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 observable 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 observable 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 particle world lines matter space spacetime 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 v y ǫ x Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Elasticity: Shear Stresses Elastic Solid ρ v y ǫ x Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Elasticity: Shear Stresses Perfect Fluid Need to include shear energy density stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) pressure More general, but still no heat flow T ab = ( e + p ) u a u b + pg ab Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Elasticity: Shear Stresses Elastic Material Need to include shear energy density stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) pressure More general, but still no anisotropic stress heat flow T ab = ( e + p ) u a u b + pg ab + π ab Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Elasticity: Shear Stresses General Need to include shear energy density heat flux stresses Add anisotropic stress term to stress-energy tensor (Karvolini and Samuelsson, 2003) pressure More general, but still no anisotropic stress heat flow T ab = ( e + p ) u a u b + pg ab + π ab 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 9 . 5 Can reproduce published Newtonian exact Riemann 9 . 0 ρ solutions (Barton et al, 8 . 5 2009) 0 . 4 Relativistic code results approach Newtonian results v x 0 . 2 in Newtonian limit 0 . 0 1 . 2 0 . 8 ǫ 0 . 4 0 . 0 − 0 . 4 − 0 . 2 0 . 0 0 . 2 0 . 4 x 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 9 . 5 9 . 0 ρ 8 . 5 Formalism works for curved 0 . 4 coordinates Riemann test in 2D v x 0 . 2 cylindrical coordinates Ring rotor in 2D cylindrical 0 . 0 coordinates 1 . 2 0 . 8 ǫ 0 . 4 0 . 0 − 0 . 10 − 0 . 05 0 . 00 0 . 05 0 . 10 x Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Interfaces: Where is the interface? level−set function x A B A B A 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? Fedkiw et al, 1999—Ghost fluid method Real Fluid (GFM): (n) P, v Continuous across contact: p , v ( n ) Real Fluid Discontinuous across contact: s , v ( t ) Ghost Fluid Calculate ρ from s and p (t) s, v Stephanie J. Erickson University of Southampton Shattering of the Neutron Star Crust
Elasticity Interfaces Shattering Conclusions Interfaces: What happens at the interface? Fedkiw et al, 1999—Ghost fluid method Real Fluid (GFM): (n) P, v Continuous across contact: p , v ( n ) Real Fluid Discontinuous across contact: s , v ( t ) Ghost Fluid Calculate ρ from s and p (t) s, v 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? Fedkiw et al, 1999—Ghost fluid method Real Fluid (GFM): (n) P, v Continuous across contact: p , v ( n ) Real Fluid Discontinuous across contact: s , v ( t ) Ghost Fluid Calculate ρ from s and p (t) s, v 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 1 . 0 relaxation Relaxed state occurs when 0 . 5 matter-space metric is proportional to spacetime 0 . 0 y metric pushed forward onto matter space − 0 . 5 SO, to shatter, reset variables to relaxed state (matter-space metric or − 1 . 0 − 1 . 0 − 0 . 5 0 . 0 0 . 5 1 . 0 x configuration gradient) 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
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