# The equation of state of nucleon matter and neutron star structure

###### Abstract

Properties of dense nucleon matter and the structure of neutron stars are studied using variational chain summation methods and the new Argonne two-nucleon interaction, which provides an excellent fit to all of the nucleon-nucleon scattering data in the Nijmegen data base. The neutron star gravitational mass limit obtained with this interaction is 1.67. Boost corrections to the two-nucleon interaction, which give the leading relativistic effect of order , as well as three-nucleon interactions, are also included in the nuclear Hamiltonian. Their successive addition increases the mass limit to 1.80 and 2.20 . Hamiltonians including a three-nucleon interaction predict a transition in neutron star matter to a phase with neutral pion condensation at a baryon number density of fm. Neutron stars predicted by these Hamiltonians have a layer with a thickness on the order of tens of meters, over which the density changes rapidly from that of the normal to the condensed phase. The material in this thin layer is a mixture of the two phases. We also investigate the possibility of dense nucleon matter having an admixture of quark matter, described using the bag model equation of state. Neutron stars of 1.4do not appear to have quark matter admixtures in their cores. However, the heaviest stars are predicted to have cores consisting of a quark and nucleon matter mixture. These admixtures reduce the maximum mass of neutron stars from 2.20 to 2.02 (1.91) for bag constant (122) MeV/fm. Stars with pure quark matter in their cores are found to be unstable. We also consider the possibility that matter is maximally incompressible above an assumed density, and show that realistic models of nuclear forces limit the maximum mass of neutron stars to be below 2.5. The effects of the phase transitions on the composition of neutron star matter and its adiabatic index are discussed.

###### pacs:

PACS numbers:21.65.+f, 26.60.+c, 97.60.Jd## I Introduction

The significant influence of nuclear forces on neutron star structure is by now firmly established by a large body of theoretical and observational evidence [4]. In the absence of these forces, the maximum possible mass of neutron stars composed of non-interacting neutrons is solar masses () [5]. Since most observed neutron star masses are above 1.3[6], they must be supported against gravitational collapse by pressure originating from nuclear forces. In the present work, we study neutron star structure using one of the most realistic models of nuclear forces currently available. A brief outline of previous calculations leading to this work is presented below.

Shortly after the discovery of pulsars, calculations of the equation of state (EOS) of neutron star matter with realistic models of the two nucleon interaction (NNI), obtained by fitting the nucleon-nucleon (NN) scattering data then available, were carried out using the lowest order constrained variational method [7, 8]. The results demonstrated that nuclear forces increase the mass limit of stable neutron stars beyond 1.4.

By the late 1970s it had become clear that the NNI alone could not account for the properties of nuclear matter or few-body nuclei. Variational [9] and Brueckner calculations [10], including higher order cluster contributions, established that nuclear matter with realistic NNI saturates at too high a density. In addition, these interactions were known to underbind H. Ignoring the latter problem, plausible density dependent terms were added to the Urbana (U14) model of NNI [11, 12] to reproduce the observed equilibrium properties of nuclear matter. The resulting density dependent (U14-DDI) model of nuclear forces predicted stable neutron stars having masses up to 1.8[13, 14].

Since nucleons are made up of quarks and have internal degrees of freedom, we can expect interactions among three (and perhaps four or more) nucleons, in addition to the NNI. The Urbana three nucleon interaction (TNI) models contain only two terms, with strengths fixed by the saturation density of nuclear matter and the binding energy of H. Wiringa, Fiks and Fabrocini (WFF) [15] used the U14 and the subsequent Argonne (A14) [16] models of NNI, together with the Urbana VII (UVII) model of TNI, to study neutron star structure and obtained mass limits of 2.19and 2.13with the U14+UVII and A14+UVII, respectively. They also found that pure neutron matter (PNM) undergoes a transition to a phase having spin-isospin order, attributed to neutral pion condensation, at a density of fm with the A14+UVII, but not with the U14+UVII. Neither of these models results in a phase transition in symmetric nuclear matter (SNM), which is composed of equal numbers of neutrons and protons.

