PROBING THE BIRTH OF POST-MERGER MILLISECOND MAGNETARS WITH X-RAY AND GAMMA-RAY EMISSION

The IC emission of GRB afterglows has been studied and detected for many GRBs (Panaitescu & Mészáros 1998; Wei & Lu 1998; Panaitescu & Kumar 2000; Wang et al. 2001a, 2001b; Harrison et al. 2001; Sari & Esin 2001; Nakar et al. 2009; Wang et al. 2010; Liu et al. 2013; Wang & Dai 2013b) . It is expected that the IC component of the reverse shock emission in our model is prominent because of the ultrarelativistic nature of the shocked electrons. The IC scattering will boost the cooling of radiative electrons so that the total power of one relativistic electron is

$$\begin{eqnarray}&&P({\gamma }_{e})={P}_{\mathrm{syn}}({\gamma }_{e})(1+Y),\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&{P}_{\mathrm{syn}}({\gamma }_{e})=\displaystyle \frac{4}{3}{\sigma }_{T}c{\gamma }^{2}{\beta }_{e}^{2}{\gamma }_{e}^{2}\displaystyle \frac{{B}^{2}}{8\pi },\end{eqnarray} \tag{ 3 }$$

Y is the Compton Y parameter, and the other parameters have their usual meanings (Sari et al. 1998). As a result, the electron cooling Lorentz factor is modified as

$$\begin{eqnarray}&&{\gamma }_{c}=\displaystyle \frac{6\pi {m}_{e}c}{{\sigma }_{T}\gamma {B}^{2}t(1+Y)}.\end{eqnarray} \tag{ 4 }$$

In determining the ejecta dynamics, we have adopted two different equations, i.e., Equations (5) and (6) in Wang & Dai (2013c) and Wang et al. (2015a), as follows,

$$\begin{eqnarray}&&{L}_{0}\mathrm{min}(t,{T}_{\mathrm{sd}})=(\gamma -{\gamma }_{\mathrm{ej},0}){M}_{\mathrm{ej}}{c}^{2}+2({\gamma }^{2}-1){M}_{\mathrm{sw}}{c}^{2},\end{eqnarray} \tag{ 5 }$$

from the energy conservation of the whole system, while

$$\begin{eqnarray}&&\displaystyle \frac{d\gamma }{{dt}}=\displaystyle \frac{\xi {L}_{\mathrm{sd}}+{L}_{\mathrm{rd}}-{L}_{\mathrm{ej},e}-\gamma { \mathcal D }\left(\tfrac{{{dE}}_{3}^{\prime }}{{dt}^{\prime} }+\tfrac{{{dE}}_{\mathrm{ej},\mathrm{int}}^{\prime }}{{dt}^{\prime} }\right)-({\gamma }^{2}-1){c}^{2}\left(\tfrac{{{dM}}_{\mathrm{sw}}}{{dt}}\right)}{{E}_{3}^{\prime }+{M}_{\mathrm{ej}}{c}^{2}+{E}_{\mathrm{ej},\mathrm{int}}^{\prime }+2\gamma {M}_{\mathrm{sw}}{c}^{2}},\end{eqnarray} \tag{ 6 }$$

for the dynamics in the differential way, where ${L}_{0}=\xi {L}_{\mathrm{sd}}$, with ξ the fraction of spin-down power of the magnetar that is caught by the ejecta, ${L}_{\mathrm{sd}}$ the spin-down power of the magnetar, ${T}_{\mathrm{sd}}$ the spin-down timescale, and ${\gamma }_{\mathrm{ej},0}$ the initial Lorentz factor of the ejecta, ${L}_{\mathrm{rd}}$ the radioactive heating power due to the r-process material, ${L}_{\mathrm{ej},e}$ the thermal luminosity of the ejecta, ${ \mathcal D }$ the Doppler factor, ${E}_{3}^{\prime }$ and ${E}_{\mathrm{ej},\mathrm{int}}^{\prime }$ the respective energy in Region 3 and in ejecta in the comoving frame. The four regions, i.e., Region 1 to 4, are defined as: unshocked medium (Region 1), shocked medium (Region 2), shocked wind (Region 3), and unshocked magnetar wind (Region 4), see Figure 1 in Wang et al. (2015a).

Equation (5) originates from Equation (1) in Gao et al. (2013a) and the fact that the energy contained in forward shock and reverse shock is comparable (Blandford & McKee 1976; Wang & Dai 2013c). Equation (6), on the other hand, is determined by accounting for all energy in different zones (Dai 2004; Yu et al. 2013; Wang et al. 2015a).

In this section we aim to evaluate the effects of IC on synchrotron spectra and light curves in an analytical manner. To this end it is affordable to adopt Equation (5) rather than (6) for the dynamical evolution of the ejecta because Wang et al. (2015a) showed that Equation (5) is quantitatively equivalent to Equation (6).

As shown in Wang & Dai (2013c), to analytically calculate the ejecta dynamics and the reverse shock light curves, several timescales have been defined (see Figures 1 and 2 in Wang & Dai 2013c, for reference): the transition times between Newtonian dynamics and relativistic dynamics ${T}_{N1}$ and ${T}_{N2}$, the ejecta deceleration time ${T}_{\mathrm{dec}}$, the spin-down time of the magnetar ${T}_{\mathrm{sd}}$, and the time ${T}_{\mathrm{ct}}$, i.e., the time at which electron cooling factor ${\gamma }_{c}$ begins to deviate from unity. The reason for defining ${T}_{\mathrm{ct}}$ is that the electrons in the reverse shock cool so efficiently at beginning that ${\gamma }_{c}=1$ before ${T}_{\mathrm{ct}}$. There are two other timescales that are important for the calculation of reverse shock light curves: T_ac and T_mc, the respective crossing time of cooling frequency ${\nu }_{c}$ with synchrotron self-absorption frequency ${\nu }_{a}$ and the typical frequency ${\nu }_{m}$.

IC scattering affects the cooling of electrons so we shall expect that the timescales ${T}_{\mathrm{ct}}$, T_ac and T_mc would be modified when IC is taken into account. The Compton Y parameter is given by (Sari & Esin 2001)

$$\begin{eqnarray}&&Y=\displaystyle \frac{1}{2}\left(\sqrt{1+4\eta \displaystyle \frac{{\epsilon }_{e}}{{\bar{\epsilon }}_{B}}}-1\right),\end{eqnarray} \tag{ 7 }$$

which can be simplified as

$$\begin{eqnarray}Y=\left\{\begin{array}{ll}\displaystyle \frac{\eta {\epsilon }_{e}}{{\bar{\epsilon }}_{B}}, & \displaystyle \frac{\eta {\epsilon }_{e}}{{\bar{\epsilon }}_{B}}\ll 1,\\ {\left(\displaystyle \frac{\eta {\epsilon }_{e}}{{\bar{\epsilon }}_{B}}\right)}^{1/2}, & \displaystyle \frac{\eta {\epsilon }_{e}}{{\bar{\epsilon }}_{B}}\gg 1,\end{array}\right.\end{eqnarray} \tag{ 8 }$$

in the limiting cases. Here η is the fraction of the electron energy that is radiated away by synchrotron and IC emission. Compared with Sari & Esin (2001), here we introduce a new parameter

$$\begin{eqnarray}{\bar{\epsilon }}_{B}=\left\{\begin{array}{lc}{\epsilon }_{B}(1+{\eta }_{\mathrm{th}}), & t\lt {T}_{\mathrm{sd}}\\ {\epsilon }_{B}, & t\gt {T}_{\mathrm{sd}}\end{array}\right.\end{eqnarray} \tag{ 9 }$$

according to the discussion given in Section 2. ${\epsilon }_{e}$ and ${\epsilon }_{B}$ are the fraction of total energy going into electrons and the magnetic field, respectively. In the following analytical calculations, we usually adopt the approximation ${\bar{\epsilon }}_{B}\approx {\epsilon }_{B}$.

