This category discusses the procedures and some of the tables used to compute a celestial line of position (LOP). Some of the tables used to resolve the LOP, including the Air Almanac, were previously mentioned. First, we will discuss the astronomical triangle upon which the tables are based. Then, we will cover how to determine the local hour angle (LHA) of Aries and the LHA of a star.

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.

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 9-3 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 9-4. 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).

LHA and the Astronomical Triangle (Part Two)

Pub. No. 249, Volume 1

Volume 1 of Pub. No. 249 deals solely with the solution concerning selected stars and is considered separately from Volumes 2 and 3. Volume 1 provides complete worldwide coverage from pole to pole for each degree of latitude. The LHA of Aries is listed in 1° increments from latitudes of 0° to 69° North and South inclusive. From 70° through 89° of latitude, the meridians are so close together that it is only necessary to tabulate the values of the LHA of Aries in even 2° increments. There are two pages devoted to each whole degree of latitude between latitudes 69° N and 69° S inclusive. From there to the pole, only one page is devoted to each whole degree of latitude. The three stars marked by diamonds on each page provide sets for fixing purposes that are favorably situated in altitude and azimuth.

The entering arguments are the assumed latitude and the LHA of Aries (to whole degrees). At any one time, the navigator has the choice of the seven listed stars for that latitude plus Polaris. The names of the stars are in capital letters if the star is of first magnitude or brighter; the second magnitude stars are printed in small letters. The names of the stars are listed every 15° of LHA of Aries (every 30° in the polar latitudes). For the time the navigator expects to make an observation, commonly called a shot, they look up the GHA of Aries and apply the approximate longitude to get a whole degree LHA of Aries. The navigator then enters Pub. No. 249, Volume 1, with the latitude closest to the dead reckoning (DR) latitude and the LHA of Aries to select the stars that will be shot.

Figure 9-5. Enter tables with LHA Aries and latitude.

Since single celestial observation results in only one LOP, it is necessary to shoot two or more bodies to obtain a fix. Suppose the navigator wants to shoot at approximately 0230 Greenwich Mean Time (GMT), he or she looks up the GHA of Aries (in the Air Almanac) and finds it to be 196°. The DR position for this time is 31° 48′ N, 075° 26′ W. A quick calculation shows the LHA of Aries is approximately 121°, and the closest latitude is 32° N. Notice in the portion of the tables reproduced in Figure 9-5 the available stars at this position are Alkaid, Regulus, Alphard, Sirius, Rigel, Aldebaran, and Capella. Using Sirius, a shot is taken at 0231 and the Ho obtained is 37° 50′.

The closest whole degree of latitude is 32° N; therefore, it is used as the assumed latitude. The assumed longitude is selected as the closest point, resulting in an LHA of Aries that is a whole degree (no minutes). The Hc of Sirius is listed as 37° 40′. The Zn is 205°.

The second shot was taken at 0234 using Regulus, the Ho being 55° 30′. A new DR position could be obtained for 0234 GMT, but the 0230Z DR position will suffice for this determination of Hc and Zn.

The assumed latitude is still 32° N and, in this case, 075° 06′ W is the assumed longitude since this is the closest longitude to the DR longitude that results in the LHA of Aries being a whole degree. The Hc of Regulus is listed as 56° 19′, and the Zn is 119°. The various corrections that must be applied, as well as the plotting of the fix, are discussed later.

Postcomputation Method

The steps in the precomputation method are as follows:

1. Determine the GHA of Aries for the time of observation from the Air Almanac.
2. Assume a position as close as possible to the DR position at the time of the shot so the latitude and LHA of Aries in whole degrees may be determined.
3. Turn to the page in Pub. No. 249 for the assumed latitude and, opposite the LHA of Aries, select the stars to be shot. In making the selection, assume the LHA of Aries will change 1° every 4 minutes of time.
4. Shoot the body and record the time, Ho, and name of the body.
5. Obtain the GHA of Aries for the time of the observation, and apply the assumed longitude to determine the LHA of Aries.
6. Turn to the pages for the assumed latitude and, opposite the LHA of Aries in the column headed by the name of the star, find and record the Hc and Zn.

