Masses of Compact (Neutron) Stars with Distinguished Cores

Abstract

In this paper, the impact of core mass on the compact/neutron-star mass-radius relation is studied. Besides the mass, the core is parameterized by its radius and surface pressure, which supports the outside one-component Standard Model (SM) matter. The core may accommodate SM matter with unspecified (or poorly known) equation-of-state or several components, e.g., consisting of admixtures of Dark Matter and/or Mirror World matter etc. beyond the SM. Thus, the admissible range of masses and radii of compact stars can be considerably extended.

1. Introduction

Strong interaction rules a variety of systems, ranging from hadrons to nuclei up to neutron stars. The related mass scales are typically ≥$m_{\pi }=0.13\mathrm{GeV}$ for mesons, ≥$m_p=0.938\mathrm{GeV}$ for baryons, $(1\cdots 250)m_p$ for nuclei, and $\mathcal{O}\left({10}^{57}\right)m_p$ for neutron stars. ($m_{\pi ,p}$ stand for pion and proton masses.) Further systems awaiting their confirmation are glueballs ($\mathcal{O}\left(\mathrm{GeV}\right)$) and non-baryon stars, such as pion stars [1]. Besides weak interaction, it is the long-range Coulomb interaction that limits the size (or baryon number) of nuclei, and gravity is the binding force of matter in neutron stars. The phenomenon of hadron mass emergence is a fundamental issue tightly related to non-perturbative effects in the realm of QCD, cf., Refs. [2,3] and citations therein. Once the masses and interactions among hadrons are understood, one can make the journey to address the masses and binding energies of nuclei and then jump to constituents of neutron stars. Despite the notion, cool neutron stars accommodate, in the crust, various nuclei immersed in a degenerate electron-muon environment (maybe as “pasta” or “spaghetti” or crystalline medium). In the deeper interior, above the neutron drip density, neutrons and light clusters begin to dominate the matter composition. These constituents and their interactions govern the mass (or energy density) of the medium. At nuclear saturation density, $n_0\approx 0.15{\mathrm{f}\mathrm{m}}^{-3}$, one meets conditions similar to the interior of heavy nuclei but with crucial impact of the symmetry energy when extrapolating from nuclear matter with comparable proton and neutron numbers to a very asymmetric proton-neutron mixture. Above saturation density, various effects hamper a reliable computation of properties of the strong-interaction medium: three-body interactions may become even more important than at $n_0$, and further baryon species become excited, e.g., strangeness is lifted from vacuum into baryons forming hyperons whose interaction could be a miracle w.r.t. the hyperon puzzle [4], and the relevant degrees of freedom become relativistic. Eventually, at asymptotically high density, the strong-interaction medium is converted into quarks and gluons; color-flavor locking and color superconductivity can essentially determine the medium’s properties. The turn of massive hadronic degrees of freedom into quark-gluon excitations is thereby particularly challenging.

While lattice QCD represents, in principle, an ab initio approach to strong-interaction systems in all their facets, the “sign problem” prevents the access to non-zero baryon number systems. Thus, the exploration of compact/neutron stars, in particular their possible mass range, by stand-alone theory is presently not feasible. Instead, an intimate connection of astrophysical data and compact-star modeling is required.

The advent of detecting gravitational waves from merging neutron stars, the related multimessenger astrophysics [5,6,7,8,9,10,11,12] and the improving mass-radius determinations of neutron stars, in particular by NICER data [13,14,15,16,17], stimulated a wealth of activities. Besides masses and radii, moments of inertia and tidal deformabilities become experimentally accessible and can be confronted with theoretical models [18,19,20,21,22,23,24]. The baseline of the latter ones is provided by non-rotating, spherically symmetric cold dense matter configurations. The sequence of white dwarfs (first island of stability) and neutron stars (second island of stability) and possibly [25] a third island of stability [26,27,28,29,30] shows up when going to more compact objects, with details depending sensitively on the actual equation of state (EoS). The quest for a fourth island has been addressed as well [31,32]. “Stability” means here the damping of radial disturbances, at least. Since the radii of configurations of the second (neutron stars) and third (hypothetical quark/hybrid stars [33,34,35,36,37,38,39,40]) islands are very similar, the notion of twin stars [31,32,33,41,42] has been coined for equal-mass configurations; “masquerade” was another related term [43].