Before T_mc, i.e., the transition time between fast cooling and slow cooling, the electrons in the reverse shock are in the fast cooling regime, so that $\eta =1$. As determined by Wang & Dai (2013c), ${\epsilon }_{B}\simeq 0.1$ so that ${\epsilon }_{e}/{\epsilon }_{B}\simeq 9\gg 1$. According to Equation (8) and the definition of ${T}_{\mathrm{ct}}$ (Wang & Dai 2013c),

$$\begin{eqnarray}&&{T}_{\mathrm{ct}}=\displaystyle \frac{3\pi {m}_{e}c}{{\sigma }_{T}\gamma {B}_{3}^{2}(1+Y)},\end{eqnarray} \tag{ 10 }$$

we find

$$\begin{eqnarray}&&{T}_{\mathrm{ct}}=7.3\times {10}^{-2}\;\mathrm{days}{L}_{0,47}^{-2/3}{M}_{\mathrm{ej},-4}^{5/6}{\epsilon }_{e}^{1/12}{\epsilon }_{B,-1}^{1/12}.\end{eqnarray} \tag{ 11 }$$

In this approximation, i.e., $Y\gg 1$, the other two timescales can also be determined

$$\begin{eqnarray}&&{T}_{{ac}}=0.141\;\mathrm{days}{L}_{0,47}^{-(8p+25)/2(6p+19)}\\ &&\times {M}_{\mathrm{ej},-4}^{(5p+16)/(6p+19)}{\epsilon }_{B,-1}^{(p+2)/2(6p+19)}{\epsilon }_{e}^{(p+3)/2(6p+19)}{\gamma }_{4,4}^{-1/(6p+19)},\end{eqnarray} \tag{ 12 }$$

$$\begin{eqnarray}&&{T}_{{mc}}=0.144\;\mathrm{days}{L}_{0,47}^{-5/7}{M}_{\mathrm{ej},-4}^{6/7}{\epsilon }_{B,-1}^{1/14}{\epsilon }_{e}^{3/14}{\gamma }_{4,4}^{1/7}.\end{eqnarray} \tag{ 13 }$$

The above numerical values are evaluated for $p=2.2$, a value preferred in Wang & Dai (2013c). Throughout this paper we evaluate all reverse shock parameters at this value of p.

Before T_mc the effect of IC on the cooling frequency ${\nu }_{c}$ is just to modify it by a constant factor so that the temporal scaling indices of ${\nu }_{c}$ is not amended. But after T_mc, i.e., when the electrons are in the slow cooling regime, not only the exact value of ${\nu }_{c}$, but also its temporal scaling indices will be changed if IC dominates the electron cooling.

Wang & Dai (2013c) showed that ${T}_{\mathrm{dec}}$ is given by (see also Gao et al. 2013a)

$$\begin{eqnarray}&&{T}_{\mathrm{dec}}=0.28\;\mathrm{days}{L}_{0,47}^{-7/10}{M}_{\mathrm{ej},-4}^{4/5}{n}^{-1/10},\end{eqnarray} \tag{ 14 }$$

which is usually larger than T_mc. Now let us calculate the evolution of ${\nu }_{c}$ in the time period ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$. Before T_mc the electrons are in the fast cooling regime and IC dominates their cooling. So we expect that IC will continue to dominate the cooling of electrons for a while.

In the time period ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$, if IC cooling is ignored, then we have ${\nu }_{c}\propto {t}^{9}$ and ${\nu }_{m}\propto {t}^{-5}$ (Wang & Dai 2013c), so that ${\nu }_{c}/{\nu }_{m}\propto {t}^{14}$. When IC cooling is taken into account, in the slow IC-dominated stage, we have (Sari & Esin 2001)

$$\begin{eqnarray}&&{\nu }_{c}/{\nu }_{m}={(t/{T}_{{mc}})}^{14}{Y}^{-2}.\end{eqnarray} \tag{ 15 }$$

Please note that, upon taking into account IC emission, T_mc is no longer the time when ${\nu }_{c}={\nu }_{m}$. Here T_mc is similar to the variable ${t}_{0}^{\mathrm{IC}}$ defined by Sari & Esin (2001). Substitution of Equation (8) into the above equation in the case of $\eta {\epsilon }_{e}/{\epsilon }_{B}\gg 1$ and knowing that

$$\begin{eqnarray}&&\eta ={({\gamma }_{c}/{\gamma }_{m})}^{-(p-2)},\end{eqnarray} \tag{ 16 }$$

for slow cooling, we have

$$\begin{eqnarray}&&Y\simeq {\left[{\left(\displaystyle \frac{t}{{T}_{{mc}}}\right)}^{14}\displaystyle \frac{1}{{Y}^{2}}\right]}^{-(p-2)/4}\sqrt{\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}},\end{eqnarray} \tag{ 17 }$$

which gives

$$\begin{eqnarray}&&Y\simeq {\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{1/(4-p)}{\left(\displaystyle \frac{t}{{T}_{{mc}}}\right)}^{-7(p-2)/(4-p)}.\end{eqnarray} \tag{ 18 }$$

We finally arrive at for the interval ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$:

$$\begin{eqnarray}&&\displaystyle \frac{{\nu }_{c}}{{\nu }_{m}}={\left(\displaystyle \frac{t}{{T}_{{mc}}}\right)}^{28/(4-p)}{\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{-2/(4-p)}.\end{eqnarray} \tag{ 19 }$$

IC emission will not dominate the electron cooling when Y = 1, which occurs at

$$\begin{eqnarray}&&{t}^{\mathrm{IC}}={T}_{{mc}}{\left(\displaystyle \frac{{\epsilon }_{e}}{{\epsilon }_{B}}\right)}^{1/7(p-2)}\simeq 0.69\;\mathrm{days}.\end{eqnarray} \tag{ 20 }$$

This time is usually later than ${T}_{\mathrm{dec}}$. Consequently, IC emission usually dominates the electron cooling in the entire stage ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$. The IC emission will gradually become less dominant for electron cooling when the temporal scaling indices of ${\nu }_{c}/{\nu }_{m}$ is positive. However, inspection of Table 1 of Wang & Dai (2013c) indicates that the temporal scaling indices of ${\nu }_{c}/{\nu }_{m}$ is non-positive when $t\gt {T}_{\mathrm{dec}}$. In other words, the IC emission becomes progressively dominant for electron cooling in the stage $t\gt {T}_{\mathrm{dec}}$.

The cooling Lorentz factor in the slow IC-dominated regime is given by

$$\begin{eqnarray}&&{\gamma }_{c}={\left[\displaystyle \frac{6\pi {m}_{e}c}{{\sigma }_{T}\gamma {B}^{2}t}{\left(\displaystyle \frac{{\epsilon }_{B}}{{\epsilon }_{e}}\right)}^{1/2}\displaystyle \frac{1}{{\gamma }_{m}^{(p-2)/2}}\right]}^{2/(4-p)},\end{eqnarray} \tag{ 21 }$$

from which we can analytically calculate ${\nu }_{c}$ for $t\gt {T}_{\mathrm{dec}}$. The calculated temporal scaling indices of ${\nu }_{c}$ are given in Table 1. It is worth mentioning that because p is very close to 2, the scaling indices of ${\nu }_{c}$ given in Table 1 are actually nearly identical to that given in Table 1 of Wang & Dai (2013c). As a result, it is usually accurate enough to use Table 1 of Wang & Dai (2013c) to assess the time evolution of the various synchrotron quantities.

Table 1.  Analytical Temporal Scaling Indices of Various Parameters of the Reverse Shock IC Emission

	${\nu }_{c}$	${\nu }_{{ca}}^{\mathrm{IC}}$	${\nu }_{{ma}}^{\mathrm{IC}}$	${\nu }_{{cc}}^{\mathrm{IC}}$	${\nu }_{{mc}}^{\mathrm{IC}}$	${\nu }_{{mm}}^{\mathrm{IC}}$	${F}_{\nu ,\mathrm{max}}^{\mathrm{IC}}$
$t\lt {T}_{N1}$	$-\tfrac{3}{2}$	$-\tfrac{3p+14}{2(p+4)}$	$-\tfrac{3p+14}{2(p+4)}$	$-\tfrac{3}{2}$	$-\tfrac{3}{2}$	$-\tfrac{3}{2}$	$-\tfrac{5}{2}$
${T}_{N1}\lt t\lt {T}_{\mathrm{ct}}$	−3	$-\tfrac{3p+14}{p+4}$	$-\tfrac{5p+22}{p+4}$	−3	−5	−7	−7
${T}_{\mathrm{ct}}\lt t\lt {T}_{{ac}}$	9	$12-\tfrac{3p+2}{p+4}$	$-\tfrac{5(p+2)}{p+4}$	21	7	−7	−7
${T}_{{ac}}\lt t\lt {T}_{{mc}}$	9	$\tfrac{6(8-3p)}{5}$	$-\tfrac{2(9p+11)}{5}$	21	7	−7	−7
${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$	$\tfrac{8+5p}{4-p}$	$\tfrac{2(p+10)}{4-p}-\tfrac{13}{5}$	$-\tfrac{23}{5}$	$\tfrac{7(p+4)}{4-p}$	$\tfrac{7p}{4-p}$	−7	−7
${T}_{\mathrm{dec}}\lt t\lt {T}_{\mathrm{sd}}$	$-\tfrac{2}{4-p}$	$-\left(\tfrac{p}{2(4-p)}+\tfrac{3}{5}\right)$	$-\tfrac{1}{10}$	$-\tfrac{p+4}{2(4-p)}$	$-\tfrac{p}{2(4-p)}$	$\tfrac{1}{2}$	$\tfrac{1}{2}$
${T}_{\mathrm{sd}}\lt t\lt {T}_{N2}$	$-\left(\tfrac{1}{4-p}+\tfrac{9}{16}\right)$	$-\left(\tfrac{1}{4-p}+\tfrac{3}{4}\right)$	$-\tfrac{3}{4}$	$-\left(\tfrac{2}{4-p}+\tfrac{9}{16}\right)$	$-\left(\tfrac{1}{4-p}+\tfrac{9}{16}\right)$	$-\tfrac{9}{16}$	$-\tfrac{17}{16}$
${T}_{N2}\lt t$	$-\tfrac{3}{5}$	$-\tfrac{18}{25}$	$-\tfrac{18}{25}$	$-\tfrac{3}{5}$	$-\tfrac{3}{5}$	$-\tfrac{3}{5}$	$-\tfrac{7}{5}$

