# Analytic Closures for M1 Neutrino Transport

## TL;DR

This paper evaluates various M1 neutrino transport closures by comparing them with Monte Carlo simulations around protoneutron stars, finding that no single closure is best in all cases, but Minerbo's maximum entropy closure performs relatively well overall.

## Contribution

It provides an extensive comparison of M1 closure relations against Monte Carlo benchmarks, highlighting their varying accuracy depending on conditions.

## Key findings

- No closure is universally best across all scenarios.
- Minerbo's maximum entropy closure performs relatively well overall.
- Closure accuracy depends on matter configuration, neutrino type, and energy.

## Abstract

Carefully accounting for neutrino transport is an essential component of many astrophysical studies. Solving the full transport equation is too expensive for most realistic applications, especially those involving multiple spatial dimensions. For such cases, resorting to approximations is often the only viable option for obtaining solutions. One such approximation, which recently became popular, is the M1 method. It utilizes the system of the lowest two moments of the transport equation and closes the system with an ad hoc closure relation. The accuracy of the M1 solution depends on the quality of the closure. Several closures have been proposed in the literature and have been used in various studies. We carry out an extensive study of these closures by comparing the results of M1 calculations with precise Monte Carlo calculations of the radiation field around spherically-symmetric protoneutron star models. We find that no closure performs consistently better or worse than others in all cases. The level of accuracy a given closure yields depends on the matter configuration, neutrino type, and neutrino energy. Given this limitation, the maximum entropy closure by Minerbo (1978) on average yields relatively accurate results in the broadest set of cases considered in this work.