We emphasize the relation of ultra-relativistic heavy-ion collision physics, probing the EoS $p(T,{\mu }_B\approx 0)$, and compact star physics, probing $p(T\approx 0,{\mu }_B)$ when focusing on static compact-star properties [44,45]. Of course, in binary or ternary compact-star merging-events, also finite temperatures T and a large range of baryon-chemical potential ${\mu }_B$ are probed, which are accessible in medium-energy heavy-ion collisions [46]. Implications of the conjecture of a first-order phase transition at small temperatures and large baryo-chemical potentials or densities [47,48,49,50] can also be studied by neutron-hybrid-quark stars [51,52,53,54]. It is known since some time [26,27,28,55,56] that a cold EoS with special pressure-energy density relation $p\left(e\right)$, e.g., a strong local softening up to first-order phase transition with a density jump, can give rise to a “third family” of compact stars, beyond white dwarfs and neutron stars. In special cases, the third-family stars appear as twins of neutron stars [43,57,58]. Various scenarios of the transition dynamics to the denser configuration as mini-supernova have also been discussed quite early [59,60].

While the Standard Model (SM) of particle physics seems to accommodate nearly all of the observed phenomena of the micro-world, severe issues remain. Among them is the ${(g-2)}_{\mu }$ puzzle or the proton’s charge radius. Another fundamental problem is the very nature of Dark Matter (DM): Astrophysical and cosmological observations seem to require inevitably its existence, but details remain elusive despite many concerted attempts, e.g., Refs. [61,62,63,64]. Supposing that DM behaves like massive particles, it could be captured gravitationally in the centers of compact stars [65,66,67], thus providing a non-SM component there. This would be an uncertainty on top of the less reliably known SM-matter state. Beyond the SM, other feebly interacting particles could also populate compact stars. A candidate scenario is provided, for instance, by Mirror World (MW) [68,69,70,71,72], i.e., a parity-symmetric complement to our SM-world with very tiny non-gravity interaction. There are many proposals of portals from our SM-world to such beyond-SM scenarios, cf., Ref. [73].

Guided by these remarks we follow here an access to static cold compact stars already launched in [74]: We describe the core by a minimum of parameters and determine the resulting compact-star masses and radii by assuming the knowledge of the equation of state of the SM-matter enveloping the core. A motivation is the quest of a mass gap between compact stars and black holes.

Our paper is organized as follows. In Section 2 we recall the Tolman–Oppenheimer–Volkoff equations, their scaling property, and introduce our core-corona decomposition. Small cores with and without MW/DM admixtures are considered in Section 3. Section 4 is devoted to the core-corona decomposition, where a specific EoS is deployed for the explicit construction. We summarize in Section 5. We supplement our paper in Appendix A by a brief retreat to the emergence of hadron masses as a key issue in understanding the typical scales of compact (neutron) star masses. Appendix B sketches a complementary approach: the construction of an EoS by holographic means, thus transporting information of a hot (quark-gluon) QCD EoS to a cool EoS and connecting heavy-ion collisions and compact star physics. These appendices survey (Appendix A) and exemplify (Appendix B) symmetry and the governing principles where the access to astrophysical objects is based upon.

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

The standard modeling of compact star configurations is based on the Tolman–Oppenheimer–Volkoff (TOV) equations

$$\begin{array}{cc}\hfill \frac{\mathrm{d}p}{\mathrm{d}r}& =-G_N\frac{[e\left(r\right)+p\left(r\right)][m\left(r\right)+4\pi r^{3}p\left(r\right)]}{r^2[1-\frac{2G_{N}m\left(r\right)}{r}]},\hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}m}{\mathrm{d}r}& =4\pi r^{2}e\left(r\right),\hfill \end{array}$$

(2)

resulting from the energy-momentum tensor of a one-component static isotropic fluid (described locally by pressure p and energy density e as only quantities relevant for the medium) and spherical symmetry of both space-time and matter, within the framework of Einstein gravity without cosmological term [75]. Newton’s constant is denoted by $G_N$, and natural units with $c=1$ are used, unless when relating mass and length and energy density, where $\hslash c$ is needed.

