Thermal properties of hot and dense matter with finite range interactions

We explore the thermal properties of hot and dense matter using a model that reproduces the empirical properties of isospin symmetric and asymmetric bulk nuclear matter, optical-model fits to nucleon-nucleus scattering data, heavy-ion flow data in the energy range 0.5–2 GeV/$A$, and the largest well-measured neutron star mass of $2M_{\odot }$. This model, which incorporates finite range interactions through a Yukawa-type finite range force, is contrasted with a conventional zero range Skyrme model. Both models predict nearly identical zero-temperature properties at all densities and proton fractions, including the neutron star maximum mass, but differ in their predictions for heavy-ion flow data. We contrast their predictions of thermal properties, including their specific heats, and provide analytical formulas for the strongly degenerate and nondegenerate limits. We find significant differences in the results of the two models for quantities that depend on the density derivatives of nucleon effective masses. We show that a constant value for the ratio of the thermal components of pressure and energy density expressed as ${\mathrm{\Gamma }}_{\text{th}}=1+(P_{\text{th}}/{\varepsilon }_{\text{th}})$, often used in simulations of proto-neutron stars and merging compact object binaries, fails to adequately describe results of either nuclear model. The region of greatest discrepancy extends from subsaturation densities to a few times the saturation density of symmetric nuclear matter. Our results suggest alternate approximations for the thermal properties of dense matter that are more realistic.

Figure 1

Neutron single-particle potentials vs momentum at $T=0$ for the densities $n$ and proton fractions $x$ as marked. Results for the MDI(A) model (solid curves) from Eqs. (10), (34), and (35). Those for the ${\mathrm{SkO}}^{\prime }$ model (dashed curves) from Eq. (40).

Figure 2

Comparison of neutron single-particle potentials vs momentum from Eqs. (34) and (35) with the variational Monte Carlo results of Ref. [81] using the UV14-TNI interaction and Bruekner-Hartree-Fock results of Ref. [82] with the inclusion of three-body interactions (labeled BHF-TBF).

Figure 3

Landau effective masses of the neutron and proton vs baryon density $n$ for the marked values of the proton fraction $x$. (a) Results for the MDI(A) model from Eqs. (37) and (38). (b) Same as (a), but for the ${\mathrm{SkO}}^{\prime }$ model from Eq. (36).

Figure 4

(a) Energy per particle $E=\mathcal{H}/n$ vs baryon number density $n \left(T=0\right)$ at the proton fractions $x$ for the MDI(A) model from Eqs. (23, 24, 25) and the ${\mathrm{SkO}}^{\prime }$ model from Eq. (39). (b) $E$ vs $n$ for matter with proton fractions $x_{\beta }$ determined from charge neutrality and $\beta$ equilibrium.

Figure 5

(a) Pressure $P$ vs baryon density $n \left(T=0\right)$ at the proton fractions $x$ shown for the MDI(A) model from Eqs. (A1, A2, A3, A4, A5, A6) and the ${\mathrm{SkO}}^{\prime }$ model from Eq. (28). (b) $P$ vs $n$ for $x_{\beta }$ determined from charge neutrality and $\beta$ equilibrium.

Figure 6

(a),(b) The neutron and proton chemical potentials vs baryon density $n$ for the MDI(A) [Eqs. (A7)–(A10)] and the ${\mathrm{SkO}}^{\prime }$ [Eq. (28)] models for values of $x$ shown. (c) $\widehat{\mu }={\mu }_n-{\mu }_p$ vs $n$.

Figure 7

(a) Mass vs radius and (b) mass vs central density of $T=0$ neutron stars for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models.

Figure 8

Momentum-dependent part of the neutron single-particle potential for the baryon densities, temperatures, and proton fractions indicated in each panel. The zeroth approximation $R^{\left(0\right)}$ is the zero-temperature $R_n$ given by the momentum-dependent terms in Eq. (34), whereas the final result $R^{\left(f\right)}$ is found via an iterative procedure using Eq. (13).

Figure 9

Thermal chemical potentials vs $n$ for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models from Eq. (7).

Figure 10

Degeneracy parameter $\eta$ in the $n\text{−}T$ plane. Results are from Eq. (54) [for MDI(A)] and Eq. (55) [for ${\mathrm{SkO}}^{\prime }]$.

Figure 11

Thermal energy per particle vs $n$ for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models from Eqs. (6) and (39).

