Nuclear Astrophysics in the New Era of Multimessenger Astronomy

J. Piekarewicz1, ∗

1Department of Physics, Florida State University, Tallahassee, FL 32306, USA(Dated: May 18, 2018)

Neutron stars are unique cosmic laboratories for the exploration of matter under extreme con-ditions of density and neutron-proton asymmetry. Due to their enormous dynamic range, neutronstars display a myriad of exotic states of matter that are impossible to recreate under normal labo-ratory conditions. In these three lectures I will discuss how the strong synergy that has developedbetween nuclear physics and astrophysics will uncover some of the deepest secrets behind thesefascinating objects. In particular, I will highlight the enormous impact that the very first detectionof gravitational waves from the binary neutron-star merger GW170817 is having in constraining thecomposition, structure, and dynamics of neutron stars.

PACS numbers: 04.40.Dg, 21.65.Ef, 24.10.Jv, 26.60.Kp, 97.60.Jd

I. PREFACE

Massive stars use the raw materials (mostly hydrogenand helium) created during the Big Bang to fuel the starsand to create via thermonuclear fusion many of the chem-ical elements found in the periodic table. However, thefusion of light nuclei into ever increasing heavier elementsterminates abruptly with the synthesis of the iron-groupelements that are characterized by having the largestbinding energy per nucleon. Once the iron core exceedsa characteristic mass limit of about 1.4 solar masses, nei-ther thermonuclear fusion nor electron degeneracy pres-sure can halt the collapse of the stellar core. The unim-peded collapse of the core and the ensuing shock waveproduce one of the most spectacular events in the Uni-verse: a Supernova Explosion. Core-collapse supernovaeleave behind exotic compact remnants in the form of ei-ther black holes or neutron stars. Neutron stars are thecentral theme of the present lectures.

The historical first detection of gravitational wavesfrom the binary neutron-star (BNS) merger GW170817by the LIGO-Virgo collaboration [1] is providing funda-mental new insights into the nature of dense matter andthe astrophysical site for the creation of the heavy ele-ments via the rapid neutron-capture process (r-process).Although GW170817 represents the very first detectionof a BNS merger, it is already furnishing answers to twoof the “eleven science questions for the next century”identified by the National Academies Committee on thePhysics of the Universe [2]: What are the new states ofmatter at exceedingly high density and temperature? andhow were the elements from iron to uranium made? Inthese three lectures I will try to illuminate the deep con-nections that exist between nuclear physics and astro-physics in understanding the composition, structure, anddynamics of neutron stars. I will discuss how the combi-nation of nuclear physics insights, modern theoretical ap-

∗Electronic address: jpiekarewicz@fsu.edu

proaches, laboratory experiments, and astronomical ob-servations using both electromagnetic and gravitationalradiation pave the way to our understanding of thesefascinating objects at the dawn of the brand new era of“multimessenger” astronomy.

The lectures were divided into three independent unitsthat were aimed to provide a coherent picture of the field.In turn, this proceedings are also divided into three chap-ters. First, I will provide a historical perspective thatintroduces some of the main actors responsible for thedevelopment of the field. Second, I will provide a de-scription of the many phases and exotic states of mat-ter that we believe “hide” in the interior of a neutronstar. Finally, I will end by discussing the deep connec-tions between “Heaven and Earth”, namely, the ongo-ing and future suite of terrestrial experiments and astro-nomical observations that—with appropriate theoreticalinsights—will unlock some of the deepest secrets lurkingwithin neutron stars.

II. HISTORICAL PERSPECTIVE

We start this chapter by highlighting the 1939 work byOppenheimer and Volkoff, a theoretical milestone in thehistory of neutron stars [3]. By then, Einstein’s generaltheory of relativity was firmly established. On the otherhand, the existence of the neutron was experimentallyconfirmed just a few years earlier, in 1932, by JamesChadwick working at the Cavendish Laboratory in theUK [4]. Yet soon after Chadwick’s discovery, the termneutron star seems to appears in writing for the first timein the 1933 proceedings of the the American Physical So-ciety by Baade and Zwicky [5]. Using what it is now com-monly referred to as the Tolman-Volkoff-Oppenheimer(TOV) equations, Oppenheimer and Volkoff concludedthat a neutron star supported exclusively by the pres-sure from its degenerate neutrons will collapse once itsmass exceeds 0.7 solar masses (0.7M�). Unbeknownstto them, this finding will eventually promote nuclearphysics to the forefront of neutron-star structure—giventhat neutron stars with masses of at least 2M� have al-

arX

iv:1

805.

0478

0v2

[nu

cl-t

h] 1

7 M

ay 2

018

2

ready been observed [6, 7]. In essence, the large discrep-ancy between observation and the theoretical predictionby Oppenheimer and Volkoff has transferred ownershipof the neutron-star problem to nuclear physics.

The Tolman-Oppenheimer-Volkoff equations, whichrepresent a generalization of Newtonian gravity to therealm of general relativity, are expressed as a coupled setof first-order differential equations of the following form:

dP

dr=−G E(r)M(r)

r2

[1+

P (r)

E(r)

] [1+

4πr3P (r)

M(r)

]×[1− 2GM(r)

r

]−1

, (1)

dM

dr= 4πr2E(r) , (2)

where G is Newton’s gravitational constant and P (r),E(r), and M(r) represent the pressure, energy density,and enclosed-mass profiles of the star, respectively. Thethree terms enclosed in square brackets encode the rele-vant corrections to Newtonian gravity. The solution tothese equations by Oppenheimer and Volkoff under theassumption that the equation of state (the relation be-tween the pressure and the energy density) is that of afree Fermi gas of neutrons yields a maximum neutronstar mass of 0.7M�. Note that the fact that the equa-tion of state (EOS) is the only input that neutron starsare sensitive to creates a unique synergy between nuclearphysics and astrophysics. The mass-versus-radius rela-tion obtained by Oppenheimer and Volkoff is displayedwith a red solid line in Fig.1, alongside the current obser-vational limit on the maximum neutron-star mass [6, 7].Also shown are predictions from more realistic modelsthat will be discussed later and that take into accountthe complicated and subtle nuclear dynamics.

I would be remiss if I did not highlight the indirect, yetpivotal, role that Subramanyan Chandrasekhar (“Chan-dra”) played in the history of neutron stars. Alreadyin 1926 R.H. Fowler—Dirac’s doctoral advisor—showedthat white-dwarf stars are supported against gravita-tional collapse by quantum degeneracy pressure, the pres-sure exerted by a cold Fermi gas by virtue of the Pauliexclusion principle. In particular, Fowler showed thatthe electron degeneracy pressure scales as the 5/3 powerof the electronic density. However, during his 1930 jour-ney to Cambridge to pursue his doctoral degree underthe supervision of no other than Fowler, Chandrasekharrealized that as the stellar density increases and the elec-trons become relativistic, the pressure support weakens,ultimately becoming proportional to the 4/3 power of theelectronic density. This weakening has dramatic conse-quences: a white-dwarf star with a mass in excess ofabout 1.4 solar masses—the so-called “Chandrasekharmass limit”—will collapse under its own weight [8]. Al-though such far-reaching result is now well accepted, atthat time it was the subject of derision, primarily byArthur Eddington. It is worth noting that Chadwick’sdiscovery of the neutron came a year after Chandra’s

6 8 10 12 14 16 18R(km)

0

1

2

3

M(M

sun)

Fermi GasFSUGoldFSUGarnetNL3

Observational limit

Oppenheimer-Volkoff

FIG. 1: Mass-Radius relation as predicted by a simple Fermi-gas model [3] and three realistic equations of state that willbe introduced in later chapters. The horizontal band indi-cates the current observational limit on the maximum stellarmass [6, 7].

prediction of the Chandrasekhar mass limit. Ultimately,however, Chandra prevailed and in recognition to hismany scientific contributions NASA launched in 1999 the“Chandra X-ray Observatory”, NASA’s flagship missionfor X-ray astronomy. We note that in a 1932 publicationLandau—independently of Chandrasekhar—predicts theexistence of a maximum mass for a white dwarf star [9].Moreover, Landau went ahead to speculate on the exis-tence of dense stars that look like giant atomic nuclei.For further historic details see Ref. [10] and chapter 14 inRef. [11].

