and now for something completely different probing the ... · and now for something completely...

And Now for Something Completely Different ...Probing the Neutron-Star Matter EOS

Neutrino Astrophysics and Fundamental Properties(INT June 2015)

Organizing Committee

Chuck Horowitz (Indiana)

Kees de Jager (JLAB)

Jim Lattimer (Stony Brook)

Witold Nazarewicz (UTK, ORNL)

Jorge Piekarewicz (FSU

Sponsors: Jefferson Lab, JSA

PREX is a fascinating experiment that uses parity

violation to accurately determine the neutron

radius in 208Pb. This has broad applications to

astrophysics, nuclear structure, atomic parity non-

conservation and tests of the standard model. The

conference will begin with introductory lectures

and we encourage new comers to attend.

For more information contact [email protected]

Topics

Parity Violation

Theoretical descriptions of neutron-rich nuclei and

bulk matter

Laboratory measurements of neutron-rich nuclei

and bulk matter

Neutron-rich matter in Compact Stars / Astrophysics

Website: http://conferences.jlab.org/PREX

The only input required to compute the structureof neutron stars is the equation of state of cold

neutron-rich matter: P=P(E ,T =0)J. Piekarewicz (FSU) Neutron-Star Matter Equation of State Neutrino Astrophysics and ... 1 / 15

My FSU Collaborators

Genaro Toledo-SanchezKarim HasnaouiBonnie Todd-RutelBrad FutchJutri TarunaFarrukh FattoyevWei-Chia ChenRaditya Utama

My Outside Collaborators

B. Agrawal (Saha Inst.)M. Centelles (U. Barcelona)G. Colò (U. Milano)C.J. Horowitz (Indiana U.)W. Nazarewicz (MSU)N. Paar (U. Zagreb)M.A. Pérez-Garcia (U.Salamanca)P.G.- Reinhard (U.Erlangen-Nürnberg)X. Roca-Maza (U. Milano)D. Vretenar (U. Zagreb)

J. Piekarewicz (FSU) Neutron-Star Matter Equation of State Neutrino Astrophysics and ... 2 / 15

Nuclear Charge and Weak-Charge Form Factors (Electroweak )

0.0 0.5 1.0 1.5 2.0q(fm-1)

10-4

10-3

10-2

10-1

100

|F c(q)|

ExperimentRskin=0.176 fmRskin=0.207 fmRskin=0.235 fmRskin=0.260 fmRskin=0.286 fm

Charge Form Factor

208Pb

Charge densities known with enormousprecision R208

ch =5.5012(13) fmStarted with Hofstadter in the late 1950’sand continues to this day in RIBFsProvides our most detailed picture of theatomic nucleus

0.0 0.5 1.0 1.5 2.0q(fm-1)

10-3

10-2

10-1

100

|F w(q)

|

PREXRskin=0.176 fmRskin=0.207 fmRskin=0.235 fmRskin=0.260 fmRskin=0.286 fm

Weak Form Factor

Weak-charge densities as fundamentalas charge densitiesWeak-charge densities are very poorlyknown R208

wk =5.826(181) fmElastic e-scattering largely insensitive tothe weak-charge distributionElastic ν-scattering very sensitive to theweak-charge distribution


Parity Violation in Elastic e-Nucleus Scattering (JLab and Mainz)Charge (proton) densities known with exquisite precision

charge density probed via parity-conserving eA scatteringWeak-charge (neutron) densities very poorly known

weak-charge density probed via parity-violating eA scattering

APV =GF Q2

2√

2πα

1− 4 sin2 θW︸︷︷︸≈0

− Fn(Q2)

Fp(Q2)

Use parity violation as Z0 couples preferentially to neutronsPV provides a clean measurement of neutron densities (R208

n )

up-quark down-quark proton neutronγ-coupling +2/3 −1/3 +1 0Z0-coupling ≈ +1/3 ≈ −2/3 ≈ 0 −1

gv=2tz − 4Q sin2 θW≈2tz−Q


CEvNS: From Dark Matter Searches to Neutron StarsCoherent elastic ν-Nucleus scattering has never been observed!Predicted shortly after the discovery of weak neutral currentsEnormously challenging; must detect exceedingly slow recoilsCEvNS (pronounced “7s” ) are backgrounds for DM searchesCEvNS is coherent (“large”) as it scales ∼N2

“Piggybacking” on the enormous progress in dark-matter searches

Z0A

Coherent Elastic ν-NucleusScattering at the Spallation NeutronSource (ORNL) may becomepossible in the “not-so-distant” future(see Kate Scholberg’s talk)


Neutron Stars: The Role of Nuclear Physics

Chandrasekhar shows that massive stars will collapse (1931)Chadwick discovers the neutron (1932)... predicted earlier by Ettore Majorana but never published!Baade and Zwicky introduce the concept of neutron stars (1933)Oppenheimer-Volkoff compute masses of neutron stars using GR (1939)Predict M?'0.7 M� as maximum NS mass or minimum black hole massDemorest/Antoniadis discover massive neutron stars (2010-2013)Observation of M?'2 M� in compact relativistic binaries

Increase from (0.7→2)M� is all Nuclear Physics!


