User: cott Date: 2011/11/11 09:00 AM
Modified: / ET.tex
Log: * some changes in TOV collapse section. ROLAND: Please check!
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File [modified]: ET.tex Delta lines: +64 -59 =================================================================== --- ET.tex 2011-11-10 23:38:02 UTC (rev 181) +++ ET.tex 2011-11-11 15:00:08 UTC (rev 182) @@ -2736,36 +2736,37 @@ \begin{figure} \label{fig:tov_ham_conv} \includegraphics[width=0.9\textwidth]{examples/tov/ham_conv} - \caption{Convergence factor of Hamiltonian constraint violation at $r=0\mathrm{M}$ and - $r=5\mathrm{M}$. The observed convergence order of about $1.5$ at the center of - the star is lower then the general second order of the hydrodynamics - evolution scheme. This is expected because the scheme's convergence rate drops to first - order at extrema or shocks, like the stellar center or surface. - Consequently, the observed convergence order about half-way between the - stellar center and surface is higher than $1.5$, but most of the time below - $2$.} + \caption{Convergence factor of Hamiltonian constraint violation at + $r=0\mathrm{M}$ and $r=5\mathrm{M}$. The observed convergence order + of about $1.5$ at the center of the star is lower then the general + second order of the hydrodynamics evolution scheme. This is + expected because the scheme's convergence rate drops to first order + at extrema or shocks, like the stellar center or surface. + Consequently, the observed convergence order about half-way between + the stellar center and surface is higher than $1.5$, but most of + the time below $2$.} \end{figure}
\subsection{Neutron star collapse\pages{2 Christian, Roland}} \label{sec:collapse_example} -The previous examples dealt either with preexisting BHs, either single or in a binary, or -with a smooth singularity free spacetime, as in the case of the TOV star. The evolution -codes in the toolkit however are -also able to handle the dynamic formation of a singularity, that is follow a neutron star -collapse -into a BH, and as a simple example of this process we study the collapse -of a non-rotating TOV star. We create initial data as in -section~\ref{sec:tov_oscillations} using $\rho_c=3.154,10^{-3}$ and $K_{\mathrm{ID}} = 100$, -$\Gamma = 2$ yielding a star model of gravitational mass $1.67,M_\odot$, that -is at the onset of instability. As is -common in these situations~\cite{Baiotti:2005vi} we trigger collapse by -reducing the pressure support after initial data has been constructed -by lowering the polytropic constant -$K_{\mathrm{ID}}$ from its initial data value to $K = 0.98 , K_{\mathrm{ID}} = 98$. To ensure -that the pressure depleted configuration remains a solution of the -constraints~\eref{eqn:analysis_hamiltonian_constraint} in the presence of -matter we rescale the rest mass density -$\rho$ such that the total energy density $T_{nn}$ +The previous examples dealt either with preexisting BHs, either single +or in a binary, or with a smooth singularity free spacetime, as in the +case of the TOV star. The evolution codes in the toolkit are, +however, also able to handle the dynamic formation of a singularity, +that is follow a neutron star collapse into a BH, and as a simple +example of this process we study the collapse of a non-rotating TOV +star. We create initial data as in section~\ref{sec:tov_oscillations} +using $\rho_c=3.154\times10^{-3}$ and $K_{\mathrm{ID}} = 100$, $\Gamma += 2$ yielding a star model of gravitational mass $1.67,M_\odot$, that +is at the onset of instability. As is common in such situations~(e.g., +\cite{Baiotti:2005vi}), we trigger collapse by reducing the pressure +support after the initial data have been constructed by lowering the +polytropic constant $K_{\mathrm{ID}}$ from its initial value to +$K = 0.98 , K_{\mathrm{ID}} = 98$. To ensure that the pressure +depleted configuration remains a solution of the Einstein constraint +equations (Eq.~\ref{eqn:analysis_hamiltonian_constraint}) in the presence +of matter we rescale the rest mass density $\rho$ such that the total +energy density $T_{nn}$ %\todo{RH: unify notation of $\rho$} %JF: Fixed, see above discussion does not change @@ -2773,20 +2774,20 @@ \rho' + K (\rho')^2 = \rho + K_{\mathrm{ID}} \rho^2. \label{eqn:collapse_rho_rescaled} \end{equation} -Compared to the initial configuration, this rescaled star possesses a higher -central density and lower pressure. -This change in $K$ speeds up the collapse and provides a -physical trigger for the collapse rather than relying on numerical noise -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 -$3.2,M_\odot$ ($2.4,M_\odot$, $1.6,M_\odot$, $0.6,M_\odot$ for higher -convergence levels). -Around