Neutron star structure in an in-medium modified chiral soliton model

We study the internal structure of a static and spherically symmetric neutron star in the framework of an in-medium modified chiral soliton model. The Equations of State describing an infinite and asymmetric nuclear matter are obtained introducing the density dependent functions into the low energy free space Lagrangian of the model starting from the phenomenology of pionic atoms. The parametrizations of density dependent functions are related to the properties of isospin asymmetric nuclear systems at saturation density of symmetric nuclear matter $\rho_0\simeq 0.16$~fm$^{-3}$. Our results, corresponding to the compressibility of symmetric nuclear matter in the range $250\,\mbox{MeV}\le K_0\le 270\,\mbox{MeV}$ and the slop parameter value of symmetry energy in the range $30\,\mbox{MeV}\le L_S\le 50\,\mbox{MeV}$, are consistent with the results from other approaches and with the experimental indications. Using the modified Equations of State, near the saturation density of symmetric nuclear matter $\rho_0$, the extrapolations to the high density and highly isospin asymmetric regions have been performed. The calculations showed that the properties of $\sim 1.4M_\odot$ and $\sim 2M_\odot$ neutron stars can be well reproduced in the framework of present approach.


I. INTRODUCTION
Analysis of the neutron star structure is an interesting topic of the modern astrophysics. In particular, in the nuclear astrophysics the structure studies of neutron stars are related to the Equations of State (EOS) of nuclear matter which describes the pressure density versus energy density relation for a broad range of density values (for example, see recent review [1] and references therein). The peculiarities of EOS are well know at the densities below the saturation density of symmetric nuclear matter ρ 0 while at the high density regions they are still remaining not clear. The high density behavior of EOS are poorly understood because of the difficulty of direct experimental accessibility in laboratories and because of the absence of ab initio theoretical calculations. Therefore, from the experimental point of view, the neutron star studies may serve as a laboratory for understanding the behavior of EOS at high densities. From the theoretical point of view, instead of ab initio calculations one can start from the phenomenological framework taking into account the well known properties of EOS at the low density region and extrapolate to the high density regions trying to describe the properties of matter under the extreme (high density and high temperature) conditions.
In this context and if one able to formulate further in-medium modifications, a chiral soliton model of Skyrme, describing the single nucleon properties in free space [2,3], or its variations including the explicit vector mesonic degrees of freedom [4] may serve as a starting point for the theoretical framework. The in-medium modifications may be expressed allowing the density dependencies of the constants entering into the initial free space Lagrangian. It is necessary to note, that in principle one should be able also to reproduce the medium dependencies in the effective Lagrangian starting from the first principles but it is not known yet the form of general low energy Lagrangian and the peculiarities of its ingredients. For this reason, in the present work we use an in-medium modified chiral soliton model [5,6] which may be considered as a truncated version of the general low energy Lagrangian.
In Refs. [5,6] the medium modifications were achieved putting a bit more phenomenology into the initial model, i.e. putting the density dependent functions into the free space Skyrme Lagrangian according to the pionic-atoms data at low energies [7] and properties of asymmetric nuclear matter at saturation density ρ 0 . Although the in-medium modified Skyrme Lagrangians is assumed to be very truncated version of the possible general Lagrangian, it must be applicable to the studies of nuclear many-body problems in the spirit of chiral effective Lagrangians. The pay for the truncation may be the possible deviations from the experimental observables in the sense of quantitative description. Nevertheless, the model has obvious virtues: i) it has the simplest Lagrangian among the same class Lagrangians, and ii) it seams have all necessary ingredients for the qualitative description of the strong interaction physics. These ideas were the basic ruling philosophy behind of the approach developed in Refs. [5,6] and we continue our model studies in the present work.
The model is phenomenological one and must pass as much as possible tests on its applicability to strong interacting systems comparing with other approaches and the experimental indications. The previous nuclear matter studies [6] showed that the in-medium modified Skyrme term is responsible for preventing the collapse of nuclear matter to the singularity at high densities in analogy to the free space case where the Skyrme term is responsibly for the stabilization of finite size solitons. The modifications showed that at some values of model parameters the properties of infinite and isospin asymmetric nuclear matter can be reproduced well near the saturation point of symmetric nuclear matter ρ 0 .