The Anatomy of a Neutron Star (Figures courtesy of Dany Page and Sanjay Reddy)

Atmosphere (10 cm): Shape of Thermal Radiation (L=4πσR2T 4)Envelope (100 m): Huge Temperature Gradient (108K ↔106K )Outer Crust (400 m): Coulomb crystal of exotic neutron-rich nucleiInner Crust (1 km): Coulomb frustrated “Nuclear Pasta”Outer Core (10 km): Neutron-rich uniform matter (n,p,e, µ)Inner Core (?): Exotic matter (Hyperons, condensates, quark matter, . . .)


Neutron Stars as Nuclear Physics Gold MinesNeutron Stars are the remnants of massive stellar explosions

Are bound by gravity NOT by the strong forceSatisfy the Tolman-Oppenheimer-Volkoff equation (vesc/c∼1/2)

Only Physics sensitive to: Equation of state of neutron-rich matterEOS must span about 11 orders of magnitude in baryon density

Increase from 0.7→2M� must be explained by Nuclear Physics!

common feature of models that include the appearance of ‘exotic’hadronic matter such as hyperons4,5 or kaon condensates3 at densitiesof a few times the nuclear saturation density (ns), for example modelsGS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quarkmatter as a component of the neutron star interior6,21, but it stronglyconstrains quark matter model parameters12. For the range of allowedEOS lines presented in Fig. 3, typical values for the physical parametersof J1614-2230 are a central baryondensity of between 2ns and 5ns and aradius of between 11 and 15 km, which is only 2–3 times theSchwarzschild radius for a 1.97M[ star. It has been proposed thatthe Tolman VII EOS-independent analytic solution of Einstein’sequations marks an upper limit on the ultimate density of observablecold matter22. If this argument is correct, it follows that our mass mea-surement sets an upper limit on this maximum density of(3.746 0.15)3 1015 g cm23, or ,10ns.Evolutionary models resulting in companion masses.0.4M[ gen-

erally predict that the neutron star accretes only a few hundredths of asolar mass of material, and result in a mildly recycled pulsar23, that isone with a spin period.8ms. A few models resulting in orbital para-meters similar to those of J1614-223023,24 predict that the neutron starcould accrete up to 0.2M[, which is still significantly less than the>0.6M[ needed to bring a neutron star formed at 1.4M[ up to theobserved mass of J1614-2230. A possible explanation is that someneutron stars are formed massive (,1.9M[). Alternatively, the trans-fer of mass from the companion may be more efficient than currentmodels predict. This suggests that systems with shorter initial orbitalperiods and lower companion masses—those that produce the vastmajority of the fully recycled millisecond pulsar population23—mayexperience even greater amounts of mass transfer. In either case, ourmass measurement for J1614-2230 suggests that many other milli-second pulsars may also have masses much greater than 1.4M[.

Received 7 July; accepted 1 September 2010.

1. Lattimer, J. M. & Prakash, M. The physics of neutron stars. Science 304, 536–542(2004).

2. Lattimer, J. M. & Prakash, M. Neutron star observations: prognosis for equation ofstate constraints. Phys. Rep. 442, 109–165 (2007).

3. Glendenning, N. K. & Schaffner-Bielich, J. Kaon condensation and dynamicalnucleons in neutron stars. Phys. Rev. Lett. 81, 4564–4567 (1998).

4. Lackey, B. D., Nayyar, M. & Owen, B. J. Observational constraints on hyperons inneutron stars. Phys. Rev. D 73, 024021 (2006).

5. Schulze, H., Polls, A., Ramos, A. & Vidana, I. Maximummass of neutron stars. Phys.Rev. C 73, 058801 (2006).

6. Kurkela, A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D 81,105021 (2010).

7. Shapiro, I. I. Fourth test of general relativity. Phys. Rev. Lett. 13, 789–791 (1964).8. Jacoby, B.A., Hotan, A., Bailes,M., Ord, S. &Kulkarni, S.R. Themassof amillisecond

pulsar. Astrophys. J. 629, L113–L116 (2005).9. Verbiest, J. P. W. et al. Precision timing of PSR J0437–4715: an accurate pulsar

distance, a high pulsarmass, and a limit on the variation of Newton’s gravitationalconstant. Astrophys. J. 679, 675–680 (2008).

10. Hessels, J.et al. inBinaryRadio Pulsars (edsRasio, F. A.&Stairs, I. H.)395 (ASPConf.Ser. 328, Astronomical Society of the Pacific, 2005).

11. Crawford, F. et al. A survey of 56midlatitude EGRET error boxes for radio pulsars.Astrophys. J. 652, 1499–1507 (2006).

12. Ozel, F., Psaltis, D., Ransom, S., Demorest, P. & Alford, M. The massive pulsar PSRJ161422230: linking quantum chromodynamics, gamma-ray bursts, andgravitational wave astronomy. Astrophys. J. (in the press).

13. Hobbs, G. B., Edwards, R. T. & Manchester, R. N. TEMPO2, a new pulsar-timingpackage - I. An overview.Mon. Not. R. Astron. Soc. 369, 655–672 (2006).

14. Damour, T. & Deruelle, N. General relativistic celestial mechanics of binarysystems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincare Phys.Theor. 44, 263–292 (1986).