In the early 1990s the Nijmegen group [17] examined carefully all NN scattering data at energies below 350 MeV published between 1955 and 1992. They extracted 1787 proton-proton (pp) and 2514 proton-neutron (np) “reliable” data, and demonstrated that these data determine all NN phase shifts and mixing parameters quite accurately. The NNI models which fit this Nijmegen data base with a are called “modern”. These include the Nijmegen models [18]: Nijmegen I, II and Reid-93, the Argonne [19], denoted here by A18, and the CD-Bonn [20]. In order to fit the pp and np data simultaneously and accurately, these models include a detailed description of the electromagnetic interactions and terms that break the isospin symmetry of nuclear forces. All include the long range one-pion exchange potential, but follow different treatments of the intermediate and short range parts of the NNI. The differences among the predictions of these models for the properties of many-body systems are much smaller than those among the predictions of older models, presumably because all modern potentials accurately fit the same scattering data. For example, the H binding energies predicted by the modern Nijmegen models and A18 are between -7.62 to -7.72 MeV [21], while that of CD-Bonn is -8.00 MeV [20]. The difference between these results and the experimental value of -8.48 MeV is used to fix one of the parameters of Urbana TNI models.

Detailed studies of the energies of dense nucleon matter were carried out recently by Engvik et al. [22] using all the modern models of NNI and the lowest order Brueckner (LOB) method. According to these studies, the results with the modern potentials are all quite similar up to densities relevant to neutron stars. For example, the energies predicted with the LOB method for the energy of neutron matter at 5, where fm is the equilibrium (saturation) density of nuclear matter, range from 80 to 93 MeV per nucleon. The spread of 13 MeV in these energies is small compared to the possible errors in the LOB method and the expected contributions of TNI at this density. This model independence results from the fact that the mean interparticle distance at is greater than 1 fm, and the predicted matter energy is therefore not sensitive to the details of the interaction at fm.

In this paper we study the structure of neutron stars with the A18 model using variational chain summation (VCS) methods, which hopefully include all leading many-body correlation effects. The Urbana model IX (UIX) [23] is used to estimate the effect of TNI. Previous studies of nucleon matter with A18 and UIX interactions have indicated the possibility of a transition to a neutral pion condensed phase for both PNM and SNM [24]. The effects of such a transition on the structure of neutron stars are studied here in detail. The effect of relativistic boost corrections [25] to the A18 interaction is also examined. At high densities, we consider the possibility of matter becoming maximally incompressible, as well as that of a transition to mixed phases of quark and nucleon matter [26, 27].

The relativistic mean field (RMF) approximation [28] has been used in many studies of high density matter and neutron stars. There exists a vast amount of literature on this topic, some of which has been reviewed by Glendenning [29]. While the RMF approximation is very elegant and pedagogically useful, it is not valid in the context of what is known about nuclear forces, which is the theme of this work. For example, using the meson parameters of the CD-Bonn S-wave potentials in the RMF approximation leads to unbound SNM. (In order to accurately fit the NN scattering data, the phenomenological scalar meson parameters in the CD-Bonn model are allowed to depend on partial wave quantum numbers.) At a density of , for SNM, the RMF approximation yields an energy per nucleon of MeV, while the LOB method gives MeV. In the mean field Hartree approximation, which is implicit in the RMF calculations, the A18 NNI gives energies per nucleon of +30 (+37) MeV at and +155 (+204) MeV at 5 for SNM (PNM), while the variational calculations presented here give -18 (+12) and +25 (+88) MeV, respectively.

The main problem is that the mean field approximation for meson fields is only valid for , where is the inverse Compton wave length of the meson and is the mean interparticle spacing. Over the 1-5 density range, estimated using a body centered cubic lattice ranges from 2 to 1.2 fm. Thus is in the range 1.4 to 0.8 for the pion and 7.8 to 4.7 for vector mesons. The mean field approximation is not applicable, since these values are obviously far from being much smaller than one. The RMF approximation can be based on effective values of the coupling constants that take into account the correlation effects. However, these coupling constants then have a density dependence, and a microscopic theory is needed to calculate them.

This paper is organized as follows. Section II contains a summary of the non-relativistic calculations with A18 and A18+UIX models of nuclear forces, while section III describes the calculations including the relativistic boost interaction, denoted by , without and with the TNI model UIX. The beta equilibrium of neutron star matter is discussed in section IV and results for neutron star structure are presented in section V, where we also discuss the effects of the possible transition to mixed nucleon and quark matter phases. The adiabatic index and sound velocities in neutron star matter are given in section VI, and conclusions are presented in section VII.

