Shattering of the Neutron Star Crust Stephanie J. Erickson - - PowerPoint PPT Presentation

Elasticity Interfaces Shattering Conclusions

Shattering of the Neutron Star Crust

Stephanie J. Erickson

University of Southampton

3 April 2012

see C. Gundlach, I. Hawke, and SJE, CQG 29 015055 (2012)

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

The Physics Problem

NS crust is only small fraction of total mass BUT, crustal modes are qualitatively different AND, crustal modes are at much lower frequencies Several models have shown that the crust can contribute to

• bservable behavior, (ie. Pulsar glitches, Ruderman 1969 and

GRB’s, Blaes et al. 1989)

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

The Physics Problem

NS crust is only small fraction of total mass BUT, crustal modes are qualitatively different AND, crustal modes are at much lower frequencies Several models have shown that the crust can contribute to

• bservable behavior, (ie. Pulsar glitches, Ruderman 1969 and

GRB’s, Blaes et al. 1989) How does shattering due to tidal forcing affect the NS and the merger system as a whole?

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Plan

What do we need to do the simulation? Elasticity Interfaces Shattering

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Relationship with relaxed state

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Relationship with relaxed state

spacetime particle world lines matter space

Use two manifolds and map between them (Carter and Quintana, 1972) Derivatives of map: configuration gradient Use commutation of partial derivatives to write evolution equation and constraint for configuration gradient

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Shear Stresses

Perfect Fluid

ρ vy x ǫ

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Shear Stresses

Elastic Solid

ρ vy x ǫ

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Shear Stresses

Need to include shear stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) More general, but still no heat flow Perfect Fluid

pressure energy density

T ab = (e + p)uaub + pgab

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Shear Stresses

Need to include shear stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) More general, but still no heat flow Elastic Material

anisotropic stress pressure energy density

T ab = (e + p)uaub + pgab + πab

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity: Shear Stresses

Need to include shear stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) More general, but still no heat flow General

anisotropic stress heat flux pressure energy density

T ab = (e + p)uaub + pgab + πab

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Elasticity Code

3D variables on 1D or 2D grid with planar symmetry Cartesian or cylindrical Minkowski metric Newtonian version of code from v ≪ c Test using Riemann problems

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Newtonian Elasticity Results

8.5 9.0 9.5 ρ 0.0 0.2 0.4 vx −0.4 −0.2 0.0 0.2 0.4 x 0.0 0.4 0.8 1.2 ǫ

Can reproduce published Newtonian exact Riemann solutions (Barton et al, 2009) Relativistic code results approach Newtonian results in Newtonian limit

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

2D Cylindrical Coordinates

Formalism works for curved coordinates Riemann test in 2D cylindrical coordinates Ring rotor in 2D cylindrical coordinates

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

2D Cylindrical Coordinates

8.5 9.0 9.5 ρ 0.0 0.2 0.4 vx −0.10 −0.05 0.00 0.05 0.10 x 0.0 0.4 0.8 1.2 ǫ

Formalism works for curved coordinates Riemann test in 2D cylindrical coordinates Ring rotor in 2D cylindrical coordinates

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Interfaces: Where is the interface?

x B A A A B level−set function

Use a level-set function to track the interface Positive in cells filled by one material, negative for other material, zero at interface Advected along with material

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Interfaces: What happens at the interface?

Ghost Fluid P, v s, v

(n) (t)

Real Fluid Real Fluid

Fedkiw et al, 1999—Ghost fluid method (GFM): Continuous across contact: p, v(n) Discontinuous across contact: s, v(t) Calculate ρ from s and p

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Interfaces: What happens at the interface?

Ghost Fluid P, v s, v

(n) (t)

Real Fluid Real Fluid

Fedkiw et al, 1999—Ghost fluid method (GFM): Continuous across contact: p, v(n) Discontinuous across contact: s, v(t) Calculate ρ from s and p Another option: Barton et al, 2010 – modified GFM uses Riemann solution to determine correct behavior and assigns cells accordingly

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Interfaces: What happens at the interface?

Ghost Fluid P, v s, v

(n) (t)

Real Fluid Real Fluid

Fedkiw et al, 1999—Ghost fluid method (GFM): Continuous across contact: p, v(n) Discontinuous across contact: s, v(t) Calculate ρ from s and p Another option: Barton et al, 2010 – modified GFM uses Riemann solution to determine correct behavior and assigns cells accordingly Progress: Fluid interfaces in 1D – can reproduce results of published Newtonian and special relativistic tests

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Shattering

Shatter = instantaneous relaxation Relaxed state occurs when matter-space metric is proportional to spacetime metric pushed forward onto matter space SO, to shatter, reset variables to relaxed state (matter-space metric or configuration gradient)

−1.0 −0.5 0.0 0.5 1.0 x −1.0 −0.5 0.0 0.5 1.0 y Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Shattering

2D homogeneous anisotropic initial data, then shatter a circular region at the center Encountered no numerical problems

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust

Elasticity Interfaces Shattering Conclusions

Conclusions

Want to find out what happens when part of the crust shatters due to tidal forcing in a binary merger system Need elasticity in GR, interfaces, and shattering Have elasticity code, shattering with no problems so far, and interfaces for fluids Still to come: solid-fluid interface, combine components

Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust