User: rhaas
Date: 2011/11/10 05:38 PM
Modified:
/
ET.tex
/examples/collapse/
H_convergence_at0.pdf, H_convergence_at0.py, radii.pdf, radii.py
Log:
change ref{eq:...} to eref to get paranthesis
cleanup in TOV collapse section, add labels and title to TOV plots.
File Changes:
Directory: /
============
File [modified]: ET.tex
Delta lines: +52 -37
===================================================================
--- ET.tex 2011-11-10 20:01:36 UTC (rev 180)
+++ ET.tex 2011-11-10 23:38:02 UTC (rev 181)
@@ -1271,7 +1271,7 @@
evolution of the shift
$\beta^i$ and thus that of the spatial coordinates $x^i$ will be exponentially
damped. This damping time scale is set by the gauge parameter $\eta$
-(see \ref{eq:eta}) which has dimension $1/T$ (inverse time).
+(see \eref{eq:eta}) which has dimension $1/T$ (inverse time).
As described, e.g., in~\cite{Muller:2009jx, Schnetter:2010cz}, this
time scale may need to be adapted in different regions of the domain
to avoid spurious high-frequency behavior in regions that otherwise
@@ -1291,7 +1291,7 @@
\end{eqnarray}
with a constant $R$ defining the transition radius between the
interior, where $q\approx1$, and the exterior, where $q$ falls off as
-$1/r$. \ref{eq:eta} describes how $q$ appears in the gauge
+$1/r$. \eref{eq:eta} describes how $q$ appears in the gauge
parameters.
@@ -1601,7 +1601,7 @@
It is these flux terms that are then used to evolve the hydrodynamic quantities.
The Roe solver~\cite{Roe:1981ar} involves linearizing the evolution system
-for the hydrodynamic evolution. \ref{eq:Riemann}, defining
+for the hydrodynamic evolution. \eref{eq:Riemann}, defining
the Jacobian matrix $A\equiv \frac{\partial f}{\partial q}$, and
working out the eigenvalues $\lambda^i$ and left and right eigenvectors,
$l_i$ and $r^j$, assumed to be orthonormalized so that
@@ -1626,7 +1626,7 @@
\subsubsection{Conservative to primitive conversion}
-In order to invert ~\ref{eq:p2c1}~--~\ref{eq:p2c3}, solving for
+In order to invert ~\eref{eq:p2c1}~--~\eref{eq:p2c3}, solving for
the primitive variables based on the values of the conservative ones,
\codename{GRHydro} uses a 1-dimensional Newton-Raphson approach that
solves for a consistent value of the pressure. Defining the (known)
@@ -1689,7 +1689,10 @@
\begin{equation}
P = K\rho^\Gamma\,\,,
\end{equation}
-where $K$ is the polytropic constant and $\Gamma$ is the adiabatic
+where $K$ is the polytropic constant
+% RH: poly.const. is correct term according to arXiv:gr-qc/0211028 below Eq.
+% (9.4)
+and $\Gamma$ is the adiabatic
index, is appropriate for adiabatic (= isentropic) evolution without
shocks. When using the polytropic EOS, one does not need to evolve the
total fluid energy equation, since the specific internal energy
@@ -1842,7 +1845,7 @@
which \codename{EHFinder} is also based.
Defining a surface as a level set $f(x^i)=r-h(\theta,\phi)=0$,
and introducing an unphysical timelike parameter $\lambda$ to
-parametrize the flow of $h$ towards a solution, \ref{eq:ah_theta}
+parametrize the flow of $h$ towards a solution, \eref{eq:ah_theta}
can be rewritten
\begin{equation}
\partial_\lambda h = - \left( \frac{\alpha}{\ell_\mathrm{max}
@@ -1865,8 +1868,8 @@
The \codename{AHFinderDirect} module~\cite{Thornburg:2003sf} is a
faster alternative to \codename{AHFinder}. Its approach is to view
-\ref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$
-using standard finite differencing methods. Rewriting \ref{eq:ah_theta}
+\eref{eq:ah_theta} as an elliptic PDE for $h(\theta,\phi)$ on $S^2$
+using standard finite differencing methods. Rewriting \eref{eq:ah_theta}
in the form
\begin{equation}
\Theta \equiv \Theta\left(h,\partial_u h,\partial_{uv}h;
@@ -2063,7 +2066,7 @@
where $x^i$ is the puncture location and $\beta^i$ is the shift. Since the
puncture location usually does not coincide with gridpoints, the shift is
interpolated to the location of the puncture.