It will be interesting to test the applicability of model to the strong interacting systems under the extreme conditions extrapolating the modified Equations of State to the high density regions. Therefore, in the present work we make further check of the basic philosophy considering an application of the model for the studies of neutron stars structure.

A. The Lagrangian
Our starting point is the in-medium modified Skyrmemodel Lagrangian described in Ref. [5] where Einstein's summation convention is always assumed (if not specified otherwise). The chiral SU (2) matrix U = exp(2iτ a π a /F π ) is defined in terms of the Cartesian isospin-components of the pion field π a (a = 1, 2, 3). The density functionals entering into the Lagrangian (α τ , α s , ζ τ , ζ s , α m and α e ) represent the influence of surrounding environment to the single soliton properties. In the single skyrmonic sector, the input parameters of the model have the following values: F π = 108.783 MeV, e = 4.854 and m π = 134.976 MeV. They correctly reproduce the experimental values of the nucleon mass m N = 938 MeV and ∆-isobara mass m ∆ = 1232 MeV in free space, i.e. if α τ = α s = α m = ζ τ = ζ s = 1 and α e = 0.
This simple Lagrangian describes the properties of nucleons in free space, the properties of nucleons in nuclear matter as well as the properties of isospin asymmetric nuclear matter starting at the same footing. The formalism for the classical solitonic solutions, the quantization method and the explanations of applications for symmetric and asymmetric matter properties can be found in Refs. [5,6] and references therein. But for seflconsistency, in the next subsection II B we briefly outline how an infinite size nuclear systems can be described in the framework of the present approach and represent some final formulas.

B. Nuclear matter
In the thermodynamic limit at zero temperature, for a system of an infinite number of baryons uniformly distributed in infinite volume but keeping the density per unit volume constant, the binding energy per nucleon can be represented as where ε V and ε A are known as volume and asymmetry energies, respectively. For the convenience, here we introduced the isoscalar λ = ρ/ρ 0 and isovector δ = δρ/ρ density parameters in terms of the isoscalar ρ = ρ neutron +ρ proton and isovector δρ = ρ neutron −ρ proton nuclear densities, and the normal nuclear matter density ρ 0 .
In the framework of present approach, using the spherically symmetric approximation for a single soliton properties via the hedgehog ansatz U = exp{iτ i r i F (r)/r} and considering an isospin asymmetric and infinite size nuclear environment, one can get the symmetric and asymmetric parts of the binding energy per nucleon [5] Here the functionals m, Λ and Λ 2 are defined as In the functionals above, using an infinite nuclear matter approximation, the initial medium functionals in Eq. (1) were rearranged by defining the new medium functions f 1,2,3,4 . The rearrangements are made in the following way and the basic principle behind the linear density dependent parametrizations of functions f i was the simplicity of form. In the last four equations C i (i = 1, 5) are the model parameters and, therefore, the present approach is a simple 5-parametric solitonic model of nuclear matter. 1 As soon as we define 5-parameters at some specific density (e.g. the normal nuclear matter density, ρ 0 ) the Equations of State for the symmetric and asymmetric nuclear matter are defined for any values of nuclear matter density if one assumes that the extrapolations by means of the linear on density functions f i are valid. 2 In the next subsection II C we concentrate our attention on the properties of baryonic systems at high densities discussing the neutron stars.

C. Neutron star
As the first approximation in describing the neutron stars, one can consider a spherically symmetric and static mass distribution. Then any part of neutron star mass M(r) inside a sphere with radius r is given by the integral Here E(r) is the mass-energy density distribution of the neutron star in the radial direction. Consequently, the total gravitational mass M of the neutron star with radius R is defined by the condition Further, in the spherically symmetric approximation, the pressure density change in radial direction inside the neutron star is given by the Tolman-Oppenheimer-Volkoff (TOV) equation [8,9] − dP (r) Here G = 6.707 × 10 −39 c GeV After solving TOV equation one can find the radius R of a star with central energy density E cent . It is defined by 1 We would like to note, although at present work we discuss the properties of an asymmetric and infinite nuclear matter the model can be applied for the studies of finite nuclei properties after fitting the density parameters C i . 2 Other details can be found in Refs. [5,6]. the pressure zero condition at the surface of the neutron star, P (r = R) = 0.
