ENERGY SOURCES AND LIGHT CURVES OF MACRONOVAE

Nagakura et al. (2014) found that the profile of ejecta obtained from simulations by Hotokezaka et al. (2013a) can be well fitted by a power-law function $\rho \propto {{v}^{-\beta }}$. The power-law index of the snapshot density β is more or less independent of the dynamics of mergers, which is in the range of $\beta \sim 3$–4 for ${{v}_{{\rm min} }}\leqslant v\leqslant {{v}_{{\rm max} }}$, where ${{v}_{{\rm max} }}$ and ${{v}_{{\rm min} }}$ are the velocities of the outer and inner edges of the ejecta, respectively. We choose the middle of this range $\beta =3.5$ in this study. We also fix the maximum velocity ${{v}_{{\rm max} }}=0.4c$ from simulation results (Hotokezaka et al. 2013a). The maximum velocity ${{v}_{{\rm max} }}$ is comparable with the escape velocity of the system. The minimum velocity ${{v}_{{\rm min} }}$ is mainly determined by complicated dynamics at the initial stage of the merger $t\ll {{10}^{2}}\;{\rm s}$. For the mass density profile at the front of the ejecta, we assume a discrete boundary and the mass density $\rho =0$ at the region $r\gt {{v}_{{\rm max} }}t$. Although this profile may be far from the actual one,⁴ our main aim is to compare two models for energy sources, so that our conclusions are not affected. In Section 4.3, we discuss the dependence on the mass density profile at the outer region of the ejecta for the observed light curve. Here, we only consider the evolution after the initial stage of the merger ($t\gg {{t}_{{\rm inj}}}$) and treat ${{v}_{{\rm min} }}$ as a model parameter. Because of homologous expansion, the density decreases as $\rho \propto {{t}^{-3}}$. Then, the density profile is described by

$$\begin{eqnarray}&&\rho \left( t,v \right)={{\rho }_{0}}{{\left( \frac{t}{{{t}_{0}}} \right)}^{-3}}{{\left( \frac{v}{{{v}_{{\rm min} }}} \right)}^{-\beta }}.\end{eqnarray} \tag{ 4 }$$

where ${{\rho }_{0}}$ and t₀ are normalization factors. The factor ${{\rho }_{0}}t_{0}^{3}$ is related to the total mass of the ejecta ${{M}_{{\rm ej}}}$ as follows,

$$\begin{eqnarray}\begin{array}{rcl} {{M}_{{\rm ej}}} & = & 4\pi \int _{{{v}_{{\rm min} }}{{t}_{0}}}^{{{v}_{{\rm max} }}{{t}_{0}}}\rho \left( {{t}_{0}},v \right){{r}^{2}}dr \\ {} & = & \frac{4\pi }{\beta -3}{{\rho }_{0}}{{\left( {{v}_{{\rm min} }}{{t}_{0}} \right)}^{3}}\left[ 1-{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-\beta }} \right], \\ \end{array}\end{eqnarray} \tag{ 5 }$$

where we use $dr\left( t={{t}_{0}} \right)={{t}_{0}}dv$ from Equation (3). We also introduce the radius of the ejecta's outer edge

$$\begin{eqnarray}&&{{r}_{{\rm out}}}={{v}_{{\rm max} }}t,\end{eqnarray} \tag{ 6 }$$

and that of the inner edge

$$\begin{eqnarray}&&{{r}_{{\rm in}}}={{v}_{{\rm min} }}t.\end{eqnarray} \tag{ 7 }$$

The inner part of the ejecta is optically thick, and therefore the propagation of radiation in the ejecta can be regarded as a diffusion process. Photons can diffusively escape from the region which satisfies the criterion that the diffusion time, ${{t}_{{\rm diff}}}$, is smaller than the dynamical time t,

$$\begin{eqnarray}&&{{t}_{{\rm diff}}}\leqslant t.\end{eqnarray} \tag{ 8 }$$

The medium in this region is considered to be effectively thin (Rybicki & Lightman 1979). For convenience, we introduce a diffusion radius ${{r}_{{\rm diff}}}\left( t \right)$ which is the radius satisfying the condition $t={{t}_{{\rm diff}}}$. Furthermore, we divide ejecta into two regions, the effectively thin ($r\geqslant {{r}_{{\rm diff}}}$) and effectively thick ($r\lt {{r}_{{\rm diff}}}$) regions. Near the diffusion radius, the optical depth is $\tau \gg 1$. We consider a random walk for photons so that the mean number of scatterings for photons to propagate the distance ${\Delta }r$ is ${{\left( {\Delta }r/{{l}_{{\rm mfp}}} \right)}^{2}}$, where ${{l}_{{\rm mfp}}}$ is the mean free path for a photon. Hence, the diffusion time ${{t}_{{\rm diff}}}$ for the propagation distance ${\Delta }r$ is

$$\begin{eqnarray}&&{{t}_{{\rm diff}}}\sim \frac{{{l}_{{\rm mfp}}}}{c}{{\left( \frac{{\Delta }r}{{{l}_{{\rm mfp}}}} \right)}^{2}}\sim \tau \frac{{\Delta }r}{c}.\end{eqnarray} \tag{ 9 }$$

In the right-hand side of Equation (9), we use $\tau \sim {\Delta }r/{{l}_{{\rm mfp}}}$.

We calculate the diffusion radius ${{r}_{{\rm diff}}}$ from the condition ${{t}_{{\rm diff}}}=t$. Since the mass density profile of the ejecta is described by a decreasing power-law function (Equation (4)), the diffusion time ${{t}_{{\rm diff}}}$ is negligible in the outer part. Thus, in order to calculate the diffusion radius ${{r}_{{\rm diff}}}$, it is a good approximation to only consider scatterings near ${{r}_{{\rm diff}}}$ (${\Delta }r\sim {{r}_{{\rm diff}}}$). However, in the early phase, the distance from the outer edge of the ejecta ${{r}_{{\rm out}}}$ to the diffusion radius ${{r}_{{\rm diff}}}$ is smaller than the diffusion radius ${{r}_{{\rm out}}}-{{r}_{{\rm diff}}}\lt {{r}_{{\rm diff}}}$. Therefore, we should take the propagation distance as

$$\begin{eqnarray}{\Delta }r\sim \left\{ \begin{array}{ccccccccccccccc} {{r}_{{\rm out}}}-{{r}_{{\rm diff}}} & \;\left( {{r}_{{\rm diff}}}\gt 0.5{{r}_{{\rm out}}} \right) \\ {} & {} \\ {{r}_{{\rm diff}}} & \;\left( {{r}_{{\rm diff}}}\leqslant 0.5{{r}_{{\rm out}}} \right). \\ \end{array} \right.\end{eqnarray} \tag{ 10 }$$

We call the first one the thin-diffusion phase and the second one the thick-diffusion phase throughout this paper. We schematically show these two phases in Figure 2. Note that in the thin-diffusion phase, since the size of the effectively thin region is much smaller than the size of the ejecta (${{r}_{{\rm out}}}-{{r}_{{\rm in}}}$), the calculation of the radiative transfer using the Monte Carlo technique (e.g., Barnes & Kasen 2013; Tanaka & Hotokezaka 2013) requires a large number of realizations to follow the temporal evolution, which do not seem to have been considered properly so far.

