The Nuclear Equation of State
and Neutron Star Structure
James M. Lattimer
lattimer@astro.sunysb.edu
Department of Physics & Astronomy
Stony Brook University
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 1/68
Outline
•
Structure
•
Dense Matter
Equation of State
•
Formation and Evolution
•
Observational Constraints
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 2/68
PSR 1919+21
sleevage.com/joy-division-unknown-pleasures/
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 3/68
Credit: Dany Page, UNAM
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 4/68
Amazing Facts About Neutron Stars
•
Densest objects this side of an event horizon: 1015 g cm−3
Four teaspoons, on the Earth’s surface, would weigh as much as the Moon.
•
Largest surface gravity: 1014 cm s−2
This is 100 billion times the Earth’s gravity.
•
Fastest spinning objects known: ν = 716 Hz
This spin rate was measured for PSR J1748-2446ad, which is
located in the globular cluster Terzan 5 located 28,000 light
years away. (33 pulsars have been found in this cluster.)
If the radius is about 15 km, the velocity at the equator
is one fourth the speed of light.
•
Largest known magnetic field strengths: B = 1015 G
The Sun’s field strength is about 1 G.
•
Highest temperature superconductor: Tc = 10 billion K
The highest known superconductor on the Earth is mercury thallium barium
calcium copper oxide (Hg12 Tl3 Ba30 Ca30 Cu45 O125 ), at 138 K.
•
Highest temperature, at birth, anywhere in the Universe since the Big Bang:
T = 700 billion K
•
Fastest measured velocity of a massive object in the Galaxy: 1083 km/s
This velocity was measured for PSR B1508+55, and is 1/300 the speed of light. It
is actually an underestimate, since only the transverse velocity can be measured.
•
The only place in the universe except for the Big Bang where neutrinos become
trapped.
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 5/68
Spherically Symmetric General Relativity
Static metric:
ds2 = eλ(r) dr2 + r2 dθ2 + sin2 θdφ2 − eν(r) dt2
Einstein’s equations:
Mass:
8πǫ(r)
=
8πp(r)
=
p′ (r)
=
m(r) = 4π
Boundaries:
r=0
Z
r
′
1 −λ(r)
−λ(r) λ (r)
,
1−e
+e
r2
r
′
1 −λ(r)
−λ(r) ν (r)
,
+e
− 2 1−e
r
r
p(r) + ǫ(r) ′
−
ν (r).
2
ǫ(r′ )r′2 dr′ ,
0
e−λ(r) = 1 − 2m(r)/r
m(0) = p′ (0) = ǫ′ (0) = 0,
r=R
m(R) = M,
p(R) = 0, eν(R) = e−λ(R) = 1 − 2M/R
Thermodynamics:
d(ln n)
=
mn(r)
=
N
=
1 dǫ
dǫ
de
dǫ
=−
dν,
h=
,
ǫ = n(m + e),
p = n2
ǫ+p
2 dp
dn
dn
(ǫ(r) + p(r))e(ν(r)−ν(R))/2 − n0 e0 ;
p = 0 : n = n 0 , e = e0
Z R
4πr2 eλ(r)/2 n(r)dr;
BE = N m − M
0
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 6/68
Neutron Star Structure
Tolman-Oppenheimer-Volkov equations of relativistic
hydrostatic equilibrium:
dp
G (m + 4πpr3 )(ǫ + p)
= − 2
dr
c r(r − 2Gm/c2 )
ǫ 2
dm
= 4π 2 r
dr
c
p is pressure, ǫ is mass-energy density
Useful analytic solutions exist:
• Uniform density
• Tolman VII
• Buchdahl
ǫ = constant
ǫ = ǫc [1 − (r/R)2 ]
√
ǫ = pp∗ − 5p
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 7/68
Uniform Density Fluid
m(r)
=
e−λ(r)
=
eν(r)
=
p(r)
=
ǫ(r)
BE
M
=
=
M
4π 3
ǫr ,
β≡
3
R
1 − 2β(r/R)2 ,
2
q
3p
1
1 − 2β −
1 − 2β(r/R)2 ,
2
2
#
" p
√
2
1 − 2β(r/R) − 1 − 2β
p
,
ǫ √
2
3 1 − 2β − 1 − 2β(r/R)
constant;
n(r) = constant
−1 √
p
3
9 2
sin
2β
3β
√
− 1 − 2β ≃
+
β + ···
4β
2β
5
14
pc < ∞ =⇒ β < 4/9
c2s = ∞
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 8/68
Tolman
VII
ǫ(r)
=
e−λ(r)
=
eν(r)
=
p(r)
=
n(r)
=
φ(r)
=
w(r)
=
ǫc 1 − (r/R)2 ≡ ǫc [1 − x]
1 − βx(5 − 3x)
(1 − 5β/3) cos2 φ,
q
1
β
3βe−λ(r) tan φ(r) − (5 − 3x) ,
2
4πR
2
ǫ(r) + p(r) cos φ(r)
mb
cos φ1
s
w1 − w(r)
β
+ φ1 ,
φ1 = φ(x = 1) = tan−1
,
2
3(1 − 2β)


s
s
#
"
−λ(r)
5
1
e
1 − 2β
,
ln  x − +
+
.
w1 = w(x = 1) = ln
6
3β
6
3β
(P/ǫ)c =
2 tan φc
15
s
1
3
− ,
β
3
c2s,c = tan φc
1
tan φc +
5
r
β
3
!
