Neutrino Heating and Transport

After the prompt shock stalls, electron-flavor neutrinos streaming from the proto-neutron star can deposit energy behind it. Shock revival becomes more favorable when accreted matter remains in the gain region long enough to absorb an appreciable fraction of its binding energy. The useful comparison is an advection time against a heating time, but both times inherit the neutrino transport, opacity, hydrodynamic, and geometric assumptions of the calculation.

Helpful background. Stress–Energy and Conservation supplies energy and lepton balances; the Einstein equations supply relativistic gravity; Relativistic Matter and Compact Objects supplies radiation moments and dense-matter opacities; Black-Hole Dynamics supplies collapse geometry; Computational Relativity supplies conservative multiphysics evolution; and Black-Hole Perturbation Theory supplies instability diagnostics.

Energy-dependent neutrino transport

For each neutrino species, the continuum input is a distribution function $f_\nu(x^\mu,p^{\hat\alpha})$ obeying a general-relativistic Boltzmann equation,

$$p^\mu\frac{\partial f_\nu}{\partial x^\mu} -\Gamma^{\hat\imath}_{\hat\alpha\hat\beta} p^{\hat\alpha}p^{\hat\beta} \frac{\partial f_\nu}{\partial p^{\hat\imath}} ={\cal C}_\nu[f],$$

written here schematically in an orthonormal momentum frame. The collision operator includes emission, absorption, and scattering with rates depending on density, temperature, composition, neutrino energy, and angle. Its moments exchange four-momentum and lepton number with the fluid. A calculation must declare the species, energy groups, interaction set, equation-of-state version, gravity model, transport closure, velocity and redshift terms, initial progenitor, and inner and outer boundary conditions.

The “neutrinosphere” is not one material surface. Optical depth depends on species, energy, and interaction, and the decoupling of number, energy, and angle can occur at different radii. Deep inside, diffusion and detailed balance dominate; far outside, free streaming requires flux factors approaching their causal limit. Mezzacappa and collaborators develop the Boltzmann problem, moment equations, closures, and numerical challenges in Mezzacappa et al. 2020, §§4.3, 4.7, and 5.2–5.3.

Heating, cooling, and the gain radius

In the region behind the shock, charged-current absorption

$$\nu_e+n\longrightarrow p+e^-, \qquad \bar\nu_e+p\longrightarrow n+e^+$$

deposits energy. A scaling for the specific heating rate is

$$\dot q_+\simeq\frac{\sigma_{\rm ref}}{4\pi r^2m_b E_{\rm ref}^2} \left[ Y_n L_{\nu_e}\langle E^2\rangle_{L,\nu_e} +Y_p L_{\bar\nu_e}\langle E^2\rangle_{L,\bar\nu_e} \right],$$

in an approximately radial, free-streaming, optically thin field. Here $\sigma(E)\simeq\sigma_{\rm ref}(E/E_{\rm ref})^2$ defines a reference cross section and energy, while $Y_n,Y_p$ count free target nucleons per baryon. The spectral moment is energy-luminosity weighted:

$$\langle E^2\rangle_L =\frac{\int E^2 L_E\,dE}{\int L_E\,dE}, \qquad L_\nu=\int L_E\,dE.$$

It is not the square of the number-weighted mean energy. The estimate follows by multiplying spectral number flux $L_E/(4\pi r^2E)$ by deposited energy $E$, absorption cross section, and targets per mass $Y_{n,p}/m_b$. For a nonradial radiation field the energy density exceeds flux divided by $c$, introducing an inverse flux-factor correction. Recoil, weak magnetism, thresholds, blocking, and redshift require the full opacity and transport model. Emission by the inverse processes cools hot matter and has a steep temperature dependence. The gain radius $r_g$ is where the angle- and energy-integrated net source changes sign: cooling dominates below it and heating above it up to the shock. Janka derives the heating scalings and gain-layer picture in Janka 2012, §IV.C.1, Eqs. (4)–(8).

The sign must be computed from the same opacity set in both directions. A homogeneous cell in thermal and weak equilibrium should show detailed balance and zero net source. This is an independent benchmark for emissivity, absorptivity, blocking factors, and energy-bin integration. In $c=\hbar=k_B=1$ units, a specific heating rate has dimension $L^{-1}$ because specific energy is dimensionless; restoring units must recover energy per mass per time. The global signature is $(-,+,+,+)$ and $G$ remains explicit.

Advection versus heating

For a transparent Newtonian or slowly moving local-frame estimate, define the gain mass and integrated net heating by

$$M_g=\int_{V_g}\rho\,dV, \qquad \dot Q_g=\int_{V_g}\rho\dot q_{\rm net}\,d^3x .$$

Here $V_g$ is the volume between gain surface and shock. A relativistic calculation instead uses conserved baryon mass $\int_{V_g}\rho W\sqrt{\gamma}\,d^3x$ and the energy-source projection in its chosen evolution equation. A comoving heating rate cannot be inserted into a coordinate-time energy budget without the corresponding lapse, velocity, and redshift factors.

