Constraints on the Dense Matter Equation ofState from Experiments, Theory andObservations
J. M. Lattimer
Department of Physics & Astronomy
Stony Brook University

October 28, 2013
Collaborators: E. Brown (MSU), K. Hebeler (OSU), D. Page (UNAM),
C.J. Pethick (NORDITA), M. Prakash (Ohio U), A. Steiner (INT),
A. Schwenk (TU Darmstadt), Y. Lim (Daegu Univ., Korea)
Supernovae and Gamma Ray Bursts in Kyoto 2013
Yukawa Institute, Supernova Conference, 28 October - 1 November, 2013
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoOutline
IGeneral Relativity Constraints on Neutron Star Structure
Maximally Compact Equation of State
Radius Limits from the Neutron Star Maximum Mass and Causality
I Constraints on Quark Matter in Hybrid Star Interiors

IIII

The Neutron Star Radius and the Nuclear Symmetry Energy
Nuclear Experimental Constraints on the Symmetry Energy
Binding Energies
Heavy ion Collisions
I Neutron Skin Thicknesses
I Dipole Polarizabilities and Giant Dipole Resonances
I Isobaric Analog States

III
I

Constraints from Pure Neutron Matter
Astrophysical Constraints
Pulsar and X-ray Binary Mass MeasurementsPhotospheric Radius Expansion Bursts
I Thermal Emission from Isolated and Quiescent Binary Sources
I Other Proposed Mass and Radius Constraints

IIJ. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Neutron Star Structure
Tolman-Oppenheimer-Volkov equations
dp
G (mc 2 + 4πpr 3 )(ε + p)
= − 4drcr (r − 2Gm/c 2 )dm
ε= 4π 2 r 2
drc
maximum mass- R∝ p-p(ε)

1/4

M(R)

-Equation of StateJ. M. Lattimer

Observations

Constraints on the Dense Matter Equation of State from Experiments, TheoExtremal Properties of Neutron Stars
The most compact and massive configurations occur when thelow-density equation of state is ”soft” and the high-density equationof state is ”stiff” (Koranda, Stergioulas & Friedman 1997).

εo−ε itsoft=⇒p⇐=

Istiff= al limusca

ε0 is the only
EOS parameter
The TOVsolutions scalewith ε0w = ε/ε0y = p/ε√0x = r √G ε0 /c 2z = m G 3 ε0 /c 2

◦p = 0 ε0oJ. M. LattimerConstraints on the Dense Matter Equation of State from Experiments, TheoExtremal Properties of Neutron Stars
The maximum mass configuration is achieved whenxR = 0.2404, wc = 3.034, yc = 2.034, zR = 0.08513.
A useful reference density is the nuclear saturation density(interior density of normal nuclei):ρs = 2.7 × 1014 g cm−3 , ns = 0.16 baryons fm−3 , εs = 150 MeV fm−3I

Mmax = 4.1 (εs /ε0 )1/2 M(Rhoades & Ruffini 1974)

2I

MB,max = 5.41 (mB c /µo )(εs /ε0 )1/2 M

IRmin = 2.82 GM/c 2 = 4.3 (M/M ) km

Iµb,max = 2.09 GeV

Iεc,max = 3.034 ε0 ' 51 (M /Mlargest )2 εsIpc,max = 2.034 ε0 ' 34 (M /Mlargest )2 εs

InB,max ' 38 (M /Mlargest )2 ns

IBEmax = 0.34 M

IPmin = 0.74 (M /Msph )1/2 (Rsph /10 km)3/2 ms =0.20 (Msph,max /M ) ms
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Causality and the Maximum Mass=⇒=⇒A lower limit to themaximum mass sets a M − R curves for maximally compact EOS
lower limit to theradius for a given mass.
Similarly, a precise(M, R) measurement
sets an upper limit tothe maximum mass.

s

1.4M stars must haveR > 8.15M .

scs2
∂p=
∂ε
c2 = s

1.4M strange quark
matter stars (and likely
hybrid quark/hadronstars) must haveR > 11 km.
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Mass-Radius Diagram and Theoretical Constraints
GR:
R > 2GM/c 2
P <∞:R > (9/4)GM/c 2

causality:2R>∼ 2.9GM/c— normal NS— SQS— R∞ =√R
1−2GM/Rc 2J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

The Radius – Pressure Correlation

km fm3/4 MeV−1/4

9.52 ± 0.49

7.06 ± 0.24

5.68 ± 0.14
Lattimer & Prakash (2001), Lattimer & Lim (2013)

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Nuclear Symmetry Energy
Defined as the difference between energies of pure neutron matter(x = 0) and symmetric (x = 1/2) nuclear matter.
S(ρ) = E (ρ, x = 0) − E (ρ, x = 1/2)
Expanding around the saturation density(ρs ) and symmetric matter (x = 1/2)

C. Fuchs, H.H. Wolter, EPJA 30(2006) 5E (ρ, x) = E (ρ, 1/2)+(1−2x)2 S2 (ρ)+. . .
L ρ − ρs+ ...
3 ρs
Sv ' 31 MeV, L ' 50 MeV
S2 (ρ) = Sv +6

symmetry energy?

Connections to pure neutron matter:
E (ρs , 0) ≈ Sv + E (ρs , 1/2) ≡ Sv − B,p(ρs , 0) = Lρs /3Neutron star matter (in beta equilibrium):"#34Sv∂(E + Ee )Lρs4 − 3Sv /L= 0, p(ρs , xβ ) '1−∂x3~c3π 2 ρsJ. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Determining Symmetry Parameters from Nuclear Masses
From liquid drop model:
Esym (N, Z ) = (Sv A − Ss A2/3 )I 2P2
χ2 = i (Eex,i − Esym,i )2 /N σD
P4 2
χvv = N2σ2i Ii Ai
D P
24 4/3
χss = N σ2i Ii Ai
D P24 5/3
χvs = N σ2i Ii AiDq2χss' 2.3 σDχvv χss −χ2sv

q2χvv' 13.2 σDχvv χss −χ2sv

σSv =σSs =σD = 2 MeV

UNEDF0 HF

2χ

vsα = 12 tan−1 χ −χ
' 9◦.8vvssχ
rvs = − √χ vsχ ' 0.997
vv

ssLiquid droplet model:
Esym (N, Z ) =Sv AI 21+(Ss /Sv )A−1/3
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Neutron Skin Thickness
rnpro 'q34πρsro =Ss I415 Sv +Ss A−1/31/3' 1.14 fm(rnp )208 = 0.175 ± 0.020 fm
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Heavy Ion Collisions
Reaction Mechanisms in Heavy Ion Collisions
Coulomb barrier toFermi energies
peripheral

Proton and neutron currents
centraljn − j p ∝ Esym ( ρ )∇I +“Diffusion”

Isospinmigration∂Esym ( ρ )∂ρI∇ρ“Drift”

Isospinfractionation,
multifragm

Sensitive to Sym. Energy andslope depending on observable

Wolter, NuSYM11J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Giant Dipole Resonances

www.tunl.duke.edurE−1 ∝Sv
5Ss −1/3
1+ 3S Av

23.3 MeV< S2 (0.1 fm−3 ) <24.9 MeV
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Dipole Polarizability
The linear response, or dynamic
polarizability, of a nuclear system
excited from its ground state toan excited state, due to an
external oscillating dipolar field.
Liquid droplet
 model:AR 2αD = 20S
1 + 35 SSvs A−1/3v

Roca-Maza et al. found anempirical correlation between
αD Sv and rnp

Data from Tamii et al. (2011)

αD,208 = 20.1 ± 0.6 fm2
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Isobaric Analog States

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Theoretical Neutron Matter Calculations

H&S: Chiral Lagrangian
GC&R: Quantum Monte CarloJ. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Consensus Experimental Constraints