15. Freire, P.C.C.&Wex,N.Theorthometricparameterisationof theShapirodelay andan improved test of general relativity with binary pulsars.Mon. Not. R. Astron. Soc.(in the press).

16. Iben, I. Jr & Tutukov, A. V. On the evolution of close binaries with components ofinitial mass between 3 solar masses and 12 solar masses. Astrophys. J Suppl. Ser.58, 661–710 (1985).

17. Ozel, F. Soft equations of state for neutron-star matter ruled out by EXO 0748 -676. Nature 441, 1115–1117 (2006).

18. Ransom, S. M. et al. Twenty-one millisecond pulsars in Terzan 5 using the GreenBank Telescope. Science 307, 892–896 (2005).

19. Freire, P. C. C. et al. Eight new millisecond pulsars in NGC 6440 and NGC 6441.Astrophys. J. 675, 670–682 (2008).

20. Freire, P. C. C.,Wolszczan, A., vandenBerg,M.&Hessels, J.W. T. Amassiveneutronstar in the globular cluster M5. Astrophys. J. 679, 1433–1442 (2008).

21. Alford,M.etal.Astrophysics:quarkmatterincompactstars?Nature445,E7–E8(2007).22. Lattimer, J. M. & Prakash, M. Ultimate energy density of observable cold baryonic

matter. Phys. Rev. Lett. 94, 111101 (2005).23. Podsiadlowski, P., Rappaport, S. & Pfahl, E. D. Evolutionary sequences for low- and

intermediate-mass X-ray binaries. Astrophys. J. 565, 1107–1133 (2002).24. Podsiadlowski, P. & Rappaport, S. CygnusX-2: the descendant of an intermediate-

mass X-Ray binary. Astrophys. J. 529, 946–951 (2000).25. Hotan, A.W., van Straten,W. &Manchester, R. N. PSRCHIVE andPSRFITS: an open

approach to radio pulsar data storage and analysis. Publ. Astron. Soc. Aust. 21,302–309 (2004).

26. Cordes, J.M. & Lazio, T. J.W.NE2001.I. A newmodel for theGalactic distribution offree electrons and its fluctuations. Preprint at Æhttp://arxiv.org/abs/astro-ph/0207156æ (2002).

27. Lattimer, J. M. & Prakash, M. Neutron star structure and the equation of state.Astrophys. J. 550, 426–442 (2001).

28. Champion, D. J. et al. An eccentric binary millisecond pulsar in the Galactic plane.Science 320, 1309–1312 (2008).

29. Berti, E., White, F., Maniopoulou, A. & Bruni, M. Rotating neutron stars: an invariantcomparison of approximate and numerical space-time models.Mon. Not. R.Astron. Soc. 358, 923–938 (2005).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio AstronomyObservatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for ScientificResearch.We thankJ. Lattimer for providing theEOSdataplotted inFig. 3, andP. Freire,F. Ozel and D. Psaltis for discussions. The National Radio Astronomy Observatory is afacility of the USNational Science Foundation, operated under cooperative agreementby Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the resultsand edited themanuscript. In addition, P.B.D. developed theMCMCcode, reduced andanalysed data, and wrote the manuscript. T.P. wrote the observing proposal andcreated Fig. 3. J.W.T.H. originally discovered thepulsar.M.S.E.R. initiated the survey thatfound the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to P.B.D. ([email protected]).

0.07 8 9 10 11

Radius (km)12 13 14 15

0.5

1.0

1.5

2.0

AP4

J1903+0327

J1909-3744

systemsDouble neutron sDouble neutron star sysy

J1614-2230

AP3

ENG

MPA1

GM3

GS1

PAL6

FSUSQM3

SQM1

PAL1

MS0

MS2

MS1

2.5 GR

Causality

Rotation

P < ∞

Mass

(M

()

Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotatingmass versus physical radius for several typical EOSs27: blue, nucleons; pink,nucleons plus exoticmatter; green, strange quarkmatter. The horizontal bandsshow the observational constraint from our J1614-2230 mass measurement of(1.976 0.04)M[, similar measurements for two other millisecond pulsars8,28

and the range of observed masses for double neutron star binaries2. Any EOSline that does not intersect the J1614-2230 band is ruled out by thismeasurement. In particular, most EOS curves involving exotic matter, such askaon condensates or hyperons, tend to predict maximum masses well below2.0M[ and are therefore ruled out. Including the effect of neutron star rotationincreases themaximum possiblemass for each EOS. For a 3.15-ms spin period,this is a=2% correction29 and does not significantly alter our conclusions. Thegrey regions show parameter space that is ruled out by other theoretical orobservational constraints2. GR, general relativity; P, spin period.

LETTER RESEARCH

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 3

Macmillan Publishers Limited. All rights reserved©2010

dMdr

= 4πr2E(r)

dPdr

= −GE(r)M(r)

r2

[1 +

P(r)E(r)

][1 +

4πr3P(r)M(r)

] [1− 2GM(r)

r

]−1

Need an EOS: P=P(E) relation

Nuclear Physics Critical


The EOS of neutron-rich matter: Where do the extra neutrons go?

