A Relativistic Toy Model for Black Hole-Neutron Star Mergers

Transcription

1 A Relativistic Toy Model for Francesco Pannarale A.Tonita, L.Rezzolla, F.Ohme, J.Read Max-Planck-Institute für Gravitationsphysik (Albert-Einstein-Institut) SFB Videoseminars, Golm - January 17, 2011

2 Introduction and Motivation Introduction and Motivation Why BH-NS binaries? Gravitational wave sources for ground-based interferometers

3 Introduction and Motivation Introduction and Motivation Why BH-NS binaries? Leading candidates as short-duration gamma-ray burst (SGRB) progenitors (along with BNSs) m pages/swift/main/index.html Francesco Pannarale SFB Videoseminars, Golm - January 17, 2011

4 Introduction and Motivation Introduction and Motivation Why BH-NS binaries? NS tidal deformations/disruptions (probing superdense matter) Francesco Pannarale SFB Videoseminars, Golm - January 17, 2011

5 Introduction and Motivation Introduction and Motivation 1. Two components GW-driven inspiral 2. Merger a.plunge 3. Final system BH PN/EOB frameworks Clean GW signal Non-linear BH horizon GW emission Ringdown QNMs

6 Introduction and Motivation Introduction and Motivation 1. Two components GW-driven inspiral PN/EOB frameworks Clean GW signal 2. Merger a.plunge b.tidal disruption Non-linear BH horizon GW emission 3. Final system BH + Torus Ringdown QNMs SGRB scenario

7 Introduction and Motivation Introduction and Motivation 1. Two components GW-driven inspiral 2. Merger a.plunge b.tidal disruption 3. Final system BH + Torus This is a difficult problem with very high computational costs Poorly explored (big) parameter space Not all results of different groups agree (e.g. torus mass) In this sense, it is desirable to study certain features of BH-NS coalescence with the use of approximations and thus a drastic reduction of required computational resources.

8 Introduction and Motivation Introduction and Motivation 1. Two components GW-driven inspiral 2. Merger a.plunge b.tidal disruption 3. Final system BH + Torus Masses BH spin NS EOS Toy Model Torus mass Pannarale, Tonita, Rezzolla, ApJ 727, 95 (2011)

9 Part I: Torus Mass Estimation Torus Mass Estimation Masses BH spin NS EOS Toy Model Torus mass

10 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid Approximations BH is unaffected by the NS NS centre of mass inspirals around the Kerr BH (2.5PN point-particle radiative losses) NS is a constant mass ellipsoid (affine prescription) Pros V free mass choices V free BH spin choice V free nuclear EOS choice V Kerr BH tidal field V effective relativistic self-gravity

11 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid fluid elements free-falling massive point particles At the onset of mass-shedding (more later) we switch to a description of the kinematic properties of the NS fluid as a set of independent, collisionless point particles in free-fall in a Kerr spacetime: build a fine grid adapted to the ellipsoidal shape of the star and divide the star into a collection of fluid elements, each one of which possesses mass, 4-position, and 4-velocity neglect pressure gradients and the self-gravity from then on

12 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid fluid elements free-falling massive point particles Torus mass calculation Mass of remnant torus = sum of bound particle masses: compute the conserved quantities for each particle for particles that have E < 1 we use root-finding techniques and look for those that have ( ) dr 2 < 0 dτ at some turning point outside the event horizon

13 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid Torus mass calculation fluid elements free-falling massive point particles

14 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid Torus mass calculation 1 Set M b,tor = M b,ns fluid elements 2 Check which fluid particles are bound/unbound 3 Accrete all the unbound particles { } { } MBH MBH ± M b,tor M b,tor free-falling massive point particles i {unbound} 4 Go to step 2 unless M b,tor /M b,tor < 10 6 M b,i

15 The Model Torus Mass Estimation Distortion and disruption approximation scheme TOV ellipsoid? Torus mass calculation 1 Set M b,tor = M b,ns fluid elements 2 Check which fluid particles are bound/unbound 3 Accrete all the unbound particles { } { } MBH MBH ± M b,tor M b,tor free-falling massive point particles i {unbound} 4 Go to step 2 unless M b,tor /M b,tor < 10 6 M b,i

16 Tuning and Validation Torus Mass Estimation The NS disruption starts when a 2 /a 1 = (a 2 /a 1 ) crit (a 2 /a 1 ) crit is our only free parameter The value of (a 2 /a 1 ) crit is identified as the one which allows us to best reproduce the available numerical relativity results

17 Tuning and Validation Torus Mass Estimation The NS disruption starts when a 2 /a 1 = (a 2 /a 1 ) crit (a 2 /a 1 ) crit is our only free parameter The value of (a 2 /a 1 ) crit is identified as the one which allows us to best reproduce the available numerical relativity results Ref. EOS C q a M b,tor /M b,ns M b,tor /M b,ns error (Γ) (toy model) (simulations) (%) Tonita et al. (2010) / Tonita et al. (2010) / Tonita et al. (2010) / < 0.01 < Tonita et al. (2010) / < 0.01 < Duez et al. (2009) / Duez et al. (2009) / Duez et al. (2009) / Etienne et al. (2008) / Etienne et al. (2008) / Etienne et al. (2008) / < 0.01 < Etienne et al. (2008) / < 0.01 < Shibata et al. (2009) / < Shibata et al. (2009) / < 0.01 < Shibata et al. (2009) / < 0.01 < Shibata et al. (2009) / < Shibata et al. (2009) / < 0.01 <

