equation of state of neutron star matter...equation of state of neutron star matter j. m. lattimer...
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Equation of State of Neutron Star Matter
J. M. Lattimer
Department of Physics & AstronomyStony Brook University
January 17, 2016
Electromagnetic Signatures of R-Process Nucleosynthesis in Neutron Star MergersINT, Seattle, Washington24 July - 18 August, 2017
J. M. Lattimer Equation of State of Neutron Star Matter
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Outline
I The Unitary Gas Constraint on the NuclearSymmetry Energy
I Constraints from Nuclear Physics and theMaximum Mass on Neutron Star UniversalStructure
I How the Possibility of Hybrid Stars LoosensConstraints
I Observational Estimates of Neutron Star Radiiand Their Problems
J. M. Lattimer Equation of State of Neutron Star Matter
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Although simpleaverage mass ofw.d. companionsis 0.23 M� larger,weighted average is0.07 M� larger
Demorest et al. 2010Fonseca et al. 2016Antoniadis et al. 2013Barr et al. 2016
Romani et al. 2012
vanKerkwijk 2010
J. M. Lattimer Equation of State of Neutron Star Matter
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Causality + GR Limits and the Maximum Mass
A lower limit to themaximum mass sets alower limit to theradius for a given mass.
Similarly, a precisionupper limit to R , witha well-measured mass,sets an upper limit tothe maximum mass.
R1.4 > 8.15 km ifMmax ≥ 2.01M�.
∂p∂ε =
c2sc2 = s
M − R curves for minimally compact EOS=⇒=⇒ssquar
kst
ars
If quark matter exists in the interior, the minimum radii aresubstantially larger.
J. M. Lattimer Equation of State of Neutron Star Matter
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The Unitary Gas
The unitary gas is an idealized system consisting of fermionsinteracting via a pairwise zero-range s-wave interaction with aninfinite scattering length:
As long as the scattering length a >> k−1F (interparticle spacing),and the range of the interaction R << k−1F , the properties of thegas are universal in the sense they don’t depend on the details ofthe interaction.
The sole remaining length scale is kF = (3π2n)1/3, so the unitarygas energy is a constant times the Fermi energy ~2k2F/(2m):
EUG = ξ03~2k2F10m
.
ξ0 ' 0.37 is known as the Bertsch parameter, measured incold-atom experiments.
J. M. Lattimer Equation of State of Neutron Star Matter
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The Unitary Gas as Analogue of the Neutron Gas
A pure neutron matter (PNM) gas differs from the unitary gas:
I |a| ' 18.5 fm; |akF |−1 ' 0.03 for n = ns .
I R ' 2.7 fm; RkF ≈ 4.5 for n = ns .
I Repulsive 3-body interactions are additionally necessary forneutron matter to fit the energies of light nuclei.
I Neutron matter has potentially atractive p-wave andhigher-order interactions.
The first three imply EPNM > EUG:
I ξ ' ξ0 + 0.6|akF |−1 + . . . |akF |−1 << 1
I ξ ' ξ0 + 0.12RkF + . . . RkF << 1
A reasonable conjecture would appear to be (u = n/ns)
EPNM(u) = E (u,Yp = 0) ≥ EUG,0u2/3 ' 12.6u2/3 MeV
J. M. Lattimer Equation of State of Neutron Star Matter
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Comparison to Neutron Matter Calculations
0 0.05 0.1 0.15 0.2
n[fm−3
]0
5
10
15
20
25
EP
NM
[ MeV
]
SCGF (2016)
Lynn et al. (2016)
Hebeler et al. (2010)
TT (2013)
GCR (2012)
GC (2010)
APR (1998)
FP (1981)
Unitary gas (ξ0 = 0.37)
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Consequences for the Nuclear Symmetry Energy
S(u) = EPNM − E (u,Yp = 1/2).
A good approximation for the Yp−dependence of E is
S(u) ' 1
8
∂2E (u,Yp)
∂Y 2p
.