The EOS of asymmetric matter[α≡(N−Z )/A, x≡(ρ−ρ0)/3ρ0

]E(ρ, α) ≈ E0(ρ) + α2S(ρ) ≈

(ε0 +

12

K0x2)+

(J + L x +

12

Ksymx2)α2

In 208Pb, 82 protons/neutrons form an isospin symmetric spherical coreWhere do the extra 44 neutrons go?

Competition between surface tension and density dependence of S(ρ)Surface tension favors placing them in the core where S(ρ0) is largeSymm. energy favors pushing them to the surface where S(ρsurf) is small

If difference S(ρ0)−S(ρsurf)∝L is large, then neutrons move to the surfaceThe larger the value of L the thicker the neutron skin of 208Pb

0 0.2 0.4 0.6 0.8 1.0 1.2ρ/ρ

0

0

10

20

30

40

50

S(ρ)

[MeV

]

Rskin208 =0.12fm

Rskin208 =0.16fm

Rskin208 =0.28fm


Heaven and Earth: Nuclear Physics Informing Neutron StarsMaximum neutron-star mass sensitive to EOS at high density

Best—perhaps unique—available constraint at ρ�ρ0

Accurate mass measurements:M =(1.97±0.04)M�M =(2.01±0.04)M�

parameters, withMCMC error estimates, are given in Table 1. Owing tothe high significance of this detection, our MCMC procedure and astandard x2 fit produce similar uncertainties.From the detected Shapiro delay, we measure a companion mass of

(0.50060.006)M[, which implies that the companion is a helium–carbon–oxygenwhite dwarf16. The Shapiro delay also shows the binary

system to be remarkably edge-on, with an inclination of 89.17u6 0.02u.This is the most inclined pulsar binary system known at present. Theamplitude and sharpness of the Shapiro delay increase rapidly withincreasing binary inclination and the overall scaling of the signal islinearly proportional to the mass of the companion star. Thus, theunique combination of the high orbital inclination and massive whitedwarf companion in J1614-2230 cause a Shapiro delay amplitudeorders of magnitude larger than for most other millisecond pulsars.In addition, the excellent timing precision achievable from the pulsarwith the GBT and GUPPI provide a very high signal-to-noise ratiomeasurement of both Shapiro delay parameters within a single orbit.The standardKeplerian orbital parameters, combinedwith the known

companionmass and orbital inclination, fully describe the dynamics of a‘clean’ binary system—one comprising two stable compact objects—under general relativity and therefore also determine the pulsar’s mass.Wemeasure a pulsar mass of (1.976 0.04)M[, which is by far the high-est preciselymeasured neutron star mass determined to date. In contrastwith X-ray-based mass/radius measurements17, the Shapiro delay pro-videsno informationabout theneutron star’s radius.However, unlike theX-ray methods, our result is nearly model independent, as it dependsonly on general relativity being an adequate description of gravity.In addition, unlike statistical pulsar mass determinations based onmeasurement of the advance of periastron18–20, pure Shapiro delay massmeasurements involve no assumptions about classical contributions toperiastron advance or the distribution of orbital inclinations.The mass measurement alone of a 1.97M[ neutron star signifi-

cantly constrains the nuclear matter equation of state (EOS), as shownin Fig. 3. Any proposed EOS whose mass–radius track does not inter-sect the J1614-2230 mass line is ruled out by this measurement. TheEOSs that produce the lowestmaximummasses tend to be thosewhichpredict significant softening past a certain central density. This is a

a

b

c

–40

–30

–20

–10

0

10

20

30

–40

–30

–20

–10

0

10

20

30

Tim

ing

resi

dual

(μs)

–40

–30

–20

–10

0

10

20

30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Orbital phase (turns)

Figure 1 | Shapiro delay measurement for PSRJ1614-2230. Timing residual—the excess delaynot accounted for by the timing model—as afunction of the pulsar’s orbital phase. a, Fullmagnitude of the Shapiro delay when all othermodel parameters are fixed at their best-fit values.The solid line shows the functional form of theShapiro delay, and the red points are the 1,752timingmeasurements in ourGBT–GUPPI data set.The diagrams inset in this panel show top-downschematics of the binary system at orbital phases of0.25, 0.5 and 0.75 turns (from left to right). Theneutron star is shown in red, the white dwarfcompanion in blue and the emitted radio beam,pointing towards Earth, in yellow. At orbital phaseof 0.25 turns, the Earth–pulsar line of sight passesnearest to the companion (,240,000 km),producing the sharp peak in pulse delay.We foundno evidence for any kind of pulse intensityvariations, as from an eclipse, near conjunction.b, Best-fit residuals obtained using an orbitalmodelthat does not account for general-relativistic effects.In this case, some of the Shapiro delay signal isabsorbed by covariant non-relativistic modelparameters. That these residuals deviatesignificantly from a random, Gaussian distributionof zero mean shows that the Shapiro delay must beincluded to model the pulse arrival times properly,especially at conjunction. In addition to the redGBT–GUPPI points, the 454 grey points show theprevious ‘long-term’ data set. The drasticimprovement in data quality is apparent. c, Post-fitresiduals for the fully relativistic timing model(including Shapiro delay), which have a root meansquared residual of 1.1ms and a reduced x2 value of1.4 with 2,165 degrees of freedom. Error bars, 1s.

