# Neutron Star Properties: Quantifying the Effect of the Crust–Core Matching Procedure

^{*}

## Abstract

**:**

## 1. Introduction

## 2. EoS Parametrization

#### Crust Matching Procedure

## 3. Results

#### 3.1. Empirical Parameters Distribution

#### 3.2. Isolating the Matching Procedure Effect

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

NS | Neutron Star |

EoS | Equation of State |

NICER | Neutron star Interior Composition Explorer |

GW | Gravitational Waves |

eXTP | enhanced X-ray Timing and Polarimetry |

$npe\mu $ | neutron-proton-electron-muon |

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**Figure 1.**Probability density functions for ${E}_{\mathrm{sym}}$, ${L}_{\mathrm{sym}}$, and ${K}_{\mathrm{sym}}$ using both crust matching procedures: $P\left({\mu}_{t}\right)$ (blue) and $P\left({\u03f5}_{t}\right)$ (red).

**Figure 2.**Crust–core transition densities, ${n}_{t}$, for the $P\left({\mu}_{t}\right)$ (

**left**) and $P\left({\u03f5}_{t}\right)$ (

**right**) matching procedures. The blue dashed line represents the mean value.

**Figure 3.**Left panel: Correlations between parameters for $P\left(\mu \right)$ (bottom triangle) and $P\left(\u03f5\right)$ (upper triangle) crust matching procedures. Right panel: Correlation difference from the crust matching procedures.

**Figure 4.**Top panels: Mean value (solid line), mean value $\pm 2\sigma $ (filled region), and maximum/minimum values (dashed lines) for the NS radius (

**left**), Love number ${k}_{2}$ (

**center**), and $\mathrm{\Lambda}$ (

**right**). The color identifies the matching procedure: Blue for $P\left(\mu \right)$ and red for $P\left(\u03f5\right)$ Middle panels: Difference between mean values, $\mathrm{\Delta}A=|{\overline{A}}_{\u03f5}-{\overline{A}}_{\mu}|$, where $A=\{R,{k}_{2},\mathrm{\Lambda}\}$. Bottom panels: Relative difference (%), defined by $\mathrm{\Delta}A=100\times \left(\right|{\overline{A}}_{\u03f5}-{\overline{A}}_{\mu}\left|\right)/\overline{A}$, where $\overline{A}=({\overline{A}}_{\u03f5}+{\overline{A}}_{\mu})/2$.

**Figure 5.**Top panels: Effective tidal deformability of the binary, $\tilde{\mathrm{\Lambda}}$, as a function of q for ${M}_{\mathrm{chirp}}=1.186{M}_{\odot}$. The mean (solid line), $\pm 2\sigma $ (filled region), and max/min values (dashed lines) are shown for each dataset, $P\left(\mu \right)$ (blue) and $P\left(\u03f5\right)$ (red). Middle panels: Difference between mean values, $\mathrm{\Delta}\tilde{\mathrm{\Lambda}}=|{\overline{\tilde{\mathrm{\Lambda}}}}_{\u03f5}-{\overline{\tilde{\mathrm{\Lambda}}}}_{\mu}|$. Bottom panels: Relative difference (%), defined by $\mathrm{\Delta}\tilde{\mathrm{\Lambda}}=100\times \left(\right|{\overline{\tilde{\mathrm{\Lambda}}}}_{\u03f5}-{\overline{\tilde{\mathrm{\Lambda}}}}_{\mu}\left|\right)/\overline{\tilde{\mathrm{\Lambda}}}$, where $\overline{\tilde{\mathrm{\Lambda}}}=({\overline{\tilde{\mathrm{\Lambda}}}}_{\u03f5}+{\overline{\tilde{\mathrm{\Lambda}}}}_{\mu})/2$.

