Strong correlations of neutron star radii with the slopes of nuclear matter incompressibility and symmetry energy at saturation

Strong correlations of neutron star radii with the slopes of nuclear matter incompressibility and symmetry energy at saturation

N. Alam    B. K. Agrawal    M. Fortin    H. Pais    C. Providência    Ad. R. Raduta    A. Sulaksono Saha Institute of Nuclear Physics, Kolkata - 700064, India
Homi Bhabha National Institute, Anushakti Nagar, Mumbai - 400094, India
N. Copernicus Astronomical Center, Polish Academy of Science, Bartycka,18, 00-716 Warszawa, Poland
CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra, Portugal
IFIN-HH, Bucharest-Magurele, POB-MG6, Romania
Departemen Fisika, FMIPA, Universitas Indonesia, Depok, 16424, Indonesia
Abstract

We examine the correlations of neutron star radii with the nuclear matter incompressibility, symmetry energy, and their slopes, which are the key parameters of the equation of state (EoS) of asymmetric nuclear matter. The neutron star radii and the EoS parameters are evaluated using a representative set of 24 Skyrme-type effective forces and 18 relativistic mean field models, and two microscopic calculations, all describing 2 neutron stars. Unified EoSs for the inner-crust-core region have been built for all the phenomenological models, both relativistic and non-relativistic. Our investigation shows the existence of a strong correlation of the neutron star radii with the linear combination of the slopes of the nuclear matter incompressibility and the symmetry energy coefficients at the saturation density. Such correlations are found to be almost independent of the neutron star mass in the range . This correlation can be linked to the empirical relation existing between the star radius and the pressure at a nucleonic density between one and two times saturation density, and the dependence of the pressure on the nuclear matter incompressibility, its slope and the symmetry energy slope. The slopes of the nuclear matter incompressibility and the symmetry energy coefficients as estimated from the finite nuclei data yield the radius of a neutron star in the range km.

pacs:
21.65.+f, 21.30.Fe, 26.60.+c

The bulk properties of neutron stars are mainly governed by the behaviour of the equation of state (EoS) of highly asymmetric dense matter. The correlations of the various EoS parameters of asymmetric nuclear matter with the different properties of neutron star, such as the crust-core transition density and pressure, radii, maximum mass and cooling rate, have been studied Glendenning (1986); Horowitz and Piekarewicz (2001); Lattimer and Prakesh (2007); Xu et al. (2009); Vidaña et al. (2009); Ducoin et al. (2010, 2011); Cavagnoli et al. (2011); Gandolfi et al. (2012); Fattoyev and Piekarewicz (2012); Newton et al. (2013); Fattoyev et al. (2014); Sotani et al. (2015); Fortin et al. (2016); Dutra et al. (2016). The crust-core transition density is strongly correlated with the slope of the symmetry energy, , at saturation density ( fm) Vidaña et al. (2009); Ducoin et al. (2010); Newton et al. (2013). However, the transition pressure is found to be strongly correlated with a linear combination of the slope and curvature of the symmetry energy at the sub-saturation density ( fm) Ducoin et al. (2011); Newton et al. (2013); Fattoyev et al. (2014). The simultaneous determination of mass and radius of low-mass neutron stars can better constrain the product of nuclear matter incompressibility and symmetry energy slope parameter Sotani et al. (2015).

The correlations of the neutron star radii of different masses with the EoS parameters have been investigated extensively. The covariance analysis, based on a single model, suggests the existence of strong correlations of the radii of low-mass neutron stars () with the symmetry energy slope parameter Fattoyev and Piekarewicz (2012), the correlations becoming weaker with the increase of the neutron star mass. Similar analysis for the correlations of the radii with the symmetry energy slope over a wider range of densities was performed for two different models, having different behaviours on the density dependence of the symmetry energy, and such correlations were found to be model dependent Fattoyev et al. (2014). Recently, correlations of neutron star radii with the symmetry energy slope parameter and the nuclear matter incompressibility coefficient have been examined using a large set of unified EoSs, based on Skyrme-type effective forces and relativistic mean field (RMF) models Fortin et al. (2016). The dependence of correlations on the neutron star mass is qualitatively similar to those obtained within the covariance analysis, but the correlations are in general somewhat weaker due to the interference of the other EoS parameters, which were kept fixed in the later case. Since the EoS for asymmetric nuclear matter is mainly governed by the nuclear matter incompressibility, symmetry energy and their slopes at saturation density, the neutron star radii may be correlated with the linear combination of these EoS parameters, rather than each parameter individually, analogous to those found in the case of the correlation between the transition pressure and the linear combination of the slope and the curvature of the symmetry energy Ducoin et al. (2011).

