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Neutron stars and quark matter
Gordon Baym
University of Illinois, Urbana
21st Century COE Workshop:
Strongly Correlated Many-Body Systems
from Neutron Stars to Cold Atoms
19 January 2006
東京大学
Cross section of a neutron star
Mass ~ 1.4 Msun
Radius ~ 10-12 km
Temperature
~ 106-109 K
Surface gravity
~1014 that of Earth
Surface binding
~ 1/10 mc2
Mountains < 1 mm
Density ~ 2x1014g/cm3
Properties of matter near nuclear matter density
Determine N-N potentials from
- scattering experiments E<300 MeV
- deuteron, 3 body nuclei (3He, 3H)
ex., Paris, Argonne, Urbana 2 body potentials
Solve Schrödinger equation by variational techniques
3
Two body potential alone:
Underbind 3H: Exp = -8.48 MeV, Theory = -7.5 MeV
4He:
Exp = -28.3 MeV, Theory = -24.5 MeV
Importance of 3 body interactions
Attractive at low density
Repulsive at high density
Various processes
that lead to three
and higher body
intrinsic interactions
(not described by
iterated nucleon-nucleon
interactions).
Stiffens equation of state at high density
Large uncertainties
Energy per nucleon in pure neutron matter
Akmal, Pandharipande and Ravenhall, Phys. Rev. C58 (1998) 1804
h p0 i
condensate
Maximum neutron star mass
2.2M¯
Mass vs. central density
Mass vs. radius
Akmal, Pandharipande and Ravenhall, 1998
Fundamental limitations of equation of state based on
nucleon interactions alone:
Accurate for n» n0.
n À n 0:
-can forces be described with static few-body potentials?
-force range » 1/2mp => relative importance of 3 (and higher)
body forces » n/(2mp)3 » 0.4n (fm3).
-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?
Well beyond nuclear matter density
Onset of new degrees of freedom: mesonic, D’s (p-N resonance),
quarks and gluons, ... .
Properties of matter in this extreme regime determine maximum
neutron star mass.
Large uncertainties!
Hyperons: S, L, ...
Meson condensates: p-, p0, KQuark matter
in droplets
in bulk
Color superconductivity
Strange quark matter
absolute ground state of matter??
strange quark stars?
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(1983)
Solid state
physics
neutron stars?
Low energy
nuclear physics
Temperature
Ultrarelativistic
heavy-ion collisions
Quark-gluon plasma
150 MeV
Hadronic matter
2SC
Nuclear
liquid-gas
0
?? CFL
Neutron stars
1 GeV
?
Baryon chemical potential
Phase diagram of quark gluon plasma
2nd order
tricritical pt.
1st order
Karsch & Laermann, hep-lat/0305025
New critical point in phase diagram:
induced by chiral condensate – diquark pairing coupling
via axial anomaly
Hatsuda, Tachibana, Yamamoto & GB, PRL 97, 122001 (2006)
Normal
Hadronic
(as ms increases)
Color SC
Predictions of phase transition at finite 
de Forcrand & Kratochvila
hep-lat/0602024
Lattice gauge theory
Strong coupling qcd
Kawamoto et al., hep-lat/0512023
Effective (NJL) theories
Ratti,Thaler,&Weise,
nucl-th/0604025
Nf=2
Onset of quark matter at low temperatures difficult
to predict via lattice gauge theory. rc»5-10rnm
Observations of massive neutron stars, M» 2M¯
=> equation of state stiff, and central density so low that
sharp transition to bulk quark liquid unlikely.
Quark droplets in nuclear matter.
Gradual onset of quark degrees of freedom.
Quark Droplets in Nuclear Matter
Glendenning; Heiselberg; Pethick, Ravenhall and Staubo
Favorable to form negatively
charged quark droplets
nu~100, nd~ns~300, R~5fm
Q~ -150|e|
at lower densities than quark-hadron transition since they
1) reduce no. of electrons in matter
2) increase fraction of protons in nuclear matter
Neutron stars likely to have such mixed phase cores, but
results are very model dependent
In fact, expect similar pasta phases of quark droplets:
Structure, neutrino emissivity?