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

The TOV equations become free of any dimension by the scalings

$$\begin{array}{c}\hfill e=\mathfrak{s}\overline{e},p=\mathfrak{s}\overline{p},r={\left(G_N\mathfrak{s}\right)}^{-1/2}\overline{r},m={\left(G_N^3\mathfrak{s}\right)}^{-1/2}\overline{m},\end{array}$$

(3)

where $\mathfrak{s}$ is a mass dimension-four quantity or has dimension of energy density. It may be a critical pressure of a phase transition or a limiting energy density, e.g., at the boundary matter-vacuum [76]. Splitting up $\mathfrak{s}=n{m}_p$ into a number density n and the energy scale $m_p$, one gets ${\left(G_N\mathfrak{s}\right)}^{-1/2}=7km/\sqrt{{10}^{-2}n/n_0}$ and ${\left(G_N^3\mathfrak{s}\right)}^{-1/2}=4.8M_{\odot }/\sqrt{{10}^{-2}n/n_0}$, where the nuclear saturation density $n_0=0.15$ fm${}^{-3}$ is used as reference density. The scales $m_{\pi ,p}$ facilitate the densities $n=m_{\pi }^3\to 2.3n_0$ and $n=m_p^3\to 833n_0$. That is, the scale solely set by the nucleon mass, i.e., $\mathfrak{s}=m_p^4$ [77], yields ${\left(G_N\mathfrak{s}\right)}^{-1/2}=2.74km$ and ${\left(G_N^3\mathfrak{s}\right)}^{-1/2}=1.86M_{\odot }$, suggesting that the nucleon mass (see Appendix A on its emergence in QCD) determines the gross properties of neutron stars, such as mass and radius (modulus factor $2\pi$) in an order-of magnitude estimate. In contrast, the density estimate via $M=\frac{4\pi }{3}\langle n\rangle m_p{R}^3$ yields, independently of $G_N$, $\langle n\rangle =2n_0\frac{M}{M_{\odot }}\frac{{10}^3}{{(R/\mathrm{k}\mathrm{m})}^3}\approx 4n_0$ for $M=2M_{\odot }$ and $R=10$ km, thus pointing to the importance of the actual numerical values of $\overline{r}$ and $\overline{m}$.

In fact, recent measurements and supplementary work report averaged mean and individual heavy compact/neutron stars masses and radii (often on 67% credible level) as follows.

PSR	$\mathit{M}\left[{\mathit{M}}_{\odot }\right]$	$\mathit{R}$ [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.

One has to add GW190814: gravitational waves from the coalescence of a 23 solar mass black hole with a $2.6M_{\odot }$ compact object [82]. An intriguing question concerns a possible mass gap between compact-star maximum-mass [83] and light black holes, cf., [84,85].

2.2. Solving TOV Equations

Given a unique relationship of pressure p and energy density e as EoS $e\left(p\right)$, in particular at zero temperature, the TOV equations are integrated customarily with boundary conditions $p(r=0)=p_c$ and $m(r=0)=0$ (implying $p\left(r\right)=p_c-\mathcal{O}\left(r^2\right)$ and $m\left(r\right)=0+\mathcal{O}\left(r^3\right)$ at small radii r), and $p\left(R\right)=0$ and $m\left(R\right)=M$ with R as circumferential radius and M as gravitational mass (acting as parameter in the external (vacuum) Schwarzschild solution at $r>R$). The quantity $p_c$ is the central pressure. The solutions $R\left(p_c\right)$ and $M\left(p_c\right)$ provide the mass-radius relation $M\left(R\right)$ in parametric form.

A great deal of effort is presently concerned about the EoS at supra-nuclear densities [86]. For instance, Figure 1 in Ref. [87] exhibits the currently admitted uncertainty: up to a factor of 10 in pressure as a function of energy density. At asymptotically large energy density, perturbative QCD constrains the EoS, though it is just the non-asymptotic supra-nuclear density region that crucially determines the maximum mass and whether twin stars may exist or quark-matter cores appear in neutron stars. Accordingly, one can fill this gap by a big number (e.g., millions [88]) of test EoSs to scan through the possibly resulting manifold of mass-radius curves, see Refs. [89,90,91,92]. However, the possibility that neutron stars may accommodate other components than Standard Model matter, e.g., exotic material as Dark Matter [93,94,95,96], can be an obstacle for the safe theoretical modeling of a concise mass-radius relation in such a manner. Of course, inverting the posed problem with sufficiently precise data of masses and radii as input offers a promising avenue towards determining the EoS [17,89,97,98,99,100,101].

