3 Collapsars

 

Now we consider briefly the collapsar model (Woosley 1993). Because a detailed description of this model has already been accepted for publication (MacFadyen & Woosley 1998), we will be brief.

A rotating massive star reaches the end of its life. Because the iron core is considerably more massive than $2\,M\hbox{$\odot$}$, it is already not far from forming a black hole. Moreover, the density gradient outside that core is shallow, so accretion occurs at a high rate. The core grows and eventually collapses to a black hole which attempts to accrete the rest of the star. However, after about 2 - 3 s, for $j = 5 - 15 \ 10^{16}$ cm² s^-1, a centrifugally supported disk forms. Matter falls into the hole along the rotational axis opening up a funnel like density structure (Fig. 1) while matter in the disk accretes more slowly at about $0.1 \,M\hbox{$\odot$}\ {\rm s}^{-1}$. Depending upon viscosity and Kerr parameter, (values $\alpha = 0.1$, a = 0.9 here for example) temperatures in the inner disk reach about 10¹¹ K at a density $\sim\! 10^{10}$ g cm^-3 and radius 10 km.

Over the next 10 s to 20 s, the black hole grows from $\sim\! 3\, M\hbox{$\odot$}$ to $5 \,M\hbox{$\odot$}$. For the chosen $\alpha$ and calculated accretion rate, most of the energy dissipated in the disk is radiated as neutrinos, $\sim\! 1 - 5 \ 10^{53}$ erg. A fraction of the neutrinos emitted by the disk encounter their anti-particles coming from the other side of the disk. The large collision angle favors neutrino annihilation and pair production. The electrons and positrons so produced retain the net momentum of the collision which has a component directed outwards along the rotational axis. The geometry of the funnel shaped density profile along the axis, the momentum of the pairs, and the large energy deposition all favor the formation of axial jets, but only after the ram pressure of the infalling matter can be reversed. This takes another few seconds (when the density declines to 10⁵ - 10⁶ g cm^-3 at 50 km), The efficiency for neutrino annihilation is not large, typically 1%, so the total energy deposited is in the range $3 -20 \ 10^{51}$ erg. These numbers are calculated using the disk solutions of Popham et al. (1998) mated smoothly to the 2D hydro calculations of MacFadyen & Woosley (1998) as described in the latter publication. Neutrino transport is calculated assuming that the disk remains optically thin to neutrinos. This is marginally violated in the inner disk for the largest values of Kerr parameter developed in the problem, hence the uncertainty in the above numbers. More accurate values can be determined in the future.

  

Figure 2: Depositing 1051 erg s-1 along the rotational axis above and below the black hole leads to jet formation, seen here 824 ms after initiation. The ratio of energy to baryonic rest mass exceeds unity. Were the matter to expand freely it would reach approximately the speed of light

Whether the energy actually comes predominantly from neutrino annihilation or from MHD processes remains unclear, but from our discussion so far, it makes sense to deposit about 10⁵¹ erg s^-1 near the inner boundary of the rotational axis of the collapsar (50 km in MacFadyen & Woosley) starting about 5 - 7 s after the iron core collapse and observe the consequences. This was done by MacFadyen & Woosley and the outcome was bipolar jets with an opening angle of degrees (Fig. 2). In the non-relativistic code they employed, it was hard to determine the final velocity of the jets (speeds considerably greater than c developed!), but it is clear that the flow will ultimately become relativistic. These calculations have been repeated recently using a special relativistic version of the 2D hydro-code "Prometheus" by Aloy et al. (1999) and velocities approaching c are indeed observed along with very high energy loading factors (up to 100 and more times the baryonic mass in the jet).

One can thus separate the collapsar phenomenon into several stages:

1.

the iron core collapses and the disk forms (2 s),

2.

accretion along the polar column declines to the point where a jet can form and move outwards (2-5 more s),

3.

the jet bores out of the star (radius about $3 \ 10^{10}$ cm; about 5 more s),

4.

the first interaction of the ejecta with the pre-explosive mass loss of the star makes the first hard X-rays; and finally,

5.

the jet continues for about 10 more s fully evacuating the polar column and reaching its asymptotic relativistic speed. Internal shocks in the jet and circumstellar interaction produce the main GRB.

What is seen will also vary with angle. Only at small angles, 10 degrees, will high $\Gamma$ matter produce a bright GRB. Mildly relativistic matter will be ejected out to a larger angle (subsequent calculations after the meeting show this), approximately 30 degrees. This matter will make a weaker GRB, but actually contains a large fraction of the explosion energy. It will contribute to the afterglow in radio, optical, and X-ray. The jet eventually ends up blowing up the entire star, leaving behind a black hole of .

The total energy in the relativistic jets is (at most) about 10⁵² erg in the neutrino powered model, perhaps an order of magnitude more if MHD processes are efficient. The beaming is about 1%. So inferred isotropic energies in bursts of up to 10⁵⁴ (neutrino) or 10⁵⁵ (MHD) erg can be accommodated (modulo the uncertain efficiency factor for turning jet energy into gamma-rays).

There are several other interesting properties of the collapsar model. First, it will occur in star forming regions in stars that have lost their hydrogen envelope. The jet always blows up the star, so that a supernova of some sort should accompany the GRB is unavoidable. In order to obtain the large helium cores and rotation rates that the collapsar model needs, it helps to have less mass loss. So collapsars may preferentially occur in low metallicity star forming regions.

Second, MacFadyen & Woosley note that the accretion rate into the black hole is not constant in time. Photodisintegration instability may modulate the accretion on a time scale $\sim\! 50$ ms. In the neutrino based collapsar model, and probably also in the MHD one, this rapid variation in accretion rate will give a highly variable $\Gamma$in the jet, conducive to the formation of internal shocks.

Finally, the wind of a Wolf-Rayet star, even a low metallicity one, is appreciable. The GRB is thus likely to be surrounded by an extended region (perhaps also asymmetrical) of high density. For a mass loss rate of $10^{-5}\, M\hbox{$\odot$}$ y^-1 and velocity 1000 km s^-1, a jet of $10^{-4} f_{\Omega} \,M\hbox{$\odot$}$ will encounter $1/\Gamma$ of its mass (for $\Gamma = 100$) in about 20 AU. Variations in mass loss and $\Gamma$ make this an approximate number. Some bursts may be made by internal shocks; some by external shocks as they run into this wind.