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The following text is extract of:

Advancing Celestial Circles of Position
THOMAS R. METCALF
Institute for Astronomy, University of Hawaii

ABSTRACT
This paper presents rigorous equations, useful with computer reduction of 
celestial sights, which correct the Greenwich Hour Angle and declination of a 
celestial body for the motion of a vessel. Advancing a circle of position in 
this way maintains the relationship between the geographical position of the 
body and the vessel, and hence is the best method for advancing an 
observation.

INTRODUCTON
When computing a running fix without the aid of a computer, one generally 
proceeds graphically by advancing a line of position (LOP) on a chart. To the 
extent that an LOP is a good approximation to the true circle of position, 
this technique works well. However, the graphical approach does not yield an 
efficient algorithm when calculating a running fix with a computer. Instead, 
to allow for the motion of the observer, the program adjusts the 
observations, not the LOP directly.
The simplest method of advancing an observation with a computer involves a 
correction to the observed altitude of a celestial body [1, 2]. However, this 
is clearly an approximation since it may alter the direction of the LOP, 
particularly for observations of bodies near the zenith. To advance the 
entire circle of position, and hence maintain the orientation of the LOP, one 
corrects the Greenwich Hour Angle (GHA) and declination of the observed body. 
By correcting the GHA and declination rather than the altitude, the circle of 
position is "dragged" along with the vessel, making the correction exact. An 
approximate technique for advancing the GETA and declination that is adequate 
for almost ah situations encountered in navigation is presented in [1]. 
However, a rigorous formulation is not much more demanding computationally 
and, with the advent of the handheld computer, may be of interest.
Table 1 indicates typical errors incurred when using the approximate 
techniques discussed aboye to advance the observation of a body near the 
zenith (85?17'). The first column gives the distance traveled from the 
starting point (45? N, 30.3? W). The second column shows the error in the 
geographical position (GP) of the fictitious body when the GHA and 
declination are advanced using the approximate formulae given in [1]. The 
error in a derived position could be substantially larger for LOPs that cross 
at small angles, as in the case of multiple sights of a single body over a 
relatively short period of time [3, 4]. The third column shows the error in 
the orientation of an LOP computed by altering the observed altitude of the 
body to correct for observer motion. Again, the error in the derived position 
could be considerable when the LOPs intersect at small angles.

METHOD
Since the geographical position of the observed body is not necessarily near 
the vessel, the correction cannot be made by simply moving the GP the same 
distance along the same course as the vessel's motion. Instead, the 
coordinate system defining GHA and declination is mathematically rotated by 
an amount equal to the vessel's change in latitude and longitude, since any 
rotation will preserve the relative position of the GP and the vessel. In 
other words, rather than treating the vessel as moving over a fixed surface, 
the vessel and the GP are considered fixed, with the earth moving underneath 
them in such a way that the vessel begins and ends at the appropriate places. 
The final position of the GP after this rotation is the advanced GP we seek.
The usual convention is to describe the GP with two angles: the GHA and the 
declination. However, the rotation takes a simpler form if the GP is 
represented as a vector directed from the center of the earth to the actual 
GP on the surface. With the axis of the rotation represented vectorally by 
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