Equation of state, neutron stars and exotic phases

Nuclear Physics A 777 (2006) 479–496

Equation of state, neutron stars and exotic phases

James M. Lattimer ∗,1, Madappa Prakash 1

Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA

Received 22 November 2004; accepted 3 January 2005

Available online 16 February 2005

Abstract

The role of the equation of state in the structure of neutron stars is summarized. At densitiesbeyond a few times the normal nuclear saturation density, the neutron-proton-electron sea can besupplemented with more exotic matter, usually involving strangeness, such as hyperons, kaon orpion condensates, and deconfined quark matter. In the case of a boson condensate or deconfinedquark matter, the exotic matter may form a mixed-phase region with the hadronic matter. Globalaspects of neutron stars, such as radii, moments of inertia and the maximum mass, can be relatedto specific properties of matter at selected density ranges. Recent observations of isolated neutronstars and pulsars in binaries are beginning to yield significant constraints on structural properties ofneutron stars and hence on the properties and composition of neutron-rich nuclear matter.© 2005 Published by Elsevier B.V.

Keywords: Neutron stars; Equation of state

1. Introduction

Recent observations have highlighted the important constraints that neutron stars canprovide for the dense matter equation of state (EOS). Precision measurements of mass androtation rates are now available, and can be coupled with estimates of radii, moments ofinertia, temperatures and ages of neutron stars to shed light on their internal properties.

* Corresponding author.E-mail address: [email protected] (J.M. Lattimer).

1 Research supported by the US DOE under grant No. DE-AC02-87ER40317.

0375-9474/$ – see front matter © 2005 Published by Elsevier B.V.doi:10.1016/j.nuclphysa.2005.01.014

480 J.M. Lattimer, M. Prakash / Nuclear Physics A 777 (2006) 479–496

Neutron stars encompass both “normal” stars, which have hadronic matter exteriorsin which the surface pressure and baryon density both vanish, and self-bound “strangequark matter” (SQM) stars [1]. The name SQM star originates from the conjecture thatdeconfined quark matter with up, down and strange quarks (the charm, bottom and topquarks are too massive to appear inside neutron stars) might have a greater binding energyper baryon at zero pressure than iron nuclei. If true, such matter is the ultimate ground stateof matter. Normal matter is then metastable, and compressed to sufficiently high density,would spontaneously convert to deconfined quark matter. SQM stars are self-bound, notrequiring gravity to hold them together, unlike normal stars. An SQM star could haveeither a bare quark matter surface with vanishing pressure but a large, supra-nuclear baryondensity, or a thin layer of normal matter supported by Coulomb forces above the quarksurface. To date, however, no conclusive observational or experimental evidence suggeststhat the SQM conjecture is true. It is more likely that if deconfined strange quark matterexists in nature at all, it is less stable at zero pressure than nuclei, and could exist at thecore of a normal neutron star. Most of this article will focus on the properties of normalstars.

Below the nuclear saturation density ns � 0.16 fm−3, and at low temperatures and en-tropies (s < 3), most nucleons are bound into heavy nuclei. Matter above ns can differ fromthe matter found in nuclei, however, because higher Fermi energies permit additional par-ticles and phases to exist. Muons already appear near ns in the neutron-rich environmentof neutron stars. It is possible that pions, kaons, and hyperons also appear. Finally, diquarkor strange quark matter could form as well, whether or not the SQM conjecture is true.Therefore, while the EOS near ns , even for asymmetric matter, is constrained by nuclearproperties, little effective laboratory constraint can be placed on high-density matter. It isprecisely this region that observations of neutron stars will have the most significance. Theappearance of additional degrees of freedom, as exhibited by pions, hyperons or quarks,generally softens matter (i.e., reduces the pressure for a given baryon density). This hasramifications for the maximum masses of neutron stars, although its effect on radii andmoments of interia of intermediate mass neutron stars is less significant [2].

2. The dense matter equation of state

The gross properties of a neutron star (such as its mass and radius) and its interior com-position (which influences the thermal evolution) chiefly depend on the nature of stronginteractions in dense matter. Investigations of dense matter can be grouped into three broadcategories: nonrelativistic potential models, effective field theoretical (EFT) models, andrelativistic Dirac–Brueckner–Hartree–Fock (DBHF) models. In addition to nucleons, thepresence of softening components such as hyperons, Bose condensates or quark matter, canbe incorporated in each of these approaches. Some general attributes, including referencesand typical compositions, of equations of state (EOS’s) in each of these approaches havebeen summarized in [2,3].

In the category of nonrelativistic potential models in which only nucleonic degreesof freedom are considered, the EOS of Akmal and Pandharipande [4] represents themost complete study to date, in which many-body and special relativistic corrections

J.M. Lattimer, M. Prakash / Nuclear Physics A 777 (2006) 479–496 481

are progressively incorporated into prior models. Effective field-theoretical approaches,beginning with the work of Walecka, are described in [5]. Calculations employing theDirac–Brueckner–Hartree–Fock approach can be found in [6].

Although neutrons dominate the nucleonic component of neutron stars, some protons(and enough electrons and muons to neutralize the matter) exist. At supra-nuclear densi-ties, perhaps, exotica such as strangeness-bearing baryons [7], condensed mesons (pion orkaon) [8,9], or even deconfined quarks [10] may appear. Fermions, whether in the formof baryons or deconfined quarks, are expected to additionally exhibit superfluidity and/orsuperconductivity.

It is possible that a transition to a mixed phase of hadronic and deconfined quark matterdevelops [11], even if SQM is not the ultimate ground state of matter. Recent years haveseen intense activity in delineating the phase structure of dense cold quark matter [12].Novel states of matter uncovered so far include color-superconducting phases with [13]and without [14] condensed mesons. Examples in the former case include a two-flavorsuperconducting (2SC) phase, a color-flavor-locked (CFL) phase, a crystalline phase [15],and a gapless superconducting phase [16]. The densities at which these phases occur areuncertain. In quark phases with finite gaps, initial estimates indicate gaps of several tensof MeV or more in contrast to gaps of a few MeV in baryonic phases. The pervasive role ofsuperfluidity and superconductivity on observable neutron star properties such as M andR [17], as well as neutrino [18] and photon luminosities [19], have also been explored.

The pressure–density relations for selected EOS’s are shown in Fig. 1. In all cases,except for PS, the pressure is evaluated assuming zero temperature and beta equilibriumwithout trapped neutrinos. PS only contains neutrons among the baryons, there being nocharged components. There are two general classes of EOS’s. First, normal EOS’s havea pressure which vanishes as the density tends to zero. Second, self-bound EOS’s have apressure which vanishes at a significant finite density.

The significant feature to note in Fig. 1 is a fairly wide range of pressures for beta-stablematter predicted even at the nuclear saturation density. This range covers about a factorof five in pressure for the EOS’s plotted, but this survey is by no means exhaustive. Thepressure in the vicinity of ns is mostly determined by the symmetry energy properties of theEOS, and it is significant that relativistic field-theoretical models generally have potentialcontributions to the symmetry energies that increase proportionately to the density whilepotential models have much less steeply rising symmetry energies.

Some normal EOS’s have considerable softening at high densities. PAL6 has an ab-normally small value of incompressibility (K = 120 MeV). GS1 and GS2 have phasetransitions to matter containing a kaon condensate, GM3 contains a large population ofhyperons appearing at high density, PS has a phase transition to neutral pion condensateand a neutron solid, and PCL2 has a phase transition to a mixed phase containing strangequark matter.

3. The relation between structure and the equation of state

Fig. 2 displays the mass-radius relation for cold, catalyzed matter using these EOS’s.Some well-established limits in the M–R plane are shown: the GR constraint R >


Fig. 1. Pressure vs. density for equations of state discussed in [2].

2GM/c2, the condition that pressure remain finite R > 9GM/4c2, the causality constraint[3,11] R � 2.94GM/c2, a limit proposed on the basis of glitches in the Vela pulsar [20], theredshift measurement z = 0.35 [21], and a restriction based upon the rotation of the fastestpulsar [22]. Contours of R∞ are also indicated in Fig. 2. EOS’s are restricted to those withmaximum masses greater than 1.4414 M�, the limit obtained from PSR 1913+16. Rhoadesand Ruffini [23] showed that causality, coupled with laboratory knowledge of the EOS upto a fiducial density ρf , limits the maximum mass to Mmax � 4.1

√ρs/ρf M�. From a the-

oretical perspective, it appears that values of R∞ in the range of 12 to 22 km are possiblefor normal neutron stars whose masses are greater than 1 M�.

Corresponding to the two general types of EOS’s, there are two general classes of neu-tron stars. Normal neutron stars are configurations with zero density at the stellar surfaceand which have minimum masses, of about 0.1 M�, that are primarily determined by theEOS below ns . At the minimum mass, the radii are generally in excess of 100 km. Thesecond class of stars are the so-called self-bound stars, which have finite density, but zero


Fig. 2. Mass-radius diagram for equations of state described in [2]. Regions excluded by GR, finite pressure,causality, and rotation are shown. Contours of R∞ , the �I/I = 0.014 glitch constraint, and redshift z = 0.35 aredisplayed.

pressure, at their surfaces. They are represented in Fig. 2 by strange quark matter stars(SQM1–3).