Figure 12

Thermal free energy for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models from Eqs. (6), (39), and (14).

Figure 13

Thermal pressure vs $n$ for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models from Eq. (16).

Figure 14

Entropy per particle vs $n$ for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models from Eq. (14).

Figure 15

Isentropes in the $n\text{−}T$ plane for the MDI(A) model in (a) and those for the ${\mathrm{SkO}}^{\prime }$ model in (b).

Figure 16

(a),(b) Specific heat at constant volume $C_V$ vs density from Eq. (18) for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models.

Figure 17

(a),(b) Specific heat at constant pressure $C_P$ vs density from Eq. (19).

Figure 18

Ratio of specific heats.

Figure 19

Neutron thermal chemical potentials [Eq. (7)] vs $n$ for the MDI(A) model in (a) and (b) and the ${\mathrm{SkO}}^{\prime }$ model in (c) and (d) compared with their limiting cases from Eqs. (58) and (B20).

Figure 20

Thermal energy per particle [Eq. (6)] and limiting cases [Eqs. (56) and (B30)] vs $n$.

Figure 21

Thermal pressure [Eq. (16)] and limiting cases [Eqs. (57) and (B35)] vs $n$.

Figure 22

Entropy per particle [Eq. (14)] vs $n$ for the MDI(A) model in (a) and (b) and the ${\mathrm{SkO}}^{\prime }$ model in (c) and (d) compared with their limiting cases from Eqs. (56) and (B34).

Figure 23

Specific heat at constant volume $C_V$ for the MDI(A) model in (a) and (b) and the ${\mathrm{SkO}}^{\prime }$ model in (c) and (d) compared with their limiting cases from Eq. (56) as $S=C_V$ in the degenerate limit, and Eq. (C23).

Figure 24

Specific heat at constant pressure $C_P$ for the MDI(A) model in (a) and (b) and the ${\mathrm{SkO}}^{\prime }$ model in (c) and (d) compared with their limiting cases from Eq. (56) as $S=C_P$ in the degenerate limit, and Eq. (19).

Figure 25

The thermal index ${\mathrm{\Gamma }}_{\text{th}}$ vs $n$ for the three models indicated in the figure. Results for the exact calculations are from Eqs. (48), (49), and (59), whereas those for the degenerate limit are from Eqs. (60, 61, 62). Results shown are for nucleons only.

Figure 26

The Dirac and Landau effective masses $M^{*}$ and $m_n^{*}$ of neutrons in a typical mean-field theoretical (MFT) model.

Figure 27

(a) Contributions from nucleons and leptons to thermal pressure, $P_{\text{th}}$, and thermal energy density, ${\varepsilon }_{\text{th}}$, for the ${\mathrm{SkO}}^{\prime }$ model at the indicated proton fraction and temperature. (b) The total $P_{\text{th}}$ and ${\varepsilon }_{\text{th}}$; the inset shows their ratio.

Figure 28

Same as Fig. 27, but at $T=50$ MeV.

Figure 29

Same as Fig. 25, but with the inclusion of leptons and photons.

Figure 30

Isentropes for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models for matter with nucleons, leptons, and photons.

Figure 31

The zero-temperature adiabatic index ${\mathrm{\Gamma }}_{S=0}$ for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models at at the indicated proton fractions. Dashed curves include only nucleons, while solid lines include nucleons and leptons.

Figure 32

Zero-temperature nucleon (dashed curves) and lepton (solid curves) pressures at the indicated proton fractions.

Figure 33

Adiabatic index at fixed entropy for the ${\mathrm{SkO}}^{\prime }$ and MDI(A) models. Results are from Eqs. (86) and (98).

Figure 34

Adiabatic index at fixed entropy from the ${\mathrm{SkO}}^{\prime }$ and MDI(A) models for SNM using Eq. (98). Results for the degenerate (left) and nondegenerate (right) limits are from Eqs. (98) and (99), respectively.

Figure 35

Adiabatic index at constant entropy for the ${\mathrm{SkO}}^{\prime }$ and MDI(A) models using Eqs. (98) and (99). Results for SNM (a),(c) and for PNM (b),(d) are shown along with their limiting cases.

Figure 36

Contributions from nucleons, leptons, and photons to the total entropy at the indicated proton fractions for the MDI(A) and ${\mathrm{SkO}}^{\prime }$ models.

Figure 37

Squared sound speed vs density for various entropy per baryon from Eq. (108).