Viscosity driven instability in rotating relativistic stars Motoyuki Saijo (University of Southampton) Eric Gourgoulhon (Observatoire de Paris) CONTENTS 1. Introduction 2. Bifurcation theory in Newtonian gravity 3. Bifurcation theory in general relativity 4. Viscosity driven instability in rotating relativistic stars 5. Summary From Geometry to Numerics, No. 1 20 November 2006 @IHP, Paris, France
1. Introduction Secular and dynamical bar mode instability Magnitude of Rotation T: Rotational kinetic energy W: Gravitational binding energy Toroidal mode analysis in Maclaurin spheroid (Chandrasekhar 69) Lagrangian ξ ( t, x ) = e i ω t ξ ( x ) displacement ω = 0 Neutral point Secular instability Star becomes unstable when we take dissipation into account or gw ω 2 = 0 Dynamical instability Dynamically unstable Star becomes unstable due to hydrodynamics (rotation) From Geometry to Numerics, No. 2 20 November 2006 @IHP, Paris, France
Secular instability in the low temperature limit Viscosity driven instability (Roberts,Stewartson 1963) Configuration transits to lower energy state due to viscosity Sets in when a mode has a zero-frequency in the frame rotating with the star ω + m Ω = 0 m=-2 mode Astrophysical scenario t ev > t vis , t mag High viscosity Maintains Strong magnetic field uniform rotation Accreting neutron star @NGSL, Michigan From Geometry to Numerics, No. 3 20 November 2006 @IHP, Paris, France
Gravitational wave driven instability (Chandrasekhar 70, Friedman Schutz 78) Configuration transits to lower energy state due to gravitational radiation Sets in when the backward going mode is dragged forward in the inertial frame ω − m Ω = 0 m=2 mode Astrophysical scenario t ev < t vis , t mag Low viscosity Leads to Magnetic braking differential rotation Newly born neutron star @Chandra web site The opponent effects competes with each other ! (Detweiler, Lindblom 1977) e.g. Viscosity driven instability is stablized by the gravitational radiation From Geometry to Numerics, No. 4 20 November 2006 @IHP, Paris, France
Gravitational waves from bar mode Mechanism Global rotational instabilities in fluids arise from nonaxisymmetric bar mode Quadrupole formula Energy Flux � 5 � T � 3 � M dE dt = 8 45 M 2 R 4 Ω 6 ∼ R W Gravitational Waveform d 2 rh + = 1 dt 2 ( I xx − I yy ) 2 = − 2 3 MR 2 Ω 2 cos 2 Ω t rh × = d 2 = − 2 3 MR 2 Ω 2 sin 2 Ω t dt 2 I xy Feature Ω / π Frequency is since the bar spins its center of mass From Geometry to Numerics, No. 5 20 November 2006 @IHP, Paris, France
Frequency band for bar instability (Cutler & Thorne 2002) From Geometry to Numerics, No. 6 20 November 2006 @IHP, Paris, France
2. Bifurcation theory in Newtonian gravity Riemann S-type ellipsoid Nonaxisymmetric body with one principal rotational axis (including a uniform vorticity) ab f ≡ ζ Ω = Const . x = a 2 + b 2 f • Maclaurin spheroid Uniformly rotating axisymmetric • incompressible body c a = b, Ω = Ω c f = 0 a • Jacobi ellipsoid b Uniformly rotating nonaxisymmetric • incompressible body Ω = Ω c f = 0 • Dedekind ellipsoid Differentially rotating nonaxisymmetric incompressible body f = ± ∞ From Geometry to Numerics, No. 7 20 November 2006 @IHP, Paris, France
(Christodoulou et al. 1995) Meaning of the secular instability E Solely L conservation Free-energy function E: ( a + bx ) 2 + ( b + ax ) 2 E = L 2 Bifurcation − 2 I ( a, b, c ) a 2 + b 2 + 2 abx Energy 2 point dissipation L: Angular momentum L 2 ( a − b ) 2 x ∂ E ∂ x = ( a 2 + b 2 + 2 abx ) 3 I: Moment of inertia L Energy minimum at x = 0 Energy contour (2nd order derivative in x is possitive) Jacobi ellipsoid Features Energy Nonaxisymmetry minimum 1. Maclaurin spheroid is the energy minimum state up to the bifurcation point b / a 2. Jacobi ellipsoid is the energy minimum state beyond the bifurcation point through the variation of circulation 3. Bifurcation point corresponds to the neutral point c / a Violation of the circulation is induced by viscous dissipation From Geometry to Numerics, No. 8 20 November 2006 @IHP, Paris, France
where Solely C conservation Free-energy function E: ( a + bx ) 2 + ( b + ax ) 2 E = C 2 C 2 ( a 2 − b 2 ) 2 x � ∂ E x � = 1 /x ∂ x � = − 2 I ( a, b, c ) ( a 2 + b 2 + 2 abx � ) 3 [2 ab + ( a 2 + b 2 ) x ] 2 2 Energy minimum at Dedekind ellipsoid (2nd order derivative in x’ is possitive) Features E 1. Maclaurin spheroid is the energy minimum state up to the bifurcation point Bifurcation 2. Dedekind ellipsoid is the energy minimum Energy point state beyond the bifurcation point through dissipation the variation of angular momentum 3. Bifurcation point corresponds to the C neutral point Violation of the angular momentum is induced by gravitational radiation From Geometry to Numerics, No. 9 20 November 2006 @IHP, Paris, France