89.1

89.12

89.14

89.16

89.18

89.2

89.22

89.24a b

0.48 0.49 0.5 0.51 0.52

Incl

inat

ion

angl

e, i

(°)

Companion mass, M2 (M()1.8 1.85 1.9 1.95 2 2.05 2.1 2.15

Pro

babi

lity

dens

ity

Pulsar mass (M()

Figure 2 | Results of theMCMCerror analysis. a, Grey-scale image shows thetwo-dimensional posterior probability density function (PDF) in theM2–iplane, computed from a histogram ofMCMC trial values. The ellipses show 1sand 3s contours based on a Gaussian approximation to the MCMC results.b, PDF for pulsar mass derived from the MCMC trials. The vertical lines showthe 1s and 3s limits on the pulsar mass. In both cases, the results are very welldescribed by normal distributions owing to the extremely high signal-to-noiseratio of our Shapiro delay detection. Unlike secular orbital effects (for exampleprecession of periastron), the Shapiro delay does not accumulate over time, sothe measurement uncertainty scales simply as T21/2, where T is the totalobserving time. Therefore, we are unlikely to see a significant improvement onthese results with currently available telescopes and instrumentation.

RESEARCH LETTER

1 0 8 2 | N A T U R E | V O L 4 6 7 | 2 8 O C T O B E R 2 0 1 0


Instead, stellar radii sensitive to EOS at intermediate densitiesNot possible to adjust EOS at ρ&2ρ0 without affecting laboratory observablesUnique synergy between laboratory experiments and astronomical observations

Neutron-star radii sensitive to one fundamental parameter of the EOSThe slope of the symmetry energy at saturation density L∝PPNM

Same pressure creates neutron-rich skin and NStar radiusCorrelation among observables differing by 18 orders of magnitude!

common feature of models that include the appearance of ‘exotic’hadronic matter such as hyperons4,5 or kaon condensates3 at densitiesof a few times the nuclear saturation density (ns), for example modelsGS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by ourresults. Our mass measurement does not rule out condensed quarkmatter as a component of the neutron star interior6,21, but it stronglyconstrains quark matter model parameters12. For the range of allowedEOS lines presented in Fig. 3, typical values for the physical parametersof J1614-2230 are a central baryondensity of between 2ns and 5ns and aradius of between 11 and 15 km, which is only 2–3 times theSchwarzschild radius for a 1.97M[ star. It has been proposed thatthe Tolman VII EOS-independent analytic solution of Einstein’sequations marks an upper limit on the ultimate density of observablecold matter22. If this argument is correct, it follows that our mass mea-surement sets an upper limit on this maximum density of(3.746 0.15)3 1015 g cm23, or ,10ns.Evolutionary models resulting in companion masses.0.4M[ gen-

erally predict that the neutron star accretes only a few hundredths of asolar mass of material, and result in a mildly recycled pulsar23, that isone with a spin period.8ms. A few models resulting in orbital para-meters similar to those of J1614-223023,24 predict that the neutron starcould accrete up to 0.2M[, which is still significantly less than the>0.6M[ needed to bring a neutron star formed at 1.4M[ up to theobserved mass of J1614-2230. A possible explanation is that someneutron stars are formed massive (,1.9M[). Alternatively, the trans-fer of mass from the companion may be more efficient than currentmodels predict. This suggests that systems with shorter initial orbitalperiods and lower companion masses—those that produce the vastmajority of the fully recycled millisecond pulsar population23—mayexperience even greater amounts of mass transfer. In either case, ourmass measurement for J1614-2230 suggests that many other milli-second pulsars may also have masses much greater than 1.4M[.

Received 7 July; accepted 1 September 2010.

1. Lattimer, J. M. & Prakash, M. The physics of neutron stars. Science 304, 536–542(2004).

2. Lattimer, J. M. & Prakash, M. Neutron star observations: prognosis for equation ofstate constraints. Phys. Rep. 442, 109–165 (2007).

3. Glendenning, N. K. & Schaffner-Bielich, J. Kaon condensation and dynamicalnucleons in neutron stars. Phys. Rev. Lett. 81, 4564–4567 (1998).

4. Lackey, B. D., Nayyar, M. & Owen, B. J. Observational constraints on hyperons inneutron stars. Phys. Rev. D 73, 024021 (2006).

5. Schulze, H., Polls, A., Ramos, A. & Vidana, I. Maximummass of neutron stars. Phys.Rev. C 73, 058801 (2006).

6. Kurkela, A., Romatschke, P. & Vuorinen, A. Cold quark matter. Phys. Rev. D 81,105021 (2010).

7. Shapiro, I. I. Fourth test of general relativity. Phys. Rev. Lett. 13, 789–791 (1964).8. Jacoby, B.A., Hotan, A., Bailes,M., Ord, S. &Kulkarni, S.R. Themassof amillisecond

pulsar. Astrophys. J. 629, L113–L116 (2005).9. Verbiest, J. P. W. et al. Precision timing of PSR J0437–4715: an accurate pulsar

distance, a high pulsarmass, and a limit on the variation of Newton’s gravitationalconstant. Astrophys. J. 679, 675–680 (2008).