BE
11
7187 2
≃
β+
β + ···
M
21
18018
π
pc < ∞ =⇒ φc < , β < 0.3862
2
c2s,c < 1 =⇒ β < 0.2698
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 9/68
Buchdahl
eν(r)
=
eλ(r)
8πp(r)
8πǫ(r)
=
=
=
mb n(r)
=
u(r)
=
r′
=
A2
=
pc =
√
ǫ = p∗ p − 5p
(1 − 2β)(1 − β − u(r))(1 − β + u(r))−1 ,
(1 − 2β)(1 − β + u(r))(1 − β − u(r))−1 (1 − β + β cos Ar′ )−2 ,
A2 u(r)2 (1 − 2β)(1 − β + u(r))−2 ,
2A2 u(r)(1 − 2β)(1 − β − 3u(r)/2)(1 − β + u(r))−2 ,
s
! 3/2
−1
r
p
1
p(r)
p
∗
−5
,
c2s (r) =
p∗ p(r) 1 − 4
p∗
2 p(r)
−1
r
1
p
β
∗
′
,
−1
sin
Ar
=
(1
−
β)
Ar′
2 p(r)
r(1 − 2β)(1 − β + u(r))−1 ,
r
π
2πp∗ (1 − 2β)−1 ,
R = (1 − β)
.
2p∗ (1 − 2β)
p∗ 2
β ,
4
ǫc =
p∗
5
β(1 − β),
2
2
nc mb =
p∗
β(1 − 2β)3/2
2
3
β
β2
BE
−1/2
−1
= (1 − β)(1 − 2β)
(1 − β)
≃ +
+ ···
M
2
2
2
c2s,c < 1 =⇒ β < 1/6
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 10/68
Tolman IV Variation: A Self-Bound Star
Finite, large surface energy density, e.g., strange quark star
Lake’s solution
ν(r)
=
eλ(r)
=
4πpR2
=
e
4πǫR
2
=
m
=
c2s
=
ǫsurf
ǫc
=
[1 − 52 β(1 − 51 x)]2
,
(1 − 2β)
(1 − 52 β(1 − 53 x))2/3
,
(1 − 25 β(1 − 53 x))2/3 − 2(1 − β)2/3 βx
#
"
5
(1 − 2 β(1 − x))
β
2/3
,
1
−
(1
−
β)
1
3
5
5
2/3
1 − 2 β(1 − 5 x)
(1 − 2 β(1 − 5 x))
3(1 − β)
2/3
β
1 − 25 β(1 − 31 x)
(1 − 52 β(1
2
β) 3 M x3/2
−
3
x))5/3
5
,
(1 −
(1 − 25 β(1 − 53 x))2/3
"
#
5/3
(2 − 5β + 3βx) (2 − 5β + 3βx)
2
2
+
(2
−
5β)
−
5β
x ,
5(2 − 5β + βx)3
22/3 (1 − β)2/3
2/3
5
5
(1 − β)−5/3 .
1− β
1− β
3
2
0.30 < c2s,c < 0.44,
0.265 <
ǫsurf
<1
ǫc
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 11/68
Mass-Radius Diagram and Theoretical Constraints
GR:
R > 2GM/c2
P <∞:
R > (9/4)GM/c2
causality:
2
R>
∼ 2.9GM/c
— normal NS
— SQS
— R∞ = √
R
1−2GM/Rc2
contours
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 12/68
Mass Measurements In X-Ray Binaries
Mass function
P (v2 sin i)3
f (M1 ) = 2πG
(M1 sin i)3
(M1 +M2 )2
=
> M1
f (M2 ) =
X-ray timing
P (v1 sin i)3
2πG
(M2 sin i)3
(M1 +M2 )2
=
> M2
In an X-ray binary, voptical has the
largest uncertainties. In some cases,
sin i ∼ 1 if eclipses are observed.
If eclipses are not observed, limits
to i can be made based on the
estimated radius of the optical star.
Optical spectroscopy
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 13/68
Pulsar Mass Measurements
Mass function for pulsar precisely obtained.
It is also possible in some cases to
17◦ yr−1 in PSR 0737
obtain the rate of periastron advance
and the Einstein gravitational redshift
+ time dilation term:
ω̇ = 3(2π/P )5/3 (GM/c2 )2/3 /(1 − e2 )
γ = (P/2π)1/3 eM2 (2M2 + M1 )(G/M 2 c2 )2/3
Gravitational radiation leads to orbit
decay:
73 2
37 4 M1 M2
192π 2πG 5/3
2 −7/2
(1 − e )
1 + 24 e + 96 e M 1/2
Ṗ = − 5c5
P
In some cases, can constrain Shapiro time delay,
r is magnitude and s = sin i is shape parameter.
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 14/68
PSR0737-3039, Kramer et al. (2007)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 15/68
Black hole? ⇒
Firm lower mass limit?⇒
M < 1.17 M⊙ (95%) ⇒
M > 1.68 M⊙ , 95% confidence{
Freire et al. 2007 {
Although simple average
mass of w.d. companions
is 0.27 M⊙ larger, weighted
average is 0.08 M⊙ smaller
companion?
} w.d.
statistics?
Champion et al. 2008
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 16/68
Roche Model for Maximal Rotation
(c.f., Shapiro & Teukolsky 1983)
ρ−1 ∇P = ∇h = −∇(ΦG + Φc ),
ΦG ≃ −GM/r,
Φc = − 21 Ω2 r2 sin2 θ
Bernoulli integral: H = h + ΦG + Φc = −GM/Rp
Rp
Enthalpy h = 0 ρ−1 dp = µn (ρ) − µn (0) in beta equilibrium
Numerical calculations show Rp is nearly
constant for arbitrary rotation
Ω2 R3
R
eq
Evaluate at equator: 2GM
= Req − 1
Req
:
Mass-shedding limit Ω2shed = GM
3
R
Req
=
3
2
GR: Cook, Shapiro & Teukolsky (1994): 1.43–1.51
q
2 3/2
GM
Ωshed = 3
R3
Pshed = 1.0 (R/10 km)3/2 (M⊙ /M )1/2 ms
GR: Lattimer & Prakash (2005): 0.96 ± 3%
Shape:
Rp
R(θ)
Limit:
Ω2 R3 sin2 θ
2GM
=
1
2
1
+
cos[
3
3
3
Rp
= 13 + 23
R(θ)
=
R
Rp
−1
sin θ 2
cos−1 (1 − 2( Ω
) )]
Ω
shed
cos[ 31 cos−1 (1 − 2 sin2 θ)] =
sin(θ)
.