Zoom In Zoom Out Reset image size

To obtain the diffusion radius, we need to calculate the optical depth τ of photons that propagate a distance ${\Delta }r$. Using Equations (4) and (10), the optical depth τ is described as

$$\begin{eqnarray}\begin{array}{rcl} \tau & = & \int _{{{r}_{{\rm diff}}}}^{{{r}_{{\rm out}}}}\kappa \rho dr \\ {} & = & \frac{\left( \beta -3 \right)\kappa {{M}_{{\rm ej}}}}{4\pi \left( \beta -1 \right)v_{{\rm min} }^{2}{{t}^{2}}}{{\left[ 1-{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-\beta }} \right]}^{-1}} \\ {} & {} & \times \left[ {{\left( \frac{{{r}_{{\rm diff}}}}{{{v}_{{\rm min} }}t} \right)}^{1-\beta }}-{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{1-\beta }} \right], \\ \end{array}\end{eqnarray} \tag{ 11 }$$

in the thin-diffusion phase, and

$$\begin{eqnarray}\begin{array}{rcl} \tau & = & \int _{{{r}_{{\rm diff}}}}^{2{{r}_{{\rm diff}}}}\kappa \rho dr \\ {} & = & \frac{\left( \beta -3 \right)\kappa {{M}_{{\rm ej}}}}{4\pi \left( \beta -1 \right)v_{{\rm min} }^{2}{{t}^{2}}}{{\left[ 1-{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-\beta }} \right]}^{-1}} \\ {} & {} & \times {{\left( \frac{{{r}_{{\rm diff}}}}{{{v}_{{\rm min} }}t} \right)}^{1-\beta }}\left( 1-{{2}^{1-\beta }} \right), \\ \end{array}\end{eqnarray} \tag{ 12 }$$

in the thick-diffusion phase, where κ is the opacity of the ejecta. For simplicity, we use a gray approximation and a spatially uniform value of the opacity κ. From the results of Tanaka & Hotokezaka (2013; see also Kasen et al. 2013) which consider the contribution from all r-process elements to the opacity of merger ejecta, the evolution of the bolometric luminosity can be approximately described by the constant value of the opacity, $\kappa \sim 3-30$ cm²g⁻¹. Following their results, we use this value for the opacity of the ejecta. Note that the exact value of the opacity of the ejecta has some uncertainties in the production efficiency of r-process elements and its spatial distribution. Moreover, if the ejecta temperature is low enough for dust formation ($T\lesssim 2000$ K), the opacity significantly increases (Takami et al. 2014). From these reasons, we consider the dependence on κ in Section 3.

Our model is based on the formulation of the light curves of supernovae (e.g., Chevalier 1992; Nakar & Sari 2010; Rabinak & Waxman 2011), but there are several differences. In the case of type II supernovae, the opacity is significantly reduced due to hydrogen recombination (e.g., Goldfriend et al. 2014). However, since the ionization potentials of the lanthanides included in the r-process elements are generally lower than those of hydrogen and the iron group, the opacity remains high at relatively low temperature (Kasen et al. 2013). Therefore, we do not consider the recombination effects for the opacity.

As far as we know, supernova studies (e.g., Chevalier 1992; Nakar & Sari 2010; Rabinak & Waxman 2011) have not taken into account the thin-diffusion phase, which is necessary for treating the thickness of the diffusion length appropriately and estimating the physical quantities using the values at the outer edge of the ejecta in the analytical formulae. This phase may also be important for the case of supernovae.

Some supernova studies consider the planar phase (Piro et al. 2010; Nakar & Sari 2010) in which the evolution of the ejecta is approximately planar as long as its radius does not double. In the case of the NS–NS merger, since the initial length scale of the merger system is small $\sim {{10}^{6}}$ cm and the velocity of the merger ejecta is subrelativistic, the planar phase is irrelevant for the observations.

2.3.1. Radioactivity

One of the two heating mechanisms we consider is nuclear heating by r-process elements. Since the beta decay products of r-process elements produced in NS binary mergers naturally heat ejecta, this mechanism is considered to power the emission of a macronova (e.g., Li & Paczyński 1998). The nuclear heating rate is calculated in several works (Metzger et al. 2010; Roberts et al. 2011; Korobkin et al. 2012; Rosswog et al. 2014; Wanajo et al. 2014). The derived heating rates per unit mass $\dot{\epsilon }\left( t \right)$ are described by the following formula

$$\begin{eqnarray}&&\dot{\epsilon }={{\dot{\epsilon }}_{0}}{{\left( \frac{t}{1\ {\rm day}} \right)}^{-\alpha }}.\end{eqnarray} \tag{ 13 }$$

In this study, we use $\alpha =1.3$ and ${{\dot{\epsilon }}_{0}}=2\times {{10}^{10}}\;{\rm erg}\;{{{\rm s}}^{-1}}{{{\rm g}}^{-1}}$ obtained by Wanajo et al. (2014). The value of $\dot{\epsilon }$ has been obtained by simulations under some simplified assumptions with only limited parameter regions. Thus, we should note that the value of ${{\dot{\epsilon }}_{0}}$ has uncertainties.

The internal energy injected by the nuclear decay is $\propto {{t}^{1-\alpha }}$ in the region $r\lt {{r}_{{\rm diff}}}$. On the other hand, the injected energy in this region is decreased by adiabatic cooling. The time evolution of  the internal energy due to adiabatic cooling is proportional to ${{t}^{-1}}$. Comparing the two temporal evolutions, the index of the adiabatic cooling is smaller than that of the increase in internal energy due to the nuclear decay for $\alpha \lt 2$. Since we use $\alpha =1.3$, we neglect the injected internal energy in the region $r\lt {{r}_{{\rm diff}}}$.

2.3.2. Engine-driven shock

Unlike the r-process model, energy injection occurs only within the time ${{t}_{{\rm inj}}}$ in the engine model. We only consider adiabatic cooling as the cooling process of ejecta after ${{t}_{{\rm inj}}}$, and therefore, the temperature distribution at time t is

$$\begin{eqnarray}&&T\left( t,v \right)={{T}_{0}}{{\left( \frac{t}{{{t}_{{\rm inj}}}} \right)}^{-1}}{{\left( \frac{v}{{{v}_{{\rm min} }}} \right)}^{-\xi }},\end{eqnarray} \tag{ 14 }$$

where the index ξ is a parameter for a snapshot distribution and T₀ is a normalization factor described later. The time dependence of ${{t}^{-1}}$ is the effect of adiabatic expansion.

The normalized value T₀ is determined by using the relation of the total injected internal energy ${{E}_{\operatorname{int}0}}$ as

$$\begin{eqnarray}\begin{array}{rcl} {{E}_{\operatorname{int}0}} & = & 4\pi \int _{{{v}_{{\rm min} }}{{t}_{{\rm inj}}}}^{{{v}_{{\rm max} }}{{t}_{{\rm inj}}}}a{{T}^{4}}\left( {{t}_{{\rm inj}}},v \right){{r}^{2}}dr \\ {} & = & \frac{4\pi }{3-4\xi }aT_{0}^{4}{{\left( {{v}_{{\rm min} }}{{t}_{{\rm inj}}} \right)}^{3}}\left[ {{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-4\xi }}-1 \right], \\ \end{array}\end{eqnarray} \tag{ 15 }$$

where we use $dr={{t}_{{\rm inj}}}dv$. For the temperature index $\xi \gt 0.75$, the innermost region of the ejecta has a dominant internal energy. As will be shown in Section 3, since the luminosity and temperature always depend on the product of ${{E}_{\operatorname{int}0}}$ and ${{t}_{{\rm inj}}}$, we treat ${{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}$ as a parameter. Thus, the engine model has two parameters, ξ and ${{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}$, instead of ${{\dot{\epsilon }}_{0}}$ and α in the r-process model.

