Embed Size (px)
Citation preview
Thermal Properties of Hot and Dense Matter
Brian MuccioliConstantinos Constantinou
Madappa PrakashJames M. Lattimer
Ohio University, Athens, OH
March 10, 2015Kent State University
Thermal Properties of Hot and Dense Matter
Outline
I Physical Settings
I The Models
I Zero Temperature Properties
I Finite Temperature Properties
I Limiting Cases
I Contribution from Leptons and Photons
I Pion Condensate (APR)
I Equation of State for a Cold Neutron Star
I Adiabatic Index
Thermal Properties of Hot and Dense Matter
Physical Setting: Core-Collapse SN matter
Credit: Freedman and Kaufmann: Universe 7th ed. Credit: R.J. Hall 2006
I Origin of core-collapse supernovae (SN)I In massive (8− 25M�) star nucleosynthesis until FeI Core contractsI Core bounce shock-waveI CBS accelerates to supersonic speedsI SN explosionI Core remains
Thermal Properties of Hot and Dense Matter
The Physical Setting: Neutron Star (NS)
Credit: Adapted from Prakash et. al. (1997)
I t = 0sI Shock stand-offI ν’s trapped in coreI Mantle:
I AccretionI β-decay and
ν-emission
I t ∼ 0.5sI Mantle collapseI Black hole (BH)?
I → ν-signal ceases
I t ∼ 15sI ν-heatingI Deleptonization
I → ν-signal ceases
Thermal Properties of Hot and Dense Matter
The Physical Setting: Binary Merger
I Binary neutron stars coalesce
I Results in single black hole
I Temperatures can reach ∼ 100 MeV
Credit: Rezzolla et. al. APJ (2011)
(n ≈ 6× 10−16ρ)
Thermal Properties of Hot and Dense Matter
Aims
Supernovae Proto-Neutron Star Neutron Star Binary MergerYp ∼ 0.4− 0.35 ∼ 0− 0.3 ∼ 0− 0.3 ∼ 0− 0.3nn0
∼ 1− 6 ∼ 0.5− 7 ∼ 0.5− 7 . 2
T (MeV) ≤ 50 ∼ 50 . 10−3 T (S = 4− 9)Data Luminosity, Neutrino specta Maximum mass, Gravitational waves,
Element synthesis of all flavors Radius, Neutrino spectraNeutrino spectra Surface temperatures,
Pulsar period
I Calculate Properties for:I SN, PNS, NS and binary mergers
I Use state-of-the-art modelsI Micro/Macro physics
I Predict observables
I Tabulate equation of state for use in large-scale simulations
Thermal Properties of Hot and Dense Matter
Skyrme Models
HSkyrme =~2
2mnτn +
~2
2mpτp
+n(τn + τp)[ t1
4
(1 +
x1
2
)+
t2
4
(1 +
x2
2
)]+(τnnn + τpnp)
[t2
4
(1
2+ x2
)− t1
4
(1
2+ x1
)]+to2
(1 +
xo2
)n2 − to
2
(1
2+ xo
)(n2
n + n2p)[
t3
12
(1 +
x3
2
)n2 − t3
12
(1
2+ x3
)(n2
n + n2p)
]nε
I Contact interactions
I Function of n, Yi and τiI Simple to use
Thermal Properties of Hot and Dense Matter
The APR model
HAPR =
[~2
2m+ (p3 + Ynp5)ne−p4n
]τn
+
[~2
2m+ (p3 + Ypp5)ne−p4n
]τp
+g1(n)[1− (1− 2Yp)2)] + g2(n)(1− 2Yp)2
g1L[1− (1− 2Yp)2] + g2L(1− 2Yp)2
= g1H [1− (1− 2Yp)2] + g2H(1− 2Yp)2
I Uses nucleon-nucleon potentials that fit properties of light nuclei
I Hamiltonian density function of n, Yi (= nin ) and τi (K.E. den.)
