Crust-core transition density and isospin-dependent parametric EOS for the core of NSs

In the present work, the crust–core transition density was estimated by adopting the thermodynamical approach under the condition that the energy per nucleon E(ρ, δ) in nuclear matter at nucleon density ρ and isospin asymmetry $\delta \equiv ({\rho }_{\text{n}}-{\rho }_{\text{p}})/\rho$ could be approximated by the isospin-parabolic expansion: $E(\rho \mathrm{,}\delta )\approx E_0(\rho )+E_{\text{sym}}(\rho )\cdot {\delta }^2\mathrm{.}$ (1) The crust-core transition point is obtained when the uniform matter starts separating into a mixture comprising single nucleons and clusters. The transition density is specifically calculated using the vanishing effective incompressibility of uniform NS matter under β equilibrium and charge-neutrality conditions: $\begin{array}{c}{K}_{\mu }={\rho }^2\frac{{\text{d}}^2{E}_0}{\text{d}{\rho }^2}+2\rho \frac{\text{d}{E}_0}{\text{d}\rho }\\ +{\delta }^2\left[{\rho }^2\frac{{\text{d}}^2{E}_{\text{sym}}}{\text{d}{\rho }^2}+2\rho \frac{\text{d}{E}_{\text{sym}}}{\text{d}\rho }-2E_{\text{sym}}^{-1}{\left(\rho \frac{\text{d}{E}_{\text{sym}}}{\text{d}\rho }\right)}^2\right].\end{array}$ (2) Therefore, the transition density ρ_t can be obtained by solving Kμ=0. Notably, the crust-core transition density can be overestimated when the parabolic approximation (Eq. (1)) is used[8]. As shown in Fig. 5 of Ref. [8], this overestimation mainly appears in the region where the parameters L and K_sym are large,namely, L > 60 MeV and K_sym > -100 MeV. However, after filtered by the NS radius data, the probability that they fall into the above-mentioned intervals is very small, according to our earlier calculations[17]. Therefore, employing the parabolic approximation hardly changes the present results. Furthermore, it can also reduce the discrepancy between the thermodynamical method used in the present work and other approaches, such as the dynamical method, when the EOS parameters are compared with the NS radius data.

Fig. 5Full-screen View

(Color online) L-K_sym correlations from Tews et al. [24] (red curves), Mondal et al. [39] (black curves), and Holt et al. [38] (green curves). The solid and dashed lines denote the mean values and boundaries of each correlation, respectively. Adapted from Ref. [12].

The transition pressure can be approximated as [18] $\begin{array}{c}{P}_{\text{t}}\approx \frac{K_0}{9}\frac{{\rho }_{\text{t}}^2}{{\rho }_0}\left(\frac{{\rho }_{\text{t}}}{{\rho }_0}-1\right)\\ +{\rho }_{\text{t}}{\delta }_{\text{t}}\left[\frac{1-{\delta }_{\text{t}}}{2}{E}_{\text{sym}}\left({\rho }_{\text{t}}\right)+{\left(\rho \frac{d{E}_{\text{sym}}\left(\rho \right)}{d\rho }\right)}_{{\rho }_t}{\delta }_t\right]\mathrm{,}\end{array}$ (3) where δ_t is isospin asymmetry corresponding to ρ_t. E₀ and $E_{\text{sym}}$ in Eqs. (1) and (2) are the energy per particle in symmetric nuclear matter (SNM) and nuclear symmetry energy, respectively. They can be parameterized as $E_0(\rho )=E_0({\rho }_0)+\frac{K_0}{2}{\left(\frac{\rho -{\rho }_0}{3{\rho }_0}\right)}^2+\frac{J_0}{6}{\left(\frac{\rho -{\rho }_0}{3{\rho }_0}\right)}^3\mathrm{,}$ (4) $\begin{array}{c}{E}_{\text{sym}}(\rho )=E_{\text{sym}}({\rho }_0)+L\left(\frac{\rho -{\rho }_0}{3{\rho }_0}\right)+\frac{K_{\text{sym}}}{2}{\left(\frac{\rho -{\rho }_0}{3{\rho }_0}\right)}^2\\ +\frac{J_{\text{sym}}}{6}{\left(\frac{\rho -{\rho }_0}{3{\rho }_0}\right)}^3\mathrm{,}\end{array}$ (5) where $E_0({\rho }_0)=-15.9\text{ MeV}$ is the energy per particle at the saturation density ρ₀ for SNM. $K_0=9{\rho }_0^2[{\partial }^2{E}_0(\rho {)/}^2{]|}_{\rho ={\rho }_0}$ and $J_0=27{\rho }_0^3[{\partial }^3{E}_0(\rho {)/}^3{]|}_{\rho ={\rho }_0}$ represent the incompressibility and skewness parameters of SNM at ρ₀, respectively. Furthermore, $E_{\text{sym}}({\rho }_0)$, $L=3{\rho }_0[\partial E_{\text{sym}}(\rho {)/]|}_{\rho ={\rho }_0}$, $K_{\text{sym}}=9{\rho }_0^2[{\partial }^2{E}_{\text{sym}}(\rho {)/}^2{]|}_{\rho ={\rho }_0}$, and $J_{\text{sym}}=27{\rho }_0^3[{\partial }^3{E}_{\text{sym}}(\rho {)/}^3{]|}_{\rho ={\rho }_0}$ denote the magnitude, slope, curvature, and skewness of nuclear symmetry energy at ρ₀, respectively. According to the systematics of terrestrial nuclear experiments and predictions of various nuclear theories, K₀, $E_{\text{sym}}({\rho }_0)$, and L have the ranges of 220– 260 MeV, 28.5– 34.9 MeV, and 30– 90 MeV[19-23], respectively. Based only on theoretical predictions [24, 25], the parameters J₀, K_sym, and J_sym characterizing the nuclear EOS at high densities have wide ranges of $-800\le J_0\le 400$ MeV, $-400\le K_{\text{sym}}\le 100$ MeV, and $-200\le J_{\text{sym}}\le 800$ MeV, respectively.

