[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 182)
cott at tapir.caltech.edu
cott at tapir.caltech.edu
Fri Nov 11 09:00:09 CST 2011
User: cott
Date: 2011/11/11 09:00 AM
Modified:
/
ET.tex
Log:
* some changes in TOV collapse section. ROLAND: Please check!
File Changes:
Directory: /
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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.
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