HIGH-ENERGY GAMMA-RAY AFTERGLOWS FROM LOW-LUMINOSITY GAMMA-RAY BURSTS

Long duration gamma-ray bursts (GRBs) are generally believed to result from the death of massive stars, and their association with core-collapse supernovae (SNe of Type Ib/c) has been observed over the last decade. The first hint for such a connection came with the discovery of a nearby SN 1998bw in the error circle of GRB 980425 (Galama et al. 1998; Iwamoto et al. 1998) at distance of only about 40 Mpc. The isotropic gamma-ray energy release is of the order of only 10⁴⁸ erg (Galama et al. 1998) and the radio afterglow modeling suggests an energy of 10⁴⁹–10⁵⁰ erg in a mildly relativistic ejecta (Kulkarni et al. 1998; Chevalier & Li 1999). Recently Swift discovered GRB 060218, which is the second nearest GRB identified so far (Campana et al. 2006; Cusumano et al. 2006; Mirabal et al. 2006; Sakamoto et al. 2006). It is also an under-energetic burst with energy in prompt γ/X-rays about 5 × 10⁴⁹ erg and is associated with SN 2006aj. Another burst, GRB 031203, is a third example of this group (Sazonov et al. 2004).

If one assumes that GRB 060218-like bursts follow the logN–logP relationship of high-luminosity GRBs then, as argued by Guetta et al. (2004), no such burst with redshifts z < 0.17 should be observed by a HETE-like instrument within the next 20 years. Therefore, the unexpected discovery of GRB 060218 may suggest that these objects form a different new class of GRBs from the conventional high-luminosity GRBs (Soderberg et al. 2006; Pian et al. 2006; Cobb et al. 2006; Liang et al. 2006; Guetta & Della Valle 2007; Dai 2009), although what distinguishes such low-luminosity GRBs (LL-GRBs) from the conventional high-luminosity GRBs (HL-GRBs) remains unknown. Their rate of occurrence may be 1 order of magnitude higher than that of the typical ones (e.g., Soderberg et al. 2008). They are rarely recorded because such intrinsically dim GRBs can only be detected from relatively short distances with present gamma-ray instruments.

The "compactness problem" of GRBs requires that the outflow of normal GRBs should be highly relativistic with a bulk Lorentz factor of Γ₀ ≳ 100 (Baring & Harding 1997; Lithwick & Sari 2001). The low-luminosity and softer spectra of LL-GRBs relax this constraint on the bulk Lorentz factor. It was suggested that the softer spectrum and low energetics of GRB060218 (or classified as X-ray flash 060218) may indicate a somewhat lower Lorentz factor of the order of ∼10 (e.g., Mazzali et al. 2006; Fan et al. 2006; Toma et al. 2006).

An alternative possibility is that LL-GRBs are driven by a trans-relativistic outflow with Γ₀ ≃ 2 (Waxman 2004; Waxman et al. 2007; Wang et al. 2007; Ando & Mészáros 2008). The flat light curve of the X-ray afterglow of the nearest GRB, GRB 980425/SN 1998bw, up to 100 days after the burst, has been argued to result from the coasting phase of a mildly relativistic shell with an energy of a few times 10⁴⁹ erg (Waxman 2004). From the thermal energy density in the prompt emission of GRB060218/SN2006aj, Campana et al. (2006) inferred that the shell driving the radiation-dominated shock in GRB 060218/SN 2006aj must be mildly relativistic. This trans-relativistic shock could be driven by the outermost parts of the envelope that get accelerated to a mildly relativistic velocity when the supernova shock accelerates in the density gradient of the envelope of the supernova progenitor (Colgate 1974; Matzner & McKee 1999; Tan et al. 2001), or it could be due to a choked relativistic jet propagating through the progenitor (Wang et al. 2007).

In this paper, we investigate the high-energy afterglow emission from LL-GRBs for both relativistic and trans-relativistic models and explore whether the Fermi Gamma-ray Space Telescope can distinguish between these two models with future observations of LL-GRBs. Since photons from the underlying supernova are an important seed photon source for inverse Compton (IC) scattering, we consider both synchrotron self-inverse Compton (SSC) and external IC scattering due to supernova photons (denoted by SNIC hereafter). At early times (within a few days after the burst), a UV–optical SN component was recently detected from SN2006aj and SN2008D (Campana et al. 2006; Soderberg et al. 2008), which has been interpreted as the cooling SN envelope emission after being heated by the radiation-dominated shock⁶ (Waxman et al. 2007; Soderberg et al. 2008; Chevalier & Fransson 2008). In our calculation, we take into account this seed photon source in addition to the late-time supernova emission, which peaks after 10 days.

Recently, Ando & Mészáros (2008) discussed the broadband emission from SSC and SNIC for a trans-relativistic ejecta in an LL-GRB at a particular time—the ejecta deceleration time. Here, we study the time evolution of the high-energy gamma-ray emission resulted from such IC processes and consider both the trans-relativistic ejecta and the highly relativistic ejecta scenarios for LL-GRBs.

The paper is organized as follows. First, we describe the dynamics of shock evolution in the two models in Section 2. In Section 3, we present the formula for the calculation of the IC emissivity. For the SNIC emission, we take into account the anisotropic scattering effect. Then we present the results of the spectra and light curves of SSC and SNIC emissions for the two different models and explore the detectability of these components by Fermi Large Area Telescope (LAT) in Section 4. Finally, we give the conclusions and discussions.

We assume that the two models have the same parameters except for the initial Lorentz factor of the ejecta. For the latter, we adopt nominal values of Γ₀ = 10 in the conventional relativistic ejecta model and Γ₀ = 2 in the trans-relativistic ejecta model, respectively. Note that in the conventional relativistic ejecta model, even if Γ₀ ≫ 10 the dynamics of the blast wave is identical to the case of Γ₀ = 10 from the time tens of seconds after the burst, because the blast wave has entered the Blandford–McKee self-similar phase since then.

We consider a spherical GRB ejecta carrying a total energy of E = 10⁵⁰E₅₀ erg expanding into a surrounding wind medium with density profile n = Kr⁻², where $K\equiv \dot{M}/(4\pi m_pv_{\rm w})=3\times 10^{35}\ {\rm cm}^{-1}\dot{m}$ with $\dot{m}\equiv (\dot{M}/10^{-5}M_{\odot }\ {{\rm yr}^{-1}})/(v_{\rm w}/10^3\ {\rm {km\,s^{-1}}})$. As the circumburst medium is swept up by the blast wave, the total kinetic energy of the fireball is (Panaitescu et al. 1998)

$$\begin{equation} E_{\rm k}=(\gamma -1)(m_{\rm ej}+m)c^{2}+\gamma U^{\prime }, \end{equation} \tag{ 1 }$$

where γ is the bulk Lorentz factor of the shell, m_ej is the ejecta mass, m is the mass of the swept-up medium, and the comoving internal energy U' can be expressed by U' = (γ − 1)mc², which is suitable for both ultrarelativistic and Newtonian shocks (Huang et al. 1999). Hereafter, superscript prime represents that the quantities are measured in the comoving frame of the shell. As usual, we assume that the magnetic field and the electrons have a fraction _B and _e of the internal energy, respectively. Following Equation (1), the differential dynamic equation can be derived as (Huang et al. 2000)

$$\begin{equation} \frac{d\gamma }{dm}=-\frac{\gamma ^{2}-1}{m_{\rm ej}+2\gamma m}, \end{equation} \tag{ 2 }$$

which describes the overall evolution of the shell from relativistic phase to non-relativistic phase. The initial value of γ is Γ₀ = E/(m_ejc²). To obtain the time dependence of γ one makes use of

$$\begin{equation} \frac{dm}{dR}=4\pi R^{2}nm_{\rm p}, \end{equation} \tag{ 3 }$$

$$\begin{equation} \frac{dR}{dt}=\beta c\gamma (\gamma +\sqrt{\gamma ^{2}-1}), \end{equation} \tag{ 4 }$$

where t is the observer time, R and $\beta =\sqrt{1-\gamma ^{-2}}$ are the radius and velocity of the shell, respectively.

