Issue 
A&A
Volume 576, April 2015



Article Number  A68  
Number of page(s)  9  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/201424800  
Published online  01 April 2015 
Neutron stars with hyperon cores: stellar radii and equation of state near nuclear density
N. Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, 00716 Warszawa, Poland
email: fortin@camk.edu.pl; jlz@camk.edu.pl; haensel@camk.edu.pl; bejger@camk.edu.pl
Received: 13 August 2014
Accepted: 23 January 2015
Context. The existence of 2 M_{⊙} pulsars puts very strong constraints on the equation of state (EOS) of neutron stars (NSs) with hyperon cores, which can be satisfied only by special models of hadronic matter. The radiusmass relation for these models is sufficiently specific that it could be subjected to an observational test with future Xray observatories.
Aims. We want to study the impact of the presence of hyperon cores on the radiusmass relation for NS. We aim to find out how, and for which particular stellar mass range, a specific relation R(M), where M is the gravitational mass, and R is the circumferential radius, is associated with the presence of a hyperon core.
Methods. We consider a set of 14 theoretical EOS of dense matter, based on the relativistic meanfield approximation, allowing for the presence of hyperons in NSs. We also discuss a recent EOS based on nonrelativistic Gmatrix theory yielding NSs with hyperonic cores and M> 2M_{⊙}. We seek correlations between R(M) and the stiffness of the EOS below the hyperon threshold needed to pass the 2 M_{⊙} test.
Results. For NS masses 1.0 <M/M_{⊙}< 1.6, we get R> 13 km, because of a very stiff prehyperon segment of the EOS. At nuclear density (n_{0} = 0.16 fm^{3}), the pressure is significantly higher than a robust upper bound obtained recently using chiral effective field theory.
Conclusions. If massive NSs do have a sizable hyperon core, then according to current models the radii for M = 1.0 − 1.6 M_{⊙} are necessarily >13 km. If, on the contrary, a NS with a radius R^{(obs)}< 12 km is observed in this mass domain, then sizable hyperon cores in NSs, as we model them now, are ruled out. Future Xray missions with <5% precision for a simultaneous M and R measurement will have the potential to solve the problem with observations of NSs. Irrespective of this observational test, present EOS allowing for hyperons that fulfill condition M_{max}> 2 M_{⊙} yield a pressure at nuclear density that is too high relative to uptodate microscopic calculations of this quantity.
Key words: dense matter / equation of state / stars: neutron
© ESO, 2015
1. Introduction
Hyperons, which are baryons containing at least one strange quark, were discovered more than 50 years ago. They are frequently studied in terrestrial laboratories. Although unstable on Earth, it is expected however that they are stably present in the dense interiors of neutrons stars (NSs). Recent measurements of 2M_{⊙} pulsars (Demorest et al. 2010; Antoniadis et al. 2013) represent a challenge for equations of state (EOS) of NSs with hyperon cores. The difficulty of reaching a high mass is related to a significant softening of the EOS associated with the hyperonization of the matter. With one exception, only EOS based on the relativistic meanfield (RMF) models, after tuning of their Lagrangians, turned out to be able to produce NS with maximum allowable mass M_{max}> 2M_{⊙} and sizable hyperon cores (see Colucci & Sedrakian 2013 and references therein). A very recent paper of Yamamoto et al. (2014) is based on the nonrelativistic Gmatrix theory of dense baryon matter. Their MPa model yields M_{max}> 2M_{⊙} for NSs with hyperon cores. As we will see, however, the properties of NS models for MPa EOS are consistent with those obtained by our set of RMF EOS.
These successful models with hyperonization of NS matter (one of them being hereafter referred to as EOS.H) merit a careful inspection. We will focus here on two specific features of these models: NS radii and EOS at prehyperon density; as we will show, both features are interrelated.
The EOS below nuclear density ρ_{0} = 2.7 × 10^{14}g cm^{3} (corresponding to the baryon number density n_{0} = 0.16 fm^{3}) is commonly believed to be rather well known (see Hebeler et al. 2013). However, to construct a complete family of NS models, up to the maximum allowable mass M_{max}, one needs the EOS for up to ~8ρ_{0}. Nuclear densities and are suitable units for the massenergy density ρ and the baryon number density n_{b} in the NS core. In what follows we will use dimensionless (reduced) densities and . Because of uncertainties in the theory of dense matter, the only chance to unveil the actual EOS of degenerate matter at supranuclear density relies on the observations of NSs.
Mathematically, both M_{max} and the radius of NS of (gravitational) mass M, R_{M} ≡ R(M), are functionals of the EOS. We expect that NS matter at (the socalled outer core) is composed mostly of neutrons, with a few percent admixture of protons, electrons, and muons. At higher densities, hyperons or even quark gluon plasma might appear, forming a strangeness carrying NS core. In the present paper, we restrict ourselves to the NSs cores where quarks are confined into baryons: nucleons and hyperons. The EOS fulfilling M_{max}> 2M_{⊙} and allowing for the presence of hyperons in NSs cores form the set EOS.H.
Independent of the uncertainties related to the structure of the cores of NSs, any theoretical EOS.H has to be consistent with the semiempirical parameters of nuclear matter at . Moreover, the hyperon component should be consistent with semiempirical estimates (potential wells, ΛΛinteraction) coming from hypernuclear physics (SchaffnerBielich 2008). Basic features of EOS.H resulting from various theoretical models and puzzles, which are as yet not resolved, are briefly summarized in Sect.2. Another constraint, related to the value of the pressure P at nuclear (saturation) density, is discussed in Sect.3.
The impact of the uncertainty in the EOS on the mass vs. central density dependence for NS is described in Sect.4. We prefer to use instead of because the former characterizes the degree of packing of baryons at the NS centre. Different segments (domains) of the EOS determine measurable global stellar parameters in different NS mass domains. The radius of a 1.4M_{⊙} NS, R_{1.4}, is mostly determined by EOS, while the value of M_{max} is to a large extent determined by EOS.
As we show in Sect.5, the NS radius for 1.0M_{⊙}<M< 1.6M_{⊙} for EOS.H is larger than 13 km. This seems to be an unavoidable consequence of M_{max}> 2.0M_{⊙} condition, which implies a very high stiffness of the prehyperon (i.e., purely nucleon) segment of EOS.H.
Section6 explores the difference in the values of NS masses at a given central pressure and connects it with the difference in pressure distribution within NS models. In Sect.7 the causallimit EOS is used to provide a bound on the NS radius, R(M). Section8 is devoted to the effects of NS rotation.
In Sect.9 we discuss a possible meaning of large radii of NSs with hyperon cores in the context of recent measurements of radii of NSs. Our conclusions are formulated in Sect.10.
Equations of state. For EOS.N_{ref} (upper part) we selected three widely used EOS, which produce standard values of NS parameters.
2. Equation of state of neutronstar matter
Except for a very outer layer, whose contribution to NS mass and radius can be neglected, the matter in NS interior is strongly degenerate, and can be approximated by that calculated at T = 0, assuming the ground state composition (cold catalyzed matter, see e.g. Haensel et al. 2007). For these EOS, pressure depends on the density only.
In general theory of relativity (GTR) the matter density is defined as ρ = ℰ /c^{2}, where ℰ is the total energy density (including rest energy of particles). Baryon density n_{b} is defined as the baryon number (baryon charge) in a unit volume. Using elementary thermodynamics one obtains the relation between P and ρ from the calculated function ℰ(n_{b}), (1)
2.1. EOS satisfying the semiempirical nuclearhypernuclear constraints and M_{max}> 2.0 M_{⊙}
The models of the EOS.H set reproduce (within some tolerance) four semiempirical values of parameters of nuclear matter at saturation: saturation density n_{s}, energy per nucleon E_{s}, symmetry energy S_{s} and incompressibility K_{s}. The semiempirical value of a fifth parameter, the density slope of symmetry energy L_{s}, is relatively poorly known (Hebeler et al. 2013) and hence it is not imposed as a constraint. Together with M_{max}> 2M_{⊙}, this amounts to five constraints imposed on an EOS belonging to EOS.H.
Apart from the five constraints described above, a given EOS.H has to reproduce additional semiempirical values of three potential wells of (zero momentum) hyperons in nuclear matter at n_{s}: these include the sixth, seventh, and eighth constraints on EOS.H. Moreover, for most of the EOS.H models, a ninth condition, fitting a semiempirical estimate of the depth of potential well of Λ in Λmatter, is also imposed. In summary, there are eight or nine constraints to be satisfied by an EOS.H. Constructing an EOS of the EOS.H set is therefore associated with a strong tuning of the dense matter models. With the exception of a very recent EOS.H of Yamamoto et al. (2014), only the EOS based on the nonlinear RMF theories can satisfy these conditions.
Selecting a very specific type of approximation, the RMF, is a first tuning of the dense matter model. Moreover, to satisfy M_{max}> 2M_{⊙}, baryon fields are not only coupled to (standard) σ, ω, and ρ meson fields. Namely, the hyperon fields are additionally coupled to the vector φ meson field. The addition of φ (and possibly also of a σ^{⋆} meson field, which provides a scalar coupling between hyperons only) is a second tuning. All EOS from EOS.H are able to satisfy M_{max}> 2.0M_{⊙} due to repulsion produced by the φmeson coupled only to hyperons. Moreover, in some cases, an amplification of the hyperon repulsion due to SU(6)symmetry breaking in the vectormeson coupling to hyperons is introduced, which is a third tuning. Finally, some models have densitydependent coupling constants of baryons to meson fields. This allows for a fourth tuning. The models included in EOS.H set are listed in Table 1 together with brief characteristics and references.
Among the eight models consistent with a 2M_{⊙} NS in Sulaksono & Agrawal (2012), we selected four of them, three with SU(6) symmetry, the stiffest, the softest and an intermediate one, and one with SU(6) symmetry broken. Lopes & Menezes (2014) obtain similar results to Weissenborn et al. (2012) and their EOS are not included in Table 1. We also restrict ourselves to EOS calculated for a zero temperature. The EOS at finite temperature, such that they are suitable for supernova and protoneutron star modelling and including a transition to hyperonic matter, have also been developed. In particular Banik et al. (2014) using a RMF model obtain a maximum mass for cold NS of M_{max} = 2.1M_{⊙}. Gulminelli et al. (2013), extending the EOS by Lattimer & Swesty (1991) to include hyperons with the model by Balberg & Gal (1997), reach M_{max} = 2.04M_{⊙}; however, they include only the Λ hyperon.
In order to have a reference set of models of widely used purely nucleon EOS, denoted as EOS.N_{ref}, we selected three models listed in the upper part of Table 1. These EOS produce “standard NS”, are consistent with all semiempirical constraints and yield M_{max}> 2.0M_{⊙}. We note that not only is L_{s} for EOS.H widely scattered (62−118 MeV) but it is also typically much higher than for EOS.N_{ref} (37−59 MeV).
The EOS of the core is supplemented with an EOS of NS crust. We assume that the crust is composed of cold catalyzed matter. For the very outer layer with ρ< 10^{8} g cm^{3} we use classical BPS EOS (Baym et al. 1971). The outer crust with ρ> 10^{8} g cm^{3} is described by the EOS of Haensel & Pichon (1994), while for the inner crust we apply the SLy EOS of Douchin & Haensel (2001). A smooth matching with an interpolation between the crust and core EOS is applied to get a complete EOS of NS interior^{1}.
3. Overpressure of EOS.H at nuclear density
We start our comparative study of EOS.H and EOS.N_{ref} by calculating the pressure at nuclear density, P^{(N)}(n_{0}) and P^{(H)}(n_{0}), respectively. Results are collected in the Appendix Table A.1. In the following P_{33} refers to the pressure P in the units of 10^{33} dyn cm^{2}. We notice a striking difference between EOS.H and EOS.N_{ref}. The values of are concentrated within 3.3 ± 0.3, while the values of are significantly higher, within 8 ± 2.5.
Hebeler et al. (2013) argue that the EOS of pure neutron matter at n_{b} ≤ n_{0} can be reliably calculated using the uptodate manybody theory of nuclear matter. Their results are in remarkable agreement with those by Gandolfi et al. (2012) using an approach completely different from that adopted by Hebeler et al. (2013). At this density, NS matter in beta equilibrium is expected to be somewhat softer than the pure neutron matter. Hebeler et al. (2013) calculate the effect of the presence of an admixture of protons and electrons in beta equilibrium on the EOS, combining the EOS of neutron matter and available semiempirical information about nuclear symmetry energy and its density dependence (slope parameter L_{s}). Interpolating between the values in their Table 5, we conclude that Hebeler et al. (2013) provide the following constraint on the pressure of NS matter at : (2)This constraint is satisfied by EOS.N_{ref}. On the contrary, it is badly violated by EOS from EOS.H, which give P_{33}(n_{0}) significantly higher than the upper bound in Eq.(2). Before considering consequences of the “overpressure” of the nucleon (prehyperon) segment of EOS.H for NS radii, we discuss two different parametrizations of NS models.
Fig. 1 Mass density ρ = ℰ /c^{2} vs. baryon density n_{b} for NS matter for the set of EOS presented in Figs. 2−4. Dotted segments correspond to the central densities of NS models, which are unstable with respect to radial oscillations. Relation ρ = ρ(n_{b}) deviates from linearity for n_{b}> 0.3fm^{3}. Nonlinearity grows with increasing n_{b} and is EOSdependent. For n_{b} ≲ 0.2fm^{3}, the linear approximation ρ ≈ n_{b}m_{n} (where m_{n} is neutron mass) is valid. (Colour online.) 