For reasons given below, it appears that the maximum radius for a neutron star withmass greater than about a solar mass is Rmax � 15 km. Coupling this with the causalityconstraint implies a limit to the neutron star maximum mass that is independent of thefiducial density:

Mmax � c2Rmax

2.94G� 3.45

(Rmax

15 km

)M�. (1)

Self-bound stars have no minimum mass, unlike normal neutron stars for which pure neu-tron matter is unbound. Unlike normal neutron stars, the maximum mass self-bound starshave nearly the largest radii possible for a given EOS. The best-known example of self-bound stars results from Witten’s conjecture [1] (see also Refs. [24,25]) that strange quarkmatter is the ultimate ground state of matter. The self-bound EOSs are represented bystrange-quark matter models SQM1–3, using perturbative QCD and a MIT-type bag model(parameter values given in [2]). The energy ceiling, mb = 939 MeV, for zero pressurematter implies B � 94.3 MeV fm−3 (SQM1), while a lower limit to B (SQM3) is set byrequiring the quark-hadron phase transition to occur above nuclear density. If the strangequark mass ms = 0 and interactions are neglected (αc = 0), the maximum mass is related


to the bag constant B by Mmax = 2.033 (56 MeV fm−3/B)1/2 M�. The addition of a fi-nite strange quark mass and/or interactions produces larger maximum masses [3,25]. Theconstraint that Mmax > 1.44 M� is thus automatically satisfied. The locus of maximummasses of self-bound stars is given by [3]

R � 2.94 × 31/4GM/c2 = 3.87GM/c2. (2)

Strange quark stars with electrostatically supported normal-matter crusts [26] have largerradii than those with bare surfaces. Thus, the non-interacting case shown in Fig. 2 withms = 0 and B = 94.3 MeV fm−3, SQM1, gives the most compact self-bound configurationpossible. Coupled with the constraint M > 1 M� from protoneutron star models [27],strange quark stars cannot have radii less than about 8.5 km or R∞ < 10.5 km. Thesevalues are comparable to the theoretical lower limits for a Bose (pion or kaon) condensateEOS, represented by models similar to GS1.

For normal stars, the pressure in the region n < 0.1 fm−3 is not important as it doesnot significantly affect the mass–radius relation. The value of the density at the core-crusttransition, and the pressure there, are, however, important ingredients for the determinationof the size of the superfluid crust of a neutron star that is believed to be involved in thephenomenon of pulsar glitches [20].

Although the M–R trajectories for normal stars can be strikingly different, in the massrange from 1 to 1.5 M� or more it is usually the case that the radius has relatively littledependence upon the stellar mass. The major exceptions illustrated are the model GS1, inwhich a mixed phase containing a kaon condensate appears at a relatively low density andthe model PAL6 which has an extremely small nuclear incompressibility (120 MeV). Bothof these have considerable softening and a large increase in central density for M > 1 M�.Pronounced softening, while not as dramatic, also occurs in models GS2 and PCL2, whichcontain mixed phases containing a kaon condensate and strange quark matter, respectively.All other normal EOS’s in this figure contain only baryons among the hadrons.

While it is often assumed that a stiff EOS implies both a large maximum mass and alarge radius, some counter examples exist. GM1 and MS3 have relatively small maximummasses but have large radii compared to most other EOS’s with larger maximum masses.Also, not all EOS’s with extreme softening have small radii for M > 1 M� (viz., GS2).Nonetheless, for stars with masses greater than 1 M�, only models with a large degree ofsoftening (including strange quark matter configurations) can have R∞ < 12 km.

4. The radius–pressure correlation

The relative insensitivity of the radius to the mass for normal neutron stars can be ex-plained by considering Newtonian polytropes, for which

R ∝ Kn/(3−n)M(1−n)/(3−n), P = Kρ1+1/n. (3)

In particular, the case n = 1 suggests that R ∝ M0, i.e., the radius is independent of themass, and R ∝ √

K , which suggests that R depends upon P at an average stellar density,not the central density. This conjecture has been verified by Ref. [2], who demonstrated the


existence of a correlation between the radius and the pressure evaluated in the vicinity ofnormal nuclear density:

R(M,n) � C(M,n)[P(n)

]0.25. (4)

Here, P(n) is the total pressure inclusive of leptonic contributions evaluated at the baryondensity n, and M is the stellar gravitational mass. The constant C(M,n), in units ofkm fm3/4 MeV−1/4, for the densities n = ns , 1.5ns and 2ns , respectively, is 9.53 ± 0.07,7.16±0.03 and 5.82±0.04 for the 1 M� case, and 9.11±0.21, 6.84±0.15 and 5.57±0.11for the 1.4 M� case. The correlation appears tighter at the higher baryon density cases. Theexponent 1/4, however, is not 1/2 as the n = 1 Newtonian polytrope predicts. The smallerpower can be understood by appealing to analytic solutions of Einstein’s equations, ofwhich only three are known that apply to normal neutron stars.