Solving Equations (2)–(4), three dynamic phases can be found: (1) Coasting phase, the shell does not decelerate significantly until it arrives at the deceleration radius, R_dec = E/(4πKΓ²₀m_pc²), where the mass of the swept-up medium m is comparable to m_ej/Γ₀ (Sari & Piran 1995). The corresponding deceleration time can be calculated from t_dec ≃ R_dec/(2Γ²₀c). For representative parameter values E₅₀ = 1 and $\dot{m}=1$, R_dec ≃ 4.4 × 10¹⁵ cm and t_dec ≃ 1.8 × 10⁴ s for a trans-relativistic ejecta with Γ₀ = 2, while R_dec ≃ 1.8 × 10¹⁴ cm and t_dec ≃ 29 s for a conventional highly relativistic ejecta with Γ₀ = 10; (2) Blandford & McKee (1976) self-similar phase (t > t_dec and γ ≳ 2), where γ ∝ t^−1/4 and R ∝ t^1/2; and (3) Non-relativistic phase (γ → 1), where β ∝ t^−1/3 and R ∝ t^2/3, i.e., the Sedov–von Neumann–Taylor solution applies (Zeldovich & Raizer 2002, p. 93; Waxman, 2004).

In the absence of radiation losses, the energy distribution of shock-accelerated electrons behind the shock is usually assumed to be a power law as dN_e/dγ_e ∝ γ^−p_e. As the electrons are cooled by synchrotron and IC radiation, the electron energy distribution becomes a broken power law, given by

1. for γ_e,c ⩽ γ_e,m,

$$\begin{equation} \frac{dN_{e}}{d\gamma _{e}}\propto \left\lbrace \begin{array}{@{}ll@{}} \gamma _{e}^{-2},&\gamma _{e,\rm c}\le \gamma _{e}\le \gamma _{e,\rm m},\\ \gamma _{e}^{-p-1},&\gamma _{e,\rm m}<\gamma _{e}\le \gamma _{e,\rm max},\end{array}\right. \end{equation} \tag{ 5 }$$

2. for γ_e,m < γ_e,c ⩽ γ_e,max,

$$\begin{equation} \frac{dN_{e}}{d\gamma _{e}}\propto \left\lbrace \begin{array}{@{}ll@{}} \gamma _{e}^{-p},&\gamma _{e,\rm m}\le \gamma _{e}\le \gamma _{e,\rm c},\\ \gamma _{e}^{-p-1},&\gamma _{e,\rm c}<\gamma _{e}\le \gamma _{e,\rm max},\end{array}\right. \end{equation} \tag{ 6 }$$

which are normalized by the total number of the electrons solved from the dynamic equations. The minimum, cooling, and maximum Lorentz factors of electrons are, respectively, given by

$$\begin{eqnarray} \gamma _{e,\rm m}&=&\epsilon _{e}\frac{p-2}{p-1}\frac{m_{\rm p}}{m_{e}}(\gamma -1)=92f_{p1}\epsilon _{e,-0.5}(\gamma -1), \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \gamma _{e,\rm c}&=&\frac{6\pi m_{e}c}{(1+Y)\sigma _{T}{B^{\prime }}^{2} (\gamma +\sqrt{\gamma ^{2}-1})t}\nonumber \\ &&\simeq \frac{1.7\times 10^3R_{15}^2}{\epsilon _{B,-3}\dot{m} t_{4} Y (\gamma +\sqrt{\gamma ^2-1})\gamma (\gamma -1)}, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} &&\hspace{-9.1pc}\gamma _{e,\rm max}=\sqrt{6\pi q_{e}\over \sigma _{T}B^{\prime }(1+Y)} \end{eqnarray} \tag{ 9 }$$

$$\begin{eqnarray} &&\hspace{2.6pc}\simeq 4.5\times 10^7R_{15}\gamma ^{-1/4}(\gamma -1)^{-1/4}Y^{-1/2}, \end{eqnarray} \tag{ 10 }$$

where f_p1 = 6(p − 2)/(p − 1), B' is the comoving magnetic field strength, and Y is the Compton parameter that is defined as the ratio of the IC luminosity (including SSC and SNIC) to the synchrotron luminosity.⁷

The accelerated electrons can be cooled by synchrotron radiation and IC scattering of seed photons (including synchrotron photons and blackbody photons emitted by the supernova). Since the IC emissivity on the basis of the Thomson cross section is inaccurate for high-energy γ-rays, we use the full Klein–Nishina cross section instead. Once the electron distribution and the flux of seed photons ($f^{\prime }_{\nu ^{\prime }_{{\rm s}}}$) (the distribution of which is isotropic) are known, the IC emissivity (at frequency ν') of electrons can be calculated by (Blumenthal & Gould 1970; Yu et al. 2007)

$$\begin{equation} {\varepsilon ^{\prime }}^{\rm IC}_{\rm iso}(\nu ^{\prime })=3\sigma _{\rm T}\int ^{\gamma _{e, \rm max}}_{\gamma _{e, \rm min}}d\gamma _{e} {dN_{e}\over d\gamma _e}\int ^{\infty }_{\nu ^{\prime }_{\rm s,\min }}d\nu ^{\prime }_{\rm s}\frac{\nu ^{\prime }f^{\prime }_{\nu ^{\prime }_{\rm s}}}{4\gamma _{e}{\nu ^{\prime }}_{\rm s}^{2}}g(x,y), \end{equation} \tag{ 11 }$$

where γ_e,min = max [γ_e,c, γ_e,m, hν'/(m_ec²)], ν'_s,min = ν'm_ec²/4[γ_e(γ_em_ec² − hν')], x = 4γ_ehν'_s/m_ec², y = hν'/[x(γ_em_ec² − hν')], and

$$\begin{equation} g(x,y)=2y\ln y+(1+2y)(1-y)+\frac{1}{2}{x^{2}y^{2}\over (1+xy)}(1-y). \end{equation} \tag{ 12 }$$