3 sin(θ/3)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 17/68
Extreme Properties of Neutron Stars
• The most compact configurations occur when the
ǫ0 is the only
EOS parameter
⇐=
soft
=⇒
ca
u
sa
l
p
=
lim ǫ −
it
ǫo
low-density equation of state is "soft" and the
high-density equation of state is "stiff".
The TOV
solutions scale
with ǫ0
stiff
p=0 ◦
ǫo
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 18/68
Maximally Compact Equation of State
Koranda, Stergioulas & Friedman (1997)
p(ǫ) = 0,
p(ǫ) = ǫ − ǫo ,
ǫ
ǫ
≤
≥
ǫo
ǫo
This EOS has a parameter ǫo , which corresponds to the surface energy density. The
structure equations then contain only this one parameter, and can be rendered into
dimensionless form using
1/2
y = mǫo
dy
dx
dq
dx
1/2
q = pǫ−1
o .
,
x = rǫo ,
=
4πx2 (1 + q)
=
(y + 4πqx3 )(1 + 2q)
−
x(x − 2y)
The solution with the maximum central pressure and mass and the minimum radius:
qmax = 2.026,
ymax = 0.0851,
xmin /ymax = 2.825
1/2
ǫs
GMmax
ǫo
MeV fm−3 , Mmax = 4.1
.
M⊙ , Rmin = 2.825
pmax = 307
ǫs
ǫo
c2
Moreover, the scaling extends to the axially-symmetric case, yielding
1/2
3/2 R
ǫs
M⊙ 1/2
Mmax 1/2
−1/2
ms = 0.76
∝ ǫo
, Pmin = 0.82
ms
Pmin ∝
3
Rmin
ǫo
10 km
M
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 19/68
Maximum Possible Density in Stars
The scaling from the maximally compact EOS yields
ǫc,max = 3.026
4.1 M⊙
Mmax
2
ǫs ≃ 13.7 × 1015
M⊙
Mmax
2
g cm−3 .
A virtually identical result arises from combining the maximum compactness constraint
(Rmin ≃ 2.9GM/c2 ) with the Tolman VII relation
2 3
15 M
c
15
1
ǫc,V II =
=
8π R3
8π 2.9G
M2
Maximum possible density:
2.2 M⊙ ⇒
ǫmax < 2.8 × 1015 g cm−3
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 20/68
Maximum Mass, Minimum Period
Theoretical limits from GR and causality
• Mmax = 4.2(ǫs /ǫf )1/2 M⊙
Rhoades & Ruffini (1974), Hartle (1978)
• Rmin = 2.9GM/c2 = 4.3(M/M⊙ ) km
Lindblom (1984), Glendenning (1992), Koranda, Stergioulas & Friedman (1997)
• ǫc < 4.5 × 1015 (M⊙ /Mlargest )2 g cm−3
Lattimer & Prakash (2005)
• Pmin ≃ 0.74(M⊙ /Msph )1/2 (Rsph /10 km)3/2 ms
Koranda, Stergioulas & Friedman (1997)
• Pmin ≃ 0.96 ± 0.03(M⊙ /Msph )1/2 (Rsph /10 km)3/2 ms
(empirical)
Lattimer & Prakash (2004)
• ǫc > 0.91 × 1015 (1 ms/Pmin )2 g cm−3
• cJ/GM 2 . 0.5
(empirical)
(empirical, neutron star)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 21/68
Constraints from Pulsar Spins
XTE J1739-285
ν = 1122 Hz
Kaaret et al. 2006
Not confirmed to
be a rotation rate
PSR J1748-2446ad
ν = 716 Hz
Hessels et al. 2006
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 22/68
BE(M, R)
Lattimer & Prakash (2001)
BE ≃ (0.60 ±
0.05) GM
Rc2
1−
GM −1
2Rc2
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 23/68
Moment of Inertia
I
=
=
where
Z R
8π
4
(λ(r)−ν(r))/2
r
[ǫ(r)
+
p(r)]
e
ω(r)dr
3c4 0
Z
dj(r)
2c2 R 3
dr,
r ω(r)
−
3G 0
dr
j(r)
dω(r)
d
r4 j(r)
dr
dr
=
j(R)
=
=
e−(λ(r)+ν(r))/2 ;
dj(r)
;
−4r3 ω(r)
dr
1,
ω(R) = 1 −
2GI
,
R3 c2
dω(0)
= 0.
dr
c2 4 dω(R)
R
I=
6G
dr
Combining these:
With φ = d ln ω/d ln r, φ(0) = 0,
dφ
dr
=
I
=
φ
d ln j
− (3 + φ) − (4 + φ)
,
r
dr
φR R3 c2
R3 φR c2
φR c2 3
R ωR =
− 2I =
.