In the framework of the minimum NS model, the non-rotating NS comprises neutrons, protons, electrons, and muons under β equilibrium and charge neutrality conditions, and the relationship between the pressure and nucleon density in the core of NSs $P(\rho \mathrm{,}\delta )={\rho }^2\frac{\text{d}ϵ\left(\rho \mathrm{,}\delta \right)/\rho }{\text{d}\rho }$ (6) is controlled by the energy density $ϵ(\rho \mathrm{,}\delta )=\rho [E(\rho \mathrm{,}\delta )+M_{\text{N}}]+{ϵ}_l(\rho \mathrm{,}\delta )$. Here, M_N and ϵ_l(ρ, δ) denote the average nucleon mass and energy densities for leptons which are calculated by the noninteracting Fermi gas model[26], respectively. δ can be obtained using the charge neutrality condition ${\rho }_{\text{p}}={\rho }_{\text{e}}+{\rho }_{\mu }$ and the β-equilibrium condition ${\mu }_{\text{n}}-{\mu }_{\text{p}}={\mu }_{\text{e}}={\mu }_{\mu }\approx 4\delta E_{\text{sym}}(\rho )$, where μ denotes the chemical potential calculated by the expression ${\mu }_i=\partial ϵ(\rho \mathrm{,}\delta )/\partial {\rho }_i$ for the ith particle.

The EOS for the core of the NSs can be constructed in terms of (4), (5), and (6). Below the crust-core transition density, the NV EOS[27] and BPS EOS[1] were employed for the inner and outer crusts, respectively.

Neutron skin and nuclear droplet model

The neutron-skin thickness of a finite nucleus in the nuclear droplet model(DM) is obtained according to the following expression[28, 3]: $\Delta R_{\text{np}}=\sqrt{\frac{3}{5}}\left[t-\frac{e^{2}Z}{70E_{\text{sym}}({\rho }_0)}+\frac{5}{2R}\left(b_{\text{n}}^2-b_{\text{p}}^2\right)\right]\mathrm{,}$ (7) where $e^{2}Z/70E_{\text{sym}}({\rho }_0)$ is a correction term owing to the Coulomb interaction. $R=r_0{A}^{1/3}$ indicates the nuclear radius, while b_n and b_p denote the surface widths of the neutron and proton density profiles, respectively. $b_{\text{n}}=b_{\text{p}}=1$ fm is usually used in the standard DM. The quantity t in Eq. (7) is calculated as[3] $t=\frac{2r_0}{3E_{\text{sym}}({\rho }_0)}L\left(1-x\frac{K_{\text{sym}}}{2L}\right)x{A}^{1/3}\left(\delta -{\delta }_C\right)$ (8) with $x=\frac{{\rho }_0-{\rho }_A}{3{\rho }_0}\mathrm{,}{\delta }_C=\frac{e^{2}Z}{20E_{\text{sym}}({\rho }_0)R}\mathrm{.}$ (9) Here, ρA=0.1 fm^-3 and ρA=0.08 fm^-3 for the ²⁰⁸Pb and ⁴⁸Ca nuclei, respectively.