Although firmly established on theoretical grounds, itwould take almost 30 years after the work by Oppen-heimer and Volkoff to discover neutron stars. The gloryof the discovery fell upon the talented young graduatestudent Jocelyn Bell, now Dame Jocelyn Bell Burnell.While searching for signals from the recently discoveredand to this day still enigmatic quasars, Bell detected a“bit of scruff” in the data arriving into her newly con-structed radio telescope. The arriving signal was “puls-ing” with such an enormous regularity, 1.337 302 088 331seconds, that both Bell and Anthony Hewish (her doc-toral advisor) were bewildered by the detection. Ini-tially convinced that the signal was a beacon from anextraterrestrial civilization, they dubbed the source as“Little Green Man 1”. Now known as radio pulsar “PSRB1919+21”, Bell had actually made the very first detec-tion of a rapidly rotating neutron star [12]. Although it iswell known that Jocelyn Bell was snubbed by the Nobelcommittee in 1974—the year that her doctoral advisor

3

Anthony Hewish shared the Nobel prize in Physics withMartin Ryle—she has always displayed enormous graceand humility in the face of this controversy. Since then,Bell has been recognized with an enormous number ofhonors and awards. Moreover, Dr. Iosif Shklovsky—a re-cipient of the 1972 Bruce Medal for outstanding lifetimecontributions to astronomy—paid her one of the highestcompliments that one can receive from a fellow scientist:“Miss Bell, you have made the greatest astronomical dis-covery of the twentieth century.”

III. ANATOMY OF A NEUTRON STAR

The structure of neutron stars is both interesting andcomplex. To appreciate the enormous dynamic range andrichness displayed by these fascinating objects, we displayin Fig. 2 what is believed to be an accurate rendition ofthe structure and composition of a neutron star. Further,to accentuate some of the extreme conditions present ina neutron star, we display in Table I some of the char-acteristic of the Crab pulsar, the compact remnant ofa supernovae explosion in the constellation Taurus thatwas observed nearly 1,000 years ago.

The outermost surface of the neutron star contains avery thin atmosphere of only a few centimeters thick thatis composed of hydrogen, but may also contain heavierelements such as helium and carbon. The detected elec-tromagnetic radiation may be used to constrain criticalparameters of the neutron star. For example, assum-ing pure blackbody emission from the stellar surface at atemperature T provides a determination of the stellar ra-dius from the Stefan-Boltzmann law: L=4πσR2T 4. Un-fortunately, complications associated with distance mea-surements and distortions of the black-body spectrummake the accurate determination of stellar radii—one ofthe most critical observables informing the equation ofstate—a challenging task. Just below the atmosphere liesthe ∼100 m thick envelope that acts as “blanket” betweenthe hot interior (with T &108 K) and the “cold” surface(with T &106 K) [13]. Further below lies the non-uniformcrust, a region characterized by fascinating exotic statesof matter that are impossible to recreate under normallaboratory conditions. The non-uniform crust sits abovea uniform liquid core that consists of neutrons, protons,electrons, and muons. The core accounts for practicallyall the mass and for about 90% of the size of a neutronstar. Finally, depending on the highest densities thatmay be attained in the inner core, there is also a pos-sibility (marked with a question mark in Fig. 2) for theemergence of new exotic phases, such as pion or kaoncondensates [14, 15], strange quark matter [16, 17], andcolor superconductors [18, 19].

FIG. 2: The left-hand panel depicts what is believed to bean accurate rendition of the fascinating structure and exoticphases that exist in a neutron star (courtesy of Dany Page).On the right-hand panel we display the assumed compositionof the crust of a neutron star—from a crystalline lattice ofexotic neutron-rich nuclei to the emergence of the nuclear-pasta phase (courtesy of Sanjay Reddy).

Name: PSR B0531+21 Constellation: TaurusDistance: 2.2 kpc Age: 960 yearsMass: 1.4M� Radius: 10 kmDensity: 1015g/cm3 Pressure: 1029 atmTemperature: 106 K Escape velocity: 0.6 cPeriod: 33 ms Magnetic Field: 1012 G

TABLE I: Characteristics of the 960 year old Crab pulsar.

A. The Outer Crust: Extreme Sensitivity toNuclear Masses

The range of the short-range nucleon-nucleon (NN) in-teraction is approximately equal to the Compton wave-length of the pion, or about 1.4 fm. In turn, in uniformnuclear matter at saturation density (ρsat≈0.15 fm3) theaverage inter-nucleon separation is about 1.2 fm. Thissuggests that at densities of about 1/2 to 1/3 of nuclear-matter saturation density, the average inter-nucleon sep-aration will be large enough that the NN interactionwill cease to be effective. Thus, to maximize the im-pact of the NN attraction it becomes energetically fa-vorable for the system to break translational invarianceand for the nucleons to cluster into nuclei. The ques-tion of which nucleus is energetically the most favor-able emerges from a dynamical competition between thesymmetry energy—which favors nearly symmetric nuclei(N&Z)—and the electronic density, which in turn favorsno electrons (Z=0).

