TThe Nuclear Physics of Neutron Stars
J. Piekarewicz Department of Physics, Florida State University, Tallahassee, FL 32306 (Dated: November 8, 2018)
Abstract
A remarkable fact about spherically-symmetric neutron stars in hydrostatic equilibrium — theso-called Schwarzschild stars — is that the only physics that they are sensitive to is the equationof state of neutron-rich matter. As such, neutron stars provide a myriad of observables that maybe used to constrain poorly known aspects of the nuclear interaction under extreme conditionsof density. After discussing many of the fascinating phases encountered in neutron stars, I willaddress how powerful theoretical, experimental, and observational constraints may be used to placestringent limits on the equation of state of neutron-rich matter.
PACS numbers: 21.65.+f,26.60.+c,21.30.Fe a r X i v : . [ nu c l - t h ] F e b . INTRODUCTION A neutron star is a gold mine for the study of the phase diagram of cold baryonic matter.While the most common perception of a neutron star is that of a uniform assembly ofneutrons packed to densities that may exceed that of normal nuclei by up to an order ofmagnitude, the reality is far different and significantly more interesting. Indeed, the merefact that hydrostatic equilibrium must be maintained throughout the neutron star, demandsa negative pressure gradient at each point in the star; otherwise the star would collapse underits own weight. This model-independent fact yields nuclear densities — at least for mostrealistic equations of states — that span over 11 orders of magnitude; from 10 to 10 g / cm .Recall that in this units nuclear-matter saturation density equals ρ = 2 . × g / cm .What novels phases of baryonic matter emerge under these conditions is both fascinatingand unknown. Moreover, most of the exotic phases predicted to exist in neutron stars cannot be realized under normal laboratory conditions. Whereas most of these phases have afleeting existence here on Earth, they become stable in neutron stars as a consequence ofthe presence of enormous gravitational fields.To establish the fundamental role played by the equation of state on the structure ofspherically-symmetric neutron stars in hydrostatic equilibrium, we start with the Tolman-Oppenheimer-Volkoff (TOV) equations — the extension of Newton’s laws to the domainof general relativity. The TOV equations may be expressed as a coupled set of first-orderdifferential equations of the following form: dPdr = − G E ( r ) M ( r ) r (cid:20) P ( r ) E ( r ) (cid:21) (cid:20) πr P ( r ) M ( r ) (cid:21) (cid:20) − GM ( r ) r (cid:21) − , (1a) dMdr = 4 πr E ( r ) , (1b)where G is Newton’s gravitational constant, while P ( r ), E ( r ), and M ( r ) represent thepressure, energy density, and enclosed-mass profiles of the star, respectively. Note that thelast three terms (enclosed in square brackets) in Eq. (1a) have a general-relativistic origin.Remarkably, the only input that neutron stars are sensitive to is the equation of state ofneutron-rich matter. Indeed, changes in pressure and enclosed mass as a function of radius(left-hand side of the equations) depend not only on the values of these quantities at r , butalso on the “unknown” energy density E ( r ) of the system. Thus, no solution of the TOVequations is possible until an equation of state ( i.e., a P vs E relation) is supplied.In this manuscript we discuss the various fascinating phases of baryonic matter that arepredicted to exist in neutron stars, but inaccessible under normal laboratory conditions.After briefly discussion the theoretical models used in this contribution, we focus on recenttheoretical, experimental, and observational constrains that place stringent limits on theequation of state of neutron-rich matter. II. ANATOMY OF A NEUTRON STAR
Neutron stars contain a non-uniform crust above a uniform liquid mantle. See Fig. 1 forwhat is believed to be an accurate rendition of the structure of a neutron star.2 . The Outer Crust
The outer crust is understood as the region of the star spanning about 7 orders of magni-tude in density; from about 10 g / cm to 4 × g / cm [1]. At these densities, the electrons— which are an essential component of the star in order to maintain charge neutrality —have been pressure ionized and move freely throughout the crust. Moreover, at these “low”densities, Fe nuclei arrange themselves in a crystalline lattice in order to minimize theiroverall Coulomb repulsion. This is the structure of the outermost layer of the crust. How-ever, as the density increases (and one moves away from the surface of the star) Fe is nolonger the most energetically favorable nucleus. This is because the electronic contributionto the energy increases faster with density than the nuclear contribution. As a result, itbecomes energetically advantageous for the energetic electrons to capture on the protonsand for the excess energy to be carried away by neutrinos. The resulting nuclear lattice isnow made of nuclei having a neutron excess larger than that of Fe. As the density con-tinues to increase, the nuclear system evolves into a Coulomb lattice of progressively moreneutron-rich nuclei until a “critical” density of about 4 × g / cm is reached. At this pointthe nuclei are unable to hold any more neutrons; the neutron drip line has been reached. FIG. 1: A rendition of the structure and phases of a neutron star (courtesy of Dany Page).
