The LIGO and Virgo detector geometry as stored in the LALSuite analysis software consists of the following for each detector on the Earth: the location of the detector and orientation of its arms. For the LIGO and Virgo detectors, these are
|X Arm||Y Arm|
|LIGO Hanford (LHO)||46°27′19″N||119°24′28″W||N36°W||W36°S|
|LIGO Livingston (LLO)||30°33′46″N||90°46′27″W||W18°S||S18°E|
(The azimuth for each arm describes the local compass direction along the arm. For instance, N36°W means a direction between local North and West, 36 degrees away from North.)
These locations and directions look like this, on a map of the Earth:
Note that the arm directions do not take into account any map projection effects, they simply indicate the direction of the arms relative to local north.
A detector records a positive strain when its X arm becomes longer, relative to its Y arm. It's useful, therefore, to represent the geometry with outward pointing arrows along the X arm and inward pointing arrows along the Y arm, indicating the sort of deformation that is associated with a positive signal in the detector:
Each detector responds optimally to a signal coming from directly overhead. The relation between the latitude and longitude of a detector and the right ascension and declination of the sky position overhead depends on the sidereal time of the observation. At the time of GW100916, which was GPS time 968654558, 2010 Sep 16, 06:42:23 UTC, the Greenwich Mean Sidereal Time was 06:22:49, so we can represent the detector locations and responses to their optimal sky positions like this:
This is not the right way to view the sky, though, since it corresponds to looking in from the outside, rather than looking out from the inside. So we flip the picture around so that the direction of increasing right ascension (eastward) is to the left rather than the right. Also, since a detector responds optimally to a signal coming either down from its zenith or up from its nadir, we also indicate the detector geometry for the antipodal sky location for each detector:
In three dimensions, each of the patterns shown above lies in a different plane, tangent to the sky at the position shown. To illustrate the response to a gravitational wave coming from an arbitrary direction, we use the fact that for any detector and any sky position, we can construct an equivalent interferometer, for which that sky position is overhead, which would register exactly the same GW strain as the actual detector, for signals coming from that sky position. In general, this artificial equivalent interferometer will have shorter arms, reflecting the less-than-optimal response of the real interferometer to gravitational waves not coming from its zenith.
For example, for the sky position directly overhead at Virgo, the responses of the three detectors look like this:
The dashed circle indicates the maximum possible response, and Virgo (magenta) lies on that circle because this is an optimal direction for Virgo. The lengths of the arrows for LHO (red) and LLO (green) indicate the reduction in response for those detectors for this non-optimal sky position.
For the actual sky position of the injection responsible for GW100916, the responses of the LIGO and Virgo detectors are the same as for the following equivalent interferometers:
Of course, the injected sky position was only known a posteriori. In the blind analysis of GW100916, what was known was that the signal arrived at LLO about 6 milliseconds before it arrived at LHO. This meant that the sky position of the signal was located somewhere along a ring in the sky corresponding to this time delay. The responses of the three detectors along this ring looked like this (note that these points are evenly spaced along the ring, although they do not look evenly spaced on this simplistic map projection):
Close up, the responses at the individual sky points along the ring look like this: