[Commits] [svn:einsteintoolkit] Paper_EinsteinToolkit_2010/ (Rev. 132)

[knarf at cct.lsu.edu]

User: knarf
Date: 2011/09/13 12:04 AM

Modified:
 /
  ET.tex
 /examples/tov/
  mode_spectrum.pdf, mode_spectrum.py

Log:
 add tov text, improve psd figure

File Changes:

Directory: /
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File [modified]: ET.tex
Delta lines: +34 -25
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--- ET.tex	2011-09-13 04:29:02 UTC (rev 131)
+++ ET.tex	2011-09-13 05:04:10 UTC (rev 132)
@@ -2554,10 +2554,11 @@
 \end{figure}
 
 \subsection{Linear oscillations of TOV stars\pages{2 Frank}}
-The previous examples did not include the evolution of matter within a relativistic
-spacetime. One interesting test of a coupled matter-spacetime evolution is
-to measure the eigenfrequencies of a stable TOV star. These eigenfrequencies
-can be compared to values known from linear perturbation theory.
+The examples in the previous subsections did not include the evolution of
+matter within a relativistic spacetime. One interesting test of a coupled
+matter-spacetime evolution is to measure the eigenfrequencies of a stable TOV
+star. These eigenfrequencies can be compared to values known from linear
+perturbation theory.
 
 This test uses an equillibrium configuration of a self-gravitating fluid
 sphere, described by a polytropic equation of state. This one-dimensional
@@ -2570,11 +2571,10 @@
 Despite the system being in equilibrium, the solution experiences oscillations
 mainly due to perturbations from two sources, the interpolation of the initial
 data onto the evolution grid, and due to the interaction of the star and the
-atmosphere. The former is damped out with time, and can be reduced by
-increasing both the resolution of the one-dimensional solution and of the
-evolution grid. For resolutions of tests presented here, this perturbation was
-small compared to perturbations from the remaining source, the interaction with
-the atmosphere. These perturbations are present during the whole evolution,
+atmosphere. The former is damped quickly, and can be reduced by increasing the
+resolution of the evolution grid. It can be seen as initial spike in the
+evolution of the central density of the star. The remaining perturbations,
+mainly from the atmosphere treatment, are present during the whole evolution,
 continuously exciting oscillations. Given enough evolution time, the
 frequencies of these oscillations can be measured with satisfactory accuracy.
 
@@ -2583,39 +2583,48 @@
 and an initial central density of $\varrho_0=1.28\times10^{-3}$. This model can
 be taken to represent a non-rotating neutron star with a mass of
 $M=1.4\mathrm{M}_\odot$.  The computational domain is a cube of length
-$640\mathrm{M}$ and a resolution of $2\mathrm{M}$ in each dimension. Four
-additional grids refine the region around the star at the origin, each doubling
-the resolution, with sizes of $120\mathrm{M}$, $60\mathrm{M}$, $30\mathrm{M}$
-and $15\mathrm{M}$, resulting in a resolution of $0.125\mathrm{M}$ across the
-entire star.
+$640\mathrm{M}$ and a base resolution of $2\mathrm{M}$ ($4\mathrm{M}$,
+$8\mathrm{M}$) in each dimension. Four additional grids refine the region
+around the star located at the origin, each doubling the resolution, with sizes
+of $120\mathrm{M}$, $60\mathrm{M}$, $30\mathrm{M}$ and $15\mathrm{M}$,
+resulting in a resolution of $0.125\mathrm{M}$ ($0.25\mathrm{M}$,
+$0.5\mathrm{M}$) across the entire star.
 
 In figure~\ref{fig:tov_rho_max} we show the evolution of the central density of
-the star, over an evolution time of~\todo{X}.
+the star, over an evolution time of $1300\mathrm{M}$. The initial spike is due
+to the perturbation of the solution resulting from the interpolation onto the
+evolution grid. These oscillations are quickly damped. Most of the remaining
+oscillations are due to perturbations resulting from the treatment of the
+atmosphere. The mean of the central density shows a drift, which converges
+to zero with increasing resolution.\todo{show}
 
 \begin{figure}
  \label{fig:tov_rho_max}
- \includegraphics[width=0.8\textwidth]{examples/tov/rho_max}
+ \includegraphics[width=0.95\textwidth]{examples/tov/rho_max}
  \caption{Evolution of the central density for the TOV star. Clearly visible is
  an initial spike, produced by the interpolation of the one-dimensional equillibirum
  solution onto the three-dimensional evolution grid. The remainder of the evolution
  however, the central density evolution is dominated by continuous excitations coming
- from the interaction of the stellar surface with the artificial atmosphere.}
+ from the interaction of the stellar surface with the artificial atmosphere. Shown
+ are three different resolutions.}
 \end{figure}
 
-In figure~\ref{fig:tov_mode_spectrum} we show the power spectrum of the central
+In figure~\ref{fig:tov_mode_spectrum} we show the power spectral density (PSD) of the central
 density oscillations computed from a full 3D relativistic hydrodynamics
 simulation, compared to the coresponding frequencies as obtained with
 perturbative techniques, kindly provided to use by Kentaro Takami and computed
-using the method described in~\cite{Yoshida01}. Clearly the 3D simulation can
-correctly represent the fundamental mode frequency and the first three
-overtones of the neutron star.
+using the method described in~\cite{Yoshida01}. The PSD was computed using the entire
+time series, removing the linear trend, averaging over Hanning windows overlapping half the
+signal length after padding the signal to five time it's length.
+The agreement of the frequencies of the fundamental mode and the first three overtones
+is clearly visible.
 
 \begin{figure}
  \label{fig:tov_mode_spectrum}
- \includegraphics[width=0.8\textwidth]{examples/tov/mode_spectrum}
- \caption{Eigenfrequency mode spectrum of TOV star. Shown is the spectrum
-   computed from a full 3D relativistic hydrodynamics simulation compared
-   to the values obtained by perturbation theory.}
+ \includegraphics[width=0.95\textwidth]{examples/tov/mode_spectrum}
+ \caption{Eigenfrequency mode spectrum of a TOV star. Shown is the power spectral density
+   of the central density, computed from a full 3D relativistic hydrodynamics simulation compared
+   to the values obtained by perturbation theory.\todo{improve caption}}
 \end{figure}
 
 

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File [modified]: mode_spectrum.pdf
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File [modified]: mode_spectrum.py
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--- examples/tov/mode_spectrum.py	2011-09-13 04:29:02 UTC (rev 131)
+++ examples/tov/mode_spectrum.py	2011-09-13 05:04:10 UTC (rev 132)
@@ -54,8 +54,8 @@
 Fy = Fy[int(start*len(Fy)): int(end*len(Fy))]
 
 # Plot basics
-fig = plt.figure()
-fig.subplots_adjust(hspace=0.45, wspace=0.3, left=0.15, bottom=0.15)
+fig = plt.figure(figsize=(15, 5))
+fig.subplots_adjust(top=0.9,bottom=0.18, left=0.08,right=0.98)
 ax = fig.add_subplot(1,1,1)
 
 # mode names and frequencies