1 Beamed gamma-ray burst afterglow models

In the Rome meeting I presented a derivation of the dynamical behavior of a beamed gamma ray burst (GRB) remnant and its consequences for the afterglow light curve. (Cf. Rhoads 1999 [Paper I]). Here, I summarize these results and apply them to test the range of beaming angles permitted by the optical light curve of GRB 970508.

Suppose that ejecta from a GRB are emitted with initial Lorentz factor $\Gamma_0$ into a cone of opening half-angle $\zeta_{\rm m}$ and expand into an ambient medium of uniform mass density $\rho$ with negligible radiative energy losses. Let the initial kinetic energy and rest mass of the ejecta be E₀ and M₀, and the swept-up mass and internal energy of the expanding blast wave be $M_{\rm acc}$ and $E_{\rm int}$. Then energy conservation implies $\Gamma E_{\rm int}\approx \Gamma^2 M_{\rm acc}c^2 \approx E_0 \approx \hbox{constant}$ so long as .

The swept-up mass is determined by the working surface area: ${\rm d} M_{\rm acc}/ {\rm d}r \approx \pi (\zeta_{\rm m}r + c_{\rm s} t_{\rm co})^2$, where $c_{\rm s}$ and $t_{\rm co}$ are the sound speed and time since the burst in the frame of the blast wave + accreted material. Once , and the dynamical evolution with radius r changes from $\Gamma \propto r^{-3/2}$ to $\Gamma \propto \exp(-r/r_{_\Gamma})$(Rhoads 1998, & Paper I). The relation between observer frame time $t_{\oplus}$ and radius r also changes, from $t_{\oplus}\propto r^{1/4}$ to $t_{\oplus} \propto \exp(r/[2 r_{_\Gamma}])$. Thus, at early times $\Gamma \propto t_{\oplus}^{-3/8}$, while at late times $\Gamma \propto t_{\oplus}^{-1/2}$. The characteristic length scale is $r_{_\Gamma}= \left(E_0 / \pi c_{\rm s}^2 \rho \right)^{1/3}$, and the characteristic observed transition time between the two regimes is $t_{\oplus,b}\approx 1.125\ (1+z) \left( E_0 c^3 / [\rho c_{\rm s}^8 \zeta_{\rm m}^2] \right)^{1/3} \zeta_{\rm m}^{8/3}$, where z is the burst's redshift.

We assume that swept-up electrons are injected with a power law energy distribution $N({\cal E}) \propto {\cal E}^{-p}$ for ${\cal E}= \gamma_{\rm e} m_{\rm e} c^2 \gt {\cal E}_{\rm min}\approx \xi_{\rm e} m_{\rm p} c^2 \Gamma$, with p > 2, and contain a fraction $\xi_e$ of $E_{\rm int}$. This power law extends up to the cooling break, ${\cal E}_{\rm cool}$, at which energy the cooling time is comparable to the dynamical expansion time of the remnant. Above ${\cal E}_{\rm cool}$, the balance between electron injection (with $N_{\rm inj} \propto {\cal E}^{-p}$) and cooling gives $N({\cal E}) \propto {\cal E}^{-(p+1)}$.

We also assume a tangled magnetic field containing a fraction $\xi_{_B}$of $E_{\rm int}$. The comoving volume $V_{\rm co}$ and burster-frame volume V are related by $V_{\rm co}\approx V/\Gamma \propto M_{\rm acc}/ \Gamma$, so that $B^2 = {8 \pi \xi_{_B}E_{\rm int}/ V_{\rm co}} \propto \Gamma^2$ and $B \propto \Gamma$.

The resulting spectrum has peak flux density $F_{\nu, \oplus, m}\propto \Gamma B M_{\rm acc}/ \max(\zeta_{\rm m}^2, \Gamma^{-2})$ at an observed frequency $\nu_{\oplus, {\rm m}}\propto \Gamma B {\cal E}_{\rm min}^2 / (1+z) \propto \Gamma^4 / (1+z)$. Additional spectral features occur at the frequencies of optically thick synchrotron self absorption (which we shall neglect) and the cooling frequency $\nu_{\oplus,{\rm cool}}$ (which is important for optical observations of GRB 970508). The cooling break frequency follows from the relations $\gamma_{\rm cool}\approx (6 \pi m_{\rm e} c)/ ( \sigma_{\rm T} \Gamma B^2 t_{\oplus})$ (Sari et al. 1998; Wijers & Galama 1998) and $\nu_{\oplus,{\rm cool}}\propto \Gamma B \gamma_{\rm cool}^2 \propto (\Gamma^4 t_{\oplus}^2)^{-1}$. In the power law regime, $F_{\nu, \oplus, m}\propto t_{\oplus}^{0}$, $\nu_{\oplus, {\rm m}}\propto t_{\oplus}^{-3/2}$, and $\nu_{\oplus,{\rm cool}}\propto t_{\oplus}^{-1/2}$; while in the exponential regime, $F_{\nu, \oplus, m}\propto t_{\oplus}^{-1}$,$\nu_{\oplus, {\rm m}}\propto t_{\oplus}^{-2}$, and $\nu_{\oplus,{\rm cool}}\propto t_{\oplus}^0$. The spectrum is approximated by a broken power law, $F_\nu \propto \nu^{-\beta}$,with $\beta \approx -1/3$ for $\nu<\nu_{\oplus, {\rm m}}$, $\beta \approx (p-1)/2$ for $\nu_{\oplus, {\rm m}}< \nu < \nu_{\oplus,{\rm cool}}$, and $\beta \approx p/2$ for $\nu \gt \nu_{\oplus,{\rm cool}}$.

The afterglow light curve follows from the spectral shape and the time behavior of the break frequencies. Asymptotic slopes are given in Table 1. For the $\Gamma \sim 1/\zeta_{\rm m}$ regime, we study the evolution of break frequencies numerically. The results for $\nu_{\oplus, {\rm m}}$ and $F_{\nu, \oplus, m}$ are given in Paper I. For $\nu_{\oplus,{\rm cool}}$, a good approximation is

  

Table 1: Light curve exponents $\alpha$ as a function of frequency and time. Here $F_{\nu, \oplus}\propto t_{\oplus}^\alpha$