Past exam of the astrophysics course of the University of Oxford / 2016 / 8

A core-collapse supernova occurs when the core of a massive evolved star loses stable pressure support and collapses. Iron-group nuclei no longer provide an exothermic fusion route, while electron capture and photodisintegration reduce electron pressure and consume energy. Collapse accelerates until nuclear-density matter stiffens the inner core and launches a shock. That shock loses energy to dissociation and neutrinos and can stall. Neutrino heating, aided by multidimensional flow, can revive it; sufficiently rapid rotation and magnetic stresses provide other possible channels. Most of the liberated compact-object binding energy escapes as neutrinos rather than optical light or ejecta kinetic energy.

Successful explosions can leave neutron stars, with additional fallback sometimes producing black holes. Failed explosions or very substantial fallback can form black holes more directly. Important theoretical uncertainties include progenitor structure and mass loss, nuclear equation of state, neutrino transport and interaction rates, convection and shock instabilities, rotation, magnetic fields and the relation between these effects and explosion asymmetry. A successful shock launch alone does not guarantee unbinding all overlying layers.

For an initially uniform-density iron core, $r(m)=[3m/(4\pi \rho ){]}^{1/3}$. Assemble spherical shells from infinity:At the supplied density $3\times 10^{10}\mathrm{kg}{\mathrm{m}}^{-3}$, the numerical coefficient is about $6.3\times 10^{43}\mathrm{J}$, consistent with the printed rounded estimate:The negative sign represents the work recovered in assembling the bound core.

The ejecta binding energy of an initially uniform core is the negative gravitational energy of the outer shells in the original core, including their interaction with the material inside them. It is not the final neutron star's entire binding energy. With $m_c=M_c/M_{\odot }=1.4$ and $d=\Delta M/M_{\odot }$, the shell integral givesAdding the supplied dissociation cost, $E_{\mathrm{diss}}=10^{45}d\mathrm{J}$, givesThe calculation assumes the initial common density for the shell binding, all the indicated outer iron encounters the dissociating shock, and dissociation is charged as the stated energy cost. It does not subtract the much larger final compact-object binding energy as though all that energy were directly available to the ejecta. Later recombination can return some nuclear binding energy, and overlying noniron layers add their own energy requirements, so the table is a simplified shock-energy budget.

Compared with a characteristic $10^{44}\mathrm{J}$ explosion, ejecting a very thin remaining core layer is relatively easy in this estimate; $0.1M_{\odot }$ is already demanding, and $1M_{\odot }$ requires substantially more energy. More massive tightly bound cores therefore tend to make shock revival and complete ejection harder, increasing the possibility of fallback or failed explosions and black-hole remnants. A thin steep-gradient outer layer may allow a rapid, relatively weak explosion and a surviving neutron star.

A natal kick requires asymmetry of ejecta, neutrino emission or subsequent gravitational interaction, not merely a large scalar energy budget. Rapid relatively symmetric low-ejecta explosions can produce small kicks; a shock that develops strong asymmetries can give larger neutron-star kicks, while direct collapse need not do so. A schematic momentum estimate is $v_{\mathrm{kick}}\sim {ϵ}_{\mathrm{asym}}\sqrt{2M_{\mathrm{ej}}{E}_{\mathrm{kin}}}/M_c$, but neither the asymmetry nor its direction is fixed by the three core masses. The source data support tendencies, not a unique mapping from iron-core mass to remnant kick.