In our case, because the radius of the supernova photosphere is much smaller than that of the GRB shock, supernova seed photons can be regarded as a point photon source locating at the center in the comoving frame of the GRB shock. These soft photons from supernova impinge on the shock region basically along the radial direction in the rest frame of the shock, so the scatterings between these photons and the isotropically distributed electrons in the shock are anisotropic. For a photon beam penetrating into the shock region where the electrons are moving isotropically, the IC scattering emissivity of the radiation scattered at an angle θ_SC relative to the direction of the photon beam in the shock comoving frame is (Aharonian & Atoyan 1981, Brunetti 2000, Fan et al. 2008):

$$\begin{equation} \begin{array}{ll}\varepsilon ^{\prime \rm AIC}(\nu ^{\prime },\cos \theta _{\rm SC})=\displaystyle\frac{3\sigma _{\rm T}c}{16\pi } \int ^{\gamma _{e,\rm max}}_{\gamma _{e,\rm min}}d\gamma _{\rm e}\frac{dN_{\rm e}}{d\gamma _{e}}\\ \displaystyle \times \int ^{\infty }_{\nu ^{\prime }_{\rm s,min}}\frac{f^{\prime {\rm SN}}_{\nu ^{\prime }_{\rm s}}d\nu ^{\prime }_{\rm s}}{\gamma _{\rm e}^2\nu ^{\prime }_{\rm s}} \left[1+\frac{\xi ^{2}}{2(1-\xi)}-\frac{2\xi }{b_{\theta }(1-\xi)}+\frac{2\xi ^2}{b_{\theta }^2(1-\xi)^2}\right]\end{array}\!, \end{equation} \tag{ 13 }$$

where ξ ≡ hν'/(γ_em_ec²), b_θ = 2(1 − cos θ_SC)γ_ehν'_s/(m_ec²), and hν'_s ≪ hν' ≪ γ_em_ec²b_θ/(1 + b_θ). On integration over θ_SC for whole solid angle (i.e., in the case that the photon distribution is also isotropic), Equation (13) reduces to Equation (11), i.e., the usual isotropic IC scattering emissivity. In the observer frame, the angle θ between the injecting photons and the scattered photons relates with the angle θ_SC in the comoving frame by cos θ_SC = (cos θ − β)/(1 − βcos θ), where β is the velocity of the GRB shock.

The observed SSC and SNIC flux densities at a frequency ν are given, respectively, by (e.g., Huang et al. 2000; Yu et al. 2007)

$$\begin{equation} F_{\nu }^{\rm SSC}=\int ^{\pi }_{0}\frac{\varepsilon ^{\prime \rm IC}_{\rm iso}(\nu /D)}{4\pi D_{\rm L}^2}D^3\sin \theta d\theta, \end{equation} \tag{ 14 }$$

and

$$\begin{equation} F_{\nu }^{\rm SNIC}=\int ^{\pi }_{0}\frac{\varepsilon ^{\prime \rm AIC}(\nu /D, \cos \theta _{\rm SC})}{4\pi D_{\rm L}^2}D^3\sin \theta d\theta, \end{equation} \tag{ 15 }$$

where D_L is the luminosity distance and D ≡ [γ(1 − βcos θ)]⁻¹ is the Doppler factor. Note that cos θ_SC in Equation (15) can be transformed to cos θ through cos θ_SC = (cos θ − β)/(1 − βcos θ).

3.1. Seed Photons from the Supernova

For seed photons from the supernova, we consider the contributions from two components. One is the early thermal UV–optical emission from the cooling supernova envelope after being heated by the radiation-dominated shock (Waxman et al. 2007; Chevalier & Fransson 2008). Such UV–optical emission has been observed recently by Swift UVOT from SN2006aj (Campana et al. 2006) and SN2008D (Soderberg et al. 2008). Another component is the supernova optical emission at later time, powered by the radioactive elements synthesized in supernovae.

The characterization of the emission from the cooling supernova envelope is expressed as follows, which is correct at t ≳ 10² s (Waxman et al. 2007):

$$\begin{equation} R_{\rm ph}(t)= 3.2\times 10^{14}\frac{E_{\rm snej,51}^{0.4}}{(M_{\rm snej}/M_{\odot })^{0.3}}t_{\rm day}^{0.8}\,\,\rm {cm}, \end{equation} \tag{ 16 }$$

$$\begin{equation} T_{\rm ph}(t)=2.2\frac{E_{\rm snej,51}^{0.02}}{(M_{\rm snej}/M_{\odot })^{0.03}}R_{0,12}^{1/4}t_{\rm day}^{-0.5}\,\,\rm {eV}, \end{equation} \tag{ 17 }$$

$$\begin{equation} L_{\rm ph}(t)=4\pi R_{\rm ph}(t)^{2}\sigma T_{\rm ph}(t)^{4}, \end{equation} \tag{ 18 }$$

where R_ph(t) and T_ph(t) are the radius and temperature of the envelope, L_ph(t) is the luminosity from the photosphere of the cooling envelope, M_snej and E_snej are the supernova ejecta mass and energy, and R₀ is the initial stellar radius of the SN progenitor. We take the following values for supernovae like SN2006aj and SN2008D: E_snej,51 = E_snej/(10⁵¹ erg) = 2, M_snej = 2 M_☉, and R_0,12 = R₀/(10¹² cm) = 0.3 (Mazzali et al. 2006; Soderberg et al. 2008). For hypernovae such as SN1998bw associated with GRB980425, modeling of the SN optical emission gives a larger kinetic energy and ejecta mass such as E_snej,51 ≃ 22 and M_snej = 6 M_☉ for SN1998bw (Woosley et al. 1999).

The luminosity of the late component is rising before ∼10 days and then decaying exponentially. For SN2008D, it is found that the rising is roughly in proportion to t^1.6 (Soderberg et al. 2008), so we can describe the luminosity as

$$\begin{equation} L_{\rm rad}(t)=\left\lbrace \begin{array}{@{}ll@{}}3\times 10^{42}\displaystyle\left(\frac{t}{10\;{\rm {days}}}\right)^{1.6}\rm {erg \, s^{-1}}, & \!\!\!t<10 \ \rm {days}\\ 3\times 10^{42}\exp \displaystyle\left(1-\frac{t}{10\;{\rm {days}}}\right)\rm {erg\, s^{-1}}, & \!\!\!t\ge 10 \ \rm {days}\end{array}.\right. \end{equation} \tag{ 19 }$$

We assumed that the radiation temperature is approximately constant, T_rad ≃ 1 eV. The bolometric luminosity of the cooling supernova envelop is dominant over that of the late component before t = 7.8 × 10⁵ s.

Through the calculation of the shock dynamic evolution, the radii of the GRB shock are R = 1.9 × 10¹⁴ cm at t = 10³ s, and R = 1.5 × 10¹⁵ cm at t = 10⁴ s for the trans-relativistic ejecta model, and for the conventional ejecta model R = 1.0 × 10¹⁵ cm at t = 10³ s and R = 4.1 × 10¹⁵ cm at t = 10⁴ s. According to Equation (16), the shell radii for supernova are 9.6 × 10¹² cm at t = 10³ s and 6.1 × 10¹³ cm at t = 10⁴ s, and the one for hypernova are 1.8 × 10¹³ cm at t = 10³ s and 1.1 × 10¹⁴ cm at t = 10⁴ s. Consequently, the shell radius is less than 10% of the GRB ejecta height, so it is reasonable to approximate that supernova photons come from a point source at the center and impinge onto the electrons in the shock from behind.

