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Universal Relations for the Moment of Inertia inRelativistic Stars
Cosima Breu
Goethe Universitat Frankfurt am Main
Astro Coffee
C. Breu Universal relations
Motivation
Crab-nebula (de.wikipedia.org/wiki/Krebsnebel)
neutron stars as laboratories forunknown nuclear physics atsupra-nuclear energy densities
neutron star propertiessensitively dependent on themodeling EOS
gravitational fielddetermined by mass, radiusand higher multipolemoments
approximately universalrelations between certainquantities
constraints on EOS andquantities which are notdirectly observable
C. Breu Universal relations
The Tolman-Oppenheimer-Volkoff Equations
first solution of Einstein’sequations for non-vacuumspacetimes
Einstein equations: Rµν − 12gµνR = 8πTµν
Metric of a spherically symmetric matterdistribution:
ds2 = −eν(r)dt2 + eλ(r)dr2 + r2(dθ2 + sin θ2dφ2)
e−λ(r) = 1 − 2M(r)r
Energy-momentum tensor of a perfect fluid:Tµν = (e + p)uµuν + pgµν
EOS needed to close system of equations
0.5
1
1.5
2
2.5
3
9 10 11 12 13 14 15
M [
Msu
n]
R [km]
APR
WFF1
WFF2
HS DD2
HS TM1
HS TMa
HS NL3
SFHo
SFHx
bhblp
L
O
N
Sly4
LS 220
stiff eos
soft eos
intermed. eos
TOV-equations
dM
dr= 4πr2e
dp
dr= −
(e + p)(M + 4πr3p)
r(r − 2M)
dν
dr= −
2
(e + p)
dp
dr
C. Breu Universal relations
The Equation of State
One-parameter EOSassumed: p = p(ρ)
neutrons in β-equilibriumwith protons and electrons(myons and hyperons athigher densities)
T = 0
33
33.5
34
34.5
35
35.5
36
36.5
14 14.2 14.4 14.6 14.8 15 15.2 15.4
log
10(p
) [d
yn
/cm
2]
log10(e) [g/cm3]
soft eos
stiff eos
intermed. eos
APR
Sly 4
L
N
O
LS220
WFF1
WFF2
bhblp
HS DD2
HS NL3
SFHo
SFHx
HS TM1
HS TMa
1 solid outer crust (e,Z )
2 inner crust (e,Z , n)
3 outer core (n,Z , e, µ)
4 inner core (here bemonsters): largeuncertainties at highdensities
C. Breu Universal relations
The Slow Rotation Approximation
www.vice.com/read/the-learning-corner-805-v18n5
Metric of a stationary, axisymmetric system:
ds2 −eν(r)dt2 +e
λ(r)dr2 + r2(dθ2 + sin θ2(dφ−Ldt)2)
Expansion of L(r, θ): L(r, θ) = ω(r, θ) + O(ω3)
local inertial frames are dragged along by the rotating fluid
expansion until first order in the angular velocity
Scale-invariant differential equation for the angular velocity relative tothe local inertial frame:
1
r4
d
dr
(r4j
ω
dr
)+
4
r
dj
drω = 0, j(r) = exp [(−ν + λ)/2]
Coordinate angular velocity: ω = Ω− ω, outside: ω = Ω− 2Jr3
C. Breu Universal relations
The Hartle-Thorne Perturbation Method
perturbative expansion up tosecond order in the angularvelocity
Second order: changes inpressure and energy density
Expansion of the metric inspherical harmonics
ds2 = −eν (1 + 2h)dt2 + e
λ[1 + 2m/(r − 2M)]dr2
+ r2(1 + 2k)[dθ2 + sin θ2(dφ− ωdt)2] + O(Ω3)
h(r, θ) = h0(r) + h2(r)P2(θ) + ...
k(r, θ) = k0(r) + k2(r)P2(θ) + ...
m(r, θ) = m0(r) + m2(r)P2(θ) + ...
P2 = (3 cos θ2 − 1)/2
Coordinate system
r = R + ξ(R, θ) + O(Ω4)
C. Breu Universal relations
Numerical Setup
Slow Rotation:
calculate the distribution of e,p and the gravitational field fora static configuration
perturbations calculated retaining only first and second orderterms
equilibrium equations become a set of ordinary differentialequations
Fourth-order Runge-Kutta algorithm
boundary conditions have to ensure metric continuity anddifferentiability
Rapid Rotation:
RNS-code for uniformly rotating stars
solve hydrostatic and Einstein’s field equations for rigidlyrotating, stationary and axisymmetric mass distributions
KEH-scheme: elliptic-type field equations converted intointegral equations
C. Breu Universal relations
The Moment of Inertia
50
100
150
200
250
300
0.5 1 1.5 2 2.5 3
I [M
su
nkm
2]
M [Msun]
APR
WFF1
WFF2
HS DD2
HS TM1
HS TMa
HS NL3
SFHo
SFHx
bhblp
L
O
N
Sly4
LS 220
stiff eos
soft eos
intermed. eos
Slow Rotation: Angularmomentum linearly relatedto moment of inertia
The moment of inertia
J =
∫T tφ
√−gd3x
I (M, ν) = J/Ω
=8π
3
∫ R
0
r4 (e + p)
1− 2M(r)/rj(r)
ω
Ωdr
C. Breu Universal relations
I-C -Relation
0.3
0.35
0.4
0.45
0.5
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
I/(M
R2)
C [Msun/km]
APRWFF1WFF2
HS DD2HS TM1HS TMAHS NL3
HS SFHoHS SFHx
bhblpLON
Sly4LS 220
Fit LS
Lattimer and Schutz 2005
I/(MR2) = a + bM
R+ c
