Gamma-Ray Bursts, part 2: short bursts and soft gamma-ray repeatersWe now move on to the short gamma-ray bursts. For these, the origin is even lesscertain than it is for the long bursts, and that’s saying something! We’ll go over some oftheir properties, then consider possible origins.Short bursts are just as diverse as long bursts, but short bursts are:• Shorter (duh!), with typical durations of a few tenths of a second.• Harder, meaning that a greater fraction of photons at higher gamma-ray energies (sev-eral hundred keV).• Found in or near multiple types of galaxies, i.e., elliptical or spiral. Long bursts areonly associated with actively star-forming galaxies.Naturally, when examined closely, there is enough heterogeneity in the population thatwe can’t be too categorical. Still, Ask class: what are some possible origins?Black hole – neutron star mergersOur first try will be mergers between black holes and neutron stars. The basic ideais that as the neutron star is disrupted, its matter forms something of an accretion disk.There is therefore a naturally clean axis along which one can imagine a jet flowing. Morespecifically, it has been proposed that in this type of merger, high-energy neutrinos andantineutrinos annihilate to form an electron-positron outflow.To get a sense for the time scales involved, let’s calculate the orbital time at the pointwhen the neutron star is disrupted. Recall that Roche lobe overflow occurs when the averagedensity inside the orbit is comparable to the average density of the donor. ThusM/r3≈ mdonor/R3donor. (1)Note, though, that the orbital frequency is ωorb=pGM/r3. As a result, the frequency atwhich mass transfer begins basically depends only on the average density of the donor. Fora neutron star, the characteristic time is thenT ∼ 2π/pGρ ≈ 1 ms (2)for a star of mass 1.5 M¯and radius 10 km (and thus an average density of 7 ×1014g cm−3).Our next calculation is to estimate the maximum mass of a black hole that can disrupta neutron star outside the horizon; obviously a disruption inside the horizon won’t yield anyobservable effect! Suppose we focus on nonrotating black holes. Then the effective averagedensity at the horizon isρeff= M/[(4π/3)(2GM/c2)3] . (3)A quick comment here about strategy for solving such equations. We could set ρeff=7 × 1014g cm−3, then cross-multiply and solve for M. However, it is easier to just pick amass (we’ll choose 1 M¯), solve for the density, then recognize ρ ∝ M−2to solve for themass. Adopting this procedure we find ρeff= 1.8 × 1016(M/M¯)−2g cm−3. This implies thatwe can go to a mass of about 5 M¯, which isn’t much! More careful consideration wouldbring you to about 10 M¯for a nonrotating black hole, and about 30 M¯for a maximallyrotating black hole for which the horizon radius is M instead of 2M .But wait Ask class: have we done this properly? Remember that we need the matterto spiral around for the process we have in mind. If the neutron star is disrupted outside thehorizon, but then plunges straight in, that doesn’t help. As a result, our condition is reallyto match the average density inside the ISCO, not the horizon. For a nonrotating black hole,that’s an extra factor of three in distance, leading to a factor of 4-5 in mass. This alreadytakes us down to about 2 M¯for nonrotating black holes. In addition, even outside theISCO, losses of angular momentum to gravitational radiation can be rapid enough to extendthe effective required radius for disruption a bit more.Adding spin can in principle improve things, because the ISCO moves in for high-spinspacetimes and prograde orbits. However, it isn’t obvious how much this helps, given thatat the relatively comparable masses of interest the neutron star itself contributes to the spinof the spacetime, in the sense of reducing the spin.With all this in mind, I published a paper in 2005 suggesting that BH-NS mergers arenot good candidates for short hard gamma-ray bursts. This is basically because I felt thatthe neutron star would be swallowed whole rather than allowing enough matter to spiralaround to produce the observed behavior. Numerical simulations of these mergers are themost challenging mergers one can do, because they involve horizons as well as complicatedhydrodynamics and an uncertain equation of state. Still, recent work tends to supportthe idea that NS-BH mergers are over and done with without much emission. The mainissue remaining involves very rapidly rotating black holes. That will still require significantdevelopment of numerical techniques.Neutron star – neutron star mergersWhat about two neutron stars? Here you know that there will be a disk of some sort,because there is no horizon (at least at first!). There are, however, two apparent problems.The first has to do with time scales. Recall that the dynamical timescale for NS dis-ruption is about 1 ms. This is still true when the other object is also a NS. By itself, thatdoesn’t mean much; after all, one might have hoped that even if the orbital time is 1 ms,the inspiral time could be much more than that, and hence be comparable to the few tenthsof a second that is observed. However, this isn’t the case. Indeed, for two neutron stars, theinspiral time due to gravitational radiation is about the same as the orbital time. This istoo short by factors of hundreds!The second issue relates to whether the combined object can survive. The lowest massNS ever inferred has M = 1.25 M¯. When two NS come together there will be some releaseof gravitational energy, but this implies that the total mass will be well above 2 M¯, andmore typically above 2.5 M¯. This is above the standard maximum mass for a neutron star.Will the object collapse directly into a black hole?The answer to both could be that when two neutron stars merge, they have a tremendousamount of angular momentum. This spin helps support the merged remnant against gravity,especially if the remnant is rotating differentially (i.e., not as a solid body). The supportisn’t unlimited, of course, but could extend to nearly 3 M¯, which is plenty for the NS pairsthat have thus far been observed. See Figure 1 for a visualization.Why, though, might this help with time scales? The key issue is in how rapidly the starcan either redistribute its angular momentum (if the support relies on differential rotation) orshed it entirely (if the mass is low enough to survive when the star has locked into