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Function: overlap()

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void overlap(float *site1_parameters, float *site2_parameters, int n, float delta_f, double *gamma12)
This function calculates the values of the overlap reduction function $\gamma(f)$, which is the averaged product of the response of a pair of detectors to an isotropic and unpolarized stochastic background of gravitational radiation.

The arguments of overlap() are:

site1_parameters: Input. site1_parameters[0..8] is an array of nine floating point variables that define the position of the central station of the first detector site and the orientation of its two arms. The three-vector site1_parameters[0..2] are the $(x,y,z)$ components (in cm) of the position vector of the central station of the first site, as measured in a reference frame with the origin at the center of the earth, the $z$-axis exiting the North pole, and the $x$-axis passing out the line of $0^\circ$ longitude. The three-vector site1_parameters[3..5] are the $(x,y,z)$ components (in cm) of a vector pointing along the direction of the first arm of the first detector (from the central station to the end station). The three-vector site1_parameters[6..8] are the $(x,y,z)$ components (in cm) of a vector pointing along the direction of the second arm of the first detector (from the central station to the end station).
site2_parameters: Input. site2_parameters[0..8] is an array of nine floating point variables that define the position of the central station of the second detector site and the orientation of its two arms, in exactly the same format as the previous argument.
n: Input. The number $N$ of discrete frequency values at which the overlap reduction function $\gamma(f)$ is to be evaluated.
delta_f: Input. The spacing $\Delta f$ (in Hz) between two adjacent discrete frequency values: $\Delta f:=f_{i+1}-f_i$.
gamma12: Output. gamma12[0..n-1] is an array of double precision variables containing the values of the overlap reduction $\gamma(f)$ for the two detector sites. These variables are dimensionless. gamma12[i] contains the value of $\gamma(f)$ evaluated at the discrete frequency $f_i=i\Delta f$, where $i=0,1,\cdots,N-1$.

The values of $\gamma(f)$ calculated by overlap() are defined by equation (3.9) of Ref. [36]:

\begin{displaymath}
\gamma(f) := {5 \over 8 \pi} \int_{S^2} d \hat \Omega \>
e^{...
...ec x/c }
\left( F_1^+ F_2^+ + F_1^\times F_2^\times \right)\ .
\end{displaymath} (11.6.226)

Here $\hat \Omega$ is a unit-length vector on the two-sphere, $\Delta
\vec x$ is the separation vector between the two detector sites, and $F_i^{+,\times}$ is the response of detector $i$ to the $+$ or $\times$ polarization. For the first detector $(i=1)$ one has
\begin{displaymath}
F_1^{+,\times} = {1 \over 2} \left( \hat X_1^a \hat X_1^b -
\hat Y_1^a \hat Y_1^b \right)e_{ab}^{+,\times}(\hat \Omega)\ ,
\end{displaymath} (11.6.227)

where the directions of the first detector's arms are defined by $\hat X_1^a$ and $\hat Y_1^a$, and $e_{ab}^{+,\times}(\hat \Omega)$ are the spin-two polarization tensors for the ``plus" and ``cross" polarizations, respectively. (A similar expression can be written down for the second detector.) The normalization of $\gamma(f)$ is determined by the following statement: For coincident and coaligned detectors (i.e., for two detectors located at the same place, with both pairs of arms pointing in the same directions), $\gamma(f)=1$ for all frequencies.
Authors: Bruce Allen, ballen@dirac.phys.uwm.edu, and Joseph Romano, romano@csd.uwm.edu
Comments: None.


next up previous contents
Next: Example: overlap program Up: GRASP Routines: Stochastic background Previous: Function: whiten()   Contents
Bruce Allen 2000-11-19