# Masses of Compact (Neutron) Stars with Distinguished Cores

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. One-Component Static Cool Compact Stars: TOV Equations

#### 2.1. Scaling of TOV Equations and Compact/Neutron Star Masses and Radii

PSR | $\mathit{M}\phantom{\rule{3.33333pt}{0ex}}\left[{\mathit{M}}_{\odot}\right]$ | $\mathit{R}$ [km] |

1.4 | $11.{94}_{-0.87}^{+0.76}$${}^{\left(1\right)}$ [12], $12.45\pm 0.65$ [13], $12.{33}_{-0.81}^{+0.76}$ [15] | |

J0030+0451 | $1.{34}_{-0.16}^{+0.15}$ | $12.{71}_{-1.19}^{+1.14}$ [16] |

$1.{44}_{-0.14}^{+0.15}$ | $13.{02}_{-1.06}^{+1.24}$ [14], $12.{18}_{-0.79}^{+0.56}$ [17] | |

J1614–2230 | $1.908\pm 0.016$ [78] | |

J0348+0432 | $2.01\pm 0.04$ [79] | |

J0740+6620 | $2.{072}_{-0.066}^{+0.067}$ | $12.{39}_{-0.98}^{+1.30}$ [15] |

$2.08\pm 0.07$ [80] | $13.{7}_{-1.5}^{+2.6}$${}^{\left(2\right)}$ [13], $11.{96}_{-0.81}^{+0.86}$ [12] | |

0952-0607 ${}^{\left(3\right)}$ | $2.35\pm 0.17$ [81] |

^{(1)}90% confidence.

^{(2)}With nuclear physics constraints at low density and gravitational radiation data from GW170817 added in, the inferred radius drops to (12.35 ± 0.75) km [13].

^{(3)}Black-widow binary pulsar PSR 0952-0607.

#### 2.2. Solving TOV Equations

## 3. Small-Core Approximation and Beyond

#### 3.1. One-Component Core

#### 3.2. Multi-Component Cores

## 4. Core-Corona Decomposition with NY$\Delta $ DD-EM2 EoS

#### 4.1. Trace Anomaly

#### 4.2. Distinguished Cores with NY$\Delta $ Envelope

#### 4.3. Example of Radial Pressure and Mass Profiles

## 5. Summary

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Emergence of Hadron Masses

#### Appendix A.1. Three Pillars of EHM

#### Appendix A.1.1. Running Quark Mass

#### Appendix A.1.2. Running Gluon Mass

#### Appendix A.1.3. Process-Independent Effective Charge

#### Appendix A.2. Hadrons in Vacuum

#### Appendix A.3. Hadrons in Cold Dense Medium

#### Appendix A.4. Supplementary Remarks

**Figure A1.**Masses within the SM. Mysterious concentration of bare SM masses (quarks $[u,d,s,c,t,b]$, leptons $[e,\mu ,\tau ]$, gauge bosons $[{W}^{\pm},{Z}^{0}]$, Higgs $\left[H\right]$) and separation of neutrinos (${\nu}^{\prime}s=\left[{\nu}_{e,\mu ,\tau}\right])$ on a large energy scale ranging from present-day cosmic background radiation ${\omega}_{2.7\phantom{\rule{3.33333pt}{0ex}}K}^{CBR}\approx 0.233\times {10}^{-3}$ eV to Planck mass ${m}_{Pl}=\sqrt{\hslash c/{G}_{N}}\approx 1.22\times {10}^{19}$ GeV. Only the QED and QCD gauge Bosons $[\gamma ,g]$ remain massless.

## Appendix B. Holographic Approach to the EoS

**Figure A2.**Trace anomaly measure $\Delta $ (

**left**panel) and ratio $e/p$ (

**middle**panel) for the holographic model with tuned parameters to describe the lattice QCD data [133] (small crosses) at ${\mu}_{B}=0$. Errors are constructed either from combining the respective maximum and minimum values (vertical error bars) or by error propagation in quadrature (blueish error bars). The scaled susceptibility ${\chi}_{2}/{T}^{2}$ is displayed in the right panel; data (symbols) from [136].

**Figure A3.**Scaled pressure $p/{T}^{4}$ (

**left**panel), ratio $e/p$ (

**middle**panel), and scaled baryon density ${n}_{B}/{T}^{3}$ (

**right**panel) as a function of temperature for ${\mu}_{B}/T=1$ (blue) and 2 (red) in comparison with the data [133] (symbols with error bars).