Here, we pursue another perspective. We parameterize the supra-nuclear core by a radius $r_x$ and the included mass $m_x$ and integrate the above TOV equations only within the corona (our notion “corona” is a synonym for “mantel” or “crust” or “envelope” or “shell”, it refers to the complete part of the compact star outside the core, $r_x\le r\le R$), i.e., from pressure $p_x$ to the surface, where $p=0$. This yields the total mass $M(r_x,m_x;p_x)$ and the total radius $R(r_x,m_x;p_x)$ by assuming that the corona EoS $e\left(p\right)$ is reliably known at $p\le p_x$ and that only SM matter occupies that region. Clearly, without knowledge of the matter composition at $p>p_x$ (may it be SM matter with an uncertainly known EoS or may it contain a Dark-Matter admixture, for instance, or monopoles or some other type of “exotic” matter) one does not get a simple mass-radius relation by such a procedure, but admissible area(s) over the mass-radius plane, depending on the core parameters $r_x$ and $m_x$ and the matching pressure $p_x$ and related energy density $e_x$. This is the price of avoiding a special model of the core matter composition.

If the core is occupied by a one-component SM medium, the region $p>p_x$ and $e>e_x$ can be mapped out by many test EoSs which locally obey the constraint $v_s^2\in [0,1]$ to obtain the corresponding region in the mass-radius plane, cf., Figure 2 in Ref. [102] for an example processed by Bayesian inference. This is equivalent, to some extent, to our core-corona decomposition for SM matter-only.

3. Small-Core Approximation and Beyond

3.1. One-Component Core

For small one-component distinguished cores one can utilize the EoS parameterization from a truncated Taylor expansion of $p\left(e\right)\ge p_x$ at $\lambda e_x$, $p\left(e\right)=p\left(\lambda e_x\right)+\frac{\partial p}{\partial e}{|}_{\lambda e_x}(e-\lambda e_x)+\cdots$,

$$\begin{array}{c}\hfill p\left(e\right)=p_x+v_s^2(e-\lambda e_x),\end{array}$$

(4)

where $\lambda >1$ is a density jump at the core boundary and $\lambda =1$ continuously continues (but may be kinky) the corona EoS at $p\ge p_x$; $p_x$ and $e_x$ mark the “end point” of the corona EoS. A Taylor expansion for small cores,

$$\begin{array}{cc}\hfill p\left(r\right)& =\sum _{i=0}^{\infty }{p}_{2i}{r}^{2i},\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill m\left(r\right)& =\sum _{i=0}^{\infty }{m}_{2i+1}{r}^{2i+1},\hfill \end{array}$$

(6)

gives by means of Equations (1) and (2) with (4) (that is $e=\lambda e_x-v_s^{-2}{p}_x+v_s^{-2}{\sum }_{i=0}^{\infty }{p}_{2i}{r}^{2i}$) and $p_{2i+1}=0$ and $m_{2i}=0$ for all $i\in {\mathbb{N}}_0$

$$\begin{array}{c}\hfill m_1=0,m_3=\frac{4\pi }{3}{e}_c,m_{2i+1}=\frac{4\pi v_s^{-2}}{2i+1}{p}_{2i-2}\mathrm{for}i\ge 2,\end{array}$$

(7)

where $e_c\equiv e\left(p_c\right)=\lambda e_x+v_s^{-2}(p_c-p_x)$, and the recurrence

$$\begin{array}{cc}\hfill p_0& =p_c,p_2=-\frac{2\pi }{3}{G}_N(e_c+p_c)(e_c+3p_c),\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill p_{2i}& =\frac{G_N}{2i}\left(2A_i-[\lambda e_x-v_s^{-2}{p}_x](m_{2i+1}+4\pi p_{2i-2})-(1+v_s^{-2})B_{i-1}\right)\mathrm{for}i\ge 1,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & A_i=\sum _{j=0}^{i}2(i-j)m_{2j}{p}_{2i-2j},\hfill \end{array}$$

(9a)

$$\begin{array}{cc}\hfill & B_i=\sum _{j=0}^i(m_{2j+3}+4\pi p_{2j})p_{2i-2j}.\hfill \end{array}$$

(9b)

