Thermal aspects of neutron star mergers

In order to extract maximal information from neutron-star merger signals, both gravitational and electromagnetic, we need to ensure that our theoretical models/numerical simulations faithfully represent the extreme physics involved. This involves a range of issues, with the finite temperature effects regulating many of the relevant phenomena. As a step toward understanding these issues, we explore the conditions for $\beta$-equilibrium in neutron star matter for the densities and temperatures reached in a binary neutron star merger. Using the results from our out-of-equilibrium merger simulation, we consider how different notions of equilibrium may affect the merger dynamics, raising issues that arise when attempting to account for these conditions in future simulations. These issues are both computational and conceptual. We show that the effects lead to, in our case, a softening of the equation of state in some density regions, and to composition changes that affect processes that rely on deviation from equilibrium, such as bulk viscosity, both in terms of the magnitude and the equilibration timescales inherent to the relevant set of reactions. We also demonstrate that it is difficult to determine exactly which equilibrium conditions are relevant in which regions of the matter due to the dependence on neutrino absorption, further complicating the calculation of the reactions that work to restore the matter to equilibrium.

Figure 1

Temperature $T$, electron fraction $Y_e$, rest-mass density with respect to nuclear saturation density $\rho /{\rho }_{\mathrm{nuc}}$, and neutron chemical potential ${\mu }_n$ for a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left column coincides with the merger time, while the right column snapshots are 5.0 ms postmerger.

Figure 2

Total entropy per baryon $\mathcal{S}$ for the simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left panel coincides with merger time, while the right panel is 5.0 ms postmerger. Note that $\mathcal{S}\le 1$ indicates degenerate matter while $\mathcal{S}\gg 1$ indicates strongly nondegenerate matter.

Figure 3

The condition number ${\mathcal{K}}_{\tau \to T}=\frac{\tau }{T}{(\frac{\partial \tau }{\partial T})}^{-1}$ which illustrates how an intrinsic numerical error in the energy $\tau$ is amplified to an error in the temperature. The behavior is robust to changes in density and velocity (as encoded in the Lorentz factor $W$ explicitly defined in Appendix Sec. 2). For the APR equation of state from [14] we see the result generically diverges as the temperature decreases, indicating serious problems with numerical accuracy at low temperatures.

Figure 4

Degeneracy parameter $T/E_{F_{\mathrm{x}}}$ for protons $p$, neutrons $n$, and electrons $e$. The baryons are assumed to be nonrelativistic while the electrons are assumed to be relativistic. The data relate to the simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left column coincides with merger time, and right column is 5.0 ms postmerger. Note that $T/E_{F_x}\ll 1$ indicates strongly degenerate matter, and $T/E_{F_x}\gg 1$ indicates strongly nondegenerate matter.

Figure 5

Deviation from cold $\beta$-equlibrium ${\mu }_{\mathrm{\Delta }}={\mu }_n-({\mu }_p+{\mu }_e)$ for a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left panel coincides with merger time, and right panel is 5.0 ms post merger.

Figure 6

Bulk viscosity regime parameter $|{\mu }_{\mathrm{\Delta }}|/T$ for the simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left panel coincides with the merger time, and the right panel is 5.0 ms postmerger. We see that there are regions in the simulation that remain subthermal (where a low-temperature expansion and a perturbative analysis of bulk viscosity would suffice), but there are also large regions where the matter is in the suprathermal regime.

Figure 7

Distribution of baryon mass $M_b$ in the ${\mu }_{\mathrm{\Delta }}-T$ plane for a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14], where ${\mu }_{\mathrm{\Delta }}={\mu }_n-{\mu }_p-{\mu }_e$. The left panel coincides with the merger time, and the right panel is 5.0 ms post merger. The dashed line denotes $T=|{\mu }_{\mathrm{\Delta }}|$. Matter above this line therefore has $\mathcal{A}<1$ and is in the sub-thermal regime, whereas matter below the line has $\mathcal{A}>1$ and is in the suprathermal regime. This distinction is replicated in the histograms to the side of and below each plot, with blue indicating subthermal, and red indicating suprathermal.

Figure 8

Illustrating the impact of the warm equilibrium on the stiffness of the equation of state. Dashed lines assume the “cold” $\beta$-equilibrium, while solid lines use the “warm” equilibrium prescription with ${\mu }_{\delta }$ take from Fig. 4 in [10]. The results show that the effect tends to soften the equation of state by up to 5% at densities up to $2{\rho }_{\mathrm{nuc}}$, but also that the impact is much less pronounced at higher temperatures.

Figure 9

Energy averaged electron neutrino and antineutrino effective opacities ${\tilde{\kappa }}_{*}$ multiplied by grid cell spacing $\mathrm{\Delta }x=400\mathrm{m}$ for a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left panel coincides with the merger time, and the right panel is 5.0 ms post merger. The neutrino opacities were calculated using NuLib [57].

Figure 10

Distribution of baryon mass $M_b$ in the ${\mu }_n-T$ plane for a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. The left panel coincides with the merger time, and the right panel is 5.0 ms post merger. The results suggest that thermal pions, which may play a role at temperature above $T\gtrsim 25\mathrm{MeV}$ [58], may impact on a significant fraction of the simulated matter.

Figure 11

Schematic of where the different equilibrium conditions discussed in Sec. 4 are expected to occur in a simulation of the merger of two $1.4M_{\odot }$ neutron stars using the APR equation of state from [14]. Left panel coincides with merger time, and right panel is 5.0 ms post-merger. The conditions on the matter are labelled as follows: “Cold” denotes matter with $T<1\mathrm{MeV}$ where the standard cold $\beta$ equilibrium condition eq. (27) is relevant, “Warm” denotes matter with $T>1\mathrm{MeV}$ and ${\tilde{\kappa }}_{*}{r}_{\mathrm{hot}}<1$ for both neutrino species where the warm $\beta$ equilibrium condition in Eq. (35) is relevant, “Hot” denotes matter where ${\tilde{\kappa }}_{*}\mathrm{\Delta }x>1$ for both neutrino species, thus the hot $\beta$ equilibrium condition in Eq. (36) is relevant, “Warm/Hot” denotes matter with $T>1\mathrm{MeV}$ but with $\mathrm{\Delta }x/r_{\mathrm{hot}}<{\tilde{\kappa }}_{*}\mathrm{\Delta }x<1$ for at least one of the neutrino species, where we cannot distinguish cleanly between hot and warm equilibrium, and “Strong” denotes matter with $T>25\mathrm{MeV}$ where pions may be relevant.