In this Rapid Communication, we examine the correlations of the neutron star radii with the key parameters governing the EoS of asymmetric nuclear matter. These EoS parameters are evaluated at the nuclear saturation density, using a representative set of RMF models, a set of Skyrme-type models, and one microscopic calculation using Brueckner-Hartree-Fock (BHF) with the Argonne force, and a three body force of Urbana type Taranto et al. (2013), and a variational approach, in particular the Akmal-Pandharipande-Ravenhall (APR) EoS Akmal et al. (1998). All models describe 2 stars. We demonstrate that the neutron star radii over a wide range of masses () are strongly correlated with the linear combination of the slopes of nuclear matter incompressibility and symmetry energy coefficients.

The EoS at a given density and asymmetry can be decomposed, to a good approximation, into the EoS for symmetric nuclear matter , and the density dependent symmetry energy coefficient as

(1)

where is the energy per nucleon at density , and the asymmetry parameter, with and the neutron and proton densities, respectively.

The isoscalar part can be expanded as

(2)

and the isovector part as

(3)
(4)

where is the symmetry energy coefficient. The incompressibility , the skewness coefficient , the slope , and the curvature of the symmetry energy are defined in, e.g., Ref. Vidaña et al. (2009).

The slope of the incompressibility, , at saturation density, and are defined as Alam et al. (2014)

(5)
(6)
Figure 1: Neutron star mass in as a function of the radius in km (left) and central density in fm (right) for a representative set of RMF (blue) and Skyrme (red) models, and microscopic (green) calculations.

We use a representative set of RMF models, a set of Skyrme-type models, and two microscopic calculations for our correlation study. The RMF models can be classified broadly into two categories: (1) models with nonlinear self and/or mixed interaction terms and constant coupling strengths and (2) models with only linear interaction terms but density-dependent coupling strengths. The type I models used in the present calculations are BSR2, BSR3, BSR6 Dhiman et al. (2007); Agrawal (2010), FSU2 Chen and Piekarewicz (2014), GM1 Glendenning and Moszkowski (1991), NL3 Lalazissis et al. (1997), NL, NL Pais and Providência (2016), NL Horowitz and Piekarewicz (2001), NL Carriere et al. (2003), TM1 Sugahara and Toki (1994), and TM1-2 Providência and Rabhi (2013). The type II models are DD2 Typel et al. (2010), DDH Gaitanos et al. (2004), DDHMod Ducoin et al. (2011), DDME1 Nikšić et al. (2002), DDME2 Lalazissis et al. (2005), and TW Typel and Wolter (1999). The Skyrme models we use in this work are SKa, SKb Kohler (1976), SkI2, SkI3, SkI4, SkI5 Reinhard and Flocard (1995), SkI6 Nazarewicz et al. (1996), Sly2, Sly9 Chabanat (1995), Sly230a Chabanat et al. (1997), Sly4 Chabanat et al. (1998), SkMP Bennour et al. (1989), SKOp Reinhard (1999), KDE0V1 Agrawal et al. (2005), SK255, SK272 Agrawal et al. (2003), Rs Friedrich and Reinhard (1986), BSk20, BSk21 Goriely et al. (2010), BSk22, BSk23, BSk24, BSk25, and BSk26 Goriely et al. (2013). The microscopic calculations include the BHF EoS from Taranto et al. (2013); Davesne et al. (2016), and the APR EoS is taken from Akmal et al. (1998); Steiner et al. (2005); Ducoin et al. (2011). The values of the EoS parameters at saturation density show a wide variation across the models. The symmetric nuclear matter properties for these models are presented in Table I of the Supplemental Material. We shall mainly focus on the correlations between the neutron star radii and the various key parameters of the EoSs: , , , , , , and , which are evaluated at saturation density.

