The Death of Giants

Section 3 of 6

Core Collapse

Part 3: Core Collapse

The Collapse Sequence

Six-panel sequence showing core-collapse supernova stages. Panel a: onion shell structure. Panel b: inward-pointing arrows showing collapse. Panel c: continued collapse with proto-neutron star forming. Panel d: red circle showing stalled shock with outward and inward arrows. Panel e: neutrino-heated region reviving the shock. Panel f: expanding red circle showing the explosion propagating outward. — Figure 8The six stages of a core-collapse supernova, from iron core to explosion. (a) Onion-shell structure with iron core. (b) Core collapses as iron photodisintegrates. (c) Inner core reaches nuclear density and bounces. (d) The bounce launches a shock wave that stalls. (e) Neutrino heating revives the shock. (f) The shock breaks through and ejects the envelope. Stages (b) to (f) take less than one second.

The collapse happens on a dynamical timescale, so before following the sequence, estimate the timescale directly using the dynamical-timescale estimate from Reading 1:

With $\bar{\rho}\sim10^{10}\ \mathrm{g\,cm^{-3}}$ for the collapsing iron core, $[G\bar{\rho}] = (\mathrm{cm^3\,g^{-1}\,s^{-2}})(\mathrm{g\,cm^{-3}}) = \mathrm{s^{-2}}$ , so $1/\sqrt{G\bar{\rho}}$ has units of seconds.

Evaluating the scale,

\tau_\text{dyn} \sim \frac{1}{\sqrt{(6.67\times10^{-8})(10^{10})}} = \frac{1}{\sqrt{6.67\times10^2}}~\text{s} \approx 0.04\ \mathrm{s}.

So once pressure support fails, collapse on a timescale of $\sim 0.04\ \mathrm{s}$ is completely reasonable. The causal sequence is then:

Electron capture ( $\sim10\ \text{ms}$ ). At densities above $\sim10^{10}\ \mathrm{g\,cm^{-3}}$ , electrons are captured by protons, $p + e^- \rightarrow n + \nu_e$ . This removes electrons that were providing degeneracy pressure and emits neutrinos — both push the core toward faster collapse.
Photodisintegration of iron ( $\sim100\ \text{ms}$ ). Above $\sim 5\times10^9\ \mathrm{K}$ , energetic photons break iron-group nuclei apart, ${}^{56}\mathrm{Fe} + \gamma \rightarrow 13\,{}^{4}\mathrm{He} + 4n$ . This photodisintegration is endothermic — it absorbs energy — so thermal pressure drops further.
Free-fall collapse ( $\sim100\ \text{ms}$ ). With both degeneracy and thermal support reduced, the inner core collapses rapidly, reaching infall speeds of order $v\sim 0.2\,c \approx 6\times10^4\ \mathrm{km\,s^{-1}}$ (free-fall onto a $1.4\,M_\odot$ core, $v\sim\sqrt{2GM/R}$ , climbs from $\sim 0.06\,c$ at $1{,}000$ km to $\sim 0.2\,c$ near $\sim 100$ km, approaching the bounce).
Core bounce ( $t=0$ ). When the inner core reaches nuclear density $\rho_\text{nuc}\sim 2.8\times10^{14}\ \mathrm{g\,cm^{-3}}$ , the equation of state stiffens sharply, neutron degeneracy pressure becomes important, and the inner core halts and rebounds, launching a shock into the still-infalling outer core.
Shock stall and neutrino heating ( $\sim100$ – $500\ \text{ms}$ ). The outgoing shock loses energy dissociating infalling nuclei and stalls. Neutrinos streaming out of the hot proto-neutron star deposit a small fraction of their energy behind the shock, which can help revive it.
Explosion ( $\sim1\ \text{s}$ ). The revived shock propagates outward, ejecting much of the star’s outer layers at $10^4$ – $3\times10^4\ \mathrm{km\,s^{-1}}$ .

Photodisintegration

The breakup of nuclei by energetic thermal photons — e.g. ${}^{56}\mathrm{Fe} + \gamma \rightarrow 13\,{}^{4}\mathrm{He} + 4n$ above $\sim 5\times10^9~\text{K}$ . It is endothermic, so it drains thermal energy from the collapsing iron core and accelerates the collapse.

The Energy Budget

The gravitational energy released by core collapse is enormous — and, as we will see, most of it never appears as light.

Worked Example 1The Energy of Core Collapse

Problem

The iron core collapses from $R_i \sim 1{,}000~\text{km}$ to $R_f \sim 10~\text{km}$ , releasing $\Delta E_\text{grav}\sim GM^2(1/R_f - 1/R_i)$ . Show the final radius dominates, estimate the energy for a $1.4\,M_\odot$ core, and compare it to a solar rest energy.

