Isospin equilibration in neutron star mergers

We analyze the isospin equilibration properties of neutrinoless nuclear ($npe$) matter in the temperature and density range that is relevant to neutron star mergers. Our analysis incorporates neutrino-transparency corrections to the isospin (“beta”) equilibrium condition which become noticeable at $T\gtrsim 1\text{MeV}$. We find that the isospin relaxation rate rises rapidly as temperature rises, and at $T\approx 5\text{MeV}$ it is comparable to the timescale of the density oscillations that occur immediately after the merger. This produces a resonant peak in the bulk viscosity at $T\approx 5\text{MeV}$, which causes density oscillations to be damped on the timescale of the merger. Our calculations suggest that isospin relaxation dynamics may also be relevant when neutrinos are treated more accurately via neutrino transport schemes.

Figure 1

Density and temperature dependence of $-{\mu }_I^{\text{eq}}$, the isospin chemical potential in isospin equilibrium [Eq. (1)], for the IUF equation of state (left panel) and QMC-RMF3 (right panel); $-{\mu }_I^{\text{eq}}$ rises with temperature because it arises from thermal blurring of the Fermi surfaces (see text). For IUF at low temperatures it is also influenced by the direct Urca threshold at $n_B\approx 4n_0$.

Figure 2

Density and temperature dependence of the isospin relaxation time $\tau =1/{\gamma }_I$ [Eq. (10)]. The shaded region shows where the relaxation time is between 0.1 ms and 25 ms, i.e., comparable to the timescale of merger dynamics. The thick line shows the temperatures and densities where the bulk viscosity of a $1\text{kHz}$ density oscillation would reach its maximum, i.e., where ${\gamma }_I=2\pi \times 1\text{kHz}$ (Sec. 2b).

Figure 3

Static (zero-frequency) bulk viscosity (21) for the IUF (left) and QMC-RMF3 (right) equations of state. The static bulk viscosity drops as temperature rises because it is inversely proportional to the relaxation rate.

Figure 4

Bulk viscosity at frequency $f=1\text{kHz}$ for the IUF (left) and QMC-RMF3 (right) equations of state. In both cases the resonant peak occurs where the relaxation rate passes through $\omega =2\pi \times 1\text{kHz}$, which occurs at $T\approx 5\mathrm{MeV}$.

Figure 5

Damping time for density oscillations of frequency 1 kHz as a function of density and temperature, for the IUF (left) and QMC-RMF3 (right) equations of state.