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

196 Generalities on the stochastic GW backgroundTable 12.3. Site and orientation of the resonant bars nearest to VIRGO. In the last columnis reported the distance with respect to VIRGO.LocationProject Latitude N (deg) Longitude E (deg) λ (deg) Distance (km)AURIGA 45.35 11.95 39.3 222.84NAUTILUS 41.80 12.67 39.3 270.12EXPLORER 46.25 6.25 39.3 443.14with good approximation inversely proportional to the distance between the twodetectors.12.3.2 Interferometer—barThe detector tensor of a resonant bar located at a given point ⃗r turns out to be (seeequation (12.41))D(ˆr) = 1 2 {cos 2λe+ (ˆr) − sin 2λe × (ˆr) + b(ˆr) − 2 3I} (12.47)where the tensorb(ˆr) = ˆm(ˆr) ⊗ˆm(ˆr) +ˆn(ˆr) ⊗ˆn(ˆr)has been introduced, and we have indicated with λ the orientation of the bar axismeasured anticlockwise from the true North. The orientation and location of theresonant bars nearest to VIRGO are reported in table 12.3.The behaviour of the overlap reduction function for the correlation of thesedetectors with VIRGO is shown in figure 12.3, where, since the resonant detectorscan be easily rotated, we considered the case in which the bars are orientedalong the direction of the first arm of VIRGO. The frequency dependence ofthe overlap reduction function is meaningless for narrow-band detectors like bars(and spheres). Therefore, the results shown in figure 12.3 have been calculated byassuming to vary the resonance frequency by varying the length of the bar.12.3.3 Interferometer—sphereThe same calculation can be repeated by replacing bars with spheres of variablesize. The overlap reduction function for the correlation of VIRGO with each ofthe five modes of a sphere (see equation (12.42)) is shown in figure 12.4. Noticethat, since the normalization factors vanish (see table 12.1) for the m = 0, ±1modes, the quantity reported in figure 12.4 for these modes is the Ɣ( f ) functiondefined in equation (12.35). It is also worthwhile to notice that the result for them = 2 case coincides exactly with that obtained with two interferometers, incomplete accordance with the quadrupole nature of the GW excitation.