To obtain the numerical solution for the profile of a star, one solves Eqs. (12) and (14) using the Equation of State To find P (E) dependence in present approach, we note that the pressure and energy dependencies on the density parameter λ for the neutron matter (δ = 1) are given by equations where ε(λ, 1) is binding energy per nucleon in neutron matter. The system of parametric equations, Eq. (17) and Eq. (18), gives the desired relation P (E) between pressure and energy densities in a neutron star.

III. RESULTS AND DISCUSSIONS
In general, the thermodynamically limiting binding energy per nucleon given in Eq. (2) must be valid at any densities and at any given isospin parameter δ. For example, it is valid also at the extreme conditions which are formed in interior regions of neutron stars. Those extreme conditions described by very high density (ρ ≃ several times of ρ 0 ) and highly isospin asymmetric (δ ≃ 1) form of nuclear matter.
From other side, it is not clear the direct relation of the liquid-drop formula of Bethe and Weizsäcker [10,11] describing the binding energy of nucleus to the binding energy of neutron star. The reason is not only due to the relativistic metric factors coming from the Einstein's equations in the calculations of binding energies of neutron stars. The reason here is rather obvious, the semiempirical liquid-drop formula describes the binding energy per nucleon near the normal nuclear matter densities (which correspond to the density profiles of the existing heavy nuclei) and its validity at high densities is not clear. Nevertheless, around the normal nuclear matter density and for the small values of asymmetry parameter δ, the thermodynamically limiting binding energy per nucleon given in Eq. (2) is well related to the Bethe and Weizsäcker's formula. Because, from one side, in the limit of small δ the binding energy formula in Eq. (2) can be approximated as where ε S (λ) is called the symmetry energy. From other side, if one ignores the Coulomb and surface effects, the binding energy per nucleon defined from the liquid-drop formula will take the form From the comparisons of the last two equations, it is seen that the density dependencies of volume and symmetry energies can be established well at the densities around ρ 0 from the stability conditions and the density variations of resonating heavy nuclei near the ground state. The symmetry energy ε S describes the energy increase in the system if the number of protons and neutrons becomes not equal relatively to the case when the neutron and proton numbers are same. Consequently, the symmetry energy is important factor in describing the properties of neutron-reach stable nuclei existing in nature as well as the properties of exotic nuclei formed under extreme conditions where the neutron-to-proton number N/Z is much smaller or much larger than one [12]. Although the definition of symmetry energy in Eq. (20) is model independent its density dependence is clearly model dependent. Therefore, during the fitting to density region near the normal nuclear matter density ρ 0 the different models are modified taking into account the properties of symmetry energy coming from the phenomenological observations. But the extrapolations of EOS to the high density (ρ >> ρ 0 ) and highly asymmetric (δ ≃ 1) regions remain not clear. In particular, this is due to the reason that in neutron stars the higher order terms in δ in Eq. (20) may be also important. Usually, it is assumed that O(δ 4 ) terms are small and the symmetry energy is mostly responsible in describing the properties of neutron stars when the neutron-to-proton number N/Z becomes infinite. 3 In the present approach, the higher order terms O(δ 4 ) come from the explicit isospin breaking symmetry in the mesonic sector and found to be negligible. 4 Consequently, we ignore them in the present work.

