The solutions of problems involving the three coordinate systems are made by means of spherical trigonometry. A figure of prime importance is the spherical triangle that lies on the celestial sphere and whose vertices are the pole, the zenith, and the celestial body involved. This is known as the astronomical or the PZS (pole-zenith-star) triangle. The astronomical triangle is shown in figure 15-3. As in the case of all spherical triangles, the sides can be expressed as the angles subtended at the center of the sphere. In the astronomical triangle, the side between the pole and the zenith is the colatitude (90 - F), between the pole and the star is the codeclination or polar distance (90 - d), and between the zenith and the star is the coaltitude, or zenith distance (90 - h). The angle at the zenith is the azimuth angle (A) of the body. The angle at the pole is the hour angle (t). The angle at the star is known as the parallactic angle and is little used in computa-tions. If the three elements of the astronomic triangle are known, the others can be found by means of spheri-cal trigonometry. The fundamental equation is the law of cosines. cos a = cos b cos c + sin b sin c cos a, in which a, b, and c are the sides of a spherical triangle, and A is the angle opposite side a (B and C are the angles opposite sides b and c, respectively). All formulas required for the solution of the astronomic triangle may be derived from this law of cosines.

Astronomical Tables Used by Surveyors

The declination and Greenwich hour angle of the sun, moon, and selected planets are given for every even hour of GMT for everyday in the year in the daily pages of the Nautical Almanac and the Ephemeris of the Sun, Polaris, and Other Selected Stars. These publications are prepared by the U.S. Naval Observatory and are available for sale at the U.S. Government Printing Office, Washington, D.C. Condensed tables of data are also available from various manufacturers of surveying equipment.

Table 15-1 shows the relevant daily page of the 1986 Nautical Almanac. You can see that for 15^h00^m00^s on 17 May the GHA listed for the sun is 45⁰54.8. For the extra 23^m18^s you turn to a table of increments and corrections in the back of the book. Table 15-2 shows the relevant page of the table. Under 23^m and beside 18s in the Sun column you find an increment of 5⁰49.5. The GHA of the sun at the time of observation, then, was 4554.8 + 549.5, or 5144.3.

On the daily page of 17 May (table 15-1), the Nautical Almanac gives a sun declination for 15^h00^m00^s GMT on N1921.3. At the foot of the column, you see a small d and the figure 0.6. In the increments and corrections table (table 15-2), you see a column of v or d corrections for declination. You go down this column to the figure 0.6, where you find that the d correction in this case is 0.2'. Whether you add this correction or subtract it depends upon whether the declination of the sun is increasing or decreasing with time. A glance at the daily page shows that in this case, it is increasing; therefore, the declination of the sun at the time of observation was N1921.3 + 0.2, or N 1921.5.

On an opposing daily page of the Nautical Almanac (table 5-3), the declinations of a select list of 57 prominent stars are given. Instead of the GHAs of these stars, however, the sidereal hour angle (SHA) of each star is given. The sidereal angle of a star is its arc distance westward from the vernal equinox or first point of Aries. The GHA of a star is its arc distance westward from the hour circle of the first point of Aries. For GHA of a star, you first determine GHA of the first point of Aries in the same manner described for the sun. You can see Aries listed in tables 15-2 and 15-3. You then add this to the SHA of the star, as given in the daily page of the Nautical Almanac (table 15-3). This can be stated as follows: GHA star= GHA Aries + SHA star. If the result is greater than 360, you subtract 360 from it.

For declination of a star, you use the declination listed on the daily page; this is good for a star at anytime of the day.

NOTE: The SHA and GHA of the vernal equinox are factors used in star observations; however, neither is applicable in observation of navigational plaints.

Table 15-1.Sun and Moon Daily Page from the Nautical Almanac

Table 15-2.- Page from the Nautical Almanac (Table of Increments and Corrections)

Table 15-3.-Star and Planet Daily Page from the Nautical Almanac

Figure 15-5.The celestial sphere.