Nonlinear Effect of R -mode Instability in Uniformly Rotating Stars - PowerPoint PPT Presentation

Nonlinear Effect of R -mode Instability in Uniformly Rotating Stars Motoyuki Saijo (Rikkyo University) CONTENTS 1. Introduction 2. Dynamical approach beyond acoustic timescale 3. Nonlinear r-mode instability 4. Summary 22nd General Relativity and Gravitation in Japan No. 1 14 November 2012 @RESCEU, The University of Tokyo, Japan

CFS instability 1. Introduction Various Instabilities in Secular Timescale (Chandrasekhar 70, Friedman & Schutz 78) • Fluid modes (f, p, g-modes) may become unstable due to gravitational radiation • Instability occurs in dissipative timescale e i ( m ϕ − ω t ) Rotating Inertial r-mode instability amplify frame frame (Andersson 98, Friedman & Morsink 98) -m J - <0 J - >0 • Fluid elements oscillate due to Coriolis force � m ��� • Instability occurs due to gravitational radiation Occurs when +m J + >0 J + >0 g-mode instability • Fluid elements oscillate due to restoring force of buoyancy • Instability occurs in nonadiabatic evolution or in convective unstable cases Kelvin-Helmholtz instability • Instability occurs when the deviation of the velocity between the different fluid layers exceeds some critical value 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 2

Dynamics of r-mode instabilities Saturation amplitude of r-mode instability 0 10 � 3D simulation • Saturation amplitude of o(1) �� 10 • Imposing large amplitude of radiation reaction potential in the system to control secular �� 10 0 10 20 30 timescale with dynamics (Lindblom et al. 00) t/P 0 1D evolution with partially included 3 wave interaction • Saturation amplitude of ~ o(0.001), which depends on interaction term (Schenk et al. 2001) Final fate of r-mode instability 3D simulation • Evolution starting from the amplitude o(1) 2 • Imposing large amplitude of radiation reaction potential • Energy dissipation of r-mode catastrophically decays to � 1 differentially rotating configuration in dynamical timescale 1D evolution including mode couplings network 0 • After reaching the saturation amplitude ~o(0.001), 0 20 40 60 t (ms) Kolmogorov-type cascade occurs (Gressman et al. 02, Lin & Suen 06) • Destruction timescale is secular 22nd General Relativity and Gravitation in Japan 3 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 3

Dynamics of r-mode instabilities Saturation amplitude of r-mode instability 0 10 � 3D simulation • Saturation amplitude of o(1) �� 10 • Imposing large amplitude of radiation reaction potential in the system to control secular �� 10 0 10 20 30 timescale with dynamics (Lindblom et al. 00) t/P 0 1D evolution with partially included 3 wave interaction Alternative approaches • Saturation amplitude of ~ o(0.001), which depends on interaction term • From linear regime to nonlinear regime (Schenk et al. 2001) Final fate of r-mode instability • From dynamical timescale to secular timescale 3D simulation • Evolution starting from the amplitude o(1) are necessary! 2 • Imposing large amplitude of radiation reaction potential • Energy dissipation of r-mode catastrophically decays to � 1 differentially rotating configuration in dynamical timescale 1D evolution including mode couplings network 0 • After reaching the saturation amplitude ~o(0.001), 0 20 40 60 t (ms) Kolmogorov-type cascade occurs (Gressman et al. 02, Lin & Suen 06) • Destruction timescale is secular 22nd General Relativity and Gravitation in Japan 3 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 3

Amplitude of r-mode instability (LIGO 10) • Isolated compact object in the supernova remnant Cassiopeia A • Compelling evident that the central compact object is neutron star • Restriction to the amplitude of the r-mode instability by not detecting gravitational waves α ≈ 0 . 14 − 0 . 005 Possibility of gravitational wave source (Bondarescu et al. 09) • Possibility of parametric resonance by nonlinear mode-mode interaction • Amplification to α ∼ 1 Necessary to obtain a common knowledge for the basic properties of r-mode instability ! 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 4

2. Dynamics beyond acoustic timescale • Timescale which cannot be reached by GR hydrodynamics Need to separate the hydrodynamics and the radiation term • Instability driven by gravitational radiation Need to impose gravitational waves “Newton gravity + gravitaional radiation reaction” are at least necessary Gravitational radiation reaction (Blanchet, Damour, Schafer 90) Quadrupole radiation metric (includes 2.5PN term) 5 c 5 � d 3 I T T h ij = − 4 G amplification factor to control the ij radiation reaction timescale dt 3 Gravitational radiation reaction potential Φ (RR) = 1 � ψ = 4 π h ij x j � i ρ 2( � ψ + h ij x j � i Φ ) 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 5