I Includes effects of pion condensate at supra-nuclear densities
Thermal Properties of Hot and Dense Matter
The MDI model
I Potential contains integralsover momentum
I Computationally tricky atfinite temperatures
HMDI =1
2m(τn + τp)
+A1
2n0(nn + np)2 +
A2
2n0(nn − np)2
+B
σ + 1
(nn + np)σ+1
nσ0
[1− y
(nn − np)2
(nn + np)2
]+
Cl
n0
∑i
∫d3pi d
3p′ifi (~ri , ~pi )f
′i (~r ′i , ~p
′i )
1 +(~pi−~p′i
Λ
)2
+Cu
n0
∑i
∫d3pi d
3pjfi (~ri , ~pi )fj(~rj , ~pj)
1 +(~pi−~pj
Λ
)2 ; i 6= j .
Thermal Properties of Hot and Dense Matter
Fit to Experiment
I Hamiltonians fit tosome symmetricnuclear propertiesat T=0
I Using fittedparametrizationcalculateproperties ofmatter at zero andfinite temperatures
Property Value Value Value Experiment[MDI(A)] [SkO′] [APR]
n0 (fm−3) 0.160 0.160 0.160 0.17±0.02E0 (MeV) -16.00 -15.75 -16.00 -16±1K0 (MeV) 232.0 222.3 266 230±30m∗0/m 0.67 0.90 0.70 0.8±0.1Sv (MeV) 30.0 31.9 32.6 30-35Lv (MeV) 65.0 68.9 58.5 40-70
Thermal Properties of Hot and Dense Matter
Zero Temperature Matter
I Nucleon energies restricted to withinthe Fermi surface
I Fermi-Dirac (FD) integrals nowanalytical:→ H = H(n,Yp)
ni =1
π2
∫ kFi
0k2i dki =
k3Fi
3π2
τi =1
π2
∫ kFi
0k4i dki =
k5Fi
5π2
I Momentum integrals in HMDI
→ Functions of kF
εk = k2∂H∂τi
+∂H∂ni
m∗i = ~2kFi
(∂εk∂k
)kF
E = H/n
P = n∂H∂n−H
µi =∂H∂ni
Thermal Properties of Hot and Dense Matter
Effective Mass Splitting (1)
I Experimental uncertainties→ different parameterizations
I Can produce models with mn∗ ≷ mp
∗
I What governs mn∗ ≷ mp
∗?
εk = k2∂H∂τi
+∂H∂ni
m∗i = ~2kFi
(∂εk∂k
)kF
0 0.1 0.2 0.3 0.4 0.5 0.6 0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n (fm-3
)
x = 0.1
0.3
0.5
neutron
proton
Sly4(b)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6
m* /m
n (fm-3
)
x = 0.1
0.3
0.5
neutron
protonMDI(A)
(a)
Thermal Properties of Hot and Dense Matter
Effective Mass Splitting (2)
I What governs mn∗ ≷ mp
∗?I mn
∗ > mp∗ →
I MDI: Cl − Cu > 0I Skyrme: t1(1 + 2x1) > t2(1 + x2)
I Experiment to determinemn∗ ≷ mp
∗
I Choose mn∗ > mp
∗
0 0.1 0.2 0.3 0.4 0.5 0.6 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n (fm-3
)
0.1
0.3
x = 0.5
neutron
proton
SkO'
(b)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6
m* /m
n (fm-3
)
x = 0.1
0.3
0.5
neutron
proton
MDI(B)(a)
Thermal Properties of Hot and Dense Matter
Effective Mass Results
0 0.1 0.2 0.3 0.4 0.5 0.6
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n (fm-3
)
Ska
proton
neutron
x = 0.1
0.3
0.5
(d)
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5
m*/
m
n (fm-3
)
APR proton
neutron
x = 0.1
0.3
0.5
