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The Neutron Star Mass-Radius Relation and the Equation of State of Dense Matter

The equation of state (EOS) of dense matter has been a long-sought goal of nuclear physics. Equations of state generate unique mass versus radius (M-R) relations for neutron stars, the ultra-dense remnants of stellar evolution. In this work, we determine the neutron star mass-radius relation and, based on recent observations of both transiently accreting and bursting sources, we show that the radius of a 1.4 solar mass neutron star lies between 10.4 and 12.9 km, independent of assumptions about the composition of the core. We show, for the first time, that these constraints remain valid upon removal from our sample of the most extreme transient sources or of the entire set of bursting sources; our constraints also apply even if deconfined quark matter exists in the neutron star core. Our results significantly constrain the dense matter EOS and are, furthermore, consistent with constraints from both heavy-ion collisions and theoretical studies of neutron matter. We predict a relatively weak dependence of the symmetry energy on the density and a value for the neutron skin thickness of lead which is less than 0.20 fm, results that are testable in forthcoming experiments.

The masses of several neutron stars have been precisely measured using pulsar timing [1]; simultaneous mass and radius measurements, however, are considerably less certain. The leading candidates for such measurements are bursting neutron stars that show photospheric radius expansion (PRE) [2] (for a review, see Lewin et al. 1993 [3]) and transiently accreting neutron stars in quiescence [4].
Observations have already begun to determine the universal M -R relation for neutron stars and to place strong constraints on the EOS of dense matter. Previously derived constraints [5][6][7][8] have several limitations, including the use of fixed parametrizations for the EOS. Those works did not show their results to be independent of their parametrizations, including the possibility of deconfined quark matter. Neither did they address the full set of sources analyzed here.Özel et al. [6] considered only PRE sources, but these may be subject to considerable systematic errors [7,9,10]. Steiner et al. [7] considered both types of sources but used a smaller data set. Here we use eight neutron stars: four which produced PRE X-ray bursts (4U 1608-522 [11], KS 1731-260 [12], EXO 1745-248 [13], and 4U 1820-30 [14]) and four in quiescent low-mass X-ray binaries (qLMXBs) in the globular clusters M13 [15], ω Cen [15], 47 Tuc [16] and NGC 6397 [17]. Refs. [8,18] took advantage of modern predictions for pure neutron matter near the saturation density to constrain the M -R relation, but ignored systematic uncertainties associated with the observations. In this paper, we demonstrate that the inferred M -R constraints are insensitive to the removal of either all the PRE burst sources or the most extreme qLMXB sources.
At the lowest energy densities (≤ 15 MeV/fm 3 or < ∼ 3 × 10 13 g/cm 3 ) the pressure-density relation is well-understood [19]. Between 15 and 200-300 MeV/fm 3 , the EOS is well-described by four parameters, the incompressibility, the skewness, the magnitude of the symmetry energy (S v ), and the parameter describing the density derivative of the symmetry energy (L), all evaluated at the nuclear saturation density (approximately 150 MeV fm −3 ). These parameters are constrained to varying degrees by experimental data [20], including nuclear masses [21], neutron skin thicknesses [22], giant dipole resonances and dipole polarizabilities [23][24][25], and heavy-ion collisions [26]. High-density matter is constrained by (i) causality (the speed of sound must not exceed the speed of light), (ii) hydrodynamical stability, and (iii) having a sufficient maximum mass (it must be greater than the largest well-determined neutron star mass, 1.97 ± 0.04 M ⊙ for PSR J1614-2230 [27]). In addition, we impose the constraint that implied neutron star masses cannot be less than what is achievable in supernova, about 0.8 M ⊙ .