3.2. Compton Parameter Y

Following Moderski et al. (2000) and Sari & Esin (2001), we define the Compton parameter Y as

$$\begin{equation} Y\equiv \frac{L_{\rm IC}}{L_{\rm SYN}}=\frac{{u^{\prime }}_{\gamma }}{{u^{\prime }}_{B}}=\frac{{u^{\prime }}_{\rm SYN}+f_{\rm a}{u^{\prime }}_{\rm SN}}{{u^{\prime }}_{B}}, \end{equation} \tag{ 20 }$$

where

$$\begin{eqnarray} {u^{\prime }}_{\rm SN}&=&\gamma ^{-2}\frac{L_{\rm SN}}{4\pi cR^2}\nonumber \\ &\simeq & 65\ {\rm erg\ cm^{-3}}t_4^{-0.4}R_{15}^{-2}\gamma ^{-2}, \end{eqnarray} \tag{ 21 }$$

$$\begin{eqnarray} {u^{\prime }}_{\rm SYN}&=&{\eta \epsilon _e{u^{\prime }}\over (1+Y)}\nonumber \\ &\simeq & 5.4\times 10^2\ {\rm erg\ cm^{-3}}\epsilon _{e,-0.5}\dot{m}R_{15}^{-2}\gamma (\gamma -1)\eta Y^{-1},\nonumber \\ \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} {u^{\prime }}_{B}&=&\epsilon _B{u^{\prime }}\nonumber \\ &\simeq & 1.8\ {\rm erg\ cm^{-3}}\epsilon _{B,-3}\dot{m}R_{15}^{-2}\gamma (\gamma -1), \end{eqnarray} \tag{ 23 }$$

are, respectively, the comoving energy densities of the blackbody supernova seed photons, synchrotron seed photons, and magnetic fields; and f_a is the factor accounting for the suppression of the photon energy density due to the anisotropic IC scattering effect (f_a = 1 corresponds to the isotropic scattering case). Here L_SN = L_ph + L_rad, which is dominated by the luminosity of the cooling envelop L_ph, u' is the comoving internal energy density, η = η_radη_KN is the radiation efficiency where η_rad is the fraction that the electron's energy radiated, and η_KN is the fraction of synchrotron photons below the KN limit frequency (Nakar 2007). For slow cooling, η_rad = (γ_e,c/γ_e,m)^2−p (Moderski et al. 2000), and

$$\begin{equation} \eta _{\rm KN}=\left\lbrace \begin{array}{@{}ll@{}} 0,&\nu ^{\prime }_{\rm KN}(\gamma _{e,\rm c})\le \nu ^{\prime }_{\rm m}\\ \left(\frac{\nu ^{\prime }_{\rm KN}(\gamma _{e,\rm c})}{\nu ^{\prime }_{\rm c}}\right)^{(3-p)/2},&\nu ^{\prime }_{\rm m}< \nu ^{\prime }_{\rm KN}(\gamma _{e,\rm c})<\nu ^{\prime }_{\rm c}\\ 1,&\nu ^{\prime }_{\rm c}\le \nu ^{\prime }_{\rm KN}(\gamma _{e,\rm c})\\ \end{array}.\right. \end{equation} \tag{ 24 }$$

For fast cooling, η_rad = 1 and

$$\begin{equation} \eta _{\rm KN}=\left\lbrace \begin{array}{@{}ll@{}} 0,&\nu ^{\prime }_{\rm KN}(\gamma _{e,\rm m})\le \nu ^{\prime }_{\rm c}\\ \left(\frac{\nu ^{\prime }_{\rm KN}(\gamma _{e,\rm m})}{\nu ^{\prime }_{\rm m}}\right)^{1/2},&\nu ^{\prime }_{\rm c}< \nu ^{\prime }_{\rm KN}(\gamma _{e,\rm m})<\nu ^{\prime }_{\rm m}\\ 1,&\nu ^{\prime }_{\rm m}\le \nu ^{\prime }_{\rm KN}(\gamma _{e,\rm m})\\ \end{array}.\right. \end{equation} \tag{ 25 }$$

Solving Equation (20), we get

$$\begin{equation} Y={1\over 2}\left[\sqrt{4\frac{\eta \epsilon _{e}}{\epsilon _{B}}+\left(1+\frac{f_{\rm a}{u^{\prime }}_{\rm SN}}{\epsilon _{B}{u^{\prime }}}\right)^{2}} +\left(\frac{f_{\rm a}{u^{\prime }}_{\rm SN}}{\epsilon _{B}{u^{\prime }}}-1\right)\right]. \end{equation} \tag{ 26 }$$

Roughly, the above expression can be simplified in three limiting cases as follows:

$$\begin{equation} Y=\left\lbrace \begin{array}{@{}ll@{}} f_{\rm a}u^{\prime }_{\rm SN}/(\epsilon _Bu^{\prime }),&f_{\rm a}{u^{\prime }}_{\rm SN}\gg u^{\prime }_{\rm SYN}\\ \sqrt{{\eta \epsilon _{e}/\epsilon _{B}}},&u^{\prime }_{B}\ll f_{\rm a}{u^{\prime }}_{\rm SN}\ll u^{\prime }_{\rm SYN} \\ (\sqrt{4{\eta \epsilon _{e}/\epsilon _{B}}+1}-1)/2,&f_{\rm a}{u^{\prime }}_{\rm SN}\ll {\rm min}[{u^{\prime }}_{\rm SYN},u^{\prime }_{B}]\\ \end{array}.\right. \end{equation} \tag{ 27 }$$

In the latter two cases, Y can be treated as a constant when the electrons are in the fast-cooling regime and ν'_m ⩽ ν'_KN(γ_e,m) with η = 1.

3.3. Pair Production Opacity for High-energy Photons

High-energy gamma-rays can be attenuated due to interaction with low-energy photons through the pair production effect. We consider the pair production opacity in the shock frame due to the absorption by low-energy photons, which include thermal photons from the supernova, synchrotron photons, SSC photons, and SNIC photons. A high-energy photon of energy E'_γ,1 in the shock frame will annihilate with a low-energy photon of E'_γ,2, provided that E'_γ,1E'_γ,2(1 − cos θ₁₂) ⩾ 2(m_ec²)², where θ₁₂ is the collision angle of the two annihilation photons. The pair creation cross section is given by

$$\begin{eqnarray} \sigma \big(E^{\prime }_{\gamma,1},E^{\prime }_{\gamma,2}\big)&=&\frac{1}{2}\pi r_{0}^{2}(1-\tilde{\beta }^{2})\nonumber\\ &&\times \left[(3-\tilde{\beta }^{4}){\rm ln}\frac{1+\tilde{\beta }}{1-\tilde{\beta }}-2\tilde{\beta }(2-\tilde{\beta }^2)\right],\quad\;\;\; \end{eqnarray} \tag{ 28 }$$

where $\tilde{\beta }\equiv v/c=\sqrt{1-2(m_{e}c^{2})^{2}/[E^{\prime }_{\gamma,1}E^{\prime }_{\gamma,2}(1-\cos \theta _{12})}]$is the velocity of electrons in the center-of-mass frame (Heitler 1954; Stecker et al. 1992) and r₀ = e²/(m_ec²) is the classic electron radius. By using the photon distribution in the shock frame, we obtain the optical depth for high-energy photons of energy,