the star, centered on the origin, we stack $5$ extra boxes of -approximate size $8\times2^\ell,M_\odot$ for $0 \le \ell \le 4$, where the -resolution on each finer +Compared to the initial configuration, this rescaled star possesses a +slightly higher central density and lower pressure. This change in +$K$ accelerates the onset of collapse that would otherwise rely on +being triggered by numerical noise, which would not be guaranteed to +converge to a unique solution with increasing 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 $3.2,M_\odot$ ($2.4,M_\odot$, $1.6,M_\odot$, +$0.6,M_\odot$ for higher convergence levels). Around the star, +centered about the origin, we stack $5$ extra boxes of approximate size +$8\times2^\ell,M_\odot$ for $0 \le \ell \le 4$, where the resolution +on each finer % RH: there is a likely typo in the paramter file which creates boxes of % radii: 2M,4M,8M,13.6M(!),32M,64M respectively. Changing it to 16M % doesn't really do any good or harm. @@ -2807,7 +2808,7 @@ $100$ times the atmosphere density. $R$ is the circumferential radius of the apparent horizon and $R_g = 2,M_\star = 2\times1.63,M_{\mathord\odot}$. An apparent horizon forms at a time roughly equal to when the mass of the star - is enclosed in its gravitational radius, forming black hole and causally + is enclosed in its gravitational radius, forming a black hole and causally disconnecting the evolution in the interior from the outside spacetime. The lower $x$-axis displays time in code units where $M_\odot=G=c=1$, and the upper $x$-axis shows the corresponding physical time using $1,M_\odot = 4.93,\mu s$.} @@ -2815,32 +2816,36 @@ \begin{figure} \label{fig:tov_collapse_H_convergence_at0} \includegraphics[width=0.9\textwidth]{examples/collapse/H_convergence_at0} - \caption{Convergence factor for the Hamiltonian constraint violation at the - center of the collapsing star. We plot convergence factors computed using - a set of 4 runs covering the diameter of the star with $\approx$ 60, 80, - 120, and 240 grid points. Units on the upper and lower $x$-axes are identical - to those of Figure~\ref{fig:tov_collapse_radii}. + \caption{Convergence factor for the Hamiltonian constraint violation + at the center of the collapsing star. We plot convergence factors + computed using a set of 4 runs covering the diameter of the star + with $\approx$ 60, 80, 120, and 240 grid points. The units of time + on the upper and lower $x$-axes are identical to those of + Figure~\ref{fig:tov_collapse_radii}. % initial star radius is: 6M, resolutions are 0.2,0.15,0.1,0.05 on the box % (radius 8M) covering the star } \end{figure} -In Figure~\ref{fig:tov_collapse_radii}, we plot the approximate coordinate -size of the star as well as the circumferential radius of the apparent horizon -that eventually forms in the simulation. -The apparent -horizon is first found at approximately the same time as when the star's size approaches -its Schwarzschild radius. In -Figure~\ref{fig:tov_collapse_H_convergence_at0} we display the convergence factor $Q$ -obtained from \todo{RH: Christian, I couldn't read your comment at this point, -so I made a guess on what it could mean [connect to equat.]. However Josh's -comment about convergence factor Eq. being required applies.} +In Figure~\ref{fig:tov_collapse_radii}, we plot the approximate +coordinate size of the star as well as the circumferential radius of +the apparent horizon that eventually forms in the simulation. The +apparent horizon is first found at approximately the time when the +star's coordinate radius approaches its Schwarzschild radius, though +one ought to keep in mind that the Schwarzschild radius is a +circumferential radius, whereas the meaning of the coordinate radius +in our BSSN calculation is closer to a radius in isotropic gauge +\todo{Roland, do you agree?}. + + In Figure~\ref{fig:tov_collapse_H_convergence_at0}, we display the + convergence factor obtained from \begin{equation} - \frac{H_{h_1}-H_{h_2}}{H_{h_2}-H_{h_3}} = \frac{h_1^Q-h_2^Q}{h_2^Q-h_3^Q} + Q = \frac{H_{h_1}-H_{h_2}}{H_{h_2}-H_{h_3}} = \frac{h_1^Q-h_2^Q}{h_2^Q-h_3^Q},, \label{eq:convergence-factor-definition} \end{equation} +where ... \todo{Roland, define H, h and subscripts} for the Hamiltonian constraint violation at the center of the collapsing star. -Up to the time when the apparent horizon forms the convergence order is -$\approx 1.5$ as expected. After that the singularity forming at the +Up to the time when the apparent horizon forms, the convergence order is +$\approx 1.5$ as expected. At later times, the singularity forming at the center of the collapsing star renders a pointwise measurement of the convergence factor at the center impossible.