H&S: Chiral Lagrangian
GC&R: Quantum Monte Carlo
D&L: IAS + neutron skinrnp = 0.179 ± 0.023 fm
white: all experimental
rnp = 0.175 ± 0.020 fm

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheovanKerkwijk 2010
Romani et al. 2012

Although simple
average mass ofw.d. companions
is 0.23 M larger,
weighted average is0.04 M smaller

Demorest et al. 2010

Antoniadis et al. 2013
Champion et al. 2008

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoBlack Widow Pulsar PSR B1957+201.6ms pulsar in circular 9.17h orbit with a ∼ 0.03 M companion.
Pulsar is eclipsed for 50-60 minutes each orbit; eclipsing object has avolume much larger than the companion or its Roche lobe.
It is believed the companion is ablated by the pulsar leading to mass loss
and an eclipsing plasma cloud. Companion nearly fills its Roche lobe.
Ablation by pulsar leads to eventual disappearance of companion.
The optical light curve does not represent the center of mass of thecompanion, but the motion of its irradiated hot spot.
pulsar radial velocityeclipse

NASA/CXC/M.Weiss
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoBlack Widow Pulsar PSR B1957+20Ki = 2π ai Psin ia∗
K∗
Pq=M
M∗ = aP = KPK∗ = Kobs 1 + Ra∗∗

0.045

K2=KobsMP > 1.8 M , sin i < 660.025◦0.020

Most probable values:
2.10M < MP < 2.75M→∗R∗

450.030KR

0.035==KsobL11RL

i=MComp (M )Modeling of light curve
shape suggests that00.040

Radiated area of companion
has a smaller orbit thanthe center of mass.
KP 3PMP = q(1 + q)2 2πGsin i

1.00i =65i =71.5

2.0MPSR (M )2.5

3.0van Kerkwijk 2010
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Simultaneous Mass/Radius Measurements
I24Measurements of flux F∞ = RD∞ σTeff−1and color temperature Tc ∝ λmax yield anapparent angular size (pseudo-BB):
R∞R
1= pD
D 1 − 2GM/Rc 2IObservational uncertainties includedistance D, interstellar absorption
NH , atmospheric composition
Best chances for accurate radius measurement:
I Nearby isolated neutron stars with parallax (uncertain atmosphere)
I Quiescent low-mass X-ray binaries (QLMXBs) in globular clusters(reliable distances, low B H-atmosperes)
I Bursting sources (XRBs) with peak fluxes close to Eddington limit(where gravity balances radiation pressure)cGM p
FEdd =
1 − 2GM/Rc 2
κD 2
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoM − R PRE Burst EstimatesFEdd,∞ , (R∞ /D)2 fc−4 , Dfrom Ozel et al.
zph = 0
Altered uncertainties
for fc , D
Lattimer & Steiner (2013)
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoM − R QLMXB Estimates
2.4

ω Cen

M13

2.2

6304

6397

2M28

M (M )

1.8
1.6
1.4
1.2
1
0.8
0.6
68

10

12

14

16

18

20

R (km)

Guillot et al. (2013)

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoM − R QLMXB Estimates
2.4

ω Cen

M132.263046397

2M28P(M, R) from H atmospheremodels of Guillot et al. (2013),
adjusted for alternate NH valuesof Dickey & Lockman (1990)

M (M )

1.8
1.6
1.4
1.2
10.8
0.6
68

101214161820R (km)Lattimer & Steiner (2013)

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Bayesian TOV InversionIε < 0.5ε0 : Known crustal EOS

II

0.5ε0 < ε < ε1 : EOS
parametrized by K , K 0 , Sv , γ

EOS parameters K , K 0 , Sv , γ, ε1 ,n1 , ε2 , n2 uniformly distributedIMmax ≥ 1.97 M , causality
enforcedIAll 10 stars equally weighted

IPolytropic EOS: ε1 < ε < ε2 : n1 ;ε > ε2 : n2
1 2.5
3

100.9

inferred p(ε)