Although subtle, the dynamics of the outer crust is en-capsulated in a relatively simple expression for the totalGibbs free energy per nucleon, which at zero temperatureequals the total chemical potential of the system. Thatis [20–24],

µ(Z,A;P ) =M(Z,A)

A+Z

Aµe −

4

3Cl

Z2

A4/3p

F. (3)

The first term is independent of the pressure—or equiv-

4

alently of the baryon density—and represents the entirenuclear contribution to the chemical potential. It de-pends exclusively on the mass per nucleon of the “opti-mal” nucleus populating the crystal lattice. The secondterm represents the electronic contribution and, as anyFermi gas, it is strongly density dependent. Finally, thelast term provides the relatively modest—although by nomeans negligible—electrostatic lattice contribution (withCl = 3.40665×10−3). Here pF is the nuclear Fermi en-ergy that is related to the baryon density through thefollowing expression:

pF

=(3π2ρ

)1/3. (4)

Finally, the connection between the pressure and thebaryon density is provided by the underlying crustalequation of state that, as anticipated, is dominated bythe relativistic electrons. That is,

P (ρ)=m4

e

3π2

(x3

FyF− 3

8

[x

FyF

(x2

F+y2

F

)−ln(x

F+y

F)])

−ρ3Cl

Z2

A4/3p

F, (5)

where xF

= peF/me and y

F= (1+x2

F)1/2 are scaled elec-

tronic Fermi momentum and Fermi energy, respectively;pe

F= (Z/A)1/3p

F. This discussion suggests that the only

unknown in the determination of the crustal compositionis the optimal nucleus, namely, the one that minimizesthe chemical potential, at a given pressure.

The search for the optimal nucleus is performed as fol-lows. For a given pressure P and nuclear species (Z,A),the equation of state is used to determine the corre-sponding baryon density of the system which, in turn,determines the Fermi momentum p

Fand the electronic

chemical potential µe. This is sufficient to compute thechemical potential of the system as indicated in Eq. (3).However, this procedure requires scanning over an entiremass table—which in some instances consists of nearly10,000 nuclei. The (Z,A) combination that minimizesµ(A,Z;P ) determines the optimal nucleus populatingthe crystal lattice at the given pressure. Naturally, if thepressure (and thus the density) is very small so that theelectronic contribution to the chemical potential is neg-ligible, then 56Fe—with the lowest mass per nucleon—becomes the nucleus of choice. As the pressure and den-sity increase so that the electronic contribution may nolonger be neglected, then it becomes advantageous to re-duce the electron fraction Z/A at the expense of increas-ing the neutron-proton asymmetry. This results in an in-crease in the mass per nucleon. Which nucleus becomesthe optimal choice then emerges from a subtle competi-tion between the electronic contribution that favors Z=0and the nuclear symmetry energy which favors (nearly)symmetric nuclei.

Even though the underlying physics is relatively sim-ple, computing the composition of the outer crust is hin-dered by the unavailability of mass measurements of ex-otic nuclei with a very large neutron-proton asymmetry.

Indeed, of the masses of the N = 82 isotones believed topopulate the deepest layers of the outer crust—such as122Zr, 120Sr, and 118Kr—none have been determined ex-perimentally [25, 26] and it is unlikely that they will everbe determined even at the most powerful rare-isotope fa-cilities. Thus, the only recourse is to resort to theoreticalcalculations which, in turn, must rely on extrapolationsfar away from their region of applicability. Whereas noclear-cut remedy exists to cure such unavoidable extrap-olations, we have recently offered a path to mitigate theproblem [27–30]. The basic paradigm behind our two-pronged approach is to start with a robust underlyingtheoretical model that captures as much physics as pos-sible, followed by a Bayesian Neural Network (BNN) re-finement of the mass residuals that aims to account forthe missing physics [27]. That is, the resulting mass for-mula is given by

M(Z,N) ≡Mmodel(Z,N) + δmodel(Z,N), (6)

where Mmodel(Z,N) is the “bare” model prediction andδmodel(Z,N) the BNN refinement to the difference be-tween the model predictions and experiment. For furtherdetails on the implementation see [27–30].

FIG. 3: Crustal composition of a canonical 1.4M� neutronstar with a 12.78 km radius as predicted by three mass models:BNN, DZ, and HFB19. See text for further explanations.

