Neutron stars and the properties of matter at high density Gordon Baym, University of Illinois Future Prospects of Hadron Physics at J-PARC and Large Scale Computational Physics’ 11 February 2012

Mass ~ M sun Radius ~ km Temperature ~ K Surface gravity ~10 14 that of Earth Surface binding ~ 1/10 mc 2 Density ~ 2x10 14 g/cm 3 Neutron star interior Mountains < 1 mm

Masses ~ 1-2 M  Baryon number ~ Radii ~ km Magnetic fields ~ G Made in gravitational collapse of massive stars (supernovae) Central element in variety of compact energetic systems: pulsars, binary x-ray sources, soft gamma repeaters Merging neutron star-neutron star and neutron star-black hole sources of gamma ray bursts Matter in neutron stars is densest in universe:  up to ~ 5-10  0 (  0 = 3 X g/cm 3 = density of matter in atomic nuclei) [cf. white dwarfs:  ~ g/cm 3 ] Supported against gravitational collapse by nucleon degeneracy pressure Astrophysical laboratory for study of high density matter complementary to accelerator experiments What are states in interior? Onset of quark degrees of freedom! Do quark stars, as well as strange stars exist?

The liquid interior Neutrons (likely superfluid) ~ 95% Non-relativistic Protons (likely superconducting) ~ 5% Non-relativistic Electrons (normal, T c ~ T f e -137 ) ~ 5% Fully relativistic Eventually muons, hyperons, and possibly exotica: pion condensation kaon condensation quark droplets bulk quark matter Phase transition from crust to liquid at n b 0.7 n fm -3 or  = mass density ~ 2 X g/cm 3 n 0 = baryon density in large nuclei 0.16 fm -3 1fm = cm

Properties of liquid interior near nuclear matter density Determine N-N potentials from - scattering experiments E<300 MeV - deuteron, 3 body nuclei ( 3 He, 3 H) ex., Paris, Argonne, Urbana 2 body potentials Solve Schrödinger equation by variational techniques Two body potential alone: Underbind 3 H: Exp = MeV, Theory = -7.5 MeV 4 He: Exp = MeV, Theory = MeV Large theoretical extrapolation from low energy laboratory nuclear physics at near nuclear matter density

Importance of 3 body interactions Attractive at low density Repulsive at high density Stiffens equation of state at high density Large uncertainties Various processes that lead to three and higher body intrinsic interactions (not described by iterated nucleon-nucleon interactions).

Three body forces in polarized pd and dp scattering 135 MeV/A (RIKEN) (K. Sekiguchi 2007) Blue = 2 body forces Red = 2+3 body forces

 0 condensate Energy per nucleon in pure neutron matter Akmal, Pandharipande and Ravenhall, Phys. Rev. C58 (1998) 1804

 0 condensate Energy per nucleon in pure neutron matter Morales, (Pandharipande) & Ravenhall, in progress AV-18 + UIV 3-body (IL 3-body too attractive) Improved FHNC algorithms. Two minima! E/A slightly higher than Akmal, Pandharipande and Ravenhall, Phys. Rev. C58 (1998) 1804

Akmal, Pandharipande and Ravenhall, 1998 Mass vs. central density Mass vs. radius Maximum neutron star mass

Equation of state vs. neutron star structure from J. Lattimer

Accurate for n ~ n 0. n >> n 0 : -can forces be described with static few-body potentials? -Force range ~ 1/2m  => relative importance of 3 (and higher) body forces ~ n/(2m  ) 3 ~ 0.4n fm-3. -No well defined expansion in terms of 2,3,4,...body forces. -Can one even describe system in terms of well-defined ``asymptotic'' laboratory particles? Early percolation of nucleonic volumes! Fundamental limitations of equation of state based on nucleon-nucleon interactions alone:

Fukushima & Hatsuda, Rep. Prog. Phys. 74 (2011)

Hatsuda, Tachibana, Yamamoto & GB, PRL 97, (2006) Yamamoto, Hatsuda, Tachibana & GB, PRD76, (2007) GB, Hatsuda, Tachibana, & Yamamoto. J. Phys. G: Nucl. Part. 35 (2008) Abuki, GB, Hatsuda, & Yamamoto,Phys. Rev. D81, (2010) New critical point in phase diagram : induced by chiral condensate – diquark pairing coupling via axial anomaly Hadronic Normal QGP Color SC (as m s increases)

BEC-BCS crossover in QCD phase diagram J. Phys.G: Nucl. Part. Phys. 35 (2008) Normal Color SC (as m s increases) BCS paired quark matter BCS-BEC crossover Hadrons Hadronic Small quark pairs are “diquarks”

