Black Hole-Neutron Star Binaries in General Relativity. Thomas Baumgarte Bowdoin College

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1 Black Hole-Neutron Star Binaries in General Relativity Thomas Baumgarte Bowdoin College 1

2 Why do we care? Compact binaries (containing neutron stars and/or black holes) are promising sources of gravitational waves for LIGO, GEO and other gravitational wave detectors Theoretical models are needed to help identify astrophysical sources in noisy output of detector 2

3 Theoretical Models of Compact Binaries Large Binary separation = post-newtonian point-mass approximations Small Binary separation = numerical relativity: initial data describing binary in quasicircular orbit subsequent dynamical evolution: coalescence and merger/tidal disruption Binary black holes initial data [Cook, 1994; Baumgarte, 2000; Gourgoulhon et.al., 2002;...] dynamical evolution [e.g. Brügmann et.al., 2004] Binary neutron stars initial data [Baumgarte et.al., 1997; Uryū et.al., 2000; Taniguchi & Gourgoulhon, 2002] dynamical evolution [Shibata & Uryū, 2000; Marronetti et.al., 2004] Mixed black hole-neutron star (BHNS) binaries so far neglected [except possibly Miller 2001] 3

4 Onset of Tidal Disruption Equate tidal force on test mass m on surface of neutron star F tid mm BHR NS s 3 with gravitational force exerted by star = tidal disruption when F grav mm NS R 2 NS s tid R NS ( MBH M NS ) 1/3 or s tid M BH ( MNS M BH ) 2/3 R NS M NS For neutron star, R NS 5M NS, so for supermassive black holes M NS M BH, so tidal effects not important for stellar mass black holes, M NS M BH, so tidal disruption possible outside of Innermost Stable Circular Orbit (ISCO) at s ISCO 6M BH Location of s tid depends on equation of state (EOS) = Observation of tidal disruption would provide wealth of astrophysical information [e.g. Vallisneri, 2000] 4

5 Introduction Overview: Quasiequilibrium Initial Data for BHNS Binaries BHNS Binaries in Newtonian Gravity Basic Equations Assumptions and Approximations Numerical Strategy BHNS Binaries in General Relativity Conformal Thin-Sandwich Decomposition Integrated Euler Equation Kerr-Schild Background Numerical Results Tests Tidal Disruption Summary and Discussion 5

6 BHNS Binaries in Newtonian Gravity z Write Newtonian potential φ as φ = φ BH + φ NS with φ BH = M BH r BH Then Poisson equation Axis of rotation Black Hole x BH x rot Neutron Star 1 1 y 2 φ = 2 φ NS = 4πρ 6

7 Matter equation Assume matter in equilibrium: stationary in corotating frame corotational fluid flow: v = Ω r rot polytropic equation of state P = κρ 1+1/n = can compute first integral of Euler equation q = 1 ( C φ + 1 ) 1 + n 2 Ω2 ϖrot 2 = n (C φ eff) where q = P/ρ = have to solve Poisson equation together with algebraic equation for matter for φ, q and eigenvalues C and Ω. 7

8 Extreme Mass Ratio To simplify problem in first step, assume M BH M NS neutron star affects spacetime only in its neighborhood, can therefore restrict numerical grid to this region Axis of rotation z y no need to excise black hole [see Cook & Pfeiffer, 2004] Black Hole Neutron Star no need to locate axis of rotation: x rot = x BH (for example, compare ADM and Komar mass [Grandclément etal, 2002; Skoge & Baumgarte, 2002] ) x BH x rot 1 1 8

9 Numerical Strategy Location of stellar surface unknown a priori z Rescale lengths with r e so that surface intersects x-axis at 1 and 1 Axis of rotation Black Hole Neutron Star y Then three eigenvalues that have to be determined: Ω, C and r e x BH x rot 1 1 Can be determined by evaluating matter equation at x = 1, x = 1 and where matter density takes maximum value q max 9

10 Iteration To construct binary with fixed q max and x BH solve Poisson equation 2 φ = 2 φ NS = 4πq n find Eigenvalues Ω, C and r e solve integrated Euler equation q = n ( C φ + 1 ) 2 Ω2 ϖrot 2 evaluate residuals of Poisson equation To construct binary with mixed M NS iterate over q max To construct sequence vary binary separation x BH 10

11 Einstein s equations for metric 3+1 decomposition of Einstein s Equations G αβ = 8πT αβ ds 2 = g ab dx a dx b = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) can be split into constraint equations (like div equations in E&M) R + K 2 K ij K ij = 16πρ D i (K ij γ ij K) = 8πj j evolution equations (like curl equations) d dt γ ij ( ) t L β (Hamiltonian constraint) (Momentum constraint) γ ij = 2αK ij d dt K ij = D i D j α + α(r ij 2K ik K k j + KK ij 8πM ij ) [Arnowitt, Deser & Misner, 1962] To construct initial data... =... solve constraint equations To study evolution... =... solve evolution equations 11