## Ii Nonrelativistic Calculations

Nonrelativistic calculations of SNM and PNM with the A18 and UIX interactions were carried out using Variational Chain Summation (VCS) techniques described in detail in [24]. Energies are calculated by evaluating the expectation value of the Urbana-Argonne Hamiltonian with a variational wavefunction composed of a product of pair correlation operators acting on a Fermi gas wavefunction. The pair correlation operators are written as a sum of eight radial correlation functions, each multiplied by one of the two-body operators: . The wavefunction depends on three variational parameters: the range of the tensor correlations, , the range of all other correlations, and a quenching parameter , meant to simulate medium effects. In this section we discuss results obtained for four cases, namely SNM and PNM with and without the three-nucleon interaction, in order to indicate their sensitivity to various terms in the nuclear force, and to extend them to higher densities, beyond the range covered in [24].

The optimum values of the parameters and in matter without and with the three-nucleon interaction are shown in Fig. 1. Some of the noise in the variation of these parameters with matter density is due to the insensitivity of the energy to their values at the variational minimum. The large increase in of PNM without at fm is due to a transition to a phase with -condensation as discussed in [24]. With there are sudden changes in the and of PNM at fm, and in the of SNM at fm. These are associated with the same phase transition. We note that the present variational wavefunction is not fully adequate to describe the long range order in the -condensed phases. However, since the change in the energy due to the -condensation appears to be small, we expect our estimates of the energies of matter with -condensation to be useful.

Our nonrelativistic Hamiltonian, , is comprised of the nonrelativisic kinetic energies and the two-body A18 and three-body UIX interactions. The NNI includes a static, long-range one-pion exchange part, with short-range cutoff, and phenomenological intermediate and short-range parts, which depend on the six static two-body operators and eight momentum-dependent (MD) two-body operators: . The A18 includes an additional isovector operator, and three isotensor operators, which distinguish between pp, np and nn interactions. These isovector and isotensor terms are small, and give zero contribution to the energy of SNM to first order. They are therefore neglected in SNM calculations. In the case of PNM all the isospin operators can be eliminated and the full A18 with isovector and isotensor terms becomes the sum of a static part with operators , and a MD part with operators and . The UIX model of contains two static terms; the two-pion exchange Fujita-Miyazawa interaction, , and a phenomenological, intermediate range repulsion . The strength of the interaction was determined by reproducing the binding energy of the triton via Greens Function Monte Carlo (GFMC) calculations [23], while that of was adjusted to reproduce the saturation density of SNM.

Expectation values of the various interactions are calculated in the VCS framework by summing terms in their cluster expansions. The one-body, two-body and many-body contributions to the kinetic and NNI energies are listed in Tables 1-4. The one-body cluster contribution includes only the Fermi gas kinetic energy, . The remainder of the kinetic energy is separated into the contribution from the two-body cluster and that coming from the many-body clusters, . The kinetic energy can be calculated using different expressions related by integration by parts. If all MB contributions are calculated, these expressions yield the same result. However, they yield different results when only selected parts of the MB clusters are summed by VCS techniques. We have calculated the many-body kinetic energy using expressions due to Pandharipande and Bethe (PB), and to Jackson and Feenberg (JF) [30]. The averages of the PB and JF results appear under the column , and the differences are listed under . In studies of atomic helium liquids the exact energies, calculated via Monte Carlo (MC) methods, lie between the PB and JF values evaluated using VCS methods [31]. Although exact MC calculations are as yet not practical for nucleon matter, we believe the average of the two expressions to be more accurate than either one, and that the difference provides a measure of the uncertainty in the many-body calculation. The is quite small in nucleon matter, due to cancellations between various many-body terms, and therefore the difference is a better indication of this uncertainty.

The two-body cluster contribution to the static and MD parts of the NNI energy are listed under , and . The is negative and large enough to bind SNM, though not PNM. The increases rapidly with density, and is proportional to at . The many-body contributions to the static and MD parts of the NNI energy are listed separately as and . Previously, great efforts were made to improve upon the accuracy of the calculation of [9, 32]; however, at large densities the grows rapidly, and becomes larger in magnitude than . The MD contribution is more difficult to calculate because of the gradients in , which may operate on the correlations of nucleons and with other nucleons. All the leading terms are calculated as discussed in [24], and the corresponding errors should therefore be much smaller than the reported values. At higher densities the becomes proportional to , as expected. An additional perturbative correction to the two-body energy is listed as . This small correction is due to an improvement in the variational wavefunction, which occurs when correlation functions are calculated separately in each channel [24].

