Supra-Kerr Axisymmetric Collapse

Introduction
Evolution of Density Profile
Evolution of Lagrangian Matter Tracers
Evolution of the Lapse Function


Introduction


Fig. 1-1 Initial Shape of the Rotating Star

Evolution is performed on a 400 x 400 grid, with outer boundaries at 13M. The initial equatorial radius of the star is Req/M = 4.2. The star is evolved from 0 < t/M < 184. The initial value of J/M2 is 1.19, so a black hole does not form in this case. The initial inward motion is halted by centrifugal forces. The inner region of the star stops collapsing and bounces and expands into the outer region, forming a strong shock. The star then expands into a torus whose radius undergoes damped oscillations.


Evolution of the Density Profile

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 approximately t/M = 46, the collapse reaches its innermost radius before bouncing (Fig. 2-3, 2-7). Then after bouncing, the torus starts to collapse again at approximately t/M = 92 (Fig. 2-4, 2-8). At the final time, the angular momentum satisfies J/M2 = 1.26 which is close to the initial value of 1.19.

Meridional 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 = 46

Fig. 2-4 Density profile at t/M = 92

Fig. 2-5 Density profile at t/M = 184

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Equatorial Plane


Fig. 2-6 Density profile at t/M = 0

Fig. 2-7 Density profile at t/M = 46

Fig. 2-8 Density profile at t/M = 92

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Evolution of Lagrangian Matter Tracers

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. The color coding is the same as the density profile clips (Fig. 2-1). At approximately t/M = 46, the collapse reaches its innermost radius before bouncing (Fig. 3-2, 3-5). Then after bouncing, the torus starts to collapse again at approximately t/M = 92 (Fig. 3-3, 3-6).


Meridional View


Fig. 3-1 Lagrangian matter tracers at t/M = 0

Fig. 3-2 Lagrangian matter tracers at t/M = 46

Fig. 3-3 Lagrangian matter tracers at t/M = 92

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Equatorial View


Fig. 3-4 Lagrangian matter tracers at t/M = 0

Fig. 3-5 Lagrangian matter tracers at t/M = 46

Fig. 3-6 Lagrangian matter tracers at t/M = 92

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Evolution of the Lapse Function

This clip shows the evolution of log(α) for the equatorial plane of the star. The minimum value of α occurs at t/M = 48 (Fig. 4-3).


Fig. 4-1 Initial star with equatorial plane

Fig. 4-2 Lapse function at t/M = 0

Fig. 4-3 Minimum α value at t/M = 48

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last updated 4 December 2014 by aakhan3

Center for Theoretical Astrophysics---University of Illinois at Urbana-Champaign

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