$$\begin{equation} \tau (E^{\prime }_{\gamma,1})=\int _{E^{\prime }_{\rm{thr}}}^{\infty }\sigma (E^{\prime }_{\gamma,1},E^{\prime }_{\gamma,2})n_{\gamma }(E^{\prime }_{\gamma,2})\frac{R}{\eta _{\rm R}}dE^{\prime }_{\gamma,2}, \end{equation} \tag{ 29 }$$

where η_R is the shock compressed ratio, which is η_R = 4γ + 3, and the threshold energy of low-energy photons is E'_thr = 2(m_ec²)²/E'_γ,1(1 − cos θ₁₂). In the calculation, we estimate the cutoff energy conservatively by assuming that colliding photons are moving isotropically in the shock frame. Such a treatment may overestimate the pair production opacity because in reality the supernova seed photons move anisotropically (i.e., moving outward in radial direction as seen by high-energy photons emitted from the shock at much larger radii).

The result of the cutoff energy in the observer frame E_γ,cut = γE'_γ(τ = 1) is given in Figure 1. Since the luminosity and peak energy of the supernova envelope emission decreases with time in general and the shock radius increases with time, the cutoff energy of the high energy spectrum increases with time, which is clearly seen in Figure 1. From this figure, we can see that the cutoff energy is above 1 GeV after the starting time of our calculation (10^2.5 s), so we can calculate the light curves at energy ∼1 GeV without considering the opacity. Since the cutoff energy is large enough, it does not affect the detectability of Fermi LAT, whose sensitive energy band is 20 MeV ∼ 300 GeV.

Zoom In Zoom Out Reset image size

4.1. Spectra and Light Curves: Numerical Results

We first compare the spectra of different IC components for the two models. The spectra at t = 10³ s are shown in Figure 2, including the synchrotron emission, the thermal emission from the supernova or hypernova, the synchrotron self-Compton emission and the SNIC emission, for parameters _e = 0.1, _B = 0.001, p = 2.2, E = 10⁵⁰ erg, and burst distance D_L = 100 Mpc. From the spectra, we can see that for the supernova case, the SSC emission dominates over the SNIC emission at energies from 1 MeV to 100 GeV for conventional relativistic ejecta model; and for trans-relativistic ejecta model, the SSC emission is higher than SNIC emission and the two components are both important at high energies below the cutoff energy. For the hypernova case, the SSC emission is dominant in the conventional relativistic ejecta model, while the SNIC emission is dominant in the trans-relativistic ejecta model at the high-energy band. This is because in the case of hypernovae and Γ₀ = 2, the energy density of hypernova photons u'_HN (multiplied by the suppressed factor f_a due to the anisotropic scattering) is significantly higher than the synchrotron radiation density, u'_SYN.

Zoom In Zoom Out Reset image size

Figure 3 shows light curves of SSC and SNIC afterglow emissions at hν = 1 GeV in the two models for E = 10⁵⁰ ergs, p = 2.2, and four different sets of parameters of _e and _B for the supernova case. In the conventional relativistic model with Γ₀ = 10, a sharp decay phase of a GeV afterglow is produced during the early hours, which is mildly dominated by the SSC emission, and a slightly flatter decay phase dominated by SNIC emission takes over at late times t > 10⁶ s. On the other hand, for the trans-relativistic ejecta model with Γ₀ = 2, a plateau of SSC emission, due to the presence of a coasting phase in the ejecta dynamic, dominates in the early time, which transits to a faster decay at later time and SNIC emission become dominant after the time 10⁵ ∼ 10⁶ s.

Zoom In Zoom Out Reset image size

Figure 4 show light curves for the hypernovae case. Light curves in the conventional relativistic ejecta model are similar to those for the supernova case except that SNIC emission become dominant at earlier time; for the trans-relativistic ejecta model, SNIC emission is always dominated with parameters, _e = 0.1 and _B = 0.001; a plateau is seen in the early time, which transits to a faster decay at a later time.

Zoom In Zoom Out Reset image size

The two models also predict different flux levels at high energies. In early hours, the total flux from the conventional relativistic ejecta model is more than 1 order of magnitude higher than that in the trans-relativistic ejecta model. This can be explained by the different amount of energy in shocked electrons. For the conventional relativistic ejecta model, the ejecta has been decelerated at this time and a great part of its energy has been converted into shocked electrons, while in the trans-relativistic ejecta model, only a small fraction of the ejecta kinetic energy has been converted to shocked electrons at this early time. A lower amount of energy in shocked electrons results in a lower flux level in the trans-relativistic ejecta model.

In addition, a comparison among the four panels in Figures 3 and 4 indicates that the flux decreases as _e decreases, which is obvious since the energy of radiating electrons E_e ∝ _e. As _B decreases, the SSC flux changes little and the SNIC flux increases for the case that SSC emission is dominant at early time. This can be understood from the following analysis. Since the SSC emission dominates and f_au'_SN ≫ u'_B at the early times, $Y\simeq \sqrt{\eta \epsilon _e/\epsilon _B}\propto \epsilon _B^{-\frac{1}{2}}\epsilon _e^{\frac{1}{2}}$. The SSC and SNIC fluxes at ν>ν_min > ν_c scale as

$$\begin{equation} \nu F_{\nu }^{\rm SSC}=\nu F_{\rm max}^{\rm SSC}\left(\frac{\nu _{\rm min}^{\rm SSC}}{\nu _{c}^{\rm SSC}}\right)^{-\frac{1}{2}}\left(\frac{\nu }{\nu _{\rm min}^{\rm SSC}}\right)^{-\frac{p}{2}}\propto \epsilon _B^{\frac{p}{4}-\frac{1}{2}}\epsilon _e^{2p-3} \end{equation} \tag{ 30 }$$

and

$$\begin{equation} \nu F_{\nu }^{\rm SNIC}=\nu F_{\rm max}^{\rm SNIC}\left(\frac{\nu _{\rm min}^{\rm SNIC}}{\nu _{c}^{\rm SNIC}}\right)^{-\frac{1}{2}}\left(\frac{\nu }{\nu _{\rm min}^{\rm SNIC}}\right)^{-\frac{p}{2}}\propto \epsilon _B^{-\frac{1}{2}}\epsilon _e^{p-\frac{3}{2}}. \end{equation} \tag{ 31 }$$

4.2. The Effect of the Anisotropic Scattering on the SNIC Emission

The incoming supernova photons are anisotropic as seen by the isotropically distributed electrons in the GRB shock, so the IC scatterings are anisotropic. In order to see how the anisotropic inverse Compton (AIC) scattering affects the SNIC flux, we compare the light curves of the SNIC emission obtained by using the isotropic scattering formula, Equation (11), and using the AIC scattering formula, Equation (13), in Figure 5. The thinner lines show the SNIC light curves obtained using the usual isotropic scattering formula, while the thicker ones correspond to the calculations with the AIC scattering effect taken into account. One can see that the flux of the SNIC emission with the AIC effect correction is reduced by a factor of about ∼0.4 compared to the isotropic scattering case. This is consistent with the calculation result obtained by Fan & Piran (2006), who studied the anisotropic IC scattering between inner optical/X-ray flare photons and electrons in the outer GRB forward shock.