Energy injection is not always a single event and the shock does not always get through the whole ejecta. It is considered that the activity of the central engine accompanies violent time variability. In this case, multiple shocks propagate into the ejecta. Some of the shock may not catch up with the outer edge of the ejecta. Current general relativistic simulations cannot calculate the evolution of ejecta for such a long time after merger (${{t}_{{\rm inj}}}\sim {{10}^{2}}$ s), so that the index ξ of the temperature distribution is highly uncertain. Therefore, we treat the temperature index ξ as a parameter.

Unlike the case of core-collapse supernova (Nakar & Sari 2010), it is difficult to determine the temperature distribution of heated ejecta from the activity of a central engine. In the case where the activity of a central engine injects the energy into the ejecta, a radiation-dominated shock (where the internal energy behind the shock is dominated by radiation) is formed in the ejecta. The ejecta are heated during the propagation of the shock. This situation is similar to the initial phase of core-collapse supernovae (e.g., Arnett 1980; Popov 1993). In the cases of core-collapse supernovae, the kinetic energy of ejecta before shock-heating is much smaller than the injected internal energy. In such ejecta, the relation between velocity and mass density was obtained by Sakurai (1960; in the non-relativistic case for the velocity of the ejecta). Using the Sakurai (1960) solution and the equipartition between the kinetic energy after the shock heating and the internal energy (Nakar & Sari 2010), the distribution of the temperature distribution is derived. However, in the case of compact binary mergers, the merger ejecta have a large velocity (∼0.01–0.1c) before shock-heating (Hotokezaka et al. 2013a). Then, injected internal energy is not always larger than the kinetic energy of ejecta so that it is not clear whether we can use the equipartition to estimate the distribution of internal energy or not.

The kinetic energy of the ejecta ${{E}_{{\rm kin}}}$ is described as

$$\begin{eqnarray}\begin{array}{rcl} {{E}_{{\rm kin}}} & = & \frac{1}{2}\times 4\pi \int _{{{v}_{{\rm min} }}}^{{{v}_{{\rm max} }}}\rho \left( t,v \right){{v}^{4}}{{t}^{3}}dv \\ {} & = & \frac{1}{2}{{M}_{{\rm ej}}}v_{{\rm min} }^{2}\frac{\left( \beta -3 \right)\left[ {{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{5-\beta }}-1 \right]}{\left( 5-\beta \right)\left[ 1-{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-\beta }} \right]}. \\ \end{array}\end{eqnarray} \tag{ 16 }$$

Note that if the injected internal energy ${{E}_{\operatorname{int}0}}$ is larger than the kinetic energy of the ejecta, it is expected that some of the internal energy is converted to the kinetic energy of the ejecta. As a result, the internal energy and the kinetic energy are equal as in the case of core-collapse supernovae. Then, the mass density distribution and the maximum velocity of the ejecta derived from simulations may change because the injection time may be long $\sim {{10}^{2}}$ s compared to that calculated by simulations $\lesssim 0.1$ s (Hotokezaka et al. 2013a). For simplicity, we only consider the case ${{E}_{\operatorname{int}0}}\leqslant {{E}_{{\rm kin}}}$.

The size of the effectively thin region gets larger with time. At the early phase of a macronova, the diffusion radius ${{r}_{{\rm diff}}}$, which is the inner radius of the effectively thin region, is near the outer edge of the ejecta ${{r}_{{\rm out}}}$. In this early phase, we take the propagation distance ${\Delta }r$ of a photon as ${\Delta }r\sim {{r}_{{\rm out}}}-{{r}_{{\rm diff}}}\left( \lt {{r}_{{\rm diff}}} \right)$. Since we assume that the density is a homologous function of the velocity $\rho \propto {{v}^{-\beta }}$, the density can be approximated as $\rho \sim \rho \left( {{v}_{{\rm max} }} \right)$ in the region ${{r}_{{\rm diff}}}\gg {\Delta }r$. Using the escape condition for the diffusing photons $t\sim {{t}_{{\rm diff}}}$, Equation (9), and approximation of the optical depth $\tau \sim {\Delta }r\kappa \rho \left( t,{{v}_{{\rm max} }} \right)$, the propagation distance ${\Delta }r$ can be estimated as

$$\begin{eqnarray}\begin{array}{rcl} {\Delta }r & \sim & \sqrt{\frac{ct}{\kappa \rho \left( t,{{v}_{{\rm max} }} \right)}} \\ {} & \propto & {{\kappa }^{-1/2}}M_{{\rm ej}}^{-1/2}v_{{\rm min} }^{\frac{3-\beta }{2}}v_{{\rm max} }^{\beta /2}{{t}^{2}}. \\ \end{array}\end{eqnarray} \tag{ 17 }$$

In the discussion of parameter dependence (Sections 3.1–3.3), we only consider the dominant term. For example, we neglect the second term in the right-hand side of Equation (5) to derive the parameter Equation (17) because the index of the mass density is $\beta \gt 3$ in our model. In Section 3.4, we include the subdominant terms to calculate the light curves numerically.

First we consider the r-process model. The evolution of temperature ${{T}_{{\rm obs}}}$ is obtained by the internal energy density $\dot{\epsilon }t\rho$ at the radius $r={{r}_{{\rm out}}}$. Using Equations (4) and (5), the parameter dependence of the density is $\rho \left( t,{{v}_{{\rm max} }} \right)\propto {{M}_{{\rm ej}}}v_{{\rm min} }^{\beta -3}v_{{\rm max} }^{-\beta }{{t}^{-3}}$. The observed temperature is

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{\left( \frac{\dot{\epsilon }t\rho \left( t,{{v}_{{\rm max} }} \right)}{a} \right)}^{1/4}} \\ {} & \propto & M_{{\rm ej}}^{1/4}v_{{\rm min} }^{\frac{\beta -3}{4}}v_{{\rm max} }^{-\beta /4}{{t}^{-\frac{2+\alpha }{4}}}. \\ \end{array}\end{eqnarray} \tag{ 18 }$$

For $\alpha =1.3$, the observed temperature evolves as ${{T}_{{\rm obs}}}\propto {{t}^{-0.875}}.$ This is because in the thin-diffusion phase the ejecta is effectively a single expanding shell with $\rho \sim \rho \left( t,{{v}_{{\rm max} }} \right)$ and the injected energy $\dot{\epsilon }t\propto {{t}^{-0.3}}$ is almost constant so that the observed temperature approximately follows adiabatic cooling T ∝ t⁻¹. Note that in this phase the observed temperature does not depend on the opacity. The bolometric luminosity ${{L}_{{\rm bol}}}$ for the radioactivity is described as the product of the mass within the thickness ${\Delta }r$ in Equation (17) and the nuclear heating rate $\dot{\epsilon }$ in Equation (13) so that

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi r_{{\rm out}}^{2}{\Delta }r\rho \left( t,{{v}_{{\rm max} }} \right)\dot{\epsilon } \\ {} & \propto & {{\kappa }^{-1/2}}M_{{\rm ej}}^{1/2}v_{{\rm min} }^{\frac{\beta -3}{2}}v_{{\rm max} }^{\frac{4-\beta }{2}}{{t}^{1-\alpha }}. \\ \end{array}\end{eqnarray} \tag{ 19 }$$

For $\alpha =1.3$, the evolution of the bolometric luminosity is ${{L}_{{\rm bol}}}\propto {{t}^{-0.3}}$.