**Figure 6.**Prediction difference in NS radius (left), Love number ${k}_{2}$ (center), and $\mathrm{\Lambda}$ (right). Top panels: Mean difference values (solid line), $\mathrm{\Delta}A=\frac{1}{N}{\sum}_{i}|{A}_{\u03f5}^{{\mathrm{EoS}}_{i}}-{A}_{\mu}^{{\mathrm{EoS}}_{i}}|$, where $A=\{R,{k}_{2},\mathrm{\Lambda}\}$ and N is the number of models, $\mathrm{\Delta}A\pm 2\sigma $ (filled region), and maximum/minimum $\mathrm{\Delta}A$ values (dashed lines); Bottom panels: Relative difference (%), defined by $\mathrm{\Delta}A=100\times (\frac{1}{N}{\sum}_{i}|{A}_{\u03f5}^{{\mathrm{EoS}}_{i}}-{A}_{\mu}^{{\mathrm{EoS}}_{i}}\left|\right)/\overline{A}$, where $\overline{A}=({\overline{A}}_{\u03f5}+{\overline{A}}_{\mu})/2$.

**Figure 7.**Effective tidal deformability of the binary, $\tilde{\mathrm{\Lambda}}$, as a function of q for ${M}_{\mathrm{chirp}}=1.186{M}_{\odot}$. The mean (solid line), $\pm 2\sigma $ (filled region), and max/min values (dashed lines) are shown for each dataset, $P\left(\mu \right)$ (blue) and $P\left(\u03f5\right)$ (red).

**Table 1.**The mean ${\overline{P}}_{i}$ and standard deviation $\sqrt{{\sigma}_{{P}_{i}}}$ of the multivariate Gaussian, where ${\sigma}_{{P}_{i}}$ is the variance of the parameter ${P}_{i}$. Our equation of states (EoSs) are sampled using the initial distribution for ${P}_{i}$ assuming that there are no correlations among the parameters. All the quantities are in units of MeV.

${\mathit{P}}_{\mathit{i}}$ | ${\mathit{E}}_{\mathrm{sym}}$ | ${\mathit{L}}_{\mathrm{sym}}$ | ${\mathit{K}}_{\mathrm{sat}}$ | ${\mathit{K}}_{\mathrm{sym}}$ | ${\mathit{Q}}_{\mathrm{sat}}$ |
---|---|---|---|---|---|

${\overline{P}}_{i}$ | 32 | 60 | 230 | −100 | 300 |

$\sqrt{{\sigma}_{{P}_{i}}}$ | 2 | 15 | 20 | 100 | 400 |

**Table 2.**The mean and standard deviation (std) of the empirical parameters. All the quantities are in units of MeV.

$\mathit{P}\left(\mathit{\mu}\right)$ | $\mathit{P}\left(\mathit{\u03f5}\right)$ | |||
---|---|---|---|---|

mean | std | mean | std | |

${K}_{\mathrm{sat}}$ | 225.32 | 17.47 | 227.75 | 19.24 |

${Q}_{\mathrm{sat}}$ | −83.20 | 30.26 | −83.69 | 34.93 |

${E}_{\mathrm{sym}}$ | 33.18 | 1.84 | 31.41 | 1.90 |

${L}_{\mathrm{sym}}$ | 61.89 | 8.14 | 77.70 | 7.47 |

${K}_{\mathrm{sym}}$ | −26.63 | 30.45 | −32.32 | 37.07 |

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Ferreira, M.; Providência, C.
Neutron Star Properties: Quantifying the Effect of the Crust–Core Matching Procedure. *Universe* **2020**, *6*, 220.
https://doi.org/10.3390/universe6110220

**AMA Style**

Ferreira M, Providência C.
Neutron Star Properties: Quantifying the Effect of the Crust–Core Matching Procedure. *Universe*. 2020; 6(11):220.
https://doi.org/10.3390/universe6110220

**Chicago/Turabian Style**

Ferreira, Márcio, and Constança Providência.
2020. "Neutron Star Properties: Quantifying the Effect of the Crust–Core Matching Procedure" *Universe* 6, no. 11: 220.
https://doi.org/10.3390/universe6110220