Proc. Neutrino Astrophysics - MPP Theory Group

98
of the engine itself). A longer time scale would characterize the collapsars where the disk is
continually fed by stellar collapse. Perhaps the long complex bursts (20 s mean) are due to
collapsars and the short bursts are merging compact objects.
For an accretion rate of 0.1 M ⊙ s −1 , the energy dissipated, mostly in neutrinos, is 1.8×10 53
F erg/s where F is 0.06 for Schwarzschild geometry (perhaps appropriate for merging compact
objects) and 0.4 for extreme Kerr geometry (perhaps appropriate for collapsars). Coupled
with their short time scale this implies that merging compact objects may give bursts of much
less total energy than collapsars, but the efficiency for getting the jet out of a collapsar may
modulate this considerably (see below). In all models, a portion of the energy goes into MHD
instabilities in the disk that may result in an energetic wind, or perhaps the rotational energy
of the black hole is tapped as in some models for AGN. But the simplest physical solution for
ejecting relativistic matter, if it works, is neutrino annihilation along the rotational axis of the
hole (Woosley 1993). The efficiency for this is uncertain, but numerical simulations (see the
paper by Ruffert, these proceedings) suggest an efficiency for converting neutrino luminosity
into pair fireball energy of ∼1%. Thus one has a total energy for black holes merging with
neutron stars of order 10 50 erg (Ruffert et al. 1997) and a total energy for collapsars that
might approach 10 52 erg.
The chief uncertainty in all these models—aside from exactly how the relativistic matter
reconverts its streaming energy into gamma-rays—is the baryon loading. One expects this
to be relatively small in neutron star and black hole mergers, but still perhaps more than
10 −5 M ⊙, depending upon the mingling of the wind from the disk, residual matter from the
merger, and the relativistic outflow. Detailed calculations remain to be done.
The problem is much more severe in the collapsar model where the jet that is formed has
to penetrate the overlying star—or what is left of it—a little like making the γ-ray burst at
the center of the sun. Still 10 52 erg is a lot of energy and we also expect that the jet from
collapsar models might be tightly beamed. This is initially the case because the disk makes a
transition from very thick (geometrically) to thin at the point where neutrino losses start to
occur on an accretion time scale. This happens at a definite radius, ∼100 km, because of the
T 9 dependence of the neutrino rates (Bob Popham, private communication). Calculations in
progress by Müller and colleagues at the MPA will show how effective this jet is at tunneling
thru the star. Our own recent 2D simulations suggest a solid angle of only about 1% for
the jet. A lot of energy gets used up ejecting all the mass above about 45 degrees from the
equator. A supernova is one consequence; a collapsar is not a “failed supernova” after all, but
this mass ejection at large angles is non-relativistic. Along the axis the jet clears a path for
more energetic matter to follow, and since the burst continues for many jet-stellar crossing
times, Γ rises. Only ∼0.01 M ⊙ initially lies in the path of the jet for the expected small
opening angle.
The energy of the ejecta and presumably the hardness of the transient that is seen thereafter
thus depends upon viewing angle. Along the axis, high Γ will overtake low Γ material
before the observable event actually commences. The start of the burst occurs when the jet
sweeps up 1/Γ times its rest mass, or about 10 −7 M ⊙. This happens after the jet has gone
about 10 15 cm and interacted with the stellar wind ejected prior to the explosion (about
10 −5 M ⊙ y −1 assumed).
In summary we expect about 10 50 erg, ∼ < 0.1% of the total energy dissipated in the disk,
to go into a jet with Γ ∼ > 100 and an opening angle of about 10 degrees. The event rate could
be 10 −4 to 10 −3 y −1 per bright galaxy.
Below we show pictures of our first 2D calculation of rotating stellar collapse about 10 s