12 BHNS Binaries in General Relativity complete solution: construct metric ds 2 = g ab dx a dx b = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) that satisfies Einstein s field equations stress energy tensor T ab = (ρ 0 + ρ i + P )u a u b + P g ab that satisfies relativistic equations of hydrodynamics (relativistic Euler equation) Quasiequilibrium initial data: only solve constraint equations R + K 2 K ij K ij = 16πρ D i (K ij γ ij K) = 8πj j (Hamiltonian constraint) (Momentum constraint) natural to adopt conformal thin-sandwich decomposition only solve integrated relativistic Euler equations 12

13 Conformal Decomposition of Hamiltonian Constraint Conformally decompose metric γ ij = ψ 4 γ ij and split extrinsic curvature into a trace and traceless part K ij = A ij γ ijk = ψ 2 Ā ij γ ijk Then Hamiltonian constraint becomes D 2 ψ 1 8 ψ R 1 12 ψ5 K ψ 7 Ā ij Ā ij = 2πψ 5 ρ [Lichnerowicz, 1944; York, 1971, 1972] 13

14 Conformal thin-sandwich decomposition Consider traceless part of t γ ij Then (from evolution equation of γ ij ), or Insert into momentum constraint = yields equation for shift β i u ij γ 1/3 t (γ 1/3 γ ij) u ij = 2αA ij + (Lβ) ij Ā ij = ψ6 2α ( ( Lβ) ij ū ij) L β i ( Lβ) ij Dj ln(αψ 6 ) αψ 6 Dj (α 1 ψ 6 ū ij ) 4 3 α D i K = 16παψ 4 j i [Isenberg, 1978; Wilson & Mathews, 1989; York 1999] Freely specifiable variables: K, α, γ ij, ū ij 14

15 Imposing Quasiequilibrium For quasiequilibrium data in corotating coordinate system have ū ij = 0 Can also set = equation for lapse α t K = 0 D 2 (αψ) = 7 8 αψ 7 Ā ij Ā ij αψ5 K αψ R + ψ 5 β i Di K + 2παψ 5 (ρ + 2S) Remaining freely specifiable variables: K, γ ij 15

16 Summary Field equation Have three coupled equations for gravitational fields ψ, α and β i D 2 ψ =... (Hamiltonian constraint) L β i =... (momentum constraint) D 2 (αψ) =... (from t K = 0) Play role of Poisson equation in Newtonian gravity Still have to choose background γ ij and K according to astrophysical situation at hand 16

17 Assume corotating fluid flow Matter equation = Equations of relativistic hydrodynamics can be integrated to yield h u t = C + 1 Find u t from normalization u a u a = 1 (in corotating coordinates have u i = 0) For polytropic equation of state P = κρ 1+1/n 0 can express enthalpy h and matter sources in field equations in terms of density function q = integrated Euler equation becomes q = n Compare with Newtonian cousin ( ) 1 + C (α 2 ψ 4 γ ij β i β j ) 1 1/2 q = n (C φ eff) = Can define relativistic effective potential 17

18 Constructing BHNS Binaries Choose black hole background solution (in Kerr-Schild coordinates) = black hole accounted for by background Solve equations in presence of equilibrium matter distribution = neutron star in presence of black hole [Baumgarte, Skoge & Shapiro, 2004] 18

19 Kerr-Schild background Compare Kerr-Schild metric ds 2 = g ab dx a dx b = (η ab + 2Hl a l b )dx a dx b where with ADM metric H = M BH r BH ds 2 = g ab dx a dx b = α 2 dt 2 + γ ij (dx i + β i dt)(dx j + β j dt) to read off background quantities γ ij and K as well as α BH and β i BH. background satisfies equations in absence of neutron star with ψ BH = 1 19

20 Numerical Strategy Split unknown field quantities into contributions from neutron star and black hole α = α NS + α BH ψ = ψ NS + ψ BH = ψ NS + 1 β i = β i NS + β i BH + β i rot = β i NS + β i BH + ( Ω r rot ) i Solve field equations for neutron star contributions α NS, ψ NS and β i NS together with integrated Euler equation for q Assume extreme mass ratios M BH M NS Adopt rescaling and iteration identical to that in Newtonian problem = yields eigenvalues Ω, C and r e together with solutions α, ψ, β i and q 20

21 Set M BH = 0 Neutron Stars in Isolation = Recover TOV solution for spherically symmetric neutron stars in isotropic coordinates For n = 1 polytrope: = 2nd order convergence to analytical solution 21

22 Construct constant rest-mass sequence Constant Rest Mass Sequences = recover Kepler s third law ΩM BH = ( xbh M BH ) 2/3 independently of neutron star mass 22

23 Location of tidal disruption n = 1 polytrope fills out Roche lobe as defined from first integral of Euler equations = onset of tidal disruption ˆx BH = 5.0 M BH = 10M NS ˆx BH =

24 Scaling with mass ratio Equate tidal force on test mass m on surface of neutron star with gravitational force exerted by star F tid G mm BHR NS s 3 F grav G mm NS R 2 NS = tidal disruption when s tid R NS ( ) 1/3 MBH M NS Here R NS = 123M NS - weakly relativistic regime 24

25 Summary Have constructed relativistic models of black hole-neutron star binaries in quasiequilibrium Adopted extreme mass ratio limit M BH M NS Located onset of tidal disruption In future relax assumption of extreme mass ratio... assume irrotational fluid flow... consider rotating black holes... study dynamical evolution 25