The expectation values of , , and for SNM and PNM are listed in Table 5. Here, is the phenomenological part, , of the NNI. Since the UIX is purely static, the error in the calculation of its expectation value is likely to be small. In SNM, with only the A18 interaction, the gives more than half of the total NNI energy at all densities considered. The corresponding calculation with A18+UIX interactions, shows a significant increase in the magnitudes of the negative contribution of between and 0.4 fm, associated with pion condensation. In PNM these pion exchange interactions make relatively small contributions at densities below the phase transition, occurring at (0.5) fm with (without) UIX. However, at the densities above the transition they make large negative contributions comparable to those in SNM. At the highest densities the contributions of and become very large, and the validity of this purely nonrelativistic approach becomes questionable. As discussed in the following section, approximately 40 % of the contribution of is due to relativistic boost corrections to the NNI , and a more plausible theory is therefore obtained by removing this boost contribution from the UIX interaction.

The total energies, calculated in the manner previously described for SNM and PNM, appear in Tables 6-7, and in Figs. 2 and 3. The pronounced kink in the of SNM with , at fm, is due to the phase transition; the corresponding feature in PNM is a somewhat more subtle change in the slope of the curve at fm.

It is evident from the figures for SNM that without the , the present calculation cannot explain the empirical saturation density , of nuclear matter. As previously noted, the strength of is adjusted to obtain the correct equilibrium density in calculations with . However, the present calculations with underbind SNM at saturation density, giving MeV per nucleon instead of the empirical value of MeV. This discrepancy is presumably due to use of imperfect variational wavefunctions, which do not include, for example, three- and higher-body correlations. It is known from comparison of the results of variational Monte Carlo (VMC) and exact GFMC calculations [33, 34], that variational wave functions of the present form underbind the light p-shell nuclei. The variational energy of Be for example, is above the exact GFMC result by 12 %, even after incorporating into the wave function some of the three-body correlations, which we have neglected here.

## Iii Relativistc Boost Correction to the Nn Interaction

In all analyses, the NN scattering data is reduced to the center of mass frame and fitted using phase shifts calculated from the NNI, , in that frame. The obtained by this procedure describes the NN interaction in that frame, in which the total momentum , is zero. In general, the interaction between particles depends upon their total momentum, and can be written as

(1) |

where is the interaction for , and is the boost interaction [25] which is zero when .

It is useful to consider a familiar example. The Coulomb-Breit electromagnetic interaction [35] between two particles of mass m and charge Q, ignoring spin dependent terms for brevity, is given by

(2) |

up to terms quadratic in the velocities of the interacting particles. In our notation it is expressed as

(3) |

with

(4) | |||||

(5) |

where is the relative momentum.

In all realistic models of , such as the A18, the dependence on is included in the momentum-dependent part of the interaction, . However, we have neglected the in the calculations presented in the previous section. Even though contributions of the boost interaction to the binding energy of SNM and H were estimated by Coester and coworkers years ago [36, 37], these contributions have been neglected in most subsequent studies of dense matter.

Following the work of Krajcik and Foldy [38], Friar [39] obtained the following equation relating the boost interaction of order to the interaction in the center of mass frame:

(6) |

The general validity of this equation in relativistic mechanics and field theory was recently discussed [25]. Incorporating the boost into the interaction yields a nonrelativistic Hamiltonian of the form:

(7) |

where the ellipsis denotes the three-body boost, and four and higher body interactions. This contains all terms quadratic in the particle velocities, and is therefore suitable for complete studies in the nonrelativistic limit.

Studies of light nuclei using the VMC method [40, 41] find that the contribution of the two-body boost interaction to the energy is repulsive, with a magnitude which is 37% of the contribution. The boost interaction thus accounts for a significant part of the in Hamiltonians which fit nuclear energies neglecting .

In the present calculations we keep only the terms of the boost interaction associated with the static part of , and neglect the last term in Eq. (6). That term is responsible for Thomas precession and quantum contributions that are negligibly small here [42]. Our is given by

(8) |

The two terms are due to the relativistic energy expression and Lorentz contraction, and are denoted and , respectively. The three-nucleon interaction used in the Eq. (7) is denoted by . Its parameters are obtained by fitting the binding energies of H and He, and the equilibrium density of SNM, including . The strength of is 0.63 times that of in UIX, while that of is unchanged. The resulting model of is called UIX.