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To quantitatively appreciate the effects of IC on the synchrotron emission, in Figure 1, we compare the evolution of synchrotron characteristic frequencies with/without IC taken into account. In this numerical calculation, we adopt Equation (6) to determine the ejecta dynamics. Before T_mc, IC cooling is to introduce an extra constant factor ${({\epsilon }_{e}/{\epsilon }_{B})}^{1/2}$ for ${\nu }_{c}$. As a result of this factor, the three times ${T}_{\mathrm{ct}}$, T_ac, and T_mc are all delayed relative to the value without IC taken into account (Wang & Dai 2013c), as can be seen from Figure 1. After T_mc, the temporal scaling indices of ${\nu }_{c}$ are only slightly changed because p is very close to 2.

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Compared with Equation (5), the adoption of Equation (6) is to increase ${\nu }_{c}$ when $t\gt {T}_{{mc}}$, as can be seen from Figure 2(b) of Wang et al. (2015a). The consideration of IC is, on the other hand, to decrease ${\nu }_{c}$ by a nearly same factor. Consequently, we find that the combination of Equation (6) and the consideration of IC cooling results in a cooling frequency that is very close to the one determined in Wang & Dai (2013c), where we showed that the optical transient PTF11agg can be accounted for by the synchrotron emission of the reverse shock powered by a millisecond magnetar.

The decrease of ${\nu }_{c}$ caused by IC cooling can be easily calculated. Substitution of ${\epsilon }_{e}/{\epsilon }_{B}\simeq 9$ and $\eta =1$ for fast cooling into Equation (7) gives $Y\simeq 2.5$. During ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$, Y declines slightly, as indicated by Equation (18). But because p is very close to 2, the decline is not significant so that we still have $Y\gt 1$. When $t\gt {T}_{\mathrm{dec}}$, Y begins to increase slowly and actually remains close to ∼1.5. Therefore, IC cooling decreases ${\nu }_{c}$ by a factor ∼7.

The photons boosted by IC scattering acquire such a high energy that they will annihilate with the low-energy synchrotron and thermal photons. It is necessary to estimate the optical depth to γ–γ collisions so that we know under what conditions γ–γ collisions can be neglected. Figures 5–8 show that the synchrotron photons are usually scattered to a maximum energy of $\sim {10}^{13}\;\mathrm{eV}$. Therefore, we set the maximum energy of the IC-scattered photons as $h{\nu }_{\mathrm{max}}={10}^{13}{e}_{13}\;\mathrm{eV}$, where h is the Plank constant. The frequency of the softest photons that can annihilate with $h{\nu }_{\mathrm{max}}$ is given by (e.g., Lithwick & Sari 2001)

$$\begin{eqnarray}{\nu }_{\mathrm{max},{an},\mathrm{dec}}^{{\rm{I}}} & = & {\left(\displaystyle \frac{{\gamma }_{\mathrm{dec}}{m}_{e}{c}^{2}}{h}\right)}^{2}\displaystyle \frac{1}{{\nu }_{\mathrm{max}}}\\ & = & 2.8\times {10}^{14}\;\mathrm{Hz}{L}_{0,47}^{3/5}{M}_{\mathrm{ej},-4}^{-2/5}{e}_{13}^{-1}{n}^{-1/5}.\end{eqnarray} \tag{ 22 }$$

This frequency is in the optical to UV bands because ${e}_{13}\lesssim 1$. The above calculation is carried out at time ${T}_{\mathrm{dec}}$ by setting the Lorentz factor as ${\gamma }_{\mathrm{dec}}$. This is based on the observation that ${T}_{\mathrm{dec}}\simeq 0.2\;\mathrm{days}$ at which the spectra of IC emission are calculated below (see Section 4.1 for the reason why we choose this time).

Comparison of ${\nu }_{\mathrm{max},{an},\mathrm{dec}}^{{\rm{I}}}$ with Equations (10) and (11) of Wang & Dai (2013c) shows that ${\nu }_{\mathrm{max},{an},\mathrm{dec}}^{{\rm{I}}}$ lies in the range ${\nu }_{m}\lt {\nu }_{\mathrm{max},{an},\mathrm{dec}}^{{\rm{I}}}\lt {\nu }_{c}$. At time ${T}_{\mathrm{dec}}$ the reverse shock is in the slow cooling regime so that ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$, and the synchrotron spectrum is ${F}_{\nu }={F}_{\nu ,\mathrm{max}}{(\nu /{\nu }_{m})}^{-(p-1)/2}\equiv {g}_{1}{\nu }^{-{\alpha }_{1}}$ for a frequency in the range ${\nu }_{m}\lt \nu \lt {\nu }_{c}$. The synchrotron photon number with frequency above ν is

$$\begin{eqnarray}&&{N}_{\gt \nu }^{{\rm{I}}}=4\pi {D}_{L}^{2}\displaystyle \frac{{g}_{1}}{h}{T}_{\mathrm{dec}}\displaystyle \frac{{\nu }^{-{\alpha }_{1}}}{{\alpha }_{1}},\end{eqnarray} \tag{ 23 }$$

where ${\alpha }_{1}=(p-1)/2\lt 1$ if $p\lt 3$.

In the center-of-mass frame of the two colliding photons, the annihilation cross section is approximately ${\sigma }_{T}$ if their energy is just enough to create an electron pair. For higher-energy photons, the cross section goes to zero as a power law of the photon energy. For a power-law distribution of the seed photons, i.e., ${F}_{\nu }={g}_{1}{\nu }^{-{\alpha }_{1}}$, the average γ–γ collision cross section can be parameterized as ${\sigma }_{\gamma \gamma }=f({\alpha }_{1}){\sigma }_{T}$ with $f({\alpha }_{1})\lt 1$. In particular, $f(1)=11/180$ (Svensson 1987; Lithwick & Sari 2001).⁹ For harder spectrum, i.e., smaller ${\alpha }_{1}$, $f({\alpha }_{1})$ is smaller. The synchrotron γ–γ annihilation optical depth is

$$\begin{eqnarray}{\tau }_{\mathrm{syn},\gamma \gamma }^{{\rm{I}}} & = & \displaystyle \frac{f({\alpha }_{1}){\sigma }_{T}{N}_{\gt {\nu }_{\mathrm{max},{an},\mathrm{dec}}^{{\rm{I}}}}}{4\pi {(4{\gamma }_{\mathrm{dec}}^{2}{{cT}}_{\mathrm{dec}})}^{2}}\\ & = & 0.97\displaystyle \frac{{(p-2)}^{p-1}}{{(p-1)}^{p}}{n}_{-1}^{7(p+1)/20}{e}_{13}^{(p-1)/2}\\ & & \times {L}_{0,47}^{(7-3p)/10}{\gamma }_{4,4}^{p-2}{\epsilon }_{e}^{p-1}{\epsilon }_{B,-1}^{(p+1)/4}{M}_{\mathrm{ej},-4}^{(p+1)/5}.\end{eqnarray} \tag{ 24 }$$

The above numerical value is obtained by setting ${\alpha }_{1}=1$, $p=2.2$. The actual value is slightly smaller because ${\alpha }_{1}\lt 1$ and therefore $f({\alpha }_{1})\lt f(1)$.