(M
R
)4
5
10
15
20
25
30
35
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
I/M
3
M/R [Msun/km]
APRWFF1WFF2
HS DD2HS TM1HS TMaHS NL3
SFHoSFHxbhblp
LON
Sly4LS 220
YYMFit
New Fit
I
M3=
a
C 2+
b
C 3+
c
C 4
C. Breu Universal relations
The Quadrupole Moment
deviation of the gravitational field away from sphericalsymmetry
extracted from the asymptotic expansion of the metricfunctions at large r
Quadrupole moment
Q(rot) = −J2
M− 8
5KM3
Q = QM/J2
Q approaches 1 for a Kerr black hole
C. Breu Universal relations
The Tidal Love Number
deformability due to tidal forces
ratio between tidally introducedquadrupole moment and tidalfield due to a companion NS
Tidal Love number
−1 − gtt
2= −
M
r−
3Qij
2r3ni nj +
εij
2r2ni nj
λ(tid) =
Q(tid)
ε(tid)
Fit:C = a + b ln λ + c(ln λ)2
detection of gravitationalwaves during the merger ofneutron star binaries
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
10 100 1000 10000
M/R
[M
su
n/k
m]
−
λ
tid
Maselli
APR
WFF1
WFF2
HS DD2
Sly4
L
N
O
bhblp
HS NL3
HS TM1
HS TMA
SFHo
SFHx
LS 220
computed in small tidaldeformation approximation
defined in buffer zoneR R R∗ with R beingthe radius of curvature of thesource of the perturbation
C. Breu Universal relations
The I-Love-Q Relations by Yagi and Yunes
10
I/M
3
Yagi Yunes
APR
WFF1
WFF2
HS DD2
Sly4
L
N
O
bhblp
HS NL3
HS TM1
HS TMA
SFHo
SFHx
LS 220
0.001
0.01
1 10
|- I-- I F
it|/- I F
it
Q/(M3j2)
Reduced Moment of inertiaI versus the Kerr factorQM/J2
10
I/M
3
Yagi Yunes
APR
WFF1
WFF2
HS DD2
Sly4
L
N
O
bhblp
HS NL3
HS TM1
HS TMA
SFHo
SFHx
LS 220
0.001
0.01
10 100 1000 10000
|- I-- I F
it|/- I F
it
−
λ
tid
Reduced Moment of inertiaI versus the tidal Lovenumber λtid
Fitting function
ln yi = ai + bi ln xi + ci (ln xi )2 + di (ln xi )
3 + ei (ln xi )4
C. Breu Universal relations
Rapid Rotation
equilibrium between gravitational, pressure and centrifugalforcesKepler angular velocity
0.5
1
1.5
2
2.5
5e+14 1e+15 1.5e+15 2e+15 2.5e+15
M [
Msu
n]
ec [g/cm3]
Slow Rot. App.
j=0.2
j=0.25
j=0.3
j=0.35
j=0.4
j=0.45
j=0.5
j=0.55
j=0.6
j=0.65
C. Breu Universal relations
Breakdown of Universal Relations
Breakdown of universal relations
6
8
10
12
14
16
18
20
1 2 3 4 5 6 7 8 9
I/(M
3)
Q/(M3j2)
APR
WFF1
WFF2
HS DD2
Sly 4
L
N
O
bhblp
HS NL3
HS TM1
HS TMA
SFHo
SFHx
LS 220
YY
I − Q-relation as a functionof the observationallyimportant (butdimensionful) rotationalangular velocity
6
8
10
12
14
16
18
20
1 2 3 4 5 6 7 8 9
I/(M
3)
Q/(M3j2)
rotation characterized bydimensionless spinparameter j = J/M2
C. Breu Universal relations
Rapidly Rotating Models
20
40
60
80
100
120
140
160
180
0.5 1 1.5 2 2.5
I [M
su
nkm
2]
Msun
Slow Rot. App.
j=0.2
j=0.25
j=0.3
j=0.35
j=0.4
j=0.45
j=0.5
j=0.55
j=0.6
j=0.65
0.25
0.3
0.35
0.4
0.45
I/(M
R2)
Fit LS
APR
Sly4
LS 220
0.001
0.01
0.1
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
|- I-- I F
it|/
- I Fit
M/R [Msun/km]
5
10
15
20
25
30
35
I/M
3
Slow Rot. Approx.
j=0.2
j=0.3
j=0.4
j=0.5
j=0.6
0.001
0.01
0.1
0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2
|- I-- I F
it|/
- I Fit
M/R [Msun/km]
dimensionless angularmomentum j = J/M2
instead of J
C. Breu Universal relations
Radius Measurement
uncertainties in the modeling of atmospheres and radiationprocesses
simultaneous measurement of mass and moment of inertia ofa radio binary pulsar (e.g. PSR J0737-3039)
determination of I up to 10 % accuracy through periastronadvance and geodetic precession
constraints on radius and EOS
1
1.5
2
2.5
3
9 10 11 12 13 14
M [
Msu
n]
R [km]
I=50 Msunkm2
I=80 Msunkm2
I=120 Msunkm2
APR
Sly 4
WFF1
LS 220
N
bhblp
0.055
0.06
0.065
0.07
0.075
0.08
0.085
0.09
0.095
0.1
0.105
0.11
1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2
δR
/R
M [Msun]
I=50 (I/M3)
I=50 (I/(MR2))
I=80 (I/M3)
I=80 (I/(MR2))
I=120 (I/M3)
I=120 (I/(MR2))
C. Breu Universal relations
Outlook
newly born NS → fast differential rotation, high temperature
Underlying physics?
approach of limiting values of Kerr-metric
dependence on internal structure far from the core → realisticEOSs are similar to each other
approximation by elliptical isodensity contours
modern EOSs are stiff → limit of an incompressible fluid
C. Breu Universal relations
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