= Solution
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:
$$
E_{\rm grav}=-\int_0^{M_{\rm Fe}}\frac{Gm}{r(m)}\,dm
=-\frac35G\left(\frac{4\pi\rho}{3}\right)^{1/3}M_{\rm Fe}^{5/3}
=-\frac{3GM_{\rm Fe}^2}{5R_{\rm Fe}}.
$$
At the supplied density $3\times10^{10}\,\mathrm{kg\,m^{-3}}$, the numerical coefficient is about $6.3\times10^{43}\,\mathrm J$, consistent with the printed rounded estimate:
$$
\boxed{E_{\rm grav}\simeq-6\times10^{43}\left(\frac{M_{\rm Fe}}{M_\odot}\right)^{5/3}\,\mathrm J.}
$$
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 gives
$$
E_{\rm bind,ej}\simeq6\times10^{43}\left[(1.4+d)^{5/3}-1.4^{5/3}\right]\,\mathrm J.
$$
Adding the supplied dissociation cost, $E_{\rm diss}=10^{45}d\,\mathrm J$, gives
$$
\begin{array}{c|rrr}
d&E_{\rm bind,ej}\ (\mathrm J)&E_{\rm diss}\ (\mathrm J)&E_{\rm required}\ (\mathrm J)\\\hline
10^{-3}&1.25\times10^{41}&1.00\times10^{42}&1.13\times10^{42}\\
0.1&1.28\times10^{43}&1.00\times10^{44}&1.13\times10^{44}\\
1&1.53\times10^{44}&1.00\times10^{45}&1.15\times10^{45}
\end{array}
$$
The 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 $v_{\rm kick}\sim\epsilon_{\rm asym}\sqrt{2M_{\rm ej}E_{\rm kin}}/M_c$, 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.
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