The only analytic solution that explicitly relates the radius, mass and pressure is thatdue to Buchdahl [28] for the EOS ρ = 12

√p∗P − 5P , where ρ is the energy density and

p∗ is a constant. In terms of the parameters p∗ and β ≡ GM/Rc2, the baryon density andstellar radius are found to be

n = 12√

Pp∗

(1 − 1

3

√P

p∗

)3/2

, R = (1 − β)c2√

π

288p∗G(1 − 2β). (5)

The exponent in Eq. (4) can thus be found:

d lnR

d lnP

∣∣∣∣n,M

= 1

2

(1 − 5

6

√P

p∗

)(1 + 1

6

√P

p∗

)−1(1 − β)(1 − 2β)

(1 − 3β + 3β2). (6)

In the limit β → 0, one has P → 0 and d lnR/d lnP |n,M → 1/2, the value characteristicof an n = 1 Newtonian polytrope. Finite values of β and P reduce the exponent. If M andR are about 1.4 M� and 15 km, respectively, for example, β � 0.14 and Eq. (5) gives p∗ =π/(288R2) ≈ 4.85 × 10−5 km−2 (in geometrized units). At a fiducial density n = 1.5ns ,this is equivalent in geometrized units to n = 2.02 × 10−4 km−2, or n/p∗ � 4.2. Eq. (5)then implies P/p∗ � 0.2 and Eq. (6) yields d lnR/d lnP � 0.28. This result, while mildlysensitive to the choices for n and R, provides a reasonable explanation for Eq. (4). The factthat the exponent is smaller than 1/2 is a GR effect.

This correlation is significant because the pressure of degenerate neutron-star matternear the nuclear saturation density ns is, in large part, determined by the symmetry proper-ties of the EOS. Studies of pure neutron matter strongly suggest that the specific energy ofnuclear matter near the saturation density may be expressed as expansions in both densityand asymmetry:

E(n,x) = −16 + K

18

(1 − n

ns

)2

+ K ′

27

(1 − n

ns

)3

+ Esym(n)(1 − 2x)2 · · · . (7)

Here, K and K ′ are the incompressibility and skewness parameters, respectively, and Esymis the symmetry energy function, approximately the energy difference at a given density be-tween symmetric and pure neutron matter. The symmetry energy parameter Sv ≡ Esym(ns).Leptonic contributions Ee = (3/4)hcx(3π2nx4)1/3 must be added. Matter in neutron stars


is in beta equilibrium, i.e., μe = μn − μp = −∂E/∂x, so the equilibrium proton fractionat ns is xs � (3π2ns)

−1(4Sv/hc)3 � 0.04. The pressure at ns is

P(ns, xs) = ns(1 − 2xs)[nsS

′v(1 − 2xs) + Svxs

] � n2s S

′v, (8)

due the small value of xs;S′v ≡ (∂Esym/∂n)ns . The pressure depends primarily upon S′

v .The equilibrium pressure at moderately larger densities similarly is insensitive to K andK ′. Experimental constraints to the compression modulus K , most importantly from analy-ses of giant monopole resonances [29] give K ∼= 220 MeV. The skewness parameter K ′has been estimated to lie in the range 1780–2380 MeV [30]. Evaluating the pressure forn = 1.5ns ,

P(1.5ns) = 2.25ns

[K

18− K ′

216+ ns(1 − 2x)2(∂Esym/∂n)1.5ns

], (9)

and it is noted that the contributions from K and K ′ largely cancel.

5. Maximum energy density inside neutron stars

Measurements of neutron star masses can set an upper limit to the maximum possibleenergy density in any compact object. If a neutron or strange quark matter star were incom-pressible, the causality constraint R � 2.94GM/c2 would imply that its central density ρc

would be

ρc,Inc = 3

4π

(c2

2.94G

)3 1

M2= 6.0 × 1015

(M�M

)2

g cm−3. (10)

However, the central pressure of any star must exceed that of the incompressible fluid. Phe-nomologically, Ref. [31] found that no causal EOS has a central density, for a given mass,greater than that for the Tolman VII [32] analytic solution. This solution corresponds to amass–energy density ρ dependence quadratic in r , ρ = ρc[1 − (r/R)2]. For this solution,

ρc,T VII = 2.5ρc,Inc � 1.5 × 1016(

M�M

)2

g cm−3, (11)

a measured mass of 2.2 M� would imply ρmax < 3.1 × 1015 g cm−3, equivalent to about8ns . A good rule of thumb for converting ρ to n, both expressed in units of their respectivesaturation densities, is

ρ/ρs ≈ 0.9(n/ns)[1 + 0.11(n/ns)

3/4]. (12)

Fig. 3 displays maximum masses and accompanying central densities for a wide varietyof neutron star EOS’s, including models containing significant softening due to “exotica”,such as strange quark matter.