We see that the optical depth ${\tau }_{\mathrm{syn},\gamma \gamma }$ at ${T}_{\mathrm{dec}}$ is close to unity for typical parameter values. At $t\lt {T}_{\mathrm{dec}}$ the synchrotron radiation in the reverse shock is intense enough that the highest energy IC photons are opaque to γ–γ collision. But in this paper we mainly focus on the IC photons with energy 100 and $1\;\mathrm{GeV}$. Equation (24) only weakly depends on ${L}_{\mathrm{0,47}}$, and ${\gamma }_{\mathrm{4,4}}$. The parameter ${\epsilon }_{e}$ does not concern us because ${\epsilon }_{e}\simeq 1$. For the $100\;\mathrm{GeV}$ IC photons, a simple extrapolation of Equation (24) indicates that an ejecta mass ${M}_{\mathrm{ej}}=7.5\times {10}^{-3}\;{M}_{\odot }$ will make the IC emission attenuated at ${T}_{\mathrm{dec}}$. However, the ejecta as massive as ${M}_{\mathrm{ej}}=7.5\times {10}^{-3}\;{M}_{\odot }$ are not subject to the above analysis because the above analysis is valid only for case I, i.e., ${M}_{\mathrm{ej}}\lt {M}_{\mathrm{ej},c}$ or equivalently ${T}_{\mathrm{sd}}\gt {T}_{\mathrm{dec}}$, where ${M}_{\mathrm{ej},c}$ is defined as (Gao et al. 2013a)

$$\begin{eqnarray}&&{M}_{\mathrm{ej},c}\sim {10}^{-3}{M}_{\odot }{n}^{1/8}{I}_{45}^{5/4}{L}_{0,47}^{-3/8}{P}_{0,-3}^{-5/2}{\xi }^{5/4}.\end{eqnarray} \tag{ 25 }$$

To find the critical ejecta mass at which the $100\;\mathrm{GeV}$ IC photons are completely attenuated by the soft photons, we need to extend the above analysis to Case III, i.e., ${M}_{\mathrm{ej}}\gt {M}_{\mathrm{ej},c}$ or equivalently ${T}_{\mathrm{sd}}\lt {T}_{\mathrm{dec}}$. In case III ${T}_{\mathrm{dec}}$ is defined as (Gao et al. 2013a; Wang et al. 2015a)

$$\begin{eqnarray}&&{T}_{\mathrm{dec}}=0.9\;\mathrm{days}{L}_{0,48}^{-7/3}{T}_{\mathrm{sd},4}^{-7/3}{M}_{\mathrm{ej},-3}^{8/3}{n}^{-1/3}.\end{eqnarray} \tag{ 26 }$$

${T}_{\mathrm{dec}}$ could be several days as long as ${M}_{\mathrm{ej}}\gt {10}^{-3}\;{M}_{\odot }$. We will evaluate the optical depth to γ–γ collisions at the time ${T}_{\mathrm{sd}}\sim 1\;\mathrm{days}$. In this case we have the following softest photons that can annihilate $h{\nu }_{\mathrm{max}}$

$$\begin{eqnarray}{\nu }_{\mathrm{max},{an},\mathrm{sd}}^{\mathrm{III}} & = & {\left(\displaystyle \frac{{\gamma }_{\mathrm{sd}}{m}_{e}{c}^{2}}{h}\right)}^{2}\displaystyle \frac{1}{{\nu }_{\mathrm{max}}}\\ & = & 2.0\times {10}^{14}\;\mathrm{Hz}{L}_{0,48}^{2}{T}_{\mathrm{sd},4}^{2}{M}_{\mathrm{ej},-3}^{-2}{e}_{13}^{-1}.\end{eqnarray} \tag{ 27 }$$

At the time ${T}_{\mathrm{sd}}$ the reverse shock is in the fast cooling regime, viz. ${\nu }_{c}\lt {\nu }_{a}\lt {\nu }_{m}$, and inspection of Equations (28)–(30) of (Wang et al. 2015a) indicates that ${\nu }_{m}\lt {\nu }_{\mathrm{max},{an},\mathrm{sd}}^{\mathrm{III}}$. The synchrotron spectrum is given by ${F}_{\nu }={F}_{\nu ,\mathrm{max}}{(\nu /{\nu }_{m})}^{-p/2}{({\nu }_{m}/{\nu }_{c})}^{-1/2}\equiv {g}_{2}{\nu }^{-{\alpha }_{2}}$ for $\nu \gt {\nu }_{m}$, where ${\alpha }_{2}=p/2$. The synchrotron photon number with frequency above ν is

$$\begin{eqnarray}&&{N}_{\gt \nu }^{\mathrm{III}}=4\pi {D}_{L}^{2}\displaystyle \frac{{g}_{2}}{h}{T}_{\mathrm{sd}}\displaystyle \frac{{\nu }^{-{\alpha }_{2}}}{{\alpha }_{2}}.\end{eqnarray} \tag{ 28 }$$

The synchrotron γ–γ annihilation optical depth is

$$\begin{eqnarray}{\tau }_{\mathrm{syn},\gamma \gamma }^{\mathrm{III}} & = & \displaystyle \frac{f({\alpha }_{2}){\sigma }_{T}{N}_{\gt {\nu }_{\mathrm{max},{an},\mathrm{sd}}^{\mathrm{III}}}}{4\pi {(4{\gamma }_{\mathrm{sd}}^{2}{{cT}}_{\mathrm{sd}})}^{2}}\\ & = & \displaystyle \frac{6.2}{p}{\left(\displaystyle \frac{p-2}{p-1}\right)}^{p-1}{e}_{13}^{p/2}{\gamma }_{4.4}^{p-2}{L}_{0,48}^{-\tfrac{11p}{4}-\tfrac{7}{2}}\\ & & \times {T}_{\mathrm{sd},4}^{-\tfrac{7p}{2}-6}{\epsilon }_{e}^{p-1}{\epsilon }_{B,-1}^{\tfrac{p+2}{4}}{M}_{\mathrm{ej},-3}^{3p+5}.\end{eqnarray} \tag{ 29 }$$

The above numerical value is evaluated at $p=2.2$ and $f({\alpha }_{2})$ is set to $f(1)$ because p is close to 2 and therefore ${\alpha }_{2}$ is close to 1. For the $100\;\mathrm{GeV}$ IC photons, adopting ${L}_{0}={10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ and ${T}_{\mathrm{sd}}={10}^{5}\;{\rm{s}}$, we find ${\tau }_{\mathrm{syn},\gamma \gamma }^{\mathrm{III}}=1$ when ${M}_{\mathrm{ej}}\;={M}_{\mathrm{ej},c}^{\mathrm{IC}}\simeq 3\times {10}^{-3}\;{M}_{\odot }$. Reducing ${\epsilon }_{B}$ can help raise the critical ejecta mass but its role is limited because ${\tau }_{\mathrm{syn},\gamma \gamma }^{\mathrm{III}}$ depends on ${M}_{\mathrm{ej}}$ very sensitively.

Now we turn to the thermal photons. At ${T}_{\mathrm{dec}}$ the electrons in the reverse shock are in slow cooling regime so that the energy is radiated dominantly by the electrons with Lorentz factor ${\gamma }_{c}$. The total synchrotron power is

$$\begin{eqnarray}&&{L}_{\mathrm{syn}}={P}_{\mathrm{syn}}({\gamma }_{c}){N}_{e},\end{eqnarray} \tag{ 30 }$$

where the total number of injected electrons is

$$\begin{eqnarray}&&{N}_{e}=\displaystyle \frac{{L}_{0}t}{{\gamma }_{4}{m}_{e}{c}^{2}}.\end{eqnarray} \tag{ 31 }$$

The thermal photons in the reverse shock with (observed) temperature T give rise to a thermal luminosity

$$\begin{eqnarray}&&{L}_{\mathrm{th}}=4\pi {R}^{2}\sigma {(T/{\gamma }_{\mathrm{dec}})}^{4}{\gamma }_{\mathrm{dec}}^{2}.\end{eqnarray} \tag{ 32 }$$

