Neutron Star instabilities in full General Relativity or An example - - PowerPoint PPT Presentation

Neutron Star instabilities in full General Relativity

• r

An example of using the Einstein Toolkit for research

Frank L¨

• ffler

Center for Computation and Technology Louisiana State University, Baton Rouge, LA

Aug 2nd 2014

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

What are neutron stars?

Mass slightly above solar mass (≈ 1.4M⊙) 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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Neutron star trivia

Gravitational constant about 1011gEarth Escape velocity about 1

3 speed of light

Object falling from 1 meter above surface:

time of fall: 1µs velocity at impact: 2000 km

s

Bending light (≈ 60% of surface visible)

Frank L¨

• ffler

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 ≈ 1011 − 1012K falls within years to ≈ 106K (neutrinos)

EOS more or less unknown

Frank L¨

• ffler

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 10−15s/rotation → 1s → 1.03s in 1My 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¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Astronomy isn’t always just still pictures

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Neutron star instabilities Equilibrium = Stability

Similarly, star models:

• ften constructed in equilibrium

might be not stable

Frank L¨

• ffler

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¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Instability types in rotating (neutron) stars

Secular (m = 2): β ≥ βsec ≈ 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 ≥ βdyn ≈ 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¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

An example study: Initial Models

Differential rotation

x(km) 0.0 0.2 0.4 e/1015 (g/cm3)

Models at M0 = 2.0M⊙

β = 0.140 β = 0.200 β = 0.250 β = 0.272 5 10 15 20 x(km) 0.0 0.5 1.0 1.5 Ω/(2π) (kHz) β = 0.140 β = 0.200 β = 0.250 β = 0.272

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Mode evolution and interaction

m=2

time ln(|P |)

m

(c) (a) (b) (d)

m=3 m=1 m=4

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Example movies

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Example movies

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Specifics of current research

1013 1014 1015 e(g/cm3) 1030 1031 1032 1033 1034 1035 P(dyne/cm2) K=30000, Γ=2.75 K=100, Γ=2 Shen(1) EOS SLy(2)EOS

Different equation of state (Γ = 2.75) Why?

more realistic for new-born NSs (temperature, compactness) more computational power (more models)

Frank L¨

• ffler

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 Newtonian: ΩC − Ω =

Ωcr2sin2θ A2r2

e +r2sin2θ

Polytropic EOS: p = KρΓ = Kρ1+1/n Evolution: Γ-law EOS: p = (Γ − 1)ρǫ Full relativistic curvature evolution Relativistic hydrodynamics No magnetic fields

Frank L¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Finding the threshold βc

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Finding the threshold

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 1/τ2

2 (ms2)

0.245 0.250 0.255 0.260 0.265 0.270 β 0.5M⊙ 1.0M⊙ 1.5M⊙ 2.0M⊙ 2.5M⊙

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Making the connection to Newtonian regime

0.0 0.5 1.0 1.5 2.0 2.5 M0/M⊙ 0.242 0.244 0.246 0.248 0.250 0.252 0.254 βc

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Dominant mode not always m=2

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

What is β?

2 4 6 8 10 Time (ms) 0.264 0.266 0.268 0.270 β dx=0.25 dx=0.30 dx=0.42 dx=0.50 dx=0.70 dx=0.84

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

How does this affect results?

1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 1/τ2

2 (ms2)

0.266 0.267 0.268 0.269 0.270 β

dx=0.25 dx=0.30 dx=0.42 dx=0.50 dx=0.70 dx=0.84 M1.5b0.268 dx=0.5 M1.5b0.272 dx=0.5

Frank L¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Things don’t run smoothly all the time...

Frank L¨

• ffler

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¨

• ffler

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¨

• ffler

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¨

• ffler

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¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Simulation details

Used components: Added analysis for hydrodynamical modes, e.g.,: Quadrupole moment Qij =

• d3x√γWi̺xixj

modulus Q ≡ 1

2

• (2Qxy)2 + (Qxx − Qyy)2

distortion parameter η ≡

2Q (Qxx+Qyy)

decomposition of density into Fourier modes: Pm ≡

• d3x̺eimφ

Do you remember your “kinetic energy” thorn? → implemented in a similar way (just a little more complicated)

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02

Summary (for real this time)

Publishable research can leverage existing software A few tweaks or additions can be neccesary, e.g.,

special initial data special analysis quantities

However, a lot of code can be re-used

modular software structure parallelism mesh refinement spacetime and hydro evolution I/O capabilities (e.g. checkpointing) supercomputer configurations (simfactory)

Frank L¨

• ffler

Neutron Star instabilities in full GR 2014-08-02