The approximate Hamiltonian , containing A18 and UIX interactions without , and the more correct Hamiltonian , containing A18, UIX and interactions, yield very similar results for light nuclei up to Be [41] and for SNM up to equilibrium density. However, the two models differ at higher densities, since the contributions of and have different density dependences.

One may also consider relativistic nuclear Hamiltonians of the type

(9) |

which require re-fitting the two-nucleon scattering data to determine the two-body interaction, , using relativistic kinetic energies [40]. In light nuclei, the contribution accounts for most of the difference between the energies obtained with and , since the difference between the contributions of the nonrelativistic and relativistic kinetic energies is largely cancelled by the difference in interaction energy contributions from and . The results obtained with are very close to those from , indicating that the former represents a significant improvement over

### iii.1 Calculation of Boost Interaction Energy

The relativistic boost contributions are calculated by evaluating terms in the cluster expansion of and . In addition to the dominant two-body cluster, we have calculated dressed three-body separable diagrams and central chain diagrams.

In the case of the two-body cluster, the gradients in the center of mass momentum operator, , can act only on the Fermi gas part of the wavefunction, since the correlations depend only on the relative coordinate. Thus, the two-body cluster contribution to is

(10) | |||||

(11) |

The quantities in the integrand of these expressions represent the spin-isospin-independent part (or “C-part”) of the operator product enclosed by the parentheses. Only the C-part of operator products appear in the cluster integrals, since the energy expectation value requires a sum over all possible and [9], which average to zero in isotropic matter. In the case of SNM the indices p,m,p run over the first six operators, as we boost only the static interactions and consider only static correlations in this calculation. The index n=1,4 comes from the exchange operator. In the case of PNM the operators are eliminated from the Hamiltonian, thus the indices and can represent only unit or spin-dependent operators. The exchange operator also contributes a factor of 1/s, where s is the degeneracy of the system (4 for SNM, 2 for PNM.)

The differs from the corresponding expression for the two-body direct contribution to by a factor , which is the expectation value of in the Fermi gas. The exchange part of the cluster, , has the same form as the corresponding expression for , with replaced by , where is the Slater function. This expression results from from gradients in acting on the plane waves

(12) | |||||

Here, A is the number of nucleons, is the normalization volume, and since we are in the thermodynamic limit, .

The two-body cluster contribution to is

(13) | |||||

(14) |

which is simply the cluster contribution of a nonrelativistic potential , with the direct term multiplied by , and with the in the exchange term replaced by . As in the case, the extra factors in result from the gradients in acting on the Fermi gas part of the wave function.

The three-body separable diagrams represent the most significant many-body cluster contributions to the boost energy. The direct term for a boost , where or , has the form:

(15) |

The plane waves are written here in terms of the relative momentum, , and the center of mass momentum, , of the interacting pair. The interacting exchange and the passive exchange expressions are obtained from the above by inserting the appropriate exchange operators ( or ) to the far left of each operator product, and replacing the first plane wave product by or .

Following the notation used in the calculation of the MD interaction energy [43, 24], separable diagrams are classified as K-diagrams and F-diagrams. The former have gradients in acting on the Fermi gas part of the wavefunction, and the latter have them acting on the correlation operators . As with the two-body cluster contribution , the K-diagrams depend linearly on the Fermi kinetic energy. While scales as , like the , the K-diagram contributions scale roughly as . The K-diagrams generally make only small contributions to , the major separable contributions coming from F-diagrams. The relatively large contribution of the F-diagrams, versus the K-diagrams, can be understood in the following way. The correlated particle in the separable diagram modifies the center of mass momentum of the interacting pair via , thus enhancing the boost correction. As the form of the F-diagram integrals suggest, we find that their contributions exhibit the same scaling behavior as , namely as .