Then Equation (1) gives the temperature at time ${T}_{\mathrm{dec}}$

$$\begin{eqnarray}&&T=6.1\times {10}^{4}\;{\rm{K}}{\eta }_{\mathrm{th}}^{1/4}{L}_{0,47}^{19/40}{\gamma }_{4,4}^{-1/4}{M}_{\mathrm{ej},-4}^{-2/5}{n}^{-3/40}{\epsilon }_{B,-1}^{-1/4},\end{eqnarray} \tag{ 33 }$$

or equivalently the frequency of the thermal photons

$$\begin{eqnarray}{\nu }_{\mathrm{th}} & \equiv & \displaystyle \frac{{k}_{B}T}{h}\\ & \approx & 1.3\times {10}^{15}\;\mathrm{Hz}{\eta }_{\mathrm{th}}^{1/4}{L}_{0,47}^{19/40}{\gamma }_{4,4}^{-1/4}{M}_{\mathrm{ej},-4}^{-2/5}{n}^{-3/40}{\epsilon }_{B,-1}^{-1/4}.\end{eqnarray} \tag{ 34 }$$

One sees that this frequency is only weakly dependent on ${\eta }_{\mathrm{th}}$, which means that a value of ${\eta }_{\mathrm{th}}=0.1$ is essentially equivalent to ${\eta }_{\mathrm{th}}=1$. This frequency is usually higher than the minimum frequency (22) to annihilate the hardest IC photons ($10\;\mathrm{TeV}$). As a result, approximately all thermal photons, whose number density in the comoving frame is denoted by ${n}_{\mathrm{th},{an}}^{\prime }$, can annihilate the $10\;\mathrm{TeV}$ photons

$$\begin{eqnarray}&&{n}_{\mathrm{th},{an}}^{\prime }=16\pi \zeta (3){\left(\displaystyle \frac{{\nu }_{\mathrm{th}}^{\prime }}{c}\right)}^{3}=16\pi \zeta (3){\left(\displaystyle \frac{{\nu }_{\mathrm{th}}}{c{\gamma }_{\mathrm{dec}}}\right)}^{3},\end{eqnarray} \tag{ 35 }$$

where $\zeta (z)$ is the Riemann zeta function and $\zeta (3)=1.20206$. Here the prime denotes the quantities in the comoving frame. The optical depth for the thermal photons to annihilate the $10\;\mathrm{TeV}$ photons is

$$\begin{eqnarray}{\tau }_{\mathrm{th},\gamma \gamma }(10\;\mathrm{TeV}) & = & {n}_{\mathrm{th}}^{\prime }s(T^{\prime} ){\sigma }_{T}{{\rm{\Delta }}}_{3}^{\prime }\\ & = & 7.1\times {10}^{3}s\;{\eta }_{\mathrm{th}}^{3/4}{L}_{0,47}^{1/8}{\gamma }_{4,4}^{-3/4}{n}^{-1/8}{\epsilon }_{B,-1}^{-3/4},\end{eqnarray} \tag{ 36 }$$

where we parameterize the effective annihilation cross section of the thermal photons by $s(T^{\prime} ){\sigma }_{T}$ with $s(T^{\prime} )\lt 1$. The comoving width of Region 3 is given by (Wang & Dai 2013c)

$$\begin{eqnarray}&&{{\rm{\Delta }}}_{3}^{\prime }=\displaystyle \frac{{N}_{e}}{4\pi {r}^{2}{n}_{3}},\end{eqnarray} \tag{ 37 }$$

$$\begin{eqnarray}&&\displaystyle \frac{{n}_{3}}{{n}_{4}}=4{\bar{\gamma }}_{3}+3,\end{eqnarray} \tag{ 38 }$$

$$\begin{eqnarray}&&{\bar{\gamma }}_{3}=\displaystyle \frac{{\gamma }_{4}}{2}\displaystyle \frac{{M}_{\mathrm{ej}}{c}^{2}}{{L}_{0}t},\end{eqnarray} \tag{ 39 }$$

$$\begin{eqnarray}&&{n}_{4}=\displaystyle \frac{{L}_{0}}{4\pi {r}^{2}{\gamma }_{4}^{2}{m}_{e}{c}^{2}}.\end{eqnarray} \tag{ 40 }$$

This thermal annihilation optical depth is large for a reasonably small ${\eta }_{\mathrm{th}}\simeq 0.1$. Consequently we expect that the $10\;\mathrm{TeV}$ photons be attenuated significantly by the thermal photons.

For the 100 and $1\;\mathrm{GeV}$ radiation that we focus on in this paper, the softest photons that annihilate the high-energy IC photons have a frequency

$$\begin{eqnarray}&&{\nu }_{\mathrm{max},{an},\mathrm{dec}}=2.8\times {10}^{16}\;\mathrm{Hz}{L}_{0,47}^{3/5}{M}_{\mathrm{ej},-4}^{-2/5}{e}_{11}^{-1}{n}^{-1/5},\end{eqnarray} \tag{ 41 }$$

which is much harder than the thermal photons (${\nu }_{\mathrm{th}}$). In this case the thermal photons that can annihilate the high-energy IC photons is just the photons at the exponential tail of the thermal distribution

$$\begin{eqnarray}&&{n}_{\mathrm{th},{an}}^{\prime }=\displaystyle \frac{8\pi }{{c}^{3}{\gamma }_{\mathrm{dec}}^{3}}{\nu }_{\mathrm{th}}{\nu }_{\mathrm{max},{an},\mathrm{dec}}^{2}\mathrm{exp}(-{\nu }_{\mathrm{max},{an},\mathrm{dec}}/{\nu }_{\mathrm{th}}).\end{eqnarray} \tag{ 42 }$$

The annihilation optical depth is therefore very small, whatever value ${\eta }_{\mathrm{th}}$ reasonably takes,

$$\begin{eqnarray}&&{\tau }_{\mathrm{th},\gamma \gamma }(100\;\mathrm{GeV})\lesssim 2.7\times {10}^{-4}.\end{eqnarray} \tag{ 43 }$$

Here we suppress its parameter dependence because the most influential factor appears in the exponent. The optical depth for $1\;\mathrm{GeV}$ photons is even smaller.

To summarize the findings in this subsection, at time ${T}_{\mathrm{dec}}$, the highest energy IC photons ($10\;\mathrm{TeV}$) are attenuated by thermal photons. The $100\;\mathrm{GeV}$ photons, on which we focus in this paper, on the other hand, are attenuated by synchrotron photons at time ${T}_{\mathrm{sd}}=1\;\mathrm{days}$ if the ejecta are more massive than $3\times {10}^{-3}\;{M}_{\odot }$. In short, the γ–γ collisions have an influence on the light curves but are usually negligible for our purpose to calculate the 1 and $100\;\mathrm{GeV}$ for ejecta mass ${10}^{-4}\;{M}_{\odot }$ and ${10}^{-3}\;{M}_{\odot }$. As a result in the following calculations we neglect γ–γ collisions but mention the caveat where appropriate.

In this subsection, we will analytically calculate the IC spectra and light curves in order to determine the best observation strategy: observational energy bands and cadence. We assume the Thomson limit in this analytical treatment and refrain from consideration of the Klein–Nishina effect until Section 4.2 where numerical calculations are performed.

When $t\lt {T}_{{ac}}$, we have ${\nu }_{c}\lt {\nu }_{a}\lt {\nu }_{m}$, for which the IC spectrum is given in the Appendix. When $t\gt {T}_{{ac}}$, whether ${\nu }_{a}\lt {\nu }_{c}\lt {\nu }_{m}$ or ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$, we adopt the IC spectrum calculated by Sari & Esin (2001). We have discussed the calculation of ${\nu }_{c}$ and ${\gamma }_{c}$ in Section 3.1. With these results, we can easily calculate the temporal evolution of IC break frequencies and IC flux density, which are given in Tables 1 and 2, respectively.