In Fig. 3 we display the composition of the outer crustas a function of depth for a neutron star with a massof 1.4M� and a radius of 12.78 km. Predictions areshown using our newly created mass model “BNN”, Du-flo Zuker, and HFB19; these last two without any BNNrefinement. The composition of the upper layers of thecrust (spanning about 100 m and depicted in yellow) con-sists of Fe-Ni nuclei with masses that are well known ex-perimentally. As the Ni-isotopes become progressivelymore neutron rich, it is energetically favorable to transi-tion into the magic N=50 region. In the particular caseof the BBN-improved model, this intermediate region ispredicted to start with stable 86Kr and then progressivelyevolve into the more exotic isotones 84Se (Z= 34), 82Ge(Z = 32), 80Zn (Z = 30), and 78Ni (Z = 28); all this inan effort to reduce the electron fraction. In this region,most of the masses are experimentally known, althoughfor some of them the quoted value is not derived from

5

purely experimental data [25]. Ultimately, it becomes en-ergetically favorable for the system to transition into themagic N =82 region. In this region none of the relevantnuclei have experimentally determined masses. Althoughnot shown, it is interesting to note that the composi-tion of the HFB19 model changes considerably after theBNN refinement, bringing it into closer agreement withthe predictions of both BNN and Duflo-Zuker. Althoughbeyond the scope of this work, we should mention thatthe crustal composition is vital in the study of certainelastic properties of the crust, such as its shear mod-ulus and breaking strain—quantities that are of greatrelevance to magnetar starquakes [31, 32] and continuousgravitational-wave emission from rapidly rotating neu-tron stars [33].

132Cd133Cd134Cd 133In 134In 135In 136In 137In 136Sn 138Sn

-2

-1

0

1

2

Mth

-Mex

p=B e

xp-B

th(M

eV) WS3 (1.12MeV)

FRDM-2012 (0.88MeV)DZ-Bare (1.21MeV)DZ-BNN (0.37MeV)

FIG. 4: Theoretical predictions for the total binding energy ofthose nuclei that have been identified as impactful in r-processnucleosynthesis [34]. All experimental values have been esti-mated from experimental trends of neighboring nuclides [26].Quantities in parentheses denote the rms deviations.

We close this section with a small comment on theimpact of our work on the historical first detection of abinary neutron star merger by the LIGO-Virgo collabora-tion [1], an event that is starting to provide fundamentalnew insights into the astrophysical site for the r-processand on the nature of dense matter. In particular, we fo-cus on a particular set of nuclear masses that have beenidentified as “impactful” in sensitivity studies of the el-emental abundances in r-process nucleosynthesis. Theseinclude a variety of neutron-rich isotopes in cadmium,indium, and tin; see Table I of Ref. [34]. In Fig. 4 theo-retical predictions are displayed for the mass of some ofthese isotopes. Predictions are provided for the WS3 [35],FRDM-2012 [36], DZ [37], and BNN-DZ [29] mass mod-els. Root-mean-square deviations of the order of 1 MeVare recorded for all models, except for the BNN-improvedDuflo-Zuker model where the deviation is only 369 keV.The figure nicely encapsulates the spirit of our two-prongapproach, namely, one that starts with a mass model ofthe highest quality (DZ) that is then refined via a BNNapproach. The improvement in the description of the ex-perimental data together with a proper assessment of the

theoretical uncertainties are two of the greatest virtuesof the BNN approach. Indeed, the BNN-DZ predictionsare consistent with all the masses of the impactful nu-clei displayed in the figure and that have been recentlyreported in the latest AME2016 mass compilation [26].

B. The Inner Crust: Coulomb Frustration andNuclear Pasta

Although not covered in the lectures because of lackof time, a few comments on the fascinating physics ofthe inner crust are pertinent. Note that there have beenvarious significant contributions from the Brazilian com-munity to this topic, both from inside [38–41] and outsideof Brazil [42–44].

The inner stellar crust comprises the region fromneutron-drip density up to the density at which unifor-mity in the system is restored; see Sec. III C. Yet thetransition from the highly-ordered crystal to the uniformliquid is both interesting and complex. This is becausedistance scales that were well separated in both the crys-talline phase (where the long-range Coulomb interactiondominates) and in the uniform phase (where the short-range strong interaction dominates) become comparable.This unique situation gives rise to “Coulomb frustra-tion”. Frustration, a phenomenon characterized by theexistence of a very large number of low-energy configura-tions, emerges from the impossibility to simultaneouslyminimize all elementary interactions in the system. In-deed, as these length scales become comparable, com-petition among the elementary interactions results in theformation of a myriad of complex structures radically dif-ferent in topology yet extremely close in energy. Giventhat these complex structures—collectively referred to as“nuclear pasta”—are very close in energy, it has beenspeculated that the transition from the highly orderedcrystal to the uniform phase must proceed through aseries of changes in the dimensionality and topologyof these structures [45, 46]. Moreover, due to the pre-ponderance of low-energy states, frustrated systems dis-play an interesting and unique low-energy dynamics thathas been captured using a variety of techniques includ-ing semi-classical numerical simulations [42–44, 47–52] aswell as quantum simulations in a mean-field approxima-tion [53–57]. For some extensive reviews on the fascinat-ing structure and dynamics of the neutron-star crust seeRefs. [58, 59], and references contain therein.