18 Tuning and Validation Torus Mass Estimation The NS disruption starts when a 2 /a 1 = (a 2 /a 1 ) crit (a 2 /a 1 ) crit is our only free parameter The value of (a 2 /a 1 ) crit is identified as the one which allows us to best reproduce the available numerical relativity results Ref. EOS C q a M b,tor /M b,ns M b,tor /M b,ns error (Γ) (toy model) (simulations) (%) Tonita et al. (2010) / Tonita et al. (2010) / Tonita et al. (2010) / < 0.01 < Tonita et al. (2010) / < 0.01 < Duez et al. (2009) / Duez et al. (2009) / Duez et al. (2009) / Etienne et al. (2008) / Etienne et al. (2008) / Etienne et al. (2008) / < 0.01 < Etienne et al. (2008) / < 0.01 < Shibata et al. (2009) / < Shibata et al. (2009) / < 0.01 < Shibata et al. (2009) / < 0.01 < Shibata et al. (2009) / < Shibata et al. (2009) / < 0.01 < Agreement within numerical relativity error!

19 Tuning and Validation The model is built to calculate the torus mass and the tuning of its only free parameter is based on tori masses; its predictions for the GW frequency at the onset of mass-shedding are in good agreement with recent numerical relativity results. Taniguchi, Baumgarte, Faber, Shapiro, PRD 77, (2008) Torus Mass Estimation 1/q Taniguchi et al. Toy Model M b,ns = 0.12R M b,ns = 0.13R M b,ns = 0.14R M b,ns = 0.15R M b,ns = 0.16R M b,ns = 0.17R GW R f tide Shibata, Taniguchi, PRD 77, (2008) q M b,ns [M ] M NS [M ] R NS [km] ftide GW [khz] GW ftide (toy model) (simulations)

20 Results Torus Mass Estimation M b,tor /M b,ns for a= Low compactness and high mass ratio favour massive torus formation q C 0.10

21 Results Torus Mass Estimation q M b,tor /M b,ns for a= C Low compactness and high mass ratio favour massive torus formation Masses attained increase with the BH spin, while the no-torus area decreases and disappears

22 Results Torus Mass Estimation q M b,tor /M b,ns for a= C Low compactness and high mass ratio favour massive torus formation Masses attained increase with the BH spin, while the no-torus area decreases and disappears For large BH spins a torus is always produced and it has non-negligible mass

23 Results Torus Mass Estimation a M b,tor /M b,ns for C= q q 0.14 C a 0.85 M b,tor /M b,ns 0.24 M b,tor 0.34 M Comparable but also smaller than predictions for unequal-mass NS-NS mergers

24 Part II: Final Spin of the BH Final Spin of the BH Masses BH spin NS EOS Toy Model Torus mass Final BH spin

25 The Model Final Spin of the BH To complete the BH + Torus picture, the final BH spin is needed. Angular momentum conservation neglecting the torus formation: a f = am 2 BH + M b,nsl z rtide (M BH + M b,ns ) 2 However, only some of the NS fluid is promptly accreted and part of its angular momentum is dissipated: a f = am 2 BH + (M b,ns M b,tor )σl z rtide (M BH + M b,ns M b,tor ) 2 (0 σ < 1) Complete agreement with numerical relativity results is achieved once the second free parameter σ is tuned

26 Results Final Spin of the BH QNM [khz] of the BH remnant for a=0.8 and the APR2 EOS q M NS

27 Will the Inspiral Tell Us About the NS EOS? Part III: Will the Inspiral Tell Us About the NS EOS? Masses BH spin NS EOS Toy Model PN GW-forms Torus mass Final BH spin EOS measurable?

28 Will the Inspiral Tell Us About the NS EOS? Will the Inspiral Tell Us About the NS EOS? 3PN+3.5PN+TaylorT4 with spin-terms for the BH l = 2, m = 0 NS tidal distortion contributions may be included in PN point-particle inspiral GW-forms One has a phase correction given in terms of a single EOS-dependent deformability parameter λ; NLO for a quasi-circular point-particle inspiral: [ dφ = 15x 3/2 λ dx T 16M 5 η 2 11η 6(1 + { η) + x 24 η2 [ (1 ] 1 4η) η (1 + } ] 1 4η) For a BH-NS binary, the integration must must be halted once the mass-shedding begins; this NS tidal disruption frequency f tide is automatically yielded by the torus mass toy model

29 Will the Inspiral Tell Us About the NS EOS? Will the Inspiral Tell Us About the NS EOS? 1.4 M NS + non-spinning BH three times as massive

