Neutron Star instabilities in full General Relativity or An example of using the Einstein Toolkit for research Frank L¨ offler Center for Computation and Technology Louisiana State University, Baton Rouge, LA Aug 2nd 2014 Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
What are neutron stars? Mass slightly above solar mass ( ≈ 1 . 4 M ⊙ ) Very small ( ≈ 12km) Very dense (about/above density of core of atoms) Rotating (often differentially at birth) Magnetic fields Environment unlike anything on or close to Earth Equation of state (EOS) not known very well → measurements Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Neutron star trivia Gravitational constant about 10 11 g Earth Escape velocity about 1 3 speed of light Object falling from 1 meter above surface: time of fall: 1 µ s velocity at impact: 2000 km s Bending light ( ≈ 60% of surface visible) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Neutron star formation Type II or Ib/c super novae (core collapses) Accretion onto white dwarf (although probably not) Initially fast rotating (ms) Likely high magnetic fields Temperature initially ≈ 10 11 − 10 12 K falls within years to ≈ 10 6 K (neutrinos) EOS more or less unknown Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Rotation of neutron stars Extremely rapid rotation after creation Slowing down (magnetic breaking, turbulence, shear viscosity, radiation) Late-time slow-down rate about 10 − 15 s/rotation → 1 s → 1 . 03 s in 1 My Glitches (star-quakes? / core-crust interactions?) Acceleration of particles near magn. poles → coherent radio emission → pulsars Most rapidly rotating star: 716/s (1.4ms) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Population ≈ 2000 known neutron stars in Milky Way and Magellanic Clouds Most radio pulsars Most in disk, but spread is large One of the closest NSs: about 130 parsecs (424 ly) 5% of neutron stars within binary systems Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Astronomy isn’t always just still pictures Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Neutron star instabilities Equilibrium � = Stability Similarly, star models: often constructed in equilibrium might be not stable Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Questions we want to answer 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? S Chandrasekhar: “Ellipsoidal figures of equilibrium” (1969) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Rotation Rotation might be differential close to maximum limit (Kepler) Characterizing quantity: β = T / | W | Theoretical limit of 0 . 5 Uniformly rotating, constant density star: limit of ≈ 0 . 11 Much higher for differential rotation or non-constant density Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Instability types in rotating (neutron) stars Secular ( m = 2): β ≥ β sec ≈ 0 . 14 needs dissipative mechanism growth time determined by dissipative time scale (tens of seconds for neutron stars) e.g., see Chandrasekhar (1970), Ou, Tohline and Lindblom (2004) Dynamical ( m = 2): β ≥ β dyn ≈ 0 . 27 hydrodynamical origin grows on dynamical timescale (tens of milliseconds for neutron stars) e.g., see Shibata, Tohline, Baiotti, Manca, ... “Low T / W instability” - Shear instability? currently not very well understood first “observed” numerically (grows on dynamical time scale) e.g., see Centrella et al (2001), Corvino (2010), ... Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Dynamical Bar-mode instability Grows within tens of milliseconds within rapidly spinning NSs Feasible using full relativistic simulations Still expensive Previous studies: Often assuming one-component fluid Usually no crust Usually no magnetic fields EOS highly uncertain Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
An example study: Initial Models Differential rotation β = 0.140 Models at M 0 = 2.0 M ⊙ e /10 15 ( g / cm 3 ) 0.4 β = 0.200 β = 0.250 0.2 β = 0.272 0.0 β = 0.140 x ( km ) 1.5 β = 0.200 Ω / ( 2 π ) ( kHz ) β = 0.250 β = 0.272 1.0 0.5 0.0 0 5 10 15 20 x ( km ) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Mode evolution and interaction m=2 m=4 ln(|P |) m m=3 m=1 (a) (b) (c) (d) time Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Example movies Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Example movies Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Specifics of current research K=30000, Γ =2.75 10 35 K=100, Γ =2 Shen ( 1 ) EOS 10 34 SLy ( 2 ) EOS P ( dyne / cm 2 ) 10 33 10 32 10 31 10 30 10 13 10 14 10 15 e ( g / cm 3 ) Different equation of state (Γ = 2 . 75) Why? more realistic for new-born NSs (temperature, compactness) more computational power (more models) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Initial models, evolution parameters Initial models: Standard axisymmetric metric Differential rotation law, A = 1 often chosen Ω c r 2 sin 2 θ Newtonian: Ω C − Ω = A 2 r 2 e + r 2 sin 2 θ Polytropic EOS: p = K ρ Γ = K ρ 1+1 / n Evolution: Γ-law EOS: p = (Γ − 1) ρǫ Full relativistic curvature evolution Relativistic hydrodynamics No magnetic fields Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Code specifics Specifics: Einstein Toolkit Cactus framework Mesh refinement (Carpet) RNS solver by Stergioulas Under the hood: Finite differences High-resolution shock-capturing hydrodynamics Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Finding the threshold β c At threshold: growth time infinity Above threshold: growth time finite Below threshold: no (clear) instability (of this type) Two methods of finding threshold Find models for which mode grows Extrapolation from unstable models, measuring growth time Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Finding the threshold 2.0 M ⊙ 2.5 M ⊙ 0.5 M ⊙ 1.0 M ⊙ 1.5 M ⊙ 0.270 0.265 0.260 β 0.255 0.250 0.245 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1/ τ 2 2 ( ms 2 ) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Making the connection to Newtonian regime 0.254 0.252 0.250 β c 0.248 0.246 0.244 0.242 0.0 0.5 1.0 1.5 2.0 2.5 M 0 / M ⊙ Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Dominant mode not always m=2 Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
What is β ? 0.270 0.268 β dx=0.25 0.266 dx=0.30 dx=0.42 dx=0.50 dx=0.70 0.264 dx=0.84 0 2 4 6 8 10 Time (ms) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
How does this affect results? M1.5b0.272 dx=0.5 0.270 dx=0.25 dx=0.30 dx=0.42 0.269 dx=0.50 0.268 β dx=0.70 0.267 M1.5b0.268 dx=0.5 0.266 dx=0.84 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 1/ τ 2 2 ( ms 2 ) Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
(Some) Results for Bar-Modes Dynamics strongly influenced by separation from critical β Genuinely nonlinear mode-coupling effects limiting life-time of instability (bar) Classical Newtonian perturbative analysis qualitatively valid Critical value decreases for more compact stars Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Things don’t run smoothly all the time... Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Computational requirements Curvature + Hydrodynamics: medium resolution: 3 × 168 × 168 × 84 grid (about 25 points across stellar radius) about 20 , 000 core-hours on supercomputer, per model about 50 GB of memory Supermike II: 1.5d on 512 cores Queenbee: 4d on 512 cores Including MHD: at least factor of four more expensive Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Summary (not quite) Neutron stars very exotic objects do exist and are observed a lot of unknowns Neutron star instabilities difficult to impossible to study analytically presence (or absence) can tell about NS interior important “tool” to constraint NS structure models Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Simulation details Used components: RNSID initial data solver for rotating neutron stars not open source, but usually available on request (Stergioulas) standalone code (not in Cactus) needed to add hdf5 support for output (ASCII default, too much data) write thorn importing these data into the Einstein Toolkit Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
Simulation details Used components: “standard” McLachlan for spacetime evolution ( ML_BSSN ) GRHydro for hydrodynamics evolution Carpet for (fixed) mesh refinement simfactory to run on about 5 different machines Frank L¨ offler Neutron Star instabilities in full GR 2014-08-02
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