(c)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
n (fm-3
)
0.1
0.3
x = 0.5
neutron
proton
SkO'
(b)
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6
m* /m
n (fm-3
)
x = 0.1
0.3
0.5
neutron
protonMDI(A)
(a)
Thermal Properties of Hot and Dense Matter
Zero Temperature Matter (1)
I Smooth pressure for Skyrme andMDI(A)
I Pion condensate creates drop inpressure
I MDI(A) and SkO’ agree
I APR exhibits a pion condensate
-20
0
20
40
60
80
100
120
140
0 0.1 0.2 0.3 0.4 0.5 0.6
E (
Me
V)
n (fm-3
)
T = 0 MeV
x = 0.1
0.3
0.5
APRSka
MDI(A)SkO'
0
20
40
60
80
100
120
140
160
0 0.1 0.2 0.3 0.4 0.5 0.6
P (
Me
V f
m-3
)
n (fm-3
)
T = 0 MeV
x = 0.1
0.5
APRSka
MDI(A)SkO'
Thermal Properties of Hot and Dense Matter
Zero Temperature Matter (2)
I Models agree atlow densities
I Differences atsupra-nucleardensities
I Pion condensate
0
100
200
300
400
0 0.1 0.2 0.3 0.4 0.5 0.6
µn (
MeV
)
n (fm-3
)
x = 0.1
0.3
0.5
T = 0 MeV
APRSka
MDI(A)SkO'
Thermal Properties of Hot and Dense Matter
APR and Skyrme at Finite Temperatures
I FD solved with JEL
I Find ψi given niI Derivatives of FD integrals
as functions of other FD’s
I Thermal properties nowfunctions of m∗ and FD
I (APR) Pion condensategoverned by density effects,not temperature
ni =1
2π2
(2m∗i T
~2
)3/2
F1/2i
τi =1
2π2
(2m∗i T
~2
)5/2
F3/2i
Fαi =
∫ ∞0
xαie−ψi exi + 1
dxi
xi =1
T
(k2i
∂H∂τi
)=
1
T
~2k2i
2m∗i
ψi =1
T
(µi −
∂H∂ni
)=µi − Vi
T
Thermal Properties of Hot and Dense Matter
MDI at Finite Temperatures (1)
I Solve self consistentset of equations
I Iterate until µi and Ri
converge
I Use final fi and Ri forenergy, pressure ...
ni =
∫d3pi fi (~ri , ~pi )
fi (~ri , ~pi ) =2
(2π~)3
1
1 + e(εpi−µi )/T
εpi =p2i
2m+ Ui (ni , nj) + Ri (ni , nj , pi )
Ri (ni , nj , pi ) =2Cl
n0
2
(2π~)3
∫d3p′i
fp′i
1 +(~pi−~p′i
Λ
)2
+2Cu
n0
2
(2π~)3
∫d3pj
fpj
1 +(~pi−~pj
Λ
)2
Thermal Properties of Hot and Dense Matter
MDI at Finite Temperatures (2)
I Convergence in 3-4 iterations
I Larger changes at high temperatures
-100
-80
-60
-40
-20
0
0 1 2 3 4
Rn (
MeV
)
MDI(A)
T = 20 MeV
x = 0.0
(a)
n = 0.5 fm-3
0.34
0.17
0.085
R(0)
R(f)
k (fm-1
)
0 1 2 3 4-100
-80
-60
-40
-20
0
MDI(A)
T = 50 MeV
x = 0.0
(b)
n = 0.5 fm-3
0.34
0.17
0.085
R(0)
R(f)
-300
-250
-200
-150
-100
-50
0
0 1 2 3 4
Rn (
MeV
)
MDI(A)
T = 20 MeVx = 0.5
(c)
n = 0.5 fm-3
0.34
0.17
0.085
R(0)
R(f)
k (fm-1
)
0 1 2 3 4-300
-250
-200
-150
-100
-50
0
MDI(A)
T = 50 MeV
x = 0.5
(d)
n = 0.5 fm-3
0.34
0.17
0.085
R(0)
R(f)
Thermal Properties of Hot and Dense Matter
Thermal Properties (1)
I Similar low-n behavior
I Differences at high-n