K-diagram contributions to have been evaluated for the direct three-body separable diagram and the interacting exchange diagram. These contributions factorize into an integral over , which is simply the corresponding two-body boost diagram, and an integral over . The latter integral is a so-called single-loop vertex correction, which is included in the more general vertex correction, , to , defined in [9]. The direct K-diagram contribution is

(16) | |||||

where the , and matrices, defined in [9], give the C-parts taking into account the non-commutativity of operators and . The corresponding interacting exchange contribution is given by

(17) | |||||

The F-diagram contributions to are evaluated for the direct, interacting exchange and passive exchange terms. These contributions also factor into separate integrals, where the integral has the form of the two-body contribution to , and the integral is a new type of vertex correction involving gradients of the . The direct and -exchange integrals are

(18) | |||||

(19) | |||||

The -exchange contribution is evaluated only to leading order, namely for the cases where at least one is a central link. In this approximation, this contribution takes the form:

(20) | |||||

In this equation, the index n runs from 2 to 4 only.

The K-diagram separable three-body contributions to have the same general structure as the corresponding contributions to . The direct term and the interacting exchange term have been evaluated and are presented below.

(21) | |||||

(22) | |||||

The separable three-body F-diagrams have a more complicated structure for the . The direct diagram has the general form

(23) | |||||

The integrand can be written as a sum of four terms having the gradients in acting on different parts of the correlations:

(24) |

The C-parts in the above expressions depend on the cosine of the azimuthal angle, , and therefore cannot be expressed exclusively in terms of the matrices, which include an implicit average over that angle. The C-parts must then be individually evaluated for each operator product.

The contribution of the direct diagram is dominated by the first term in the above sum. In evaluating the interacting exchange and passive exchange diagrams, only the term corresponding to that dominant part has been included. For example, the interacting exchange contribution is approximated by

(25) |

The corresponding passive exchange contribution was calculated with the additional simplification of considering only leading term contributions, having at least one central link.

We have also evaluated central-chain diagram contributions, denoted by , to the boost interaction expectation values. These diagrams are obtained by dressing two-body cluster diagrams with hypernetted central chains. Their contribution was found to be significantly smaller than the contribution from separable three-body terms.

The combined results for the two-body clusters and for the many-body clusters appear in Table 8. The first four columns contain the boost contributions to the energy of SNM and PNM, calculated using the optimal wavefunctions for the A18 interaction alone, while the next four columns contain the corresponding boost contributions, calculated with the optimal wavefunctions for the A18+UIX. As expected, the many-body contributions to the boost interaction energy are comparable to the two-body contributions, because and have similar magnitudes.

Detailed breakdowns of the boost contributions in the A18+UIX model are presented in Tables 9 and 10 for and . In all cases the bulk of the contributions come from the direct terms. Since we can integrate by parts to obtain plus additional terms, their expectation values should be similar in magnitude. The ratio of the contributions of and in SNM is found to be 0.7 at all densities, while in PNM it is 0.75 at fm, and at higher densities. Results of VMC calculations [40] have shown that this ratio is 0.5 in H and He.

The has two factors, and , suggesting the approximation

(26) | |||||

Our results show that this approximation has errors of only 10 % in SNM, but can be wrong by 50 % in PNM. Nevertheless it may be used to estimate the order of magnitude of the contribution of relativistic boost interactions.

### iii.2 Nucleon Matter Energies

We evaluate the boost interaction contributions as a first order perturbation. Thus the energies of nucleon matter with the A18+ interactions are obtained by simply adding the contributions listed in Table 8 to the A18 energies. The results are listed in Tables 6 and 7, and shown in Figs. 2 and 3.

The three-body interaction, UIX, to be used with A18+, contains the term that is 0.63 times the in the UIX. Since the boost effects are treated only in first order, the energies for the A18++UIX interaction are obtained by adding to the energies for A18+UIX interactions. The results are listed in Tables 6 and 7, and shown in Figs. 2 and 3. At low densities the A18+UIX and A18++UIX interactions yield similar results. However the energies predicted by the latter model are lower than those of the former at higher densities, where 0.37 is much larger than . The difference between the energies predicted by A18 + and A18++UIX interactions at higher densities is due to three-body forces; it is smaller than that between the energies obtained from A18 and A18+UIX by almost a factor of two.

The results obtained with the Urbana density-dependent interaction (U14-DDI) [13] are also shown in Figs. 2 and 3 for comparison. Since Skyrme-type interactions based on the U14-DDI explain nuclear binding energies quite accurately, it is likely to provide a reliable representation of phenomena at .