6G
6
G
G(6 + 2φR )
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 24/68
I(M, R)
Lattimer & Prakash (2001)
M km
+ 90
I ≃ (0.237 ± 0.008)M R2 1 + 4.2 R
M
⊙
M km
R M⊙
4 J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 25/68
The Uncertain Nuclear Force
⇐==⇒
The density dependence of Esym (n) is crucial but poorly
constrained. Although the second density derivative, the
incompressibility K, for symmetric matter is known well,
the third density derivative, the skewness, is not.
Esym (n)
K ≃ 220 MeV
⇐= Skewness
C. Fuchs, H.H. Wolter, EPJA 30(2006) 5
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 26/68
Schematic Energy Density
n: number density; x: proton fraction; T : temperature
ns ≃ 0.16 ± 0.01 fm−3 : nuclear saturation density
B ≃ −16 ± 1 MeV: saturation binding energy
K ≃ 220 ± 15 MeV: incompressibility parameter
Sv ≃ 30 ± 6 MeV: bulk symmetry parameter
a ≃ 0.065 ± 0.010 MeV−1 : bulk level density parameter
ǫ(n, x, T )
=
P
=
µn
=
=
µ̂
=
s
=
"
2
#
n
n
s
+ Sv
T2
(1 − 2x)2 + a
ns
ns
n
n2 K
n
2an ns 2/3 2
2 ∂(ǫ/n)
2
n
=
− 1 + Sv (1 − 2x) +
T
∂n
ns 9 ns
3
n
x ∂ǫ
∂ǫ
−
∂n
n ∂x
K
n
n
a ns 2/3 2
n
B+
1−3
+ 2Sv
(1 − 2x) −
T
1−
18
ns
ns
ns
3 n
n
1 ∂ǫ
= µn − µp = 4Sv
(1 − 2x)
−
n ∂x
ns
n 2/3
1 ∂ǫ
s
= 2a
T
n ∂T
n
n B+
K
18
1−
n 2/3
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 27/68
Phase Coexistence
Schematic energy density
ǫ
=
P
=
µn
=
µ̂
=
"
K
18
2
#
n
n
s
+ Sv
T2
(1 − 2x)2 + a
ns
ns
n
n
2an ns 2/3 2
n2 K
2
T
− 1 + Sv (1 − 2x) +
ns 9 ns
3
n
K
n
n
a ns 2/3 2
n
B+
T
1−3
+ 2Sv
(1 − 2x) −
1−
18
ns
ns
ns
3 n
n 2/3
n
s
µn − µp = 4Sv
T
(1 − 2x) ,
s = 2a
ns
n
n B+
1−
n 2/3
Free Energy Minimization With Two Phases
F = ǫ−nT s = uFI +(1−u)FII ,
∂F
= 0,
∂nI
∂F
= 0,
∂xI
n = unI +(1−u)nII ,
∂F
=0
∂u
=⇒ µnI = µnII ,
Critical Point (Ye = 0.5) ∂P 5
ns ,
nc =
12
Tc =
5
12
∂n
1/3 =
T
5K
32a
∂2P
∂n2
1/2
,
nYe = uxI nI +(1−u)xII nII
µpI = µpII ,
PI = PII
=0
T
sc =
12
5
1/3 5Ka
8
1/2
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 28/68
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 29/68
The Pressure of Neutron Star Matter
Expansion of cold nucleonic matter energy near ns and isospin symmetry x = 1/2:
E(n, x)
≃
P (n, x)
≃
µe
=
Beta Equilibrium:
xβ
≃
Pβ
=
3~c
E(n, 1/2) + Esym (n)(1 − 2x)2 +
x(3π 2 nx)1/3 ,
4
dE
~c
dE(n,
1/2)
sym
+
(1 − 2x)2 +
nx(3π 2 nx)1/3 ,
n2
dn
dn
4
2
K
n
~c(3π 2 nx)1/3 ,
E(n, 1/2) ≃ −B +
.
1−
18
ns
∂E
∂x
n
= µp − µn + µe = 0 .
3
4E
sym
,
(3π 2 n)−1
~c
Kn2
dEsym
n
+ Esym nxβ (1 − 2xβ )
− 1 + n2 (1 − 2xβ )2
9n0
ns
dn
Esym (ns ) ≡ Sv ≃ 30 MeV, ~c ≃ 200 MeV/fm,
xβ → 0.04 ,
n → ns =⇒
dE
sym Pβ → n2s
.
dn ns
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 30/68
The Uncertain Esym (n)
C. Fuchs, H.H. Wolter, EPJA 30(2006) 5
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 31/68
Neutron Star Matter Pressure
p ≃ Knγ
γ = d ln p/d ln n ∼ 2
Wide variation:
p(ns )
1.2 < MeV
<7
fm−3
⇓
⇑
n↓
s
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 32/68
Polytropes
Polytropic Equation of State: p = Knγ
n is number density, γ is polytropic exponent.
Hydrostatic Equilibrium in Newtonian Gravity:
Gm(r)n(r)
dp(r)
=−
,
dr
r2
dm(r)
= 4πnr2
dr
Dimensional analysis:
3
M ∝ nc R ,
When γ ∼ 2:
M2
p∝ 4,
R
R ∝ K 1/(3γ−4) M (γ−2)/(3γ−4)
1/2 −1
nf M 0
R ∝ K 1/2 M 0 ∝ pf
General Relativistic analysis using Buchdahl’s solution:
p
r
1 (1 − β)(2 − β) 1 − 10 p/p∗
d ln R π
p
=
,
R = (1 − β)
.