B. The Inner Crust
The inner crust of the neutron star comprises the region from neutron-drip density upto the density at which uniformity in the system is restored (approximately 1 / / outercore , these length scales are well separated and the system organizes itself into a crystallinelattice of neutron-rich nuclei. In contrast, at a much higher density of the order of halfof nuclear-matter saturation density, uniformity in the system is restored and the systembehaves as a uniform Fermi liquid. Yet the transition region from the highly-ordered crystal3o the uniform liquid mantle is complex and not well understood. Length scales that werewell separated in both the crystalline and uniform phases are now comparable, giving rise toa universal phenomenon known as “Coulomb frustration” . It has been speculated that thetransition to the uniform phase must go through a series of changes in the dimensionality andtopology of these complex structures, colloquially known as “nuclear pasta” [2, 3]. In Fig. 2 asnapshot obtained from Monte-Carlo/Molecular-Dynamics simulations of a nuclear systemat densities relevant to the inner crust are displayed [4, 5]. The figure displays how thesystem organizes itself into neutron-rich clusters ( i.e., “nuclei”) of complex topologies thatare surrounded by a vapor of (perhaps superfluid) neutrons. Such complex pasta structuresmay have a significant impact on various transport properties, such as neutrino and electronpropagation. FIG. 2: (color online) A snapshot of a Monte Carlo simulation for a configuration of 4,000 nucleonsat a baryon density of n = 0 .
025 fm − (a sixth of normal nuclear matter saturation density), a protonfraction of Y p = Z/A = 0 .
2, and a temperature of T = 1 MeV. C. The Stellar Core
As the density continues to increase, the neutron-rich nuclei will “melt” and uniformityin the system will be restored. At these densities (of the order of 1 / / what is the maximum mass of a neutron star? Or equivalently, whatis the minimum mass of a black hole?
Note that if the equation of state is “soft”, very high4ensities may be reached in the stellar core. At such high densities new states of matter maydevelop as the quarks within the hadrons become deconfined. Such an exciting possibilitywill not be considered further in this manuscript.
III. CONSTRAINTS ON THE EQUATION OF STATE
Before addressing the role that recent observables play in constraining various theoreticaldescription of the equation of state, we introduce the relativistic mean-field models that areused to compute these observables.Relativistic mean-field descriptions of the ground-state properties of medium to heavynuclei have enjoyed enormous success. These highly economical descriptions encode a greatamount of physics in a handful of model parameters that are calibrated to a few ground-stateproperties of a representative set of medium to heavy nuclei. An example of such a successfulparadigm is the relativistic NL3 parameter set of Lalazissis, Ring, and collaborators [6, 7].The Lagrangian density employed in this work is rooted on the seminal work of Walecka,Serot, and their many collaborators (see Refs. [8, 9, 10] and references therein). Sincefirst published by Walecka more than three decades ago [8], several refinements have beenimplemented to improve the quantitative standing of the model. In the present work weemploy an interacting Lagrangian density of the following form [11, 12, 13]: L int = ¯ ψ (cid:104) g s φ − (cid:16) g v V µ + g ρ τ · b µ + e τ ) A µ (cid:17) γ µ (cid:105) ψ − κ
3! ( g s φ ) − λ
4! ( g s φ ) + ζ (cid:16) g V µ V µ (cid:17) +Λ v (cid:16) g ρ b µ · b µ (cid:17)(cid:16) g V µ V µ (cid:17) . (2)The original Lagrangian density of Walecka consisted of an isodoublet nucleon field ( ψ )together with neutral scalar ( φ ) and vector ( V µ ) fields coupled to the scalar density ( ¯ ψψ )and conserved nucleon current ( ¯ ψγ µ ψ ), respectively [8]. In spite of its simplicity (indeed,the model contains only two dimensionless coupling constants), symmetric nuclear mattersaturates even when the model was solved at the mean-field level [8]. By adding addi-tional contributions from a single isovector meson ( b µ ) and the photon ( A µ ), Horowitz andSerot [14] obtained results for the ground-state properties of finite nuclei that rivaled someof the most sophisticated non-relativistic calculations of the time. However, whereas thetwo dimensionless parameters in the original Walecka model could be adjusted to repro-duce the nuclear saturation point, the incompressibility coefficient (now a prediction of themodel) was too large ( K (cid:38)