The low-density part of all the EOS parametrizations (except for strange quark stars) are described as in Steiner et al. [7]. All of the parameters in the lowdensity EOS, as well as the parameters in the highdensity EOS models, are described with uniform prior distributions. Our fiducial EOS model (A) parametrizes the high-density EOS as a set of two piecewise continuous power laws defining pressure P = ε 1+1/n as a function of energy density ε. Model A has four high-density parameters: the transition energy density between the lowdensity EOS and the first polytrope, the transition energy density between the first and second polytrope, and the two polytropic indices, n 1 and n 2 . We also employ an EOS model (B) which is similar to model A, except that the exponents in the two polytropes are parametrized with uniform priors in Γ i instead of n i . A third EOS model (C), parametrizes the EOS at high densities with a uniform prior in the pressure at four fixed energy densities, 400, 600, 1000, and 1400 MeV/fm 3 . The low-density EOS is used up to energy densities of 200 MeV/fm 3 and the EOS is assumed to be linear between that point and 400 MeV/fm 3 . The linear relation between energy densities of 1000 and 1400 MeV/fm 3 is extrapolated to higher energy densities when necessary. All parameters are chosen with large enough ranges to ensure the results do not change significantly when the range is increased. A fourth EOS model (D) assumes that matter is a polytrope at intermediate densities and quark matter at high densities. The pressure of quark matter is described by [28] where µ is the quark chemical potential. The quantity B is the bag constant and simulates confinement. The parameter a 4 describes corrections to the leading coefficient from a non-interacting Fermi gas (for which a 4 = 1). Corrections from perturbative quantum chromodynamics at high density suggest 0.6 < a 4 < 1. The parameter a 2 approximately describes corrections from the finite strange quark mass m s and the quark superfluid gap ∆ and is given by a 2 = m 2 s − 4∆ 2 (see also Ref. [8]). In order to include the effects of a possible mixed phase in a modelindependent way, a polytrope is added in between the low-density EOS and quark matter. In this model, there are five parameters in total: the transition energy density between low densities and the polytrope, the polytropic index, the transition energy density between the polytrope and quark matter (used to fix the value of B), a 2 , and a 4 . To describe bare strange quark stars, Model E applies Equation 1 at all densities, with neither a lowdensity EOS nor an intermediate polytrope. Model E thus has three parameters, a 2 , a 4 , and B.
Our baseline data set includes all eight astrophysical sources, interprets the PRE burst sources with an extended photosphere, and assumes the same distribution of color correction factors and distances as in Steiner, et al. [7]. We consider several modifications to the baseline scenario. Suleimanov et al. [10] have suggested that the X-ray spectra for PRE sources are affected by accretion and the eclipse of the neutron star by the disk. This affects the normalization at late times and we take this into account by increasing the color correction factor f C , taking 1.45 < f C < 1.8 (modification I). Alternatively, Boutloukos et al. [9] have suggested that the color correction factors are, instead, smaller as a result of magnetic confinement of the X-ray burst; this is considered in modification II in which we assume 1 < f C < 1.35. Some previous works assumed the photosphere of PRE bursts is coincident with the neutron star surface [6], and  we also consider this scenario (III). Finally, we test the sensitivity of our results to the removal of any one source or class of sources by removing X7 (IV) or M13 (V), and by removing all PRE sources (VI). To summarize, the standard models we examine include the baseline case (A), three variations of the EOS (B-D), and six other modifications of the baseline model varying the included data or its interpretation (A I-A VI). In addition, we also examine several more speculative scenarios which are described below. We use the Bayesian method of Ref. [7], using marginal estimation to determine the posterior probability densities of quantities of interest. The marginal estimation integrals are performed using Markov Chain Monte Carlo.
In our baseline analysis, we find strong constraints on the M -R curve and on the dense matter EOS: the radius of a 1.4 M ⊙ neutron star is between 11.2 and 12.3 km (95% confidence). The permissible radius range encompassing all variations of the EOS and interpretations of the astrophysical data, but not including the more speculative scenarios, is 10.4-12.9 km (95% confidence), only moderately larger than the baseline result. The 68% and 95% confidence ranges are displayed in the upper and middle portions of Table I. We determine the M -R relation for a range of neutron star masses (see Figure 1). We also determine the 68% and 95% confidence intervals of the EOS of dense matter (Figure 2). The estimated uncertainty of the pressure is approximately 30-50% at all  Table I) but not the more speculative scenarios. The lines give the 95% confidence regions for the eight neutron stars in our data set. densities achievable in neutron star interiors. In addition, the posterior distributions of the central energy density of the maximum mass star imply that the highest central density is ∼ 1200 MeV/fm 3 [29].
Producing significantly different neutron star radii requires extreme assumptions regarding the EOS and the data. We now consider more speculative scenarios, which are presented in the bottom portion of Table I (see also Figure 3). To achieve significantly smaller radii, we must assume both that the color correction factor is anomalously small (≤ 1.3) for all of the PRE sources and that the EOS has strong phase transitions (model C). In this case, we get radii as small as 9 km. Increasing the maximum mass constraint, as would be the case if the estimated most-likely mass of the pulsar B1957+20 is 2.4 M ⊙ [32], slightly increases radii (modification VII). We obtain even larger radii if we add the long PRE burst source 4U 1724-307 [10] and further assume, as suggested in Suleimanov et al. [10] (modification VIII), that the short PRE bursts and the qLMXBs M13 and ω Cen not be considered because of modifications to their spectra due to accretion. This scenario cannot yet explain, however, why short PRE burst cooling tails are observed to have constant normalizations.