2p∗ (1 − 2β)
d ln p n,M
2 (1 − 3β + 3β 2 ) 1 + 2 p/p∗
For M = 1.4M⊙ , R = 14 km, n = 1.5ns , ǫ = 1.5mb ns ≃ 3 × 10−4 km−2 :
β = 0.148,
p∗ = 0.00826,
p/p∗ = 0.00221.
d ln R ≃ 0.234
d ln p n,M
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 33/68
The Radius – Pressure Correlation
R ∝ p1/4
Lattimer & Prakash (2001)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 34/68
Nuclear Symmetry Energy
The density dependence of Esym (n) is crucial. Some information is available from
nuclei (for n < ns ). Heavy ion collisions have potential for constraining it for n > ns .
It is common to expand Esym (n) as
L
Esym (n) ≃ J +
3
J = Esym (ns ),
L = 3ns
2
Ksym
n
n
+ ···
−1 +
−1
ns
18
ns
∂Esym
∂n
,
Ksym = 9n2s
ns
∂ 2 Esym
∂n2
ns
Almost no information is available for Ksym .
What information can constrain Esym (n) from the laboratory?
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 35/68
Nuclear Mass Formula
Bethe-Weizsäcker (neglecting pairing and shell effects)
E(A, Z) = −av A + as A2/3 + aC Z 2 /A1/3 + Sv AI 2 .
Myers & Swiatecki introduced the surface asymmetry term:
E(A, Z) = −av A + as A2/3 + aC Z 2 /A1/3 + Sv AI 2 − Ss (N − Z)2 /A4/3 .
Droplet extension: consider the neutron/proton asymmetry of the nuclear surface.
E(A, Z) = (−av + Sv δ 2 )(A − Ns ) + as A2/3 + aC Z 2 /A1/3 + µn Ns .
Ns is the number of excess neutrons associated
with the surface, I = (N − Z)/(N + Z),
δ = 1 − 2x = (A − Ns − 2Z)/(A − Ns ) is the
asymmetry of the nuclear bulk fluid, and
µn is the neutron chemical potential.
From thermodynamics,
∂as A2/3
I−δ
Ss δ
Ns = −
=A
,
=
∂µn
Sv 1 − δ
1−δ
−1
Ss
δ =I 1+
,
1/3
Sv A
E(A, Z) = −av A + as A2/3 + aC Z 2 /A1/3 + Sv AI 2
Ss
1+
Sv A1/3
−1
.
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 36/68
Nuclear Structure Considerations
Information about Esym can be extracted from nuclear binding energies and models for
nuclei. For example, consider the schematic liquid droplet model (Myers & Swiatecki):
Sv
A + ac Z 2 A−1/3
−1/3
1 + (Ss /Sv )A
Fitting binding energies results in a strong correlation between Sv and Ss , but not
definite values.
E(A, Z) ≃ −av A + as A2/3 +
Blue: ∆E < 0.01 MeV/b
Green: ∆E < 0.02 MeV/b
Gray: ∆E < 0.03 MeV/b
Circle: Moeller et al. (1995)
Crosses: Best fits
Dashed: Danielewicz (2004)
Solid: Steiner et al. (2005)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 37/68
Schematic Dependence
Nuclear Hamiltonian:
Q
H = HB + n′2 ,
2
"
HB ≃ n −B +
K
18
1−
n
ns
2 #
+ Esym (1 − 2x)2
Lagrangian minimization of energy with respect
(symmetric matter):
to n K
n 2
Q ′2
,
µ0 = −av
n 1−
HB − µ0 n = n =
2
18
ns
Ss = 4πr02 σδ
Liquid Droplet surface parameters: as = 4πr02 σ0 ,
Z +∞
Z ns
q
dn
4
QKn3s
σ0 =
[H − µ0 n]dz =
(HB − µ0 n) ′ =
n
45
−∞
0
r
r
Z
Z 0.9ns
du
Qns 0.9
Qns
dn
≃
9
=
3
t90−10 =
√
′
K 0.1
u(1 − u)
K
0.1ns n
r Z
Sv
Q ns
σ δ = Sv
n
− 1 (HB − µ0 n)−1/2 dn
2 0
Esym
√ Z
u
Sv t90−10 ns 1
Sv
− 1 du
=
3
Esym
0 1−u
p
Z
2
1
n
=⇒
→ 0.28 (p = ) , 0.93 (p = ) , 2.0 (p = 1)
Esym ≃ Sv
ns
2
3
s
r
−1
Z
L
S
L
n
3Sv
v
Esym ≃ Sv +
1+
− 1 tan−1
=⇒
→2−
≃1+
3 ns
L
3L
3Sv
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 38/68
Schematic Dependence
Ss
t90−10
≃
Sv
r0
Z
≃ 2.05
Z
=⇒ 0.57
1.91
4.1 .
For Pb208 :
δR =
r
3
t90−10 A − 2Z
(Rn − Rp ) ≃
5
6
Z(1 − Z/A)
Z
=⇒ 0.05
0.16
0.35
PREX experiment (E06002) at Jefferson Lab to measure
the neutron radius of lead to about 1% accuracy (current
accuracy is about 5%) using the parity violating asymmetry
in elastic scattering due to the weak neutral interaction.
Requires corrections for Coulomb distortions (Horowitz).
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 39/68
=⇒
~
′
K~
~
~
~
~
From T. Klähn
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 40/68
Flow Constraint From Heavy Ions
K ′ = 0 MeV
K ′ = 100 MeV
K ′ = 200 MeV
K ′ = 300 MeV
Consistent with
0 < K ′ /MeV < 300
Danielewicz, Lacey & Lynch 2002
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 41/68
Nuclei in Dense Matter
Liquid Droplet Model, Simplified
F = u(FI + fLD /VN ) + (1 − u)FII ,
fC
=
fT
=
fS
=
fLD = fS + fC + fT
u
3 1/3
3 Z 2 e2
3 Z 2 e2
+
1− u
D(u)
=
5 RN
2
2
5 RN
u
mT 3/2
T ln
− T = µT − T ,
nQ =
2π~2
nQ VN A3/2
2
4πRN
σ(µs )
n = unI + (1 − u)nII ,
nYe = unI xI + (1 − u)nII xII + u
Ns
VN
Free Energy Minimization
∂F
= 0,
∂zi
µn,II
=
PII
=
zi = (nI , xI , RN , u, νs , µs )
µT
3σ
2 ∂σ
,
µ̂II = µ̂I −
= −µs ,
Ns = −4πRN
A
RN nI xi
∂µs
!