-Equation~(\ref{eq:puncturetracking}) is implemented with a simple first order
+Equation~(\eref{eq:puncturetracking}) is implemented with a simple first order
Euler scheme, which seems to be accurate enough for controlling the location
of the mesh refinement hierarchy.
%\todo{ES: I think this paragraph provides too much detail for this paper.}
@@ -2455,7 +2458,7 @@
Table~\ref{table:BHB_ID} provides more details about
the initial binary parameters used to generate the initial data.
The \codename{TwoPunctures} module uses these initial parameters
-to solve \ref{eq:twopunc_u}, the elliptic Hamiltonian constraint for
+to solve \eref{eq:twopunc_u}, the elliptic Hamiltonian constraint for
the regular component of the conformal factor (see section~\ref{sec:twopunctures}).
The spectral solution for this example was
determined by using $[n_A,n_B,n_{\phi}]=[28,28,14]$ collocation
@@ -2743,36 +2746,37 @@
$2$.}
\end{figure}
-\subsection{Collapse\pages{2 Christian, Roland}}
+\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 as a star collapses
+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 $\rho_c=3.154e-3$ and $K_{ID} = 100$,
-$\Gamma = 2$ yielding a star of mass $1.67\,M_\odot$. As is
+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
-$K_{ID}$ from its initial data value to $K = 0.98 \, K_{ID} = 98$. To ensure
+$K_{\mathrm{ID}}$ 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~\ref{eqn:analysis_hamiltonian_constraint} in the presence of
+constraints~\eref{eqn:analysis_hamiltonian_constraint} in the presence of
matter we rescale the rest mass density
$\rho$ such that the total energy density $T_{nn}$
%\todo{RH: unify notation of $\rho$}
%JF: Fixed, see above discussion
does not change
\begin{equation}
- \rho' + K (\rho')^2 = \rho + K_{ID} \rho^2.
+ \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 relying on random numerical
-fluctuations
+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
@@ -2788,46 +2792,57 @@
% doesn't really do any good or harm.
level is twice that of the surrounding level. In order to resolve the large
density gradients developing 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
+$4\,M_\odot$ and $2\,M_\odot$ are present inside 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
-edge of the star we expect the observed convergence rate to be somewhat lower
+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_radii}
\includegraphics[width=0.9\textwidth]{examples/collapse/radii}
- \caption{Stellar radius of the collapsing star and radius of the forming
+ \caption{Coordinate radius of the surface of the collapsing star and radius
+ of the forming
apparent horizon. The stellar surface is defined as the point where $\rho$ is
$100$ 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
- disconnecting the evolution in the interior from the outside spacetime.}
+ 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 $1\,M_\odot = 4.93\,\mu s$.}
\end{figure}
\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 evaluated at the
+ \caption{Convergence factor for the Hamiltonian constraint violation at the
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.
+ 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}.
% 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 star surface and the circumferential radius of the
-apparent
-horizon once a horizon is found in the simulation. Clearly the apparent
+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
-for the Hamiltonian constraint at the center of the collapsing star.
+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.}
+\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}
+ \label{eq:convergence-factor-definition}
+\end{equation}
+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 which forms at the
-center of the black hole prevents a clear measurement of the convergence
-factor.
+$\approx 1.5$ as expected. After that the singularity forming at the
+center of the collapsing star renders a pointwise measurement of the
+convergence factor at the center impossible.