In leading order one obtains

$$\begin{array}{cc}\hfill r_x& \approx \frac{{\delta }^{1/2}(1-\mathcal{O}\left(\delta \right))}{\sqrt{\frac{2\pi }{3}{G}_N{p}_{x}W}}\approx \frac{60.1\mathrm{k}\mathrm{m}}{\sqrt{W}}\sqrt{\delta \frac{100{\mathrm{M}\mathrm{e}\mathrm{V}/\mathrm{f}\mathrm{m}}^3}{p_x}},\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill m_x& \approx 2\lambda \frac{e_x}{p_x}\frac{{\delta }^{3/2}(1-\mathcal{O}\left(\delta \right))}{\sqrt{\frac{2\pi }{3}{G}_N^3{p}_x{W}^3}}\approx \frac{81.2M_{\odot }}{W^{3/2}}\lambda \frac{e_x}{100{\mathrm{MeV}/\mathrm{f}\mathrm{m}}^3}\times {\left(\delta \frac{100{\mathrm{MeV}/\mathrm{f}\mathrm{m}}^3}{p_x}\right)}^{3/2},\hfill \end{array}$$

(11)

where $\delta :=p_c/p_x-1$ and $W:=3+4\lambda \frac{e_x}{p_x}+{\lambda }^2{\left(\frac{e_x}{p_x}\right)}^2$. The scale setting is by $p_x$ and $\lambda e_x/p_x$. The core-mass–core-radius relation is $m_x\left(\delta \right)\approx \frac{4\pi }{3}{p}_x\lambda \left(\frac{e_x}{p_x}\right)r_x{\left(\delta \right)}^3(1+\mathcal{O}\left(\delta \right))$.

To control and extend the above approximations we numerically solve the scaled TOV equations by assuming that Equation (4) holds true in the core. (Of course, this is an ad hoc assumption aimed at providing an explicit example of mass-radius relations of the core.) The core-corona matching is at $p_x$, i.e., the maximum (minimum) pressure of the corona (core) EoS. The related energy density is $e_x=3p_x/(1-3{\Delta }^{corona})$, where ${\Delta }^{corona}$ denotes the trace anomaly measure discussed below in Equation (12). It is used here for a suitable parameterization of the double $(e_x,p_x)$ by means of the corona EoS. The core EoS Equation (4), $e\left(p\right)=\lambda e_x+v_s^{-2}(p-p_x)$, enters, after scaling according to Equation (3), the dimensionless TOV equations

$$\begin{array}{cc}\hfill \frac{\mathrm{d}\overline{p}}{\mathrm{d}\overline{r}}& =-\frac{[\overline{e}\left(\overline{r}\right)+\overline{p}\left(\overline{r}\right)][\overline{m}\left(\overline{r}\right)+4\pi {\overline{r}}^3\overline{p}\left(\overline{r}\right)]}{{\overline{r}}^2[1-\frac{2\overline{m}\left(\overline{r}\right)}{\overline{r}}]},\hfill \end{array}$$

(11a)

$$\begin{array}{cc}\hfill \frac{\mathrm{d}\overline{m}}{\mathrm{d}\overline{r}}& =4\pi {\overline{r}}^2\overline{e}\left(\overline{r}\right),\hfill \end{array}$$

(11b)

for $\overline{p}\in [{\overline{p}}_c,{\overline{p}}_x]$ to get ${\overline{r}}_x\left({\overline{p}}_c\right)$ and ${\overline{m}}_x\left({\overline{p}}_c\right)$. The corresponding scaled core mass vs. core radius relations are exhibited in Figure 1, where the scaling quantity $\mathfrak{s}=p_x$ is employed, i.e., ${\overline{p}}_x=1$. The figure offers a glimpse on the systematic of the parameter dependence. Note that it applies to all values of $p_x>0$. The finite pressure and energy density at the core boundary facilitates a pattern of mass-relations and dependence on sound velocity as known from bag model EoS, cf., Figure 7 in Ref. [76]. $\lambda >1$ causes an overall shrinking of the pattern plus a slight up-shift; see the right panel for $\lambda =1.5$. Considering, e.g., $e_x\in [150,1500]$ MeV/fm${}^3$ and the scaling $\propto 1/\sqrt{e_x}$, cf., (3), the core masses and radii change by a factor up to three, depending on the actual value of $e_x$.