Table 2.  Analytical Temporal Scaling Indices of IC Flux Density of the Reverse Shock

$({\nu }_{c}\lt {\nu }_{a}\lt {\nu }_{m})$	$\nu \lt {\nu }_{{ca}}^{\mathrm{IC}}$	${\nu }_{{ca}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{mm}}^{\mathrm{IC}}$	${\nu }_{{mm}}^{\mathrm{IC}}\lt \nu$
$t\lt {T}_{N1}$	$-\displaystyle \frac{2p+5}{2(p+4)}$	$-\displaystyle \frac{13}{4}$	$-\displaystyle \frac{3p+10}{4}$
${T}_{N1}\lt t\lt {T}_{\mathrm{ct}}$	$-\displaystyle \frac{4p+13}{p+4}$	$-\displaystyle \frac{17}{2}$	$-\displaystyle \frac{7p+10}{2}$
${T}_{\mathrm{ct}}\lt t\lt {T}_{{ac}}$	$-\displaystyle \frac{5(2p+11)}{p+4}$	$\displaystyle \frac{7}{2}$	$-\displaystyle \frac{7(p-2)}{2}$
$({\nu }_{a}\lt {\nu }_{c}\lt {\nu }_{m})$	$\nu \lt {\nu }_{{ca}}^{\mathrm{IC}}$	${\nu }_{{ca}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{cc}}^{\mathrm{IC}}$	${\nu }_{{cc}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{mm}}^{\mathrm{IC}}$	${\nu }_{{mm}}^{\mathrm{IC}}\lt \nu$
${T}_{{ac}}\lt t\lt {T}_{{mc}}$	$-\displaystyle \frac{6(17-2p)}{5}$	−14	$\displaystyle \frac{7}{2}$	$-\displaystyle \frac{7(p-2)}{2}$
$({\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c})$	$\nu \lt {\nu }_{{ma}}^{\mathrm{IC}}$	${\nu }_{{ma}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{mm}}^{\mathrm{IC}}$	${\nu }_{{mm}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{cc}}^{\mathrm{IC}}$	${\nu }_{{cc}}^{\mathrm{IC}}\lt \nu$
${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$	$-\displaystyle \frac{8}{5}$	$-\displaystyle \frac{14}{3}$	$-\displaystyle \frac{7(p+1)}{2}$	$\displaystyle \frac{7p(p-2)}{2(4-p)}$
${T}_{\mathrm{dec}}\lt t\lt {T}_{\mathrm{sd}}$	$\displaystyle \frac{2}{5}$	$\displaystyle \frac{1}{3}$	$\displaystyle \frac{p+1}{4}$	$-\displaystyle \frac{p(p-2)}{4(4-p)}$
${T}_{\mathrm{sd}}\lt t\lt {T}_{N2}$	$-\displaystyle \frac{3}{8}$	$-\displaystyle \frac{7}{8}$	$-\displaystyle \frac{9p+25}{32}$	$-\left(\displaystyle \frac{9p+34}{32}+\displaystyle \frac{1}{4-p}\right)$
${T}_{N2}\lt t$	$-\displaystyle \frac{18}{25}$	$-\displaystyle \frac{6}{5}$	$-\displaystyle \frac{3p+11}{10}$	$-\displaystyle \frac{3p+14}{10}$

Note. We have ${\nu }_{c}\lt {\nu }_{a}\lt {\nu }_{m}$ if $t\lt {T}_{{ac}}$, ${\nu }_{a}\lt {\nu }_{c}\lt {\nu }_{m}$ if ${T}_{{ac}}\lt t\lt {T}_{{mc}}$, and ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$ if $t\gt {T}_{{mc}}$.

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Provided that the observational bands lie in the range $\nu \gt \mathrm{max}({\nu }_{{mm}}^{\mathrm{IC}},{\nu }_{{cc}}^{\mathrm{IC}})$, which is usually always true if we aim in detecting $\;\mathrm{GeV}$ emission, we can read off the evolution of IC flux density directly from Table 2. We find that the flux density always declines except in the period ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$. This fact can be clearly seen from the numerical calculation results, i.e., Figures 2 and 3. But owing to the fact that p is very close to 2, the light curves in the time period ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}}$ are almost flat. The rapid increase in the flux density at the very beginning results from the ejecta becoming progressively transparent for γ-ray emission. Wang et al. (2015a) showed that the opacity at X-ray and γ-ray bands is caused by the elastic scattering of photons off free electrons in the shocked wind. Consequently the time for the ejecta becoming transparent for γ-ray emission is given by Equation (43) in Wang et al. (2015a):

$$\begin{eqnarray}&&{T}_{\gamma ,\mathrm{thin}}=8.2\times {10}^{-3}\;\mathrm{days}{M}_{\mathrm{ej},-4}^{2/3}{L}_{0,47}^{-1/3}.\end{eqnarray} \tag{ 44 }$$

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The IC component of the reverse shock emission can be observed if it dominates over synchrotron emission. We therefore need to determine the frequency ${\nu }^{\mathrm{IC}}$ at which the synchrotron component crosses the IC component.

When $t\lt {T}_{{ac}}$, i.e., for ${\nu }_{c}\lt {\nu }_{a}\lt {\nu }_{m}$, Appendix shows that the IC flux density has a spectral index of $-1/2$ for a wide frequency range ${\nu }_{{ca}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{mm}}^{\mathrm{IC}}$. By assuming that ${\nu }^{\mathrm{IC}}$ lies in the synchrotron spectrum segment with spectral index $-p/2$ and between the IC break frequencies ${\nu }_{{ca}}^{\mathrm{IC}}\lt \nu \lt {\nu }_{{mm}}^{\mathrm{IC}}$ we have

$$\begin{eqnarray}&&{\nu }^{\mathrm{IC}}\\ &&=\ \left\{\begin{array}{l}1.3\times {10}^{18}\;\mathrm{Hz}{\epsilon }_{B,-1}^{1/2}{\epsilon }_{e}^{2}{M}_{\mathrm{ej},-4}^{-\tfrac{5-p}{2(p-1)}}{\gamma }_{4,4}^{\tfrac{2p}{p-1}}{t}_{3}^{\tfrac{4}{p-1}-\tfrac{3}{2}},\\ \quad t\lt {T}_{N1};\\ 1.5\times {10}^{17}\;\mathrm{Hz}{\epsilon }_{B,-1}^{1/2}{\epsilon }_{e}^{2}{L}_{0,47}^{\tfrac{6}{p-1}-\tfrac{7}{2}}{M}_{\mathrm{ej},-4}^{-\tfrac{4(3-p)}{p-1}}{\gamma }_{4,4}^{\tfrac{2p}{p-1}}{t}_{3}^{\tfrac{5(3-p)}{p-1}},\\ \quad {T}_{N1}\lt t\lt {T}_{\mathrm{ct}};\\ 7.7\times {10}^{16}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\tfrac{p+1}{2(p-1)}}{\epsilon }_{e}^{\tfrac{2p-1}{p-1}}{L}_{0,47}^{-\tfrac{7p-3}{2(p-1)}}{M}_{\mathrm{ej},-4}^{\tfrac{2(2p-1)}{p-1}}{\gamma }_{4,4}^{\tfrac{2p}{p-1}}{t}_{3}^{-\tfrac{5p-3}{p-1}},\\ \quad {T}_{\mathrm{ct}}\lt t\lt {T}_{{ac}}.\end{array}\right.\end{eqnarray} \tag{ 45 }$$

We see that ${\nu }^{\mathrm{IC}}$ lies in the soft X-ray bands for the typical parameters if $t\lt {T}_{{ac}}$.

When ${T}_{{ac}}\lt t\lt {T}_{{mc}}$, i.e., for ${\nu }_{a}\lt {\nu }_{c}\lt {\nu }_{m}$, if ${\nu }^{\mathrm{IC}}\lt {\nu }_{{cc}}^{\mathrm{IC}}$, we have

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 2.5\times {10}^{15}\;\mathrm{Hz}{\epsilon }_{B,-1}^{-\displaystyle \frac{3(4-p)}{2(3p+2)}}{\epsilon }_{e}^{\tfrac{6p-13}{3p+2}}{L}_{0,47}^{\displaystyle \frac{7(22-3p)}{2(3p+2)}}\\ & & \times {M}_{\mathrm{ej},-4}^{-\tfrac{12(8-p)}{3p+2}}{\gamma }_{4,4}^{\tfrac{6p}{3p+2}}{t}_{4}^{\tfrac{3(38-5p)}{3p+2}},\end{eqnarray} \tag{ 46 }$$

otherwise we have

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 3.2\times {10}^{18}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{p+1}{2(p-1)}}{\epsilon }_{e}^{\tfrac{2p-1}{p-1}}\\ & & \times {L}_{0,47}^{-\tfrac{7p-3}{2(p-1)}}{M}_{\mathrm{ej},-4}^{\tfrac{2(2p-1)}{p-1}}{\gamma }_{4,4}^{\tfrac{2p}{p-1}}{t}_{4}^{-\tfrac{5p-3}{p-1}},\end{eqnarray} \tag{ 47 }$$

if ${\nu }^{\mathrm{IC}}\gt {\nu }_{{cc}}^{\mathrm{IC}}$.