The upper limit to the density could be lowered if the causal constraint is not approachedin practice. For example, at high densities in which quark asymptotic freedom is realized,the sound speed is limited to c/

√3. Using this as a strict limit at all densities, the Rhoades

and Ruffini [23] mass limit is reduced by approximately 1/√

3 and the compactness limit


Fig. 3. Maximum mass vs. central density for normal and SQM stars. Predictions for an incompressible fluid andTolman VII solutions are shown. The dashed lines for 1.44 and 2.2 M� serve to guide the eye.

GM/Rc2 = 1/2.94 is reduced by a factor 3−1/4 to 1/3.8 [3]. In this extreme case, themaximum density would be reduced by a factor of 3−1/4 from that of Eq. (11). A 2.2 M�measured mass would imply a maximum density only about 4.2ns .

6. Mass–radius limits from rotation

The maximum spin frequency is the mass-shedding limit (Keplerian frequency), ΩK =√GM/R3 in Newtonian gravity, where M and R refer to the spinning star. In general

relativity, frame dragging must be taken into account, but the Newtonian relation is stillapproximately correct. The maximum rotation rate for a given normal matter EOS includ-ing GR can be approximated as [3,33,34]

νEOSmax � 1225 ± 40

(Mmax

M�

)1/2(10 km

Rmax

)3/2

Hz, (13)

where Mmax and Rmax refer to the maximum mass non-rotating configuration for that EOS.This formula includes the fact that rotation stabilizes 10% to 20% more mass than in thenon-rotating case. It is amazing that the same coefficient holds for all EOS’s, except forstrange quark matter stars. Nevertheless, this formula is not useful for estimating the max-


imum spin rate of a star with a given mass. A similar expression, valid for the Keplerianfrequency of any normal star whose non-rotating mass and radius are M and R, is [22]

νM,RK � 1045 ± 30

(M

M�

)1/2(10 km

R

)3/2

Hz, (14)

This relation excludes a region in the M–R diagram for any star with a measured ν. Themost rapidly rotating star, PSR B1937+21, has ν = 641 Hz, giving rise to the excludedregion displayed in Fig. 2.

7. Observations of neutron stars

7.1. Masses

The most accurately measured neutron star masses are from timing observations of theradio binary pulsars [35,36]. As shown in Fig. 4, these include pulsars orbiting anotherneutron star, a white dwarf or a main-sequence star. Ordinarily, observations of pulsars inbinaries yield orbital sizes and periods from Doppler phenomenon, from which the totalmass of the binary can be deduced. But the compact nature of several binary pulsars permitsdetection of relativistic effects, such as Shapiro delay or orbit shrinkage due to gravitationalradiation reaction, which constrains the inclination angle and permits measurement of eachmass in the binary. A sufficiently well-observed system can have masses determined toimpressive accuracy. The textbook case is the binary pulsar PSR 1913+16, in which themasses are 1.3867 ± 0.0002 and 1.4414 ± 0.0002 M�, respectively [37].

One particularly significant development is mass determinations in binaries with whitedwarf companions, which show a broader mass range than binary pulsars having neutronstar companions. It has been suggested that a rather narrow set of evolutionary circum-stances conspire to form double neutron star binaries [38], leading to a restricted range ofneutron star masses. This restriction is relaxed for other neutron star binaries. Evidence isaccumulating that a few of the white dwarf binaries may contain neutron stars larger thanthe canonical 1.4 M� value, including the fascinating case of PSR J0751+1807 in whichthe estimated mass with 1σ error bars is 2.2 ± 0.2 M� [39]. For this neutron star, a massof 1.4 M� is about 4σ from the optimum value. In addition, the mean observed value ofthe white dwarf-neutron star binaries exceeds that of the double neutron star binaries by0.25 M�. However, the 1σ errors of all but one of these systems extends into the rangebelow 1.45 M�. Continued observations guarantee that these errors will be reduced. Rais-ing the limit for the neutron star maximum mass could eliminate entire families of EOS’s,especially those in which substantial softening begins around 2 to 3ns . This could be ex-tremely significant, since exotica (hyperons, Bose condensates, or quarks) generally reducethe maximum mass appreciably.

Masses can also be estimated for another handful of binaries which contain an accretingneutron star emitting X-rays. Some of these systems are characterized by relatively largemasses, but the estimated errors are also large. The system of Vela X-1 is noteworthybecause its lower mass limit (1.6 to 1.7 M�) is at least mildly constrained by geometry[40]. Mass estimates of selected X-ray binaries are also shown in Fig. 4.


