Onset of Iron-Core Collapse

An iron core begins dynamical collapse when its pressure response can no longer keep pace with self-gravity. Relativistic electron degeneracy is already close to the marginal adiabatic index $4/3$; electron capture lowers the number of pressure-carrying electrons, while nuclear dissociation and general-relativistic gravity further reduce the margin. The result is not simply “the core exceeds one mass”: instability depends on the evolving entropy, electron fraction, composition, and mass profile.

Helpful background. Stress–Energy and Conservation supplies conserved currents; the Einstein equations supply relativistic gravity; Relativistic Matter and Compact Objects supplies equations of state and stellar support; Black-Hole Dynamics supplies generic collapse geometry; Computational Relativity supplies conservative evolution methods; and Black-Hole Perturbation Theory supplies stability language.

Marginal support in an iron core

For an ultrarelativistic, strongly degenerate electron gas, the electron number density is $n_e=Y_e n_b$, with $Y_e$ the electron fraction and $n_b$ the baryon number density. The leading pressure scales as

$$P_e=K\,(Y_e\rho)^{4/3},$$

where $K$ contains $\hbar$ and the baryon-mass conversion. Hydrostatic scaling $P_c\sim GM^2/R^4$ and $\rho\sim M/R^3$ then eliminates $R$ and gives

$$M_{\rm Ch}\propto \frac{Y_e^2}{G^{3/2}m_b^2}, \qquad M_{\rm Ch}\simeq 5.83\,Y_e^2 M_\odot$$

for the idealized cold, uniform-composition model. Entropy, Coulomb effects, nonuniform composition, general relativity, and the finite iron-core boundary alter the coefficient. The scaling is the important check: reducing $Y_e$ reduces the supported mass quadratically.

Define the Lagrangian pressure response

$$\Gamma_{\rm eff}= \left(\frac{d\ln P}{d\ln\rho}\right)_{\rm evolution}.$$

A Newtonian, nearly homologous sphere becomes dynamically vulnerable when $\Gamma_{\rm eff}<4/3$. Relativistic gravity and nonuniform structure raise the critical value slightly, so equality at $4/3$ is not a universal finite-core threshold. Janka summarizes the collapse sequence and its microphysical drivers in Janka 2012, §II.B and §IV.B.

Electron capture removes pressure and lepton number

During late burning and collapse, captures on nuclei and free protons convert electrons into electron neutrinos,

$$e^-+p\longrightarrow n+\nu_e .$$

Before neutrinos are trapped, they escape and carry lepton number and energy. The falling $Y_e$ reduces electron degeneracy pressure directly. Capture also changes nuclear composition, and the entropy response depends on neutrino escape and nuclear statistical equilibrium. It is therefore misleading to insert a fixed “capture correction” into an otherwise barotropic equation of state: the state needs at least $\rho$, temperature or entropy, and $Y_e$, with weak rates consistent with the nuclear ensemble.

At larger optical depth, neutrino diffusion becomes slower than collapse. Electron-type neutrinos then approach trapping, and the more nearly conserved quantity is total electron-lepton fraction $Y_l=Y_e+Y_{\nu_e}-Y_{\bar\nu_e}$ rather than $Y_e$ alone, with $Y_\nu=n_\nu/n_b$. Electron antineutrinos are scarce in the highly degenerate inner core, giving $Y_l\simeq Y_e+Y_{\nu_e}$. Capture continues toward weak equilibrium but no longer removes lepton number freely from each fluid element. Burrows discusses this transition and the roles of nuclear capture and neutrino opacity in Burrows 2013, §§II.B–II.D.

Dissociation and the effective adiabatic response

As compression heats the core, energetic photons and collisions break heavy nuclei into lighter nuclei and nucleons. Iron-group nuclei have binding energy of order several MeV per baryon, so dissociation absorbs internal energy that would otherwise raise temperature and pressure. In a Lagrangian compression this latent-energy sink lowers the effective pressure response. Dissociation is especially consequential for matter processed by the emerging shock, but the same nuclear-composition physics contributes to softening during infall.

The first-law bookkeeping for a fluid element can be written schematically as

$$T\,ds=d\!\left(\frac{\epsilon}{n_b}\right) +P\,d\!\left(\frac{1}{n_b}\right) -\sum_i\mu_i\,dY_i .$$

Here $\epsilon$ is total energy density and $s$ is entropy per baryon, so every term is an energy per baryon. The composition potentials $\mu_i$ are the conjugates of the independent fractions $Y_i$ in this choice of variables. The composition term shows why $\Gamma_{\rm eff}$ along the evolving trajectory differs from a partial derivative at frozen composition. An equation-of-state table and weak-rate set must use the same nuclear masses and binding-energy zero. Otherwise apparent cooling or heating can be a normalization error. Energy per baryon has dimension of inverse length in $c=\hbar=1$ units; $G$ remains explicit in gravitational estimates.

Homologous inner core and sonic separation

Once instability is dynamical, the characteristic collapse time is

$$t_{\rm dyn}\sim(G\rho)^{-1/2}.$$

The central region remains in approximate sonic communication and collapses nearly homologously, $v(r,t)\propto r$ at fixed time. Farther out, infall becomes supersonic. A sonic point separates the homologous inner core from the outer core, which continues to accrete. The mass inside that point matters at bounce because only the inner core reverses coherently when nuclear matter stiffens.

Useful independent checks follow from this picture. A homologous profile must approach $v=0$ linearly at the center and remain regular. The inferred inner-core mass should decrease when the prescribed deleptonization lowers $Y_e$. Baryon number must close, while changes in electron lepton number must match the emitted or trapped neutrino contribution. In the limit of frozen composition and negligible relativistic correction, the calculation should recover the $4/3$ marginal response and Chandrasekhar scaling.

This page stops at the infalling configuration. Nuclear stiffening, rebound, shock formation, and subsequent multidimensional explosion dynamics require time-dependent hydrodynamics and transport and are developed on the following pages.

Check the electron-fraction scaling

An idealized core deleptonizes from $Y_e=0.50$ to $Y_e=0.42$ while the other assumptions behind the Chandrasekhar scaling are held fixed. By what factor does the supported mass change?

Solution

Because $M_{\rm Ch}\propto Y_e^2$,

$$\frac{M_{\rm Ch}(0.42)}{M_{\rm Ch}(0.50)} =\left(\frac{0.42}{0.50}\right)^2 =0.7056 .$$

The idealized supported mass falls to about $70.6\%$ of its original value, a $29.4\%$ reduction. This calculation isolates the electron-fraction effect; a real core also changes entropy, composition, density profile, and relativistic compactness, so the number is a scaling diagnostic rather than a collapse prediction.

References

• Burrows, Adam. “Colloquium: Perspectives on Core-Collapse Supernova Theory.” Reviews of Modern Physics 85 (2013): 245–261. DOI.
• Janka, Hans-Thomas. “Explosion Mechanisms of Core-Collapse Supernovae.” Annual Review of Nuclear and Particle Science 62 (2012): 407–451. arXiv.