**Figure A4.**Curves of constant pressure over the T-${\mu}_{B}$ plane. In such a way, the pressure data on the T axis are directly “transported” towards the ${\mu}_{B}$ axis, in particular $p(T=0,{\mu}_{B}^{\left(0\right)})={p}_{0}:=p({T}_{0},{\mu}_{B}=0)$ along the constant-pressure curve $T\left({\mu}_{B}\right){|}_{p={p}_{0}}$ starting at $T({\mu}_{B}=0)={T}_{0}$ and terminating at $T\left({\mu}_{B}^{\left(0\right)}\right)=0$. The energy density, with $s(T,{\mu}_{B})$ and ${n}_{B}(T,{\mu}_{B})$ given, then follows from $e=-p+Ts+{\mu}_{B}{n}_{B}$ (Gibbs–Duhem).

**Left**panel: A simple toy model is employed here for the purpose of demonstration ($s=4a{T}^{3}+2bT{\mu}_{B}^{2}$, ${n}_{B}=4c{\mu}_{B}^{3}+2b{T}^{2}{\mu}_{B}$, and numerical values $b/a=0.027384$, $c/a=0.000154$ referring to a two-flavor ideal quark-gluon plasma).

**Right**panel: For the holographic model (A1), (A3) and (A4) in a region (grey hatched) controlled by lattice QCD data [133] on the dark-grey beam sections.

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**Figure 1.**Scaled core masses ${\overline{m}}_{x}$ as a function of the scaled core radii ${\overline{r}}_{x}$ for various values of ${\Delta}^{corona}=\Delta $ and ${v}_{s}^{2}$ for a one-component medium with EoS (4). The

**left**(

**right**) panel is for $\lambda =1$ (1.5). The scaling quantity is $\mathfrak{s}={p}_{x}$. For central pressures ${\overline{p}}_{c}=1.{1}^{n}$, $n=1\cdots 55$. The displayed curves are limited by ${\overline{m}}_{x}<{\overline{r}}_{x}/2$ (black hole limit) and ${\overline{m}}_{x}>\frac{4\pi}{3}\lambda \frac{3}{1-3{\Delta}^{corona}}{\overline{r}}_{x}^{3}$. The latter expression is for the respective asymptotic curve in the small-${\overline{r}}_{x}$ region. To convert to usual dimensions, one employs ${r}_{x}={\overline{r}}_{x}\frac{86.9\phantom{\rule{3.33333pt}{0ex}}\mathrm{k}\mathrm{m}}{\sqrt{{p}_{x}/100\phantom{\rule{3.33333pt}{0ex}}{[\mathrm{MeV}/\mathrm{f}\mathrm{m}}^{3}]}}$ and ${m}_{x}={\overline{m}}_{x}\frac{58.8\phantom{\rule{3.33333pt}{0ex}}{M}_{\odot}}{\sqrt{{p}_{x}/100\phantom{\rule{3.33333pt}{0ex}}{[\mathrm{MeV}/\mathrm{f}\mathrm{m}}^{3}]}}$, where “${p}_{x}/100\phantom{\rule{3.33333pt}{0ex}}\left[{\mathrm{MeV}/\mathrm{f}\mathrm{m}}^{3}\right]$” denotes the scaling pressure ${p}_{x}$ in units of 100 MeV/fm${}^{3}$. The approximations (10) and (11) apply only in the small-${\overline{m}}_{x}$ and small-${\overline{r}}_{x}$ regions.

**Figure 2.**

**Left**panel: As the left panel of Figure 1, but for a two-component medium of SM + MW matter with ${p}_{\left(1\right)}\left(r\right)={p}_{\left(2\right)}\left(r\right)$ and EoS (4) for both components. Note the difference to the two-fluid core-shell construction in Ref. [103].

**Right**panel: As a left panel but for a three-component medium.

**Figure 3.**

**Left**panel: Trace anomaly measure $\Delta =1/3-p/e$ as a function of $\eta \equiv lne/\left({n}_{0}{m}_{p}\right)$ for the NY$\Delta $ DD-EM2 EoS of [31] (black curve with symbols) and the adjusted parameterization by Equation (7) in Ref. [105] (red curve; ${\eta}_{c}=1.1$, other parameters as in Ref. [105], see Equation (14) below). NY$\Delta $ DD-EM2 shows a softening at $p\approx 10$ MeV/fm${}^{3}$, $\eta \approx 0.5$, caused by the onset of $\Delta $ proliferation, similar to Ref. [108].

