# Forms of the Symmetry Energy Relevant to Neutron Stars

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{208}Pb [2], frequency of isovector giant dipole resonance in

^{208}Pb, isospin diffusion in heavy ion reactions, excitation energies of isobaric analog states, isoscaling of fragments from intermediate energy heavy-ion collisions used in ground-based experiments to obtain data that allow constraining the symmetry energy below and in the vicinity of the saturation point ${n}_{0}$. Analysis of these data led to relatively consistent results for the form of the symmetry energy for sub-saturation densities [3,4]. Heavy-ion collisions (HICs) represent a class of experiments that allow one to study dense nuclear matter under conditions changing in a controlled manner. Although the experimental creation and then the analysis of nuclear matter in a wide temperature, density, and neutron-proton asymmetry range become feasible, the results strongly depend on the achievable composition of projectile and target nuclei and the energy of beams. Recent investigations of the asymmetric nuclear matter mostly focused on the symmetry energy. Nevertheless, the symmetry energy is still unconstrained for densities higher then the saturation density. This applies to both the experimental and theoretical aspects of this issue. The direct measurement of the symmetry energy is not possible, and therefore, many theoretical assumptions are necessary for the interpretation of the experimental data. The basic problem that influences the interpretation of the existing data is the fact that HICs are highly non-equilibrium processes, and their analysis necessarily involves a transport model. Another problem is the identification of observables sensitive to the symmetry energy in the presence of many uncertainties including those introduced by transport models. Analyses based on the theory of transport suggest using isospin ratio observables for extracting information about the symmetry energy [5,6]. The measurements of the observables in question are conditional on the energies achievable during experiments. The choice of observables suitable for obtaining information on the properties of asymmetric nuclear matter in the high-density range is connected with the phenomena of collective flows and meson production. The upper limit of the density that can be achieved in central heavy-ion collisions at beam energies of several hundred MeV per nucleon in the range $2\xf73\phantom{\rule{4pt}{0ex}}{n}_{0}$ [7]. A description of collisions during subsequent phases of the system evolution is carried out based on the transport model, which typically uses the power-law function that represents the symmetry energy dependence on density. The potential part of the symmetry energy has the form:

## 2. Symmetry Energy: The Taylor Series

- the binding energy at saturation density ${n}_{0}$: $E({n}_{0},0)$
- the symmetry energy ${E}_{2,sym}\left({n}_{b}\right)$ with the symmetry energy coefficient ${E}_{2,sym}\left({n}_{0}\right)$
- the slope of the symmetry energy ${E}_{2,sym}$: $L=3{n}_{0}\frac{d{E}_{2,sym}}{d{n}_{b}}\left({n}_{0}\right)$
- the incompressibility of symmetric nuclear matter $E({n}_{b},0)$: ${K}_{0}=9{n}_{0}^{2}\frac{{d}^{2}E({n}_{b},0)}{d{n}_{b}^{2}}\left({n}_{0}\right)$
- the curvature of the symmetry energy: ${K}_{sym}=9{n}_{0}^{2}\frac{{d}^{2}{E}_{2,sym}\left({n}_{b}\right)}{d{n}_{b}^{2}}\left({n}_{0}\right)$.

## 3. Symmetry Energy: The Padé Approximation

## 4. Isospin-Sensitive Properties of Neutron Stars

## 5. The Formalism

#### 5.1. Nuclear Matter

#### 5.2. Matter of a Neutron Star

## 6. Results

#### 6.1. The Effect of the Symmetry Energy for Neutron Star Matter

#### 6.2. The Effect of the Approximation Method for Neutron Star Matter

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**The ratio ${E}_{4,sym}/{E}_{2,sym}$ in the discussed model and its linear approximation as a function of baryon number density ${n}_{b}$.

**Figure 2.**The EoSs obtained for the TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0$, $0.0165$, and $0.03$, respectively.

**Figure 3.**The equilibrium relative baryon fractions ${Y}_{n}$ and ${Y}_{p}$ as a function of baryon number density ${n}_{b}$ obtained for the TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.0165$, and $0.03$, respectively. The red dotted line represents the level of critical proton fraction ${Y}_{p}^{crit}$.

**Figure 4.**The equilibrium relative lepton fractions ${Y}_{e}$ and ${Y}_{\mu}$ as a function of baryon number density ${n}_{b}$ obtained for the TM1 parametrization for ${\mathsf{\Lambda}}_{V}=0,\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}0.0165$, and $0.03$, respectively.

**Figure 5.**The mass-radius relations calculated for the TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0$ and for ${\mathsf{\Lambda}}_{V}=0.03$. Dots represent the selected neutron star configurations: the maximum mass configuration and the one characterized by the central density equal to $2{n}_{0}$.

**Figure 6.**Dependence of isospin asymmetry ${\delta}_{a}$ on baryon density $u={n}_{b}/{n}_{j}$, where ${n}_{j}$ denotes the saturation density (left panel), and on stellar radius r (right panel). The radial dependence includes two selected configurations of stars: the maximum mass configuration and the one with the central density $2{n}_{0}$. The results were obtained for TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0$ and ${\mathsf{\Lambda}}_{V}=0.03$.