Learning about dense matter from
neutron star observations
a) Masses of neutron stars: equation of state
b) Glitches: probe n,p
superfluidity and crust
c) Cooling of n-stars: search for exotica
d) Burst oscillations: probe
nuclear physics to ~109g/cm3
Infer masses from periods and Doppler shifts
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 ± 0.23M¯
Vela X-1: M=1.86 ± 0.15M¯ allow some softening
PSR J0751+1807: M » 2.1 M¯
no softening
QPO 4U1820-30: M » 2.2-2.3 M¯ challenge microscopic e.o.s.
Measured neutron star masses in radio pulsars
Thorsett and Chakrabarty, Ap. J. 1998
Hulse-Taylor
neutron star
- neutron star
binaries
M=1.35±0.04M¯
1.18M¯ < M < 1.44M¯
Measured neutron star
masses in radio pulsars
(from I. Stairs)
Hulse-Taylor binary
Possible path to compact
binary system (Bart & Kalogera)
NICE
NEW BINARY PULSAR SYSTEM
Lyne et al., Science 303, 1153 (2004)
22-ms pulsar J0737-3039A
+2.7-sec pulsar J0737-3039B companion
orbital period = 2.4 hours!
Highly-relativistic double-neutron-star system
See eclipsing of A by B
Laboratory for gravitational physics!
See orbit almost edge-on:
Mass determinations:
Stellar masses A=1.337(5)M¯ , B=1.250(5)M¯
Vela X-1 (LMXB) light curves
Serious deviation from Keplerian radial velocity
Excitation of (supergiant) companion atmosphere?
M=1.86 ± 0.33 (2s)M¯
M. H. van Kerkwijk,
astro-ph/0403489
1.4M¯
1.75M¯<M<2.44M¯
Quaintrell et al.,
A&A 401, 313 (2003)
1.4M¯
PSR J0751+1807
3.4 ms. pulsar in circular 6h binary w. He white dwarf
Nice et al., Ap.J. 634, 1242 (2005)
Pulsar slowing down due to gravitational radiation: dP/dt = 6.4£10-14
Shapiro delay of signal due to gravitational field of companion:
D t = - (2Gm2/c3) ln(1-cosq)
q
q = angle between ns and wd seen by observer
Measurements free (?) of uncertainties from possible atmospheric
distortion in companion
M=2.1M¯
Additional physics that allows one to pin down the masses
from D. Nice
Neutron star (pulsar) - white dwarf binaries
Nice et al., Ap.J. 634, 1242 (2005), Splaver et al., Ap.J 620, 405 (2005).
Observations of white dwarf companion
C. G. Bassa, van Kerkwijk, & Kulkarni, astro-ph/0601205
Companion is very red: Teff » 4000K.
Implies w.d. has He or He-H atmosphere.
Two mysteries:
1) Evolutionary models suggest companion should have
hot (burning) H atmosphere.
2) The pulsar does not seem to heat the w.d. atmosphere.
Absorbed and re-emitted radiation < 15%.
Need more detailed observations of spectra of white dwarf
Neutron-Star Low Mass X-ray Binaries
Kilohertz quasiperiodic oscillations (QPOs)
in accreting neutron stars
Detected in ~ 25 neutron stars
Sco X-1
QPOs remarkably coherent (Q
=n/dn ~ 30–200)
Large amplitude
Usually see 2 simultaneous kHz
QPOs (never 3)
Frequencies of the two QPOs
can vary by hundreds of Hz in few
hundred seconds, but
X-ray flux power density spectrum
Wijnands et al. (1998)
Separation nQPO = nQPO2 -nQPO1
of the two QPOs fairly constant
≈ nspin or ≈ nspin/2
innermost circular stable orbit (ISCO)
in GR:
R=6MG/c2
Strong evidence that higher frequency nQPO2 is the ISCO frequency,
Then have direct measurement of neutron star mass:
M*= c3/(63/2£ 2pnQPO2 G )
=(2198/ nQPO2(Hz) )M¯
R* =
c/(61/3£
2pnQPO2)
(Miller, Lamb, & Psaltis 1997)
Ex.: QPO 4U1820-30, nQPO2 = 1170 Hz => M~ 2.2-2.3 M¯
Implies very stiff equation of state.
