Neutrino Transfer in Three Dimensions for Core-Collapse Supernovae. I. Static Configurations

We develop a numerical code to calculate the neutrino transfer with multi-energy and multi-angle in three dimensions (3D) for the study of core-collapse supernovae. The numerical code solves the Boltzmann equations for neutrino distributions by the discrete-ordinate (S_n) method with a fully implicit differencing for time advance. The Boltzmann equations are formulated in the inertial frame with collision terms being evaluated to the zeroth order of v/c. A basic set of neutrino reactions for three neutrino species is implemented together with a realistic equation of state of dense matter. The pair process is included approximately in order to keep the system linear. We present numerical results for a set of test problems to demonstrate the ability of the code. The numerical treatments of advection and collision terms are validated first in the diffusion and free streaming limits. Then we compute steady neutrino distributions for a background extracted from a spherically symmetric, general relativistic simulation of 15Msun star and compare them with the results in the latter computation. We also demonstrate multi-D capabilities of the 3D code solving neutrino transfers for artificially deformed supernova cores in 2D and 3D. Formal solutions along neutrino paths are utilized as exact solutions. We plan to apply this code to the 3D neutrino-radiation hydrodynamics simulations of supernovae. This is the first article in a series of reports on the development.

💡 Research Summary

The paper presents the development and validation of a three‑dimensional neutrino‑radiation transport code designed for core‑collapse supernova studies. The authors solve the time‑dependent Boltzmann equation for neutrino distribution functions using the discrete‑ordinate (Sₙ) method, which discretizes both angular and energy spaces, and they advance the solution with a fully implicit time differencing scheme to ensure numerical stability across the wide range of dynamical timescales present in a supernova core. The transport equations are formulated in the inertial frame, and collision terms are retained to zeroth order in v/c, an approximation justified by the relatively low fluid velocities compared with the speed of light in the dense core. A basic set of weak interaction processes—absorption, iso‑energetic scattering on nucleons, electron‑positron pair creation/annihilation, and neutrino‑neutrino pair processes—is implemented for all three neutrino species, together with a realistic high‑density equation of state. To keep the overall system linear, the pair‑process source term is treated in an approximate, linearized form.

The code is first exercised in two limiting regimes. In the diffusion limit, the numerical solution reproduces the expected Laplacian behavior of the neutrino flux, while in the free‑streaming limit it correctly yields straight‑line propagation with appropriate vacuum boundary conditions. These tests confirm that both advection and collision operators are handled accurately.

Subsequently, the authors import a static background profile (density, temperature, electron fraction) from a spherically symmetric, general‑relativistic 15‑M⊙ progenitor simulation and compute steady‑state neutrino distributions with the new 3D code. The resulting energy spectra, angular moments, and lepton‑number fluxes agree closely with those obtained from the original 1D transport calculation, demonstrating that the three‑dimensional implementation does not introduce spurious artifacts when applied to a spherically symmetric configuration.

To showcase genuine multidimensional capability, the authors construct artificially deformed core models in both two and three dimensions (e.g., ellipsoidal and irregular shapes) and solve for the stationary neutrino field. They compare the numerical results with exact solutions obtained by formal integration along neutrino trajectories (ray‑tracing). The agreement remains excellent even in the presence of strong geometric asymmetries, confirming that the code can handle complex boundary shapes and non‑radial fluxes.

The paper concludes by outlining plans to embed this transport solver into fully coupled 3D radiation‑hydrodynamics simulations of core‑collapse supernovae. By providing a robust, high‑fidelity neutrino transport module that operates directly on the Boltzmann equation without resorting to moment closures or ray‑by‑ray approximations, the work represents a significant step toward resolving the long‑standing problem of accurately modeling neutrino‑matter feedback in realistic, three‑dimensional supernova explosions.