Fig. 4. Measured and estimated masses of neutron stars in radio binary pulsars (gold, silver and blue regions) andin X-ray accreting binaries (green). Letters in parentheses refer to references cited in [34]. (For interpretation ofthe references to colour in this figure legend, the reader is referred to the web version of this article.)

7.2. Radii

Most known neutron stars are observed as pulsars and have photon emissions from radioto X-ray wavelengths dominated by non-thermal emissions believed to be generated in aneutron star’s magnetosphere. Such emissions are difficult to utilize in terms of constrain-ing the star’s global aspects, such as mass, radius and temperature. However, approximatelya dozen neutron stars with ages up to a million years old have been identified [41] with sig-nificant thermal emissions. Stars of these ages are expected to have surface temperatures


in the range of 3 × 105 K to 106 K, i.e., they are predominately X-ray sources. If their totalphoton fluxes were that of a blackbody, they would obey

F∞ = L∞/4πd2 = σT 4∞(R∞/d)2, (15)

where, d is the distance, and T∞, F∞ and L∞ refer to the temperature, flux and lu-minosity redshifted relative to their values at the neutron star surface. The redshift isz = (1 − 2GM/Rc2)−1 − 1. (For example, T∞ = T/(1 + z),F∞ = F/(1 + z)2.) As aresult, the so-called radiation radius, R∞ = R(1 + z), is a quantity that can be estimated ifF∞, T∞ and d are known. R∞ is a function of both M and R, but if redshift information isavailable, M and R could be determined. Contours of R∞ are displayed in Fig. 2. A valueof R∞ requires both that R < R∞ and M < (3

√3G)−1c2R∞ � 0.13(R∞/km)M−1� .

A serious problem in determining R∞ and T∞ is that the star’s atmosphere rearrangesthe spectral distribution of emitted radiation, i.e., they are not blackbodies [42]. Neu-tron star atmosphere models are mostly limited to non-magnetized atmospheres, althoughpulsars are thought to have intense magnetic fields � 1012 G [35]. Strongly magnetized hy-drogen is relatively simple, but magnetized heavy element atmospheres are still in a stateof infancy.

A useful constraint is provided by a few cases in which the neutron star is sufficientlyclose for detection of optical radiation. These stars are observed to have optical fluxes fac-tors of 3 to 5 times greater than a naive blackbody extrapolation from the X-ray rangewould imply. This optical excess is a natural consequence of the neutron star atmosphere,and results in an inferred R∞ substantially greater than that deduced from the X-rayblackbody. In many cases a heavy-element atmosphere appears to fit the global spec-tral distributions from X-ray to optical energies while also yielding neutron star radii ina plausible range. However, the observed absence of narrow spectral features, predictedby heavy-element atmosphere models, is puzzling [43]. The explanation could lie withbroadening or elimination of spectral features caused by intense magnetic fields or highpressures.

The estimation of radii from isolated neutron stars is also hampered by uncertaintiesof source distance. Distances to pulsars can be estimated by their dispersion measures[35], but in three cases (Geminga, RX J185635-3754 and PSR B0656+14) [44] parallaxdistances have been obtained, although errors are still large. As a consequence, values ofR∞ determined from thermally-emitting neutron stars, while in a plausible range, are notsufficiently precise at present to usefully restrict properties of dense matter.

7.3. Estimates from quiescent X-ray bursters in globular clusters

In this context, the recent discovery of thermal radiation from quiescent X-ray burstersin globular clusters is particularly interesting [45]. These systems contain rejuvenated 10billion year-old neutron stars heated by recent episodes of mass accretion from their com-panions. Since the accreted material was dominated by hydrogen, and accretion is knownto quench magnetic fields, these stars may have the simplest of all possible atmospheres:non-magnetic hydrogen. Current results are consistent with radiation radii R∞ ∼ 12 km,but accuracy is limited by systematic uncertainties in the intervening interstellar hydrogencolumn density, since this material obscures 50% or more of the X-ray flux. Interestingly,


the distances to these sources will likely become relatively well known in the near future,reducing a source of error that plagues interpretations of isolated neutron stars.

7.4. Estimates from active X-ray bursters

Neutron stars that are actively accreting material from companion stars produce X-raybursts. The resulting light curves can be modelled taking into account light-bending whichconstrains the star’s compactness. For the source XTE J1814-338, Ref. [46] found z <

0.38. For the source 4U 1820-30, Ref. [47] found 0.20 < z < 0.30 with a small dependenceupon assumed source distance. These estimates are based on a geometric technique and areargued to be insensitive to spectral modelling.

