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Function: qn_qring()

int qn_qring(float psi0, float eps, float M, float a,
             float dt, float atten, int max, float **strainPtr)
The routine qn_qring() is a quick ringdown generator which constructs a damped sinusoid with a frequency and quality approximately equal to that of the $\ell=m=2$ quasinormal mode of a Kerr black hole and an amplitude approximately equal to angle-averaged strain expected for black hole radiation at a distance $GM_\odot/c^2=T_\odot c\simeq1.4766\,{\mathrm{km}}$. To obtain the waveforms at a distance $r$, multiply the result by  $GM_\odot/c^2r=T_\odot c/r$. The arguments to the routine are:
psi0: Input. The initial phase (in radians) of the waveform (see below).
eps: Input. The fractional mass loss in quadrupolar ($\ell=m=2$) radiation. ( $0<\texttt{eps}\ll1$.)
M: Input. The mass of the black hole in solar masses.
a: Input. The dimensionless angular momentum parameter of the Kerr black hole, $\vert\hat{a}\vert\le1$, which is negative if the black hole is spinning clockwise about the $\iota=0$ axis (see figure [*]).
dt: Input. The time interval, in seconds, between successive data points in the returned waveform.
atten: Input: The attenuation level, in dB, at which the routine will terminate calculation of the waveforms.
max: Input. The maximum number of data points to be returned in the waveforms.
strainPtr: Input/Output. A pointer to an array which, on return, contains the angle-averaged waveform sampled at intervals dt. If the array has the value NULL on input, the routine allocates an amount of memory to *strainPtr to hold max elements.

The routine qn_ring() returns the number of data points that were written to the array (*strainPtr)[]; this is either the number specified by the input parameter max or the number of points computed when the waveform was attenuated by the threshold atten. The array contains the angle averaged waveform

\begin{displaymath}
H_{\mathrm{\scriptstyle ave}}(t_{\mathrm{\scriptstyle ret}}...
...mathcal{H}}(t_{\mathrm{\scriptstyle ret}}) e^{i\psi_0} \bigr],
\end{displaymath} (8.9.178)

where ${\mathcal{H}}(t_{\mathrm{\scriptstyle ret}})$ is given by equation ([*]), sampled at time intervals $dt$. The constant $\psi_0$ defines the initial phase of the waveform. The amplitude factor is set by the following argument: The gravitational strain (at a distance $GM_\odot/c^2=T_\odot c\simeq1.4766\,{\mathrm{km}}$) that would be observed by an interferometer is given by  $H(t_{\mathrm{\scriptstyle ret}})=F_+(\theta,\phi,\psi)
H_+(t_{\mathrm{\scripts...
...F_\times(\theta,\phi,\psi)
H_\times(t_{\mathrm{\scriptstyle ret}},\iota,\beta)$ where $F_+$ and $F_\times$ represent the antenna patterns of the interferometer. When averaged over $\theta$, $\phi$, and $\psi$, one finds $\langle F_+^2\rangle=\langle F_\times^2\rangle=\frac{1}{5}$ and  $\langle F_+F_\times\rangle=0$. Thus,
$\displaystyle \langle H^2(t_{\mathrm{\scriptstyle ret}})
\rangle_{\theta,\phi,\psi,\iota,\beta}$ $\textstyle =$ $\displaystyle {\textstyle\frac{1}{5}} \langle
H_+^2(t_{\mathrm{\scriptstyle ret...
...)
+ H_\times^2(t_{\mathrm{\scriptstyle ret}},\iota,\beta)
\rangle_{\iota,\beta}$  
  $\textstyle =$ $\displaystyle {\textstyle\frac{1}{5}} \langle
\vert (H_+ - iH_\times)(t_{\mathrm{\scriptstyle ret}},\iota,\beta) \vert^2
\rangle_{\iota,\beta}$  
  $\textstyle =$ $\displaystyle {\textstyle\frac{1}{10}} \vert{\mathcal{H}}(t_{\mathrm{\scriptstyle ret}})\vert^2$  
  $\textstyle \approx$ $\displaystyle \overline{H_{\mathrm{\scriptstyle ave}}^2}$ (8.9.179)

where the overbar indicates a time average over a single cycle; approximate equality becomes exact in the limit of a high quality ringdown. It is in this sense that the quantity  $H_{\mathrm{\scriptstyle ave}}(t_{\mathrm{\scriptstyle ret}})$ can be viewed as an angle-averaged waveform.

Rather than compute the eigenfrequency using the routine qn_eigenvalues(), this routine uses the analytic fits to the eigenfrequency found by Echeverria [25]. These expressions are:

\begin{displaymath}
\hat{\omega} \simeq f(\hat{a}) \bigl( 1 - {\textstyle\frac{1}{4}}ig(\hat{a})
\bigr)
\end{displaymath} (8.9.180)

with
$\displaystyle f(\hat{a})$ $\textstyle =$ $\displaystyle 1 - 0.63(1-\hat{a})^{3/10}$ (8.9.181)
$\displaystyle g(\hat{a})$ $\textstyle =$ $\displaystyle (1-\hat{a})^{9/20}.$ (8.9.182)

Author: Jolien Creighton, jolien@tapir.caltech.edu
Comments: Since this routine does not need to compute the spheroidal wave function and uses an analytic approximation to the eigenfrequency, it is much simpler than the routine qn_ring(). The approximate eigenfrequencies are typically accurate to within $\sim5\%$, so this routine is to be preferred when computing quadrupolar $(\ell=m=2)$ quasinormal waveforms unless accuracy is critical.


next up previous contents
Next: Function: qn_filter() Up: GRASP Routines: Black hole Previous: Example: ringdown program   Contents
Bruce Allen 2000-11-19