Celestial Precomputation

Celestial precomputation methods have been brought to the forefront with the proliferation of high-speed aircraft. Aircraft speeds make it necessary to minimize the time between shooting and fixing. Since the sextant may be the only means of viewing the body, it is necessary to precompute the altitude and azimuth of a body in order to locate it. Remember corrections may be applied to the Hc, Ho, or intercept, and pay close attention to the sign of the correction. In addition to precomputation, the fix may be resolved faster by preplotting the true azimuths of the bodies.

Introduction to Celestial Precomputation

Celestial precomputation is neither new nor revolutionary. The tables necessary to do precomputation have been available since 1940; however, there was no operational requirement for precomputation at that time. With present day high-speed aircraft, however, the picture has changed radically. By postcomping, a great deal of work must be done after the last celestial observation. The fix could easily be 15 minutes old by the time it is plotted on the chart. At 450 knots groundspeed (GS), a fix that is 15 minutes old is over 100 miles behind the aircraft and is of questionable value. Another factor necessitating precomputation in high-speed aircraft lies in the method of shooting celestial. With the limited field of view of the sextant, the correct star is difficult to find unless you know where to look.

Precomputation greatly reduces both of the problems just mentioned. By completing most of the computations before shooting, the navigator reduces the time necessary to plot the fix after the last observation. Also, the problem of finding the star in the optics of the sextant is simplified. The procedure for finding the star is similar to the heading check performed with the periscopic sextant, using the true bearing (TB) method as explained in the Special Celestial Techniques category. In this case, the true azimuth (Zn) is set into the sextant mount and the computed altitude (Hc), which approximates the sextant altitude (Hs), is set into the sextant. Now, instead of sighting the body to determine the true heading (TH), set the TH under the vertical crosshair to find the selected body, hopefully very close to the crosshairs in the sextant field of view. Use the inverse relative bearing (IRB) method to avoid erroneous settings in the azimuth window and to increase speed in setting up the sextant. In this method, the azimuth window remains permanently at 000.0° and the IRB is computed by the formula: IRB = TH – Zn. The body should be found at its computed altitude when its IRB appears under the crosshairs.

Precomputation Techniques (Part One)

There are many acceptable methods of precomputation in general usage. However, these methods are basically either graphical, mathematical, or a combination of both methods. Selection is largely based on individual navigator preference. Celestial corrections that are used in precomputation include atmospheric refraction, parallax of the moon, instrument and acceleration errors, Coriolis and rhumb line, precession and nutation, motion of the observer, and wander. With precomputation, new corrections and terminology are introduced that include fix time, solution time, observation time, scheduled time, and motion of the body adjustment.

Fix time is the time for which the lines of position (LOP) are resolved and plotted on the chart. Solution time is the time for which the astronomical triangle is solved. Observation time is the midtime of the actual observation for each celestial body. Scheduled time is the time for which the astronomical triangle is solved for each LOP in the graphic method. Motion of the body correction is used to correct for the changing altitude of the selected bodies from shot to fix time and may be applied either graphically or mathematically.

Motion of the Body Correction

Motion of the body correction can be applied graphically by moving the assumed position eastward or westward for time. This is possible because the Greenwich hour angle (GHA) and the subpoint of the body move westward at the rate of 1° of longitude per 4 minutes of time. In the graphic method, a scheduled time of observation is given to each body. If shooting is off schedule, the following rules apply:

1. For every minute of time that the shot is taken early, move the assumed position 15′ of longitude to the east; for every minute of time that the shot is taken late, move the assumed position 15′ of longitude to the west.
2. When the latitude of the assumed position and the Zn of the body are known, the motion of the body can be computed mathematically. For 1 minute, the formula is: 15(cos lat)/(sin Zn). This correction is shown in tabular form in Figure 10-1.

Figure 10-1. Correction for motion of the body.

The National Imagery and Mapping Agency has published the Sight Reduction Tables for Air Navigation in a publication referred to as Pub. No. 249. These tables are published in three volumes. Volume 1, used by both the marine and air navigator, contains the altitude and azimuth values of seven selected stars for the complete ranges of latitude and hour angle of Aries. These seven stars represent the best selection for observation at any given position and time, and provide the data for presetting instruments before observation and for sight reduction afterwards. Volumes 2 and 3 cover latitudes 0-40 and 39-89 respectively and are primarily used by the air navigator in conjunction with observations of celestial bodies to calculate the geographic position of the observer.

In Publication No. 249, the local hour angle (LHA) increases 1° in 4 minutes of time. Thus, the Hc for an LHA that is 1° less than the LHA used for precomputation is the Hc for 4 minutes of time earlier than the solution time. The difference between the two Hcs is the value to apply to the Hc or Hs to advance or retard the LOP for 4 minutes of time. If the Hc decreases (Zn greater than 180°), the body is setting and the sign is minus (–) to advance the LOP if the value is applied to the Hs. If the Hc increases (Zn less than 180°), the body is rising and the sign is plus (+) to advance the LOP if the value is applied to the Hs.

In addition, motion corrections may be determined by using a modified MB-4 computer. This modification allows for greater accuracy and speed in computation of combined motions (motion of the observer and motion of the body) than the Pub. No. 249 tables.

Special Celestial Techniques

The main difference between the basic methods of precomputation is the manner in which the motion of the observer and the motion of the body corrections are applied. In the graphic method, both corrections are applied graphically by movement of the assumed position or the LOP. In the mathematical method, both corrections are applied mathematically to the Hc, the Hs, or the intercept after being obtained from tables, a modified MB-4 computer, or the Pub. No. 249.

