Equations of state for supernovae and compact stars

A review is given of various theoretical approaches for the equation of state (EoS) of dense matter, relevant for the description of core-collapse supernovae, compact stars, and compact star mergers. The emphasis is put on models that are applicable to all of these scenarios. Such EoS models have to cover large ranges in baryon number density, temperature, and isospin asymmetry. The characteristics of matter change dramatically within these ranges, from a mixture of nucleons, nuclei, and electrons to uniform, strongly interacting matter containing nucleons, and possibly other particles such as hyperons or quarks. As the development of an EoS requires joint efforts from many directions, different theoretical approaches are considered and relevant experimental and observational constraints which provide insights for future research are discussed. Finally, results from applications of the discussed EoS models are summarized.

Figure 1

Temperatures and densities reached during a CCSN simulation within 1 s post bounce. The color coding shows the electron fraction $Y_e$. From T. Fischer.

Figure 2

2D mass histograms for (left panel) mass density $\rho$ and electron fraction $Y_e$ or (right panel) entropy per baryon $s$, of a NS merger remnant at a time of 85.4 ms after the first contact. The color coding is a measure of the amount of matter experiencing the specific thermodynamical conditions. Adapted from [721].

Figure 3

Feynman diagrams illustrating the lowest-order contributions to the self-energy in Dyson’s equation (5) for the single-particle Green’s function in the ladder approximation (without exchange contributions). Solid lines represent fermion propagators and wavy lines an interaction.

Figure 4

Nuclear phase diagram in the temperature–baryon density plane. Solid and dash-dotted blue lines indicate boundaries of the liquid-gas coexistence region for symmetric and asymmetric matter calculated with TM1 interactions ([902]). The shaded area corresponds to typical conditions for nuclear multifragmentation reactions ([120]). The dashed black lines are isentropic trajectories characterized by constant entropy per baryon, $s=1$, 2, 4, and 6 calculated with the statistical model for supernova matter (SMSM) ([120]). The dotted red lines show results of a CCSN simulation from [910] just before bounce (BB), at core bounce (CB), and postbounce (PB). From [148].

Figure 5

Equilibrium constants of $\alpha$ particles extracted from HIC experiments (black diamonds) in comparison with those of various theoretical models, which are all adapted for the conditions in HICs, as far as possible. The gray band is the experimental uncertainty in the temperature determination. The black line shows the equilibrium constant of the ideal gas model. From [440].

Figure 6

Comparison of results for the energy per baryon of neutron matter at $T=0\mathrm{MeV}$ from different ab initio approaches. The vertical bar represents the range given at saturation density in [539]. For details see text.

Figure 7

Summary of recent NS radius estimations from observations for a star with the canonical mass of $1.4M_{\odot }$. Shown are $2\sigma$ error bars. For details see Table 2 of [318] and the text. From M. Fortin.

Figure 8

Probability distribution of the symmetry energies at saturation $J$ (top panel) and of the symmetry energy slope parameter $L$ (bottom panel) from various studies. See text for details.

Figure 9

Graphical representation of the structure and composition of the crust of a $1.44M_{\odot }$ NS. Each subpanel shows in color coding (see legend at right) the mean local density of the nucleons, for the position in the NS as indicated in the bottom part of the figure. The illustration uses the SLy4 EDF, the EoS of [818] for the outer crust, and the EoS of [257] for the inner crust and the core.

Figure 10

Composition of matter in the center of a CCSN 6 ms before bounce at thermodynamic conditions taken from a simulation of [720]. The color map shows the distribution of nuclei (mass fractions) in the (a) SMSM ([147]), the (b) EoS of [394], and the (c) HS(DD2) model. (c) The black cross indicates the average heavy nucleus. The black diamond and triangle show the representative heavy nucleus of STOS and LS220, respectively, calculated within the SNA.

Figure 11

The lines delimit regions in temperature and baryon number density where the number density fractions of $\mathrm{\Lambda }$ hyperons exceed ${10}^{-4}$ for the different models. From left to right corresponds to a fixed hadronic charge fraction $Y_q=0.1$, 0.3, and 0.5. $\mathrm{\Lambda }$ hyperons appear at high densities and temperatures, i.e., in the upper right part of each panel.

Figure 12

Thermodynamic quantities as functions of temperature for $n_B=0.15$ (left) and $0.3{\mathrm{fm}}^{-3}$ (right), corresponding roughly to 1 and 2 times nuclear matter saturation density, and a charge fraction of $Y_q=0.1$. The upper panels show the pressure, the middle ones the scaled internal energy per baryon with respect to the proton mass, and the lower ones the number fractions of additional particles. These are $\mathrm{\Lambda }$ hyperons for “$\mathrm{STOS}\mathrm{\Lambda }$,” “$\mathrm{LS}220\mathrm{\Lambda }$,” “$\mathrm{BHB}\mathrm{\Lambda }$,” and “$\mathrm{BHB}\mathrm{\Lambda }\varphi$”; all hyperons for “STOSY30” and “$\mathrm{STOSY}30\pi$”; and the ${\pi }^{-}$ fraction for “$\mathrm{LS}220\pi$.” See the text for an explanation of the acronyms.

Figure 13

The slope parameter of the symmetry energy $L$ vs the value of the symmetry energy $J$ at normal nuclear matter density. The light gray region is the constraint of [561], and the dark gray region is taken from Fig. 8. The different symbols show the values of the nucleon interactions of Table 4 that are applied in the general purpose EoSs of Table 3. FSU2.1 gives the same value as FSUgold.

Figure 14

Mass-radius relations of spherically symmetric NSs for the different EoSs that cover the full thermodynamic parameter range and include only nucleonic degrees of freedom; cf. Table 3. The two horizontal bars indicate the two recent precise NS mass determinations, PSRJ1614-2230 ([235]) (hatched blue) and PSR $\mathrm{J}0348+0432$ ([29]) (yellow). The curve labeled “SkA,” although based on the same nuclear interaction, does not represent the H&W EoS, but the model by [394].

Figure 15

Same as Fig. 14 for EoS models including additional degrees of freedom. The onset of additional degrees of freedom is visible as a change in the slope.

Figure 16

Pressure as a function of baryon density within the EoS models listed in Table 3 for symmetric nuclear matter (right) and pure neutron matter (left). The results are compared with the theoretical calculation for neutron matter from [426] and the experimental flow constraint from [220] for symmetric matter.

Figure 17

Relation between the dominant GW frequency in the postmerger phase of a NS-NS binary merger event and the radius of a nonrotating NS with a gravitational mass of $M=1.6M_{\odot }$. The frequency has been normalized with respect to the total mass of the system. Different symbols refer to different total masses, where equal-mass binaries are assumed. From [80] with kind permission of The European Physical Journal (EPJ).

Figure 18

A comparison of the values of charge chemical potential ${\mu }_Q$ for different EoSs and typical thermodynamic conditions in the infall epoch. The proton fraction has been varied at constant entropy per baryon $s=1$ and constant baryon number density $n_B={10}^{-3}{\mathrm{fm}}^{-3}$.

Figure 19

The top panel shows the evolution of the PNS’s radius for a two-dimensional CCSN simulation employing the LS180, the H&W, and the STOS EoSs. The bottom panel shows the evolution of the shock radius. From [490].