StepThe small final radius dominates

$\frac{1}{R_f}-\frac{1}{R_i} = \frac{1}{10^6}-\frac{1}{10^8} = 10^{-6}(1-0.01)~\text{cm}^{-1}\approx 10^{-6}~\text{cm}^{-1},$ so the $1/R_i$ term contributes only $1\%$ — the final radius sets the scale.

StepEvaluate (M = 1.4 Msun = 2.8e33 g, R_f = 10 km = 1e6 cm)

$\Delta E_\text{grav}\sim\frac{GM^2}{R_f}\sim\frac{(6.67\times10^{-8})(2.8\times10^{33})^2}{10^6}\approx 5\times10^{53}~\text{erg}.$

Dimensional check

$\mathrm{cm^3\,g^{-1}\,s^{-2}}\cdot\mathrm{g^2}/\mathrm{cm} = \mathrm{g\,cm^2\,s^{-2}} = \mathrm{erg}$ ✓.

Result

With a solar rest energy $M_\odot c^2 = (2.0\times10^{33})(3\times10^{10})^2 = 1.8\times10^{54}~\text{erg}$ , $\Delta E_\text{grav}\approx\frac{5\times10^{53}}{1.8\times10^{54}}\,M_\odot c^2 \approx 0.28\,M_\odot c^2$ — about a fifth of the $1.4\,M_\odot$ core’s own rest energy, released in under a second. (General relativity softens this Newtonian estimate; the realistic neutron-star binding energy is $\sim 0.1$ – $0.15\,M_\odot c^2$ , still staggering.) For comparison, the Sun radiates only $\sim 10^{51}~\text{erg}$ over its entire $10~\text{Gyr}$ life — one collapse releases hundreds of solar lifetimes of energy in a heartbeat.

Because this is an order-of-magnitude estimate, it is standard to summarize the release as a few $\times10^{53}\ \mathrm{erg}$ . Where does it go? A representative energy budget is:

Channel	Typical energy	Approximate fraction	Timescale
Neutrinos	$\sim 3\times10^{53}\ \mathrm{erg}$	$\sim 99.7\%$	$\sim 10\ \mathrm{s}$
Kinetic energy of ejecta	$\sim 10^{51}\ \mathrm{erg}$	$\sim 0.3\%$	days–months
Photons (light)	$\sim 10^{49}\ \mathrm{erg}$	$\sim 0.003\%$	weeks–months

So the most luminous optical event in the universe is not where most of the energy goes. The optical display is spectacular because photons are easy to detect, not because they carry the dominant share of the energy. The visible light curve is powered largely by radioactive decay, especially the chain ${}^{56}\mathrm{Ni}\rightarrow{}^{56}\mathrm{Co}\rightarrow{}^{56}\mathrm{Fe}$ .

Quick check

A supernova can briefly outshine its host galaxy in visible light. Does that imply visible light carries most of the explosion energy? Use the table above to justify your answer quantitatively.

The More You Know: Enrichment: SN 1987A — The Neutrino Confirmation

On February 23, 1987, a supernova was detected in the Large Magellanic Cloud ( $d \approx 50~\text{kpc}$ ) — the nearest naked-eye supernova since Kepler’s of 1604. About three hours before the optical brightening, Kamiokande II in Japan and IMB in Ohio detected a burst of about 20 neutrinos within a $\sim 13~\text{s}$ window (about 24 including Baksan).

Figure 9Supernova 1987A, before and after. Left: the bright supernova in the Large Magellanic Cloud, naked-eye visible in February 1987. Right: the same field beforehand, with the blue supergiant progenitor (Sanduleak -69 202) arrowed. SN 1987A was the closest supernova since Kepler's of 1604 and the first from which neutrinos were detected, confirming the core-collapse mechanism.Anglo-Australian Observatory

Twenty neutrinos sounds modest, but the numbers are stunning: the total neutrino energy was $\sim 3 \times 10^{53}~\text{erg}$ , emitted as $\sim 10^{58}$ neutrinos, passing through Earth at $\sim 10^{10}~\text{neutrinos}/\text{cm}^2$ . Neutrinos interact so weakly that the $\sim 3{,}000$ -tonne water detectors caught only a tiny fraction. Those detections confirmed the core-collapse theory: the energy scale, timing, and burst duration all matched the prediction that most of the collapse energy escapes as neutrinos before the optical peak.