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4. Two densities and two parametrizations of neutron star models
When investigating the EOS of NS matter, we have to consider two distinct densities, ρ = ℰ /c^{2} and n_{b} (Sect.2). While ρ is the relevant quantity for GTR calculations of the NS structure, it is n_{b} that is associated with an average distance between baryons (treated as pointlike objects), . Therefore, knowing n_{b}, we can compare an actual r_{b} with the average distance between nucleons in nuclear matter at normal nuclear density, , . At subnuclear densities, ρ of NS matter can be very well approximated by n_{b}m_{n}, where m_{n} is neutron rest mass. However, at supranuclear densities ρ grows nonlinearly with n_{b}. This nonlinear dependence is model dependent, see Fig. 1, and actually determines the EOS, see Eq. (1).
Fig. 2 Gravitational mass M vs. central baryon density n_{c} for nonrotating NS models based on the sets EOS.H (blue lines – H) and EOS.N_{ref} (black lines – N_{ref}). In N_{ref}: A is APR, B is BSk20, C is DH; in H: a is SA.BSR2, b indicates BM165, c indicates GM1Z0, d is UU1, e is G.TM1C. EOS labels from Table 1. Solid lines: stable NS configurations. Dotted lines: configurations unstable with respect to small radial perturbations. Vertical lines crossing the M(n_{c}) curves indicate configurations with n_{c}/n_{0} = 2,3,.... Hatched strip correspond to M = 1.4 ± 0.05M_{⊙}, and the observational constraints for J16142230 and J0348+0432 are marked in blue and magenta, respectively (1σ errors). (Colour online.) 