500 MeV) as compared with existing data on breathing-modeenergies [15]. To overcome this problem, Boguta and Bodmer introduced cubic ( κ ) andquartic ( λ ) scalar meson self-interactions that accounted for a significant softening of theequation of state ( K = 150 ±
50 MeV) [16]. Two parameters of the Lagrangian density ofEq. (2) remain to be discussed, namely, ζ and Λ v . Both of these parameters are set tozero in the enormously successful NL3 model, suggesting that the experimental data used inthe calibration procedure is insensitive to the physics encoded in these parameters. Indeed,M¨uller and Serot found possible to build models with different values of ζ that reproducethe same observed properties at normal nuclear densities, but which yield maximum neutronstar masses that differ by almost one solar mass [12]. This result indicates that observa-tions of massive neutron stars — rather than laboratory experiments — may provide theonly meaningful constraint on the high-density component of the equation of state. Finally,5he isoscalar-isovector coupling constant Λ v was added in Ref. [13] to modify the densitydependence of the symmetry energy. It was found that models with different values of Λ v reproduce the same exact properties of symmetric nuclear matter, but yield vastly differentvalues for the neutron skin thickness of heavy nuclei and for the radii of neutron stars [17].The Parity Radius Experiment (PREx) at the Jefferson Laboratory promises to measurethe skin thickness of Pb accurately and model independently via parity-violating electronscattering [18, 19]. PREx will provide a unique experimental constraint on the density de-pendence of the symmetry energy due its strong correlation to the neutron skin of heavynuclei [20].
A. Theoretical Constraints
One of the most stringent constraints on the equation of state of low density neutron-rich matter emerges from theoretical considerations, namely, from the universality of diluteFermi gases with an “infinite” scattering length ( a ). In this limit the only energy scale inthe problem is the Fermi energy ( ε F ), so the energy per particle is constrained to be that ofthe free Fermi gas up to a dimensionless universal constant ( ξ ) that is independent of thedetails of the two-body interaction [21]. That is, FIG. 3: (color online) Equation of state of pure neutron matter as a function of the Fermi momen-tum. Predictions are shown for the accurately calibrated NL3 [6, 7] (green line) and FSUGold [11](blue line) parameter sets. Shown also are various microscopic descriptions — including a model-independent result based on the physics of resonant Fermi gases by Schwenk and Pethick [25] (redregion). EN = ξ ε F . (3)6o date, the best theoretical estimates place the value of the universal constant around ξ ≈ . a nn = − . r e = +2 . k F ∼ r − (cid:39) . − . Suchcorrections have been recently computed by Schwenk and Pethick [25], with their results dis-played as the red hatched region in Fig 3. Also shown are the predictions of two microscopicmodels based on realistic two-body interactions, one of them being the venerated equation ofstate of Friedman and Pandharipande [26]. Finally, the predictions of NL3 and FSUGold arealso shown. It is gratifying that the softening of the symmetry energy of FSUGold — causedby incorporating constraints from breathing-mode energies [11] — appears consistent withthe physics of resonant Fermi gases. Such a powerful universal constraint should be routinelyand explicitly incorporated into future determinations of density functionals. Indeed, sucha constrain appears to rule out many of the models displayed in Fig. 2 of Ref. [20]. B. Experimental Constraints
FIG. 4: (color online) Binding energy per nucleon as a function of baryon density (expressed in unitsof the saturation density ρ = 0 .