Next we consider the engine model. We should take into account the freedom of the temperature index ξ in the temperature distribution (Equation (14)). Since we only consider a dominant term in the right-hand side of Equation (15) (the first term for $\xi \lt 0.75$ or the second term for $\xi \gt 0.75$) in this subsection, the parameter dependence of the temperature T₀ is described as

$$\begin{eqnarray}{{T}_{0}}\propto E_{\operatorname{int}0}^{1/4}t_{{\rm inj}}^{-3/4}\times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm min} }^{-\xi }v_{{\rm max} }^{\frac{4\xi -3}{4}} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{-3/4} & \left( \xi \gt 0.75 \right) \\ \end{array} \right..\end{eqnarray} \tag{ 20 }$$

Substituting $v={{v}_{{\rm max} }}$ into Equation (14), the observed temperature is described as

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{T}_{0}}{{\left( \frac{t}{{{t}_{{\rm inj}}}} \right)}^{-1}}{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{-\xi }} \\ {} & \propto & E_{\operatorname{int}0}^{1/4}t_{{\rm inj}}^{1/4}{{t}^{-1}}\times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm max} }^{-3/4} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{\frac{4\xi -3}{4}}v_{{\rm max} }^{-\xi } & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 21 }$$

Since the observed temperature ${{T}_{{\rm obs}}}$ approximately equals the temperature at the outer edge of the ejecta, ${{T}_{{\rm obs}}}\sim T\left( t,{{v}_{{\rm max} }} \right)$, the evolution of the observed temperature and luminosity are also determined by the adiabatic cooling. The bolometric luminosity in the effectively thin region is equal to the total radiation created by thermal emission in this region (Rybicki & Lightman 1979). Using Equation (21), the bolometric luminosity is described as

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi r_{{\rm out}}^{2}{\Delta }r\frac{aT_{{\rm obs}}^{4}}{t} \\ {} & \propto & {{\kappa }^{-1/2}}M_{{\rm ej}}^{-1/2}{{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}{{t}^{-1}} \\ {} & {} & \times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm min} }^{\frac{3-\beta }{2}}v_{{\rm max} }^{\frac{\beta -2}{2}} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{\frac{-3-\beta +8\xi }{2}}v_{{\rm max} }^{\frac{4+\beta -8\xi }{2}} & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 22 }$$

The time evolution of bolometric luminosity for the engine model does not depend on the temperature index ξ in the thin-diffusion phase.

Comparing the engine model with the r-process model in the thin-diffusion phase, the bolometric luminosity and the observed temperature decrease faster in the engine model than those in the r-process model. These time-dependence do not depend on the indices of the density and temperature.

Note that the light curve may depend on the detailed profile of the front of the ejecta in this thin-diffusion phase. The profile of the ejecta front is difficult to calculate by the numerical simulation due to its low density, and hence has large uncertain (Kyutoku et al. 2014). We discuss its dependence in Section 4.

We consider diffusion to evaluate the diffusion radius in the thick-diffusion phase. We take the propagation distance ${\Delta }r\sim {{r}_{{\rm diff}}}$ after the time when the difference between the radius of the outer edge of the ejecta ${{r}_{{\rm out}}}$ and the diffusion radius ${{r}_{{\rm diff}}}$ is larger than the diffusion radius, ${{r}_{{\rm out}}}-{{r}_{{\rm diff}}}\gt {{r}_{{\rm diff}}}$, since the optical depth of the outer part is negligible for the density profile in Equation (4). In this thick-diffusion phase, the mass density significantly deviates from $\rho \left( {{v}_{{\rm max} }} \right)$. Substituting Equations (9) and (12) into $t={{t}_{{\rm diff}}}$, the diffusion radius ${{r}_{{\rm diff}}}$ is calculated as

$$\begin{eqnarray}\begin{array}{rcl} {{r}_{{\rm diff}}} & \sim & {{\left[ \frac{\left( \beta -3 \right)\kappa {{M}_{{\rm ej}}}v_{{\rm min} }^{\beta -3}{{t}^{\beta -4}}}{4\pi \left( \beta -1 \right)c} \right]}^{\frac{1}{\beta -2}}} \\ {} & \propto & {{\kappa }^{\frac{1}{\beta -2}}}M_{{\rm ej}}^{\frac{1}{\beta -2}}v_{{\rm min} }^{\frac{\beta -3}{\beta -2}}{{t}^{\frac{\beta -4}{\beta -2}}}, \\ \end{array}\end{eqnarray} \tag{ 23 }$$

where we use the relations ${\Delta }r\sim {{r}_{{\rm diff}}}$ and $v\sim {{r}_{{\rm diff}}}/t$. The latter is obtained from the assumption of homologous expansion. Regarding the optical depth τ, the second term is neglected in the right-hand side of Equation (11) to focus only on the dominant term to study parameter dependence in Sections 3.1–3.3. For $\beta =3.5$, the diffusion radius decreases with time (${{r}_{{\rm diff}}}\propto {{t}^{-1/3}}$). Then, emission from the region with relatively high mass density can be observed progressively in this phase ($\rho \propto {{t}^{-3}}{{\left( {{r}_{{\rm diff}}}/t \right)}^{-\beta }}\propto {{t}^{\left( 4\beta -9 \right)/3}}={{t}^{1.667}}$).

We introduce the transition time ${{t}_{\times }}$ between the thin- and thick-diffusion phases, which satisfies the relation ${{r}_{{\rm diff}}}=0.5{{r}_{{\rm out}}}.$ Substituting Equation (23) and ${{r}_{{\rm out}}}={{v}_{{\rm max} }}t$ into the relation ${{r}_{{\rm diff}}}=0.5{{r}_{{\rm out}}}$, we can obtain the transition time ${{t}_{\times }}$ as

$$\begin{eqnarray}\begin{array}{rcl} {{t}_{\times }} & \sim & \sqrt{\frac{{{2}^{\beta -4}}\left( \beta -3 \right)\kappa {{M}_{{\rm ej}}}}{\pi \left( \beta -1 \right)c{{v}_{{\rm max} }}}{{\left( \frac{{{v}_{{\rm max} }}}{{{v}_{{\rm min} }}} \right)}^{3-\beta }}} \\ {} & \sim & 4.1\;\kappa _{10}^{1/2}M_{{\rm ej},0.1}^{1/2}v_{{\rm min} ,0.1}^{\frac{\beta -3}{2}}v_{{\rm max} ,0.4}^{\frac{2-\beta }{2}}\;{\rm day}, \\ \end{array}\end{eqnarray} \tag{ 24 }$$