10. Hessels, J.et al. inBinaryRadio Pulsars (edsRasio, F. A.&Stairs, I. H.)395 (ASPConf.Ser. 328, Astronomical Society of the Pacific, 2005).

11. Crawford, F. et al. A survey of 56midlatitude EGRET error boxes for radio pulsars.Astrophys. J. 652, 1499–1507 (2006).

12. Ozel, F., Psaltis, D., Ransom, S., Demorest, P. & Alford, M. The massive pulsar PSRJ161422230: linking quantum chromodynamics, gamma-ray bursts, andgravitational wave astronomy. Astrophys. J. (in the press).

13. Hobbs, G. B., Edwards, R. T. & Manchester, R. N. TEMPO2, a new pulsar-timingpackage - I. An overview.Mon. Not. R. Astron. Soc. 369, 655–672 (2006).

14. Damour, T. & Deruelle, N. General relativistic celestial mechanics of binarysystems. II. The post-Newtonian timing formula. Ann. Inst. Henri Poincare Phys.Theor. 44, 263–292 (1986).

15. Freire, P.C.C.&Wex,N.Theorthometricparameterisationof theShapirodelay andan improved test of general relativity with binary pulsars.Mon. Not. R. Astron. Soc.(in the press).

16. Iben, I. Jr & Tutukov, A. V. On the evolution of close binaries with components ofinitial mass between 3 solar masses and 12 solar masses. Astrophys. J Suppl. Ser.58, 661–710 (1985).

17. Ozel, F. Soft equations of state for neutron-star matter ruled out by EXO 0748 -676. Nature 441, 1115–1117 (2006).

18. Ransom, S. M. et al. Twenty-one millisecond pulsars in Terzan 5 using the GreenBank Telescope. Science 307, 892–896 (2005).

19. Freire, P. C. C. et al. Eight new millisecond pulsars in NGC 6440 and NGC 6441.Astrophys. J. 675, 670–682 (2008).

20. Freire, P. C. C.,Wolszczan, A., vandenBerg,M.&Hessels, J.W. T. Amassiveneutronstar in the globular cluster M5. Astrophys. J. 679, 1433–1442 (2008).

21. Alford,M.etal.Astrophysics:quarkmatterincompactstars?Nature445,E7–E8(2007).22. Lattimer, J. M. & Prakash, M. Ultimate energy density of observable cold baryonic

matter. Phys. Rev. Lett. 94, 111101 (2005).23. Podsiadlowski, P., Rappaport, S. & Pfahl, E. D. Evolutionary sequences for low- and

intermediate-mass X-ray binaries. Astrophys. J. 565, 1107–1133 (2002).24. Podsiadlowski, P. & Rappaport, S. CygnusX-2: the descendant of an intermediate-

mass X-Ray binary. Astrophys. J. 529, 946–951 (2000).25. Hotan, A.W., van Straten,W. &Manchester, R. N. PSRCHIVE andPSRFITS: an open

approach to radio pulsar data storage and analysis. Publ. Astron. Soc. Aust. 21,302–309 (2004).

26. Cordes, J.M. & Lazio, T. J.W.NE2001.I. A newmodel for theGalactic distribution offree electrons and its fluctuations. Preprint at Æhttp://arxiv.org/abs/astro-ph/0207156æ (2002).

27. Lattimer, J. M. & Prakash, M. Neutron star structure and the equation of state.Astrophys. J. 550, 426–442 (2001).

28. Champion, D. J. et al. An eccentric binary millisecond pulsar in the Galactic plane.Science 320, 1309–1312 (2008).

29. Berti, E., White, F., Maniopoulou, A. & Bruni, M. Rotating neutron stars: an invariantcomparison of approximate and numerical space-time models.Mon. Not. R.Astron. Soc. 358, 923–938 (2005).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio AstronomyObservatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for ScientificResearch.We thankJ. Lattimer for providing theEOSdataplotted inFig. 3, andP. Freire,F. Ozel and D. Psaltis for discussions. The National Radio Astronomy Observatory is afacility of the USNational Science Foundation, operated under cooperative agreementby Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the resultsand edited themanuscript. In addition, P.B.D. developed theMCMCcode, reduced andanalysed data, and wrote the manuscript. T.P. wrote the observing proposal andcreated Fig. 3. J.W.T.H. originally discovered thepulsar.M.S.E.R. initiated the survey thatfound the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to P.B.D. ([email protected]).

