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Critical Phenomena in 2D Axisymmetric Collapse of Neutron Stars

Critical Phenomena in 2D Axisymmetric Collapse of Neutron Stars. Can Critical Collapse Occur in Nature ? Ke Jian Jin. Introduction GR equations for simulation Prior work of critical phenomena in general relativity Axisymmetrical Code Idea, implement, advantage

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Critical Phenomena in 2D Axisymmetric Collapse of Neutron Stars

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  1. Critical Phenomena in 2D Axisymmetric Collapse of Neutron Stars

  2. Can Critical Collapse Occur in Nature ? Ke Jian Jin • 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

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

  4. 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

  5. Sec IIAxisymmetrical 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)

  6. Sec II-2 Convergence Test of the Axisymmetric Code • 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.

  7. Convergence Test Convergence of static TOV star Section II-2

  8. Convergence Test Convergence of boosted TOV star Section II-2

  9. Convergence TestConvergence of headon process (1)Sec II-2

  10. Convergence TestConvergence of headon process (2) Sec II-2

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

  12. 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.

  13. 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

  14. Boundary (Grid Size) Effect Momentum vs. hamiltonian at t=324Sec II-2 Momentum Obvious, Convergent Hamiltonian Nearly Independent

  15. Sec IIICritical 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

  16. 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,...

  17. Another view:0.786 – 0.793 Mu

  18. Critical PhenomenaUniversality (2) Sec III-1Minimum lapse vs. time (varying velocity, fixed density & separation)

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

  20. Critical PhenomenaGet the departure (from the critical curve) time Sec III-1 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%.

  21. Critical Phenomena Calculate the IndexSection III-1 The departure time

  22. Convergent Critical Index & Universality (1) Section III-1varying density, and showing the convergence

  23. Convergent CriticalIndex & Universality (2) Section III-1varying velocity, and showing the convergence

  24. Convergent CriticalIndex & 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

  25. 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 ?

  26. Varying Gamma of the EOS: 10.87+-0.06

  27. 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 A short summary

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