3.2. Multi-Component Cores

In a multi-component medium ( note that our considerations apply to components which do not mutually interact in a direct microscopical manner, but are coupled solely by the common gravity field) with known EoSs one has to add the contributions by $[m\left(r\right)+4\pi r^{3}p\left(r\right)]\to {\sum }_i[m_{\left(i\right)}\left(r\right)+4\pi r^3{p}_{\left(i\right)}\left(r\right)]$ and $[1-\frac{2G_{N}m\left(r\right)}{r}]\to [1-\frac{2G_N{\sum }_i{m}_{\left(i\right)}\left(r\right)}{r}]$ and solve in parallel the multitude of TOV equations for the components i. A particular minimum-parameter model is Mirror World matter (component $i=2$) which is completely symmetric to SM matter (component $i=1$), i.e., $p_{\left(1\right)}\left(r\right)=p_{\left(2\right)}\left(r\right)$ etc., turning ${\sum }_{\left(i\right)}\to 2$. The results are exhibited in Figure 2, left. The increased energy density by the two superimposed fluids cause a shrinking of the core-mass–core-radius curves similar to the increase of $\lambda$ in Figure 1. The extension to three components is exhibited in Figure 2, right. Here, ${\sum }_{\left(i\right)}\to 3$ applies. Due to further increased energy density and pressure, the configurations become even more “compact”, i.e., core masses and core radii shrink further on.

One may quantify the core compactness by ${\overline{C}}_x:=2{\overline{m}}_x^{max}/{\overline{r}}_x{|}_{{\overline{m}}_x^{max}}$ and note that, within the scanned parameter patch, it (i) is independent of the number of fluids, (ii) increases slightly with ${\Delta }^{corona}$ at $v_s^2=const$ and (iii) decreases with decreasing $v_s^2=const$ at ${\Delta }^{corona}=const$, see

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

. Note that the usual definition of compactness is without the factor of two.

Of course, more general is an asymmetric Mirror World component facilitating $p_{\left(1\right)}\left(r\right)\ne p_{\left(2\right)}\left(r\right)$, especially $p_{\left(1\right)}(r=0)\ne p_{\left(2\right)}(r=0)$, even for the common EoS (4), dealt with in Appendix A in Ref. [74]. We do not dive into various conceivable scenarios here and refer the interested reader to Refs. [103,104].

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

4.1. Trace Anomaly

An example of an EoS largely compatible with neutron star data is NY$\Delta$ based on the DD-ME2 density functional [31]. Despite some peculiarities (see the left panel in Figure 3), it shares features recently advocated as essential w.r.t. QCD tracy anomaly and conformality [105,106]. A suitable measure of the trace anomaly is

$$\begin{array}{c}\hfill \Delta :=\frac{1}{3}-\frac{p}{e}\end{array}$$

(12)

which is related to the sound velocity squared

$$\begin{array}{c}\hfill v_s^2:=\frac{\partial p}{\partial e}=\frac{n_B}{{\mu }_b{\chi }_B}=\frac{1}{3}-\Delta -e\frac{\partial \Delta }{\partial e},\end{array}$$

(13)

where $n_B$ stands for the baryon density, ${\mu }_B$ the baryo-chemical potential, and ${\chi }_B={\partial }^{2}p/\partial {\mu }_B=\partial n_B/\partial {\mu }_B$ is the second-order cumulant of the net-baryon number density. Thermodynamic stability and causality constrain $\Delta \in [-2/3,1/3]$. Restoration of scale invariance means $\Delta \to 0$ and $v_s^2\to 1/3$. Although $\Delta$ approaches monotonically to zero, $v_s^2$ can develop a peak at lower energy densities. In fact, $v_s^2$ as a function of $\eta \equiv lne/\left(n_0{m}_p\right)$ displays such a peak at $\eta \approx 1.3$ (see Figure 2 in Ref. [105] or Figure 1 in Ref. [88] and discussion in Ref. [107]), which should be considered a signature of conformality even at strong coupling. Continuing with the adjustment of $\Delta$ beyond the values tabulated in Ref. [31] (see the symbols in Figure 3, left), $v_s^2$ drops first slightly below $1/3$ and approaches then slowly $1/3$ from below, in agreement with QCD asymptotic.