In closing this section, we display in Fig. 5 a MonteCarlo snapshot obtained from a numerical simulation of asystem containing Z=800 protons andN=3200 neutronsthat nicely illustrates how the system organizes itself intoneutron-rich clusters of complex topologies that are im-mersed in a dilute neutron vapor [47, 48]. We note thata great virtue of these numerical simulations is that itclearly illustrates how pasta formation is very robust.Indeed, our numerical simulations proceed in an unbi-ased manner without assuming any particular shape. In-

6

FIG. 5: A snapshot of a Monte-Carlo simulation for a systemconsisting of 4000 nucleons at a baryon density of ρ= ρsat/6, aproton fraction of Z/A= 0.2, and an “effective” temperatureof T =1 MeV [47, 48].

stead, the system evolves dynamically into these com-plex shapes from a simple underlying two-body interac-tion consisting of a short-range nuclear attraction and along-range Coulomb repulsion.

C. The Liquid Core: Uniform Neutron-RichMatter

At densities of about 1014g/cm3 the common percep-tion of a neutron star as a uniform assembly of extremelyclosed packed neutrons is finally realized in the stellarcore. As we articulate below, the liquid core is responsi-ble for the most salient structural features of a neutronstar, such as its mass and its radius. Given the uniquesynergy between laboratory experiments and astrophys-ical observations, we devote the entire next section toillustrate these connections. However, nowhere in thesediscussions we examine in detail the possibility of ex-otic states of matter harboring the stellar core. Rather,we push our formalism consisting of conventional con-stituents (nucleons and charged leptons) to the extremesof density and neutron-proton asymmetry. Evidence infavor of exotic degrees of freedom may then emerge as ouraccurately calibrated models show serious discrepancieswhen compared against observations.

IV. HEAVEN AND EARTH

At densities of about a half to a third of nuclear-mattersaturation density the nuclear-pasta phase will “melt”and uniformity in the system will be restored. How-ever, in order to maintain both chemical equilibrium and

charge neutrality a small fraction of about 10% of pro-tons and charged leptons (electrons and muons) is re-quired. Although the stellar crust is driven by uniqueand intriguing dynamics, its structural impact on thestar is rather modest. Indeed, more than 90% of thesize and essentially all the stellar mass reside in the core.However, the equation of state of neutron-rich matter atthe highest densities attained in the core is poorly con-strained by laboratory observables. Thus, the cleanestconstraint on the EOS at high density is likely to emergefrom astrophysical observations of massive neutron stars.In this regard, enormous progress has been made withthe observation of two massive neutron stars by Demor-est [6] and Antoniadis [7]. For example, the measurementof the mass of PSR J164-2230 (1.97± 0.04M�) by it-self has ruled out EOS that are too soft to support a2M� neutron star—such as those with exotic cores. Un-doubtedly, the quest to find even more massive neutronstars will continue with the deployment of new missions,such as the Neutron Star Interior Composition Explorer(NICER) and with the imminent detection of more bi-nary neutron-star mergers. Predictions for the mass-vs-radius relation for a variety of relativistic models consis-tent with the 2M� limit, with the exception of FSUG-old [60] (“FSU” in the figure) are displayed in Fig. 6. Wenote that all our calculations have been done using mod-els that yield an accurate description of the propertiesof finite nuclei, while providing a Lorentz covariant ex-trapolation to dense matter [61]. This implies that byconstruction, the EOS remains causal at all densities.

8 10 12 14 16 18R(km)

0

1

2

3

M/M

sun

NL3

FSUIU-FSU

TFa TFc

TFb

4U 1608-52EXO 1745-2484U 1820-30

Demorest

Antoniadis

SuleimanovR>14km forM<2.3Msun

Ozel

Steiner

FIG. 6: Predictions for the mass-vs-radius relation for a va-riety of relativistic models of the EOS [61]. Photometric con-straints on stellar masses and radii extracted from variousanalyses of X-ray bursts are shown [62–64]. Also shown areconstraints obtained from the measurement of two massiveneutron stars by Demorest [6] and Antoniadis [7].