**Figure 7.**Neutron star masses and radii as a function of ${\Delta}_{a}$ (

**a**). The radial dependence of ${\Delta}_{a}$ calculated for two selected configurations: the maximum mass configuration and the one characterized by the central density equal to $2{n}_{0}$. The results were obtained for TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0$ and ${\mathsf{\Lambda}}_{V}=0.03$ (

**b**). Dots represent the locations of masses and the radii of the presented configurations.

**Figure 8.**The equilibrium proton fraction obtained for the TM1 parameterization for ${\mathsf{\Lambda}}_{V}=0.0165$ and for ${\mathsf{\Lambda}}_{V}=0.03$. The results show differences that arise when different methods of approximation are applied (parabolic approximation and Padé approximants).

**Figure 9.**The equilibrium EoS. (

**a**) ${\mathsf{\Lambda}}_{V}=0$, (

**b**) ${\mathsf{\Lambda}}_{V}=0.0165$, and (

**c**) ${\mathsf{\Lambda}}_{V}=0.03$.

**Figure 10.**The sensitivity of the equilibrium proton concentrations ${Y}_{p}^{eq}$ (

**left panel**) and ${E}_{sym}$ (

**right panel**) to the change of L for ${n}_{b}=2\phantom{\rule{0.166667em}{0ex}}{n}_{0}$ for the Padé approximation and the truncated Taylor series cases.

**Table 1.**The TM1 parameter set described in [66].

$\sigma $ | ${m}_{\sigma}=511.198$ MeV | ${g}_{\sigma}=10.029$ | ${g}_{2}=$ 7.2327 fm${}^{-1}$ | ${g}_{3}=0.6183$ | - |

$\omega $ | ${m}_{\omega}=783$ MeV | ${g}_{\omega}=12.614$ | ${c}_{3}=71.308$ | - | - |

$\rho $ | ${m}_{\rho}=770$ MeV | ${\mathsf{\Lambda}}_{V}$ | 0.0 | 0.0165 | 0.03 |

${g}_{\rho}\left({\mathsf{\Lambda}}_{V}\right)$ | 9.2644 | 10.037 | 11.10 |

${\mathsf{\Lambda}}_{\mathit{V}}$ | ${\mathbf{\Delta}}_{\mathit{a}}$ | ${\mathit{n}}_{\mathit{c}}/{\mathit{n}}_{0}$ | M[M${}_{\odot}$] | R (km) | ${\mathit{\delta}}_{\mathit{a}}\left(0\right)$ | |
---|---|---|---|---|---|---|

M${}_{max}$ | 0.03 | 0.77 | 6.05 | 2.11 | 11.6 | 0.69 |

$2\times {n}_{0}$ | 0.03 | 0.85 | 2.08 | 1.13 | 12.42 | 0.79 |

M${}_{max}$ | 0.0 | 0.6 | 5.71 | 2.17 | 12.09 | 0.41 |

$2\times {n}_{0}$ | 0.0 | 0.79 | 2.05 | 1.33 | 13.05 | 0.63 |

**Table 3.**Crust-core transition density ${n}_{tr}$ and corresponding values of equilibrium proton fraction ${Y}_{p}^{eq}\left({n}_{tr}\right)$ and pressure ${P}^{eq}\left({n}_{tr}\right)\left[MeVf{m}^{-3}\right]$ obtained for the parabolic approximation (PA) and Padé approximants.

PA | ${\mathsf{\Lambda}}_{\mathit{V}}=0$ | ${\mathsf{\Lambda}}_{\mathit{V}}=0.0165$ | ${\mathsf{\Lambda}}_{\mathit{V}}=0.03$ | Padé | ${\mathsf{\Lambda}}_{\mathit{V}}=0$ | ${\mathsf{\Lambda}}_{\mathit{V}}=0.165$ | ${\mathsf{\Lambda}}_{\mathit{V}}=0.03$ |
---|---|---|---|---|---|---|---|

n${}_{tr}$ (fm${}^{-3})$ | 0.0876 | 0.0932 | 0.0947 | n${}_{tr}$ (fm${}^{-3}\left]\right)$ | 0.0877 | 0.0931 | 0.0946 |

Y${}_{p}^{eqT}\left({n}_{tr}\right)$ | 0.037 | 0.041 | 0.0430 | Y${}_{p}^{eqP}\left({n}_{tr}\right)$ | 0.041 | 0.047 | 0.05 |

P${}^{eqT}\left({n}_{tr}\right)$ | 1.256 | 0.378 | 0.410 | P${}^{eqP}\left({n}_{tr}\right)$ | 1.285 | 0.400 | 0.423 |

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Bednarek, I.; Sładkowski, J.; Syska, J.
Forms of the Symmetry Energy Relevant to Neutron Stars. *Symmetry* **2020**, *12*, 898.
https://doi.org/10.3390/sym12060898

**AMA Style**

Bednarek I, Sładkowski J, Syska J.
Forms of the Symmetry Energy Relevant to Neutron Stars. *Symmetry*. 2020; 12(6):898.
https://doi.org/10.3390/sym12060898

**Chicago/Turabian Style**

Bednarek, Ilona, Jan Sładkowski, and Jacek Syska.
2020. "Forms of the Symmetry Energy Relevant to Neutron Stars" *Symmetry* 12, no. 6: 898.
https://doi.org/10.3390/sym12060898