Bayesian inference approach

The Bayesian theorem is expressed as $P\left(\mathcal{M}|D\right)=CP\left(D|\mathcal{M}\right)P(\mathcal{M})$ (10) where C is a normalization constant. $P(\mathcal{M}|D)$ denotes the obtained posterior probability distribution function (PDF) of the model $M$ when the dataset D is provided. $P(D|\mathcal{M})$ is a likelihood function obtained by comparing the theoretical results from model $\mathcal{M}$ with data D, and $P(\mathcal{M})$ is the prior probability of model $\mathcal{M}$ representing knowledge of the theoretical parameters of $M$ before comparison with data D.

In the present work, we randomly sampled the transition density in the range of 0.03 fm ^-3 $\le {\rho }_{\text{t}}\le$ 0.2 fm ^-3, and the matched six-parameter set was determined by solving the equation Kμ=0 using Eq. (2). Two methods were adopted to generate the posterior PDF of the transition density. One was based only on the observed data of the NS radii, while the other was based on both the NS radius data and neutron-skin thickness data. In the first method, we used the six matched parameters to construct the NS EOS in the framework of the minimum NS model, as described above, and substitute them in the TOV equation to compute the theoretical values of the NS radii. We then obtained the likelihood of the transition density or the matched set of parameters by comparing the theoretical values of the NS radii with the observed values using the following likelihood function: $P\left(D|\mathcal{M}\right)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left[-\frac{{\left(R_{\text{th}}-R_{\text{obs}}\right)}^2}{2{\sigma }^2}\right]\mathrm{.}$ (11) Here, $R_{\text{th}}$ is the theoretical values while $R_{\text{obs}}$ and σ represent the observed values and 1σ error bar for the NS radii, respectively. The NS EOS generated above should satisfy the thermodynamical stability condition ($\text{d}P/\text{d}ϵ\ge 0$, where P is the pressure inside NSs) and the causality condition ($0\le v_{\text{s}}^2\le c^2$, v_s and c denote the speed of sound and light, respectively) at all densities, and should be stiff enough to support the maximum mass of the NS observed thus far. A sharp cut-off of 1.97 $M_{\odot }$ was used in this analysis.

In the second method, the matched six-parameter set was obtained using Eq. (2) by sampling the transition density. We then set as the input to the DM to calculate the theoretical values of the neutron-skin thickness for ²⁰⁸Pb and ⁴⁸Ca. We discarded these parameter sets and transition densities when the calculated values for the neutron-skin thickness were far from the experimental values using the following likelihood function: $P\left(D|\mathcal{M}\right)=\frac{1}{\sqrt{2\pi }{\sigma }^{\prime }}\exp \left[-\frac{{\left(\Delta R_{\text{np}}^{\text{th}}-\Delta R_{\text{np}}^{\text{exp}}\right)}^2}{2{{\sigma }^{\prime }}^2}\right]\mathrm{.}$ (12) Here $\Delta R_{\text{np}}^{\text{th}}$ and $\Delta R_{\text{np}}^{\text{exp}}$ are the theoretical and experimental values, respectively, and σ’ represents the 1σ error bars for the experimental data. Subsequently, the remaining parameter sets were used to construct the NS EOS and estimate the posterior PDFs of the transition density, as described above.

