Neutron star interiors: are we there yet?
Gordon Baym, University of Illinois
Workshop on Supernovae and Gamma-Ray Bursts
YIPQS Kyoto’
October 28, 2013
Neutron star interior
Mass ~ 1.4-2 Msun
Radius ~ 10-12 km
Temperature
~ 106-109 K
Mountains < 1 mm
Surface gravity
~1014 that of Earth
Surface binding
~ 1/10 mc2
Density ~ 2x1014g/cm3
Nuclei before neutron drip
e-+p
n
n + n : makes nuclei neutron rich
as
_ electron Fermi energy increases with depth
p+ e- + n : not allowed if e- state already occupied
Beta equilibrium: mn = mp + me
Shell structure (spin-orbit forces) for very neutron rich nuclei?
Do N=50, 82 remain magic numbers?
Being explored at rare isotope accelerators, RIKEN, GSI, FRIB, KORIA
Valley of b stability
in neutron stars
! " #$%
&' () " *+&, - +
. " ' (/ - (0() " *+(, +
&1/ 2 *+&, - +, $0%
" (+
64+
Separation energy S2n (MeV)
65+
ionization energy (eV)
neutron drip line
B" +
< "+
20
A' +
@" +
!,+
10
34+
10
20
30
40
50
60
Z (atomic number)
56+
28
?' +
70
768+
19
10
< (+
=&+
9: +
;, +
1
>+
10
20
30
40
50
60
70
80
90
100
110
120
N (neutron number)
130
140
150
160
170
180
80
RIKEN, H. Sakurai 2013
Loss of shell structure for N >> Z
even
No shell effect for Mg(Z=12), Si(14), S(16), Ar(18) at N=20 and 28
Instability of bcc lattice in the inner crust
D. Kobyakov and C. J. Pethick, ArXiv 1309.1891
BCC: Lower energy than FCC or simple cubic.
Predicted (pre-pasta) Coulomb structure at
crust-liquid interface
GB, H. A. Bethe, C. J. Pethick, Nucl. Phys. A 175, 225 (1971)
But effective finite wavenumber proton-proton interaction strongly
modified by screening:
kFT = Thomas-Fermi screening length
For k > kFT screening by electron
less effective.
Critical wavenumber above which pp
interaction is attractive:
Most unstable direction determined from
modification of elastic constants
J. Cahn, Acta Metallurgica 10, 179 (1962)
Possibly leads to BaTiO3
-like structure:
Similar to pasta phases of nuclei,
rearrangement of lattice structure affects thermodynamic and
transport properties of crust; effects on cooling and glitches,
pinning of n vortices, crust bremsstrahlung of neutrinos.
Modifies elastic properties of crust (breaking strains, modes, ...)
affect on precursors of γ-ray bursts in NS mergers, and generation of
gravitational radiation.
Pasta Nuclei
over half the mass of the crust !!
onset when nuclei fill ~ 1/8 of space
Lorentz, Pethick and Ravenhall. PRL 70 (1993) 379
Iida, Watanabe and Sato, Prog Theo Phys 106 (2001) 551; 110 (2003) 847
Important effects on crust bremsstrahlung of neutrinos,
pinning of n vortices, ...
QMD simulations of pasta phases
Sonoda, Watanabe, Sato, Yasuoka and Ebisuzaki, Phys. Rev. C77 (2008) 035806
0.10
0.20
0.39
0.49
0.58
T=0
T>0
Pasta phase diagram
Properties of liquid interior
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
Large theoretical extrapolation from low energy laboratory
nuclear physics at near nuclear matter density
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
p0
condensate
Neutron star models using static interactions between nucleons
Maximum neutron star mass
Mass vs. central density
Mass vs. radius
Akmal, Pandharipande and Ravenhall, 1998
Equation of state vs. neutron star structure
from J. Lattimer
Well beyond nuclear matter density
Onset of new degrees of freedom: mesonic, D’s, 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?
Hyperons in dense matter
Produce hyperon X of baryon no. A and charge eQ when
Amn - Qme > mX (plus interaction corrections).
mn = baryon chemical potential and me = electron chemical potential
Ex. Relativistic mean field
model w. baryon octet +
meson fields, w. input from
double-Λ hypernuclei.
Bednarek et al.,
Astron & Astrophys 543 (2012) A157
Y = number fraction vs. baryon density
Significant theoretical uncertainties in forces!
Hard to reconcile large mass neutron stars with softening of e.o.s
due to hyperons -- the hyperon problem. Requires stiff YN interaction.
Fundamental limitations of equation of state based on
nucleon-nucleon interactions alone:
Accurate for n~ n0.
n >> n0:
-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 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!
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
Challenges to nuclear theory!!
High mass neutron star, PSR J1614-2230
-- in neutron star-white dwarf binary
Demorest et al., Nature 467, 1081 (2010); Ozel et al., ApJ 724, L199 (2010).
