LHA and the Astronomical Triangle (Part Three)

Entering Argument

Volumes 2 and 3 are entered with the LHA of the body, in contrast to Volume 1, which is entered with the LHA of Aries. The range extends from 0° through all LHAs applicable within the altitude limits of the body. Between latitude 70° and the pole, the LHA interval is 2°; for latitudes below 70°, the interval is 1°. Arguments of LHA of the body less than 180° appear on the left margin, and arguments greater than 180° appear on the right.

Several pages are devoted to each degree of latitude. Each page has 15 declination (Dec) columns and is labeled with its value at the top and bottom. Each page is also marked Declination Contrary Name to Latitude or Declination Same Name as Latitude.


The entering arguments of LHA of the body, for declination of contrary name to latitude, always increase from the bottom of the page on the left side and decrease on the right. The opposite arrangement exists on pages where Dec and latitude has the same name. Occasionally, one page is blank in the middle and the top half covers Declination Same Name as Latitude, while the bottom half is Declination Contrary Name to Latitude.

Azimuth angle (Z) is listed instead of true azimuth (Zn). Since Zn is used for plotting, it is necessary to convert Z to Zn. The rules for conversion are listed on the left-hand side at the top and bottom of every page. Notice that LHA and Zn will never occur on the same side of 180°.

In addition to Hc and Z, a value of d is also listed. This d-value is the change in altitude (Hc) with a 1° increase in Dec. If the LHA and Dec of the body and the latitude of the assumed position are each a whole number of degrees, the Hc and Z are found in the correct Dec column opposite the LHA of the body on the page marked by the proper latitude value.

For example, refer to the portion of the table shown in Figure 9-6. At the latitude 40° N, if the LHA of a body is 86° and its Dec is 5° N, the Hc is 06° 16′ and the azimuth angle (Z) is 089°. The rule in the upper left-hand corner of the page applies for the conversion of Z to Zn. Zn = 360° – Z or 360° – 089° = 271°. Here again the position is assumed so that latitude and LHA are whole numbers.

Figure 9-6. Enter tables with latitude, Dec, and LHA.

Figure 9-6. Enter tables with latitude, Dec, and LHA. [click image to enlarge]

Interpolation for Declination (Dec)

When the Dec of a body is a number of minutes in addition to a whole number of degrees, the altitude (Hc) is extracted for the whole number of degrees and corrected by interpolation for the additional minutes. There is rarely a need for interpolation of Z, which is given only to the nearest degree.

Interpolation for Hc should always be made in the direction of increasing Dec in accordance with the sign of the d-value. Not all of the signs are printed; the sign is given at least once in each block of five entries and can always be found by looking either up or down the column from the value of d in question. The correction to altitude for additional minutes of Dec is proportional to d and proportional to the number of additional minutes.

In the previous example, the latitude was 40°, the LHA of the body was 086°, and the Dec was 5° N. Suppose the Dec had been 5° 17′ N. The basic figures obtained would be 06° 16′ Hc and 089° Z as before, and the true azimuth (Zn) would still be 271°. The Hc of 06° 16′ is not correct for a Dec of 5° 17′ N, but is correct for 5° N. The Hc change for an additional 1° of Dec (d-value) is +39 minutes of altitude. However, the correction needed in this case is for 17 minutes of Dec, not a whole degree. Consequently, the additional correction is 17/60 of 39′. To the closest whole number, this would be +11 minutes of altitude.


This multiplication can be done on the slide rule face of the DR computer or by means of a table found in back of Pub. No. 249, Volumes 2 and 3. A portion of this table is shown in Figure 9-7. Notice that there are no signs listed. The proper sign for the answer from this table is the same sign as the basic d-value. A rule of thumb is the correction is a plus (+) for Declination of Same Name as Latitude and a negative (–) for Declination of Contrary Name as Latitude. Values of d are given across the top of the table and additional minutes of Dec are given down the side of the table. In the table, the correction 11′ is found by looking across 17′ for Dec and down 39′ for d to their intersection at 11′. Since the sign of the d-value is plus, this correction is added to the tabulated Hc. The correct Hc value then becomes 06° 16′ + 11′ or 06° 27′.

Figure 9-7. Table performs the multiplication.

Figure 9-7. Table performs the multiplication. [click image to enlarge]

Following is a sample problem illustrating the solution. Refer to the portion of the tables in Figure 9-8 for the solution.

Figure 9-8. Declination 0–14° Contrary Name to Latitude.

Figure 9-8. Declination 0–14° Contrary Name to Latitude. [click image to enlarge]

Suppose the sun is observed at 1005 GMT. The DR position is 38° 12′ N, 101° 47′ E, and the Ho of the sun is 10° 52′.

Dec of the Sun for 1000Z S7° 37′
GHA Sun for 1000Z 326° 53′
Correction to GHA for 5 minutes +1° 15′
GHA Sun for l005Z 328° 08′
Closest longitude for whole degree LHA
(assumed longitude)
+E101° 52′
430° 00′
–360° 00′
LHA Sun for 1005Z 070° 00′

The closest whole degree of latitude is 38° N and is used as the assumed latitude. Since the assumed latitude is north and Dec is south, the navigator must use Pub. 249, Volume 2, page for 38° latitude, which is headed Declination (0°–14°) Contrary Name to Latitude. Following LHA 070° across the page to 7° Dec, the navigator extracts:

Tab Hc 11° –06′
d-value –40′
Z 108°
d-correction from Pub. No. 249, Volume 2 –25′
Corrected Hc 10° –41′
Zn using rule in the upper left-hand corner
of the page

Postcomputation Summary

Before proceeding, review the procedures for finding the Hc and Zn of a body whose Dec lies between 30° N and 30° S, using Pub. No. 249, Volume 2 or 3.

  1. Shoot the body and record the time of observation, the body’s name, and the Ho.
  2. From the Air Almanac, extract GHA and Dec of the body for the time of the observation.
  3. Assume a position close to the DR position so that the latitude is a whole number of degrees and the longitude combined with the GHA of the body gives a whole number of degrees of LHA of the body. Find the LHA of the body for this position.
  4. Select the correct volume (2 or 3) and page that contains the correct arguments of Dec and LHA of the body, temporarily disregarding the odd additional minutes of Dec. Thus, if the Dec were N 19° 55′, use the column for 19°. Select the table labeled Declination Same Name as Latitude, if Dec and latitude are both north or both south, or select the table labeled Declination Contrary Name to Latitude, if one is north and the other south. Opposite the LHA of the body, read the tabulated altitude, d-value, and Zn in the column headed by the whole degrees of Dec.
  5. If the Dec is not a whole number of degrees, determine the correction for the additional minutes of Dec. Enter the table in the Pub. No. 249 volume with the value d and the number of additional minutes of Dec. Apply the correction to the tabulated altitude (Hc) according to the sign of d. This is the corrected Hc.
  6. Convert azimuth angle (Z) to true azimuth (Zn) by means of the rule at the top or bottom of the page.
  7. This completes the solution for the Dec tables. Keep in mind this solution is computed after the observation. Because of the speeds involved in air navigation, we will explain a way to compute the solution before the shot in the next category.