The observed data for the NS radii and experimental data for the neutron-skin thickness used in this study are summarized in Table 1. These NS data include: (i) $R_{1.4}{=11.9}_{-1.4}^{+1.4}$ km at 90% confidence level (CFL) by analyzing the GW170817 source reported by the LIGO/Virgo Collaboration[29]; (ii) $R_{1.4}{=10.8}_{-1.6}^{+2.1}$ km at 90% CFL by analyzing the same source of GW170817[30]; (iii) $R_{1.4}{=11.7}_{-1.1}^{+1.1}$ km at 90% CFL reported earlier by analyzing the quiescent low-mass X-ray binaries (QLMXBs)[31]; (iv) Reported by the NICER Collaboration[33, 36] at 68% CFL, $R{=12.71}_{-1.19}^{+1.14}$ with mass of ${1.34}_{-0.16}^{+0.15}{\text{M}}_{\odot }$[32] and $R{=13.02}_{-1.06}^{+1.24}$ with mass of ${1.44}_{-0.14}^{+0.15}{\text{M}}_{\odot }$[36] for the source of PSR J0030+0451, $R{=13.7}_{-1.5}^{+2.6}$ with mass of ${2.08}_{-0.07}^{+0.07}{\text{M}}_{\odot }$ for the source of PSR J0740+6620. The data for the neutron-skin thickness were $\Delta R_{\text{np}}{=0.121}_{-0.026}^{+0.026}$ fm and $\Delta R_{\text{np}}{=0.283}_{-0.071}^{+0.071}$ fm for ⁴⁸Ca and ²⁰⁸Pb, respectively, which were obtained from Refs. [34, 35] recently reported by the CREX and PREX-2 Collaborations.

The Metropolis-Hastings algorithm within a Markov chain Monte Carlo (MCMC) approach was adopted to generate the posterior PDFs of the model parameters. The PDFs of the individual parameters and the PDFs for the two-parameter correlations were calculated by integrating over all other parameters using the marginal estimation approach. The initial samples in the so-called burn-in period were discarded [37] so that the MCMC process started from an equilibrium distribution. In the present analysis, 40,000 steps and the remaining one million steps were used for the burn-in progress and for calculating the PDF of the transition density, respectively.

Exploring the crust-core transition density via NS observations

The posterior PDFs of the crust-core transition density ρ_t, corresponding transition pressure P_t, and their correlations with the EOS coefficients are plotted in Fig. 1. Two types of priors for ρ_t were adopted in the calculations. The first form is a uniform distribution, which is a better choice because we have no information about ρ_t. This is illustrated by the black curves in Fig. 1. The second form, which is indicated by the purple curve in Fig. 1, is a Gaussian distribution with an average value of 0.078 fm^-3 and standard deviation of 0.04, as in. [3]. The panels located in the two upper rows show the posterior PDFs of the correlations mentioned above using uniform priors for ρ_t. The panels in the two bottom rows show the results obtained using the Gaussian priors. These results are based only on the observed data of the NS radii, as summarized in Table 1.

Fig. 1Full-screen View

(Color online)Posterior PDFs of the crust-core transition density and pressure as well as their correlations with the EOS parameters. The calculations are performed by adopting the uniform (blue curves) and Gaussian (red curves) priors for the transition density based on the NS radius data. For comparison, the uniform (black curve) and Gaussian (purple curve) forms for the prior distributions of the transition density are included. The correlations in the two upper (bottom) rows are calculated by using the uniform (Gaussian) prior for the transition density.

A two-humped posterior distribution for ρ_t was observed for both the uniform and Gaussian priors used in the calculations. The first peak which is located at ρ_t= 0.08 fm^-3, is often used as a fiducial value in the literature. The second peak is located at ρ_t= 0.1 fm^-3. The 68 % and 90 % credible intervals calculated using the highest posterior density interval approach for ρ_t and P_t are listed in Table 2. Relative to the prior distributions, the posterior PDFs of ρ_t narrow down to small intervals, which indicates that the crust-core transition density is sensitive to the NS radius. It has a higher probability of falling into the region where the values of ρ_t exceed 0.1 fm^-3 when an uninformative prior is used. This can be attributed to the correlations between ρ_t and some EOS parameters, which will be discussed later.

A better constraint on ρ_t was found when the Gaussian prior was used in comparison with those using the uniform prior. It is easy to understand that more information is available for the Gaussian prior than for the uniform prior before comparing them with the NS radius data. The generated ranges of P_t, namely, 0.05 ${}_{-0.04}^{+1.25}$ MeV/fm³ using the uniform prior and 0.1 ${}_{-0.1}^{+0.28}$ MeV/fm³ using the Gaussian prior at the 68% confidence level, as listed in Table 2, covered those calculated using the meta-modelling, dynamical, and thermodynamical models, as shown in Table I in Ref. [15]. However, the most probable values are smaller than them. Our results for ρ_t are consistent with those in literature.