When $t\gt {T}_{{mc}}$, i.e., for ${\nu }_{a}\lt {\nu }_{m}\lt {\nu }_{c}$, if ${\nu }^{\mathrm{IC}}\lt {\nu }_{{mm}}^{\mathrm{IC}}$, we have

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 2.0\times {10}^{16}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{3{p}^{2}-10p+4}{2(p-4)(3p+2)}}{\epsilon }_{e}^{\tfrac{2(3{p}^{2}-8p-7)}{(p-4)(3p+2)}}\\ & & \times {L}_{0,47}^{-\tfrac{21{p}^{2}-98p+176}{2(p-4)(3p+2)}}{M}_{\mathrm{ej},-4}^{\tfrac{12({p}^{2}-5p+10)}{(p-4)(3p+2)}}\\ & & \times {\gamma }_{4,4}^{\tfrac{2(p+2)(3p-11)}{(p-4)(3p+2)}}{t}_{4}^{\tfrac{15{p}^{2}-76p+148}{(4-p)(3p+2)}},\end{eqnarray} \tag{ 48 }$$

for ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}};$

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 2.7\times {10}^{18}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{3{p}^{2}-10p+4}{2(p-4)(3p+2)}}{\epsilon }_{e}^{\tfrac{2(3{p}^{2}-8p-7)}{(p-4)(3p+2)}}\\ & & \times {L}_{0,47}^{-\tfrac{7p-34}{2(p-4)(3p+2)}}{n}^{\displaystyle \frac{3({p}^{2}-5p+10)}{2(p-4)(3p+2)}}\\ & & \times {\gamma }_{4,4}^{\displaystyle \frac{2(p+2)(3p-11)}{(p-4)(3p+2)}}{t}_{4}^{\tfrac{p+2}{(p-4)(3p+2)}},\end{eqnarray} \tag{ 49 }$$

for ${T}_{\mathrm{dec}}\lt t\lt {T}_{\mathrm{sd}};$

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 6.9\times {10}^{18}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{3{p}^{2}-10p+4}{2(p-4)(3p+2)}}{\epsilon }_{e}^{\tfrac{2(3{p}^{2}-8p-7)}{(p-4)(3p+2)}}\\ & & \times {L}_{0,47}^{-\tfrac{7p-34}{2(p-4)(3p+2)}}{n}^{\displaystyle \frac{3({p}^{2}-5p+10)}{2(p-4)(3p+2)}}\\ & & \times {T}_{\mathrm{sd},5}^{\tfrac{27{p}^{2}-122p+104}{16(p-4)(3p+2)}}{\gamma }_{4,4}^{\tfrac{2(p+2)(3p-11)}{(p-4)(3p+2)}}{t}_{5}^{-\tfrac{3(9{p}^{2}-46p+24)}{16(p-4)(3p+2)}},\end{eqnarray} \tag{ 50 }$$

for ${T}_{\mathrm{sd}}\lt t\lt {T}_{N2};$ and

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 1.2\times {10}^{17}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{3{p}^{2}-10p+4}{2(p-4)(3p+2)}}{\epsilon }_{e}^{\tfrac{2(3{p}^{2}-8p-7)}{(p-4)(3p+2)}}\\ & & \times {L}_{0,47}^{\tfrac{3{p}^{2}-338p+1688}{80(p-4)(3p+2)}}{n}^{\tfrac{117{p}^{2}-542p+872}{80(p-4)(3p+2)}}\\ & & \times {T}_{\mathrm{sd},5}^{\tfrac{69{p}^{2}-334p+424}{40(p-4)(3p+2)}}{\gamma }_{4,4}^{\tfrac{2(p+2)(3p-11)}{(p-4)(3p+2)}}{t}_{7}^{-\tfrac{3(3{p}^{2}-18p+28)}{5(p-4)(3p+2)}},\end{eqnarray} \tag{ 51 }$$

for ${T}_{N2}\lt t$. On the other hand, if ${\nu }^{\mathrm{IC}}\gt {\nu }_{{mm}}^{\mathrm{IC}}$, we have

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 9.0\times {10}^{13}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{p}{2(p-4)}}{\epsilon }_{e}^{-\tfrac{2(p-5)(p-1)}{p-4}}{L}_{0,47}^{\displaystyle \frac{4{p}^{2}-15p-44}{2(p-4)}}\\ & & \times {M}_{\mathrm{ej},-4}^{-\tfrac{2({p}^{2}-3p-16)}{p-4}}{\gamma }_{4,4}^{-\tfrac{2(p-5)(p-2)}{p-4}}{t}_{4}^{\tfrac{2{p}^{2}-5p-40}{p-4}},\end{eqnarray} \tag{ 52 }$$

for ${T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}};$

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 7.9\times {10}^{23}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{p}{2(p-4)}}{\epsilon }_{e}^{-\tfrac{2(p-5)(p-1)}{p-4}}\\ & & \times {L}_{0,47}^{\displaystyle \frac{{p}^{2}-9p+24}{4(p-4)}}{n}^{-\tfrac{{p}^{2}-3p-16}{4(p-4)}}{\gamma }_{4,4}^{-\tfrac{2(p-5)(p-2)}{p-4}}{t}_{4}^{-\tfrac{(p-5)p}{2(p-4)}},\end{eqnarray} \tag{ 53 }$$

for ${T}_{\mathrm{dec}}\lt t\lt {T}_{\mathrm{sd}};$

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 8.8\times {10}^{24}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{p}{2(p-4)}}{\epsilon }_{e}^{-\tfrac{2(p-5)(p-1)}{p-4}}{L}_{0,47}^{\displaystyle \frac{{p}^{2}-9p+24}{4(p-4)}}\\ & & \times {n}^{-\tfrac{{p}^{2}-3p-16}{4(p-4)}}{\gamma }_{4,4}^{-\tfrac{2(p-5)(p-2)}{p-4}}{T}_{\mathrm{sd},5}^{-\tfrac{8{p}^{2}-33p-12}{16(p-4)}}{t}_{5}^{\tfrac{7}{16}-\displaystyle \frac{1}{4-p}},\end{eqnarray} \tag{ 54 }$$

for ${T}_{\mathrm{sd}}\lt t\lt {T}_{N2};$ and

$$\begin{eqnarray}{\nu }^{\mathrm{IC}} & = & 6.4\times {10}^{20}\;\mathrm{Hz}{\epsilon }_{B,-1}^{\displaystyle \frac{p}{2(p-4)}}{\epsilon }_{e}^{-\tfrac{2(p-5)(p-1)}{p-4}}{L}_{0,47}^{\displaystyle \frac{20{p}^{2}-195p+588}{80(p-4)}}\\ & & \times {n}^{-\tfrac{20{p}^{2}-75p-212}{80(p-4)}}{\gamma }_{4,4}^{-\tfrac{2(p-5)(p-2)}{p-4}}\\ & & \times {T}_{\mathrm{sd},5}^{-\tfrac{20{p}^{2}-75p-84}{40(p-4)}}{t}_{7}^{\tfrac{5p-24}{5(p-4)}},\end{eqnarray} \tag{ 55 }$$

for ${T}_{N2}\lt t$. The above expressions are similar to Equation (5.1) in Sari & Esin (2001).

To determine the maximum energy of IC photons, we need to calculate the maximum Lorentz factor of the electrons in the shocked wind, which is given by

$$\begin{eqnarray}&&{\gamma }_{M}=\displaystyle \frac{3}{2}\displaystyle \frac{{m}_{e}{c}^{2}}{\sqrt{{q}^{3}{B}_{3}(1+Y)}},\end{eqnarray} \tag{ 56 }$$

if the IC cooling effect is taken into account. Here B₃ is the magnetic field of Region 3, i.e., the shocked wind.

$$\begin{eqnarray}&&{\gamma }_{M}\\ &&=\ \left\{\begin{array}{l}2.4\times {10}^{5}{M}_{\mathrm{ej},-4}^{-1/4}{\epsilon }_{e}^{-1/4}{t}_{3}^{3/4},\\ \quad t\lt {T}_{N1},\\ 1.9\times {10}^{5}{M}_{\mathrm{ej},-4}^{-3/2}{\epsilon }_{e}^{-1/4}{L}_{0,47}^{5/4}{t}_{3}^{2},\\ \quad {T}_{N1}\lt t\lt {T}_{{mc}},\\ 1.8\times {10}^{7}{L}_{0,47}^{\tfrac{5p}{4(4-p)}}{M}_{\mathrm{ej},-4}^{-\tfrac{3p}{2(4-p)}}{\gamma }_{4,4}^{-\tfrac{p-2}{2(4-p)}}{\epsilon }_{B,-1}^{-\tfrac{p-2}{4(4-p)}}{\epsilon }_{e}^{-\tfrac{p-1}{2(4-p)}}{t}_{4}^{\tfrac{3p+2}{2(4-p)}},\\ \quad {T}_{{mc}}\lt t\lt {T}_{\mathrm{dec}},\\ 1.8\times {10}^{8}{L}_{0,47}^{-\tfrac{p}{16(4-p)}}{n}^{-\tfrac{3p}{16(4-p)}}{\gamma }_{4,4}^{-\tfrac{p-2}{2(4-p)}}{\epsilon }_{B,-1}^{-\tfrac{p-2}{4(4-p)}}{\epsilon }_{e}^{-\tfrac{p-1}{2(4-p)}}{t}_{4}^{\tfrac{8-3p}{8(4-p)}},\\ \quad {T}_{\mathrm{dec}}\lt t\lt {T}_{\mathrm{sd}}.\end{array}\right.\end{eqnarray} \tag{ 57 }$$

The maximum energy of the IC photons is given by $\gamma {\gamma }_{M}{m}_{e}{c}^{2}$, which is $\gtrsim 10\;\mathrm{TeV}$ for the most likely observational window, i.e., ${T}_{{mc}}\lt t\lt {T}_{\mathrm{sd}}$.

