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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 , 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 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 -axis exiting the North pole, and the -axis passing out the line of longitude. The three-vector site1_parameters[3..5] are the 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 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 of discrete frequency values at which the overlap reduction function is to be evaluated.
delta_f: Input. The spacing (in Hz) between two adjacent discrete frequency values: .
gamma12: Output. gamma12[0..n-1] is an array of double precision variables containing the values of the overlap reduction for the two detector sites. These variables are dimensionless. gamma12[i] contains the value of evaluated at the discrete frequency , where .

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

 (11.6.226)

Here is a unit-length vector on the two-sphere, is the separation vector between the two detector sites, and is the response of detector to the or polarization. For the first detector one has
 (11.6.227)

where the directions of the first detector's arms are defined by and , and 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 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), for all frequencies.
Authors: Bruce Allen, ballen@dirac.phys.uwm.edu, and Joseph Romano, romano@csd.uwm.edu