Model calculations of phase diagram with axial anomaly, pairing, chiral symmetry breaking & confinement NJL alone: H. Abuki, GB, T. Hatsuda, & N. Yamamoto, PR D81, (2010). NPL with Polyakov loop description of confinement: P. Powell & GB, arXiv: Couple quark fields together with effective 4 and 6 quark interactions: At mean field level, effective couplings of chiral field φ and pairing field d: K and K’ from axial anomaly PNJL phase diagram

Spatially ordered chiral transition = quarkyonic phase Kojo, Hidaka, Fukushima, McLerran, & Pisarski,arXiv: Structure deduced in limit of large number of colors, N c

Well beyond nuclear matter density Hyperons: , ,... Meson condensates:  -,  0, K - Quark matter in droplets in bulk Color superconductivity Strange quark matter absolute ground state of matter?? strange quark stars? Onset of new degrees of freedom: mesonic,  ’s, quarks and gluons,... Properties of matter in this extreme regime determine maximum neutron star mass. Large uncertainties!

Hyperons in dense matter Produce hyperon X of baryon no. A and charge eQ when A  n - Q  e > m X (plus interaction corrections) Djapo, Schaefer & Wambach, PhysRev C81, (2010)

Pion condensed matter Softening of collective spin-isospin oscillation of nuclear matter Above critical density have transition to new state with nucleons rotated in isospin space: with formation of macroscopic pion field

Strangeness (kaon) condensates Analogous to  condensate Chiral SU(3) X SU(3) symmetry of strong interactions => effective low energy interaction Kaplan and Nelson (1986), Brown et al. (1994) “Effective mass” term lowers K energies in matter => condensation =>

Lattice gauge theory calculations of equation of state of QGP Not useful yet for realistic chemical potentials

Learning about dense matter from neutron star observations

Masses of neutron stars Binary systems: stiff eos Thermonuclear bursts in X-ray binaries => Mass vs. Radius, strongly constrains eos Glitches: probe n,p superfluidity and crust Cooling of n-stars: search for exotica Learning about dense matter from neutron star observations

Dense matter from neutron star mass determinations Softer equation of state => lower maximum mass and higher central density Binary neutron stars ~ 1.4 M  consistent with soft e.o.s. Cyg X-2: M=1.78 ± 0.23 M  Vela X-1: M=1.86 ± 0.33M  allow some softening PSR J M=1.97 ± 0.04M  allows no softening; begins to challenge microscopic e.o.s.

Neutron Star Masses ca. 2007

Vela X-1 (LMXB) light curves Serious deviation from Keplerian radial velocity Excitation of (supergiant) companion atmosphere? 1.4 M  M=1.86 ± 0.33 M  ¯ M. H. van Kerkwijk, astro-ph/ M  <M<2.4 M  Quaintrell et al., A&A 401, 313 (2003)

Highest mass neutron star, PSR J in neutron star-white dwarf binary Spin period = 3.15 ms; orbital period = 8.7 day Inclination = 89:17 o ± 0:02 o : edge on M neutron star =1.97 ± 0.04M  ; M white dwarf = ±006M  (Gravitational) Shapiro delay of light from pulsar when passing the companion white dwarf Demorest et al., Nature 467, 1081 (2010); Ozel et al., ApJ 724, L199 (2010.

Highest mass neutron star M= 1.97±0.04 M 

Akmal, Pandharipande and Ravenhall, 1998

Can M max be larger? Larger M max requires larger sound speed c s at lower n. For nucleonic equation of state, c s -> c at n ~ 7n 0. Further degrees of freedom, e.g., hyperons, mesons, or quarks at n ~ 7n 0 lower E/A => matter less stiff. Stiffer e.o.s. at lower n => larger M max. If e.o.s. very stiff beyond n = 2n 0, M max can be as large as 2.9 M . Stiffer e.o.s. => larger radii.

Measuring masses and radii of neutron stars in thermonuclear bursts in X-ray binaries Time (s) Measurements of apparent surface area, & flux at Eddington limit (radiation pressure = gravity), combined with distance to star constrains M and R. Ozel et al., l2 Apparent Radius Eddington Luminosity

EXO in globular cluster Terzan 5, D = 6.3  0.6 kpc (HST ) Mass vs. radius determination of neutron stars in burst sources Ozel et al., ApJ U in NGC U in globular cluster NGC 6624, D = kpc KS Galactic bulge source

M vs R from bursts, Ozel at al, Steiner et al.