I Pion condensate effects limited to zerotemperature properties
0
5
10
15
20
25
30
0.1 0.2 0.3 0.4 0.5
Eth
(M
eV
)
T = 20 MeV
x = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0.1 0.2 0.3 0.4 0.5 0
10
20
30
40
50
60
70
80
T = 50 MeV
x = 0.5
APRSka
MDI(A)SkO'
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0.1 0.2 0.3 0.4 0.5
Pth
(M
eV
fm
-3)
T = 20 MeVx = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0.1 0.2 0.3 0.4 0.5 0
2
4
6
8
10
12
14
16
18
T = 50 MeVx = 0.5
APRSka
MDI(A)SkO'
Thermal Properties of Hot and Dense Matter
Thermal Properties (2)
I Similar low-n behavior
I Differences at high-n
-50
-40
-30
-20
-10
0
10
0 0.1 0.2 0.3 0.4 0.5
µn,th (
Me
V)
T = 20 MeV
x = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0 0.1 0.2 0.3 0.4 0.5-200
-150
-100
-50
0
50
T = 50 MeV
x = 0.5
APRSka
MDI(A)SkO'
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1 0.2 0.3 0.4 0.5
S (
kB)
T = 20 MeV
x = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0.1 0.2 0.3 0.4 0.5 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
T = 50 MeV
x = 0.5
APRSka
MDI(A)SkO'
Thermal Properties of Hot and Dense Matter
Thermal Properties (3)
CV =∂E
∂T
∣∣∣∣n
I Similar trends
CP = CV +T
n2
(∂P∂T
∣∣n
)2
∂P∂n
∣∣T
(1)
I Peak due to ∂P∂n → 0
0
0.5
1
1.5
2
0.1 0.2 0.3 0.4 0.5
Cv (
kB)
T = 20 MeV
x = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0.1 0.2 0.3 0.4 0.5 0.5
1
1.5
2
2.5
T = 50 MeV
x = 0.5
APRSka
MDI(A)SkO'
0
0.5
1
1.5
2
2.5
3
3.5
4
0.1 0.2 0.3 0.4 0.5
Cp (
kB)
T = 20 MeV
x = 0.5
n (fm-3
)
APRSka
MDI(A)SkO'
0.1 0.2 0.3 0.4 0.5 0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
T = 50 MeV
x = 0.5
APRSka
MDI(A)SkO'
Thermal Properties of Hot and Dense Matter
Limiting Cases
I Degenerate Limit (low-T, high-n)I Landau’s Fermi Liquid Theory
I Non-degenerate (high-T, low-n)I Taylor expand FD integralsI Invert series F 1
2 ito find zi (ni ,T )
I Both:I Model dependence through m∗
I Provides check for resultsI Simple expressions
a =π2
2
m∗
kF2
S = 2aT
Eth = aT 2 (2)
Fαi ' Γ(α + 1)
(zi −
z2i
2α+1+ . . .
)zi =
niλ3i
γ+
1
23/2
(niλ
3i
γ
)2
λi =
(2π~2
m∗i T
)1/2
γ = 2 (the spin orientations) (3)
Thermal Properties of Hot and Dense Matter
Limiting Cases
0
5
10
15
20
25
30
0.1 0.2 0.3 0.4 0.5
Eth
(M
eV
)
T = 20 MeV
x = 0.5
0.0
Deg.
Exact
MDI(A)
n (fm-3
)
Non-
(a)
Deg.
0.2 0.4 0.6 0.8 0
20
40
60
80
T = 50 MeV
0.5
x = 0
Deg.
Exact
MDI(A)
Non-
(b)
Deg.
0
5
10
15
20
25
30
0.1 0.2 0.3 0.4 0.5E
th (
MeV
)
T = 20 MeV
0.5
x = 0
Deg.
Exact
SkO'
n (fm-3
)
Non-
(c)
Deg.
0.2 0.4 0.6 0.8 0
20
40
60
80
T = 50 MeV
0.5
x = 0
Deg.
Exact
SkO'
Non-
(d)
Deg.
