3 Jetted GRBs from NS collapse

Relativistic jets seem to be emitted when mass is accreted at a high rate onto black holes and neutron stars. They have been resolved by VLA radio observations (Rodriguez & Mirabel 1998) into highly magnetized clouds of plasma (plasmoids) that are emitted in injection episodes which are correlated with sudden removal of the accretion disk material. After initial expansion these plasmoids seem to retain a constant radius ($R_{\rm p}\sim 2\ 10^{15}~{\rm cm}$) until they slow down and spread. Their formation is not well understood yet. But, it seems very likely that highly relativistic jets are also ejected in NS collapse because the accretion rates and the magnetic fields in NS collapse are much larger. If momentum imbalance in the ejection of two opposite relativistic jets is responsible for the large mean velocity ($V_{\rm NS}\approx 450\pm 90~{\rm km~s}^{-1}$; Lyne & Lorimer 1994) of slowly spinning pulsars (presumably QSs), then momentum conservation implies that the kinetic energy of the jets satisfies

$$E_{\rm jet} \geq cP_{\rm NS}\sim cM_{\rm NS}V_{\rm NS}\sim 4\ 10^{51}~{\rm erg}.$$ (1)

Thus, the total jet energy may be $E_{\rm jet}\sim 10^{52}~{\rm erg}$. Such highly relativistic jets/plasmoids are strong emitters of beamed $\gamma$-rays through synchrotron emission, inverse Compton scattering and resonance scattering of interstellar light. When they point in our direction in external galaxies, they produce the observed GRBs and their afterglows (e.g. Shaviv & Dar 1995; Dar 1998a). If the true rate of GRBs is comparable to the birth rate of NSs, $\dot N_{\rm NS}\simeq 2\ 10^{-2}~{\rm y}^{-1}$ in galaxies like our own (van den Bergh & Tamman 1991), then the inferred rate of observable GRBs in galaxies like our own, $\dot N_{\rm GRB}\sim 10^{-8}~{\rm y}^{-1}$ (Wijers et al. 1997), implies that the GRBs must be narrowly beamed into a solid angle

$$\Delta\Omega\simeq 2\pi \dot N_{\rm GRB}/\dot N_{\rm NS}\simeq \pi\ 10^{-6}.$$ (2)

Emission from narrow jets with bulk motion Lorentz factor $\Gamma=10^3$ is beamed into $\Delta\Omega\sim \pi/\Gamma^2\simeq\pi\ 10^{-6}$. Such strong beaming implies that we observe only a very small fraction of the events that produce GRBs.

If the highly relativistic plasmoid consists of a pure ${\rm e^+e^-}$plasma, then inverse Compton scattering of stellar light ($h\nu=\epsilon_{\rm ev}\times 1~{\rm eV}$) by the plasmoid can explain the observed typical $\gamma$ energy ($\epsilon_{\gamma}\sim 4\Gamma_3^2 \epsilon_{\rm eV} /3(1+z)~{\rm MeV}$), GRB duration ($T\sim R_{\rm SFR}/ 2c\Gamma^2\sim 50~{\rm s}$), pulse duration ($t_{\rm p}\sim R_{\rm p}/ 2c\Gamma^2\sim 150~{\rm ms}$), fluence $(F_\gamma\sim 10^{-5}~{\rm erg~cm}^{-2})$, light curve and spectral evolution of GRBs (Shaviv & Dar 1995; Shaviv 1996; Dar 1998). For instance,

$$F_\gamma\simeq \frac{\sigma_{\rm T} N\epsilon_\gamma}{\Gamma... ..._3\epsilon_{\rm ev}E_{52}}{D_{29}^2} \frac{\rm erg}{{\rm cm}^2}$$ (3)

where $D=D_{29}\ 10^{29}~{\rm cm}$ is the luminosity distance of the GRB at redshift z, z₂=(1+z)/2, $N=N_{22}\ 10^{22}~{\rm cm}^{-2}$ is the column density of photons along the jet trajectory in the star burst region, $\sigma_{\rm T}=0.65\ 10^{-24}~{\rm cm}^2$ is the Thomson cross section, $E_{\rm jet}=E_{52}\ 10^{52}~{\rm erg}$ and $\Gamma=\Gamma_3\ 10^3$.

If the plasmoid consists of normal NS crustal material, then photoabsorption of stellar light by partially ionized iron and its reemission as $\gamma$ rays yield $\epsilon_\gamma\sim \Gamma\epsilon_{\rm x}/(1+z)\sim~{\rm MeV}$ in the observer frame (Shaviv 1996) and

$$F_\gamma \simeq \frac{\sigma_{\rm a} N\epsilon_\gamma}{\Gamm... ...n}_{\rm x}\Gamma_3E_{52}} {D_{29}^2} \frac{\rm erg}{{\rm cm}^2}$$ (4)

where $\sigma_{\rm a}= \sigma_{19}\ 10^{-19}~{\rm cm}^2$ is the mean photoabsorption cross section of X-rays by partially ionized iron.