Ozel, GB, & Guver Steiner, Lattimer, & Brown PRD 82 (2010) Ap. J. 722 (2010) - little stiffer Results ~ consistent with each other and with maximum mass ~ 2 M  Pressure vs. mass density of cold dense matter inferred from neutron star observations Fit equation of state with polytropes above

Quark matter cores in neutron stars Canonical picture: compare calculations of eqs. of state of hadronic matter and quark matter. Crossing of thermodynamic potentials => first order phase transition. Typically conclude transition at  ~ 10  nm -- would not be reached in neutron stars given observation of high mass PSR J with M = 1.97M  => no quark matter cores ex. nuclear matter using 2 & 3 body interactions, vs. pert. expansion or bag models. Akmal, Pandharipande, Ravenhall 1998

More realistically, expect gradual onset of quark degrees of freedom in dense matter Hadronic Normal Color SC n perc ~ 0.34 (3/4  r n 3 ) fm -3 Quarks can still be bound even if deconfined. Calculation of equation of state remains a challenge for theorists New critical point suggests transition to quark matter is a crossover at low T Consistent with percolation picture that as nucleons begin to overlap, quarks percolate [GB, Physics 1979]

More realistically, expect gradual onset of quark degrees of freedom in dense matter Hadronic Normal Color SC n perc ~ 0.34 (3/4  r n 3 ) fm -3 Quarks can still be bound even if deconfined. Calculation of equation of state remains a challenge for theorists New critical point suggests transition to quark matter is a crossover at low T Consistent with percolation picture that as nucleons begin to overlap, quarks percolate [GB, Physics 1979]

Present observations of high mass neutron stars M ~ 2M begin to confront microscopic nuclear physics. High mass neutron stars => very stiff equation of state, with n c < 7n 0. At this point for nucleonic equation of state, sound speed c s = (  P/  ) 1/2  c. Naive theoretical predictions based on sharp deconfinement transition would be inconsistent with presence of (soft) bulk quark matter in neutron stars. Further degrees of freedom, e.g., hyperons, mesons, or quarks at n matter less stiff. Quark cores would require very stiff quark matter. Expect gradual onset of quark degrees of freedom.

どうもありがとう

Nuclei before neutron drip e - +p n +  makes nuclei neutron rich as electron Fermi energy increases with depth n p+ e - + : not allowed if e - state already occupied _ Beta equilibrium:  n =  p +  e Shell structure (spin-orbit forces) for very neutron rich nuclei? Do N=50, 82 remain magic numbers? To be explored at rare isotope accelerators, RIKEN, GSI, FRIB, KORIA

Valley of  stability in neutron stars neutron drip line

No shell effect for Mg(Z=12), Si(14), S(16), Ar(18) at N=20 and 28 Loss of shell structure for N >> Z even

Neutron Star Models = mass within radius r E = energy density =  c 2 n b = baryon density P(  ) = pressure = n b 2  (E/n b )/  n b Equation of state: Tolman-Oppenheimer-Volkoff equation of hydrostatic balance: general relativistic corrections 1) Choose central density:  (r=0) =  c 2) Integrate outwards until P=0 (at radius R) 3) Mass of star

Outside material adds ~ 0.1 M ¯ Maximum mass of a neutron star Say that we believe equation of state up to mass density   but e.o.s. is uncertain beyond Weak bound: a) core not black hole => 2M c G/c 2 < R c b) M c = s 0 R c d 3 r  (r)  (4  /3)  0 R c 3 => c 2 R c /2G  M c  (4  /3)  0 R c 3 M c max = (3M ¯ /4  0 R s ¯ 3 ) 1/2 M ¯  R c ) =  0 M max  13.7 M ¯ £ (10 14 g/cm 3 /  0 ) 1/2 R s ¯ =2M ¯ G/c 2 = 2.94 km 4  0 R c 3 /3

Strong bound: require speed of sound, c s, in matter in core not to exceed speed of light: Maximum core mass when c s = c Rhodes and Ruffini (PRL 1974) c s 2 =  P/   c 2 WFF (1988) eq. of state => M max = 6.7M ¯ (10 14 g/cm 3 /  0 ) 1/2 V. Kalogera and G.B., Ap. J. 469 (1996) L61  0 = 4  nm => M max = 2.2 M ¯ 2  nm => 2.9 M ¯

Neutron drip Beyond density  drip ~ 4.3 X g/cm 3 neutron bound states in nuclei become filled. Further neutrons must go into continuum states. Form degenerate neutron Fermi sea. Neutrons in neutron sea are in equilibrium with those inside nucleus Protons never drip, but remain in bound states until nuclei merge into interior liquid. Free neutrons form 1 S 0 BCS paired superfluid

J. Negele and D. Vautherin, Nucl. Phys. A207 (1973) 298 neutron drip n p p Hartree-Fock nuclear density profiles

Energy per nucleon vs. baryon density in symmetric nuclear matter Akmal, Pandharipande and Ravenhall, Phys. Rev. C58 (1998) 1804

Phase diagram of equilibrated quark gluon plasma Karsch & Laermann, 2003 Critical point Asakawa-Yazaki st order crossover