EARLY THERMAL X-RAY EMISSION FROM LONG GAMMA-RAY BURSTS AND THEIR CIRCUMSTELLAR ENVIRONMENTS

We perform hydrodynamical calculations of the propagation of an ultrarelativistic jet in a massive star and the subsequent interaction with the CSM by using the special relativistic hydrodynamics code in 2D spherical coordinates (r, θ) developed by one of the authors. In this code, we adopt a mapping procedure, in which the width of the radial zones is doubled as the jet head reaches a fraction (∼0.9) of the maximum of the radial coordinate, in order to calculate the propagation of the jet until t ∼ 1800 s. Thus, the radial resolution becomes coarser as the time elapses. At t = 0, the radial coordinate ranges from r = 10⁹ cm to r = 10¹¹ cm. At the end of the calculations, the maximum of the radial coordinate reaches r ∼ 6 × 10¹³ cm. The radial zone is divided into N_r uniform cells and the number N_r = 1024 is fixed. The angular coordinate θ ranges from θ = 0 to θ = π/2 and is composed of N_θ = 256 uniform cells.

As a presupernova model, we adopt 16TI model in Woosley & Heger (2006), which is commonly used in calculations of collapsar jets. In this study, we consider several models to clarify the effect of the ejecta–CSM interaction. Since the spatial distribution of the CSM is highly uncertain, we adopt the simplest steady wind model whose density profile is given by

$$\begin{equation} \rho _\mathrm{w}(r)=\frac{\dot{M}}{4\pi r^2v_\mathrm{w}}. \end{equation} \tag{ 1 }$$

The density profile is uniquely determined for a given ratio of the mass-loss rate $\dot{M}$ and the wind velocity v_w. In this study, the wind velocity v_w is fixed to be 1000 km s⁻¹. We performed calculations with the mass-loss rates of $\dot{M}=10^{-7}$, 10⁻⁶, 10⁻⁵, 10⁻⁴, and 10⁻³ M_☉ yr⁻¹. In the following, we especially focus on the two extreme cases, the models with $\dot{M}=10^{-7}$ and 10⁻³ M_☉ yr⁻¹ (hereafter they are referred to as the dense and dilute CSM models).

The jet is injected from the inner boundary r = 10⁹ cm from t = 0 to t = 60 s at a constant energy injection rate by using the same method as the previous works (e.g., Zhang et al. 2003; Morsony et al. 2007; Mizuta et al. 2011). The parameters specifying the jet injection condition are as follows: the total energy E_tot = 3 × 10⁵² erg, the energy injection rate $\dot{E}=5\times 10^{50}$ erg s ⁻¹, the opening angle θ_j = 10°, the initial Lorentz factor Γ₀ = 5, and the specific internal energy ₀/c² = 20.

Several previous works on an ultrarelativistic jet emanating from the progenitor star have been carried out and unveiled the dynamical evolution of the jet, such as the formation of the recollimation shock and the realization of the well-known fireball solution. (e.g., Zhang et al. 2003; Morsony et al. 2007; Mizuta et al. 2011). Our calculations successfully reproduce and confirm their findings. Some snapshots of the spatial distributions of the Lorentz factor and the density of the dilute CSM model are shown in Figure 1. The jet propagates in the interior of the progenitor star and then breaks out, and ejects stellar materials into the circumstellar space. As seen in the top right panel, the emergence of a hot material from the jet cavity follows the breakout of the collimated jet. The ejecta rapidly expand to form a spherical cocoon as seen in the bottom left panel of Figure 1. It is noteworthy that the cocoon expands at mildly relativistic speeds. The appearance and the subsequent expansion of the cocoon have also been reported and investigated by several previous works (see, e.g., Aloy et al. 2000; Ramirez-Ruiz et al. 2002; Zhang et al. 2003; Lazzati & Begelman 2005).

