Transcript 3-D simulations of Phase-transition

Phase transition induced
collapse of Neutron stars
Kim, Hee Il
Astronomy Program, SNU
13th Haengdang Symposium, 11/30/2007
Neutron star collapse
Howto?
 Find a (initial) equilibrium star
 Perturb the equilbrium star
 Follow the evolution
 Various instability modes


Showing the inner structure of a star, e.g.
helioseismology
Gravitational wave emission
Equilibrium Star




Stars are stable in most of their lifetime
= Hydrostatic solution
= initial data for evolution
Non-rotating star


1D problem. Trivial to Integrate
Rotating star


2D elliptic Differential eqs.
Unknow boundary surface
p      2
 2  4G
Hachisu’s self consistent field method (HSCF 86)






Newtonian star
Integral representation instead of the differential
eq.  define entalphy H and some constants
Iteration method (     H with b.c. (H=0) 
)
Parameters: axis ratio & central density
Solutions for almost all parameter ranges
New ring-like sequence (Dyson-Wong ring)
Rotating equilibrium star for GR
(Komatsu, Eriguchi, & Hachisu, KEH89)
ds2  e2 dt2  e2 (dr2  r 2d 2 )  e2 r 2 sin 2  (d  dt)2
•
•
•
•
Perfect fluid e.m. tensor, Tab
Fluid four-velocity, ua
Proper velocity w.r.t zamo, v
angular velocity measured
from infinity, 
• Hydrostatic eq is integrable if
• j() is given by hands
• H is entalphy


Iteration:   metrics  H with b.c. (H=0)  
Parameters: axis ratio, maximum density, rotation parameter A
HSCF in Special Relativistic regime

Why needs SR?

Newtonian approach breaks down



Full GR is too expensive



If the motion is relativistic, e.g. rapid rotation
If the equation of state is relativistic, e.g. quark matter
If the gravitiy is very weak, we don’t want to know the
spacetime structure even in weak field limit
Low resolution due to the limited computational resources
SR + pseudo-Newtonian approach

Modified Poisson equation:
2  4Gactive where active  T00  Ti i
Phase transition to a Quark Star

Quark Star



Observations




Stable  (strange) quark star
Metastable  mixed phase quark star
Millisecond pulsar (XTE J1739-285, 1122 Hz, 2006)
Long duration supernova (SN2006gy): Quark nova
after SN explosion
GRB…
Recent works


Lin, et al (2006): GWs, Newtonian Hydro
Yasutake, et al (2007): GWs during the core
collapse, Newtonian-hydro
Collapse of neutron stars
induced by the phase transition





Not implemented yet
Nuclear matter  quark matter
~ softened EOS
~ instant change of polytropic EOS: stiff  soft
Expected results and Questions
 weak transition  GW emission during the
stabilization period
 strong transition  BH formation in the end
 Comparison with the Newtonian results ?
 Especially non-axisymmetric instability ?
GR Hydro simulations with
Cactus/Carpet/Whisky





Cactus provides Einstein equation solvers
Carpet is a mesh-refinement driver for
Cactus
Whisky is a GRHydro code based on
Cactus
Oriented for 3-D simulations
Free softwares but partly unavailable
Initial neutron star model
& other numerical setup


Neutron star
G
 Polytropic EOS: P=K & P=(G1)e :  mass
density, e specific internal energy density
 _center = 6x1014gcm-3 ~ 2 nucleon
 Axis ratio = 0.83   
 G2, K100  90
NR setup




Evolution: BSSN
Lapse: 1+log
Shift: static
Unigrid (PUGH) & Fixed Mesh Refinement (Carpet)
Tests on Starbucks:
max 10 cpus and 10 giga ram with 1gigabit ethernet
Lapse
Mass density
Evolution of central mass density
• Seems to be converging as the resolution increases
• Showing the stabilization
PUGH :
64x64x32
128x128x64
180x180x84
Carpet :
With 3 levels
64x64x64
128x128x64
Density profile
Rho_center
Gravitational wave extraction:
Q_even (l=2 & m=0) at 40M & 60M
• GWs become stronger at larger distances ???
• too coarse to extract GWs
• too close extraction points
• Unfortunately, extraction code for Carpet is not available yet
Unigrid 128x128x64
Unigrid 180x180x84
Quadrupole moment
I ij   d xDx x
3
i
j
Gravitational Wave???
h 
Ixx (t ' )  Iyy (t ' )
r
h  2
Ixy (t ' )
r
h  2
h 
Ixx (t ' )  Iyy (t ' )
r
Ixy (t ' )
r
Psi_4_Re : outgoing waves???
Concluding Remarks

It requires more expensive and elaborated
studies to get meaningful numbers and
results





Wave extraction
Detectability (# of events, …)
Microphysics (details of the transition,
realistic EOSs, …)
Instabilities (non-axisymmetric modes)
BH formation?