With the above calculations, we can now design the observational strategy. To clearly detect the IC component, viz. avoid the contamination from synchrotron component, it is best to observe in γ-ray bands. From Table 2 and also from Figures 2 and 3, it can be seen that the IC emission declines rapidly when $t\gt {T}_{\mathrm{sd}}$. In other words, the IC emission can be detected only in the period $t\lesssim {T}_{\mathrm{sd}}\sim 1\;\mathrm{days}$ for typical millisecond magnetars with ${B}_{p}={10}^{15}\;{\rm{G}}$.

To accurately calculate the high-energy spectrum of the IC emission of the reverse shock, the Klein–Nishina effect should be taken into account. Unfortunately, the analytical treatment of this effect is complicated and consists of several breaks (Nakar et al. 2009). We therefore adopt a numerical approach.

The IC volume emissivity in the full precision of the Klein–Nishina scattering cross section for an electron distribution $N(\gamma )$ is given by (Blumenthal & Gould 1970)

$$\begin{eqnarray}&&{j}_{\nu }^{\mathrm{IC}}=3{\sigma }_{T}\int d\gamma N(\gamma ){\displaystyle \int }_{{\nu }_{s,\mathrm{min}}}^{\infty }\displaystyle \frac{\nu }{4{\gamma }^{2}{\nu }_{s}^{2}}g(x,y){F}_{{\nu }_{s}}d{\nu }_{s},\end{eqnarray} \tag{ 58 }$$

where ${F}_{{\nu }_{s}}$ is the synchrotron seed flux density, ${\nu }_{s,\mathrm{min}}=\nu /4{\gamma }^{2}$, $g(x,y)$ is given by

$$\begin{eqnarray}&&g(x,y)=2y\mathrm{ln}y+1+y-2{y}^{2}+\displaystyle \frac{1}{2}\displaystyle \frac{{x}^{2}{y}^{2}}{1+{xy}}(1-y),\end{eqnarray} \tag{ 59 }$$

and x, y are defined as

$$\begin{eqnarray}&&x=\displaystyle \frac{4\gamma h{\nu }_{s}}{{m}_{e}{c}^{2}},y=\displaystyle \frac{h\nu }{x(\gamma {m}_{e}{c}^{2}-h\nu )}=\displaystyle \frac{\nu }{4{\gamma }^{2}{\nu }_{s}-x\nu }.\end{eqnarray} \tag{ 60 }$$

We calculate the IC emission of both the reverse shock and the forward shock numerically, assuming that the NS mergers are located at a luminosity distance ${D}_{L}={10}^{27}\;\mathrm{cm}$. In Figures 2 and 3, we choose the fiducial model (solid lines) as the one with parameters ${M}_{\mathrm{ej}}={10}^{-4}\;{M}_{\odot }$, ${L}_{\mathrm{sd}}={10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, and ${T}_{\mathrm{sd}}={10}^{5}\;{\rm{s}}$. To see the effect of varying ${M}_{\mathrm{ej}}$, we show the light curves for ${M}_{\mathrm{ej}}=5\times {10}^{-4}\;{M}_{\odot }$ as dashed lines. It can be seen that the more massive the ejecta, the easier for it to be observed. The figure also shows that the IC emission from forward shock is unlikely to be observed.

To assess the influence of a different value of ${L}_{\mathrm{sd}}$, we compare the light curves for ${L}_{\mathrm{sd}}=5\times {10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ with the fiducial model in Figure 3. What should be noted is that the rotational energy of the magnetars is fixed to be ${10}^{52}\;\mathrm{erg}$. As a result, with a spin-down power of ${L}_{\mathrm{sd}}=5\times {10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, its spin-down timescale is reduced to ${T}_{\mathrm{sd}}=2\times {10}^{4}\;{\rm{s}}$. From Figure 3 we see that the IC emission power is enhanced by the more powerful magnetars, but its duration is shorten significantly. Consequently, only high-cadence observations can reveal the formation of post-merger magnetars with a larger spin-down power.

In the above calculations we set the Lorentz factor of the unshocked wind (Region 4) as ${\gamma }_{4}={10}^{4}$, as determined in the literature (Atoyan 1999; Dai 2004; Wang & Dai 2013c). The ambient medium density is set as $n=0.1\;{\mathrm{cm}}^{-3}$ (Berger et al. 2005; Soderberg et al. 2006; Berger 2007; Wang & Dai 2013c). We will first observe the reverse shock emission at time $\sim {T}_{\gamma ,\mathrm{thin}}$ and then the forward shock emission at time $\sim {T}_{\mathrm{dec}}$ if the forward shock γ ray can be observed.

Figure 4 shows the effects of varying ${\gamma }_{4}$ on the resulting IC light curves. We see that increasing ${\gamma }_{4}$ will suppress the early IC emission but enhance it later on. This behavior can be understood as follows. At beginning, the ejecta velocity is low and the electrons' individual energy in the reverse shock is high enough to inverse scatter the photons to high energy. As a result, the IC emission intensity is enhanced because of the lower Lorentz factor of the unshocked wind and hence more numerous electrons in the shocked wind (the wind power ${L}_{\mathrm{sd}}$ is fixed here). At later time, however, the ejecta become relativistic and the electrons' individual energy in the reverse shock is reduced. In this case, a larger Lorentz factor of the unshocked wind is prone to produce more intense IC emission.

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Here we choose the observational bands at energies of 1 and $100\;\mathrm{GeV}$. The energy band at $100\;\mathrm{GeV}$ is chosen here because this is in the sensitive region for CTA and far below the high-energy cut of the reverse shock IC emission $\gamma {\gamma }_{M}{m}_{e}{c}^{2}$. We choose the energy band at $1\;\mathrm{GeV}$ since this is the most sensitive observational band for Fermi/LAT and it is also near the peak energy of the IC spectrum. These points can be easily seen from Figures 5–8.

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To appreciate the IC spectrum and its evolution, we show the spectra at times $t=0.2\;\mathrm{days}$ and $t=1\;\mathrm{day}$ in Figures 5 (the fiducial model, ${M}_{\mathrm{ej}}={10}^{-4}\;{M}_{\odot }$, ${L}_{\mathrm{sd}}={10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$, and ${T}_{\mathrm{sd}}={10}^{5}\;{\rm{s}}$), 6 (${M}_{\mathrm{ej}}=5\times {10}^{-4}{M}_{\odot }$), 7 (${L}_{\mathrm{sd}}=5\times {10}^{47}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$), and 8 (${\gamma }_{4}={10}^{6}$). From Figure 2 we see that the second peak occurs at $t\simeq 0.2\;\mathrm{days}$ for the fiducial model. This is why we choose to calculate one of the IC spectra at $t=0.2\;\mathrm{days}$. The reason we do not choose the time when the first peak occurs is that it happens too early and is therefore not easy to be caught by a regular-cadence observation. We choose to calculate the other spectrum at $t=1\;\mathrm{day}$ because ${T}_{\mathrm{sd}}\simeq 1\;\mathrm{day}$ for a typical magnetar so that its emission is still very strong to be caught by telescopes. These figures also show that the frequencies ${\nu }^{\mathrm{IC}}$ at which IC emission dominates over synchrotron emission usually occur in the X-ray and γ-ray bands for typical parameters.

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