Supra-Kerr 3D Collapse

This case evolves the same star as in the Supra-Kerr axisymmetric case except axisymmetry is relaxed and only pi symmetry about the rotation axis is adopted. Evolution is performed on a 280 x 140 x 200 grid with boundaries at [-13M, 13M] x [0, 13M] x [0, 13M] and runs for 0 < t/M < 7. The initial angular momentum parameter is J/M² = 1.19. The collapse, formation of the torus, and subsequent flattening occurs as in the 2D runs, but the torus quickly fragments into four clumps symmetrically located about the origin, roughly 90° apart, at t/M = 5.7. This system will emit a substantial burst of gravity waves due to the oscillation of the torus and the rotation of the clumps ("splash radiation"). The dominant mode of emission is quadrupole l=2, m=0 radiation from the global collapse and bounce of the torus. The second largest mode is octupole l=4, m=0 radiation from the global collapse and l=4, m=+/-4 radiation from the rotation of the 4 clumps. With current computational resources, we were unable to determine the final fate of the four clumps. It is possible they could continue to collapse to black holes, or the collapse may be halted by shock-induced thermal pressure.

In the meridional clip, the density is plotted on a logarithmic scale normalized to the central density of the star at the current time (Fig 2-1). The clumps first form around t/M = 5.7 (Fig. 2-3). At the final time t/M = 7, the angular momentum satisfies J/M² = 1.15 which is close to the original value of 1.19 (Fig. 2-5).

Equatorial Plane

Fig. 2-1 Color code for density profile	Fig. 2-2 Density profile at t/M = 0
Fig. 2-3 Density profile at t/M = 5.7	Fig. 2-4 Density profile at t/M = 7.0
Fig. 2-5 Measurement of the spin parameter at t/M = 7.0

**A zoom occurs after the clumps form so visual inspection should not be used to compare sizes.

In these clips, we track 100,000 Lagrangian matter tracers (particles) that represent fluid elements. The initial distribution of Lagrangian tracers is proportional to the initial mass density. We then calculate the trajectories of the tracers by integrating fluid velocities. The particles are also color-coded according to the density at their current location. Unlike the density profile clip, the densities used to color the particles are normalized to the central density of the star at the initial time. In addition, a separate color coding is used (Fig. 3-1).

Equatorial View

Fig. 3-1 Color code for Lagrangian matter tracers	Fig. 3-2 Lagrangian matter tracers at t/M = 0
Fig. 3-3 Lagrangian matter tracers at t/M = 5.7	Fig. 3-4 Lagrangian matter tracers at t/M = 7

**A zoom occurs after the clumps form so visual inspection should not be used to compare sizes.

Meridional View

Fig. 3-5 Lagrangian matter tracers at t/M = 0	Fig. 3-6 Lagrangian matter tracers at t/M = 5.7
Fig. 3-7 Lagrangian matter tracers at t/M = 7