Two lines observed in X-ray burst spectra of EXO 0748-676 have been suggested to beH- and He-like Fe lines and imply z � 0.35 [21]. This inference received additional credi-bility by the measurement [48] of a 45 Hz neutron star spin frequency. This low spin rate isconsistent with the observed widths of these lines if they are due to Fe and further implies9.5 < R < 15 km (corresponding to 1.5 < M < 2.3 M�). Since this star is a member ofan eclipsing binary, an independent mass measurement might yet be possible, which couldfix R. These techniques could potentially be extended to other X-ray bursters.

7.5. Moments of inertia

Pulsars provide several sources of information concerning neutron star properties. Thefastest pulsars provide constraints on neutron star radii. Their spins and spin-down rateprovide estimates of magnetic field strengths and ages. A potentially rich source of dataare pulsar glitches, the occasional disruption of the otherwise regular pulses. While theorigin of glitches is unknown, their magnitudes and stochastic behavior suggests they areglobal phenomena [20]. The leading glitch model involves angular momentum transfer inthe crust from the superfluid to the normal component [49]. Both are spinning, but thenormal crust is decelerated by the pulsar’s magnetic dipole radiation. The superfluid isweakly coupled with the normal matter and its rotation rate is not diminished. But whenthe difference in spin rates becomes too large, something breaks and the spin rates arebrought closer in alignment. The angular momentum observed to be transferred betweenthe components, in the case of the Vela pulsar, amounts to at least 1.4% of the star’s totalangular momentum [20].

If this also corresponds to the fraction of the moment of inertia residing in the neutronstar crust, limits can be set on the star’s mass and radius. In terms of the density andpressure nt and Pt , at the base of the crust, this fraction is

�I

I� 28π

3

PtR4

GM2

1 − 1.67β − 0.6β2

1 + (2Pt/ntβ2)(1 + 5β − 14β2). (16)

The dependence on nt is weak. Pt and nt depend on the symmetry energy’s density depen-dence as well as on the incompressibility. Hence, there is a range 0.25 < Pt/(MeV fm−3) <

0.65 among published EOS’s. For given values of �I/I and M , the smallest R compatiblewith Eq. (16) is obtained employing the largest Pt value in this range. For a 1.4 M� star,this leads to the limit shown in Fig. 2. This limit has a relatively weak dependence on the


EOS (R ∝ P−1/4t ), however, it applies only to the Vela pulsar, and it depends upon the

crustal superfluid coupling hypothesis.

7.6. Prospects from relativistic binaries

It might be possible to measure the entire moment of inertia of a neutron star usingpulsar timing in relativistic binaries. Spin–orbit coupling produces two relativistic effectsthat could be measured: a small extra advance of the periastron of the orbit beyond thestandard post-Newtonian advance, and the precession of the pulsar spins about the directionof the total angular momentum of the system, an effect also known as geodetic precession[50,51]. Since the total angular momentum remains fixed, the precession of the pulsarspins produces a compensating change in the orientation of the orbital plane. Since theorbital angular momentum dominates the spin angular momenta, the geodetic precessionamplitude is very small while the associated spin precession amplitudes are substantial.The spin–orbit effects usually simplify because one star (hereafter called A) spins muchfaster than the other star B . Then, all observable spin–orbit effects are proportional to themoment of inertia of pulsar A, IA. To lowest post-Newtonian order, the spin SA and orbital L angular momenta evolve according to [50],

SA = 2π

PpA

L × SA

| L| ,˙ LSO = 2π

PpA

( SA

| L| − 3 L · SA

| L|3 L)

, (17)

where a and e are the orbital semimajor axis and the eccentricity, respectively, and theprecession period PpA and | L| are

PpA = 2aP (MA + MB)c2(1 − e2)

GMB(4MA + 3MB), | L| = 2πMAMBa2(1 − e2)1/2

P(MA + MB), (18)

where P is the binary orbital period. If θA is the angle between SA and L, the amplitudeof the change in the orbital inclination i due to A’s precession is

δi = | SA|| L| sin θA � IA(MA + MB)

a2MAMB(1 − e2)1/2

P

PA

sin θA, (19)

where PA is A’s spin period. This will cause a periodic departure from the expected time-of-arrival of pulses from pulsar A of amplitude (for e � 0)

δta = MB

MA + MB

a

cδi cos i = a

c

IA

MAa2

P

PA

sin θA cos i. (20)

The ratio of the periastron advances due to spin–orbit coupling and to first-order postNewtonian contributions is [51]

ApA

A1PN

= IA(4MA + 3MB)P

6(1 − e2)1/2Ma2MAPA

(2 cos θA + cot i sin θA sinφA), (21)

where φA is the angle between the line of sight to pulsar A and the projection of SA on theorbital plane.

