$\beta$ equilibrium in neutron-star mergers

We show that the commonly used criterion for $\beta$ equilibrium in neutrino-transparent dense nuclear matter becomes invalid as temperatures rise above 1 MeV. Such temperatures are attained in neutron-star mergers. By numerically computing the relevant weak-interaction rates we find that the correct criterion for $\beta$ equilibrium requires an isospin chemical potential that can be as large as 10–20 MeV, depending on the temperature at which neutrinos become trapped.

Figure 1

Energy relative to their Fermi energy (defined as “$\gamma$”) for particles participating in direct Urca reactions in APR nuclear matter obeying the standard low-temperature condition (2) for $\beta$ equilibrium. At each density we choose the momenta and energies of participating particles (consistent with energy and momentum conservation) that maximize the product of their Fermi–Dirac factors. Above threshold, all particles can have $\gamma =0$. Below threshold, the plot shows the least-Boltzmann-suppressed processes. The circles indicate the particles that cause the Boltzmann suppression.

Figure 2

Exact direct Urca and approximate modified Urca rates at $T=500\text{keV}$, obeying the low-temperature $\beta$-equilibrium condition (2). Above threshold, the two direct Urca rates balance each other, and they also match the approximate direct Urca rate. Below threshold, the direct Urca rates exponentially fall off, and modified Urca dominates. However, the two direct Urca rates have different exponential falloffs below threshold. In the lower plot, the deviation ${\mu }_{\delta }$ from low-temperature $\beta$ equilibrium needed to achieve true $\beta$ equilibrium is plotted as a function of density.

Figure 3

Exact direct Urca and approximate modified Urca rates at $T=5\mathrm{MeV}$, obeying the low-temperature $\beta$-equilibrium condition (2). Above threshold, electron-capture direct Urca dominates and agrees with the approximate direct Urca rate calculation. At densities greater than $6n_{\text{sat}}$, we expect neutron decay direct Urca to match electron-capture direct Urca. Below threshold, electron-capture direct Urca dominates over all modified Urca processes, which contradicts the conventional wisdom. In the lower plot, the deviation ${\mu }_{\delta }$ from low-temperature $\beta$ equilibrium needed to achieve true $\beta$ equilibrium is plotted as a function of density.

Figure 4

The isospin-coupled chemical potential ${\mu }_{\delta }$ needed to achieve true $\beta$ equilibrium in APR matter at various temperatures. For a given temperature, the upper and lower curves indicate the range of values of ${\mu }_{\delta }$ that are consistent with our estimates of the theoretical uncertainty. Further details on the uncertainty are given in the text.

Figure 5

Proton fraction in neutrino-transparent nuclear matter in true $\beta$ equilibrium, for several different temperatures. As temperature increases, the $\beta$-equilibrated nuclear matter becomes more neutron-rich than predicted by the low-temperature $\beta$-equilibrium condition.

Figure 6

Urca rates in true $\beta$ equilibrium, ${\mu }_n={\mu }_p+{\mu }_e+{\mu }_{\delta }$. Above threshold, the two direct Urca processes dominate. Below threshold, direct Urca electron capture balances against modified Urca neutron decay $(n$ spectator) and, to a lesser extent, direct Urca neutron decay.

Figure 7

Fractional change in the six Urca rates at $T=5\mathrm{MeV}$ when we change ${\mu }_{\delta }$ from zero to the value for true $\beta$ equilibrium. Below threshold, the change is the most prominent, and as density increases to far above the threshold density, the true $\beta$-equilibrium condition approaches the behavior of the low-temperature $\beta$-equilibrium condition (2).

Figure 8

Mean-free path of neutrinos in nuclear matter described by the HS(DD2) equation of state (EoS). As temperature rises above 5 MeV it becomes more likely that neutrinos will be trapped inside the merger region.