User: rhaas
Date: 2011/10/27 08:52 PM
Modified:
/
ET.tex
/examples/collapse/
H_convergence_at0.pdf, H_convergence_at0.py
Log:
work on text for tov collapse section
update tov collpase plot with nicer tick marks
remove some todos of Roland's
point out conflict in usage of \\rho
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +60 -33
===================================================================
--- ET.tex 2011-10-24 19:14:38 UTC (rev 156)
+++ ET.tex 2011-10-28 01:52:58 UTC (rev 157)
@@ -1352,6 +1352,10 @@
\subsection{Hydrodynamics Evolution}
\label{sec:GRHydro}
+\todo{RH: there is a conflict of notation between the rest mass density $\rho$
+as used by GRHydro and the $T_{nn}$ component used by the spacetime codes
+(McLachlan, TOVSolver, TwoPunctures if we described it). Ways around it would
+be $\rho_0$ and $\rho_{ADM}$ for examples.}
Hydrodynamic evolution in the Einstein Toolkit is designed so that it
interacts with the metric curvature evolution through a small set of
variables, allowing for maximum modularity in implementing, editing,
@@ -2011,7 +2015,7 @@
base modules described in Sec.~\ref{sec:base_modules}, explicitly written
as:
\begin{eqnarray}
- H &=& R - K^i{}_j K^j{}_i + K^2 - 16 \pi \rho \\
+ H &=& R - K^i{}_j K^j{}_i + K^2 - 16 \pi \rho \label{eqn:analysis_hamiltonian_constraint}\\
M_i &=& \nabla_j K_i{}^j - \nabla_i K - 8 \pi S_i
\end{eqnarray}
where $S_i=-\frac{1}{\alpha} \left( T_{i0} - \beta^j T_{ij} \right)$.
@@ -2121,7 +2125,7 @@
\end{center}
\caption{Grid layout of a simulation using \codename{Cartoon2D}. The
$z$-axis is the axis of rotational symmetry. Image courtesy of
- Denis Pollney.\todo{RH: ask Denis if we can use his figure}}
+ Denis Pollney.}
\label{fig:cartoon-plane}
\end{figure}
@@ -2141,7 +2145,7 @@
point $x''$ for which there is actual data stored. In this
example, two reflection symmetries along the horizontal and vertical axis
are present. Notice how the vector components change in
- transformations $A$ and $B$. Image courtesy of Steve White.}
+ transformations $A$ and $B$.} % Image courtesy of Steve White.}
\label{fig:faces}
\end{figure}
@@ -2687,33 +2691,49 @@
$2$.}
\end{figure}
-\subsection{Collapse\pages{2 Christian}}
-Show TOV collapse and BH formation
-\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 resolution run to show what happens.}
+\subsection{Collapse\pages{2 Christian, Roland}}
The previous examples dealt either with a preexisting black hole (BBH) or
-with a smooth singularity free spacetime (TOV oscillations). The ET however is
+with a smooth singularity free spacetime (TOV oscillations). The evolution
+codes in the toolkit however are
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-rotating TOV solution into a black hole. We create initial data as in
+of a non-rotating TOV star 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 mass $1.67\,M_\odot$. 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
+$\Gamma = 2$ yielding a star of mass $1.67\,M_\odot$. 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
+$K_{ID}$ from its initial data value to $K = 0.98 \, K_{ID} = 98$. To ensure
+that the pressure depleted configuration remains a solution of the
+constraints~\cite{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
+$\rho$} does not change
+\begin{equation}
+ \rho' + K (\rho')^2 = \rho + K_{ID} \rho^2.
+ \label{eqn:collapse_rho_rescaled}
+\end{equation}
+Compared to the initial configuration, this rescaled star posseses a higher
+central density and lower pressure.
+This change in $K$ speeds up the collapse and provides a
+physical trigger for the collapse rather relying on 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 convergence levels).