148 fm − ) for symmetric nuclear matter. Theoretical predictionsare shown for the NL3 [6, 7] (green line) and FSUGold [11] (blue line) models. Shown in the insetis a comparison between the equation of state extracted from energetic nuclear collisions [27] andthe predictions of these two models. Energetic nuclear collisions may be used to constrain the high-density behavior of nucle-onic matter. To illustrate this point we display in Fig. 4 the binding energy per nucleonof symmetric nuclear matter as a function of the baryon density as predicted by both the7L3 and FSUGold models. Note that both models reproduce the equilibrium properties ofsymmetric nuclear matter and display the same quantitative behavior at densities below thesaturation point. Yet their high-density predictions are significantly different. This emergesfrom a combination of two factors. First, FSUGold predicts an incompressibility coefficient K considerably lower than NL3, namely, 230 MeV vs
271 MeV. Second, and more impor-tantly, FSUGold includes an omega-meson self-energy coupling [labeled by ζ in Eq. (2)]that is responsible for a significant softening at high density. We now compare the pre-dictions of these two models against results obtained from energetic nuclear collisions thatcan compress baryonic matter to densities as high as those predicted to exist in the core ofneutron stars. The inset in Fig. 4 provides us with such a comparison. By analyzing themanner in which matter flows after the collision of two energetic gold nuclei, the equation ofstate of symmetric nuclear matter was extracted up to densities of 4-to-5 times saturationdensity [27]. Figure 4 seems to rule out overly stiff equations of state (such as NL3). Andwhile it is gratifying that FSUGold is consistent with this analysis, one must stress thatthe connection between energetic nuclear collisions and the equation of state of cold nuclearmatter is model dependent. C. Observational Constraints
A recent observation that seems to suggest a hard equation of state is that of the low-mass X-ray binary EXO 0748-676. Note that such a binary system consists of a neutronstar accreting mass from a normal (non-compact) companion. The first constraint on theequation of state from such an object came from the detection of gravitationally redshiftedabsorption lines in Oxygen and Iron by Cottam and collaborators [28]. By measuring agravitational redshift of z = 0 .
35, the mass-to-radius ratio of the neutron star gets fixed at
M/R (cid:39) .
15 (with M expressed in solar masses and R in kilometers). By incorporatingadditional constraints arising from Eddington and thermal fluxes, a recent analysis by ¨Ozelseems to place simultaneous limits on the mass and radius of the neutron star in EXO 0748-676. That is, M ≥ . ± . M (cid:12) and R ≥ . ± .
80 km [29]. These limits are indicatedby the black solid line in Fig. 5. An earlier determination of the spin frequency of thesame neutron star by Villarreal and Strohmayer [30], when combined with the rotationalbroadening of surface spectral lines, yields an independent determination of the stellar radiusof R ≈ . +3 . − . km. This estimate, when combined with the gravitational redshift, yieldsthe orange line in Fig. 5. Finally, mass-vs-radius predictions from the NL3 and FSUGoldmodels are displayed in Fig. 5. The results clearly indicate the significantly harder characterof the equation of state predicted by NL3 relative to FSUGold. This, even when both modelspredict practically identical properties for existent ground-state observables of finite nuclei.A critical observation that would have impacted significantly on the high-density com-ponent of equation of state is the one by Nice and collaborators at the Arecibo radio tele-scope [31]. Such (initial) observation of a neutron-star–white-dwarf binary system suggesteda neutron-star mass of M (PSR J0751+1807) = 2 . ± . M (cid:12) . This was the largest neutron-star mass ever reported and promised, provided that the errors could be tighten further, topractically pin down the high-density component of the equation of state. However, at avery recent conference celebrating the 40th anniversary of the discovery of pulsars in Mon-treal, Nice reported a significantly reduced value for the mass of PSR J0751+1807, namely, M (PSR J0751+1807) ≈ . ± . M (cid:12) . This revised result is denoted by the red hatchedregion in Fig. 5 and no longer invalidates any of the theoretical models under consideration.8 IG. 5: (color online) Constraints on the mass-vs-radius relationship of neutron stars. Displayedin red is the recently revised region allowed by the analysis of Nice and collaborators [31]. Theblack and orange solid lines result from the analyzes of EXO 0748-676 by ¨Ozel [29], and Villarrealand Strohmayer [30], respectively. Also shown are the theoretical predictions from the NL3 [6, 7](green line) and FSUGold [11] (blue line) models.
