Universal Relations for the Moment of Inertia in ... · Universal Relations for the Moment of Inertia in Relativistic Stars Cosima Breu Goethe Universit at Frankfurt am Main Astro

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


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


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


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


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)


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


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


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


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


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


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


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


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


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


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))


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


Thank you!


Documents

Universal Relations for the Moment of Inertia in ... · Universal Relations for the Moment of Inertia in Relativistic Stars Cosima Breu Goethe Universit at Frankfurt am Main Astro