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Detailed explanation of chirp_filters() routine

0 The routine chirp_filters() calls phase_frequency() to find out the how the orbital phase and frequency evolve in accordance with the input parameters. It then makes a single pass through that phase and frequency ephemeris, computing the chirps as it goes, and storing the information in the space already allocated for the phase and frequency. Most of the fault checking and computations are done in the phase_frequency() routine, and all the errors messages and warnings come from there.

The routine chirp_filters() computes

\begin{displaymath}
h_c(t) = 2 \biggl ({\mu \over M_\odot} \biggr )
\biggl [ { 2...
...{\rm tot} f(t) \over M_\odot } \biggr ]^{2/3}
\cos 2 \phi (t)
\end{displaymath} (6.6.36)

and the other orbital-phase chirp which is $\pi/2$ out of phase with $h_c (t)$
\begin{displaymath}
h_s(t) = 2 \biggr ( {\mu \over M_\odot} \biggr )
\biggl [ { ...
...tot} f(t) \over M_\odot } \biggr ]
^{2/3} \sin 2 \phi (t) \; ,
\end{displaymath} (6.6.37)

with all the leading numerical factors we display.

If the so called ``restricted'' post$^2$-Newtonian polarizations [leading order in the amplitude, but post$^2$-Newtonian phase corrections] are desired, they can be easily assembled from $h_c$ and $h_s$. The ``$+$'' (plus) polarization is given by

\begin{displaymath}
h_{+}(t) = - { T_\odot c \over D} (1+\cos^2 i ) h_c(t) \; ,
\end{displaymath} (6.6.38)

and the ``$\times$" (cross) polarization is given by
\begin{displaymath}
h_\times (t) = -2 { T_\odot c \over D} ( \cos i ) \; h_s(t) \; .
\end{displaymath} (6.6.39)

Here $D$ is the (luminosity) distance to the source in centimeters, c is the speed of light in centimeters/second, and $i$ is the inclination angle (radians) of the of the angular momentum axis of the source relative to the line-of-sight. See Will and Wiseman [8] figure 7 for the precise definition of the inclination angle.

The restricted post$^2$-Newtonian strain amplitude impinging on the detector can also be calculated from the output of chirp_filters() by

\begin{displaymath}
h(t) = F_+ h_+(t) + F_\times h_{\times}(t) \; ,
\end{displaymath} (6.6.40)

where $F_+$ and $F_\times$ are the detector beam-pattern functions.

In the remainder of this section we will clarify some technical issues involving the orbital phase. First, in computing $\phi (t)$ in phase_frequency() we have arbitrarily set the constant $\phi_c$ in Eq.([*]) such that $\phi=0$ at the beginning of the chirp. The astrophysical convention for defining the orbital phase angle $\phi$ given in [8] measures $\phi$ in the plane of the orbit from the ascending node. [The ascending node of the orbit is where body-1 passes through the plane of the sky going away from the observer.] Choosing $\phi_c$ in this way we have assumed that body-1 is passing through the ascending node of the orbit at the instant we start our chirp. Detailed information about the overall phase is not needed for many purposes (i.e. matched filters), therefore our choice is of little consequence. If this information needs to be included for some application, chirp_filters() can be modified to do so; thus one can leave the computational engine phase_frequency() untouched.

The second issue involving the phase is a bit more delicate. We have used the true orbital phase $\phi (t)$ to compute oscillatory part of the chirp in Eqs.([*]) and ([*]). But should we use the logarithmically modulated phase variable

\begin{displaymath}
\psi (t) = \phi - { 4 G m_{\rm tot} \pi f(t) \over c^3} \ln[f(t)/f_o]
\end{displaymath} (6.6.41)

in our computation of the chirp? After all, the true phase of the gravitational-wave signal impinging on the detector is $2\psi$. Let us examine the effect on our signal replacing $\sin 2 \phi$ in Eq.([*]) with the logarithmically corrected $\sin 2 \psi$
$\displaystyle \sin 2 \psi$ $\textstyle =$ $\displaystyle \sin \biggl ( 2\phi-{8\pi m_{\rm tot} f G \over c^3} \ln(f(t)/f_o) \biggr)$  
    $\displaystyle = \sin 2 \phi \cos \biggl ( {8\pi m_{\rm tot} f G \over c^3} \ln(...
...os 2 \phi \sin \biggl ( {8\pi m_{\rm tot} f G \over c^3} \ln(f(t)/f_o) \biggr )$  
    $\displaystyle \approx \biggl ( 1 +O(1/c^6) \biggr ) \sin 2 \phi
- \biggl ( {8\pi m_{\rm tot} f G \over c^3} \ln(f(t)/f_o) \biggr )\cos 2\phi \; .$ (6.6.42)

The $O[1/c^6]$ is a post$^3$-Newtonian term and can be neglected in the present post$^2$-Newtonian analysis. However the coefficient of the $\cos 2 \phi$ is a post$^{3/2}$-Newtonian order correction to the waveform, and must be included in any full post$^2$-Newtonian analysis. This logarithmic term is included in the waveform calculation in the strain() routine. However, the last line of Eq.([*]) also shows that the logarithmic phase correction can be considered a post$^{3/2}$-Newtonian correction to the amplitude. In our present restricted post-Newtonian chirp calculation we neglect these higher order amplitude corrections, so we are justified in neglecting the logarithmic correction to the phase.

The advantage of neglecting the logarithm is that it speeds up the calculation of the chirps: we don't have to compute a logarithm at each time step. However, this may be at expense of accurately tracking the signal phase of a strongly relativistic source. After all much research has gone into computing the gravitational wave phase from these sources and we shouldn't willy-nilly discard these phase corrections. Is it difficult to modify our code to include this term? Not at all. In fact, the inclusion of the logarithmic correction to the gravitational wave phase would not affect phase_frequency(), at all. The fact that this logarithmic propagation effect only enters the chirp_filters() routine and not the phase_frequency() routine may seem like a computational quirk, but this actually has a physical origin: The routine phase_frequency() computes the local orbital phase of the binary; whereas, the physical origin of the logarithmic term is a propagation effect and has nothing to do with the orbital phase,

This is not say that no log terms will ever be needed in phase_frequency(). Note that at post$^4$-Newtonian order there are log terms which do affect the local instantaneous orbital motion of the binary, so if phase_frequency() is ever modified to incorporate that order, then log terms will appear there also.

Another issue involving the log term in the phase is the presence of the ``arbitrary'' scale factor $f_o$ entering the definition of $\psi (t)$ in Eq.([*]). The net effect of adjusting this constant is to change the value of another arbitrary constant in our phase and frequency equations; it shifts the value of $t_c$ in Eq.([*]). In order to to facilitate swift computation, we choose $f_o$ to be the minimum frequency of the requested chirp. This insures that the ratio in the logarithm is of order unity during the chirp computation.


next up previous contents
Next: Example: filters program Up: GRASP Routines: Gravitational Radiation Previous: Function: chirp_filters()   Contents
Bruce Allen 2000-11-19