where ${{\kappa }_{10}}\equiv \kappa /10$ cm² g⁻¹, ${{M}_{{\rm ej},0.1}}\equiv {{M}_{{\rm ej}}}/0.1\;{{M}_{\odot }}$, ${{v}_{{\rm min} ,0.1}}\equiv {{v}_{{\rm min} }}/0.1c$, and ${{v}_{{\rm max} ,0.4}}\equiv {{v}_{{\rm max} }}/0.4c$. As seen above, the transition time ${{t}_{\times }}$ is typically several days. This timescale is expected to allow follow-up observations (Aasi et al. 2014). Thus, we should consider both phases to predict something useful for follow-up observations. If we fix κ and ${{v}_{{\rm max} }}$ and use $\beta =3.5$, Equation (24) gives ${{t}_{\times }}\propto M_{{\rm ej}}^{1/2}v_{{\rm min} }^{1/4}$. If we increase the total mass of the ejecta ${{M}_{{\rm ej}}}$ and the velocity at the inner edge of the ejecta ${{v}_{{\rm min} }}$, the mass density of the ejecta ρ and hence the optical depth are increased. As a result, the transition time ${{t}_{\times }}$ becomes large.

First, we consider the r-process model. Here, we introduce the velocity ${{v}_{{\rm diff}}}={{r}_{{\rm diff}}}/t$ based on the homologous relation. Using the velocity ${{v}_{{\rm diff}}}$ and Equation (23) for the mass density (Equation (4)), the evolution of temperature is

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{\left( \frac{\dot{\epsilon }t\rho \left( t,{{v}_{{\rm diff}}} \right)}{a} \right)}^{1/4}} \\ {} & \propto & {{\kappa }^{\frac{\beta }{4\left( 2-\beta \right)}}}M_{{\rm ej}}^{\frac{1}{2\left( 2-\beta \right)}}v_{{\rm min} }^{\frac{\beta -3}{2\left( 2-\beta \right)}}{{t}^{\frac{1}{\beta -2}-\frac{\alpha }{4}}}. \\ \end{array}\end{eqnarray} \tag{ 25 }$$

For $\alpha =1.3$ and $\beta =3.5$, the evolution is described by ${{T}_{{\rm obs}}}\propto {{t}^{0.341}}$ so that the observed temperature increases with time. The bolometric luminosity is described as the product of the mass between ${{r}_{{\rm diff}}}$ and $2{{r}_{{\rm diff}}}$ and the nuclear heating rate $\dot{\epsilon }$. Using Equations (13) and (23), we obtain the bolometric luminosity as

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi r_{{\rm diff}}^{3}\rho \left( t,{{v}_{{\rm diff}}} \right)\dot{\epsilon } \\ {} & \propto & {{\kappa }^{\frac{3-\beta }{\beta -2}}}M_{{\rm ej}}^{\frac{1}{\beta -2}}v_{{\rm min} }^{\frac{\beta -3}{\beta -2}}{{t}^{\frac{2\left( \beta -3 \right)}{\beta -2}-\alpha }}. \\ \end{array}\end{eqnarray} \tag{ 26 }$$

For $\alpha =1.3$ and $\beta =3.5$, the evolution of the bolometric luminosity is ${{L}_{{\rm bol}}}\propto {{t}^{-0.633}}$.

Next, we consider the engine model. Using Equations (14) and (23), the evolution of the observed temperature is described as

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{T}_{0}}{{\left( \frac{t}{{{t}_{{\rm inj}}}} \right)}^{-1}}{{\left( \frac{{{v}_{{\rm diff}}}}{{{v}_{{\rm min} }}} \right)}^{-\xi }} \\ {} & \propto & {{\kappa }^{-\frac{\xi }{\beta -2}}}M_{{\rm ej}}^{-\frac{\xi }{\beta -2}}E_{\operatorname{int}0}^{1/4}t_{{\rm inj}}^{1/4}{{t}^{\frac{-\beta +2\xi +2}{\beta -2}}} \\ {} & {} & \times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm min} }^{\frac{\xi \left( \beta -3 \right)}{2-\beta }}v_{{\rm max} }^{\frac{4\xi -3}{4}} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{\frac{3\beta -6-4\xi }{4\left( 2-\beta \right)}} & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 27 }$$

For $\beta =3.5$, the value $\xi =0.75$ is the boundary marking whether the observed temperature increases with time ($\xi \gt 0.75$) or not ($\xi \lt 0.75$). The evolution of the luminosity equals the total radiation created by thermal emission in the sphere with radius ${{r}_{{\rm diff}}}$. Using the relation $v\left( {{r}_{{\rm diff}}} \right)\sim {{r}_{{\rm diff}}}/t$ and Equations (23) and (27), we obtain

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi r_{{\rm diff}}^{3}\frac{aT_{{\rm obs}}^{4}}{t} \\ {} & \propto & {{\kappa }^{\frac{3-4\xi }{\beta -2}}}M_{{\rm ej}}^{\frac{3-4\xi }{\beta -2}}{{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}{{t}^{\frac{2\left( \beta +1-4\xi \right)}{2-\beta }}} \\ {} & {} & \times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm min} }^{\frac{\left( 3-4\xi \right)\left( \beta -3 \right)}{\beta -2}}v_{{\rm max} }^{4\xi -3} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{\frac{3-4\xi }{2-\beta }} & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 28 }$$

If we take $\beta =3.5$ and $\xi =1.0$, the evolution of the bolometric luminosity is ${{L}_{{\rm bol}}}\propto {{t}^{-0.666}}$. This is almost the same dependence as in the r-process model. Note that even if the inner part of the ejecta has larger internal energy ($\xi \gt 0.75$), the bolometric luminosity does not always increase with time. Using the relation ${{E}_{\operatorname{int}}}\left( v,t \right)\propto {{t}^{-1}}$, from adiabatic cooling, the evolution of bolometric luminosity for a given mass shell with v is ${{L}_{{\rm bol}}}\sim {{E}_{\operatorname{int}}}\left( v \right){{t}^{-1}}\propto {{t}^{-2}}$, where ${{E}_{\operatorname{int}}}\left( v,t \right)$ is the total internal energy for the mass shell with a given expanding velocity v. Since ${{E}_{\operatorname{int}}}\left( {{v}_{{\rm difff}}} \right)\propto {{v}^{3-4\xi }}$ and ${{v}_{{\rm diff}}}={{r}_{{\rm diff}}}/t\propto {{t}^{\frac{2}{2-\beta }}}$, the bolometric luminosity increases with time for the value of the temperature index $\xi \gt \left( \beta +1 \right)/4=1.125$.

Once the diffusion radius reaches the inner edge of the ejecta $\left( {{r}_{{\rm diff}}}={{r}_{{\rm in}}} \right)$, all photons emitted from the ejecta can diffuse out within  dynamical timescale. If energy is not injected into the ejecta in this transparent phase, the internal energy in the ejecta runs out immediately. The transition time from the thick-diffusion phase to the transparent phase ${{t}_{{\rm tr}}}$ is described as

$$\begin{eqnarray}\begin{array}{rcl} {{t}_{{\rm tr}}} & \sim & \sqrt{\frac{\left( \beta -3 \right)\kappa {{M}_{{\rm ej}}}}{4\pi \left( \beta -1 \right)c{{v}_{{\rm min} }}}} \\ {} & \sim & 6.9\;\kappa _{10}^{1/2}M_{{\rm ej},0.1}^{1/2}v_{{\rm min} ,0.1}^{-1/2}\;{\rm day}, \\ \end{array}\end{eqnarray} \tag{ 29 }$$

where we use the diffusion radius ${{r}_{{\rm diff}}}={{r}_{{\rm in}}}$.