1/3
uD′
15σ
3σ
1+
,
RN =
PI +
2RN
D
8πn2I x2I e2 D
µn,I +
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 42/68
Cold Catalyzed Matter
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 43/68
Cold Catalyzed Matter
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 44/68
Supernova Matter
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 45/68
Proto-Neutron Stars
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 46/68
Proto-Neutron Star Evolution
dYL
dt
ds
nT
dt
n
=
=
dYe
dYν
1 ∂ 2
+
= 2
r Fν
dt
dt
r ∂r
X
1 ∂Lν
dYi
−
−
n
µ
.
i
4πr2 ∂r
dt
n,p,e,ν
n
In the diffusion approximation, fluxes are driven by density gradients:
Z ∞ c
∂nν̄ (Eν )
∂nν (Eν )
Fν = −
− λν̄
λν
dEν ,
3
∂r
∂r
0
Z ∞
X cλi ∂ǫi (Eν )
2
E
dEν .
Lν = −
4πr
3
∂r
0
i
i
λν and λE ’s are mean free paths for number and energy transport, respectively. nν (Eν )
and ǫi (Eν ) are the number and energy density of species i = e, µ at neutrino energy Eν .
There are two main sources of opacity:
1. ν-nucleon absorption. Affects only e−types.
2. Neutrino-electron scattering. Inelastic scattering affects all types of neutrinos.
Mean free paths for these processes are approximately:
1. λν ≃ λν̄ ≃ 5 cm, λν ∝ Eν−2 ;
2. λiE ≃ 100 cm, λiE ∝ T −1 Eν−2 .
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 47/68
Model Simulations
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 48/68
Model Simulations
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 49/68
Model Signal
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 50/68
Effective Minimum Masses
Strobel, Schaab & Weigel (1999)
2.4
CNS
2.2
s1
2.0
s1
s2
LPNSYL04, LPNSYL04
1.8
s4s1
EPNS YL04
1.6
MG [ Msun ]
s2
HNS , HNS
s5s1
EPNS YL04
1.4
Yℓ = 0.4, sin = 1, sout = 4 − 5 ⇐ M = 1.18 M⊙
1.2
1.0
0.8
Yℓ = 0.4, s = 1 − 2
0.6
0.4
Yν = 0, s = 1 − 2
0.2
0.0
T =0
8
12
16
20
24
28
32
Rinf [ km ]
36
40
44
48
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 51/68
Neutron Star Cooling
Gamow & Schönberg proposed the direct Urca process: nucleons at the top of the Fermi
sea beta decay. c
p → n + e+ + ν̄e
n → p + e− + νe ,
Energy conservation guaranteed by beta equilibrium
µn − µp = µe
Momentum conservation requires |kF n | ≤ |kF p | + |kF e |.
Charge neutrality requires kF p = kF e ,
therefore |kF p | ≥ 2|kF n |.
3 , thus x ≥ x
Degeneracy implies ni ∝ kF
DU = 1/9.
i
With muons (n > 2ns ), xDU =
2
2+(1+21/3 )3
≃ 0.148
If x < xDU , bystander nucleons needed:
modified Urca process is then dominant.
(n, p) + p → (n, p) + n + e+ + ν̄e
(n, p) + n → (n, p) + p + e− + νe ,
Neutrino emissivities:
ǫ̇M U RCA ≃
T
µn
2
ǫ̇DU RCA .
Beta equilibrium composition:
xβ ≃ (3π 2 n)−1
4Esym
~c
3
≃ 0.04
n
ns
0.5−2
.
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 52/68
Direct Urca Threshold
Klähn et al., Phys. Rev. C74 (2006) 035802
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 53/68
Neutron Star Cooling
Page, Steiner, Prakash & Lattimer (2004)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 54/68
Light element
envelopes
Neutron 3 P2 gap = 0
Light element
envelopes
Neutron 3 P2 gap = 0
Heavy
element envelopes
Heavy
element envelopes
Neutron 3 P2 gap = "a"
Neutron 3 P2 gap = "b"
Neutron 3 P2 gap = "b"
Minimal Cooling Paradigm
Neutron 3 P2 gap = "a"
Page, Lattimer, Prakash & Steiner (2009)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 55/68
Minimal Cooling Paradigm
• If some observations are inconsistent with the MCP,
then according to Sherlock Holmes, rapid cooling must
occur for these exceptions.
• All sources are consistent with the MCP only IF
• tight conditions are placed on the magnitude and
density dependence of the neutron 3 P2 gap, AND
• some neutron stars have heavy Z envelopes and
others have light Z envelopes, AND
• ALL core-collapse supernova remnants with no
observable thermal emission contain black holes.
• Highly suggestive that rapid cooling occurs in some
neutron stars (of higher masses?)
• A possible constraint on Esym (n), i.e., it’s not supersoft?
