Results on the classical high-! bar-mode instability in relativistic star models for polytropic EoS with adiabatic index!=2.75.

Transcription

1 Results on the classical high-! bar-mode instability in relativistic star models for polytropic EoS with adiabatic index!=2.75 Luca Franci (1) in collaboration with Roberto De Pietri (1), Alessandra Feo (1) and Frank Loeffler (2) (1) Department of Physics and Earth Sciences, University of Parma and INFN, Parma (2) Center for Computation and Technology Louisiana State University, Baton Rouge, LA

2 3D Numerical relativity Among other things, numerical relativity aims at: solve the Einstein equations without approximations(!) investigate the physics of gravitational collapse investigate structure and stability of the most relativistic astrophysical objects: neutron stars model the most catastrophic events in the Universe (GRBs, magnetars, etc.) model real sources of gravitational waves: core collapse in supernova and binary mergers: NS-NS, BH-NS, BH-BH) unkwowns about NS: what is the correct EoS? how to deal with neutrinos?... 2

3 Neutron Stars may be unstable Neutron stars in nature are rotating and subject to nonaxisymmetric rotational instabilities Which type of instabilities will develop? Does a fully developed instability persist for long? If not, what induces its decay? Would it radiate gravitational waves and how much? What is the threshold of instabilities? (dependence on EOS, rotation rate and profile) Are dynamics affected by vicinity to threshold? 3

4 Instability types in rotating stars Secular (m = 2):!! sec 0.14 growth time determined by dissipative time scale (tens of seconds for neutron stars)! Axisymmetric configurations e.g., see Chandrasekhar (1970), Ou, Tohline and Lindblom (2004) Dynamical (m = 2):!! dyn 0.27 grows on dynamical timescale (tens of milliseconds for neutron stars) e.g., see Shibata, Tohline, Baiotti, Manca,... e Dedekind Ω=0, ζ>0 Maclaurin β=0.27 Jacobi Ω>0, ζ=0 Low T/W instability - Shear? first observed numerically (grows on dynamical time scale) e.g., see Centrella et al (2001), Corvino (2010),... eigenvalue of the m=2 mode Real part β=0.14 NON-Axisymmetric configurations a /a 2 1 axes ratio in the x-y plane [*] Chandrasekhar, Elipsoidal figures of Equilibrium (Yale Univ. Press, 1969) Imaginary part β β 4

5 Dynamical bar-mode instability Dynamics of the global modes m=2 m=4 ln( P m ) m=3 m=1 (a) (b) (c) (d) time Eigenfrequency of the m=2 bar-mode Luca Franci - Università di Parma Parma, 12 dicembre Pagina 5

6 Dinamical bar-mode instability in full GR L. Baiotti, R. De Pietri, G. M. Manca and L. Rezzolla, "Dynamical non-axisymmetric instabilities in rotating relativistic stars", Class. Quantum Grav. 24(2007) S171-S186, arxiv: t = 12 ms t = 17 ms L. Franci, R. De Pietri, K. Dionysopoulou and L. Rezzolla, "Dynamical bar-mode instability in rotating and magnetized relativistic stars", arxiv: B = G -> unstable B = G -> stable 6

7 What s new: EoS polytropic EoS with! = 2 ->

8 Code + Computational set up Cactus framework for parallel high performance computing (Grid computing, parallel I/O) Einstein Toolkit open set of over 100 Cactus thorns for computational relativity along with associated tools for simulation management and visualization Mesh refinement with Carpet RNS solver by Stergioulas 3D cartesian grids with 3 refinement levels resolution: 0.5 M 0.75 km (for the finest grid) grid size: 150 M 225 km (for the corsest grid) 8

9 Initial models and Evolution Initial models standard axisymmetric metric ds 2 = e + dt 2 + e r 2 sin 2 (d'!dt) 2 + e 2 (dr 2 + r 2 d 2 ) highly differential rotation (with  = 1) Ω c Ω = r2 e  2 ˆ polytropic EoS (with " = 2.75) p = Kρ Γ, = + Evolution #-law EOS: p = (# 1)"% Full relativistic curvature evolution Relativistic hydrodynamics No magnetic fields (yet) [ (Ω ω)r 2 sin 2 θe 2ρ 1 (Ω ω) 2 r 2 sin 2 θe 2ρ ] 9

10 The bar-mode instability in action 10

11 Instability diagram Baiotti et al. (2007) present work 11

12 Finding the treshold At threshold: growth time = Above threshold: growth time within dynamical timescale Below threshold: no (clear) instability (of this type) M M 1.5M 2.0M 2.5M b bc /t2 2(ms2 ) M? /M Extrapolation from unstable models, measuring the growth time 12

13 Conclusions Preliminary results Past studies confirmed with current codes (current codes validated using published results) Found value of! c for more realistic EOS parameters (#= ) (! c = ) What comes next? Effect of magnetic fields (thresold for the suppression of instability) Effect of other equations of state Other instabilities, e.g., low T/W 13

14 Thank you for your attention! 14

15 Overview of a model 15