0.07 8 9 10 11

Radius (km)12 13 14 15

0.5

1.0

1.5

2.0

AP4

J1903+0327

J1909-3744

systemsDouble neutron sDouble neutron star sysy

J1614-2230

AP3

ENG

MPA1

GM3

GS1

PAL6

FSUSQM3

SQM1

PAL1

MS0

MS2

MS1

2.5 GR

Causality

Rotation

P < ∞

Mas

s (M

()

Figure 3 | Neutron star mass–radius diagram. The plot shows non-rotatingmass versus physical radius for several typical EOSs27: blue, nucleons; pink,nucleons plus exoticmatter; green, strange quarkmatter. The horizontal bandsshow the observational constraint from our J1614-2230 mass measurement of(1.976 0.04)M[, similar measurements for two other millisecond pulsars8,28

and the range of observed masses for double neutron star binaries2. Any EOSline that does not intersect the J1614-2230 band is ruled out by thismeasurement. In particular, most EOS curves involving exotic matter, such askaon condensates or hyperons, tend to predict maximum masses well below2.0M[ and are therefore ruled out. Including the effect of neutron star rotationincreases themaximum possiblemass for each EOS. For a 3.15-ms spin period,this is a=2% correction29 and does not significantly alter our conclusions. Thegrey regions show parameter space that is ruled out by other theoretical orobservational constraints2. GR, general relativity; P, spin period.

LETTER RESEARCH

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 3



The Enormous Reach of the Neutron Skin: Covariance Analysis

Neutron skin as proxy for neutron-star radii . . . and more!Calibration of nuclear functional from optimization of a quality measureNew era: predictability typical – uncertainty quantification demandedNeutron skin strongly correlated to a myriad of neutron star properties:

Radii, Enhanced Cooling, Moment of Inertia, . . .

40 50 60 70 80L (MeV)

12

12.5

13

13.5

14

14.5

R NS [

M/M

sun]

(km

) RNS[0.8]RNS[1.4]

CAB=0.988

FSUGold

CAB=0.946

0 0.2 0.4 0.6 0.8 1Correlation with skin of 208Pb

skin 132Sn

RNS[0.8]RNS[1.4]

skin 48Ca

lt

skin 208Pb

L

Ypt

MDUrca

Icrust[0.8]St

ruct

ure

Pasta

Cooling

Glitches


PREX: The Lead Radius EXperiment Abrahamyan et al., PRL 108, (2012) 112502

Ran for 2 months: April-June 2010First electroweak observation of a neutron-rich skin in 208PbPromised a 0.06 fm measurement of R208

n ; error 3 times as large!

0 0.4 0.8 1.2 1.6q(fm-1)

0

0.2

0.4

0.6

0.8

1

F w(q)

SV-minFSUGoldPREX

qCREX

48Ca

208Pb

6Fw×10

Fw=0

“One of the main science drivers of FRIB is the study ofnuclei with neutron skins 3-4 times thicker than iscurrently possible ... Studies of neutron skins at JLab andFRIB will help pin down the behavior of nuclear matter atdensities below twice typical nuclear density”

A Physics case for PREX-II, CREX, and ...Coherent ν-nucleus scattering


“Heaven and Earth” Guillot et al., ApJ, 772:7 (2013)

Same pressure creates neutron skin and NS radiusCorrelation among observables differing by 18 orders of magnitude!

“Using conservative assumptions, we found: RNS =9.1+1.3−1.4km”

... theory of dense nuclear matter may need to be revisited

Very difficult to reconcile small stellar radii with large R 208skin

May be evidence of a softening due to phase transition (quark matter?)

Very difficult to reconcile small stellar radii with large limiting massEOS must stiffen again to account for large neutron-star massesThe Neutron Star Radius

9.1+1.3�1.4 km

(90%conf.)

Guillot et al (2013)

<11 km (99% conf).

M-R by J. Lattimer

WFF1=Wiring, Fiks

and Fabrocini (1988)

Contains uncertainties from:

DistanceAll spectral parametersCalibration

0.15 0.20 0.25 0.30 0.35 0.40Rskin

208 (fm)

8

9

10

11

12

13

14

15

R 1.4

(km

)phase transition?

FSUGold

NL3Models

qLM

XB

PREX-II

Tension between theory/experiment/observationJ. Piekarewicz (FSU) Neutron-Star Matter Equation of State Neutrino Astrophysics and ... 13 / 15

Addressing the Tension ... W.-C. Chen and JP (arXiv:1505.07436)

Guillot et al., assumes all neutron stars have a common radius!Assumption on observable MR rather than on EOS

One-to-one correspondence between MR and EOSTOV equation + EOS→ MR

“Lindblom’s inversion algorithm” proves the inverse [APJ 398, 569 (1992)]

TOV equation + MR→ EOSTension in reconciling NS with large masses and small radii

Is the resulting EOS causal or superluminal?Stellar radius of 1.4 M� must exceed 10.7 km!

Astrophysical observations are imposing similar upper limits!

6 8 10 12 14 16 18 20R(km)

0

0.5

1

1.4

2

2.5

3

M/M

sun

8 9 10 11 13

R8R9R10R11AV14+UVIIFSUGoldFSUGold2FSUGarnetNL3

this work

L&PBH

Causal

ity

0 250 500 750 1000 1250 1500 1750ε(MeV/fm3)

0

250

500

750

1000

1250

1500

1750

P(MeV/fm

3 )

R8R9R10R11AV14+UVIINL3Superluminal

P=ε


Conclusions and Outlook: The Physics of Neutron StarsAstrophysics: What is the minimum mass of a black hole?Atomic Physics: Pure neutron matter as a Unitary Fermi GasCondensed-Matter Physics: Signatures for the liquid to crystalline transition?General Relativity: Rapidly rotating neutrons stars as a source of gravitational waves?Nuclear Physics: What are the limits of nuclear existence and the EOS of nuclear matter?Particle Physics: QCD made simple — the CFL phase of dense quark matter

22 AUGUST 2000 PHYSICS TODAY © 2000 American Institute of Physics, S-0031-9228-0008-010-8

Quantum chromodynamics,familiarly called QCD, is

the modern theory of thestrong interaction.1 Historic-ally its roots are in nuclearphysics and the description ofordinary matter—understand-ing what protons and neu-trons are and how they inter-act. Nowadays QCD is used todescribe most of what goes on at high-energy accelerators.