Unfortunately, the extraction of stellar radii by pho-tometric means has been notoriously challenging, as ithas been plagued by large systematic uncertainties, of-ten revealing discrepancies as large as 5-6 km [62–64]; seeFig. 6. It appears, however, that the situation has im-

7

proved through a better understanding of systematic un-certainties, important theoretical developments, and theimplementation of robust statistical methods [65–72]. Aswe will show later, stellar radii also leave their imprinton the gravitational wave form measured in the mergerof two neutron stars and will play a critical role as detec-tions of these mergers become more plentiful.

Unlike massive neutron stars, stellar radii are sensitiveto the density dependence of the symmetry energy inthe immediate vicinity of nuclear-matter saturation den-sity [73]. This may prove advantageous, as the symme-try energy at moderate densities may be constrained byterrestrial experiment. A fundamental property of theEOS that has received considerable attention over thelast decade is the slope of the symmetry energy at satu-ration density. The symmetry energy is an essential com-ponent of the equation of state that strongly impacts thestructure, dynamics, and composition of neutron stars.Before chemical equilibrium is enforced, the equation ofstate depends on the conserved baryon density ρ= ρn+ρpand the neutron-proton asymmetry α≡(ρn−ρp)/(ρn+ρp).As it is customarily done, the energy per nucleon may beexpanded at zero temperature in even powers of α:

E

A(ρ, α)−M ≡ E(ρ, α) = ESNM(ρ)+α2S(ρ)+O(α4) , (7)

where ESNM(ρ)=E(ρ, α≡0) is the energy per nucleon ofsymmetric nuclear matter (SNM) and S(ρ) is the symme-try energy, which represents the first-order correction tothe symmetric limit. More intuitively, the symmetry en-ergy is nearly equal to the energy cost required to convertsymmetric nuclear matter, with α=0, into pure neutronmatter (PNM) with α=1:

S(ρ)≈E(ρ, α=1)−E(ρ, α=0) . (8)

Note that no odd powers of α appear in the expansionas the nuclear force is assumed to be isospin symmet-ric and (for now) electroweak contributions have been“turned off”. Finally, it is customary to characterize thebehavior of both symmetric nuclear matter and the sym-metry energy near saturation density in terms of a fewbulk parameters. To do so, one performs a Taylor seriesexpansion around nuclear matter saturation density ρsat.That is [74],

ESNM(ρ) = ε0 +1

2Kx2 + . . . , (9a)

S(ρ) = J + Lx+1

2Ksymx

2 + . . . , (9b)

where x=(ρ−ρsat)/3ρsat is a dimensionless parameter thatquantifies the deviations of the density from its value atsaturation. Here ε0 and K represent the energy per nu-cleon and the incompressibility coefficient of SNM; J andKsym are the corresponding quantities for the symmetryenergy. However, unlike symmetric nuclear matter whosepressure vanishes at ρsat, the slope of the symmetry en-ergy L does not vanish at saturation density. Indeed,

assuming the validity of Eq. (8), L is directly propor-tional to the pressure of PNM (P0) at saturation density,namely,

P0 ≈1

3ρsatL . (10)

Given that neutron-star radii are sensitive to the den-sity dependence of the symmetry energy in the vicinityof nuclear-matter saturation density [73], laboratory ex-periments play a critical role in constraining stellar radii.Indeed, L is strongly correlated to both the thicknessof the neutron skin in 208Pb [75–78] and the radius of aneutron star [79–83]. Note that the neutron-skin thick-ness is defined as the difference between the neutron andproton root-mean-square radii. The thickness of the neu-tron skin emerges from a competition between the surfacetension, which favors placing the excess neutrons in theinterior, and the difference between the value of the sym-metry energy at the surface relative to that at the center;namely, L. If such a difference is large, then it is favor-able to move the extra neutrons to the surface, therebycreating a thick neutron skin. Similarly, if the pressure ofpure neutron matter at saturation is large, a large stel-lar radius develops. This suggests a powerful correlation:the larger the value of L the thicker the neutron skin andthe larger the radius of a neutron star [80]. In this way,the neutron-skin thickness in 208Pb is identified as a lab-oratory observable that may serve to constrain the radiusof a neutron star—despite a difference in size of 19 ordersof magnitude!