3. Bifurcation theory in general relativity Nonaxisymmetric spacetime lapse N : N i : correspond to shift ds 2 = − N 2 dt 2 + A 2 ( dr − N r dt ) 2 + r 2 A 2 ( d θ − N θ ) function of the + r 2 sin 2 θ B 2 ( d ϕ − N ϕ dt ) 2 A, B : spatial metric Only true when the azimuthal variable is separable Equilibrium state N r , N θ = 0 ( r, θ ) and all the functions only depend on Axisymmetric spacetime in quasi-isotropic coordinate Nonaxisymmetric perturbation ln N = ln N eq (1 + � sin 2 θ cos 2 ϕ ) Treatment of the spacetime ( r, θ , ϕ ) 1. Lapse and shift depend on ( r, θ ) 2. A and B depend on � M � ∼ 10 − 6 � × Fully constraint equations to be solved R Consistent up to 1/2 PN order satisfactory approximation! From Geometry to Numerics, No. 10 20 November 2006 @IHP, Paris, France
Iterative evolution approach (Bonazzolla, Frieben, Gourgoulhon 96, 98) Investigate nonaxisymmetric instability in quasi-static evolution in general relativity Spherical Advantage No restriction to the iteration (time) step Fully constraint scheme Equilibrium Coincidence of a bifurcation point in Newtonian incompressible star (Gondek-Rosinska, Gourgoulhon 03) Impose bar-mode perturbation Disadvantage The direction of time evolution is Iteration Step not clear in a strict sense (Evolution) Unstable Stable Restriction to the axisymmetric fluid flow From Geometry to Numerics, No. 11 20 November 2006 @IHP, Paris, France
Diagnostics ∞ q = max | ln ˆ � ln ˆ N 2 | where N m e im ϕ ln N 2 = Illustration m =0 q q N N Unstable: q grows exponentially Stable: q decays exponentially through iteration through iteration Precise measurement 0.05 Investigate the logarithmic -5 � � =25, � amp =10 derivative of q q / q Unstable: Positive value 0 Stable: Negative value -0.05 From Geometry to Numerics, 200 300 No. 12 20 November 2006 @IHP, Paris, France N
4. Viscosity driven instability in rotating relativistic stars Incompressible stars (Gondek Rosinska, Gourgoulhon 2002) M/R --- 0.0227 0.0438 0.0984 0.1556 0.2000 0.2430 T/W 0.1375 0.1412 0.1446 0.1539 0.1642 0.1729 0.1822 Relativistic gravitation stabilizes the system from viscosity driven instability The above statement also agrees with the pN results in incompressible (Di-Girolamo, Vietri 02) stars Rigidly rotating polytropic stars (Bonazzola, Frieben, Gourgoulhon 98) Investigate the stability at mass-shedding limit Relativistic gravitation stabilizes the system from viscosity driven instability From Geometry to Numerics, No. 13 20 November 2006 @IHP, Paris, France
Rigidly rotating stars in Newtonian gravity (Bonazzola, Frieben, Gourgoulhon 1996) Compressible stars have T/W ≈ 0 . 135 slightly lower criterion of T/W than in the incompressible star mass-shedding limit From Geometry to Numerics, No. 14 20 November 2006 @IHP, Paris, France
Rigidly rotating stars in general relativity (MS, Gourgoulhon 06) Polytropic equation of state Relativistic gravitation stabilizes viscosity driven instability The bifurcation point is not so sensitive to the stiffness of the equation of state From Geometry to Numerics, No. 15 20 November 2006 @IHP, Paris, France
Differentially rotating stars in general relativity Due to viscous friction, the angular momentum distribution should be changed We assume that it is small and still remains the present angular momentum distribution M/R = 0.20 0.25 1. Fixed rotation profile Γ = 2 0.24 Rotation raw (equilibrium state) A 2 Ω 0 0.23 Ω ≈ A = R e − → Ω 0 / Ω eq ∼ 2 � 2 + A 2 (T/W) crt 0.22 Relaxes the restriction of the mass-shedding limit in rigid rotation 0.21 Differentially rotation also stabilizes the sytem 0.2 But do you believe this current result? 0.19 0.18 From Geometry to Numerics, No. 16 20 November 2006 @IHP, Paris, France
2. Varied rotation profile Helical Killing vector ξ µ : timelike Killing vector k µ = ξ µ + Ω χ µ χ µ : rotational Killing vector Ω = Constant stationary But lazy physicist ... otherwise static configuration Variation of rotation profile ε omg / ε rot Adjust to rot = A − 1(eq) A − 1 [1 − ε rot ( N − N ptb )] rot maintain J conservation Ω c = Ω (eq) [1 − ε omg ( N − N ptb )] throughout the iteration c From Geometry to Numerics, No. 17 20 November 2006 @IHP, Paris, France
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