Thermal Properties of Hot and Dense Matter
Leptons and Photons
I Leptons:I Electromagnetic interactions
negligibleI Treat as free Fermi gasI FD integrals solved with JEL
I Photons:I Described adequately as a blackbodyI Constant with baryon density
εγ =π2
15
T 4
(~c)3
Pγ =εγ3
sγ =4
3
εγT
(4)
0
50
100
150
200
250
300
350
400
450
0 0.1 0.2 0.3 0.4 0.5
µe (
Me
V)
n (fm-3
)
T = 0 MeV
x = 0.5
x = 0.1
50 MeV
Rel.JEL
0
20
40
60
80
100
120
140
160
180
0 0.1 0.2 0.3 0.4 0.5
Ee (
Me
V)
n (fm-3
)
T = 50 MeV
T = 50 MeV
x = 0.5
x = 0.1
0 MeV
0 MeV
Rel.JEL
Thermal Properties of Hot and Dense Matter
EOS with a Pion Condensate (1)
I Drop in pressure ⇒Mechanical instability
dP
dn≤ 0
I Maxwell construction toestablish mixed-phaseregion where
dP
dn= 0
I Solve system for nL and nH :
P(nL) = P(nH)
µ(nL) = µ(nH)
0
5
10
15
20
25
30
35
40
0.1 0.2 0.3
P (
MeV
fm
-3)
T = 20 MeV
APR
n (fm-3
)
Yp = 0.1
0.3
0.5
0.1 0.2 0.3 0
10
20
30
40
50
60
T = 50 MeV
APR
Yp = 0.1
0.3
0.5
Thermal Properties of Hot and Dense Matter
EOS with a Pion Condensate (2)
I Maxwell construction toestablish mixed-phaseregion where
dP
dn= 0
I Solve system for nL and nH :
P(nL) = P(nH)
µ(nL) = µ(nH)
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
0 0.1 0.2 0.3 0.4 0.5
n (
fm-3
)
Yp
nc
nH
nL
T = 0 MeV
50 MeV
APR
0
5
10
15
20
25
30
35
40
0.1 0.2 0.3
P (
MeV
fm
-3)
T = 20 MeV
APR
n (fm-3
)
Yp = 0.1
0.3
0.5
0.1 0.2 0.3 0
10
20
30
40
50
60
T = 50 MeV
APR
Yp = 0.1
0.3
0.5
Thermal Properties of Hot and Dense Matter
EOS with a Pion Condensate (3)
I No drop in pressure ⇒Mechanical stability
dP
dn≥ 0
0.15
0.17
0.19
0.21
0.23
0.25
0.27
0.29
0.31
0.33
0.35
0 0.1 0.2 0.3 0.4 0.5
n (
fm-3
)
Yp
nc
nH
nL
T = 0 MeV
50 MeV
APR
0
5
10
15
20
25
30
35
40
0.1 0.2 0.3
P (
MeV
fm
-3)
T = 20 MeV
APR
n (fm-3
)
Yp = 0.1
0.3
0.5
0.1 0.2 0.3 0
10
20
30
40
50
60
T = 50 MeV
APR
Yp = 0.1
0.3
0.5
0
5
10
15
20
25
30
35
40
0.1 0.2 0.3
P (
MeV
fm
-3)
T = 20 MeV
APR
n (fm-3
)
Yp = 0.1
0.3
0.5
0.1 0.2 0.3 0
10
20
30
40
50
60
T = 50 MeV
APR
Yp = 0.1
0.3
0.5
Thermal Properties of Hot and Dense Matter
EOS of a Cold Pure Neutron Star
Model Mmax ( M�) Rmax ( km) ncent ( fm−3)APR 2.20 10.15 1.11Ska 2.24 11.35 0.96
MDI(A) 2.00 10.51 1.15
SkO′ 1.99 10.47 1.16
PSR M( M�) Ref.J1614-2230 1.97± 0.04 [14]J0348+0432 2.01± 0.04 [15]
I T = 0 MeV and Yp = 0 obtain E (n) and P(n)
I Solve Tolman-Oppenheimer-Volkoff (TOV)equations and extract M(R) and ncent
0e+00
2e-04
4e-04
6e-04
8e-04
1e-03
0 0.2 0.4 0.6 0.8
ε (
MO•
km
-3)
x = 0
APR
Ska
MDI(A)
SkO'
0 0.2 0.4 0.6 0.8 10e+00
1e-04
2e-04
3e-04
4e-04
5e-04
6e-04
7e-04
P (
MO•
km
-3)
x = 0
n (fm-3
)
APR
Ska
MDI(A)
SkO'
0
0.5
1
1.5
2
2.5
8 10 12 14 16 18 20
MG
/MO•
R (km)
APR
Ska
x = 0
MDYIN
SkO’
Thermal Properties of Hot and Dense Matter
Adiabatic Index at Zero Temperature (1)
ΓS =∂ lnP
∂ ln n
∣∣∣∣S
=n
P
∂P
∂n
∣∣∣∣S