Near ns ,S(u) ' S0 +
L
3(u − 1) +
Ksym
18(u − 1)2 + · · ·
E (u,Yp = 1/2) ' −B +Ks
18(u − 1)2 + · · ·
In this case, the unitary gas conjecture is
S(u) > EUG,0u2/3 −
[−B +
Ks
18(u − 1)2 + · · ·
]≡ SLB(u)
Thus, the symmetry energy parameters S0 and L must satisfy
S(u = 1) = S0 ≥ SLB0 = EUG,0 + B ' 28.5 MeV
L(u = 1) = L0 = 3 (udS/du)u=1 = 2EUG,0 ' 25.2MeV
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S0 > SLB0 :
(S = SLB
)ut,(dS/du = dSLB/du
)ut
S(u
) (M
eV
)
u=n/n0
0
10
20
30
40
50
0.2 0.4 0.6 0.8 1 1.2 1.4
Excluded
Allowedut=0.5-1.1
ut=1 (S0LB
,L0)
SLB
(u)
Tews et al. (2017)
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Symmetry Parameter Exclusions
L (
Me
V)
S0 (MeV)
0
20
40
60
80
100
120
24 26 28 30 32 34 36 38 40
Excluded
Allowed
(S0LB
,L0)ut=1
SFHx
SFHoIUFSU
FSUgold
DD2,
STOS,TM1NL3
TMA
LS220
NLρ
NLρδ
DBHF
DD,D3C,DD-F
KVR
KVOR
TKHS
HSGCR
ut=1/2
MKVOR
Tews et al. (2017)
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Symmetry Parameter Correlations
Compilations from Dutra et al. (2012, 2014)
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More Realistic Exclusion Region
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Analytic Approximation for the Boundary
S(ut) = SLB(ut),
(dS
du
)
ut
=
(dSLB
du
)
ut
gives
S0 +L
3(ut − 1) +
Ksym
18(ut − 1)2 = EUG,0u
2/3t + B − Ks
18(ut − 1)2
L +Ksym
3(ut − 1) = 2EUG,0u
−1/3t − Ks
3(ut − 1)
Assume Kn = 3L (i.e., Ksym ≈ 3L− Ks). Then
S0 =EUG,0
3u4/3t
(1 + 2u2t )− E0, L =2EUG,0
u4/3t
or after eliminating ut ,
S0 =L
6
[1 + 2
(2EUG,0
L
)3/2]− E0
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Experimental Constraints
24 26 28 30 32 34 36 38 40
S0 [MeV]
0
20
40
60
80
100
120
L[ M
eV]
Allowed
Excluded
Mas
ses
Sn neutron skin
Pbdi
polepola
rizab
ility
HIC
GDR
IAS
+∆
R
UGUG analytic
Isovector Skins andIsobaric Analog Statesfrom Danielewicz et al. (2017)
Other experimental constraintsfrom Lattimer & Lim (2013)
Unitary gas constraints fromTews et al. (2017)
Experimental and neutronmatter constraints are compatiblewith unitary gas bounds.
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Piecewise PolytropesCrust EOS is known: n < n0 = 0.4ns .
Read, Lackey, Owen & Friedman (2009)found high-density EOS can be modeledas piecewise polytropes with 3 segments.
They found universal break points(n1 ' 1.85ns , n2 ' 3.7ns) optimized fitsto a wide family of modeled EOSs.
For n0 < n < n1, assume neutron matterEOS. Arbitrarily choose n3 = 7.4ns .
For a given p1 (or Γ1):0 < Γ2 < Γ2c or p1 < p2 < p2c .0 < Γ3 < Γ3c or p2 < p3 < p3c .
Minimum values of p2, p3 set by Mmax ;maximum values set by causality.
◦n3, p3
nm
◦n0, p0
◦n1, p1
◦n2, p2
13.8 14.814.3 15.3 15.7
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Maximum Mass and Causality Constraints
p 3<
p 2
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Mass-Radius Constraints from Causality
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Photospheric Radius Expansion X-Ray Bursts
Galloway, Muno, Hartman, Psaltis & Chakrabarty (2006)
⇐ FEdd = GMcκD2
√1− 2GM
Rphc2⇐ FEdd
A = f −4c (R∞/D)2 A = f −4c (R∞/D)2
EXO 1745-248
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PRE Burst Model
Observations measure:
FEdd ,∞ =GMc
κD2
√1− 2β, β =
GM
Rc2
A =F∞σT 4∞
= f −4c
(R∞D
)2
Determine parameters:
α =FEdd ,∞√
A
κD
f 4c c3
= β(1− 2β)
γ =Af 4c c
3
κFEdd ,∞=
R∞α.
Solution:
β =1
4±√
1− 8α
4, α ≤ 1
8for real solutions.
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PRE M − R Estimates
Ozel & Freire (2016)
0.164± 0.0240.153± 0.0390.171± 0.0420.164± 0.0370.167± 0.0450.198± 0.047
αmin
��������������������
�������������������
β=
1/4
caus
ality
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PRE Burst Models – Effect of Source Redshift
Ozel et al. zph = z β = GM/Rc2 Steiner et al. zph << z
FEdd =GMc
κD
√1− 2β
A =F∞σT 4∞
= f −4c
(R∞D
)2
α =FEdd√
A
κD
F 2c c
3= β(1− 2β)
γ =Af 4c c
3
κFEdd
=R∞α
β =1
4± 1
4
√1− 8α
α ≤ 1
8required.
FEdd =GMc
κD
α = β√
1− 2β
θ =1
3cos−1
(1− 54α2
)
β =1
6
[1 +√
3 sin θ − cos θ]
α ≤√
1
27' 0.192 required.
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QLMXB M − R Estimates
Ozel & Freire (2016)
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Combined R fits
Assume P(M) is that measured from pulsar timing(M = 1.4M�).
