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previously mentioned. First, we'll discuss the astronomical triangle upon which the tables are based.
Then we'll cover how to determine the local hour angle (LHA) of Aries and the LHA of a star.
Section 9B— LHA and the Astronomical Triangle
9.2. Basics. The basic principle of celestial navigation is to consider yourself to be at a certain assumed
position at a given time; then, by means of the sextant, determining how much your basic assumption is
in error. At any given time, an observer has a certain relationship to a particular star. The observer is a
certain number of nautical miles away from the subpoint, and the body is at a certain true bearing called
true azimuth (Zn), measured from the observer's position (Figure 9.1).
Figure 9.1. Subpoint of a Star.
AFPAM11-216 1 MARCH 2001 217
9.3. Intercept. Assume yourself to be at a given point (called the assumed position). At a given time,
there exists at that instant a specific relationship between your assumed position and the subpoint. The
various navigational tables provide you with this relationship by solving the astronomical triangle for
you. From the navigational tables, you can determine how far away your assumed position is from the
subpoint and the Zn of the subpoint from the assumed position. This means, in effect, that the tables give
you a value called computed altitude (Hc) which would be the correct observed altitude (Ho) if you were
anywhere on the circle of equal altitude through the assumed position. Any difference between the Hc
determined for the assumed position and the Ho as determined by the sextant for the actual position is
called intercept. Intercept is the number of NM between your actual circle of equal altitude and the circle
of equal altitude through the assumed position. It is by means of the astronomical triangle that you can
solve for Hc and Zn in the Pub. No. 249 tables.
9.4. Construction of the Astronomical Triangle. Consider the solution of a star as it appears on the
celestial sphere. Start with the Greenwich meridian and the equator. Projected on the celestial sphere,
these become the celestial meridian and the celestial equator (called equinoctial) as shown in Figure 9.2.
Notice also in the same illustration how other known information is derived, namely the LHA of the star
Aries—equal to the Greenwich hour angle (GHA) of Aries minus longitude west. You can also see that
if the LHA of Aries and sidereal hour angle (SHA) of the star are known, the LHA of the star is their
sum. It should also be evident that the GHA of Aries plus SHA of the star equals GHA of the star. Also,
the GHA of the body minus west longitude (or plus east longitude) of the observer's zenith equals LHA
of the body. These are important relationships used in the derivation of the Hc and Zn.
Figure 9.2. Astronomical Triangle.
218 AFPAM11-216 1 MARCH 2001
9.4.1. Figure 9.3 shows part of the celestial sphere and the astronomical triangle. Notice that the known
information of the astronomical triangle is the two sides and the included angle; that is, Co-Dec, Co-Lat
and LHA of the star. Co-Dec, or polar distance, is the angular distance measured along the hour circle of
the body from the elevated pole to the body. The side, Co-Lat, is 90o minus the latitude of the assumed
position. The included angle in this example is the LHA. With two sides and the included angle of the
spherical triangle known, the third side and the interior angle at the observer are easily solved. The third
side is the zenith distance, and the interior angle at the observer is the azimuth angle (Z). Instead of
listing the zenith distance, the astronomical tables list the remaining portion of the 90o from the zenith,
or the Hc. Hc equals 90o minus zenith distance of the assumed position, just as zenith distance of the
assumed position equals 90o – Hc. Note that when measured with reference to the celestial horizon,
zenith distance is synonymous with co-altitude. Figure 9.4 is a side view of this solution.
Figure 9.3. Celestial— Terrestial Relationship.
9.4.2. So far, the astronomical triangle has been defined only on the celestial sphere. Refer again to
Figure 9.3 and notice the same triangle on the terrestrial sphere (earth). The same triangle with its
corresponding vertices may be defined on the earth as follows: (1) celestial pole—terrestrial pole; (2)
zenith of assumed position—assumed position; and (3) star—-subpoint of the star. The three interior
angles of this triangle are exactly equal to the angles on the celestial sphere. The angular distance of
each of the three sides is exactly equal to the corresponding side on the astronomical triangle. Celestial
and terrestrial terms are used interchangeably. For example, refer to Figure 9.3 and notice that Co-Lat on
　
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