The comparison of the resulting mass-radius relations is exhibited in Figure 3, right. We follow the parameterization [105]

$$\begin{array}{c}\hfill \Delta =\frac{1}{3}-\frac{1}{3}\left(1-\frac{A}{B+{\eta }^2}\right)\frac{1}{1+e^{-\kappa (\eta -{\eta }_c)}},\end{array}$$

(14)

however, with parameters adjusted to [31]: $\kappa =3.45$, ${\eta }_c=1.1$, $A=2$ and $B=20$, see Figure 3. Despite the tiny differences of the red curve (for adjusted $\Delta$) and the solid black curve with markers (for NY$\Delta$ DD-EM2 EoS of [31]), the overall shapes of $M\left(R\right)$ differ in detail, see Figure 3, right. While the maximum masses agree, the radii for NY$\Delta$ DD-EM2 EoS of [31] are greater by about 1 km at $M\approx 1.4M_{\odot }$. A small dropping of the parameter ${\eta }_c$ from 1.2 to 1.0 in the $\Delta$ EoS lets us increase the radius by 1 km. All these differences can be traced back essentially to differences in the low-density part of the employed EoSs.

4.2. Distinguished Cores with NY$\Delta$ Envelope

Focusing now on NY$\Delta$ DD-EM2 EoS of Ref. [31] and the core-corona decomposition, one obtains the resulting mass-radius radius relation exhibited in Figure 4. We find it convenient to keep the respective core radius constant and vary the core masses as $m_x={10}^{-4}1.5^n{M}_{\odot }$, $n=1,2,3\cdots$. The very small core masses (of course, a large core with several-km radius and small included mass is a very exotic thing; it may be referred to as bubble or void with surface pressure $p_x$ and the stable interface to SM matter) occupy the right end of the solid blue curves, where the dots refer to the increasing values of n. Heavy core masses occupy the left sections of the blue curves, i.e., larger values of n. For considerably smaller values of the core radii (not displayed), the blue curves approach the conventional mass-radius curve of the NY$\Delta$ DD-EM2 EoS of Ref. [31] (fat black curve). The limit $r_x\to 0$ and $m_x\to 0$ of the core-corona decomposition curve is depicted by the asterisk, which also agrees with the mass and radius of NY$\Delta$ DD-EM2 EoS of Ref. [31] with $p_c=p_x$. Increasing values of $p_x$ make an up-shift of the core-corona mass-radius curves for a constant value of the core radius $r_x$, and a larger range of masses is occupied.

All the features discussed in Section 2 in Ref. [74] are recovered, e.g., the convex shape of the curves $r_x=const$, which however becomes visible only by extending the plot towards smaller values of R (not shown). The right end points of the core-corona curves refer to very small values of $m_x$, while the not displayed left end points are on the limiting black hole curve $2G_{N}M=R$. However, the core-corona decomposition shows that the maximum-mass region of NY$\Delta$-DD-ME2 (see fat solid curves) is easily uncovered too, interestingly with sizeable core radii and noticeably smaller up to larger total radii.

We refrain from displaying the core-corona mass-radius curves for $m_x=const$. They can be easily inferred by connecting the points $n=const$ in each of the panels in Figure 4.

To stress the relation to the usual mass-radius relation $M\left(R\right)$ obtained from the parametric representations $M\left(p_c\right)$ and $R\left(p_c\right)$, let us mention that the respective dotted curve sections above the asterisk on the fat solid curves are just an example—here simply NY$\Delta$ DD-EM2 EoS of Ref. [31] for $p>p_x$. Other continuations of the EoS at $p>p_x$ are within the region filled by the blue core-corona curves, which however may not be completely mapped out by many conceivable EoSs: The core can contain more than just SM matter, such as Dark Matter or Mirror Matter or other exotic material. In the simplest case, a multi-component composition with hypothesized EoS for each component can be used for the explicit construction. All that counts in our core-corona decomposition is that a core with radius $r_x$ and included mass $m_x$ is present and supports the pressure $p_x$ at its boundary. By definition, outside the core, only SM matter with known EoS is there. (For a dedicated study of crust properties, cf., Ref. [109].)