A. Nuclear matter and Symmetry energy
Let us first discuss the properties of the symmetric nuclear matter. The value of the first experimental parameter in the liquid-drop formula Eq. (19) is well known, a V ≈ 16 MeV. It is also well established that the saturation density of symmetric nuclear matter, where the 3 Although the neutron star is initially formed from the ordinary matter which has the finite neutron-to-proton number with well separated electrons, due to the gravitational collapse and due to further formed electron degenerate states, and due to the following nucleon degenerate states at even higher densities the frequent collisions lead to the intensive nuclear reactions. As a result "an effective" neutron-to-proton number N/Z becomes infinite. 4 See the discussions in subsection V.C of Ref. [6].  Table I. pressure becomes zero P sym = 0, is around 0.16 fm −3 . Further, the compressibility of symmetric nuclear matter within the various approaches is found to be K 0 ∼ 290±70 MeV at the saturation density ρ 0 [13][14][15][16][17][18]. Therefore, these three quantities may be used to fix some of the model parameters (e.g. C 1 , C 2 and C 3 ) if one expands the volume energy into the Taylor series around the saturation density, λ = 1. After fitting the values of parameters, the EOS of symmetric matter is defined for any values of density. 5 The density dependencies of the binding energy per nucleon in symmetric matter are shown in Fig. 1 for the three possible values of the compressibility K 0 . The corresponding parameters and the volume energy coefficients at saturation density ρ 0 are given in Table I. It is seen, that our results become consistent with the results from other approaches and the phenomenological indications at some values of density parameters C i . For example, the density dependence of volume energy from the Set II is close to the APR (Akmal-Pandharipande-Ravenhall) predictions (see the model A18+δv+UIX * in Ref. [19]) made on the basis of Argonne v 18 two-nucleon interactions [20]. Similarly, the results from the Set III is very close to the results from the model A18+UIX [19].
The quantity in the last column, referred as the skewness parameter, is proportional to the third derivative of the volume term at saturation density ρ 0 and defined as Its values presented in Table I are outcome results from the present approach. There is a nonlinear correlation between Q and compressibility K 0 , Q increases if the value of K 0 increases. Our predictions for Q is qualitatively similar to the results from the Hartree-Fock approach based on Skyrme interactions [21], and to the result from the MDI (isospin and momentum-dependent interaction) model [22]. Another example, the phenomenological momentum-independent model (MID) also predicts the similar results [23]. For comparison, in the present model one has Q/K 0 ≈ −1.71 at K 0 = 240 MeV while MID model gives the result Q/K 0 ≈ −1.6 at that value of K 0 . Now let us discuss the properties of the asymmetric matter. As we said above, in the approximation that the higher order terms in δ are small (see Eq. (20)), the asymmetric matter properties are completely determined by the density dependence of the symmetry energy. The properties of the symmetry energy can be studied again by expansion into the Taylor series around the saturation density, λ = 1. While the first coefficient ε S (1) in Eq. (24) is known to be more or less well defined phenomenologically, ε S (1) ∼ 29 to 34 MeV, the values of slop parameter L S and compressibility of asymmetric matter K S remain unclear. The reason is that EOS of asymmetric matter is highly sensitive to those parameters and the different models give the different predictions. For example, in relativistic mean field approaches (see Ref. [24], for the recent optimized versions) there are mainly two classes: i) small ε S (ρ 0 ) ∼ 30 MeV and small L S ∼ 50 MeV (BSP, IUFSU * , IUFSU) and ii) large ε S (ρ 0 ) ∼ 37 MeV and large L S ∼ 110 MeV (G1, G2, TM1 * , NL3). The recent analyses of the data from heavy-ion collisions [25] and neutron-skin experiments [26] give the prediction L S ∼ 70 ± 20 MeV.
During the modifications of EOS in the present work we choose the value of slop parameter L S in the interval 30−50 MeV which is in agreement with various empirical constraints [25][26][27].
The symmetry energies, calculated using two different sets of parameters, are shown in Fig. 2 and the parame-  Table II. ters are determined in Table II. Here we present only two representatives among the many sets producing the symmetry energy parameters in commonly adopted range. Depending on the compressibility K 0 value of the symmetric nuclear matter we classify the results into two models, Model II with relatively smaller value of the compressibility K 0 = 250 MeV (more soft EOS) and Model III with relatively bigger value of the compressibility K 0 = 270 MeV (more stiff EOS).