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Fig. 3 Same as in Fig.2, except with n_{c} replaced by ρ_{c}. (Colour online.) 

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In Fig.2 we plot the relations between M and the central baryon density n_{c} for nonrotating NS models. Several conclusions result from this figure. First, the central density in a 2M_{⊙} NS is typically , so that at the star’s centre r_{b}/r_{0} ≈ 0.6. Second, the N segment of EOS.H, corresponding to , is so much stiffer than a similar segment of the EOS.N_{ref}, that . In other words, to yield M_{max}> 2.0 M_{⊙} despite the hyperon softening, the prehyperon (nucleon) segment of EOS.H has to be very stiff.
Figure 3 shows the curves. The in 2M_{⊙} stars can be as high as 6−7, significantly larger than the corresponding values of . For the M_{max} configurations, the difference is even larger. However, it is n_{c} and not ρ_{c} that determines the mean interbaryon distance at the centre of the star.
The value of for 2M_{⊙} stars can be used to evaluate the importance of relativistic effects in the relevant manybody problem. The number density n_{i} of baryon species i = n,p,Λ,... with a mass m_{i} can be related to their mean velocity ⟨ v_{i} ⟩. In the free Fermi gas approximation, , where x_{i} = n_{i}/n_{b} and m_{n} is the neutron mass. At the centre of a 2M_{⊙} star we expect so that . The multicomponent character of dense baryon matter implies lower ⟨ v_{i} ⟩ as compared to a pure neutron matter case and, consequently, smaller relativistic effects in the manybody system, as stressed in the classical paper of Bethe & Johnson (1974) to justify the use of a nonrelativistic manybody theory of dense matter. Therefore, we might expect that not only RMF but also some nonrelativistic models, consistent with semiempirical nuclear and hypernuclear matter constraints, could yield 2M_{⊙} stars with hyperonic cores. However, as far as we know, there is only one such a nonrelativistic dense matter model satisfying these conditions (Yamamoto et al. 2014).
A very recent calculation of Katayama & Saito (2014) is performed using a relativistic formulation of Gmatrix theory (DiracBruecknerHartreeFock approximation). Some models from this work give M_{max}> 2M_{⊙}, but they do not satisfactorily reproduce the semiempirical parameters of nuclear matter (see Table 1 of Katayama & Saito 2014).
Fig. 4 Gravitational mass M vs. circumferential radius R for nonrotating NS models. For the labels, see details in the caption of Fig. 2. (Colour online.) 

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5. Radii of neutron stars with hyperon cores
The radius R is a measurable NS parameter and therefore large radii of NS with hyperon cores could be subject to an observational test. The M(R) lines for selected EOS from EOS.H are plotted in Fig.4. By construction, selected EOS.H include those producing an envelope of a complete Hbundle of M(R) curves.
Fig. 5 Pressure at nuclear density vs. NS radius for 1.4M_{⊙} for EOS.H and EOS.N_{ref} (see Table 1 for the labels). Grey horizontal strip – range of P(n_{0}) determined by Hebeler et al. (2013). Circles correspond to EOS with the TM1, squares with NL3, pentagons with GM1 model for the nucleon sector, respectively. Triangles indicate EOS built with RMF models with densitydependent couplings. (Colour online.) 

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In the mass range 1 <M/M_{⊙}< 1.6, the Hbundle is centred around ~14.2 km. There is a wide >1 km gap between the H and N_{ref} bundles in this mass range. More specifically, we find a lower bound R^{(H)}> 13 km in the considered mass range. In Fig. 5 we plot the points calculated for EOS.H in the P_{33}(n_{0}) − R_{1.4} plane, where R_{1.4} ≡ R(1.4M_{⊙}). Large values of are correlated^{2} with a high P(n_{0}) violating the upper bound of Hebeler et al. (2013).
Fig. 6 NS radius for 2.0M_{⊙} vs. radius at maximum allowable mass. EOS labels from Table 1. (Colour online.) 

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Consider the largest, uptodate, measured pulsar mass, 2.0M_{⊙}, and EOSdependent (theoretical) maximum allowable mass M_{max}. The EOSdependent radius of 2.0M_{⊙} star and the radius at M_{max} are denoted by R_{2.0} and R_{Mmax}, respectively. Calculated points in the R_{Mmax}R_{2.0} plane are shown in Fig.6. Those for EOS.N_{ref} are tightly grouped (within less than 1 km) around R_{2.0} = 11 km. The points calculated for EOS.H are loosely distributed along the diagonal of the bounding box, with ranging within 10.5−12.5 km, and within 11.5−14 km.
Fig. 7 Dependence of M on the central pressure, for nonrotating NS models, for M< 1.6M_{⊙}. Dashed and dasheddotted lines describe the fits for N and H families, respectively. For details, see the text. (Colour online.) 

