User: rhaas Date: 2011/10/10 11:41 PM

Modified: / ET.tex

Log: add some text for TOV collapse section mostly just setup and some generic stuff, not spellchecke or anything. Just to give an idea of the lenght of the section

File Changes:

Directory: / ============

File [modified]: ET.tex Delta lines: +42 -0 =================================================================== --- ET.tex 2011-10-04 19:58:29 UTC (rev 145) +++ ET.tex 2011-10-11 04:41:31 UTC (rev 146) @@ -2692,6 +2692,33 @@ \todo{RH: there are three runs for this for convergence testing. All three will be used eventually. These plots~\ref{fig:tov_collapse_rho_central}, \ref{fig:tov_collapse_radii} currently only use data from the highest resoluton run to show what happens.} +The previous examples dealt either with a pre-exisitng black hole (BBH) or +with a smooth singularity free spacetime (TOV oscilations). The ET however is +also able to handle the dynamic formation of a singularity as a star collapses +into a black hole. As a simple example of this process we study the collapse +of a non-roating TOV solution into a black hole. We create initial data as in +section~\ref{sec:tov_oscilations} using $\rho_c=3.154e-3$ and $K_{ID} = 100$, +$\Gamma = 2$ yielding a star of masss $1.67,M_\odot$. As is common in these +situations we trigger collapse by reducing the pressure during the evolution +by reducing the polytropic constant $K_{ID}$ from its initial data value to $K += 0.98 , K_{ID} = 98$. \todo{RH: cite some papers, eg. Whisky and ref +therein} +Doing so speeds up the collapse and provides a +physical trigger for the collapse rather than random numerical fluctuations +which would not be guaranteed to converge to a unique value with higher +resolution. In order to resolve the star as well as push the outer boundary +far enough away so that the star and the numerical outer boundary are not in +causal contact during the simulation we employ a fixed mesh refinement scheme. +The outermost box has a radius of $R_0 = 204.8,M_\odot$ and a resolution of +$3.2,M_\odot$ ($1.6,M_\odot$, $0.6,M_\odot$ for higher covnergence levels). +Around the star which is centered on the origin we stack $5$ extra boxes of +size $4\times2^\ell$, $0 \le \ell \le 4$ where the resolution on each finer +level is twice that of the surrounding level. We use the PPM +reconstruction method and the HLLE Riemann solver to obtain second +order convergent results in smooth regions. Due to the presence of the +density maximum at the center of the star and the non-smooth atmosphere at the +edge of the star we expect the observed convergence rate to be somewhat lower +than second order, but higher than first order. \begin{figure} \label{fig:tov_collapse_rho_central} \includegraphics{examples/collapse/tov_collapse_rho_central} @@ -2709,7 +2736,22 @@ \includegraphics{examples/collapse/rho_maximum_convergence} \caption{Convergence factor (based on three runs) for the central density.} \end{figure} +In Figure~\ref{fig:tov_collapse_radii} we plot on the same graph the +approximate location of the edge of star and the areal radius of the apparent +horizon once a horizon is found in the simultion. Clearly the apparent +horizon is found at approzimately the same time as the star's size approaches +its Schwarzschild radius.\todo{RH:replace AH radius with coordinate radius} In +Figure~\ref{fig:tov_collapse_H_convergence} we display the convergence factor +for the maximum of the Hamiltion constraint outside of the apaperent horizon +and at $r=2,M_\odot$. Hopefully we +see that the convergence order is between $1$ and $2$. Just before the +apparent horizon forms the density at the center of the star increases rapidly +and we are unable to properly resolve the small central region with the +available computational resoureces. For this reason the maximum of the +Hamilation constraint rises rapidly just before the the interior of the star +is encapsulated in the horizon.

+ \subsection{Cosmology\pagesdone{1}} The Einstein Toolkit is not only designed to evolve compact-object spacetimes, but it is also capable of solving the initial-value