where is the distance between S1 and S2, and the speed of light. Since is known and is measured, is estimated; the solution to eq. is represented by a ring, or annulus, whose width depends on the timing uncertainties ( ) and on the mutual distances (): the farther apart the detectors, the more precise the localization. The number of independent couples of detectors (and, therefore, the number of independent annuli) is two, when taking into account S3 as well; thus, the burst direction must be inside one of the two intersection regions, that may be sometimes many degrees far from one another. When another spacecraft, S4, detected the burst too, then there is no ambiguity; otherwise, when at least one among S1, S2 and S3 has some rough localization capabilities, this can be conclusive for rejecting the false position (actually, also Earth blockage can be used to this aim, as happened in some cases, and even a non-detection by a satellite around the Earth can help in limiting the number of possible solutions). The annulus width is obtained by propagating the uncertainty on the time delay ; thus, from eq. it follows:
In eq. , expressing in radians, the uncertainty on has been neglected, as the main contribution comes from the timing uncertainties: actually, one has to take into account not only the time resolution of each detectors, but also the difficulty of comparing different light curves, often derived from different energy bands. For example, when spans the typical range: few - few light seconds, then from eq. it comes out that a minimum time resolution of the order of s is required, in order to have few arcminutes or less (otherwise useless for a precise localization).
The third IPN started in 1990 when the NASA/ESA Ulysses mission spacecraft, carrying a GRB experiment, was launched: its orbit plane is normal to the ecliptic; since its aphelion is beyond Jupiter's orbit, it is periodically the farthest spacecraft (fig. ) joining the IPN, and, therefore, the most important for the above reasons.
In 1991 the Compton Gamma Ray Observatory (CGRO) joined the IPN with the BATSE experiment till its demise (June 2000).
Missions like Pioneer Venus Orbiter , Mars Observer , and the Near Earth Asteroid Rendezvous mission (NEAR) joined the network while they were operating ([Laros, 1997,Laros, 1998]).
Apart from Ulysses, after the landing of NEAR on the Eros asteroid (February 2001), presently the second farthest spacecraft contributing to the IPN triangulations, is WIND with the Konus experiment, located near the sunward Sun-Earth equilibrium point (L1); for a more detailed description of this experiment, see below.
Currently (December 2001), in addition to Ulysses and WIND, the main near-Earth spacecrafts operating in the IPN are BeppoSAX, HETE-II and RossiXTE: other members include the Indian SROSS-C2 spacecraft, the Air Force's Defense Meteorological Satellites, the Japanese Yohkoh spacecraft, the recently launched Chinese SZ-2 mission, and the 2001 Mars Odyssey mission. In the near future, the INTEGRAL and the AGILE will join the IPN as well.