Universal Relations for the Moment of Inertia in Relativistic Stars

Transcription

1 Universal Relations for the Moment of Inertia in Relativistic Stars Cosima Breu Goethe Universität Frankfurt am Main Astro Coffee

2 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

3 The Tolman-Oppenheimer-Volkoff Equations first solution of Einstein s equations for non-vacuum spacetimes M [M sun ] WFF2 HS DD2 HS TM1 HS TMa HS L3 SFHo SFHx L O Sly4 stiff eos soft eos intermed. eos Einstein equations: R µν 2 1 gµν R = 8πTµν Metric of a spherically symmetric matter distribution: ds 2 = e ν(r) dt 2 + e λ(r) dr 2 + r 2 (dθ 2 + sin θ 2 dφ 2 ) e λ(r) = 1 2M(r) r Energy-momentum tensor of a perfect fluid: T µν = (e + p)u µu ν + pg µν EOS needed to close system of equations R [km] TOV-equations dm = 4πr 2 e dr 3 dp (e + p)(m + 4πr p) = dr r(r 2M) dν dr = 2 dp (e + p) dr

4 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 = soft eos stiff eos intermed. eos Sly 4 1 solid outer crust (e, Z) log 10 (p) [dyn/cm 2 ] log 10 (e) [g/cm 3 ] L O LS220 WFF2 HS DD2 HS L3 SFHo SFHx HS TM1 HS TMa 2 inner crust (e, Z, n) 3 outer core (n, Z, e, µ) 4 inner core (here be monsters): large uncertainties at high densities

5 The Slow Rotation Approximation Metric of a stationary, axisymmetric system: ds 2 e ν(r) dt 2 + e λ(r) dr 2 + r 2 (dθ 2 + sin θ 2 (dφ Ldt) 2 ) Expansion of L(r, θ): L(r, θ) = ω(r, θ) + O(ω 3 ) 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 d r 4 r 4 j ω ) + 4 dj ω = 0, dr dr r dr Coordinate angular velocity: ω = Ω ω, j(r) = exp [( ν + λ)/2] outside: ω = Ω 2J r 3

6 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 + 2h)dt 2 + e λ [1 + 2m/(r 2M)]dr 2 + r 2 (1 + 2k)[dθ 2 + sin θ 2 (dφ ωdt) 2 ] + O(Ω 3 ) h(r, θ) = h 0 (r) + h 2 (r)p 2 (θ) +... k(r, θ) = k 0 (r) + k 2 (r)p 2 (θ) +... m(r, θ) = m 0 (r) + m 2 (r)p 2 (θ) +... Coordinate system r = R + ξ(r, θ) + O(Ω 4 ) P 2 = (3 cos θ 2 1)/2

7 umerical 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: RS-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

8 The Moment of Inertia I [M sun km 2 ] M [M sun ] WFF2 HS DD2 HS TM1 HS TMa HS L3 SFHo SFHx L O Sly4 stiff eos soft eos intermed. eos The moment of inertia J = Tφ t gd 3 x I (M, ν) = J/Ω = 8π 3 R 0 r 4 (e + p) 1 2M(r)/r j(r) ω Ω dr Slow Rotation: Angular momentum linearly related to moment of inertia

9 I-C -Relation WFF2 HS DD2 HS TM1 HS TMA HS L3 HS SFHo HS SFHx L O Sly4 Fit LS WFF2 HS DD2 HS TM1 HS TMa HS L3 SFHo SFHx L O Sly4 YYM Fit I/(MR 2 ) I/M C [M sun /km] M/R [M sun /km] Lattimer and Schutz 2005 I /(MR 2 ) = a + b M R + c ( M R ) 4 ew Fit I M 3 = a C 2 + b C 3 + c C 4

10 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 5 KM3 Q = QM/J 2 Q approaches 1 for a Kerr black hole

11 The Tidal Love umber deformability due to tidal forces ratio between tidally introduced quadrupole moment and tidal field due to a companion S Tidal Love number M/R [M sun /km] Maselli WFF2 HS DD2 Sly4 L O HS L3 HS TM1 HS TMA SFHo SFHx 1 gtt 2 = M r λ (tid) = Q(tid) ε (tid) Fit:C = a + b ln λ + c(ln λ) 2 3Q ij 2r 3 ni n j + ε ij 2 r 2 n i n j detection of gravitational waves during the merger of neutron star binaries λ tid 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

12 The I-Love-Q Relations by Yagi and Yunes Yagi Yunes L HS TMA Yagi Yunes L SFHo O SFHx O WFF2 WFF2 HS DD2 HS L3 HS DD2 HS L3 Sly4 HS TM1 Sly4 HS TM1 HS TMA SFHo SFHx I/M 3 10 I/M 3 10 Ī-Ī Fit /Ī Fit 0.01 Ī-Ī Fit /Ī Fit Q/(M 3 j 2 ) λ tid Reduced Moment of inertia Ī versus the Kerr factor QM/J 2 Reduced Moment of inertia Ī versus the tidal Love number 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

13 Rapid Rotation equilibrium between gravitational, pressure and centrifugal forces Kepler angular velocity M [M sun ] 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= e+14 1e e+15 2e e+15 e c [g/cm 3 ]

14 Breakdown of Universal Relations Breakdown of universal relations I/(M 3 ) WFF2 HS DD2 Sly 4 L O HS L3 HS TM1 HS TMA SFHo SFHx YY I/(M 3 ) Q/(M 3 j 2 ) Q/(M 3 j 2 ) Ī Q-relation as a function of the observationally important (but dimensionful) rotational angular velocity rotation characterized by dimensionless spin parameter j = J/M 2

15 Rapidly Rotating Models I [M sun km 2 ] 180 Slow Rot. App. j= j=0.25 j= j=0.35 j=0.4 j= j=0.5 j= j=0.6 j= Ī-Ī Fit /Ī Fit I/M Slow Rot. Approx. j=0.2 j=0.3 j=0.4 j=0.5 j= M sun M/R [M sun /km] I/(MR 2 ) Fit LS Sly4 dimensionless angular momentum j = J/M 2 instead of J Ī-Ī Fit /Ī Fit M/R [M sun /km]

16 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 J ) determination of I up to 10 % accuracy through periastron advance and geodetic precession constraints on radius and EOS I=50 M sun km 2 I=80 M sun km 2 I=120 M sun km 2 Sly I=50 (I/M 3 ) I=50 (I/(MR 2 )) I=80 (I/M 3 ) I=80 (I/(MR 2 )) I=120 (I/M 3 ) I=120 (I/(MR 2 )) 0.09 M [M sun ] δr/r R [km] M [M sun ]

17 Outlook newly born S 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

18 Thank you!