The results show that the symmetry energy is less sensitive to the different model parameters presented in Table II. In the Fig. 2 we have shown the density dependencies of symmetry energy corresponding to two boundary regions referred as more soft (II-a) and more stiff (III-f) Equations of State. The other sets reproduce the density dependence of symmetry energy corresponding to somewhere in between of these two boundary curves. The relatively more stiff EOS (III-f) in the present approach reproduce the density dependence of symmetry energy which is close to the APR predictions [19], to the result from the MDI model with the parameter x = 0 [22] as well as to the results from MID model with the parameter y = −0.73 [23].
The quantities in the last five columns in Table II are outcome from our model calculations. In particular, the quantities K τ and K 0,2 are related to the compressibility of asymmetric matter and defined as They describe the correlations between the symmetry energy coefficients and important for estimating the shift in compressibility value in asymmetric matter The condition for the lowering of saturation density value in asymmetric matter leads to the constraint K τ < 0. This constraint is fulfilled in the present work.
The calculated values of K τ and K 0,2 are consistent with the results from other approaches. For example, the phenomenological momentum-independent model predicts the range for the values of K 0,2 : −477 MeV ≤ K 0,2 ≤ −241 MeV [23]. Our predictions, made using the different sets, are also belong to that range (see Table II).
It is interesting also to compare the low density behavior of the symmetry energy in the present model with other model predictions. For example, an analysis of the giant dipole resonance (GDR) of 208 Pb using a series of microscopic Hartree-Fock plus Random phase approximation calculations predicts the following values of symmetry energy at subnormal nuclear matter density: 20 MeV < ε S (ρ = 0.1 fm −3 ) < 25.4 MeV [28]. The recent analysis of the properties of double magic nuclei [29] puts more constraints, ε S (ρ = 0.1 fm −3 ) = 25.4 ± 0.8 MeV. An analysis of data on the neutron skin thickness of Sn isotopes and the isotope binding energy difference for heavy nuclei at slightly higher density value gave the result ε S (ρ = 0.11 fm −3 ) = 26.65 ± 0.2 MeV [30]. For comparison, our results are presented in the last two columns of Table II. One can see that they are mainly consistent with the results in Refs. [28][29][30].
The discussions of density dependencies of the pressure in symmetric and asymmetric matter within the present approach can be found in Ref. [6]. Here we only would like to note, that the modifications are also consistent with the results from other approaches. Moreover, the density dependencies of pressure in symmetric and asymmetric matter are weakly sensitive to the change of model parameters and the reproduced values correspond to the allowed region deduced from the experimental flow data and simulations studies by Danielewicz et al. [31].
In summary, the properties of nuclear matter at saturation density ρ 0 and the extrapolations to the lower than ρ 0 regions are found to be satisfactory and consistent with the results from other approaches. Therefore, we now concentrate on the extrapolations to the high density regions and discuss the properties of neutron stars.

B. Properties of neutron stars
Let us start from the mass-radius relations of neutron stars. There are dramatic differences in predictions from the different approaches as concerned the mass-radius relation of neutron stars. But most of the models with Equations of State of normal (nonstrange) nuclear matter predict existence of the regions around the one solar mass where the radius of neutron star is independent from the mass.
In the Fig. 3 we present our results corresponding to the two possible values of the compressibility of symmetric matter K 0 = 250 MeV in the left panel and K 0 = 270 MeV in the right panel, respectively. One can see that the mass independent regions exist also in the framework of the present approach. Another the common for many models peculiarity, the existence of the maximal mass of the neutron star, is also reproduced well in this model. As we mentioned above, the results are sensitive to the compressibility value, if the compressibility value decreases it leads to decreased values of the max-  Table II).
imal mass and radius of neutron star leading in such a way to more compact neutron stars. From other side, the results are not sensitive to the change in the value of ε S (1) and, therefore, we presented only the results corresponding to the value of ε S (1) = 32 MeV. In general, our results are in qualitative agreement with the results from other approaches.