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 56/68
Possible Kinds of Observations
• Maximum and Minimum Mass (binary pulsars)
• Minimum Rotational Period∗
• Radiation Radii or Redshifts from X-ray Thermal
∗
Emission∗
• Crustal Cooling Timescale from X-ray Transients∗
• X-ray Bursts from Accreting Neutron Stars∗
• Seismology from Giant Flares in SGR’s∗
• Neutron Star Thermal Evolution (URCA or not)∗
• Moments of Inertia from Spin-Orbit Coupling∗
• Neutrinos from Proto-Neutron Stars (Binding Energies,
Neutrino Opacities, Radii)∗
• Pulse Shape Modulations∗
• Gravitational Radiation from Neutron Star Mergers∗
(Masses, Radii from tidal Love numbers)
Significant dependence on symmetry energy
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 57/68
Potentially Observable Quantities
•
•
Apparent angular diameter from flux and temperature measurements
r
β ≡ GM/Rc2
R
1
1
R∞
F∞
√
=
=
2 T2
D
D 1 − 2β
σ f∞
∞
Redshift
z = (1 − 2β)−1/2 − 1
•
Eddington flux
•
Crust thickness
FEDD =
mb c2
ln H ≡ ht =
2
GM c
(1 − 2β)1/2
2
2
κc D
Z
pt
0
dp
= µn,t − µn,t (p = 0)
n
∆
R − Rt
(H − 1)(1 − 2β)
≡
≃
≃ (H − 1)
R
R
H + 2β − 1
•
Moment of Inertia
•
Crustal Moment of Inertia
•
Binding Energy
1
−1 .
2β
I ≃ (0.237 ± 0.008)M R2 (1 + 2.84β + 18.9β 4 ) M⊙ km2
∆I
8π R6 pt
≃
I
3 IM c2
B.E. ≃ (0.60 ± 0.05)
β
1 − β/2
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 58/68
Radiation Radius
• Combination of flux and temperature measurements
yields apparent angular diameter (pseudo-BB):
R∞
R
1
= p
d
d 1 − 2GM/Rc2
• Observational uncertainties include distance,
interstellar H absorption (hard UV and X-rays),
atmospheric composition
• Best chances for accurate radii are from
• Nearby isolated neutron stars (parallax measurable)
• Quiescent X-ray binaries in globular clusters
(reliable distances, low B H-atmosperes)
• X-ray pulsars in systems of known distance
• CXOU J010043.1-721134 in the SMC: R∞ ≥ 10.8
km (Esposito & Mereghetti 2008)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 59/68
RX J1856-3754
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 60/68
Radiation Radius: Nearby Neutron Star
RX J1856-3754:
Walter & Lattimer 2002
Braje & Romani 2002
Truemper 2005
D=120 pc
BUT
D=140 pc
Kaplan, van Kerkwijk
& Anderson 2002
Raises R∞
limit to 19.5 km
Magnetic H atmosphere
R∞ ≈ 17 km
Ho et al. 2007
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 61/68
Radiation Radius: Globular Cluster Sources
Webb & Barret 2007
47 Tuc
M 13
ω Cen
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 62/68
The Neutron Star Crust
Hydrostatic equilibrium in the crust:
dp(r)
dµ
GM
=
dr.
=− 2
mb n
mb
r − 2GM r/c2
rt (R − 2GM/c2 )
1
µt − µ0
= ln
.
2
2
mb c
2
R(rt − 2GM/c )
Defining ln H = 2(µt − µ0 )/mb c2 ,
∆
R − Rt
(H − 1)(1 − 2β)
R2
≡
=
≃ (H − 1)(1 − 2β)
.
R
R
H + 2β − 1
2M
Crustal moment of inertia
8π
∆I = 4
3c
Approximately,
R0
Z
R
8πω(R)
r4 (ǫ + p)e−λ jωdr ≃
3M c2
R−∆
p(R−∆)
r6 dp ≃ R6 pt e−4.8∆/R .
∆I
8πR4 pt
≃
I
3M 2 c2
M R2
− 2β
I
Z
0
r6 dp
p(R−∆)
pt < 0.65 MeV fm−3 .
e−4.8∆/R
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 63/68
Pulsar Glitches
Pulsars occasionally undergo glitches, when the spin rate "hiccups".
Each glitch changes the angular momentum of the star by ∆J = Iliquid ∆Ω.
The glitches are stochastic, but total angular momentum transfer is regular.
X ∆Ω
J(t) = Iliquid Ω̄
,
Ω
d(∆Ω/Ω)
J˙ = Iliquid Ω̄
.
dt
A leading model is that as the crust slows due to pulsar’s dipole radiation, the interior
acquires an excess differential rotation. The acquired excess is limited: J˙ ≤ Ω̇Icrust .
Icrust
Icrust
Ω̄ d
=
≥
Iliquid
I − Icrust
Ω̇
Crab Glitch
P
(∆Ω/Ω)
≃ 0.014.