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6. Mass vs. central pressure for neutron stars with hyperon cores
The difference between M − R relations for EOS.N_{ref} and EOS.H and in particular an Rgap for 1.0 <M/M_{⊙}< 1.6 reflects a difference in the pressure distributions within NS models of a given M. This is visualized in Fig.7 where we show M(P_{c}) plots for EOS.N_{ref} and EOS.H families. For M ≲ 1.6M_{⊙} we find that M = M(P_{c}) is well approximated by (3)where P_{34} = P/ 10^{34}erg cm^{3} and β_{H} ≈ β_{N} = 0.52, while prefactor for the N_{ref}bundle is significantly smaller than for the Hbundle.
There is a sizable gap between M^{(N)}(P_{c}) and M^{(H)}(P_{c}) bundles. The Mgap ranges from ~0.2M_{⊙} at P_{c,34} = 3 to ~0.7M_{⊙} at P_{c,34} = 8. The fit with β = const. breaks down for the Hfamily for M> 1.6M_{⊙} because of the hyperon softening of the EOS. For M ~ 2M_{⊙} (not shown in Fig.7), the gap between the N_{ref} and H bundles disappears.
7. On the causal bounds on R(M)
The radius of a neutron star of given mass M, based on a causal EOS, cannot exceed a limit that is calculated by replacing this EOS above some “fiducial density” n_{⋆} by the causallimit (speed of sound = c) continuation. The causallimit (CL) segment of the EOS.CL is: (4)for n_{b} ≥ n_{⋆}, whereas P_{⋆} and ρ_{⋆} are calculated from the original EOS, P_{⋆} = P(n_{⋆}), ρ_{⋆} = ρ(n_{⋆}).
Fig. 8 Solid lines: massradius relations for an EOS.N(DH) and an EOS.H (DS08), respectively. Longdash line: causallimit upper bound on R_{M} for these EOS, assuming n_{⋆} = n_{0}. Dotted line: similar upper bound on , except with n_{⋆} = 1.8n_{0}, as in Hebeler et al. (2013). (Colour online.) 

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We choose n_{⋆} = n_{0}. Our choice is more conservative, for the reasons explained later in this section, than 1.8n_{0} used by Hebeler et al. (2013). We fix the model for the crust segment of all EOS, i.e. for n_{b}< 0.5n_{0}, we use the EOS described in the final fragment of Sect. 2.1.
In Fig.8 we show M(R) curves for selected models from EOS.N_{ref} and EOS.H (DH and DS08 EOS, respectively), together with their causallimit (CL) bounds. The CL curves bifurcate from the actual M(R) curves at n_{c} = n_{⋆}, corresponding to M = M_{⋆}. For a given M>M_{⋆}, the CLbound is denoted by R_{max}(M).
Consider first . We get and . As is significantly higher than , is larger than , e.g. by about 1 km at 1.4M_{⊙}. The difference decreases to 0.5 km at 2M_{⊙}.
Consider now chosen by Hebeler et al. (2013). We then get and . At 1.4M_{⊙} the actual R^{(H)}(M) is larger than by about 1.4 km. The H configuration there violates the condition P_{c} ≫ P^{(H)} needed for a weak dependence of R_{max}(M) on the EOS below n_{⋆}. Therefore, a CLbound derived for EOS.N_{ref} should not be applied to EOS.H stars. Even at 2M_{⊙} the CLbound for H is larger by 1 km than the Nbound. We conclude that chosen by Hebeler et al. (2013) is too large to yield a unique CLbound for radii of the N_{ref} and H families simultaneously.
Fig. 9 M(R_{eq}) for two selected EOS: N (DH) and H (DS08). The R_{eq} is defined in the text. Solid curves: nonrotating configurations. Dashed lines: configurations rotating uniformly at f = 300 and 600 Hz. (Colour online.) 

