Critical Phenomena in 2D Axisymmetric Collapse of Neutron Stars

● Introduction

●GR equations for simulation

●Prior work of critical phenomena in general relativity

● Axisymmetrical Code

●Idea, implement, advantage

●Convergence tests: resolution, boundary position

● Critical Phenomena in axisymmetric Collapse of Neutron Stars

●Universality, convergence, index

●Possibility of existence in the universe

● Conclusion

Can Critical Collapse Occur in Nature ?Ke Jian Jin

Sec I-1 (3+1) Formulation of Einstein Equations, hydradynamics and gauge choices

Sec I-2 Prior work on critical phenomena in GR

• Begin with Matthew W. Choptuik • Most studies spherical (with few exceptions)

– Scalar field, YM, MP, Wave;– Perfect fluid: Wave pocket

• Limitations:– System not realistic– Do not know much beyond spherical

Sec II Axisymmetrical Code

For a system of axisymmetry, only need to develop one radial slice Based on 3D cartesian coordinate code

Very stable Evolution On A Quasi-2D Grid Lots of utilities available

Implement as Boundary Condition after Evolution Advantage

323^3 (3D) = 2560x5x2560 (2D)

3 R K 2 K ijKij 16 ADM 0,

j Kij ij

j K 8 j i 0.Hamiltonian constraint equation

Momentum constraint equations

Practically they should be convergent to zero in certain order with raising resolution:

For IVP(t=0), there should be 2nd order convergent (~h2);For evolution(t>0), it should be 1st order convergent (~h1) in the peak due to TVD, and so for the long time run.

Read the order from Fig.:Raise resolution: h2 = (1/2)*h1, error e2 = (1/2)*e1;Scaling: e2 -> 2*e2 , => e2 = e1, the two curves will overlap.

Sec II-2 Convergence Test of the Axisymmetric Code

Convergence Test Convergence of static TOV star Section II-2

Convergence Test Convergence of boosted TOV star Section II-2

Convergence Test Convergence of headon process (1) Sec II-2

Convergence Test Convergence of headon process (2) Sec II-2

Convergence of headon process (3) Momentum x (x, short time) Sec III-2

Convergence of headon process (4) Boundary Effect Sec II-2

The momentum constraints is convergent over step length, but not over boundary effect. The boundary effect will be bigger than the interior value for higher resolution run. For small sized grid, the boundary effect will propagate in and ruin the convergence over long time.

Convergence of headon process (5) Boundary (Grid Size) Effect Sec II-2

Short time, nearly same Momentum Constraints Longer time, small grid size one worse

Boundary (Grid Size) Effect Momentum vs. hamiltonian at t=324 Sec II-2

MomentumObvious, Convergent

HamiltonianNearly Independent

Sec III Critical Phenomena in Axisymmetric Collapse of Neutron Stars

Movie: density oscillates with time in one collision which is near critical point

Density as a heightmovie on wugrav

Critical Phenomena Universality (1) Sec III-1Minimum lapse vs. time (varying density, falling from infinity)

Two heavy stars collide into a black hole; the lapse in collision center dips into zero. On the other hand, two light stars collide, the lapse dips, rebounds up. When two stars with the critical density collide, the lapse will dip, rebound, dip, rebound,...

Another view:0.786 – 0.793 M

Critical Phenomena Universality (2) Sec III-1

Minimum lapse vs. time (varying velocity, fixed density & separation)

Critical Phenomena Universality (3) Sec III-1Minimum lapse vs. time (varying density, fixed velocity & separation)

Critical Phenomena Get the departure (from the critical curve) time Sec III-1

vs. time vs. time

When α(t,ρ) or α(t,v) departures from the critical one α*? We made several criterions, that are: 5%, 10%, 15%, 20%. The following figure is for (α-α*)/α* = 5%.

Critical Phenomena Calculate the Index Section III-1

logv v0

v0

The departure time

Convergent Critical Index & Universality (1) Section III-1

varying density, and showing the convergence

t const

1 0

0

lo g .

Convergent Critical Index & Universality (2) Section III-1

varying velocity, and showing the convergence

tv v

vconst

1 0

0

lo g .

Convergent Critical Index & Universality (3) Section III-1

Evidence showing universality (from dx=0.12):

vary density falling from infinity: 10.87+-0.04 vary velocity with fixed d & s: 10.78+-0.05 vary density with fixed v & s: 10.67+-0.06

Sec III-2 Possibility of Existence in the Universe

● Fine tuning of parameter hard to realize● EOS could vary continuously● The duration of EOS varying longer than collapse

– T of EOS varying: 10 sec.– T of critical collapse: 0.05 millisec.

● Is there Critical Phenomena for EOS varying ?

Varying Gamma of the EOS: 10.87+-0.06

A short summary

First, we constructed & tested the GRAstro-2D code,

2nd, we showed the universality, convergence of the

critical collapse,

3rd, we might find critical collapse in the universe