It was shown in Ref. Fortin et al. (2016) that non-unified EoSs may introduce a large uncertainty on the determination of low-mass star radii, i.e. , mainly if the behaviours of the symmetry energy slope for the EoSs of the inner crust and core are very different. For the RMF models, the EoSs for -equilibrated matter are built according to the following procedure. The EoS for the outer crust region is taken from the work of Baym-Pethick-Sutherland Baym et al. (1971). For the inner crust region, we use the EoS including the pasta phases, if they exist, obtained within a Thomas Fermi calculation Grill et al. (2014) up to the crust-core transition density, . At the crust-core transition, the inner crust EoS is matched to the corresponding homogeneous EoS. The fraction of the particles at a given density is determined imposing -equilibrium and charge neutrality. The model used for the outer crust is not the same as the one used for the inner crust and the core regions. However, the use of different EoSs for the outer crust has been shown to barely affect the radius of a star for masses above Fortin et al. (2016). For the Skyrme models, the same functional is used for the crust and the core. In the crust, a compressible liquid-drop model (CLDM) and a variational approach, detailed in Gulminelli and Raduta (2015); Fortin et al. (2016), are employed to describe the nuclei. Finally, for the BHF and APR EoSs, the outer and inner crusts are described by the EoSs Haensel et al. (1989) and Negele and Vautherin (1973), respectively.

The mass and radius of static neutron stars are obtained by solving the Tolman-Oppenheimer-Volkoff equations Weinberg (1972), using all of these 44 EoSs. The mass-radius relations are plotted in Fig. 1 (left panel), where the horizontal strips indicate the masses of the two heaviest neutron stars observed so far: PSR J0348+0432 Antoniadis and et. al (2013) and PSR J1614-2230 Demorest et al. (2010); Fonseca et al. (2016). For the BSk models, the relations obtained with EoSs based on a simplified CLDM used in this work are close to the ones calculated with a full microscopic model in Fantina et al. (2014); Pearson et al. (2014), see in discussion in Fortin et al. (2016). To facilitate our discussion, we also display the mass as a function of central density in the right panel of the same figure. The EoSs give rise to different neutron star properties. The spread in the maximum mass is , and the spread in the radius of neutron star with canonical mass () is km. In Table II of the Supplemental Material, we provide the maximum masses together with the radii for different neutron star masses, calculated for all the models used in this study.

The values of the EoS parameters and neutron star radii, obtained for these models, will be used to study the correlations between these quantities. A linear correlation between any two quantities, and , can be quantitatively studied by the Pearson’s correlation coefficient, , given by

(7)

with the covariance, , written as

(8)

where the index runs over the number of models Brandt (1997). In what follows, and correspond to the neutron star radius for a fixed mass and a EoS parameter, respectively, obtained for the different models. A correlation coefficient equal to 1 in absolute value indicates a perfect linear relation between the two quantities that are considered.

Figure 2: (Color online) Radii (left) and (right) of a and neutron star versus the EoS parameters , and , obtained using our sets of RMF (blue triangles) and Skyrme (red diamonds) models, together with the BHF and the APR (green stars) calculations.

In Fig. 2, we plot the radii of 1.0 and stars, and , versus some of these EoS parameters for our representative sets of RMF (blue triangles) and Skyrme (red diamonds) models, together with the BHF and the APR (green stars) calculations. The solid lines in the figures are obtained by linear regression and the correlation coefficients are indicated for each case considered. The correlations between the neutron star radius and the isoscalar parameters and increase with the increase of the neutron star mass, however, they are not significantly strong to make a meaningful prediction. The correlation is weaker than the correlation, which is opposite to the trend observed for the cases of and . In Table 1, we list all the correlation coefficients between the EoS parameters , , , , , and , and the radii of neutron stars with different masses.