Spin period = 3.15 ms; orbital period = 8.7 day
Inclination = 89:17o ± 0:02o : edge on
Mneutron star =1.97 ± 0.04M ; Mwhite dwarf = 0.500 ±006M
(Gravitational) Shapiro delay of light from pulsar
when passing the companion white dwarf
Second high mass neutron star, PSRJ0348+0432
-- in neutron star-white dwarf binary
Antonidas et al., Science 340 1233232 (April 26, 2013)
Spin period = 39 ms; orbital period = 2.46 hours
Inclination =40.2o
Mneutron star =2.01 ± 0.04M ; Mwhite dwarf = 0.172
±0.003M
Significant gravitational radiation
400 Myr to coalescence!
A third high mass neutron star, PSR J1311-3430 -in neutron star - flyweight He star binary
Romani et al., Ap. J. Lett., 760:L36 (2012)
Mneutron star > 2.0 M ;
Mcompanion ~ 0.01-0.016M
Uncertainties arising from internal dynamics of companion
Akmal, Pandharipande and Ravenhall, 1998
Mass vs. radius determination of neutron stars in burst sources
M vs R from bursts, Ozel at al, Steiner et al.
J. Poutanan, at Trento workshop on
Neutron-rich matter and neutron stars, 30 Sept. 2013
Or perhaps overestimated, since
R. Rutledge, at Trento workshop on
Neutron-rich matter and neutron stars, 30 Sept. 2013
Temperature
Ultrarelativistic
heavy-ion collisions
Quark-gluon plasma
150 MeV
Hadronic matter
2SC
Nuclear
liquid-gas
0
Neutron stars
1 GeV
?
Baryon chemical potential
CFL
Phase diagram of equilibrated quark gluon plasma
Critical point
Asakawa-Yazaki 1989.
1st order
crossover
Karsch & Laermann, 2003
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.
ex. nuclear matter using 2 & 3 body
interactions, vs. pert. expansion or bag
models. Akmal, Pandharipande, Ravenhall 1998
Typically conclude transition at ~10nm -- would not be reached
in neutron stars given observation of high mass PSR
J1614-2230 with M = 1.97M
=> no quark matter cores
Fukushima & Hatsuda, Rep. Prog. Phys. 74 (2011) 014001
K. Fukushima (IPad)
BEC-BCS crossover in QCD phase diagram
GB, T.Hatsuda, M.Tachibana, & Yamamoto. J. Phys. G: Nucl. Part. 35 (2008) 10402
H. Abuki, GB, T. Hatsuda, & N. Yamamoto,Phys. Rev. D81, 125010 (2010)
Normal
Hadronic
(as ms increases)
Color SC
Hadrons
BCS-BEC
crossover
BCS paired
quark matter
Small quark pairs are “diquarks”
Continuous evolution from nuclear to quark matter
K. Masuda, T. Hatsuda, & T. Takatsuka, Ap. J.764, 12 (2013)
Hadron-quark crossover equation of state
K. Masuda, T. Hatsuda, &T. Takatsuka, Ap. J.764, 12 (2013)
Neutron matter at low density with smooth interpolation to
Nambu Jona-Lasinino model of quark matter at high density
quark content vs. density
E.o.s. with interpolation
between 2 - 4
Model calculations of phase diagram with axial anomaly,
pairing, chiral symmetry breaking & confinement
NJL alone: H. Abuki, GB, T. Hatsuda, & N. Yamamoto, PR D81, 125010 (2010).
NPL with Polyakov loop description of confinement: P. Powell & GB PR D 85, 074003 (2012)
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:
PNJL phase diagram
K and K’ from axial anomaly
Model calculations of neutron star matter
and neutron stars within NJL model
NJL Lagrangian supplemented with universal repulsive
quark-quark vector coupling
K. Masuda, T. Hatsuda, & T. Takatsuka, Ap. J.764, 12 (2013)
GB, T. Hatsuda, P. Powell, ... (to be published)
Include up, down, and strange quarks with realistic masses
and spatially uniform pairing wave functions
Smoothly interpolate from nucleonic equation of state (APR) to quark
equation of state:
pressure
5
1.5
1
0
baryon density
mass density
Neutron star equation of state vs. phenomenological
fits to observed masses and radii
Lines from bottom
to top:
gV /G = 0, 1, 1.5, 5
Cross-hatched
region = Ozel et al.
(2010)
Shaded region =
Steiner et al.
(2010)
Masses and radii of neutron stars vs. central mass density
from integrating the TOV equation
2.5
5
2
2.5
1.5
2
1
gv/G=
M/M
1
1.5
0
S
M/M
S
1.5
1
0.5
0.5
0
0
10
20
30
rc/r0
40
50
60
0
Mass vs. central density:
only stars on rising curves
are stable
6
7
8
9
10
11
R (km)
M vs. R:
only stars on rapidly
rising curves are stable
12
Maximum neutron star mass vs. gV
2.6
2.4
2
M
max
/M
S
2.2
1.8
1.6
1.4
0
2
4
6
8
10
12
14
16
18
20
g /G
V
PNJL accomodates large mass neutron stars as well
as strange quarks -- avoiding the “hyperon problem”
-- and is consistent with observed masses and radiii
But we are not quite there yet:
Uncertainties in interpolating from nuclear matter to quark
matter lead to errors in maximum neutron star masses and radii.
Uncertainties in the vector coupling gv
The NJL model does not treat gluon effects well, which leads
to uncertainties in the “bag constant” B of quark matter:
At very high baryon density, the energy density is
E = B + Cpf4 ,
with B ~ 100-200 MeV/fm3. Then the pressure is
P = -B + Cpf4 /3.
Effect of B on maximum neutron star mass?
Need to calculate gluon contributions accurately to pin down B.
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