I Spinodalinstability when∂P∂n → 0
I Does not occurwhen leptons areincluded
1
2
3
4
0.001 0.01 0.1 1
Yp,e = 0
(d)
1
2
3
4
0.001 0.01 0.1 1
Yp,e = 0.1
(c)
1
2
3
4
0.001 0.01 0.1 1
Yp,e = 0.3
(b)
1
2
3
4
0.001 0.01 0.1 1
MDI(A)
T = 0 MeV
Γ
Yp,e = 0.5
n (fm-3
)
(a)
0.001 0.01 0.1 1
1
2
3
4
Ye = 0
(h) 0.001 0.01 0.1 1
1
2
3
4
Ye = 0.1
(g) 0.001 0.01 0.1 1
1
2
3
4
Ye = 0.3
(f) 0.001 0.01 0.1 1
1
2
3
4
SkO'
T = 0 MeV
Γ
Ye = 0.5
n (fm-3
)
(e)
Thermal Properties of Hot and Dense Matter
Adiabatic Index at Zero Temperature (2)
I Leptons dominate the totalpressure at low densities
-1
-0.5
0
0.5
1
0.001 0.01 0.1 1
P (
MeV
fm
-3)
MDI(A)T = 0 MeV
0.10.3
x = 0.5
(a)
NucleonLepton
0.01 0.1 1-1
-0.5
0
0.5
1
SkO'T = 0 MeV
0.10.3
x = 0.5
(b)
n (fm-3
)
NucleonLepton
Thermal Properties of Hot and Dense Matter
Adiabatic Index at Finite Entropies
I Interpolate to find T(n,S)
I Single ΓS crossing
I Interesting combination of (n,T)
1
10
100
0.001 0.01 0.1 1
T (
MeV
)
n (fm-3
)
Ye = 0.5
MDI(A)SkO'
S = 0.25
0.5
1
2
3
4
5
1.5
2
2.5
0.01 0.1 1
SkO'
(c)
Ye = 0.5
S = 2
3 4
ExactDeg
0.01 0.1 1
1.5
2
2.5
SkO'
(d)
Ye = 0.5
S = 2
3 4
ExactND
1.5
2
2.5
0.01 0.1 1
MDI(A)
(a)
Ye = 0.5
S = 2
34
ExactDeg
0.01 0.1 1
1.5
2
2.5
MDI(A)
(b)
Ye = 0.5
S = 2
34
n (fm-3
)
ΓS
ExactND
Thermal Properties of Hot and Dense Matter
ΓS Limiting Cases
I Degenerate→ (high-T, low-n)
I Non-Degenerate→ (low-T, high-n)
1.2
1.6
2
2.4
0.001 0.01 0.1 1
S = 2
1.2
1.6
2
2.4
0.001 0.01 0.1 1
S = 3
1.2
1.6
2
2.4
0.001 0.01 0.1 1
MDI(A)
S = 4
ΓS
Yp = 0.5
n (fm-3
)
ExactDegND
1.2
1.6
2
2.4
0.001 0.01 0.1 1
S = 2
1.2
1.6
2
2.4
0.001 0.01 0.1 1
S = 3
1.2
1.6
2
2.4
0.001 0.01 0.1 1
SkO'
S = 4
ΓS
Yp = 0.5
n (fm-3
)
ExactDeg.Non-Deg.
Thermal Properties of Hot and Dense Matter
Photon Entropies
I Photon entropynegligible at lowStotal
I Photons play arole at highStotal and n
I Sγ has T 3
dependence
0
0.3
0.6
0.9
1.2
0.001 0.01 0.1 1
Ye = 0.5 STot = 1
SkO'Nucleons
LeptonsPhotons
0.01 0.1 1 0
0.3
0.6
0.9
1.2Ye = 0.1 STot = 1
SkO'Nucleons
LeptonsPhotons
1
2
3
4
0.001 0.01 0.1 1
Ye = 0.5 STot = 4
SkO'Nucleons
LeptonsPhotons
0.01 0.1 1
1
2
3
4Ye = 0.1 STot = 4
n (fm-3
)
S (
kB)
SkO'Nucleons
LeptonsPhotons
Thermal Properties of Hot and Dense Matter
Future Work
Credit: Matthew W. Carmell, Stony Brook dissertation
Credit: Newton, William G. (2013)
I The Inhomogeneous PhaseI Low density structures and nucleiI Effects of nuclear deformationI Pasta phases
Thermal Properties of Hot and Dense Matter
Summary
I Hamiltonians fit to nuclear matter properties
I Effective mass splitting
I Models have similar properties for T=0
I APR distinguished by pion condensate
I Finite temperature properties governed byeffective mass
I Both leptons and photons play significant roles
I Limiting cases provide compact equations thatgive physical insight
Thermal Properties of Hot and Dense Matter