Ozel & Freire (2015)
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Role of Systematic Uncertainties
Systematic uncertainties plague radius measurements.
I Assuming uniform surface temperatures leads tounderestimates in radii.
I Uncertainties in amounts of interstellar absorption
I Atmospheric composition: In quiescent sources, He or Catmospheres predict about 50% larger radii than Hatmospheres.
I Non-spherical geometries: In bursting sources, the use ofthe spherically-symmetric Eddington flux formula leads tounderestimate of radii.
I Disc shadowing: In bursting sources, leads tounderprediction of A = f −4c (R∞/D)2, overprediction ofα ∝ 1/
√A, and underprediction of R∞ ∝
√α.
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Piecewise-Polytrope Radius Distributions
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Folding Observations with Piecewise Polytropes
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Other Studies
Hebeler, Lattimer, Pethick & Schwenk 2010 Ozel & Freire 2016
↑R1.4,min = 10.6 kmMmax = 1.97M�
R1.4,min = 9.25 kmMmax = 1.66M�
↑
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Radius - p1 Correlation
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Binding Energy - Mass Correlations
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Binding Energy - Mass - Radius Correlations
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Moment of Inertia - Mass - Radius Correlations
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Moment of Inertia - Radius Constraints
PSR 0737-3039A
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Tidal Deformatibility - Moment of Inertia
See also Yagi & Yunes (2013)
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Tidal Deformatibility - Mass
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Tidal Deformatibility - Mass - Radius
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Binary Tidal Deformability
In a neutron star merger, both stars are tidally deformed. Themost accurately measured deformability parameter is
Λ =16
13
[λ1q
4(12q + 1) + λ2(1 + 12q)]
where
q =M1
M2< 1
For S/N ≈ 20− 30, typical measurement accuracies areexpected to be (Rodriguez et al. 2014; Wade et al. 2014):
∆Mchirp ∼ 0.01− 0.02%, ∆Λ ∼ 20− 25%
∆(M1 + M2) ∼ 1− 2%, ∆q ∼ 10− 15%
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Binary Tidal Deformatibility - Λ
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Binary Tidal Deformatibility - Mchirp
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Binary Tidal Deformatibility - Mchirp - R1.4
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Hybrid Stars With First-Order Phase TransitionTianqi Zhou
Follows scheme of Alford, Burgio,
Han, Taranto & Zappala (2015)
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Mass-Radius ComparisonsTianqi Zhou
Green = hadronic; Blue = hybrid
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Binding Energy - Compactness ComparisonsTianqi Zhou
Green = hadronic; Blue = hybrid
J. M. Lattimer Equation of State of Neutron Star Matter
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Moment of Inertia - Compactness ComparisonsTianqi Zhou
Green = hadronic; Blue = hybrid
J. M. Lattimer Equation of State of Neutron Star Matter
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Tidal Deformability Comparisons
Tianqi Zhou
Green = hybrid c2s = 1
Blue = hadronic
Red = hybrid c2s = 1/3
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Future Observations
I Twin stars with different radii:Evidence for phase transitions
I Neutron star seismology and r-modes from GWobservations:
νellipticity = 2f , νr−mode ≈ (4/3)f
I Compactness from νr−mode .I Temperature if r-modes dominate heating.I Moment of inertia if r-modes dominate spindown.I Require factor of 3–10 improvement in sensitivity over
aLIGO.I Potential sources would be very young.
I What else?
J. M. Lattimer Equation of State of Neutron Star Matter
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Additional Proposed Radius and Mass Constraints
I Pulse profiles Hot or cold regionson rotating neutron stars alterpulse shapes: NICER and LOFTwill enable X-ray timing andspectroscopy of thermal andnon-thermal emissions. Lightcurve modeling → M/R;phase-resolved spectroscopy → R.
I Moment of inertia Spin-orbitcoupling of ultra- relativisticbinary pulsars (e.g., PSR0737+3039) vary i and contributeto ω: I ∝ MR2.
I Supernova neutrinos Millions ofneutrinos detected from aGalactic supernova will measureBE= mBN −M, < Eν >, τν .
I QPOs from accreting sourcesISCO and crustal oscillations
NASA
Neutron star Interior Composition ExploreR
Large Observatory For x-ray Timing
ESA/NASA
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”Is Grandpa in the rocket ship?”
NICER successfully launchedaboard a Falcon rocket,
June 3.
Was powered up June 13.
Now taking data:
J0437-4715 (1.44M�)J0030+0451J1231-1411J1614-2230 (1.93M�)
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X-ray Timing of RXJ0720.4-3125
Hambaryan et al. (2017) undertook phase-resolved spectroscopy ofthe isolated neutron star RXJ0720.4-3125, one of ”magnificent 7”.Spin period is 16.79s.Tbb ∼ 90 eV,TFe ∼ 105 eV.R1.4 ' 13.2± 0.3 km.
J. M. Lattimer Equation of State of Neutron Star Matter