Twenty or even fifteen years ago, this activity wascommonly called “testing QCD.” Such is the success of thetheory, that we now speak instead of “calculating QCDbackgrounds” for the investigation of more speculativephenomena. For example, discovery of the heavy W and Zbosons that mediate the weak interaction, or of the topquark, would have been a much more difficult and uncer-tain affair if one did not have a precise, reliable under-standing of the more common processes governed byQCD. With regard to things still to be found, searchstrategies for the Higgs particle and for manifestations ofsupersymmetry depend on detailed understanding of pro-duction mechanisms and backgrounds calculated bymeans of QCD.

Quantum chromodynamics is a precise and beautifultheory. One reflection of this elegance is that the essenceof QCD can be portrayed, without severe distortion, in thefew simple pictures at the bottom of the box on the nextpage. But first, for comparison, let me remind you that theessence of quantum electrodynamics (QED), which is ageneration older than QCD, can be portrayed by the sin-gle picture at the top of the box, which represents theinteraction vertex at which a photon responds to the pres-ence or motion of electric charge.2 This is not just ametaphor. Quite definite and precise algorithms for calcu-lating physical processes are attached to the Feynmangraphs of QED, constructed by connecting just such inter-action vertices.

In the same pictorial language, QCD appears as anexpanded version of QED. Whereas in QED there is justone kind of charge, QCD has three different kinds ofcharge, labeled by “color.” Avoiding chauvinism, we mightchoose red, green, and blue. But, of course, the colorcharges of QCD have nothing to do with physical colors.Rather, they have properties analogous to electric charge.In particular, the color charges are conserved in all phys-ical processes, and there are photon-like massless parti-cles, called color gluons, that respond in appropriate ways

to the presence or motion ofcolor charge, very similar tothe way photons respond toelectric charge.

Quarks and gluonsOne class of particles thatcarry color charge are thequarks. We know of six differ-ent kinds, or “flavors,” of

quarks—denoted u, d, s, c, b, and t, for: up, down,strange, charmed, bottom, and top. Of these, only u and dquarks play a significant role in the structure of ordinarymatter. The other, much heavier quarks are all unstable.A quark of any one of the six flavors can also carry a unitof any of the three color charges. Although the differentquark flavors all have different masses, the theory is per-fectly symmetrical with respect to the three colors. Thiscolor symmetry is described by the Lie group SU(3).

Quarks are spin-1/2 point particles, very much likeelectrons. But instead of electric charge, they carry colorcharge. To be more precise, quarks carry fractional elec-tric charge (+ 2e/3 for the u, c, and t quarks, and – e/3 forthe d, s, and b quarks) in addition to their color charge.

For all their similarities, however, there are a fewcrucial differences between QCD and QED. First of all,the response of gluons to color charge, as measured by theQCD coupling constant, is much more vigorous than theresponse of photons to electric charge. Second, as shownin the box, in addition to just responding to color charge,gluons can also change one color charge into another. Allpossible changes of this kind are allowed, and yet colorcharge is conserved. So the gluons themselves must beable to carry unbalanced color charges. For example, ifabsorption of a gluon changes a blue quark into a redquark, then the gluon itself must have carried one unit ofred charge and minus one unit of blue charge.

All this would seem to require 3 ! 3 = 9 differentcolor gluons. But one particular combination of gluons—the color-SU(3) singlet—which responds equally to allcharges, is different from the rest. We must remove it ifwe are to have a perfectly color-symmetric theory. Thenwe are left with only 8 physical gluon states (forming acolor-SU(3) octet). Fortunately, this conclusion is vindicat-ed by experiment!

The third difference between QCD and QED, which isthe most profound, follows from the second. Because glu-ons respond to the presence and motion of color chargeand they carry unbalanced color charge, it follows thatgluons, quite unlike photons, respond directly to oneanother. Photons, of course, are electrically neutral.Therefore the laser sword fights you’ve seen in Star Warswouldn’t work. But it’s a movie about the future, so maybethey’re using color gluon lasers.

We can display QCD even more compactly, in terms of

FRANKWILCZEK is the J. Robert Oppenheimer Professor of Physics atthe Institute for Advanced Study in Princeton, New Jersey. Next monthhe moves to Cambridge, Massachusetts, to take up the Herman FeshbachChair of Physics at the Massachusetts Institute of Technology.

QCD MADE SIMPLEQuantum chromodynamics is

conceptually simple. Its realizationin nature, however, is usuallyvery complex. But not always.

Frank Wilczek

Neutron Stars are the natural meeting place forfundamental and interesting Physics


and now for something completely different probing the ... · and now for something completely...

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