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8. Effect of neutron star rotation
The fact that NSs rotate should be taken into account when comparing their radii; the values will differ from a nonrotating solution because of the effect of the centrifugal force. Mature NSs, and especially extremely stably spinning millisecond pulsars (spin frequencies larger than 100 Hz) may be, to an excellent approximation, described using the assumption of uniform rotation^{3}. Nonaxial deformations supported by the elastic strain accumulated in the crust, as well as nonaxiality generated by the internal magnetic field, are small for the rotating NSs that we are modelling. Therefore, in what follows we will assume that the configurations are stationary and axisymmetric, and we will define the circumferential equatorial radius R_{eq} as the proper length of the equator divided by 2π. For a given mass M, the value for EOS.Hbased NS increases with rotation rate more rapidly than for EOS.N_{ref}. This is a direct consequence of larger radii of hyperonic NSs. In Newtonian terms, for fixed mass M and spin frequency f, the gravitational pull at the equator is weaker, while the centrifugal force is stronger in the case of EOS.H stars: both effects increase the value of R_{eq}. Figure 9 compares M(R_{eq}) curves calculated for f = 300Hz and f = 600Hz with the nonrotating (f = 0) case. Frequencies ~300 Hz are measured for many radio and Xray millisecond pulsars, while f ~ 600 Hz is characteristic of a group of the fastest (Lorimer 2008; Patruno & Watts 2012).
For the EOS in Fig.9, the rotational increase of ΔR_{eq} is roughly twice as large for EOS.H compared with EOS.N. It can be well approximated, for a fixed EOS and mass M, as a quadratic function in f: (5)For M = 1.4M_{⊙}, one has ℬ_{H} = 3.1km / kHz^{2} and ℬ_{N} = 1.6km / kHz^{2}. At a given f, ΔR_{eq} decreases with increasing M. For an astrophysically motivated range of masses between 1.2M_{⊙} and 1.8M_{⊙}, the decrease of ΔR_{eq} with an increasing mass is approximately linear: ℬ(M) changes from 2.0km / kHz^{2} to 1.2km / kHz^{2} for EOS.N, and from 3.8km / kHz^{2} to 2.2km / kHz^{2} for EOS.H. These linear approximations for ℬ and relation (5) may be used to estimate the increase of R_{eq} for rotating nucleonic and hyperonic NSs.
Radius determination: most recent publications.
9. Observational determination of radii of neutron stars
The radius R can be extracted from the analysis of Xray spectra emitted by the NS atmosphere. Recent attempts are based on the modelling of Xray emission from four different types of objects (see Potekhin 2014, for a review):

isolated NSs (INSs);
quiescent Xray transients (QXTs) in lowmass Xray binaries (LMXBs), i.e. NSs in a binary system observed when the accretion of matter from their binary companion has stopped or is strongly reduced;
bursting NSs (BNSs) i.e. NSs from which recurring and very powerful bursts, socalled photospheric radius expansion (PRE) bursts, are observed;
rotationpowered radio millisecond pulsars (RPMSPs).
While the magnetic field and the chemical composition of the atmosphere of INSs are unknown and difficult to determine, QXTs have a low magnetic field, as a result of its decay due to the accretion of matter, and an atmosphere likely to be composed of light elements (H, possibly He; see Catuneanu et al. 2013; Heinke et al. 2014). In addition, QXTs in globular clusters are promising sources for the massradius determination since their distances are well known. BNSs that also have low magnetic fields and a lightelement atmosphere are interesting sources, all the more if the distance to them is known. However the modelling of the PRE bursts is still subject to uncertainties and debates (see Poutanen et al. 2014; Steiner et al. 2013; Güver & Özel 2013). Finally, constraints can be derived from the modelling of the shape of the Xray pulses observed from RPMSPs, in particular, if the mass is known from radio observations (Bogdanov 2013).
Conflicting constraints on the radius from the Xray emission of QXTs, BNSs, and RPMSPs have been obtained by different groups. In what follows, we restrict ourselves to the most recent publications, indicated in Table 2. Figure 10 shows the constraints with 2σ error bars derived in these papers for the radius R_{1.4} of a NS with a mass of 1.4 M_{⊙}.
Performing a simultaneous spectral analysis of six QXTs assuming unmagnetized hydrogen atmosphere, Guillot & Rutledge (2014) derive a constraint suggesting a small radius: R = 9.4 ± 1.2 km (90% confidence level) consistent with their previous results (Guillot et al. 2013). Heinke et al. (2014), using new and archival Xray data, reperformed an analysis of the spectrum of the one of these QXTs, NGC 6397 and argue that a helium atmosphere is favoured over a hydrogen atmosphere such as the one used in Guillot et al. (2013), which favours a larger radius.
For all papers in Table 2, if not available, constraints on R_{1.4} with a 2σ error bars were derived assuming a Gaussian distribution for the radius, which makes the comparison between the results easier. The constraint from Güver & Özel (2013) is extracted from their Fig. 3.
Fig. 10 Observational constraints on the radius R_{1.4} of a 1.4 M_{⊙} NS from the most recent publications. See Table 2 for the labels and text for details. The 2σ errorbars are plotted. 