Zoom In Zoom Out Reset image size

Figure 5 shows that the AIC effect suppresses the SNIC flux only slightly. The anisotropic photon distribution results in more head-on scatterings, i.e., the photon beam scatter preferentially with those electrons that move in the direction antiparallel to the photon beam, so one can expect that the scattered IC emission power has a maximum at θ_SC = π and goes to zero for small scattering angles (e.g., Brunetti 2000). The photons scattered into the angles 0 ≲ θ_SC ≲ π/2 relative to the shock moving direction in the shock comoving frame will fall into the cone of angle 1/Γ in the observer frame, according to the transformation formula cos θ_SC = (cos θ − β)/(1 − βcos θ). Therefore, the AIC scatterings decrease the IC emission in the 1/Γ cone along the direction of the photon beam, but meanwhile they enhance the emission at larger angles (about half of the emission falling into angles between 1/Γ and 2/Γ, see Wang & Mészáros 2006). For a spherical outflow, as we consider here, the IC emission after integration over angles should have the same flux in every direction in the observer frame, with a flux level only slightly reduced comparable to the isotropic scattering case.

4.3. Analytical Light Curves

As a comparison, we derive here approximate analytical expressions for afterglow light curves, which provide an explanation for the physical origin of the behavior. Since the anisotropic SNIC emission are depressed by a factor of 0.4, which is almost constant, relative to the isotropic seed photons case, we can consider the isotropic seed photons case for the approximate analytic treatment of the afterglow light curves. The blackbody photons from the supernova can be approximated as mono-energetic photons with hν_SN = 2.7kT_SN. Thus, similar to the description for synchrotron emission of a single electron (Sari et al. 1998), the radiation power and characteristic frequencies of SNIC from a single electron scattering supernova photons in the observer frame can be described by

$$\begin{equation} P(\gamma _{e})=\frac{4}{3}\sigma _Tc\gamma ^2\gamma _e^2\frac{L_{\rm {SN}}}{\gamma ^2\pi R^2c} \end{equation} \tag{ 32 }$$

and

$$\begin{equation} \nu (\gamma _{e})=2\gamma \gamma _e^2\nu _{\rm {SN}}/\gamma =2\gamma _e^2\nu _{\rm {SN}}, \end{equation} \tag{ 33 }$$

respectively. Similar to the analysis in Sari and Esin (2001), the maximum flux of the SNIC spectrum is

$$\begin{equation} F_{\rm max}^{\rm SNIC}=\frac{N_e}{4\pi D_{\rm L}^2}\frac{P(\gamma _{e})}{\nu (\gamma _{e})}. \end{equation} \tag{ 34 }$$

Characteristic SNIC frequencies are

$$\begin{equation} \nu _{\rm min}^{\rm SNIC}=2\gamma _{e,\rm m}^2\nu _{\rm SN} \end{equation} \tag{ 35 }$$

and

$$\begin{equation} \nu _{c}^{\rm SNIC}=2\gamma _{e,\rm c}^2\nu _{\rm SN}, \end{equation} \tag{ 36 }$$

respectively.

By adopting the broken power-law approximation for the IC spectral component (Sari & Esin 2001) and the dynamics of the shock discussed in Section 2, one can derive the analytic light curves in an approximate way.

4.3.1. Light Curves in the Conventional Relativistic Ejecta Model

In the conventional relativistic ejecta model, at time t_dec < t < t_tran, where t_tran is defined as the transition time when γ_e,m = γ_e,c, we have η = 1 and u'_B ≪ f_au'_SN ≪ u'_SYN, so $Y\simeq \sqrt{\eta \epsilon _e/\epsilon _B}\propto t^{0}$. The shock dynamic follows the Blandford & McKee (1976) self-similar solution in the wind medium, i.e., $R\propto t^{\frac{1}{2}}$ and $\gamma \propto t^{-\frac{1}{4}}$. Then we can obtain the evolution of the break frequencies of the SSC and SNIC spectral components and their peak flux in the following way:

$$\begin{equation} \nu _{\rm min}^{\rm SSC}=2\gamma _{e,\rm m}^2\nu _{\rm min}\propto t^{-2}, \quad \nu _{\rm c}^{\rm SSC}=2\gamma _{e,\rm c}^2\nu _{\rm c}\propto t^{2}, \end{equation} \tag{ 37 }$$

$$\begin{equation} F_{\nu,\rm max}^{\rm SSC}=\frac{\sigma _TN_e}{4\pi R^2}F_{\rm max}^{\rm SYN}\propto t^{-1}; \end{equation} \tag{ 38 }$$

and

$$\begin{equation} \nu _{\rm min}^{\rm SNIC}\propto t^{-1}, \quad \nu _{\rm c}^{\rm SNIC}\propto t, \quad F_{\nu,\rm max}^{\rm SNIC}\propto t^{-0.4}, \end{equation} \tag{ 39 }$$

where ν_SN ∝ T_SN ∝ t^−1/2 has been used for the cooling envelope emission (see Equation (13)). The SSC and SNIC fluxes at an observed frequency ν higher than characteristic frequencies vary as νF^SSC_ν ∝ t^{−p + 1} and $\nu F_{\nu }^{\rm SNIC}\propto t^{-\frac{p}{2}+0.6}$, respectively, for t < t_tran.

At t > t_SNIC, where t_SNIC is the time when f_au'_SN = u'_SYN, we take Y ≃ f_au'_SN/(_Bu') because the SNIC emission becomes dominated. At such time, the shock is likely to enter the non-relativistic phase, so we take the Sedov–von Neumann–Taylor solutions $R\propto t^{\frac{2}{3}}$ and $\beta \propto t^{-\frac{1}{3}}$, which induce that $Y\propto t^{\frac{2}{3}-0.4}$. So the minimum and cooling Lorentz factors of electrons vary as $\gamma _{e,\rm m}\propto \beta ^{2}\propto t^{-\frac{2}{3}}$ and $\gamma _{e,\rm c}\propto \beta ^{-2}R^2t^{-1}Y^{-1}\propto t^{0.4+\frac{1}{3}}$. Thus, the break frequencies of the SSC and SNIC spectral components and their peak fluxes evolve with time in the following way:

$$\begin{equation} \nu _{\rm min}^{\rm SSC}\propto t^{-\frac{11}{3}}, \quad \nu _{\rm c}^{\rm SSC}\propto t^{\frac{1}{3}+1.6}, \quad F_{\nu,\rm max}^{\rm SSC}\propto t^{-1}; \end{equation} \tag{ 40 }$$

and

$$\begin{equation} \nu _{\rm min}^{\rm SNIC}\propto t^{-\frac{11}{6}}, \;\nu _{\rm c}^{\rm SNIC}\propto t^{\frac{1}{6}+0.8}, \;F_{\nu,\rm max}^{\rm SNIC}\propto t^{-\frac{2}{3}+0.1}. \end{equation} \tag{ 41 }$$

Then the SSC and SNIC fluxes at high-energy ν vary as $\nu F_{\nu }^{\rm SSC}\propto t^{1.8-\frac{11}{6}p}$ and $\nu F_{\nu }^{\rm SNIC}\propto t^{\frac{5}{6}-\frac{11}{12}p}$ at t > t_SNIC.