Pub. No. 249, Volumes 2 and 3

Volume 1 consists of tables of Hc and Zn for selected stars. Because the Dec and SHA of each star change slowly, these tables may be used for many years with only small corrections. The Dec and SHA of a nonstellar body change rapidly, making a permanent format similar to Volume 1 impossible for the sun, moon, and planets.

Volumes 2 and 3 have Dec tables adequate for determining the Hc and Zn of any celestial body within the Dec range of 30° N to 30° S. They are intended primarily for use when observing nonstellar (solar system) bodies. Volume 2 provides latitudes between 39° N and 39° S, and Volume 3 provides for latitudes from 40° N or S to the poles.

Provision is made for observed altitudes from 90° above to 3° below the horizon (7° from latitudes 70° to the pole). In view of refraction and of possible long intercepts, the tables are actually extended 2° below these limits.

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.

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.

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.

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′.

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:

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.

Precession and Nutation

The earth’s axis does not maintain a fixed direction in space. Actually, the earth is like a slow running gyro that is wobbling. There are several separate patterns that the wobble makes. Some of those patterns have short cycles, while others take hundreds of years to complete. Two of the many patterns are shown in Figure 9-9. One involves small nodding motions while at the same time completing a larger circular path. You must use a correction called precession and nutation to account for these variations in the apparent position of the stars. This correction is applied only to celestial LOPs determined with Pub. No. 249, Volume 1.

Figure 9-9. Earth’s axis wobble.

Precession

Because of the equatorial bulge, the attractive forces of other solar system bodies, principally the moon, are unbalanced about the center of the earth. The imbalance is directed toward aligning the equator with the plane of the ecliptic. However, the rotation of the earth transforms this force into an effect acting 90° away in the direction of rotation—a precessional effect. The result is that the poles travel in a conical path westward around the ecliptic poles, as shown in Figure 9-10 (the point 90° from the ecliptic). Consequently, the points of intersection of the equator with the ecliptic, or the equinoxes, travel in a westerly direction along the ecliptic. This travel is called precession of the equinoxes, and it amounts to approximately 5/6 of a minute (50.26″) annually. The equinoxes complete one revolution along the ecliptic in approximately 25,800 years. The equator is used as a reference for Dec and its movement, due to precession of the equinoxes, causes slight changes in the celestial coordinates of the stars that otherwise appear fixed in space.

Figure 9-10. Precession of the equinoxes. 

Nutation

As the relative positions and distances from the earth to the sun, moon, and planets vary, so does the rate of precession. The only variation of importance in navigation is nutation. Nutation is a nodding of the poles, one oscillation occurring in about 18.6 years.

In Figure 9-11, you can see that if the stars remain fixed and the equinoctial moves up and down, the Dec of these bodies is changing.

Figure 9-11. Nutation changes the declination.

Nutation, being approximately perpendicular to the ecliptic, has an appreciable influence on Dec. It is caused by complex gravitational forces among the sun, moon, and earth because the moon’s orbit does not always lie in the plane of the ecliptic. The change in Dec of the celestial bodies caused by the resulting wobble of the earth’s axis is called nutation.

Position Corrections

Because of precession and nutation, Hc and Zn for a star are accurate only at the instant, or epoch, at which the LHA and Dec for the computations are correct. A position obtained at any other time with that Hc and Zn requires a correction. Pub. No. 249, Volume 1, contains Hc and Zn calculated for an epoch year (midnight, 1 January, of that year) so, if the volume is used in any other year, the resultant position must be corrected. The precession and nutation corrections are combined and given in Pub. No. 249, Volume 1, Table 5.

Entering arguments for the table is year, latitude, and LHA of Aries, and the correction is presented in the form of a distance and direction to move the fix. The tabulated values show the distance, parallel to the ecliptic, between the observer’s position in the year of the fix and the position in the epoch year at the latitude and LHA of Aries.

Directions for using Table 5 are printed in the introduction of Pub. No. 249, Volume 1. One point needs emphasis here: the table is to be used only for observations plotted with the aid of Volume 1, never in conjunction with Volumes 2 or 3.