**Right**panel: Mass-radius relation for NY$\Delta $ DD-EM2 EoS of Ref. [31] (solid black curve) in comparison with the parameterization by Equation (7) in Ref. [105] (see Equation (14) below for three values of the parameter ${\eta}_{c}$ (read dashed/solid/dotted curves for ${\eta}_{c}=1.2$, 1.1 and 1.0).

**Figure 4.**Mass-radius plane and its occupancy by compact stars with given core radii ${r}_{x}$ (blue curves with dots at core masses ${m}_{x}={10}^{-4}$$1.{5}^{n}{M}_{\odot}$, $n=1,2,3\cdots $ from right to left; the open circles depict points for $n=15$, and the right-most endpoints are for $n=1$). The values of ${p}_{x}$ are 50 (

**left**panel), 100 (

**middle**panel), and 150 MeV/fm${}^{3}$ (

**right**panel). The fat solid curve is obtained by standard integration of the TOV equations using the NY$\Delta $ EoS tabulated in Ref. [31] with linear interpolation both in between the mesh and from the tabulated minimum energy density to the $p=0$ point at ${e}_{0}=1$ MeV/fm${}^{3}$. The asterisks display the mass-radius values for ${p}_{c}={p}_{x}$. That is, the sections above the asterisks (dotted curves) are for a particular continuation of NY$\Delta $ at $p>{p}_{x}$, which is, (trivially) in this case, NY$\Delta $ itself. One could instead employ the parameterization Equation (4) which would deliver another dotted curve. For other examples, in particular the small-R region near black hole and Buchdahl limits, the interested reader is referred to Ref. [74].

**Figure 5.**Pressure $p\left(r\right)$ (

**left**panel) and mass $m\left(r\right)$ (

**right**panel) as a function of the radius r for the special value ${p}_{x}=100$ MeV/fm${}^{3}$, selected here as the end point of the “reliable EoS” NY$\Delta $ at $p\le {p}_{x}$. Assuming a possible continuation at $p>{p}_{x}$ by NY$\Delta $ itself as a particular example, it yields the fat solid curves for the (ad hoc) choice ${p}_{c}=200$ MeV/fm${}^{3}$. Keeping the resulting value ${r}_{x}$ from $p\left({r}_{x}\right)={p}_{x}$ and integrating the TOV equations in the corona, $r\ge {r}_{x}$ with ${p}_{x}$ and ${m}_{x}^{(1,2)}$ as initial values, one gets the dashed blue (${m}_{x}^{\left(1\right)}=0.5{m}_{x}$) and dotted blue (${m}_{x}^{\left(2\right)}=2{m}_{x}$) curves, where the respective values of R and M can be read off. Using ${m}_{x}$ as core mass, the blue circles (on top of the fat black curve at $r\ge {r}_{x}$) are obtained. Using a multitude of values ${m}_{x}^{\left(n\right)}$ would generates one additional blue curve in Figure 4. In the present example, for ${r}_{x}=6$ km.

**Table 1.**Core compactness ${\overline{C}}_{x}=2{\overline{m}}_{x}^{max}/{\overline{r}}_{x}{|}_{{\overline{m}}_{x}^{max}}$ for various values of ${v}_{s}^{2}$ (sound velocity squared in the core) and ${\Delta}^{corona}=\Delta $. For $\lambda =1$.

$\mathsf{\Delta}$ | 0.1 | 0.2 | 0.3 | |
---|---|---|---|---|

${\mathit{v}}_{\mathit{s}}^{2}$ | ||||

1 | 0.64 | 0.67 | 0.70 | |

1/3 | 0.49 | 0.51 | 0.53 | |

1/6 | 0.37 | 0.38 | 0.40 |

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**MDPI and ACS Style**

Zöllner, R.; Ding, M.; Kämpfer, B. Masses of Compact (Neutron) Stars with Distinguished Cores. *Particles* **2023**, *6*, 217-238.
https://doi.org/10.3390/particles6010012

**AMA Style**

Zöllner R, Ding M, Kämpfer B. Masses of Compact (Neutron) Stars with Distinguished Cores. *Particles*. 2023; 6(1):217-238.
https://doi.org/10.3390/particles6010012

**Chicago/Turabian Style**

Zöllner, Rico, Minghui Ding, and Burkhard Kämpfer. 2023. "Masses of Compact (Neutron) Stars with Distinguished Cores" *Particles* 6, no. 1: 217-238.
https://doi.org/10.3390/particles6010012