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Results of the dense and dilute CSM models are compared in Figures 2 and 3. Figure 2 represents the spatial distribution of the Lorentz factor (left) and the pressure (right) at t ∼ 200 s for the dense CSM model (lower panel) and the dilute CSM model (upper panel).

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Near the jet axis (θ < 10°–20°), no difference between the two models is recognized. On the other hand, in the region with large inclination angles (θ > 20°), we can see differences between the models. Denser CSM reduces the size of the cocoon in comparison with dilute CSM. In addition, the pressure distribution shows a shell-like structure in the dense CSM.

This difference can also be seen in the radial profiles of some physical variables of the cocoon, as illustrated in Figure 3. In both cases, the expansion velocities are mildly relativistic as seen in the top panel. From the bottom panel showing the pressure profiles, one can see that the reverse shock forms as a result of the cocoon–dense-CSM interaction. On the other hand, in dilute CSM, the rarefaction wave propagates toward the center in the cocoon.

This is due to the aspherical distribution of the energy deposited into the ejecta. Near the jet axis, the energy carried by the jet is too enormous for the CSM to affect the jet propagation. On the other hand, the energy deposited into the cocoon component is much smaller than that of the jet. The kinetic energy and mass of the cocoon component, which are now defined as those confined in the region outside the star and θ > 10°, can be obtained from the results of the simulation. They are found to be 3 × 10⁵⁰ erg and 2 × 10⁻³ M_☉ at t = 10 s and 10⁵¹ erg and 2 × 10⁻² M_☉ at t = 20 s. These values are almost independent of the mass-loss rate, because the dissipation of the kinetic energy of the jet to form the cocoon takes place in the star. The kinetic energy and mass of the cocoon increase due to the continuous energy and mass injection by the jet. In our calculations, the injection of the jet is terminated at t = 60 s, which means that the injection of the mass and kinetic energy into the cocoon lasts even after the cocoon begins to expand. The kinetic energy of the cocoon is up to a few percent of the total injected energy and thus has a potential for producing thermal X-ray photons with the observed luminosity ∼10⁴⁵–10⁴⁸ erg s⁻¹ for several hundred seconds. On the other hand, the mass of the ultrarelativistic jet component, which is defined as the material with the Lorentz factor larger than 100, is 3 × 10⁻⁶ M_☉, while a substantial fraction (∼10⁵¹ erg) of the injected energy is carried by this component.

As the radial profiles along θ = 45° at t = 200 s in Figure 3 shows, a reverse shock is formed in the dense CSM model. The other model with the mass-loss rates 10⁻⁴ M_☉ yr⁻¹ also form a reverse shock. The energy of the matter ejected immediately after the breakout of the jet from the surface results from the dissipation of a part of the kinetic energy of the jet for the initial several seconds. Denoting the dissipated kinetic energy by E_dis and the fraction of the internal energy to the total by , the pressure of the cocoon scales as

$$\begin{equation} P_\mathrm{c}\sim \frac{\epsilon E_\mathrm{dis}}{4\pi (v_\mathrm{exp}t)^3}, \end{equation} \tag{ 2 }$$

where we have assumed that the cocoon is spherically expanding at the velocity v_exp. On the other hand, the ram pressure of the CSM behind the forward shock is given by

$$\begin{equation} \rho _\mathrm{w}\Gamma ^2c^2\sim \frac{\dot{M}\Gamma _\mathrm{exp}^2c^2}{4\pi v_\mathrm{w}(v_\mathrm{exp}t)^2}, \end{equation} \tag{ 3 }$$

where Γ_exp = (1 − v²_exp/c²)^−1/2. The reverse shock forms when the pressure P_c of the cocoon becomes comparable to the ram pressure ρ_wΓ²c² of the shocked CSM. The balance between the pressure of the cocoon and the ram pressure yields the following expression for the time of the reverse shock formation:

