Star I

Evolution is performed on a 128x128 grid, with outer boundaries at r/M=4.3, where M is the initial mass of the star. In this simulation, the spin parameter is J/M²=1.0, and M₀/M_0,sup=1.47 where M_0,sup is the rest-mass limit for uniformly rotating polytropes. Figure 1.1 shows the initial shape of the star.

In the meridional and equatorial clips, the density is plotted on a logarithmic scale normalized to the central density of the star at the start of evolution (Fig. 2-1).

At t/M=0, the star has a toroidal density profile due to rapid differential rotation (Fig. 2-2). Viscosity destroys differential rotation and transfers angular momentum to the outer layers. At time t/M=972 (25.3P_rot), the star becomes dynamically unstable and collapses to a rotating black hole. An apparent horizon (dark grey) appears at t/M=1030 (26.8P_rot) as show in Figure 2-3. Excising the black hole interior (white area) and evolving, we find that the system settles down to a black hole surrounded by a massive ambient disk (Fig. 2-4).

In these clips, we place 50,000 Lagrangian matter tracers 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. The color-coding is the same as the density profile clips (Fig. 2-1). The dark grey volume is the apparent horizon and the white volume is the area excised.

The Lagrangian matter tracers show quite clearly the formation of a disk and ISCO (Fig. 3-6).

This clip shows the evolution of log(α) for the equatorial plane of the star. The excision boundary is denoted by a white ring on the surface (Fig. 4-3). Anything inside of the white ring is excised. Note that in the adopted time coordinate log(α) stays finite on the excision boundary (Fig. 4-4), gauranteeing a "horizon penetrating" lapse, necessary for excision.

The star collapses to form a rotating black hole surrounded by a massive disk. Listed in the table below are the relative parameters including the spin of the black hole and disk, and the rest mass of the black hole and disk. These parameters indicate the formation of a substantially massive disk.

In the meridional and equatorial clips, the density is plotted on a logarithmic scale normalized to the central density of the star at the start of evolution (Fig. 2-1).

As in the no cooling limit, the inner core contracts and the outer layers expand in a quasi-stationary manner. The core then collapses dynamically to a black hole and leaves behind a massive accretion disk. Since there is no net viscous heating, the whole process occurs quicker than in the no cooling limit. The dynamical collapse occurs at t/M=495.

**A zoom occurs just prior to excision so visual inspection should not be used to compare sizes.

In these clips, we place 50,000 Lagrangian matter tracers 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. The color-coding is the same as the density profile clips (Fig. 5-1). The dark grey volume is the apparent horizon and the white volume is the area excised.

Meridional View

Fig. 6-1 Lagrangian matter tracers at t/M = 0	Fig. 6-2 Black hole excision at t/M = 495
Fig. 6-3 Lagrangian matter tracers at t/M = 517

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Equatorial View

Fig. 6-4 Lagrangian matter tracers at t/M = 0	Fig. 6-5 Black hole excision at t/M = 495
Fig. 6-6 Lagrangian matter tracers at t/M = 517

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Evolution of the Lapse Function (Rapid Cooling Limit)

Fig. 7-1 Initial star with equatorial plane	Fig. 7-2 Lapse function at t/M = 0
Fig. 7-3 Black hole excision at t/M = 495	Fig. 7-4 Black hole excision at t/M = 517

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Black Hole and Disk Parameters (Rapid Cooling Limit)

JD/J	0.55
JBH/J	0.45
M0,D/M0(0)	0.21
M0,BH/M0(0)	0.79
JBH/M0,BH2	0.60
JBH/Mirr2	0.87

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