First we consider the r-process model. The observed temperature equals to the temperature at the inner edge of the ejecta, ${{T}_{{\rm obs}}}\sim {{\left[ [\dot{\epsilon }t\rho \left( {{v}_{{\rm min} }} \right)/a] \right]}^{1/4}}$. Using Equation (4), we obtain

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{\left( \frac{\dot{\epsilon }t\rho \left( t,{{v}_{{\rm min} }} \right)}{a} \right)}^{1/4}} \\ {} & \propto & M_{{\rm ej}}^{1/4}v_{{\rm min} }^{-3/4}{{t}^{-\frac{2+\alpha }{4}}}. \\ \end{array}\end{eqnarray} \tag{ 30 }$$

Since the energy is continuously injected due to nuclear heating in the r-process model, the bolometric luminosity from the entire ejecta is described as ${{L}_{{\rm bol}}}\sim {{M}_{{\rm ej}}}\dot{\epsilon }$. However, the outer part of the ejecta emits photons with lower temperature and/or X-rays and γ-rays produced directly by radioactive decays. Although such emission contributes to the bolometric luminosity, we here focus only on the optical and infrared emissions. In the thick-diffusion phase, the observed emission comes from the region between $\sim {{r}_{{\rm diff}}}$ and $\sim 2{{r}_{{\rm diff}}}$. In the transparent phase, we assume that the time evolution of the diffusion radius ${{r}_{{\rm diff}}}$ is the same as the thick-diffusion phase until $2{{r}_{{\rm diff}}}={{r}_{{\rm in}}}$ and the observed luminosity comes from the region from ${{r}_{{\rm in}}}$ to $2{{r}_{{\rm diff}}}$ for simplicity. Then, the bolometric luminosity is described as

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi r_{{\rm in}}^{3}\rho \left( t,{{v}_{{\rm min} }} \right)\dot{\epsilon } \\ {} & \propto & {{M}_{{\rm ej}}}{{t}^{-\alpha }}. \\ \end{array}\end{eqnarray} \tag{ 31 }$$

Although it appears that this time evolution directly reflects the nuclear decay rate, when we calculate the mass between ${{r}_{{\rm in}}}$ and $2{{r}_{{\rm diff}}}$ the evolution of the upper limit of the integration $2{{r}_{{\rm diff}}}$ makes the decrease of the luminosity faster than $\propto {{t}^{-\alpha }}$ (see the dashed line in the middle panel of Figure 3). In addition, the evolution of ${{r}_{{\rm diff}}}$ depends on the index β (see Equation (23)), so that the mass between ${{r}_{{\rm in}}}$ and $2{{r}_{{\rm diff}}}$ also depends on the index β.

Zoom In Zoom Out Reset image size

Next we consider the engine model. We assume that the internal energy is exhausted when the diffusion radius reaches $2{{r}_{{\rm diff}}}={{r}_{{\rm in}}}$. For the observed temperature ${{T}_{{\rm obs}}}$, we assume the relation ${{T}_{{\rm obs}}}=T\left( t,{{v}_{{\rm min} }} \right)$ and use Equation (14),

$$\begin{eqnarray}\begin{array}{rcl} {{T}_{{\rm obs}}} & \sim & {{T}_{0}}{{\left( \frac{t}{{{t}_{{\rm inj}}}} \right)}^{-1}} \\ {} & \propto & E_{\operatorname{int}0}^{1/4}t_{{\rm inj}}^{1/4}{{t}^{-1}} \\ {} & {} & \times \left\{ \begin{array}{ccccccccccccccc} v_{{\rm min} }^{-\xi }v_{{\rm max} }^{\frac{4\xi -3}{4}} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ v_{{\rm min} }^{-3/4} & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 32 }$$

The bolometric luminosity is described as

$$\begin{eqnarray}\begin{array}{rcl} {{L}_{{\rm bol}}} & \sim & 4\pi \int _{{{r}_{{\rm in}}}}^{2{{r}_{{\rm diff}}}}\frac{aT_{{\rm obs}}^{4}}{t{\rm dr}} \\ {} & \propto & {{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}{{t}^{-2}} \\ {} & {} & \times \left\{ \begin{array}{ccccccccccccccc} {{\kappa }^{\frac{3-4\xi }{\beta -2}}}M_{{\rm ej}}^{\frac{3-4\xi }{\beta -2}}v_{{\rm min} }^{\frac{\left( \beta -3 \right)\left( 3-4\xi \right)}{\beta -2}} & {} \\ \;\times v_{{\rm max} }^{4\xi -3}{{t}^{\frac{2\left( 3-4\xi \right)}{2-\beta }}} & \left( \xi \lt 0.75 \right) \\ {} & {} \\ 1 & \left( \xi \gt 0.75 \right). \\ \end{array} \right. \\ \end{array}\end{eqnarray} \tag{ 33 }$$

Since the internal energy at the innermost region almost equals the total internal energy ${{E}_{\operatorname{int}}}\left( {{v}_{{\rm min} }} \right)\sim {{E}_{\operatorname{int}0}}{{\left( t/{{t}_{{\rm inj}}} \right)}^{-1}}$ and determines the bolometric luminosity ${{L}_{{\rm bol}}}\sim {{E}_{\operatorname{int}}}\left( {{v}_{{\rm min} }} \right)/t$ for the temperature index $\xi \gt 0.75$, the bolometric luminosity does not depend on the mass ${{M}_{{\rm ej}}}$ and velocities ${{v}_{{\rm max} }}$ and ${{v}_{{\rm min} }}$. This luminosity always corresponds to the maximum luminosity for $\xi \gt 0.75$, so that we can impose the lower limit on the parameter ${{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}$.

We show the temporal evolution of the diffusion radius ${{r}_{{\rm diff}}}$, the bolometric luminosity ${{L}_{{\rm bol}}}$, and the observed temperature ${{T}_{{\rm obs}}}$ in Figure 3 under the fiducial parameter set. The parameters are summarized in the first column of Table 1. Here, we do not use the approximations $\rho \left( v \right)\sim \rho \left( {{v}_{{\rm max} }} \right)$ and $T\left( v \right)\sim T\left( {{v}_{{\rm max} }} \right)$ at the thin-diffusion phase as in Section 3.1. Instead, the diffusion radius ${{r}_{{\rm diff}}}$ is calculated from Equations (9)–(11) without approximations. Using the obtained diffusion radius ${{r}_{{\rm diff}}}$ and the relation ${{v}_{{\rm diff}}}={{r}_{{\rm diff}}}/t$, we calculate the observed temperatures in the thin- and thick-diffusion phases, ${{T}_{{\rm obs}}}\sim {{\left[ [\dot{\epsilon }t\rho \left( t,{{v}_{{\rm diff}}} \right)/a] \right]}^{1/4}}$ (Equation (25)), and ${{T}_{{\rm obs}}}\sim {{T}_{0}}{{\left( t/{{t}_{{\rm inj}}} \right)}^{-1}}{{\left( {{v}_{{\rm diff}}}/{{v}_{{\rm min} }} \right)}^{-\xi }}$ (Equation (27)) for the r-process and the engine models, respectively. In the transparent phase, the temperature in Equations (30) and (32)are evaluated with $v={{v}_{{\rm min} }}$. Equations on observed temperature and bolometric luminosity for both models are summarized in appendix. The set of parameters we choose here explains the observed optical and infrared light curves of GRB 130603B (see next section). The vertical dash-dotted lines in Figure 3 show the time $t={{t}_{\times }}$ (Equation (24)). The diffusion radius is plotted only up to the transition time $t={{t}_{{\rm tr}}}$ (Equation (29)).

