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 C. Breu Universal relations

Motivation gravitational field determined by mass, radius and higher multipole moments approximately universal relations between certain quantities constraints on EOS and Crab-nebula (de.wikipedia.org/wiki/Krebsnebel) quantities which are not neutron stars as laboratories for directly observable unknown nuclear physics at supra-nuclear energy densities neutron star properties sensitively dependent on the modeling EOS C. Breu Universal relations

The Tolman-Oppenheimer-Volkoff Equations 3 APR first solution of Einstein’s WFF1 WFF2 HS DD2 HS TM1 HS TMa 2.5 HS NL3 equations for non-vacuum SFHo SFHx bhblp L 2 O spacetimes M [M sun ] N Sly4 LS 220 stiff eos soft eos 1.5 intermed. eos 1 Einstein equations: R µν − 1 2 g µν R = 8 π T µν 0.5 Metric of a spherically symmetric matter 9 10 11 12 13 14 15 distribution: R [km] ds 2 = − e ν ( r ) dt 2 + e λ ( r ) dr 2 + r 2 ( d θ 2 + sin θ 2 d φ 2 ) TOV-equations e − λ ( r ) = 1 − 2 M ( r ) r Energy-momentum tensor of a perfect fluid: T µν = ( e + p ) u µ u ν + pg µν dM = 4 π r 2 e dr EOS needed to close system of equations ( e + p )( M + 4 π r 3 p ) dp = − dr r ( r − 2 M ) d ν 2 dp = − dr ( e + p ) dr C. Breu Universal relations

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 36.5 soft eos 1 solid outer crust ( e , Z ) stiff eos intermed. eos 36 APR Sly 4 L 2 inner crust ( e , Z , n ) N 35.5 O log 10 (p) [dyn/cm 2 ] LS220 WFF1 WFF2 3 outer core ( n , Z , e , µ ) 35 bhblp HS DD2 HS NL3 34.5 SFHo 4 inner core (here be SFHx HS TM1 HS TMa 34 monsters): large 33.5 uncertainties at high 33 14 14.2 14.4 14.6 14.8 15 15.2 15.4 densities log 10 (e) [g/cm 3 ] C. Breu Universal relations

The Slow Rotation Approximation Metric of a stationary, axisymmetric system: ds 2 − e ν ( r ) dt 2 + e λ ( r ) dr 2 + r 2 ( d θ 2 +sin θ 2 ( d φ − Ldt ) 2 ) Expansion of L ( r , θ ): L ( r , θ ) = ω ( r , θ ) + O ( ω 3 ) www.vice.com/read/the-learning-corner-805-v18n5 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 j ¯ ω � + 4 d dj dr ¯ ω = 0 , j ( r ) = exp [( − ν + λ ) / 2] r 4 dr dr r ω = Ω − 2 J Coordinate angular velocity: ¯ ω = Ω − ω , outside: ¯ r 3 C. Breu Universal relations

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 ds 2 = − e ν (1 + 2 h ) dt 2 + e λ [1 + 2 m / ( r − 2 M )] dr 2 + r 2 (1 + 2 k )[ d θ 2 + sin θ 2 ( d φ − ω dt ) 2 ] + O (Ω 3 ) Coordinate system h ( r , θ ) = h 0 ( r ) + h 2 ( r ) P 2 ( θ ) + ... r = R + ξ ( R , θ ) + O (Ω 4 ) k ( r , θ ) = k 0 ( r ) + k 2 ( r ) P 2 ( θ ) + ... m ( r , θ ) = m 0 ( r ) + m 2 ( r ) P 2 ( θ ) + ... P 2 = (3 cos θ 2 − 1) / 2 C. Breu Universal relations

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 C. Breu Universal relations

The Moment of Inertia 300 APR The moment of inertia WFF1 WFF2 250 HS DD2 HS TM1 HS TMa 200 HS NL3 � √− gd 3 x SFHo I [M sun km 2 ] T t J = SFHx φ bhblp 150 L O N I ( M , ν ) = J / Ω 100 Sly4 LS 220 � R stiff eos = 8 π 1 − 2 M ( r ) / r j ( r ) ¯ ( e + p ) ω soft eos 50 r 4 intermed. eos Ω dr 3 0 0.5 1 1.5 2 2.5 3 M [M sun ] Slow Rotation: Angular momentum linearly related to moment of inertia C. Breu Universal relations

I- C -Relation 0.5 APR 35 WFF1 APR WFF2 WFF1 HS DD2 WFF2 HS TM1 HS DD2 HS TMA HS TM1 HS NL3 HS TMa 30 HS SFHo HS NL3 0.45 HS SFHx SFHo bhblp SFHx L bhblp O L N O Sly4 25 N LS 220 Sly4 Fit LS LS 220 0.4 YYM Fit I/(MR 2 ) I/M 3 20 0.35 15 0.3 10 5 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 C [M sun /km] M/R [M sun /km] Lattimer and Schutz 2005 New Fit M 3 = a I C 2 + b C 3 + c � 4 � M C 4 I / ( MR 2 ) = a + bM R + c R C. Breu Universal relations

