can see that for *t* < *δt*, because of a constant magnetic field, the spectral slope of *νF<sup>ν</sup>* is 1/2 as described by [35]. However, when *t* > *δt*, the spectral slope will deviate from 1/2 and become a larger value (even 4/3). If the electron index (*dNe*/*dγ<sup>e</sup>* ∝ *γ* −*u e* ) is *u*, the *F<sup>ν</sup>* slope of synchrotron radiation (*F<sup>ν</sup>* ∝ *ν* <sup>−</sup>*w*) would be *w* = (*u* − 1)/2 and the photon spectral index (defined as *dNγ*/*dE<sup>γ</sup>* = *E* −*α γ* , where *E<sup>γ</sup>* is the photon energy, and *N<sup>γ</sup>* is the photon number flux) would be *α* = −(*w* + 1). Due to the decaying magnetic field, *u* tends to be zero, and thus *w* = −1/2 and *α* = −1/2. However, when *α* > −2/3, because of the overlying effect, the low energy photon index of the electrons with ∼ *γe*,*<sup>m</sup>* is −2/3 and will cover the emission of electrons with smaller Lorentz factors. So, due to the effect of the low-energy radiation tail of electrons with Lorentz factor *γe*,*m*, *α* is at most equal to −2/3, and we can get −3/2 < *α* < −2/3. This is consistent with the observations [2,3], which suggest *α* ∼ −1.

**Figure 3.** The top panel corresponds to time-resolved spectra in four different *t* as in Figure 2 and the bottom panel shows corresponding synchrotron spectral slopes. The same *δt*, *tcrs*, and dynamics parameters as in Figure 1 are taken in numerical calculations.