Central density ~ 1.0 fm-3 ~ 6rnm
EXO0748-676: low mass x-ray binary
thermonuclear burst source
z=redshift of Fe and O lines
M ' 2.1§ 0.28 M¯
R ' 13.8 § 1.8 km
F. Özel, astro-ph//0605106
hypothetical star: 1.8M¯, R=10km
Akmal, Pandharipande and Ravenhall, 1998
Present observations of neutron stars masses M ' Mmax ' 2.2
M¯
beginning to confront microscopic nuclear physics.
High mass neutron stars => very stiff equation of state,
with nc < 7n0. At this point for nucleonic equation of state,
»
sound speed cs = ( P/r)1/2  c.
Naive theoretical predictions based on sharp
deconfinement transition seemingly inconsistent with
presence of (soft) bulk quark matter in neutron stars.
Further degrees of freedom, e.g., hyperons, mesons, or
quarks at n < 7n0 lower E/A => matter less stiff.
»
Quark cores possible, only if quark matter is very stiff.
Maximum mass of a neutron star
Say that we believe equation
of state up to mass density r0
but e.o.s. is uncertain beyond
r(Rc) = r0
Weak bound:
a) core not black hole => 2McG/c2 < Rc
b) Mc = s0Rc d3r r(r)  (4p/3) r0Rc3
=> c2Rc/2G  Mc  (4p/3) r0Rc3
Rs¯=2M¯ G/c2 = 2.94 km
Mcmax = (3M¯/4pr0Rs¯3)1/2M¯
4pr0Rc3/3
Mmax  13.7 M¯ £(1014g/cm3/r0)1/2
Outside material adds ~ 0.1 M¯
Strong bound: require speed of sound, cs, in
matter in core not to exceed speed of light:
cs2 = P/r  c2
Maximum core mass when cs = c
Rhodes and Ruffini (PRL 1974)
WFF (1988) eq. of state => Mmax= 6.7M¯(1014g/cm3/r0)1/2
V. Kalogera and G.B., Ap. J. 469 (1996) L61
r0 = 4rnm => Mmax = 2.2 M¯
2rnm =>
2.9 M¯
Can Mmax be larger?
Larger Mmax requires larger sound speed cs at lower n.
For nucleonic equation of state, cs -> c at n » 7n0.
Further degrees of freedom, e.g., hyperons, mesons, or
quarks at n » 7n0 lower E/A => matter less stiff.
Stiffer e.o.s. at lower n => larger Mmax. If e.o.s. very stiff
beyond n ' 2n0, Mmax can be as large as 2.9 M¯ .
Stiffer e.o.s. => larger radii (cf. EXO0748-676).
Gradual onset of quark degrees of freedom
Quarks degrees of freedom -- not accounted for by
nucleons interacting via static potentials -- expected to
play role.
As nucleons begin to overlap,
matter percolates at
nperc » 0.34 (3/4p rn3)
(Quarks can still be bound even if deconfined!)
Transition to quark matter likely crossover at low T
Normal
Hadronic
Color SC
Hatsuda, Tachibana, Yamamoto & GB,
PRL 97, 122001 (2006)
Gradual onset of quark degrees of freedom
Quarks degrees of freedom -- not accounted for by
nucleons interacting via static potentials -- expected to
play role.
As nucleons begin to overlap,
matter percolates at
nperc » 0.34 (3/4p rn3)
(Quarks can still be bound even if deconfined!)
Transition to quark matter likely a crossover at low T
Normal
Hadronic
Color SC
Hatsuda, Tachibana, Yamamoto & GB,
PRL 97, 122001 (2006)

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