We explored the correlations among ρ_t, P_t, and the EOS parameters. Here, we did not consider the correlations between the EOS parameters, which were consistent with those reported in our previous publications[17, 40]. Low-order parameters such as $E_{\text{sym}}({\rho }_0)$ and K₀ were barely correlated with the transition. However, as mentioned in Refs. [12], the condition Kμ=0 required a larger ρ_t as K₀ increased. This phenomenon is observed in Fig. 1, where a weak positive correlation is observed between ρ_t and K₀ when both the uniform and Gaussian priors are used.

Strong correlations among ρ_t and the isovector compressibility K_sym, skewness J_sym were discovered, which was consistent with the results reported in Refs. [16]. The negative correlation between ρ_t and K_sym was inconsistent with the results of Ref. [16], in which the EOS parameters were filtered by the predictions from the effective field theory and surface coefficients were determined by the nuclear masses in the framework of the extended Thomas Fermi approximation method. The positive correlations between ρ_t and J_sym and ρ_t and P_t (shown in Fig. 1) are consistent with those reported in Ref. [16, 15]. The transition was unaffected by the skewness of the symmetric nuclear matter, J₀.

L exhibited a negative correlation with ρ_t. Interestingly, a positive correlation appeared in the region ρ_t>0.1 fm^-3 when the uniform prior was used; however, this did not occur when the Gaussian prior was used. Is this related to the shoulder indicated in the posterior PDF of ρt using the uniform prior in Fig. 1, ? To answer this question, we plot the L–K_sym correlations from the three types of calculations, as indicated in Fig. 2, namely, using the uniform and Gaussian priors based on the NS radius data, and the uniform prior based on both the NS radius and neutron-skin thickness data. Two phenomena are observed in Fig. 2. They are the anti-correlation shown in the left and right panels, and the very weak correlation shown in the middle panel between L and K_sym, as shown in the left panel in Fig. 2, K_sym has a high probability to stay in the region where K_sym is extremely negative, i.e. K_sym< -200 MeV. The latter is clearly responsible for the shoulder because the shoulder for the posterior PDF of ρt disappears, as shown in Fig. 3 although the L-K_sym anti-correlations are the same in their calculations in the left and right panels. Therefore, K_sym plays a more important role in constraining the crust-core transition density of NSs than the L–K_sym correlation, as studied in Ref. [12]. The transition pressure was weakly correlated with the EOS parameters.

Fig. 2Full-screen View

(Color online)Correlations between L and K_sym calculated from the NS radius data using the uniform (left) and Gaussian (middle) priors, and from both the neutron-star radius and neutron-skin thickness data (right) using the uniform prior.

Fig. 3Full-screen View

(Color online)Posterior PDFs of the crust-core transition density and pressure and their correlations with the EOS parameters. In the calculations, both the neutron-skin thickness and NS radius data are adopted. The prior distribution (black curve) for the transition density is included.

Effect of the neutron-skin thickness and comparison with other calculations

The neutron-skin thickness is known to be an effective probe for nuclear symmetry energy, especially its slope parameter [41, 42]. The latter plays an important role in determining the crust-core transition density of NSs. The results presented in Fig. 3 are the same as those in Fig. 1 but based on both the NS radius data and neutron-skin thickness data. In the calculations, we first performed a Bayesian inference of the coefficients $E_{\text{sym}}({\rho }_0)\mathrm{,}L$ and K_sym within the framework of the nuclear droplet model, as described in Sec. 2.2 based on the neutron-skin thickness data reported by the CREX [34] and PREX-2 [35] Collaborations as listed in Table 1. Subsequently, we regarded the obtained distributions of these parameters and the matched transition density as their priors to infer the posterior PDF of the transition density based on the NS radius data within the minimum NS model. Notably, there is a significant controversy regarding constraining the slope of the symmetry energy through the neutron-skin thickness data of ⁴⁸Ca and ²⁰⁸Pb reported by the CREX and PREX-2 Collaborations. A recent study demonstrated that the ranges of the slope L obtained from CREX and PREX-2 were completely inconsistent [43]. Therefore, more accurate measurements are required. Fortunately, there are methods for determining the neutron-skin thickness, such as the configurational information entropy analysis [44, 45].