The variation of the kinetic and interaction energies with nucleon density in our most realistic model, with A18++UIX, interactions is shown in Figs. 4 and 5. Due to a large cancellation between the contributions of and , the total contribution is now smaller than that of , and the contribution is also small. An order of magnitude estimate of the boost correction to the three body interaction is given by , using generalizations of Eq. (26). This estimate is less than 10 MeV at the highest density considered.

The most significant remaining problem appears to be our neglect of the boost corrections to the , the unboosted contribution of which is quite large at high densities. Such corrections involve terms of order (velocity), which are beyond the scope of the present work. The kinetic energy also has corrections of that order, which we cannot include because the A18 interaction is fitted to data using nonrelativistic kinetic energy. However, the correction to the Fermi gas kinetic energy is , which is less than 10 MeV in PNM at the density of 6.

The Dirac-Brueckner approximation provides another way to estimate relativistic effects using realistic models of nuclear forces. The energies of our A18++UIX model are lower than those of lowest order Dirac-Bruekner calculations with the Bonn A potential [44]. For example, at 4 we obtain 58 and 128 MeV per nucleon, while the Dirac-Bruekner calculation gives 76 and 164 MeV per nucleon, for SNM and PNM respectively. However, the difference between the results of the two methods is significantly smaller than that between either of the two and the uncorrected A18 energies of 3 and 59 MeV per nucleon. (Tables 6 and 7).

In the last column of Table 6 we list SNM energies obtained by adding a correction to the A18++UIX results. The empirical binding energy and density of SNM are reproduced with MeV fm, and fm. This correction has a maximum value of 4.5 MeV at 0.11 fm, and the “corrected” of SNM is a better representation of known nuclear properties at lower densities. It becomes identical to the obtained using the A18++UIX model at higher densities.

## Iv Cold Catalyzed Nucleon Matter

In this section we use the results of the earlier sections to calculate the equation of state and composition of cold, catalyzed matter, i.e. matter at zero temperature in its lowest energy state. Here the matter is assumed to be made up of neutrons, protons and leptons; the possible admixture of quark matter is considered later. Since the boost interaction is clearly an integral part of the two-nucleon interaction, we regard the models A18+ and A18++UIX as realistic, and discuss their results in detail. The difference between the two models demonstrates the effect of the three-nucleon interaction. Some of the results obtained with the less realistic models without the boost interaction are also presented for comparison.

Matter at zero pressure, at the surface of a neutron star, is made up of atoms of Fe, just as in terrestrial iron. This is the most stable form of electrically neutral matter composed of neutrons, protons and electrons. Below the neutron star surface, as a result of the increased pressure of the matter caused by the gravitational attraction, the atoms become completely ionized and the electrons form a relativistic Fermi gas, whose Fermi energy becomes competitive in magnitude with nuclear energies [45]. Consequently, electron capture by the protons can occur. As a result, with increasing depth below the stellar surface the nuclei become more neutron rich, and cross the neutron-drip line. At this point the most energetic neutron orbitals have become unbound, and the matter consists of neutron-rich nuclei immersed in a neutron gas, whose density also increases as the pressure is increased further. The baryon number density at which this transition occurs is about fm. As the pressure and density continue to increase, the charge number of the nuclei remains in the range but the mass number grows steadily, and the distance between nuclei decreases. In the density range above about fm, where the volumes occupied by nuclei and by the surrounding neutron gas become comparable, the matter may undergo an inversion to bubbles of neutron gas surrounded by nuclear matter by going through a progression of phases involving non-spherical shapes, although this aspect of the crustal structure is somewhat model dependent [14, 46]. At a density above about fm there no longer occur nuclei or other clumps of proton-containing matter, and cold, catalyzed matter becomes a uniform fluid of neutrons with a small fraction of protons.

A reliable discussion of the properties of matter over the crustal density range requires a nuclear model that can describe inhomogeneous matter in the geometries occurring there. The difficulties encountered cause the model dependence mentioned. However, for neutron stars with masses 1.4, the mass fraction contained in the crust of the star is less than about 2%. We have therefore used results of earlier work [14, 46] for matter at densities fm. Since over that density range our present matter energies agree well with ones used earlier, this substitution causes negligible inconsistency in our conclusions about the total mass of the neutron star.

At densities of 0.1 fm and greater, we require properties of charge neutral uniform matter made up of neutrons, protons, electrons and muons in beta equilibrium. At a given baryon number density , the conditions to be satisfied by the components are charge neutrality,