The study of the correlations clearly indicates that the radius of low-mass neutron stars is more sensitive to the isovector EoS parameters ( and ), but, as the mass of the neutron star increases, the sensitivity to the isoscalar parameters ( and ) tend to dominate. A similar conclusion was drawn in Ref. Fortin et al. (2016), where the behaviour of the radius of stars with mass for 33 models, including 9 RMF models and 24 Skyrme forces, were plotted as a function of and . The correlation coefficient was 0.87, while was 0.64. The value of was 0.63, whereas the values of and were found to be . These values are quite in agreement with the values we are finding in this work, especially for the correlation coefficients of the low-mass neutron star radii.

0.565 0.383 0.548 0.815 0.887 0.581
0.617 0.416 0.597 0.743 0.881 0.658
0.655 0.461 0.646 0.680 0.850 0.698
0.684 0.514 0.695 0.621 0.803 0.714
0.704 0.571 0.743 0.562 0.745 0.711
0.718 0.628 0.787 0.502 0.674 0.691
0.725 0.686 0.828 0.438 0.590 0.653
Table 1: Correlation coefficients between the neutron star radii and the different EoS parameters obtained for a representative set of RMF, Skyrme and microscopic calculations. The EoS parameters are the nuclear matter incompressibility coefficient , its skewness , and slope , the symmetry energy coefficient , its slope , and curvature , and the parameter , calculated at the saturation density. denotes the neutron star radius for a given mass in units of .
Figure 3: (Color online) Neutron star radii (left) and (right) versus the linear correlations (top) and (bottom), using a set of RMF (blue triangles), Skyrme (red diamonds), and BHF+APR (green stars) calculations.

Next we look into the correlations of neutron star radii with selected combinations of isoscalar and isovector EoS parameters. In Fig. 3, we plot the neutron star radii for (left) and 1.4 (right) as a function of the linear combinations, (top), and (bottom). We can see that the neutron star radii are better correlated with these combinations, than with the each of the parameter separately, as seen in Fig. 2. Further, the strongest correlations occur between the neutron star radii and the linear combination . In Table 2, we list again all the correlation coefficients of neutron star radii with and , for different neutron star masses. The values of and , also listed, are obtained in such a way that the correlations of these quantities with the neutron star radii are maximum.


Corr. Coeff. Corr. Coeff.
2.970 0.905 43.115 0.936
2.111 0.914 35.575 0.949
1.564 0.902 28.370 0.945
1.177 0.879 22.189 0.935
0.883 0.850 17.089 0.924
0.643 0.817 12.781 0.913
0.432 0.782 8.970 0.903


Table 2: The correlation coefficients of neutron star radii with and , along with the values of and .

In the top panel of Fig. 4, we show the variation of the correlation coefficients of neutron star radii with , , and , as a function of the mass of the star. The correlation of neutron star radii with is better than those with and individually. However, for , the correlations of neutron star radii decrease gradually with the increase of the neutron star mass, even considering . In the bottom panel of Fig. 4, we repeat the same exercise for , , and . Again, contrary to the individual parameters and , the neutron star radii are strongly correlated with over a wide range of neutron star masses ().

In order to interpret the correlations obtained, we consider the dependence of the pressure on the isoscalar coefficients, , and on the slope of the symmetry energy, . Taking the expansions given in (2) and (4), the pressure is given by

(9)

with , or expressing in terms of and , by

(10)

These two equations and the empirical relation , identified in Ref. Lattimer and Prakash (2001); Lattimer and Prakesh (2007), where is the star radius and the pressure, calculated for some fiducial density, , allow an interpretation of the above correlations of with and .

Figure 4: (Color online) Correlation coefficients between the neutron star radii and several EoS parameters as a function of the neutron star mass. The EoS parameters ’’ denote , , and the linear combination in the top panel, and , , and in the bottom panel.