As seen in Fig. 3, the correlations are roughly same as those in Fig. 1. A weak anti-correlation between the symmetry energy magnitudes $E_{\text{sym}}({\rho }_0)$ and ρ_t was observed, owing to the constraint from the neutron-skin thickness data on $E_{\text{sym}}({\rho }_0)$. The range for K_sym is smaller than that obtained using only the NS radius data, which is also because of the effect of the neutron-skin thickness data. The constraint on ρt obtained is improved in comparison with those based only on NS data, as listed in Table 2 because the parameters L and K_sym are better constrained when the neutron-skin thickness data are considered.

In Fig. 4 we compare our results with those inferred using a compressible liquid-drop model within a Bayesian framework[15], which employs two filters, namely, the low density (LD) behavior of the energy functionals that should be rigorously limited in the uncertainty intervals of the effective field theory calculations for symmetric and pure neutron matter [46] and the high density (HD) behavior of the functionals that should obey the conditions such as positive symmetry energy at all densities and causality [15]. Our results were consistent with theirs. There P_t had a long tail for when the uniform prior was used in the calculations, because a large probability of ρ_t existed in the region at ρ_t> 0.1 fm^-3. The most probable values of P_t obtained in this study were smaller than those in Ref. [15].

Fig. 4Full-screen View

(Color online)Posterior PDFs of the crust-core transition density and pressure based on only the NS radius data using the uniform (blue curve) and Gaussian (red curve) priors, and based on both the NS radius and neutron-skin thickness data using the uniform prior indicated by NS+NST (black curve). The results presented in Ref. [15] shown by LD-EPJA2019 and HD-EPJA2019 are included for comparison.

Effect of L-K_sym correlations

As L-K_sym correlations significantly affect both the crust-core transition density and pressure[12]. To further explore this effect, we considered three typical L-K_sym correlations predicted in the literature as the priors to infer the posterior PDF of ${\rho }_t$. For completeness and ease of discussions, we briefly describe the three correlations. The first one is based on the theoretical predictions from 240 Skyrme Hartree-Fock and 263 relativistic mean-field calculations[39] by Mondal et al. and written as $K_{\text{sym}}=\left(-4.97\pm 0.07\right)\left(3E_{\text{sym}}({\rho }_0)-L\right)+66.80\pm 2.14\text{MeV}\mathrm{.}$ (13) The second was obtained from Ref. [24] by Tews et al. and written as $K_{\text{sym}}=3.50L-305.67\pm 24.26\text{MeV}\mathrm{.}$ (14) In the framework of the Fermi liquid theory, Holt and Lim[38] derived the expressions $L=6.70E_{\text{sym}}({\rho }_0)-148.60\pm 4.37$ MeV and $K_{\text{sym}}=18.50E_{\text{sym}}({\rho }_0)-613.18\pm 9.62$ MeV. Thus, we can formulate the last relationship between L and K_sym as[12] $K_{\text{sym}}=2.76L-203.07\pm 21.69\text{MeV}\mathrm{.}$ (15) The correlations are shown in Fig. 5. The discrepancy between the correlations obtained by Mondal et al. and Tews et al. is not very large because they are from the same sets of theoretical predictions. However, the results shown in Figs. 2 and 5 differ significantly. This illustrates that the L–K_sym correlations are strongly model-dependent, and constraining them has a long way to go.

In the Bayesian inference approach, after considering the abovementioned correlations, L and K_sym were no longer independent when we randomly sampled them between their specific ranges. The uniform prior for ρ_t and only the NS radius data were employed in the calculations performed in this subsection. The generated posterior PDFs for ρ_t and P_t are presented in Fig. 6, and the corresponding confidence intervals are summarized in Table 2. As stated in Refs. [12], the L-K_sym correlations play a significant role in constraining both the transition density and pressure. We observe that: (i) The results from the relations by Mondal et al. and Tews et al. were completely consistent primarily because these two correlations largely overlapped within the allowed error limits. (ii)The most probable values obtained from the relations by Holt et al. differed significantly from the other two cases. For the latter, the transition density and pressure were larger than those for the former because K_sym was not extremely negative in the relationship by Holt et al., that is, K_sym, with values higher than approximately -115 MeV as shown by the green curves in Fig. 5. The present results are consistent with those reported in Ref. [12], in which the authors studied the effects of L-K_sym correlations on the crust-core transition density and pressure by adopting fixed values of the other parameters in Eqs. (4) and (5).

Fig. 6Full-screen View

(Color online)Posterior PDFs of the crust-core transition density and pressure as the three correlations of L-K_sym as in Fig. 5 are used in the calculations.