$$\begin{eqnarray} t &\sim & \frac{\epsilon E_\mathrm{dis}v_\mathrm{w}}{\dot{M}v_\mathrm{exp}\Gamma _\mathrm{exp}^2c^2}\sim 10^2 \left(\frac{\epsilon }{5.0\times 10^{-4}}\right) \left(\frac{E_\mathrm{dis}}{10^{51}\ \mathrm{erg}}\right) \nonumber\\ && \times \left(\frac{v_\mathrm{w}}{10^8\ \mathrm{km\ s}^{-1}}\right) \left(\frac{\dot{M}}{10^{-4}\, M_\odot \, \mathrm{yr}^{-1}}\right)^{-1}\ \mathrm{s}, \end{eqnarray} \tag{ 4 }$$

where we have derived the final expression by assuming v_exp = 0.9 c. The value of the fraction is found from the result of hydrodynamical calculations. This rough estimation is consistent with the fact that a reverse shock is observed for models with $\dot{M}=10^{-3}$ and 10⁻⁴ M_☉ yr⁻¹ at t = 200 s and no reverse shock for models with lower mass-loss rates.

In the following, we investigate whether thermal X-ray emission from GRBs can probe their circumstellar environments. We derive the expected light curve and the spectra of thermal X-ray emission from our models by calculating the photospheric emission.

According to Starling et al. (2012), thermal emission with the isotropic luminosity of the order of 10⁴⁷ erg s⁻¹ and the photon temperature ∼0.1–0.9 keV is observed. Illuminated by the radiation, heavy atoms, such as oxygen and carbon, are rapidly photo-ionized. The recombination timescale, which is much longer than the ionization timescale, keeps those ions fully ionized and the dominant opacity source becomes electron scattering. Thus, we calculated the Thomson photosphere from a distant observer along the axis of the jet (θ = 0). In deriving light curves and spectra, we have assumed that the matter and radiation are strongly coupled and the internal energy density on the photosphere is dominated by that of radiation.

At first, we briefly consider the properties of the emission. Since the ejecta move at mildly relativistic velocities with the Lorentz factor of a few, the relativistic beaming effect strengthens the emission, especially, in the early phase. From the top and bottom panels of Figure 3, the radiation temperature of the shocked region for the dense CSM can be estimated to be

$$\begin{equation} \Gamma k_\mathrm{B} T_\mathrm{ph}=\Gamma k_\mathrm{B}\left(\frac{3p}{a_\mathrm{r}}\right)^{1/4}\sim 0.1\hbox{--}0.2\ \mathrm{keV}, \end{equation} \tag{ 5 }$$

in the observer frame. This is consistent with observed values.

The resultant light curves and time-integrated νF_ν spectra of the photospheric emission for both models are presented in Figures 4 and 5. At first, for both models, the photospheric emission is very bright for the first ∼200 s. This is because the cocoon is hot immediately after the emergence from the stellar or wind photosphere. After the early phase, the radial velocity at the photosphere decreases as the photosphere moves inward and the cocoon gradually cools. This corresponds to the decrease of the luminosity. The cocoon cools in different ways for the dense and the dilute CSM. As seen in Figure 3, the cocoon component is shocked in the dense CSM model. The shock converts the kinetic energy of the cocoon into the thermal energy and keeps the shocked region hot. As a result, the photospheric emission remains luminous even at t ∼ 1000 s. In the dilute CSM model, on the other hand, the cocoon adiabatically cools.

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Figure 5 shows the time-integrated νF_ν spectra for the dense (upper panel) and dilute (lower panel) CSM models. In each panel, νF_ν spectra integrated over t = 0–200 s (solid line), t = 200–1500 s (dashed line), and t = 0–1500 s (dotted line) are plotted. If we fit a blackbody spectrum with a single temperature to each of these spectra, the temperature is found to be k_BT = 0.16, 0.077, and 0.13 keV for the 0–200 s, 200–1500 s, and 0–1500 s spectra of the dense CSM model and k_BT = 0.061, 0.038, and 0.051 keV for the dilute CSM model. In high-energy part, however, a deviation from the planck function is prominent. This shows that each spectrum is actually superposition of blackbody spectra with different temperatures.