A transparent pair of estimates is

$$\tau_{\rm adv}\sim\frac{M_g}{\dot M}, \qquad \tau_{\rm heat}\sim\frac{|E_{\rm bind,g}|}{\dot Q_g}.$$

The ratio $\tau_{\rm adv}/\tau_{\rm heat}\gtrsim1$ indicates that matter can absorb energy comparable to its binding before being advected through the gain layer. It is a favorable diagnostic, not a theorem of explosion. Definitions based on tracer residence times, radial integrals of inverse velocity, or volume-integrated masses need not agree in a turbulent flow. Likewise, “binding energy” depends on whether rest mass, nuclear recombination, general relativity, and the overlying envelope are included.

Burrows discusses critical conditions and the coupled accretion, luminosity, and shock-radius feedback in Burrows 2013, §IV.A.4. Heating expands the shock, which can increase gain mass and residence time; that feedback lowers the additional heating needed. But a transient ratio above unity does not guarantee positive asymptotic diagnostic energy or sustained expansion.

Closures and multidimensional limitations

Solving the full distribution in space, energy, and angle is expensive. Moment schemes evolve energy and flux and close the pressure tensor with a prescription such as M1; flux-limited diffusion evolves a more restricted angular description; Monte Carlo methods sample packets; and ray-by-ray methods approximate lateral transport. All can be accurate in controlled regimes, but no local closure reconstructs arbitrary crossing beams. M1 can merge beams artificially, and ray-by-ray treatment can correlate local hydrodynamic variations with radial neutrino flux too strongly.

A closure comparison must use matched opacities, progenitor, equation of state, gravity, energy grid, resolution, and run duration. Heating efficiency, gain mass, shock radius, luminosity, spectral moments, and the timescale ratio should be compared along with total energy and lepton-number residuals. Differences cannot be assigned to “transport” if microphysics or resolution also changed.

Benchmark, convergence, and rerun boundary

A numerical benchmark should first reproduce cooling and heating in a homogeneous cell, including detailed balance, and then a published spherical gain-region model with a verified sign change at $r_g$. The environment must pin code commit, compiler or interpreter, dependencies, floating-point precision, equation-of-state and opacity files with checksums, solver tolerances, random seeds where applicable, and hardware-relevant transport settings.

Convergence studies refine radial and angular grids, time step, neutrino energy groups, momentum-angle representation or moment closure, and nonlinear and gravity tolerances. Report error norms and uncertainty in luminosity, mean squared energy, gain mass, heating efficiency, shock radius, and $\tau_{\rm adv}/\tau_{\rm heat}$. A result is stronger when an independent implementation agrees on a matched spherical reference.

Each failure mode needs a visible diagnostic: opacity inconsistency can violate detailed balance; omitted velocity or redshift terms bias energy exchange; closure failure distorts intersecting beams; ray-by-ray coupling imprints angular artifacts; atmosphere or inner-boundary tuning can manufacture an explosion; and poor energy resolution can drift total energy or lepton number.

No transport calculation was run for this page. A reproducible companion would provide one documented rerun command that regenerates machine-readable cell, spherical, and refinement results from its pinned software environment and fails when conservation, realizability, benchmark, or convergence tolerances are missed.

Check the timescale diagnostic

A gain layer has $M_g=0.010\,M_\odot$, accretion rate $\dot M=0.050\,M_\odot\,{\rm s}^{-1}$, binding magnitude $|E_{\rm bind,g}|=2.0\times10^{50}\,{\rm erg}$, and net heating $\dot Q_g=2.0\times10^{51}\,{\rm erg\,s}^{-1}$. Evaluate the two timescales and interpret their ratio.

Solution

The estimates are

$$\tau_{\rm adv}=\frac{0.010}{0.050}\ {\rm s}=0.20\ {\rm s}, \qquad \tau_{\rm heat}=\frac{2.0\times10^{50}}{2.0\times10^{51}}\ {\rm s} =0.10\ {\rm s}.$$

Thus $\tau_{\rm adv}/\tau_{\rm heat}=2$. Matter has, in this averaged model, enough residence time to absorb twice its gain-layer binding scale, so shock revival is favored. The calculation is not an explosion verdict: turbulence, continued accretion, recombination, the envelope binding energy, and the time evolution of all four inputs still matter.

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.
• Mezzacappa, Anthony, Eirik Endeve, O. E. Bronson Messer, and Stephen W. Bruenn. “Physical, Numerical, and Computational Challenges of Modeling Neutrino Transport in Core-Collapse Supernovae.” Living Reviews in Computational Astrophysics 6 (2020): 4. DOI.