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Formally, at 2σ level, all constraints on R_{1.4} but those by Poutanen et al. (2014) and Guillot & Rutledge (2014) are consistent with one another and give the radius of ~12.8km for M = 1.4M_{⊙}. All present EOS.H have a substantially larger radius R_{1.4} ≥ 13.51 km as indicated in Table A.1 and Fig. 5 and thus violate this upper bound. They are consequently consistent only with the lower bounds on R_{1.4} obtained by Poutanen & Suleimanov (2013) and Bogdanov (2013).
However, the determination of the radius of NSs is subject to many assumptions, uncertainties, and systematic effects, (see e.g. Table 1 in Potekhin 2014). In addition, the inclusion of rotation strongly complicates the analysis of the collected Xray spectra from both QXTs and BNSs, which are likely to rotate at a frequency of few hundreds of Hz. Rotation is expected to affect the radius determination by ~10% according to Poutanen et al. (2014) and Baubock et al. (2015).
10. Summary, discussion, and conclusions
The observational constraint that M_{max}> 2.0M_{⊙} for NS with hyperon cores imposes a fine tuning of the dense matter model, which has to be simultaneously consistent with semiempirical nuclear and hypernuclear parameters. With one very recent exception (Yamamoto et al. 2014), which will be discussed separately, only specific types of nonlinear RMF models are able to satisfy this constraint. The EOS of NS matter for these models, forming the set EOS.H, have the features that are described below.
Overpressure at . Pressures P^{(H)}(n_{0}) are significantly higher than the robust upper bound obtained by Hebeler et al. (2013). Introducing a densitydependence for the coupling constants in the RMF Lagrangian does not help in this respect.
Large radii. For 1.0 <M/M_{⊙}< 1.6, one obtains 13 <R^{(H)}/ km < 15. These radii are consistent at 2σ confidence level only with two lower bounds out of the five most recent constraints, derived by analyzing and modelling the Xray emission from NSs in quiescent LMXBs, from those exhibiting photospheric expansion bursts, and from radio millisecond pulsars. Future simultaneous determinations of M and R through analysis of the Xray spectra with a ~5% precision, thanks to the forthcoming NICER (Gendreau et al. 2012), Athena+ (Motch et al. 2013) and possibly LOFT (Feroci et al. 2012) missions, could either rule out hyperon cores in NS or leave open the possibility of sizable hyperon cores.
Overpressure at and large radii are likely to be interrelated. To get M_{max}> 2M_{⊙}, an EOS.H should necessarily have unusually stiff prehyperon segment, and this results in large radii for M ≲ 1.6M_{⊙}. However, in our opinion the upper bound on P(n_{0}) by Hebeler et al. (2013) is sufficiently robust to be respected. Therefore, we propose including this constraint in the procedure of determination of the RMF Lagrangian coupling constants, in addition to the standard fitting four parameters E_{s},n_{s},E_{sym},K_{s}. In the weakest form, this constraint would read (6)where P^{(PNM)} is the pressure of pure neutron matter (PNM) calculated for a given RMF model, and is the upper bound to this pressure obtained by Hebeler et al. (2013). We prefer to use constraint (6) instead of fitting the very uncertain L_{s}, referring to weakly asymmetric nuclear matter. We are now studying the possibility of imposing both the PNM constraint (6) and getting M_{max}> 2M_{⊙} to narrow the EOS.H family of RMF models.
A very recent model of EOS.H fulfilling M_{max}> 2M_{⊙} (Yamamoto et al. 2014) deserves a separate discussion. This model does not rely on the RMF approximation, rather it is calculated using the Gmatrix theory. A crucial new element is a strong three and fourbaryon repulsion resulting from the multipomeron exchanges between baryons. The manybody theory is applied to a number of terrestrial nuclear and hypernuclear data, which are sufficient to fit three sets of parameters of the models. The set MPa yields the stiffest hyperon NScores and is the only set satisfying M_{max}> 2M_{⊙}. The MPa curves in Figs. 8, 9 and 11 of Yamamoto et al. (2014) indicate that the MPa EOS has similar basic features as those characteristic of the RMF models of our EOS.H set. Namely, R^{(MPa)} ≳ 13.5 km for 1.2 <M/M_{⊙}< 1.6 and , which is twice the value characteristic of our EOS.N_{ref} set. This indicates that the prehyperon segment of the MPa EOS is very stiff, in particular, with a large overpressure at . In conclusion, these features of the MPa EOS coincide with those of our RMFEOS.H set, and should be subject to the same tests.
As far as observations are concerned, owing to the current large uncertainties on the radius determination that exist because of assumptions in the models and systematic effects, no stringent conclusion on the radius of a 1.4M_{⊙} NS can be derived.
In our opinion, a robust observational upper bound on R will become available only with advent of highprecision Xray astronomy, like that promised by the NICER, Athena+, and LOFT projects. A simultaneous measurement of M and R within a few percent error is expected to be achieved, and then used in combination with a maximum measured pulsar mass (at present 2.01 ± 0.04M_{⊙}) as a robust criterion in our quest to unveil the structure of neutron star cores.
The nonuniqueness of the crustcore matching in the EOS introduces some indeterminacy, which only disappears for unified EOS where the crust and core EOS are based on the same nuclear manybody model, starting from the same nucleon interaction (see e.g. Douchin & Haensel 2001; Grill et al. 2014).
To a good approximation , consistently with Lattimer & Prakash (2001) (Blaschke, priv. comm.).
At the NS birth, the temperature is high enough to allow for differential rotation, but as soon as the star cools, physical processes such as convection, turbulent motions, and shear viscosity promptly enforce uniform rotation. At sufficiently low temperatures, the NS interior becomes superfluid. Total stellar angular momentum is then represented by a sum of quantized vortices, but on scales much larger than the vortex separation (which is typically much smaller than 1 cm for known pulsars’ rotation rates) fluid motion averages out and may be considered uniform (see e.g. Sonin 1987).
Acknowledgments
We thank V. Dexheimer, M. Oertel, A. Sulaksono, H. Uechi, and Y. Yamamoto for providing us with the EOS tables. We are grateful to H. Uechi and Y. Yamamoto for helpful comments concerning their EOS. Correspondence with N. Chamel and A. Fantina about the BSk EOS was very helpful. We are grateful to J.M. Lattimer for his comments after a talk by one of the authors (PH) during the EMMI meeting a FIAS (Frankfurt, Germany, November 2013). We thank M. Oertel and M. Hempel for useful discussions. We are also grateful to D. Blaschke for his helpful remarks on the meaning of Fig. 5. One of the authors (MF) was supported by the FrenchPolish LIA HECOLS and by the Polish NCN HARMONIA grant DEC2013/08/M/ST9/00664. This work was partially supported by the Polish NCN grant No. 2011/01/B/ST9/04838.
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Appendix A: Collected numerical results for EOS.N and EOS.H
Parameters of the EOS and of NS models based on them.
All Tables
Equations of state. For EOS.N_{ref} (upper part) we selected three widely used EOS, which produce standard values of NS parameters.
All Figures
Fig. 1 Mass density ρ = ℰ /c^{2} vs. baryon density n_{b} for NS matter for the set of EOS presented in Figs. 2−4. Dotted segments correspond to the central densities of NS models, which are unstable with respect to radial oscillations. Relation ρ = ρ(n_{b}) deviates from linearity for n_{b}> 0.3fm^{3}. Nonlinearity grows with increasing n_{b} and is EOSdependent. For n_{b} ≲ 0.2fm^{3}, the linear approximation ρ ≈ n_{b}m_{n} (where m_{n} is neutron mass) is valid. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 2 Gravitational mass M vs. central baryon density n_{c} for nonrotating NS models based on the sets EOS.H (blue lines – H) and EOS.N_{ref} (black lines – N_{ref}). In N_{ref}: A is APR, B is BSk20, C is DH; in H: a is SA.BSR2, b indicates BM165, c indicates GM1Z0, d is UU1, e is G.TM1C. EOS labels from Table 1. Solid lines: stable NS configurations. Dotted lines: configurations unstable with respect to small radial perturbations. Vertical lines crossing the M(n_{c}) curves indicate configurations with n_{c}/n_{0} = 2,3,.... Hatched strip correspond to M = 1.4 ± 0.05M_{⊙}, and the observational constraints for J16142230 and J0348+0432 are marked in blue and magenta, respectively (1σ errors). (Colour online.) 