IV. CONCLUSIONS
Neutron stars are unique laboratory for the study of cold baryonic matter over an enor-mous range of densities. After an introduction to the “anatomy” of a neutron star, I reliedon recent theoretical, experimental, and observational constraints to elucidate importantfeatures of the equation of state of neutron-rich matter. As mentioned in the Introduction,the only physics that spherically-symmetric neutron stars in hydrostatic equilibrium are sen-sitive to is the equation of state of neutron-rich matter [see Eqs. (1)]. This makes neutronstars gold mines for the study of baryonic matter. The various constraints utilized in thiscontribution emerged from the universal behavior of dilute Fermi gases with large scatteringlengths [25], heavy-ion experiments that probe the high-density domain of the equation ofstate [27], and astronomical observations that place limits on masses and radii of neutronstars [29, 31]. On the basis of these comparisons, it was concluded that FSUGold meets allthe challenges, even when no attempt was ever made to incorporate these constraints intothe calibration procedure. The promise of new terrestrial laboratories (such as
Facilitiesfor Rare Ion Beams ) together with improved observations with existent and future missions(such as
Constellation X ) offers the greatest hope for determining the equation of state ofcold baryonic matter in the near future. 9 cknowledgments
The author is grateful to the organizers of the
XXXI Symposium on Nuclear Physics fortheir kind invitation and hospitality. The author also wishes to acknowledge his many collab-orators that were involved in this work — especially Prof. C.J. Horowitz. This work was sup-ported in part by United States Department of Energy under grant DE-FD05-92ER40750. [1] G. Baym, C. Pethick, and P. Sutherland, Astrophys. J. , 299 (1971).[2] D. G. Ravenhall, C. J. Pethick, and J. R. Wilson, Phys. Rev. Lett. , 2066 (1983).[3] M. Hashimoto, H. Seki, and M. Yamada, Prog. Theor. Phys. , 320 (1984).[4] C. J. Horowitz, M. A. Perez-Garcia, and J. Piekarewicz, Phys. Rev. C69 , 045804 (2004),astro-ph/0401079.[5] C. J. Horowitz, M. A. Perez-Garcia, J. Carriere, D. K. Berry, and J. Piekarewicz, Phys. Rev.
C70 , 065806 (2004), astro-ph/0409296.[6] G. A. Lalazissis, J. Konig, and P. Ring, Phys. Rev.
C55 , 540 (1997), nucl-th/9607039.[7] G. A. Lalazissis, S. Raman, and P. Ring, At. Data Nucl. Data Tables , 1 (1999).[8] J. D. Walecka, Annals Phys. , 491 (1974).[9] B. D. Serot and J. D. Walecka, Adv. Nucl. Phys. , 1 (1986).[10] B. D. Serot and J. D. Walecka, Int. J. Mod. Phys. E6 , 515 (1997), nucl-th/9701058.[11] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett , 122501 (2005), nucl-th/0504034.[12] H. Mueller and B. D. Serot, Nucl. Phys. A606 , 508 (1996), nucl-th/9603037.[13] C. J. Horowitz and J. Piekarewicz, Phys. Rev. Lett. , 5647 (2001), astro-ph/0010227.[14] C. J. Horowitz and B. D. Serot, Nucl. Phys. A368 , 503 (1981).[15] D. H. Youngblood, C. M. Rozsa, J. M. Moss, D. R. Brown, and J. D. Bronson, Phys. Rev.Lett. , 1188 (1977).[16] J. Boguta and A. R. Bodmer, Nucl. Phys. A292 , 413 (1977).[17] C. J. Horowitz and J. Piekarewicz, Phys. Rev.
C64 , 062802 (2001), nucl-th/0108036.[18] C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels, Phys. Rev.
C63 , 025501 (2001),nucl-th/9912038.[19] R. Michaels, P. A. Souder, and G. M. Urciuoli (2005), URL http://hallaweb.jlab.org/parity/prex .[20] B. A. Brown, Phys. Rev. Lett. , 5296 (2000).[21] J. Carlson, S.-Y. Chang, V. R. Pandharipande, and K. E. Schmidt, Phys. Rev. Lett. ,050401 (2003).[22] G. A. Baker, Phys. Rev. C60 , 054311 (1999).[23] H. Heiselberg, Phys. Rev.
A63 , 043606 (2002), cond-mat/0002056.[24] Y. Nishida and D. T. Son, Phys. Rev. Lett. , 050403 (2006), cond-mat/0604500.[25] A. Schwenk and C. J. Pethick, Phys. Rev. Lett. , 160401 (2005), nucl-th/0506042.[26] B. Friedman and V. R. Pandharipande, Nucl. Phys. A361 , 502 (1981).[27] P. Danielewicz, R. Lacey, and W. G. Lynch, Science , 1592 (2002), nucl-th/0208016.[28] J. Cottam, F. Paerels, and M. Mendez, Nature , 51 (2002), astro-ph/0211126.[29] F. Ozel, Nature , 1115 (2006).[30] A. R. Villarreal and T. E. Strohmayer, Astrophys. J. , L121 (2004), astro-ph/0409384.[31] D. J. Nice et al., Astrophys. J. , 1242 (2005), astro-ph/0508050., 1242 (2005), astro-ph/0508050.