where N¡=0 if there are no observations in the ith azimuth bin and N¡= 1 otherwise; Mj=0 if there are no catalog stars in theyth longitude bin and Mj= 1 otherwise; and B is the number of bins.

Rotate the observation frame by 8 by adding 8 to each observation azimuth and compute a score for the new configuration. Repeat this process for a complete 360-deg circuit. The highest score corresponds to the correct phase for the observations. Alternatively, the process can be stopped when a score is attained which the analyst feels is sufficiently large to ensure that the correct phase has been found.

A major limitation of the phase match technique is that it fails if either the N¡ or Af, values in Eq. (7-129) are mostly 1.

To ensure that Mj=0 often enough, the mean number of catalog stars per longitude bin, <S>, must be <2; <S> is given by

where p is the density of stars brighter than the limiting magnitude of the sensor (see Section 5.6). If <5>>1. those bins where there are no stars become the important ones; if the star catalog is complete, these "holes" will never contain observations when the correct phase is found.

Because the catalog must contain all or nearly all stars to the limiting magnitude of the sensor, the only way to control <S> is by adjusting the sensor sensitivity. If the threshold is sufficiently high (i.e., only fairly bright stars are detected), the star density, p, will be low; thus, (S) will be low and Mj will be zero sufficiently often. This procedure also ensures that will be zero often enough because the sensor cannot observe more stars than are in the catalog, if the catalog is nearly complete.

Several refinements to this technique are possible. If the star catalog is complete, we can assume that a single selected observation will match some star in the catalog. By matching the observation with each catalog star in turn, a set of possible phase angles is generated. Because the number of catalog stars is normally far less than the number of bins, this reduces the number of scores which must be calculated.

A second refinement makes use of elevation and azimuth information. Because the maximum elevation error for an observation is S2, the elevation information is useful if

To include elevation information, redefine the score given in Eq. (7-129) as

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