\subsection{Cosmology}\label{sec:cosmology}
@@ -2864,13 +2879,13 @@
\end{widetext}
In Figure~\ref{fig:kasner}, we show the full evolution of the $t=1$ slice
-of spacetime~\ref{eq:kasner}, along with the associated error for a sequence of
+of spacetime~\eref{eq:kasner}, along with the associated error for a sequence of
time resolutions.
\begin{figure}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/cosmology/figs/kasner.pdf}
\includegraphics[width=0.33\textwidth,angle=-90]{examples/cosmology/figs/err.pdf}
- \caption{Left: the evolution of a vacuum spacetime of the type~\ref{eq:gowdyT3},
+ \caption{Left: the evolution of a vacuum spacetime of the type~\eref{eq:gowdyT3},
with $P=Q=\lambda=0$; the initial data are chosen as
$\gamma_{ij}=\delta_{ij}$ and $K_{ij}={\rm diag}(-2/3,-2/3,1/3)$.
Right: the numerical error for a sequence of four time resolutions $dt=[0.0125,0.025,0.05,0.1]$;
Directory: /examples/collapse/
==============================
File [modified]: H_convergence_at0.pdf
Delta lines: +0 -0
===================================================================
(Binary files differ)
File [modified]: H_convergence_at0.py
Delta lines: +5 -3
===================================================================
--- examples/collapse/H_convergence_at0.py 2011-11-10 20:01:36 UTC (rev 180)
+++ examples/collapse/H_convergence_at0.py 2011-11-10 23:38:02 UTC (rev 181)
@@ -16,14 +16,16 @@
ax = fig.add_subplot(1,1,1)
# plot
-ax.plot(t, ord1, linestyle='-', color='black', label='$C(H_{3.2},H_{2.4},H_{1.6})$')
-ax.plot(t, ord2, linestyle='--', color='blue', label='$C(H_{3.2},H_{1.6},H_{0.8})$')
+ax.plot(t, ord1, linestyle='-', color='black', label='$Q(H_{3.2},H_{2.4},H_{1.6})$')
+ax.plot(t, ord2, linestyle='--', color='blue', label='$Q(H_{3.2},H_{1.6},H_{0.8})$')
# plot properties
#ax.set_xlim(xlim)
#ax.set_ylim(ylim)
#plt.title('PSD', fontsize=fontsize)
+plt.text(30, 2.65, 'NS collapse',ha='center')
+plt.text(30, 2.4, 'convergence factor of the\nHamiltonian constraint violation',ha='center',va='top')
ax.set_xlabel(r't [M]')
ax.xaxis.set_major_locator(mticker.MaxNLocator(7))
@@ -34,7 +36,7 @@
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.AutoMinorLocator(2))
-ax.set_ylabel(r'$C(H)$')
+ax.set_ylabel(r'$Q(H)$')
ax.yaxis.set_major_locator(mticker.MaxNLocator(5))
ax.yaxis.set_minor_locator(mticker.MaxNLocator(10))
ax.yaxis.grid(False)
File [modified]: radii.pdf
Delta lines: +0 -0
===================================================================
(Binary files differ)
File [modified]: radii.py
Delta lines: +3 -2
===================================================================
--- examples/collapse/radii.py 2011-11-10 20:01:36 UTC (rev 180)
+++ examples/collapse/radii.py 2011-11-10 23:38:02 UTC (rev 181)
@@ -17,7 +17,7 @@
ax = fig.add_subplot(1,1,1)
# plot
-ax.plot(Fx, Fy/(2*1.63), linestyle='-', color='black', label='outer star edge')
+ax.plot(Fx, Fy/(2*1.63), linestyle='-', color='black', label='radius of stellar surface')
ax.plot(Bx, By/(2*1.63), linestyle='--', color='blue', label='apparent horizon radius')
# plot properties
@@ -25,8 +25,9 @@
#ax.set_ylim(ylim)
#plt.title('PSD', fontsize=fontsize)
+plt.text(30, 1.4, r'NS collapse',ha='center')
-ax.set_xlabel(r't [M]')
+ax.set_xlabel(r't [$M_\odot$]')
ax.xaxis.set_major_locator(mticker.MaxNLocator(7))
ax.xaxis.set_minor_locator(mticker.MaxNLocator(14))
ax.xaxis.grid(False)