To summarize, the temporal evolution of the SSC and SNIC afterglow emissions at high energies are

$$\begin{equation} \nu F_{\nu }^{\rm SSC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{-1.2},&t\lesssim t_{\rm tran}\\ t^{-2.2},&t>t_{\rm SNIC}\end{array}\right. \end{equation} \tag{ 42 }$$

and

$$\begin{equation} \nu F_{\nu }^{\rm SNIC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{-0.5},&t\lesssim t_{\rm tran}\\ t^{-1.2},&t>t_{\rm SNIC}\end{array}\right. \end{equation} \tag{ 43 }$$

in the two asymptotic phases for p = 2.2.

4.3.2. Light Curves in the Trans-relativistic Ejecta Model

In the trans-relativistic ejecta model, one would expect a much flatter light curve of high-energy gamma-ray emission at early times, which could be dominated by both the SSC emission and the SNIC emission, depending on the properties of the underlying supernova and the shock parameter _e and _B. For the supernova case, the SSC emission is dominant before the deceleration time and the transition time for most parameters, while for the hypernova case, the SSC emission is dominant at earlier time with _e = 0.3 and _B = 0.01 and the SNIC emission is always dominant for _e = 0.1 and _B = 0.001.

For cases where SSC emission dominated in trans-relativistic ejecta model, we have η = 1 in fast cooling regime and u'_B ≪ f_au'_SN ≪ u'_SYN, so $Y\simeq \sqrt{\eta \epsilon _e/\epsilon _B}\propto t^{0}$. Since t < t_dec, we adopt the approximation R ∝ t, γ ∝ t⁰. So the SSC and SNIC fluxes vary as $\nu F_{\nu }^{\rm SSC}\propto t^{1-\frac{p}{2}}$ and $\nu F_{\nu }^{\rm SNIC}\propto t^{0.1-\frac{p}{4}}$, respectively, at t < min(t_dec, t_tran). At later time, when t > t_dec and γ → 1, the SNIC emission become dominant, and the evolution of light curves is the same as that in conventional relativistic ejecta model, i.e., the SSC and SNIC fluxes vary as $\nu F_{\nu }^{\rm SSC}\propto t^{1.8-\frac{11}{6}p}$ and $\nu F_{\nu }^{\rm SNIC}\propto t^{\frac{5}{6}-\frac{11}{12}p}$ at t > t_dec. Therefore, for the supernova case and the hypernova case with early dominated SSC emission, the temporal evolution of the SSC and SNIC emissions at high frequency ν for p = 2.2 are given by

$$\begin{equation} \nu F_{\nu }^{\rm SSC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{-0.1},&t\lesssim {\rm min}(t_{\rm dec}, t_{\rm tran})\\ t^{-2.2},&t\gg t_{\rm dec}\end{array}\right. \end{equation} \tag{ 44 }$$

and

$$\begin{equation} \nu F_{\nu }^{\rm SNIC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{-0.45},&t\lesssim {\rm min}(t_{\rm dec}, t_{\rm tran})\\ t^{-1.2},&t\gg t_{\rm dec}\end{array}.\right. \end{equation} \tag{ 45 }$$

.

For cases where SNIC emission dominated in trans-relativistic ejecta model, we have f_au'_SN ≫ u'_SYN, so Y ≃ f_au'_SN/(_Bu'). At the time t < t_dec, we adopt the approximation R ∝ t, γ ∝ t⁰ to yield Y ∝ t^−0.4. So the SSC and SNIC fluxes vary as $\nu F_{\nu }^{\rm SSC}\propto t^{1.8-\frac{p}{2}}$ and $\nu F_{\nu }^{\rm SNIC}\propto t^{0.5-\frac{p}{4}}$, respectively, at t < min(t_dec, t_tran). At later time, when t > t_dec and γ → 1, the evolution of light curves is the same as that in conventional relativistic ejecta model, i.e., the SSC and SNIC fluxes vary as $\nu F_{\nu }^{\rm SSC}\propto t^{1.8-\frac{11}{6}p}$ and $\nu F_{\nu }^{\rm SNIC}\propto t^{\frac{5}{6}-\frac{11}{12}p}$ at t > t_dec. Therefore, for the supernova case and the hypernova case with early dominated SSC emission, the temporal evolution of the SSC and SNIC emissions at high frequency ν for p = 2.2 are given by

$$\begin{equation} \nu F_{\nu }^{\rm SSC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{0.7},&t\lesssim {\rm min}(t_{\rm dec}, t_{\rm tran})\\ t^{-2.2},&t\gg t_{\rm dec}\end{array}\right. \end{equation} \tag{ 46 }$$

and

$$\begin{equation} \nu F_{\nu }^{\rm SNIC}\propto \left\lbrace \begin{array}{@{}ll@{}} t^{-0.05},&t\lesssim {\rm min}(t_{\rm dec}, t_{\rm tran})\\ t^{-1.2},&t\gg t_{\rm dec}\end{array}.\right. \end{equation} \tag{ 47 }$$

.

4.4. Detectability by the Fermi LAT

We explore here whether Fermi LAT can detect the high-energy gamma-ray emission from LL-GRBs in the two models considered above. Following Zhang & Mészáros (2001), Gou & Mészáros (2007), and Yu et al. (2007), the fluence threshold for long-duration observations is F_thr = [ϕ₀(t/t_eff)^1/2]E_pht_eff which is in proportion to t^1/2 due to the limitation by the background, where we take the average energy of the detected photons as E_ph = 400 MeV and the effective time as t_eff = 0.5 yr. ϕ₀ is the integral sensitivity above 100 MeV for LAT for a steady source after a year sky survey, which is ϕ₀ ∼ 3 × 10⁻⁹ phs cm⁻² s⁻¹ (Atwood et al. 2009) and is improved by a factor of 3 by keeping the GRB position at the center of the LAT field of view as long as possible (Gou & Mészáros 2007). For short-time observation, the fluence threshold is calculated by F_thr = 5E_ph/A_eff under the assumption that at least five photons are collected. Taking the effective area A_eff = 6000 cm², we can obtain the fluence threshold of Fermi LAT,

$$\begin{equation} F_{\rm thr}=\left\lbrace \begin{array}{@{}ll@{}} 5.3\times 10^{-7}\, \rm {erg\, cm^{-2}},&t\le 4.4\times 10^{4}\,\,\rm {s},\\ 2.5\times 10^{-9}t^{1/2}\, \rm {erg \,cm^{-2}},&t>4.4\times 10^{4}\,\,\rm {s}.\end{array}\right. \end{equation} \tag{ 48 }$$

With this fluence threshold, the detectability of high-energy emission (with supernova seed photons luminosity given above and a total energy E = 10⁵⁰ erg) by the Fermi LAT is shown in Figure 6. The time-integrated fluence shown in the plot is defined as an integration of the flux density (F_ν) over the Fermi LAT energy band [20MeV, 300GeV] and the time interval [0.5t, t] as used in Gou & Mészáros (2007), which is $\int _{0.5t}^{t}\int _{\nu _{1}}^{\nu _{2}}F_{\nu }d\nu dt$.