Dynamics beyond the acoustic timescale • Shortest timescale in the system restricts the maximum timestep for evolution Acoustic timescale in Newtonian gravity (control acoustic timescale) • Relax the restriction from the rotation of the background star Introduce rotating reference frame Anelastic approximation Kill the degree of freedom of the sound wave propagation v j r j h + ( Γ � 1) h r j v j = 0 Linear regime No shocks (Villain & Bonazzolla 02) r j ( ρ v j ) = 0 r j ( ρ eq v j ) = 0 Propagation of the sound wave ∂ t = 1 ∂ρ ∂ P ∂ t = �r j ( ρ v j ) ✓ ◆ ∂ 4 � 1 c 2 P = S s ∂ c 2 ∂ t ∂ t r j ( ρ v j ) + 4 P = S s Imposing the anelastic approximation changes the structure of the pressure equation 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 6

Basic equations in rotating reference frame (Lie derivative) Spatial component of the Time evolution momentum velocity ∂ρ γ ij ( v j ∂ t = 0 (eq) + v j ) u i (eq) + u i = ˜ γ ij = δ ij + h ij ˜ spatial metric ∂ ∂ t ( ρ u i ) + r j ( ρ u i v j ) = �r i p � ρ r i ( Φ + Φ (RR) ) � ρ ( v j (eq) + v j ) r j u i (eq) + ρ u j r i v j (eq) up to 1st order of � Pressure poisson equation Boundary condition: � p = S p P=0 at the stellar surface Anelastic approximation (constraint) r j ( ρ v j ) = 0 Need a special technique to satisfy constraints throughout the evolution 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 7

Similar procedure to SMAC method, which is used to Procedure solve Navie-Stokes incompressible fluid (McKee et al. 08) 1. Time update the linear momentum ( ρ ∆ u i ) ( ∗ ) = ( ρ ∆ u i ) ( n ) � ∆ t [ � i p + · · · ] Note that the velocity does not automatically satisfy anelastic condition φ 2. Introduce an auxiliary function and solve the following Poisson’s equation ( � φ ) ( ∗ ) = ∂ j ( ρ ∆ v j ) ( ∗ ) φ = 0 Boundary Condition: at the stellar surface 3. Adjust the 3-velocity in order to satisfy the anelastic condition ( ρ ∆ v i ) ( n +1) = ( ρ ∆ v i ) ( ∗ ) − ( ∂ i φ ( ∗ ) ) 4. Introduce another auxiliary function and solve the following ψ Poisson’s equation � ψ = δ ij [ ∂ j ( ρ ∆ u i ) ( ∗ ) � ∂ j ( ρ ∆ u i ) ( n +1) ] ψ = 0 Boundary Condition: at the stellar surface 5. Time update the pressure p ( n +1) = p ( n ) + ψ ( ∗ ) ∆ t 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 8

3. Nonlinear r-mode instability r p /r e T/W 0.55 0.102 Equilibrium configuration of the star 0.65 0.088 • Rapidly rotating neutron star 0.70 0.076 • Uniformly rotating, n=1 polytropic 0.75 0.062 equation of state Eigenfunction and eigenvector of r-mode in incompressible star Eigenfunction of the velocity � r � l Y ( B ) α = 1 × 10 − 4 δ v = α Ω R ll R Impose eigenfunction type perturbation on the equilibrium velocity to trigger r-mode instability Eigenfrequency (rotating reference frame) 2 m ω = l ( l + 1) Ω Incompressible star case Check the excitation of the eigenfrequency 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 9

Spectrum 0.0008 2 2 ) (z) (R / M Eigenfrequency of the 0.0006 r-mode from slow rotation approximation (Yoshida & Lee 01) 0.0004 Our excitation 2 S + frequency 0.0002 ��� r g-mode ？ �� 0 -10 0 10 -5 5 �� P c Due to • slow rotation approximation • anelastic approximation the eigenfrequency does not perfectly agree 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 10

Gravitational Waveform 3e-05 2 ��� r h + R / M 2e-05 1e-05 0 -1e-05 -2e-05 �� -3e-05 2 ��� r h x R / M 2e-05 1e-05 0 -1e-05 -2e-05 � -3e-05 0 100 200 300 400 t / P c Saturation amplitude is around α ≈ 10 − 3 22nd General Relativity and Gravitation in Japan 14 November 2012 @RESCEU, The University of Tokyo, Japan No. 11