30 Will the Inspiral Tell Us About the NS EOS? Will the Inspiral Tell Us About the NS EOS? The reduction in accumulated GW phase φ φ 3.5,PP φ 3.5,λ 1 is greater for bigger tidal Love numbers; 2 grows with the binary mass ratio M NS /M BH ; 3 decreases as the NS mass grows; 4 shows a very weak dependence on the BH spin BH-NS, a = 0 BH-NS, a = 1 BNS EOS f tide φ f tide φ δφ [khz] [rad] [khz] [rad] [rad] APR BBB BBS BPAL GM GM GNH PS Sly

31 Will the Inspiral Tell Us About the NS EOS? Will the Inspiral Tell Us About the NS EOS? Using the PS EOS (has the lowest f tide s)

32 Will the Inspiral Tell Us About the NS EOS? Will the Inspiral Tell Us About the NS EOS? An EOS measurement is unlikely for next generation detectors

33 Summary and Conclusions Summary and Conclusions We built a relativistic toy model to calculate the mass of the torus remnant of BH-NS mergers, for a given BH spin, binary mass ratio, and NS EOS Using available numerical relativity results, we tuned the only free parameter and validated the model Correct GW frequencies at the onset of mass-shedding are automatically obtained A simple extension yields also the spin of the BH remnant Massive torus formation requires: low NS compactness, high mass ratio, and large BH spin For a canonical binary, M b,tor 0.34 M By combining the output of the model with PN GW-forms, we showed that the next generation of detectors will not be able to infer the NS EOS from GWs emitted by BH-NS binaries

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35 Further Insight The Model: Neutron Star Deformation The NS Lagrangian includes a tidal interaction term (L T ) and internal dynamics terms (L I ), i.e. internal kinetic energy K I, stellar fluid internal energy U, self-gravity potential (V ): L NS = L T + L I L I = K I U V Replace the Newtonian V with an effective relativistic self-gravity scalar potential V TOV. Assume the EOS to be barotropic and determine all EOS-dependent quantities numerically.

36 Further Insight The Model: Neutron Star Deformation How do we build V TOV? Rampp, Janka, A&A 396, 361 (2002) dp dr = (ɛ+p)(m TOV+4πr 3 P) r(r 2m TOV ) dm TOV dr = 4πɛr 2 Relativistic stellar structure EOS dφ TOV dr = 1 dp ρ dr Newtonian equilibrium V TOV = 4π ρr 3 dφ TOV dr dr Pseudo-relativistic self-gravity

37 SGRBs from Compact Binaries Further Insight Short gamma-ray burst observations Time scales suggest accretion onto a stellar-mass compact object Identification of SGRBs with elliptical galaxies No recent star formation, rules out collapse of massive stars as central engines Suggests compact binaries (BH-NS and BNS) as progenitors

38 SGRBs from Compact Binaries Compact binary engine ingredients: 1 Hot massive accretion disk Further Insight 2 Baryon-free funnel (no baryon contamination) around the rotation axis Ruffert, Janka, Eberl (1998) m Encyclopedia of Science

39 Equations of State Used Further Insight APR2: n, p, e, µ; v 18 + δv 18 + UIX ; the nuclear many-body Schrödinger equation is solved using a variational approach; M max = 2.20 M. BBB2: Paris+TBF; Brueckner-Bethe-Goldstone based; M max = 1.92 M. BBS1: n, p, e, µ; v 18 + UVII ; the ground state energy is calculated using G-matrix perturbation theory; M max = 2.01 M. BPAL32: n, p, e, µ with a density dependent nucleon-nucleon effective interaction (as for Skyrme nuclear interactions); incompressibility k = 240 MeV and M max = 1.93 M. GM1: n, p, e, µ, and hyperons in beta equilibrium; special case of the Walecka model; k = 300 MeV and M max = 2.36 M. GM3: k = 240 MeV and M max = 2.03 M. GNH3: n, p, e, µ up to ρ H 2ρ s, then also, Σ, Λ, Ξ and π, σ, ρ, ω, K, K ; M max = 1.96 M. PS ( Liquid ): Only n s exchanging charged particles; M max = 2.66 M. Sly4: Skyrme Lyon effective N-N interactions; M max = 2.05 M. Crust: Douchin-Haensel (SLy4) crust for ρ d ρ 0.5ρ s ; Haensel-Pichon (HP94) for 10 8 g/cm 3 ρ ρ d ; Baym-Pethick-Sutherland (BPS) ρ 10 8 g/cm 3.

40 Bibliography Further Insight Effective Relativistic Self-Gravity Scalar Potential Rampp, Janka, A&A 396, 361 (2002) BH-NS Binaries in Numerical Relativity Etienne, Liu, Shapiro, Baumgarte, PRD 79, (2009) Shibata, Kyutoku, Yamamoto, Taniguchi, PRD 79, (2009) Duez, Foucart, Kidder, Ott, Teukolsky, CQG 27, (2010) Foucart, Duez, Kidder, Teukolsky, arxiv: v1 Taniguchi, Baumgarte, Faber, Shapiro, PRD 77, (2008)