dt
8
spin rate
10 ∆Ωc/Ωc
5
liquid
0
glitch
-5
0
5
time (days)
crust
10
time
cumulative angular momentum
instability
10
-5
1970
−5
2X10
1980
1990
Vela
−5
1X10
Link, Epstein & Lattimer (1999)
40000
45000
50000
MJD
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 64/68
Moment of Inertia
• Spin-orbit coupling of same magnitude as
post-post-Newtonian effects (Barker & O’Connell 1975,
Damour & Schaeffer 1988)
• Precession alters inclination angle and periastron
advance
• More EOS sensitive than R: I ∝ M R2
• Requires extremely relativistic system to extract
• Double pulsar PSR J0737-3037 is a marginal
candidate
• Even more relativistic systems should be found, based
on dimness and nearness of PSR J0737-3037
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 65/68
Moments of Inertia and Precession
•
~˙ A = −L
~˙ =
Spin-orbit coupling: S
G(4MA +3MB ) ~
L
2MA a3 c2
~A
×S
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Moments of Inertia and Precession
•
•
B) ~
~˙ A = −L
~˙ = G(4MA +3M
~A
Spin-orbit coupling: S
L×S
2MA a3 c2
Precession Period:
2(MA + MB )ac2
Pp =
P (1 − e2 ) ≃ 74.9 years
GMB (4MA + 3MB )
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Moments of Inertia and Precession
•
•
B) ~
~˙ A = −L
~˙ = G(4MA +3M
~A
Spin-orbit coupling: S
L×S
2MA a3 c2
Precession Period:
2(MA + MB )ac2
Pp =
P (1 − e2 ) ≃ 74.9 years
GMB (4MA + 3MB )
•
~A × L:
~
Precession Amplitude ∝ S
|S~A |
IA (MA + MB ) P
−5
sin
θ
≃
(3.6
−
7.2)
sin
θ
×
10
sin θ ≃
δi =
~
a2 M A M B
PA
|L|
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Moments of Inertia and Precession
•
•
B) ~
~˙ A = −L
~˙ = G(4MA +3M
~A
Spin-orbit coupling: S
L×S
2MA a3 c2
Precession Period:
2(MA + MB )ac2
Pp =
P (1 − e2 ) ≃ 74.9 years
GMB (4MA + 3MB )
•
~A × L:
~
Precession Amplitude ∝ S
|S~A |
IA (MA + MB ) P
−5
sin
θ
≃
(3.6
−
7.2)
sin
θ
×
10
sin θ ≃
δi =
~
a2 M A M B
PA
|L|
•
Delay in Time-of-Arrival:
MB
a
∆t =
δi cos i ≈ 0.4 − 4.0 sin θ µs
MA + MB c
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Moments of Inertia and Precession
•
•
B) ~
~˙ A = −L
~˙ = G(4MA +3M
~A
Spin-orbit coupling: S
L×S
2MA a3 c2
Precession Period:
2(MA + MB )ac2
Pp =
P (1 − e2 ) ≃ 74.9 years
GMB (4MA + 3MB )
•
~A × L:
~
Precession Amplitude ∝ S
|S~A |
IA (MA + MB ) P
−5
sin
θ
≃
(3.6
−
7.2)
sin
θ
×
10
sin θ ≃
δi =
~
a2 M A M B
PA
|L|
•
Delay in Time-of-Arrival:
MB
a
∆t =
δi cos i ≈ 0.4 − 4.0 sin θ µs
MA + MB c
•
~A · L:
~ AP /AP N =
Periastron Advance ∝ S
!r
2πIA
2 + 3MB /MA
MA + MB
cos θ ≃ (2.2−4.3)×10−4 cos θ
2
2
PA
3MA + 3MB + 2MA MB
Ga
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Moments of Inertia and Precession
•
•
B) ~
~˙ A = −L
~˙ = G(4MA +3M
~A
Spin-orbit coupling: S
L×S
2MA a3 c2
Precession Period:
2(MA + MB )ac2
Pp =
P (1 − e2 ) ≃ 74.9 years
GMB (4MA + 3MB )
•
~A × L:
~
Precession Amplitude ∝ S
|S~A |
IA (MA + MB ) P
−5
sin
θ
≃
(3.6
−
7.2)
sin
θ
×
10
sin θ ≃
δi =
~
a2 M A M B
PA
|L|
•
Delay in Time-of-Arrival:
MB
a
∆t =
δi cos i ≈ 0.4 − 4.0 sin θ µs
MA + MB c
•
~A · L:
~ AP /AP N =
Periastron Advance ∝ S
!r
2πIA
2 + 3MB /MA
MA + MB
cos θ ≃ (2.2−4.3)×10−4 cos θ
2
2
PA
3MA + 3MB + 2MA MB
Ga
•
Moment of Inertia – Mass – Radius
"
I ≃ (0.237 ± 0.008)M R2 1 + 4.2
M km
+ 90
R M⊙
M km
R M⊙
4 #
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 66/68
Comparison of Binary Pulsars
References
a/c (s)
P (h)
e
MA (M⊙ )
MB (M⊙ )
TGW (M yr)
i
PA (ms)
θA
φA
PpA (yr)
δta /IA,80 (µs)
ApA /(A1P N IA,80 )
A2P N /A1P N
PSR 0707-3039
a, b, c
PSR 1913+16
d
PSR 1534+12
e, f
2.93
2.45
0.088
6.38
7.75
0.617
7.62
10.1
0.274
1.337 ± 0.005
1.250 ± 0.005
1.4414 ± 0.0002
1.3867 ± 0.0002
87.9 ± 0.6◦
245
1.333 ± 0.001
1.345 ± 0.001
47.2◦
77.2◦
59
37.9
85
22.7
◦
10◦
◦
13 ±
246◦ ± 5
74.9
0.7 ± 0.6
+0.2
× 10−5
3.4−0.1
5.2 × 10−5
21.1◦
◦
9.7
297.2
11.2
1.0 × 10−5
4.7 × 10−5
2250
25.0◦ ± 3.8◦
290◦ ± 20◦
700
7.9 ± 1.1
+0.04
× 10−5
1.15−0.03
2.3 × 10−5
a: Lyne et al. (2004); b: Solution 1, Jenet & Ransom (2004); c: Coles et al. (2004)
d: Weisberg & Taylor (2002, 2004); e: Stairs et al. (2002, 2004); f: Bogdanov et al. (2002)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 67/68
EOS Constraint
Lattimer & Schutz (2005)
Bejger & Haensel (2003)
J.M. Lattimer, CompSchool2009, NBIA, 17-21 August 2009 – p. 68/68