Function: qn_ring()

int qn_ring(float iota, float beta,
            float eps, float M, float a, int l, int m,
            float dt, float atten, int max,
            float **plusPtr, float **crossPtr)

This routine is used to compute the ``$+$'' and ``$\times$'' polarizations of the gravitational waveform, $H(t_{\mathrm{\scriptstyle ret}})$, produced by a black hole ringdown 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 are:

iota: Input. The polar angle (inclination), $\iota$ (in radians), of the sky position of the observer with respect to the (positive) spin axis of the black hole, $0\le\texttt{iota}\le\pi$.

beta: Input. The azimuth, $\beta$ (in radians), of the sky position of the observer with respect to the axis of the perturbation at the start time. ( $0\le\texttt{beta}\le2\pi$.)

eps: Input. The fraction of the total mass lost in gravitational radiation from the particular mode. ( $0<\texttt{eps}\ll1$.)

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 ).

l: Input. The mode integer $\ell$. ( $\texttt{l}\ge2$)

m: Input. The mode integer $m$. ( $\vert\texttt{m}\vert\le\texttt{l}$)

dt: Input. The time interval, in seconds, between successive data points in the returned waveforms.

atten: Input. The attenuation level, in dB, at which the routine will terminate calculation of the waveforms. I.e., the routine will terminate when the amplitude, $A=A_0\exp(-{\mathrm{Im}}\,\omega t_{\mathrm{\scriptstyle ret}})$, falls below the level  $A_{\mathrm{\scriptstyle cutoff}}=A_0\,{\mathrm{alog}}_{10}(-0.1\times \texttt{atten})$.

max: Input. The maximum number of data points to be returned in the waveforms.

plusPtr: Input/Output. A pointer to an array which, on return, contains the waveform $H_+$ sampled at intervals dt. If the array has the value NULL on input, the routine allocates an amount of memory to *plusPtr to hold max elements.

crossPtr: Input/Output. A pointer to an array which, on return, contains the waveform $H_\times$ sampled at intervals dt. If the array has the value NULL on input, the routine allocates an amount of memory to *crossPtr to hold max elements.

The routine qn_ring() returns the number of data points that were written to the arrays (*plusPtr)[] and (*crossPtr)[]; 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 eigenvalues are obtained from the function qn_eigenvalues(). The waveform is then computed using $H_+-iH_\times={\mathcal{H}}(t_{\mathrm{\scriptstyle ret}}) {}_{-2}S_{\ell m}(\mu)e^{im\beta}$ with  ${\mathcal{H}}(t_{\mathrm{\scriptstyle ret}})$ given by equation (). The spheroidal wave function is obtained from the function sw_spheroid().

Author: Jolien Creighton, jolien@tapir.caltech.edu