LHA and the Astronomical Triangle (Part One)

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 (NM) away from the subpoint, and the body is at a certain true bearing (TB) called true azimuth (Zn), measured from the observer’s position. [Figure 9-1]

Figure 9-1. Subpoint of a star.

Figure 9-1. Subpoint of a star.

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 Sight Reduction Tables for Air Navigation found in Pub. No. 249.

 

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.

Figure 9-2. Astronomical triangle.

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 90° 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 90° from the zenith, or the Hc. Hc equals 90° minus zenith distance of the assumed position, just as zenith distance of the assumed position equals 90° – 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.

Figure 9-3. Celestial-terrestial relationship.

Figure 9-4. Co-altitude equals 90 minus Hc.

Figure 9-4. Co-altitude equals 90 minus Hc.

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 the terrestrial triangle is also called Co-Lat on the celestial triangle. To be perfectly correct, the term on the celestial sphere corresponding to latitude on the earth is declination (Dec); therefore, the celestial side could well be called codeclination of the zenith of the assumed position.

 

Rather than have this confusion, the terrestrial term Co-Lat is also used with reference to the celestial sphere, just as latitude of the subpoint is considered to be the Dec amount from the equator. Latitude is used when referring to the observer or zenith, and Dec is used when referring to the star or its subpoint. The distance between the subpoint and the assumed position is generally referred to as zenith distance (Co-Alt) rather than the segment of the vertical circle joining the subpoint and the assumed position. These angular distance terms are interchangeable on the celestial and terrestrial spheres.

The values of the Zn and the interior angle (Z) are listed in the Pub. No. 249 tables depending upon whether or not a Dec solution is desired. Pub. No. 249, Volume 1, lists the Zn rather than the interior angle. Pub. No. 249, Volumes 2 and 3, list the interior angle (Z). It is necessary to follow rules printed on each page to convert the interior angle (Z) to true azimuth (Zn).