+$3.2\,M_\odot$ ($2.4\,M_\odot$, $1.6\,M_\odot$, $0.6\,M_\odot$ for higher
+convergence 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
+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.
+level is twice that of the surrounding level. In order to resolve the large
+density gradients developping during the collapse, two more levels with radii
+$4\,M_\odot$ and $2\,M_\odot$ are present inside of the star. 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
@@ -2722,30 +2742,37 @@
\begin{figure}
\label{fig:tov_collapse_radii}
\includegraphics[width=0.9\textwidth]{examples/collapse/radii}
- \caption{Location on x axis where $\rho = 100\,\mathtt{whisky\_rho\_min}$ and
- the areal radius of the apparent horizon ($R_g = 2\,M_\star =
- 2\,1.63\,M_{\mathord\odot}$).}
+ \caption{Stellar radius of the collapsing star and radius of the forming
+ apparent horizon. The stellar surface is defined as the point where $\rho$ is
+ $100$ time 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
+ disconnecting the evolution in the interior from the outside spacetime.}
\end{figure}
\begin{figure}
\label{fig:tov_collapse_H_convergence_at0.pdf}
\includegraphics[width=0.9\textwidth]{examples/collapse/H_convergence_at0}
\caption{Convergence factor for the Hamiltonian constraint evaluated at the
- center of the collapsing star.}
+ center of the collapsing star. We plot convergence factors computed using
+ a set of 5 runs covering the diameter of the star with $\approx 60, 80,
+ 120, 240$ grid points.
+ % 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 on the same graph the
-approximate location of the edge of star and the areal radius of the apparent
+approximate location of the star surface and the circumferential radius of the
+apparent
horizon once a horizon is found in the simulation. Clearly the apparent
horizon is found at approximately the same time as the star's size approaches
-its Schwarzschild radius.\todo{RH:replace AH radius with coordinate radius} In
+its Schwarzschild radius. In
Figure~\ref{fig:tov_collapse_H_convergence_at0} we display the convergence factor
for the Hamiltonian constraint at the center of the collapsing star.
-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 resources. For this reason the maximum of the
-Hamiltonian constraint rises rapidly just before the the interior of the star
-is encapsulated in the horizon.
+Up to the time when the apparent horizon forms the convergence order is
+$\approx 1.5$ as expected. After that the singularity which forms at the
+center of the black hole prevents a clear measurement of the convergence
+factor.
\subsection{Cosmology\pagesdone{1}}
Directory: /examples/collapse/
==============================
File [modified]: H_convergence_at0.pdf
Delta lines: +0 -0
===================================================================
(Binary files differ)
File [modified]: H_convergence_at0.py
Delta lines: +2 -2
===================================================================
--- examples/collapse/H_convergence_at0.py 2011-10-24 19:14:38 UTC (rev 156)
+++ examples/collapse/H_convergence_at0.py 2011-10-28 01:52:58 UTC (rev 157)
@@ -27,13 +27,13 @@
ax.set_xlabel(r't [M]')
ax.xaxis.set_major_locator(mticker.MaxNLocator(7))
-ax.xaxis.set_minor_locator(mticker.MaxNLocator(14))
+ax.xaxis.set_minor_locator(mticker.AutoMinorLocator(2))
ax.xaxis.grid(False)
ax2 = ax.twiny()
ax2.set_xlabel(r't [ms]')
ax2.set_xlim((ax.get_xlim()[0]/M_to_ms, ax.get_xlim()[1]/M_to_ms))
ax2.xaxis.set_major_locator(mticker.MaxNLocator(7))
-ax2.xaxis.set_minor_locator(mticker.MaxNLocator(14))
+ax2.xaxis.set_minor_locator(mticker.AutoMinorLocator(2))
ax.set_ylabel(r'$C(H)$')
ax.yaxis.set_major_locator(mticker.MaxNLocator(5))
ax.yaxis.set_minor_locator(mticker.MaxNLocator(10))