In the thick-diffusion phase, the diffusion radius (${{r}_{{\rm diff}}}\propto {{t}^{\frac{\beta -4}{\beta -2}}}={{t}^{-1/3}}$ for $\beta =3.5$) moves inward in the ejecta ($r\propto t$). Since the observed luminosity and temperature are determined at the diffusion radius ${{r}_{{\rm diff}}}$, the time evolution of luminosity and temperature strongly depends on the indices of the profile, β and ξ. For the r-process model, the bolometric luminosity decreases with time (${{L}_{{\rm bol}}}\propto {{t}^{\frac{2\left( \beta -3 \right)}{\beta -2}-\alpha }}={{t}^{-0.633}}$) in the thick-diffusion phase (see Equation (26)), which is more rapid than that in the thin-diffusion phase (${{L}_{{\rm bol}}}\propto {{t}^{1-\alpha }}={{t}^{-0.3}}$; see Equation (19)). Since the index of the mass density $\beta =3.5$ is close to 3, in which the mass of each shell with a certain size $\delta r$ is the same value in logarithmic scale, the mass between the diffusion radius ${{r}_{{\rm diff}}}$ and its doubled value $2{{r}_{{\rm diff}}}$ does not significantly change with time. The luminosity is mainly determined by that mass, so that the evolution of the luminosity is slow compared with the evolution of nuclear heating rate ($\propto {{t}^{-\alpha }}$) in the thick-diffusion phase. On the other hand, bolometric luminosity and observed temperature increase with time in the engine model with the parameter set of the fiducial model. These mainly reflect the profile of the temperature distribution ($\xi =1.6$). In fact, using Equation (28), the index of the time t for the bolometric luminosity is $-2\left( \beta +1-4\xi \right)/\left( \beta -2 \right)\sim 2.53$ for the engine model.

After the transition time $t\geqslant {{t}_{{\rm tr}}}$, the luminosity and temperature are almost determined by the quantities at the inner edge of the ejecta. Then, the evolution of the luminosity and temperature does not significantly depend on the indices of profile β and ξ as in the case of the thin-diffusion phase (except for the case $\xi \lt 0.75$ of the engine model, Equation (33)). Since our used profile of mass density has an artificially steep cut-off at the inner edge of the ejecta (Figure 2), bolometric luminosity in both models rapidly declines after the time $t\geqslant {{t}_{{\rm tr}}}$. In the bottom panel of Figure 3, the observed temperature in both models has a steep cutoff at $2{{r}_{{\rm diff}}}={{r}_{{\rm in}}}$. For comparison, we also consider the time evolution of bolometric luminosity from the whole ejecta ${{L}_{{\rm bol}}}={{M}_{{\rm ej}}}{{t}^{-\alpha }}$ in the r-process model. Time evolution is shown in the middle panel of Figure 3 as a blue long-dashed line. This luminosity evolution (${{L}_{{\rm bol}}}\propto {{t}^{-\alpha }}={{t}^{-1.3}}$) is significantly slower than that of the engine model in the transparent phase. In Section 4.4, we discuss the implication for discriminating the r-process model and the engine model using these temporal behaviors.

We compare the results with the optical and infrared observations of the short GRB 130603B in Figure 4. The fiducial parameter set in Table 1 is adopted. The r-process model and the engine model result in similar light curves at the optical and infrared bands. Both of them satisfy the observational data of GRB 130603B. Note that the detection point at the F606W band at $\sim {{10}^{5}}$ s is consistent with the afterglow of GRB 130603B modeled as a smoothly broken power law (blue dashed line; Tanvir et al. 2013). We regard this detected value as an upper limit for the luminosity of emission from the ejecta. The detection point at F160W band at $\sim {{10}^{6}}$ s exceeds the extrapolation of the afterglow emission (red dashed line, Tanvir et al. 2013), so that we regard this detected emission as a thermal radiation from the ejecta.

Zoom In Zoom Out Reset image size

The range of the model parameters ${{v}_{{\rm min} }}$ and ${{M}_{{\rm ej}}}$ to satisfy the constraints obtained from the observation of GRB 130603B is shown in Figure 5 as colored areas (red area for the r-process model and blue area for the engine model). Note that the red area has a complete overlap with the blue area. We fix the other model parameters ${{v}_{{\rm max} }}=0.4c$, $\beta =3.5$, ${{\dot{\epsilon }}_{0}}=2\times {{10}^{10}}$ erg s⁻¹ g⁻¹, and $\alpha =1.3$ as in the fiducial model. We take into account the uncertain range of the opacity $\kappa =3-30$ cm² g⁻¹ to constrain the parameters, ${{v}_{{\rm min} }}$ and ${{M}_{{\rm ej}}}$. In the engine model, ξ and ${{E}_{\operatorname{int}0}}{{t}_{{\rm inj}}}$ are additionally treated as free parameters to derive the allowed area in Figure 5.

Zoom In Zoom Out Reset image size

4.1.1. Limits on Ejecta Mass

4.1.2. Limits on the Minimum Velocity

4.1.3. Dependence on Opacity

4.1.4. Dependence on Engine Parameters

Several deep optical observations of short GRBs give stringent upper limits on the luminosity of macronovae (Kann et al. 2011). We compare the results with two deep optical observations of short GRBs, GRB 050509B and GRB 080905A. For the fiducial parameter set, the luminosity exceeds the observational upper limits on these two observations. In the engine model, we can reduce the luminosity in the early phase, ≲10⁵ s, without reducing the luminosity in the late phase, $\sim {{10}^{6}}$ s, by utilizing the steep temperature profile (large ξ).

Here, we introduce the hot interior parameter set with value of index ξ larger than that of the fiducial parameter set. Since emission from the inner part of the ejecta is observed at a later time, the luminosity at the early phase decreases and avoids observational limits if most of the internal energy is injected to the inner part of the ejecta. We show the light curve of the hot interior parameter set in Figure 6. We choose the parameters ${{M}_{{\rm ej}}}=0.08\;{{M}_{\odot }}$, ${{v}_{{\rm min} }}=0.18c$, ${{t}_{{\rm inj}}}={{10}^{2}}\;{\rm s}$, ${{E}_{\operatorname{int}0}}=0.8\times {{10}^{51}}\;{\rm erg}$, and $\xi =2.7$ (the right column of Table 1). From Figure 6, the light curves are consistent with all three observations using the same model parameters. A possible scenario for the hot interior parameter set is that the shock produced by the activity of the central engine may not be able to catch up with the outer part of the ejecta because the velocity of the ejecta is close to the light speed (${{v}_{{\rm max} }}=0.4c$). Then, only the inner part of the ejecta will be heated. For comparison, we also show the light curves in the r-process model with the parameter set of the hot interior in Figure 6 as dashed lines. The luminosity of the r-process model exceeds the observed upper limits in two observations, GRB 050509B and GRB 080905A (left and middle panels of Figure 6), if we choose the parameter set of the hot interior model. We are not able to find any parameter set in the r-process model that simultaneously satisfies the observed limits of the three observations. Note that we do not argue that the r-process model is excluded from these results because we need to take into account the variations of the model parameters for each event.