4.3. Example of Radial Pressure and Mass Profiles

To illustrate that feature in some detail let us consider an example and select $p_x=100$ MeV/fm${}^3$. Assuming that the continuation of the EoS above this $p_x$ is, hypothetically, by NY$\Delta$ itself, one obtains the pressure and mass profiles as displayed by black solid curves in Figure 5 for $p_c=200$ MeV/fm${}^3$. Clearly, taking $r_x$, $m_x$ and $p_x$ as start values and integrating in the corona up to the surface at $p=0$, one obtains the same values of M and R as for the standard integration from $p(r=0)=p_c$ to $p\left(R\right)=0$ (see the blue circles on top of the black solid curve sections). However, keeping $r_x$ but using other core masses $m_x$, e.g., $m_x^{\left(1\right)}=0.5m_x$ or $m_x^{\left(2\right)}=2m_x$, one obtains different pressure and mass profiles and, consequently, also different values of M and R, see Figure 5. This is the very construction of the core-corona decomposition leading to the results exhibited in Figure 4.

5. Summary

The core-corona decomposition relies on the assumption that the EoS of compact/ neutron star matter (i) is reliably known up to energy density $e_x$ and pressure $p_x$ and (ii) occupies the star as the only component at radii $r\ge r_x$. The base line for static, spherically symmetric configurations is then provided by the TOV equations, which are integrated, for $r\in [r_x,R]$, to find the circumferential radius R (where $p\left(R\right)=0$) and the gravitational mass $M=m\left(R\right)$. We call that region $r\in [r_x,R]$ “corona”, but “crust” or “mantle” or “envelope” or “shell” are also suitable synonyms. The region $r\in [0,r_x]$ is the “core”, which is parameterized by the included mass $m_x$. The core must support the corona pressure at the interface, i.e., $p\left(r_x^{-}\right)=p\left(r_x^{+}\right)$. The core can contain any material compatible with the symmetry requirements. In particular, it could be modeled by a multi-component fluid with SM matter plus Dark Matter and/or Mirror World matter or anything beyond the SM. The TOV equations then deliver $R(r_x,m_x,p_x)$ and $M(r_x,m_x,p_x)$.

It helps our intuition to think of a SM matter material in the core with assumed fiducial EoS, which determines, for given value of $p_x$, $r_x\left(p_c\right)$ and $m_x\left(p_c\right)$, thus $R\left(p_c\right)$ and $M\left(p_c\right)$ resulting in $M\left(R\right)$, as conventionally done. Using many fiducial test EoSs at $p>p_x$, one maps out a certain region in the M-R plane accessible by SM matter. Our core-corona decomposition extends this region, since the core can contain much more than SM matter only. With reference to a special EoS, applied tentatively in core and corona, the accessible masses and radii become smaller (larger) for heavy (light) cores. That effect is noticeable for large (km size) and heavy (fraction of solar mass) cores. Small and light cores hardly have an impact.

This analysis should be refined in follow-up work by employing improved EoSs in the low-density region and by taking care of mass-radius values for the heaviest neutron star(s), still keeping precise radius values for the $1.4M_{\odot }$ neutron stars. In addition, the holographic model in Appendix B, which is aimed at mapping hot QCD thermodynamics to a cool EoS, deserves further investigations, before arriving quantitatively at an EoS suitable for compact star properties and (merging) dynamics. The emphasis in Appendix B is the illustration of the extrapolation scheme beyond a patch of QCD thermodynamic state space.

Taking up the suggestion in Ref. [77] that the scaling property of the TOV equations with a mass dimension-4 quantity, e.g., by $m_p^4$, where $m_p$ is the proton mass, essentially determines the gross parameters of compact/neutron stars, we supply in Appendix A a brief contemplation on the emergence of hadron masses within QCD-related approaches, thus bridging from micro to macro physics: $M_{PSRJ0740+6620}\approx 2M_{\odot }\approx 2.38\times {10}^{57}{m}_p$, $R_{PSRJ0740+6620}\approx 12.3\mathrm{k}\mathrm{m}\approx 1.5\times {10}^{19}{r}_p$ with proton charge radius $r_p\approx 0.88\mathrm{f}\mathrm{m}\approx 4.2m_p^{-1}$ which is somewhat larger than its “natural” scale $\hslash c/m_p$. Our focus, however, is here on the emergence of the gravitational mass (the gravitational mass in our core-corona decomposition is $M=m_x+4\pi {\int }_{r_x}^R\mathrm{d}r{r}^{2}e\left(r\right)$ which, due to gravitational binding, is different from the total mass $\int {\mathrm{d}}^{3}Ve\left(r\right)$ where the integral measure d${}^{3}V$ accounts for the gravitational spatial deformation, cf., Ref. [110]) determining the trajectories of light rays and test particles outside of compact astrophysical objects.