(ed.). Gravitational waves (IOP, 2001)(422s).

The overlap reduction function 193Table 12.1. The normalization factor F for three different geometries of the detectors:interferometer (ITF), cylindrical bar (BAR) and sphere (SPH). A ⋆ denotes entries that canbe obtained from the symmetry of the table.SPHITF BAR m = 0 m =±1 m =±2ITF 2/5 ⋆ ⋆ ⋆ ⋆BAR 2/5 8/15 ⋆ ⋆ ⋆m = 0 0 2 √ 3/15 2/5 ⋆ ⋆SPH m =±1 0 0 0 2/5 ⋆m =±2 2/5 2/5 0 0 2/5where[ ρ0]ρ 1 (η) = 1[ 2η2 ][ ]−4η 2 j0ρFη 2 −4η 2 16η −20 j 1 (η), (12.45)2 η 2 −10η 35 j 2with j k (η) the standard spherical Bessel functions:j 0 (η) = sin ηη , j 1(η) = j 0(η) − cos η, j 2 (η) = 3 j 1(η)− j 0 (η).ηηIn the following, this formalism will be applied to treat the case in whichthe first detector is an interferometer and the other is, in turn, an interferometer, acylindrical bar and a sphere.12.3.1 Two interferometersFollowing equation (12.40), the detector tensor of the interferometer located at agiven point ⃗r k can be written asD k = 1 4 {(cos 2ξ k − cos 2ζ k )e + (ˆr k ) − (sin 2ξ k − sin 2ζ k )e × (ˆr k )} (k = 1, 2)(12.46)where ξ k and ζ k are the orientations of the kth interferometer arms measuredanticlockwise from the true north. The value of these angles, together withthe location of the central (corner) station on the Earth’s surface, for differentinterferometers are reported in table 12.2. By inserting this expression inequation (12.44) it is possible to compute the overlap reduction function for ageneric pair of interferometers. The results of this calculation in the case in whichone of the interferometers is VIRGO are shown in figure 12.1.Some important conclusions can be drawn from the inspection of this figure.First of all, owing to the oscillating behaviour with η ∝ fdof the spherical Bessel