It is also interesting to compare our results in quantitative level too. Measurements of the thermal spectra from the quiescent low-mass X-ray binaries inside globular clusters gave the possibility to fit the data sets with a neutron star radius R NS = 9.1 +1.3 −1.5 km at 90% confidence level [32]. Determinations of the mass-radius relation, based on recent observations of both transiently accreting and bursting sources, gave the radius range between 10.4 and 12.9 km for 1.4 solar mass neutron stars [33].
In the present approach some class of parameters (Model III) defined in Table II give very good agreement with the above mentioned estimations in quantitative level. The properties of neutron stars reproduced using some subclasses of Model III is presented in Table III. One can see that, our results corresponding to more than 2M ⊙ as well as ∼ 1.4M ⊙ neutron stars are very similar to the estimations from the Refs. [32,33].
For comparison, in Table III we present two of the possible neutron stars parametrizations from the Ref. [21] as the representatives of the Skyrme effective forces used in the density functional approach. One can see that our results in qualitative agreement with the results from the Ref. [21]. It is also necessary to note, that the compressibility value used in Ref. [21] is smaller K 0 = 230 MeV in comparison with the compressibility value in Model III K 0 = 270 MeV. Therefore, we have more stiff EOS leading to slightly higher maximal mass and larger radius neutron stars. But all of the models presented in Table III give more or less similar results as concerned 1.4M ⊙ mass neutron stars.
The central number densities of maximal mass neutron stars in the present approach are around 6.5ρ 0 and the corresponding central mass-energy densities are around 2.44 × 10 15 gr/cm 3 (≈ 1300 MeV fm −3 ). Our results are close to the results from nonrelativistic potential model approaches discussed in Ref. [34].
It is interesting also to compare the total baryon number of neutron star in the present approach with the results from other approaches. General Relativistic formula for the total baryon number of the neutron star is given by the integral One can find the radial dependence in the number density ρ(r) from the relation P = P (ρ/ρ 0 ) after finding the radial dependence of the pressure P = P (r). Our results are in qualitative agreement with results from Ref. [21] (see Table III). After calculations of the total baryon number, one can also estimate the binding energy E b of the neutron star. Due to the decrease of the gravitational mass E b is defined by the formula where we used the mass of nucleon m N in free space. 6 One can see that the calculated binding energies corresponding to 1.4 solar mass neutron stars are consistent with the estimations made on the analysis of detected neutrinos from SN1987A: E b = 3.8 ± 1.2 × 10 53 erg [35].
In addition we present the binding energy per unit mass versus GM/Rc 2 relation in Fig. 4 for some representatives of neutron star models given in Table III.  Table II. Thin and black solid curve represents an approximate relation Eq. (29) presented in Ref. [34].
approximate formula suggested by Lattimer and Prakash [34]. It is seen that in the present approach the small values of slop parameter L s ∼ 30 MeV of symmetry energy give the results close to the parametrization in Eq. (29).

IV. SUMMARY
In summary, we discussed the application of the inmedium modified chiral soliton model to the studies of asymmetric matter properties and neutron star structure. The symmetric and asymmetric matter equations of state where reproduced by very simple 5-parametric density approach to the single nucleon properties in the nuclear environment introducing the isospin breaking effects in the mesonic sector of the model. After reproducing the asymmetric matter properties near the saturation density of symmetric matter ρ 0 we extrapolated the Equations of State to the high density and highly isospin asymmetric regions. Our primary goal was a crude qualitative analysis of neutron star properties. Nevertheless, at some given set of parameter values, our results are very close to the predictions from analysis of the data compiled during the observation of neutron stars [32,33]. In particular, the calculations showed that the properties of ∼ 1.4M ⊙ and ∼ 2M ⊙ neutron stars can be well reproduced within the present approach.
As an outlook for further studies we note that the inmedium chiral soliton model presented here can easily be extended to the studies of finite nuclei properties. This task can be realized in the local density approach for the density of environment surrounding the soliton under the consideration. Those studies are under the way.