Table 1 compares properties of the relativistic binaries for which the spin orientationof pulsar A can be estimated. It is noteworthy that the net timing delays due to inclination


Table 1Comparison of binary pulsars (for explanation of symbols, see text)

PSR J0707-3039 PSR B1913+16 PSR B1534+12

References [53] [37,54] [55]a/c (s) 2.93 6.38 7.62P (h) 2.45 7.75 10.1e 0.088 0.617 0.274MA (M�) 1.34 1.441 1.333MB (M�) 1.25 1.387 1.345i 90.26 ± 0.13◦ 47.2◦ 77.2◦PA (ms) 22.7 59 37.9θA 13◦ ± 10◦ 21.1◦ 25.0◦ ± 3.8◦φA 246◦ ± 5◦ 9.7◦ 290◦ ± 20◦PpA (yr) 74.9 297.2 700δta/IA,80 (µs)† 0.17 ± 0.16 7.5 5.2 ± 0.8ApA/(A1PN IA,80) 6.6+0.2

−0.6 × 10−5 1.0 × 10−5 1.1 × 10−5

†IA,80 = IA/80 M� km2.

shifts due to precession are more than an order of magnitude larger for PSR B1913+16 andPSR B1534+12 than for PSR J0737-3039, due to their smaller inclinations. In particular,PSR 0737-3039 has a nearly edge-on orbit, i � 90◦, and a small misalignment angle θA thatmake the inclination timing delays extremely small. However, these same attributes renderthe spin–orbit contribution to the periastron advance about 6 times larger than for the othertwo systems. Coupled with 0737’s shorter precession period, this produces a factor 24 inobservability for this effect, which has remained undetectable in other systems. It is hopedthat IA can be determined to about 10% after a few years observations. (We note that anglesquoted for PSR J0737-3039 [53] are tentative and are used for illustration.)

The importance of a measurement of I to within ±10% is illustrated in Fig. 5. It isclear that relatively few equations of state would survive these constraints. Those familiesof models lying close to the measured values would have their parameters limited cor-respondingly. Ref. [2] demonstrated that unless the EOS has a maximum mass of order1.7 M� or less, or unless the star is a strange quark matter star, a relatively unique relationexists between I/MR2 and M/R. For M � 1 M�, this relation is well approximated by[52]

I � (0.237 ± 0.008)MR2[

1 + 4.2M km

M�R+ 90

(M km

M�R

)4]. (22)

Simultaneous mass and moment of inertia measurements will usefully constrain the radiusthrough Eq. (22). For a 1.4 M� star, this would result in a radius estimate with about 6%to 7% uncertainty [52].

7.7. QPO’s

QPO’s are accreting neutron stars that display quasi-periodic behavior in their X-rayemissions. Their power spectra contain a number of features, the most prominent of whichare twin high frequency peaks near 1 kHz separated by about 400 Hz. One explanation for


Fig. 5. The moment of inertia scaled by M3/2. EOS labels are described in [2]. The shaded band illustrates a

±10% error on a hypothetical I/M3/2 measurement of 50 km2 M−1/2� ; the error bar is for M = 1.34 M�. The

dashed curve (Crab) is a lower limit derived by [56] for the Crab pulsar.

these peaks is a variation of the beat-frequency model [57], which holds that the higherpeak frequency is the orbital frequency of the inner edge of the accretion disc and that theseparation of the peaks is either one or two times the neutron star’s spin rate. In generalrelativity, a particle orbiting a non-rotating compact object has an innermost stable circularorbit at a radius rISCO = 6GM/c2 with an orbital frequency

νISCO = (2π)−1√

GM/r3ISCO � 2200 (M�/M) Hz. (23)

The highest peak frequency observed to date is 1329 Hz, for the system 4U 1636-53 [58],implying M � 1.7 M� and R < 14.6 km [59]. If the inner edge of the accretion disc is, infact, located near rISCO, then the mass exceeds 1.5 M�, as does the maximum mass. Forcompeting models, see [60].

8. Epilogue

This summary has focused on new astronomical measurements of neutron star masses,radii, redshifts, rotation frequencies and moments of inertia, which have the potentialof constraining several properties of the EOS, including its symmetry behavior and thecomposition of matter at high densities. We have not discussed several other kinds of obser-vations, such as those dealing with neutron star magnetic fields, cooling, neutrino emission,


gravitational waves, etc. that may also yield important information. We also omitted a sum-mary of new and proposed experiments for the skin thickness of Pb, the properties of giantresonances, and masses of neutron-rich nuclei from rare-isotope accelerators that can pro-vide complementary information [61].

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Equation of state, neutron stars and exotic phases