Zoom In Zoom Out Reset image size

For _e = 0.3 and other representative parameter values, high-energy gamma-ray emission in the conventional relativistic ejecta model stays detectable up to ∼10⁶ s, while the high-energy gamma-ray emission in the trans-relativistic ejecta model can only be detected in a short period around 10^4.5 s. For a lower value such as _e = 0.1, the high-energy gamma-ray emission can still be detected in the conventional relativistic ejecta model, while it becomes undetectable for the trans-relativistic ejecta model.

In order to compare with earlier results in Ando & Mészáros (2008), we increase the total energy to E = 2 × 10⁵⁰ erg, which yields a kinetic energy E_k = (Γ₀ − 1)/Γ₀E = 10⁵⁰ erg in the trans-relativistic ejecta model, the same as that used in Ando & Mészáros (2008). This will increase the fluence by a factor of 2. We also increase the SN luminosity from that of a normal SN (shown in Section 3.1) to SN1998bw-like hypernovae. In Figure 7, we show the detectability of high-energy emission by Fermi LAT in this case. By comparing the flux of light curves between the supernova case and the hypernova case, which is shown in Figures 3 and 4, we can see that increasing the SN luminosity can hardly enhance the total IC flux for the two models. Our IC flux is still lower than that obtained by Ando & Mészáros (2008). The main difference between our calculation and that of Ando & Mészáros (2008) is the different minimum Lorentz factors used in the calculations. Ando & Mészáros (2008) may overestimate the minimum Lorentz factor of electrons by taking γ_e,m = _e(m_p/m_e)γ, which is a factor of (p − 1)/(p − 2) larger than ours. From the formula describing the high energy flux νF^SNIC_ν ∝ ν^{(p − 1)/2}_min,SNIC ∝ γ^{p − 1}_e,m, one expects that the flux is increased by a factor of $\big(\frac{p-1}{p-2}\big)^{p-1}$, which is about 8 for p = 2.2.

Zoom In Zoom Out Reset image size

The external stellar wind provides a source of Thomson opacity to scatter the supernova emission, thus a quasi-isotropic, back-scattered SN radiation field is present. Let us study whether this component is important. For a GRB shock locating at radius R and moving with a Lorentz factor γ, the Thomson scattering optical depth of the wind is $\tau _{\rm w}=\sigma _TRn=\sigma _TKR^{-1}=2.0\times 10^{-4}\dot{m}R_{15}^{-1}$. The scattered SN energy density by the wind in the comoving frame of the blast wave is $u^{\prime \rm w}_{\rm SN}=(L_{\rm SN}/{4\pi c R^2}) \tau _{\rm w} \gamma ^2$ due to the relativistic boosting effect. On the other hand, the energy density of the supernova photons impinging the shock from behind is u'_SN = γ⁻²(L_SN/4πcR²) (see Equation (21)). The ratio between these two energy densities is $u^{\prime \rm w}_{\rm SN}/u^{\prime }_{\rm SN}=2.0\times 10^{-4}\dot{m}R_{15}^{-1} \gamma ^4$. For the trans-relativistic ejecta model with Γ₀ = 2, the radius of the GRB ejecta is R = 1.9 × 10¹⁴ cm at t = 10³s, so u'^w_SN ≪ u'_SN for typical wind parameters. For the conventional relativistic ejecta model, the Lorentz factor and radius of the GRB ejecta are, respectively, γ ∼ 4 and R = 1.0 × 10¹⁵ cm at t = 10³s, so we also have u'^w_SN ≪ u'_SN. This means that the wind-scattered supernova radiation field is subdominant compared to the direct impinging supernova photon field, and hence we neglect its contribution to the high-energy gamma-ray emission.

Our estimate of the pair production opacity for high-energy photons in Section 3.3 is based on the common assumption that the colliding photons are isotropic in the rest frame. However, as we have shown above, the low-energy photons from the supernova essentially move radially outward before colliding with high-energy photons (e.g., Wang et al. 2006). Therefore, the collision process between the high-energy photons and the soft supernova photons is anisotropic, which would decrease the pair production opacity. This will be the subject of a more detailed future calculation and would be useful to explore whether TeV photons can escape from the source, which is important for checking the detectability by ground-based Cherenkov detectors such as Magic, VERITAS, Milago, H.E.S.S., ARGO, etc. Additionally, besides the high-energy gamma-ray emission discussed in this work, the high-energy neutrino emission arising from pγ interactions between shock-accelerated protons and photons from the supernova may also provide a constraint on the model for LL-GRBs (e.g., Yu et al. 2008).

Trans-relativistic ejecta may also exist in the usual high luminosity long GRBs, besides in low luminosity ones, since accelerating shocks are expected to accompany the supernova. Berger et al. (2003) and Sheth et al. (2003) found that the radio and optical afterglow indicates a low-velocity component more than 1.5 days after the explosion in GRB030329/SN2003dh. However, since the highly relativistic ejecta are much more energetic than the trans-relativistic component, high-energy gamma-ray emission from the latter component could easily remain hidden.

In summary, we have calculated the spectra and light curves of the high-energy gamma-ray afterglow emission from LL-GRBs for the two main models in the literature, i.e., the trans-relativistic ejecta model (Γ₀ ≃ 2) and the conventional highly relativistic ejecta model (Γ₀ ≳ 10), considering both the SSC and the external IC due to photons from the underlying supernova/hypernova. Our analysis takes into account a full Klein–Nishina cross section for IC scatterings, the anisotropic scattering of supernova photons and the opacity for high-energy photons due to annihilation with low-energy photons (mainly from the supernova/hypernova).

We find that, for the supernova case, the conventional relativistic outflow model predicts a relatively high gamma-ray flux from SSC at early times (<10⁵–10⁶ s for typical parameters) with a rapidly decreasing flux; while in the trans-relativistic outflow model, a much flatter light curve of high-energy gamma-ray emission dominated by the SSC emission is expected at early times. For the hypernova case, the SSC emission also dominates and decays sharply at early time in the conventional relativistic ejecta model, while for the trans-relativistic ejecta model, both the SSC emission and the SNIC emission could be dominant at early time, depending on the shock parameters _e and _B. The main difference between these two models arises from their different initial Lorentz factors, which induces a different dynamical evolution of the shock at early times. As a result, different observational features arise, such as different light curve shapes, different flux levels, and different dominant components (as detailed in Section 4.1). As shown in Section 4.4, high-energy gamma-ray emission can be detected in both models as long as _e is large enough, although detection from the conventional relativistic ejecta is much easier. Thus, with future high-energy gamma-ray observations by Fermi LAT, one can expect to be able to distinguish between the two models based on the above differences in observational features.

We thank Z. G. Dai, Y. F. Huang, and D. M. Wei for useful discussions. This work is supported by the National Natural Science Foundation of China under grants 10973008 and 10403002, the National Basic Research Program of China (973 program) under grant No. 2009CB824800, the Foundation for the Authors of National Excellent Doctoral Dissertations of China, the Qing Lan Project, and NASA grant NNX 08AL40G.