Universal Relations for the Moment of Inertia in Relativistic Stars - - PowerPoint PPT Presentation

Universal Relations for the Moment of Inertia in Relativistic Stars

Cosima Breu

Goethe Universit¨ at Frankfurt am Main

Astro Coffee

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Universal relations

Motivation

Crab-nebula (de.wikipedia.org/wiki/Krebsnebel)

neutron stars as laboratories for unknown nuclear physics at supra-nuclear energy densities neutron star properties sensitively dependent on the modeling EOS

gravitational field determined by mass, radius and higher multipole moments approximately universal relations between certain quantities constraints on EOS and quantities which are not directly observable

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The Tolman-Oppenheimer-Volkoff Equations

first solution of Einstein’s equations for non-vacuum spacetimes

Einstein equations: Rµν − 1

2 gµνR = 8πTµν

Metric of a spherically symmetric matter distribution: 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 [Msun] 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

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The Equation of State

One-parameter EOS assumed: p = p(ρ) neutrons in β-equilibrium with protons and electrons (myons and hyperons at higher 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 log10(p) [dyn/cm2] 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 be

monsters): large uncertainties at high densities

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The Slow Rotation Approximation

www.vice.com/read/the-learning-corner-805-v18n5 Metric of a stationary, axisymmetric system: ds2 −eν(r)dt2 +eλ(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 to the local inertial frame: 1 r 4 d dr

• r 4j ¯

ω dr

• + 4

r dj dr ¯ ω = 0, j(r) = exp [(−ν + λ)/2] Coordinate angular velocity: ¯ ω = Ω − ω,

• utside: ¯

ω = Ω − 2J

r 3

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The Hartle-Thorne Perturbation Method

perturbative expansion up to second order in the angular velocity Second order: changes in pressure and energy density Expansion of the metric in spherical harmonics

ds2 = −eν(1 + 2h)dt2 + eλ[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)

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Numerical Setup

Slow Rotation: calculate the distribution of e,p and the gravitational field for a static configuration perturbations calculated retaining only first and second order terms equilibrium equations become a set of ordinary differential equations Fourth-order Runge-Kutta algorithm boundary conditions have to ensure metric continuity and differentiability Rapid Rotation: RNS-code for uniformly rotating stars solve hydrostatic and Einstein’s field equations for rigidly rotating, stationary and axisymmetric mass distributions KEH-scheme: elliptic-type field equations converted into integral equations

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The Moment of Inertia

50 100 150 200 250 300 0.5 1 1.5 2 2.5 3 I [Msunkm2] 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: Angular momentum linearly related to moment of inertia The moment of inertia

J =

• T t

φ

√−gd3x I(M, ν) = J/Ω = 8π 3 R r 4 (e + p) 1 − 2M(r)/r j(r) ¯ ω Ωdr

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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/(MR2) C [Msun/km]

APR WFF1 WFF2 HS DD2 HS TM1 HS TMA HS NL3 HS SFHo HS SFHx bhblp L O N Sly4 LS 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/M3 M/R [Msun/km]

APR WFF1 WFF2 HS DD2 HS TM1 HS TMa HS NL3 SFHo SFHx bhblp L O N Sly4 LS 220 YYM Fit

New Fit I M3 = a C 2 + b C 3 + c C 4

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The Quadrupole Moment

deviation of the gravitational field away from spherical symmetry extracted from the asymptotic expansion of the metric functions at large r Quadrupole moment Q(rot) = −J2 M − 8 5KM3 ¯ Q = QM/J2 ¯ Q approaches 1 for a Kerr black hole

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The Tidal Love Number

deformability due to tidal forces ratio between tidally introduced quadrupole moment and tidal field due to a companion NS

Tidal Love number − 1 − gtt 2 = − M r − 3Qij 2r3 ni nj + εij 2 r2ni nj λ(tid) = Q(tid) ε(tid) Fit:C = a + b ln ¯ λ + c(ln ¯ λ)2

detection of gravitational waves during the merger of neutron star binaries

0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 10 100 1000 10000 M/R [Msun/km]

− 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 tidal deformation approximation defined in buffer zone R ≫ R ≫ R∗ with R being the radius of curvature of the source of the perturbation

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The I-Love-Q Relations by Yagi and Yunes

10 I/M3

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-- IFit|/- IFit Q/(M3j2)

Reduced Moment of inertia ¯ I versus the Kerr factor QM/J2

10 I/M3

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-- IFit|/- IFit

− λ tid

Reduced Moment of inertia ¯ I versus the tidal Love number ¯ λtid Fitting function ln yi = ai + bi ln xi + ci(ln xi)2 + di(ln xi)3 + ei(ln xi)4

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Rapid Rotation

equilibrium between gravitational, pressure and centrifugal forces Kepler angular velocity

0.5 1 1.5 2 2.5 5e+14 1e+15 1.5e+15 2e+15 2.5e+15 M [Msun] 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

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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/(M3) 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 function

• f the observationally

important (but dimensionful) rotational angular velocity

6 8 10 12 14 16 18 20 1 2 3 4 5 6 7 8 9 I/(M3) Q/(M3j2)

rotation characterized by dimensionless spin parameter j = J/M2

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Rapidly Rotating Models

20 40 60 80 100 120 140 160 180 0.5 1 1.5 2 2.5 I [Msunkm2] 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/(MR2)

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-- IFit|/- IFit M/R [Msun/km] 5 10 15 20 25 30 35 I/M3

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-- IFit|/- IFit M/R [Msun/km]

dimensionless angular momentum j = J/M2 instead of J

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Radius Measurement

uncertainties in the modeling of atmospheres and radiation processes simultaneous measurement of mass and moment of inertia of a radio binary pulsar (e.g. PSR J0737-3039) determination of I up to 10 % accuracy through periastron advance and geodetic precession constraints on radius and EOS

1 1.5 2 2.5 3 9 10 11 12 13 14 M [Msun] 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))

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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 → realistic EOSs are similar to each other approximation by elliptical isodensity contours modern EOSs are stiff → limit of an incompressible fluid

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Thank you!

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Universal relations