Zoom In Zoom Out Reset image size

Note that the extended emission was not detected in three short GRBs. However, it is not unreasonable to miss the extended emission of these bursts. One possibility is a selection effect. Observationally, the fraction of short GRBs with extended emission is significantly larger at softer energy bands: ∼25% in the Swift BAT samples (>15 keV; Norris et al. 2010) and ∼7% in the BATSE samples (>20 keV; Bostanci et al. 2013). This suggests that observations with a low energy threshold may dramatically increase the number of short GRBs with extended emission (Nakamura et al. 2014). This will be further tested by future soft X-ray survey facilities such as Wide-Field MAXI (0.7–10 keV; Kawai et al. 2014). The three mentioned short GRBs were detected by Swift BAT and therefore Swift BAT could not detect extended emission by chance. Alternatively, the outflow following the main short GRB jet could not break out from the ejecta. Nagakura et al. (2014) and Murguia-Berthier et al. (2014) investigated the propagation of jets in merger ejecta. They found the cases where relativistic jets can penetrate merger ejecta and produce the prompt emission of short GRBs, but in the late energy injection cases, outflow fails to break out from the ejecta. Therefore, some extended emission may not be observed, although the central engine works actively.

In the thin-diffusion phase, the light curve strongly depends on the density profile of the ejecta surface. The density profile is determined by complex merger dynamics (Hotokezaka et al. 2013a), so that the density profile of the ejecta cannot be analytically derived as mentioned in Section 2.3. Since the outer part of the density profile is difficult to calculate precisely, little attention has been paid to the mass profile at the outer region in current numerical simulations. In order to investigate the dependence of the light curve on the mass profiles in the thin-diffusion phase, we consider other forms of the mass profile and compare the light curve with that of Equation (4). We adopt an exponential profile

$$\begin{eqnarray}\begin{array}{rcl} \rho \left( t,v \right) & = & {{\rho }_{0}}{{\left( \frac{t}{{{t}_{0}}} \right)}^{-3}}{{\left( \frac{v}{{{v}_{{\rm min} }}} \right)}^{-\beta }} \\ {} & {} & \times {\rm exp} \left( -\frac{v-0.5{{v}_{{\rm max} }}}{v_{{\rm max} }^{\prime }-v} \right). \\ \end{array}\end{eqnarray} \tag{ 34 }$$

We introduce an additional free parameter $v_{{\rm max} }^{\prime }\left( \geqslant {{v}_{{\rm max} }} \right)$ and the ejecta expands with ${{v}_{{\rm min} }}\leqslant v\leqslant v_{{\rm max} }^{\prime }$. The calculations are the same except ${\Delta }r\sim v_{{\rm max} }^{\prime }t-{{r}_{{\rm diff}}}$ is used. We fix the mass with velocity larger than $0.5{{v}_{{\rm max} }}$ and calculate three models for $v_{{\rm max} }^{\prime }=0.4c,0.5c,0.6c$. For the other parameters, we adopt from the fiducial parameter set in Table 1. We show the results in the r-process model in Figure 7. Since we fix the mass with velocity larger than $0.5{{v}_{{\rm max} }}$, the bolometric luminosities at the time $t\sim {{t}_{\times }}$ are almost the same values. In the thin-diffusion phase ($t\ll {{t}_{\times }}$), both the luminosity and the temperature are smaller than the fiducial model. This is because the density at the front of the ejecta is reduced in this mass profile. Since the maximum velocity effectively becomes large and the adiabatic cooling becomes efficient, these effects for luminosity and temperature should also be seen in the engine model. We conclude that the luminosities in the thin-diffusion phase ($t\ll {{t}_{\times }}$) have uncertainties of at least ∼1–2 mag, which originates from the uncertainty of the outermost mass profile.

Zoom In Zoom Out Reset image size

Note that the emission from the ejecta with the mass profile discussed here reduces the tension between the light curve in the r-process model and the upper limits of the deep optical observations (GRB 050509 and GRB 080905A) as discussed in Section 4.2. Especially in the case $v_{{\rm max} }^{\prime }=0.6c$ (thin-dotted–dashed line), the optical luminosity significantly decreases after t ≳ 10⁵ s. Therefore, the front of the ejecta, which has a relatively shallow mass distribution, is able to explain the current optical follow-up observations in the r-process model.

In the fiducial parameter set, the light curves for two models in the optical and infrared bands are similar (Figure 4). In Figure 5, the allowed parameter region to explain the observation of GRB 130603B for both models is also overlapped. Recall that emission from ejecta is described as blackbody radiation for the two models, whose spectrum is narrow in bands. Therefore, no excess at other wavelengths is expected and the prediction is also consistent with radio observations (Fong et al. 2014). Therefore, it is difficult to discriminate between the two models from the currently available observational data.

In the r-process model, the light curve with mass profile $\rho \propto {{v}^{-\beta }}$ ($3\lesssim \beta \lesssim 4$) and a parameter set which explains the infrared excess detected from GRB 130603B cannot explain the upper limits obtained from the deep optical observations of some short GRBs. Therefore, if both stringent optical upper limits at $\sim {{10}^{5}}$ s and bright infrared emission at $\sim {{10}^{6}}$ s are simultaneously obtained from a single event (with a difference larger than two magnitudes ${{M}_{{\rm optical}}}\left( \sim {{10}^{5}}\;{\rm s} \right){\mkern 1mu} -{{M}_{{\rm infrared}}}$ $\left( \sim {{10}^{6}}\;{\rm s} \right)\gtrsim 2$ mag ), the r-process model is significantly restricted. For the engine model, these observations give a constraint for the temperature distribution in the ejecta, which may give new insights into the activity of the central engine.

As shown in the middle panel of Figure 3, the bolometric luminosity in the r-process model from the whole ejecta (blue long-dashed line), including low temperature and/or X-rays and γ-rays produced directly in radioactive decays (Churazov et al. 2014), declines more gradually than that for the engine model. This is because there is no energy injection after the time $t\gt {{t}_{{\rm inj}}}$ for the engine model. Then, the luminosity significantly decreases when photons at the inner edge of the ejecta begin to diffuse out (see the middle panel of Figures 3 and 4). The luminosity from the whole ejecta can be described as ${{L}_{{\rm bol}}}\sim {{M}_{{\rm ej}}}\dot{\epsilon }$ in the transparent phase. The index of time t is determined by the nuclear heating rate, $\alpha \sim 1.3$. Therefore, the two models are distinguishable by observing the temporal evolution of bolometric luminosity from the whole ejecta in this phase.