Open with DEXTER  
In the text 
Fig. 3 Same as in Fig.2, except with n_{c} replaced by ρ_{c}. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 4 Gravitational mass M vs. circumferential radius R for nonrotating NS models. For the labels, see details in the caption of Fig. 2. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 5 Pressure at nuclear density vs. NS radius for 1.4M_{⊙} for EOS.H and EOS.N_{ref} (see Table 1 for the labels). Grey horizontal strip – range of P(n_{0}) determined by Hebeler et al. (2013). Circles correspond to EOS with the TM1, squares with NL3, pentagons with GM1 model for the nucleon sector, respectively. Triangles indicate EOS built with RMF models with densitydependent couplings. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 6 NS radius for 2.0M_{⊙} vs. radius at maximum allowable mass. EOS labels from Table 1. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 7 Dependence of M on the central pressure, for nonrotating NS models, for M< 1.6M_{⊙}. Dashed and dasheddotted lines describe the fits for N and H families, respectively. For details, see the text. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 8 Solid lines: massradius relations for an EOS.N(DH) and an EOS.H (DS08), respectively. Longdash line: causallimit upper bound on R_{M} for these EOS, assuming n_{⋆} = n_{0}. Dotted line: similar upper bound on , except with n_{⋆} = 1.8n_{0}, as in Hebeler et al. (2013). (Colour online.) 

Open with DEXTER  
In the text 
Fig. 9 M(R_{eq}) for two selected EOS: N (DH) and H (DS08). The R_{eq} is defined in the text. Solid curves: nonrotating configurations. Dashed lines: configurations rotating uniformly at f = 300 and 600 Hz. (Colour online.) 

Open with DEXTER  
In the text 
Fig. 10 Observational constraints on the radius R_{1.4} of a 1.4 M_{⊙} NS from the most recent publications. See Table 2 for the labels and text for details. The 2σ errorbars are plotted. 

Open with DEXTER  
In the text 
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