In the following, we present some arguments that explain the correlations: a) if , only the term survives and this may explain why the radius of low-mass neutron stars is well correlated with ; b) from eq. (10), it is seen that, for , the pressure depends only on and . This behavior explains that the correlation of with the star radius shown in Fig. 2 is better for than for , and also that the correlation of with is so strong; c) the contribution of the term in (9) is more important than the term for , which explains the correlation of with ; d) d) the asymmetry parameter monotonically decreases from its maximum value 0.95, obtained at densities of the order of to at . If the term in is neglected in eq. (10), the pressure satisfies Taking for the upper and lower value of in Table 2, we get, respectively, and , above and just below the value , when the relation is exact. Therefore, it seems the relation is being applied within the valid range of density. On the other hand, from eq. (9) and neglecting the term in , the relation is obtained. We now take for the upper and lower values of from Table 2, and we get, respectively, and . The last value is already out of the validity of the approximation, and even for , this approximation is not very good. This might be the plausible reason that only the low-mass neutron star radii are strongly correlated with .

Figure 5: (Color online) as a function of for , and MeV, as obtained from the multiple linear regression. The gray shaded region indicates the constraint on derived in Ref. De et al. (2015).

The knowledge of the slopes of nuclear matter incompressibility and symmetry energy at saturation density can constrain the neutron star radii as these radii are strongly correlated with . An overall variation in MeV is obtained from the analysis of the giant dipole resonance of Pb Trippa et al. (2008); Roca-Maza et al. (2013a), the giant quadrupole resonance in Pb Roca-Maza et al. (2013b), the pigmy dipole resonance in Ni and Sn Carbone et al. (2010), and nuclear ground state properties, using the standard Skyrme Hartree-Fock approach Chen (2011). A fit of the EoS for asymmetric nuclear matter, or pure neutron matter Chabanat et al. (1998); Roca-Maza et al. (2011), or the binding energies of large number of deformed nuclei Nikšić et al. (2008); Zhao et al. (2010) within different mean field models, constrains the value of in the range of MeV. The combined results of nuclear structure and heavy ion collisions lead to the central value of MeV Tsang et al. (2012). We adopt MeV, which has a good overlap with these investigations. The value of MeV De et al. (2015) at the saturation density seems to be consistent with its value at fm, deduced from the energies of the isoscalar giant monopole resonance in the Sn and Pb nuclei Khan et al. (2012); Khan and Margueron (2013). In Fig. 5, we plot the incompressibility slope parameter as a function of , for different values of the symmetry energy slope , and MeV. The gray shaded region corresponds to the constraint on as obtained in Ref. De et al. (2015). These values of and suggest that should be in the range of km, which is consistent with the results of the Ref. Steiner et al. (2016). Let us remark that if we had only taken the RMF models, the above correlations would have been slightly stronger, as expected, because all models in the study would have had a similar underlying framework, and larger radii would have been obtained for a 1.4 star, namely would have come out in the range of km. Indeed Fig. 1 shows that, on average, RMF models lead to larger radii than the other types of calculations.

In conclusion, we have studied the possible existing correlations between neutron star radii at different masses and the nuclear coefficients of the nuclear matter EoSs, calculated at the saturation density. The neutron star radii are obtained using unified EoSs, fully for the Skyrme models, and partially for inner-crust-core EoSs for the RMF models, except for the two microscopic EoSs. All EoSs are consistent with the existence of neutron stars. The radii of the low-mass neutron stars are better correlated with the symmetry energy coefficient and its slope . As the neutron star mass increases, the correlations of the radii with the nuclear matter incompressibility and its slope grow stronger. Our investigation reveals that the neutron star radii are better correlated with the linear combinations and than with the individual EoS parameters. In particular, noticeable improvement is seen in the correlations of the radii with these linear combinations, for neutron stars. The correlations of the radii with are stronger, and almost independent of the neutron star mass, in the range . A plausible interpretation for the existence of such correlations is traced back to the correlations between the pressure and similar linear combinations of the EoS parameters in the relevant density range. The values of and , as currently deduced from finite nuclei data, constrain in the range km.

H.P. is supported by FCT under Project No. SFRH/BPD/95566/2013. Partial support comes from “NewCompStar”, COST Action MP1304. The work of M.F. has been partially supported by the NCN (Poland) Grant No. 2014/13/B/ST9/02621.

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