Using a purely electroweak reaction—parity-violatingelectron scattering—the pioneering Lead Radius Experi-ment (“PREX”) at the Jefferson Laboratory provided thefirst model-independent evidence in favor of a neutron-rich skin in 208Pb [84, 85]: R208

skin = 0.33+0.16−0.18 fm. Un-

fortunately, the larger than anticipated statistical errorhas hindered a meaningful comparison against theoreti-cal predictions. Yet, in an effort to impose meaningfultheoretical constraints, a follow-up experiment (PREX-II) is scheduled to run in 2019 that is envisioned to reachthe original 0.06 fm sensitivity.

Finally, constraints on the density dependence of thesymmetry energy have emerged from an unexpectedsource: the historical first detection of the binaryneutron-star merger GW170817 [1]. The tidal polarizabil-ity of a neutron star, namely, the tendency to developa mass quadrupole as a response to the tidal field in-duced by its companion [86, 87], is imprinted in the grav-itational wave form associated with the binary inspiral.Indeed, the gravitational wave form maintains its point-mass (black-hole-like) behavior longer for compact starsthan for stars with larger radii. In a recent publicationthat examined the impact of GW170817 on the tidal po-larizability, we inferred a limit on the stellar radius of a1.4M� neutron star of R1.4

? < 13.76 km [88]. In the con-text of Fig. 6, this is a highly significant result. With theexception of IU-FSU [89], all models displayed in the fig-ure are ruled out either because they predict a maximum

8

mass that is too low or a stellar radius that is too large.Further, assuming that one can extrapolate our findingsdown to saturation density, constraints from GW170817also provide limits on the neutron-skin thickness of 208Pbof R208

skin.0.25 fm, well below the current upper limit ob-tained by the PREX collaboration [84, 85]. This suggestan intriguing possibility: If PREX-II confirms that R208

skinis large, this will suggest that the EOS at the typical den-sities found in atomic nuclei is stiff. In contrast, the rel-atively small neutron-star radii suggested by GW170817implies that the symmetry energy at higher densities issoft. The evolution from stiff to soft may be indicativeof a phase transition in the neutron-star interior.

V. CONCLUSIONS AND OUTLOOK

Neutron stars are gold mines for the study of physicalphenomena across a variety of disciplines ranging fromthe very small to the very large, from elementary-particlephysics to general relativity. From the perspective ofhadronic and nuclear physics, the main topic of the XIVInternational Workshop, neutron stars hold the answer toone of the most fundamental questions in the field: Howdoes subatomic matter organize itself and what phenom-ena emerge? [90]. Although the most common perceptionof a neutron star is a uniform ensemble of neutrons, weshowed that the reality is far different and much moreinteresting. In particular, during our journey through aneutron star we uncovered a myriad of exotic states ofmatter that are speculated to exist in a neutron star,such as Coulomb crystals, pasta phases, and perhapseven deconfined quark matter. As exciting, we discussedthe fundamental role that nuclear astrophysics will playin the new era of multimessenger astronomy. Althoughbinary pulsars—such as the Hulse-Taylor pulsar—havebeen used to infer the existence of gravitational waves,

the evidence was indirect. Now, however, we have thefirst direct evidence of gravitational waves from a binaryneutron-star merger. In a testament to human ingenu-ity, many of the observed phenomena associated with thebinary neutron-star merger were predicted by earlier the-oretical simulations. Surprisingly, this very first observa-tion has provided a treasure trove of insights into thenature of dense matter and the site of the r-process.

Yet the era of multimessenger astronomy is in its in-fancy and much excitement is in store. Electromag-netic, gravitational and, hopefully soon, neutrino radi-ation from spectacular neutron-star mergers will revealsome of nature’s most intimate secrets. This new era isof particular significance for the Hadron Physics serieswhose summer school format provides an ideal venue toeducate and motivate the next generation of scientists.Undoubtedly, nuclear and hadronic physics will play afundamental role in elucidating the physics underlyingthese spectacular events. And it is the new generation ofscientists that will reap the benefits from this scientificrevolution and who will make the new discoveries. I hopethat through this set of lectures that I was privileged todeliver, I was able to inspire many young scientists tojoin this fascinating field.

Acknowledgments

I am very grateful to the organizers of the XIV Inter-national Workshop in Hadron Physics, particularly Prof.Menezes, Prof. Benghi, and Dr. Oliveira, for their kind-ness and hospitality. The financial support of the CNPqis greatly appreciated. This material is based upon worksupported by the U.S. Department of Energy Office ofScience, Office of Nuclear Physics under Award NumberDE-FG02-92ER40750.

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