0.8
100.8

20.7


RX J1856

M (M )

0.6
1.5
0.5

10

1

Fid. EOS, PREs+QLMXBs w/adj. NH
-1

100.9

Lattimer & Steiner 2013

0.7

2P (MeV/fm3)

1
Fiducial EOS, PREs, QLMXBs w/adjusted NH

200

4006008001000 1200 1400 1600

ε (MeV/fm3)

0.6
0.5

Ho et al. (2007)

0.4

0.4

0.3 1

0.3

0.2

0.2

0.10.50inferred M(R)68100.1
121416180R (km)J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Astronomy vs. Astronomy vs. Physics
2.5Ozel et al., zph = z:
R = 9.74 ± 0.50 km.

Fiducial EOS, PREs, QLMXBs w/adjusted NH

Guillot et al. (2013):
All stars have same radius:

2Lattimer & Steiner (2013):
TOV, crust EOS, causality,maximum mass > 2M ,but no Sv − L constraints.
Lattimer & Lim (2013):
Nuclear experiments:
29 MeV < Sv < 33 MeV,
40 MeV < L < 65 MeV:
R1.4 = 12.0 ± 1.4 km.

M (M )

R = 9.1+1.3−1.5 km.
1.5

10.5
681012141618R (km)J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Additional Proposed Radius and Mass Constraints
II

Pulse profiles
Hot or cold regions on rotating
neutron stars alter pulse shapes:
NICER and LOFT will enabletiming and spectroscopy ofthermal and non-thermal emissions.
Light curve modeling → M/R;phase-resolved spectroscopy → R.
Moment of inertia
Spin-orbit coupling of ultrarelativistic binary pulsars(e.g., PSR 0737+3039) vary i andcontribute to ω̇: I ∝ MR 2 .

ISupernova neutrinos
Millions of neutrinos detected froma Galactic supernova will measure
BE= mB N − M, < Eν >, τν .

I

QPOs from accreting sources
ISCO and crustal oscillations
J. M. Lattimer

Neutron star Interior Composition ExploreR

NASA

Large Observatory For x-ray Timing

ESA/NASAConstraints on the Dense Matter Equation of State from Experiments, TheoConclusions
INuclear experiments set reasonably tight constraints on symmetryenergy parameters and the symmetry energy behavior near thenuclear saturation density.

I

Theoretical calculations of pure neutron matter predict very similar
symmetry constraints.

I

These constraints predict neutron star radii R1.4 in the range
12.0 ± 1.4 km.

I

Combined astronomical observations of photospheric radius
expansion X-ray bursts and quiescent sources in globular clusters
suggest R1.4 ∼ 12.1 ± 0.6 km.

I

The nearby isolated neutron star RX J1856-3754 appears to have aradius near 12 km, assuming a solid surface with thin H atmosphere(Ho et al. 2007).

IThe observation of a 1.97 M neutron star, together with the radiusconstraints, implies the EOS above the saturation density isrelatively stiff; any phase transition is small.
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, TheoMaximum Energy Density in Neutron Stars

p = s(ε − ε0 )J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Constraints from Observations of Gravitational Radiation
Mergers:Chirp mass M = (M1 M2 )3/5 M −1/5 andtidal deformability λ ∝ R 5 (Love number)are potentially measurable during inspiral.
λ̄ ≡ λM −5 is related to Ī ≡ IM −3 by an
EOS-independent relation (Yagi & Yunes2013). Both λ̄ and Ī are also related toM/R in a relatively EOS-independent way(Lattimer & Lim 2013).
INeutron star - neutron star: Mcrit for
prompt black hole formation, fpeak
depends on R.

I

Black hole - neutron star:ftidal disruption depends on R, a, MBH .
Disc mass depends on a/MBH and on
MNS MBH R −2 .

Rotating neutron stars: r-modes
J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo

Consistency with Neutron Matter and Heavy-Ion Collisions

J. M. Lattimer

Constraints on the Dense Matter Equation of State from Experiments, Theo