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 ) = − J 2 M − 8 5 KM 3 Q = QM / J 2 ¯ ¯ Q approaches 1 for a Kerr black hole C. Breu Universal relations

The Tidal Love Number 0.2 Maselli deformability due to tidal forces APR WFF1 0.18 WFF2 HS DD2 ratio between tidally introduced 0.16 Sly4 L M/R [M sun /km] quadrupole moment and tidal N 0.14 O bhblp field due to a companion NS HS NL3 0.12 HS TM1 HS TMA SFHo 0.1 SFHx LS 220 Tidal Love number 0.08 0.06 10 100 1000 10000 1 − g tt M 3 Q ij ε ij 2 r 3 n i n j + r 2 n i n j − tid = − − − λ 2 r 2 Q ( tid ) λ ( tid ) = computed in small tidal ε ( tid ) deformation approximation λ ) 2 Fit: C = a + b ln ¯ λ + c (ln ¯ defined in buffer zone R ≫ R ≫ R ∗ with R being the radius of curvature of the detection of gravitational source of the perturbation waves during the merger of neutron star binaries C. Breu Universal relations

The I-Love-Q Relations by Yagi and Yunes Yagi Yunes L HS TMA Yagi Yunes L HS TMA APR N SFHo APR N SFHo WFF1 O SFHx WFF1 O SFHx WFF2 bhblp LS 220 WFF2 bhblp LS 220 HS DD2 HS NL3 HS DD2 HS NL3 Sly4 HS TM1 Sly4 HS TM1 I/M 3 10 I/M 3 10 I Fit I Fit I Fit |/ - 0.01 I Fit |/ - 0.01 I- - I- - | - | - 0.001 0.001 1 10 10 100 1000 10000 Q/(M 3 j 2 ) − tid λ Reduced Moment of inertia Reduced Moment of inertia ¯ ¯ I versus the Kerr factor I versus the tidal Love number ¯ QM / J 2 λ tid Fitting function ln y i = a i + b i ln x i + c i (ln x i ) 2 + d i (ln x i ) 3 + e i (ln x i ) 4 C. Breu Universal relations

Rapid Rotation equilibrium between gravitational, pressure and centrifugal forces Kepler angular velocity Slow Rot. App. j=0.2 2.5 j=0.25 j=0.3 j=0.35 j=0.4 2 j=0.45 j=0.5 M [M sun ] j=0.55 j=0.6 j=0.65 1.5 1 0.5 5e+14 1e+15 1.5e+15 2e+15 2.5e+15 e c [g/cm 3 ] C. Breu Universal relations

Breakdown of Universal Relations Breakdown of universal relations 20 20 18 18 16 16 14 14 APR I/(M 3 ) I/(M 3 ) WFF1 WFF2 12 12 HS DD2 Sly 4 L N 10 10 O bhblp HS NL3 8 HS TM1 8 HS TMA SFHo SFHx 6 6 LS 220 YY 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 Q/(M 3 j 2 ) Q/(M 3 j 2 ) ¯ I − ¯ Q -relation as a function rotation characterized by of the observationally dimensionless spin parameter j = J / M 2 important (but dimensionful) rotational angular velocity C. Breu Universal relations

Rapidly Rotating Models 180 35 Slow Rot. App. Slow Rot. Approx. j=0.2 j=0.2 30 160 j=0.25 j=0.3 j=0.3 j=0.4 25 140 j=0.35 j=0.5 j=0.4 I/M 3 j=0.6 20 j=0.45 120 I [M sun km 2 ] j=0.5 15 j=0.55 100 j=0.6 10 j=0.65 5 80 0.1 60 I Fit I Fit |/ - 0.01 40 I- - | - 20 0.001 0.5 1 1.5 2 2.5 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 M sun M/R [M sun /km] 0.45 dimensionless angular Fit LS APR Sly4 0.4 momentum j = J / M 2 LS 220 I/(MR 2 ) 0.35 instead of J 0.3 0.25 0.1 I Fit I Fit |/ - 0.01 I- - | - 0.001 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 M/R [M sun /km] C. Breu Universal relations

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 3 0.11 I=50 M sun km 2 I=50 (I/M 3 ) I=80 (I/(MR 2 )) I=80 M sun km 2 0.105 I=120 M sun km 2 I=50 (I/(MR 2 )) I=120 (I/M 3 ) 0.1 APR 2.5 Sly 4 I=80 (I/M 3 ) I=120 (I/(MR 2 )) 0.095 WFF1 LS 220 0.09 N bhblp M [M sun ] 2 0.085 δ R/R 0.08 0.075 1.5 0.07 0.065 1 0.06 0.055 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 9 10 11 12 13 14 M [M sun ] R [km] 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 → realistic EOSs are similar to each other approximation by elliptical isodensity contours modern EOSs are stiff → limit of an incompressible fluid C. Breu Universal relations

Thank you! C. Breu Universal relations