While we are able to significantly constrain the P -ε relation, determination of the composition of neutron star cores is not yet possible. To this end, we consider EOS The predicted pressure as a function of baryon density of neutron-star matter as obtained from astrophysical observations. The region labeled "NS 68%" gives the 68% confidence limits and the region labeled "NS 95 %" gives the 95% confidence limits. Results for neutron-star matter from effective field theory [18] (see inset), from quantum Monte Carlo [30], and from constraints inferred from heavy-ion collisions [31] are also shown for comparison. model E, which describes the entire star by the highdensity quark matter EOS used in model D, i.e a selfbound strange quark star. In the mass range 1.4-2 solar masses, the radii are not significantly different from our baseline model so that there is no strong preference for either strange quark or hadronic stars; however, model E predicts radii significantly less than 10 km for low masses (≤ 1.2 M ⊙ ).
Our neglect of rotation is unlikely to affect our conclusions. Rotation increases the radius at the equator and decreases the radius at the poles, and this could be relevant for the interpretation of some PRE X-ray bursts: the rotation rate of 4U 1608-522 is 619 Hz, more than half of the rate for which the equatorial radius is increased by about 50%. However, this is likely to produce a systematic uncertainty smaller than that due to variations in f C , which we have already taken into account. The rotation rates for the qLMXBs in our sample are unknown. Assuming they are similar to other qLMXBs, however, means that the effect of rotation is smaller than that of their distance uncertainties.
The relationship between pressure and energy density ( Figure 2) that we determine from our baseline analysis from observations is consistent with effective field theory [18] and quantum Monte Carlo [8,30] calculations of low-density neutron matter. Note that these neutron matter results are incompatible with Suleimanov's interpretation of 4U 1724-307 [10] and suggested exclusion of short PRE bursts and qLMXLBs M13 and ω Cen. Our results are also consistent with the high-density constraints from heavy-ion collisions [31]. Our results imply that over one third of the modern Skyrme models studied in Stone et al. [33] are inconsistent with observations. Covariant field-theoretical models that have symmetry energies which increase nearly linearly with density, such as the model NL3 [34], are also inconsistent with our results, although they may still adequately describe isospin-symmetric matter in nuclei.
Our models do not place effective constraints on the symmetry parameter S v , but do place significant constraints on the symmetry energy parameter L; these are summarized in Figure 4. The probability distribution for each model is renormalized to fix the maximum probability at unity and are then shifted upwards by an arbitrary amount. The range which encloses all of the models and modifications to the data is 43.3 to 66.5 MeV to 68% confidence and 41.1 to 83.4 MeV to 95% confidence. The allowed values of L are substantially larger for Model C, which allows strong phase transitions at high densi- ties, because the parametrization decouples the low-and high-density behaviors.
Our preferred range for L is similar to that obtained from other astrophysical studies [8,20,35] and experimental studies, e.g., Refs. [24,35]. Our results suggest that the neutron skin thickness of 208 Pb [36,37] is less than about 0.20 fm. This is compatible with experiment [38] and also with measurements of the dipole polarizability of 208 Pb [39].
While we have endeavored to take into account some systematic uncertainties in our analysis, we cannot rule out corrections due to the small number of sources and to possible drastic modifications of the current understanding of low-mass X-ray binaries. Nevertheless, it is encouraging that these astrophysical considerations agree not only with nuclear physics experiments but also with theoretical studies of neutron matter at low densities and heavy-ion experiments at higher densities.
We thank G. Bertsch, J. Linnemann and S. Reddy for useful discussions. This work is supported by Chandra grant TM1-12003X (A.W.S.), by the Joint Institute for Nuclear Astrophysics at MSU under NSF PHY grant 08-22648 (A.W.S. and E.F.B.), by NASA ATFP grant NNX08AG76G (A.W.S. and E.F.B.), and by DOE grants DE-FG02-00ER41132 (A.W.S.) and DE-AC02-87ER40317 (J.M.L.). J.M.L. and E.F.B thank the Institute for Nuclear Theory at the University of Washington for partial support during this work. E.F.B. is a member of an International Team in Space Science on type I X-ray bursts sponsored by the International Space Sci-ence Institute (ISSI) in Bern, and thanks the ISSI for hospitality during part of this work.