Precomputation Techniques (Part Two)

Celestial Computation Sheets

The format in Figure 10-2 is a typical celestial precomputation and illustrates one acceptable method of completing a precomputation. The explanation is numbered to help locate the various blocks on the celestial sheets. [Figure 10-2]

Figure 10-2. Typical celestial precomputation format.

NOTE: Not all blocks apply on every precomputation.

1. DATE—place the Zulu date of the Air Almanac page used in this block.
2. FIX TIME—GMT (coordinated universal time) of the computation.
3. BODY—the celestial body being observed.
4. DR LAT LONG—the dead reckoning (DR) position for the time of the observation.
5. GHA—the value of GHA extracted from Air Almanac (10-minute intervals).
6. CORR—the GHA correction for additional minutes of time added to the GHA in block 5 and, if necessary, the 360° addition required establishing the LHA. SHA–When a star is precomped with Pub. No. 249, Volume 2 or 3, SHA is placed in this block.
7. GHA—corrected GHA (sum of blocks 5 and 6).
8. ASSUM LONG (–W/+E)—the assumed longitude required to obtain a whole degree of LHA.
9. LHA—LHA of the body (or Aries).
10. ASSUME LAT—the whole degree of latitude nearest the DR position.
11. DEC—the declination of the celestial body (not used with Pub. No. 249, Volume 1).
12. TAB Hc—the Hc from the appropriate page of Pub. No. 249, Volume 2 or 3.
13. D—the d correction factor found with previous Hc. Include + or –, as appropriate. The value is used to interpolate between whole degrees of Dec.
14. DEC—minutes of declination from block 11.
15. CORR—the correction from the Correction to Tabulated Altitude for Minutes of Declination table in Volume 2 or 3, using blocks 13 and 14 for entering arguments.
16. CORR Hc—this is the corrected Hc—sum of blocks 12 and 15 or extracted from Pub. No. 249, Volume 1.
17. Zn—true azimuth of the celestial body from the formula in Pub. No. 249, Volume 2 or 3, or directly from Volume 1.
18. TRACK—the true course (track) of the aircraft.
19. GS—the groundspeed of the aircraft.
20. ALT MSL—aircraft altitude.
21. CORIOLIS—the Coriolis correction extracted from Pub. No. 249, the Air Almanac, or a Coriolis/rhumb line table.
22. PREC/NUT—precession and nutation correction computed from the table in Pub. No. 249, Volume 1.
23. REL Zn or Zn—the difference between Zn and track, used to determine motion of the observer correction.
24. MOTION OF OBSERVER (MOO)—motion of the observer correction for either 1 minute (using 1-minute motion correction table) or 4 minutes (using 4-minute correction table in Pub. No. 249) of time.
25. MOTION OF BODY (MOB)—motion of the body correction for either 1 minute (using 1-minute motion correction table) or 4 minutes (using tabulated Hc change for 1° of LHA or 4-minutes correction table in Pub. No. 249) of time.
26. 4-MINUTE ADJUST—algebraic sum of 24 and 25; for use of 4-minute motion corrections extracted from Pub. No. 249.
27. X-Time—time in minutes between planned shot time and fix time.
28. TOTAL MOT ADJUST/ADV/RET—correction based on combined motion of observer and body, for the difference between the time of the shot and fix time. The sign of this correction is the same as the sign in block 26 if the observation was taken prior to the computation time. If it was taken later, the sign is reversed.
29. REFR—correction for atmospheric refraction.
30. PERS/SEXT—sextant correction or personal error.
31. SD—semidiameter correction for Sun or Moon.
32. PA—parallax correction for Moon observation.
33. POLARIS/Q CORR—the Q correction for the time of the Polaris observation (extracted from Pub. No. 249 or the Air Almanac).
34. Total ADJ—algebraic sum of blocks 28–33 as applicable.
35. OFF-TIME MOTION—motion adjustment for observation other than at planned time.
36. Ho—height observed (sextant reading).
37. INT—intercept distance (NM) is the difference between the final Hc and Ho. Apply the HOMOTO rule to determine direction (T or A) along the Zn.
38. LAT—polaris latitude.
39. CONV ANGLE (W/–E)—convergence angle used in grid navigation.
40. GRID Zn—the sum of blocks 17 and 39.

Corrections Applied to Hc

In some methods of precomputation, corrections are applied in advance to the Hc to derive an adjusted Hc. When using corrections that are normally applied to Hs, the signs of the corrections are reversed if applied to Hc. For example:

Corrections Applied to Hs

Corrections Applied to Hc

This demonstrates that corrections may be applied to either Hs or Hc. As long as they are applied with the proper sign, the intercept remains the same. The following sample precomp uses a common fix time (though computation times are different) and common observation times to facilitate comparison.

NOTE: Atmospheric refraction correction must be extracted for the actual Hs. It may then be applied to either Hc or Hs using the proper sign. Extracting the value for Hc may cause large errors, especially when the body is near the horizon. Figure 10-3 is a sample three-star precomputation using the mathematical format. Corrections to altitude of the body are applied to the Hc and the sign of the correction has been reversed in this process, so the fix can be plotted prior to the computation time. All shots are early shots, allowing the navigator to resolve the fix and alter at fix time. However, any minor errors in interpolation for motions are multiplied for the two earliest shots and may cause inaccuracies in the fix.

Figure 10-3. Mathematical solution.

Figure 10-4 shows a three-star precomputation using a three-LHA or graphical solution. The assumed position is then moved for track and GS to accommodate LOPs shot off time. Each observation is taken on time and then plotted out of its own plotting position. This precomp is easier and faster to accomplish with relatively few opportunities for math errors to occur. The three assumed positions required for this solution, on the other hand, often cause large intercepts and may make star identification difficult if care is not taken in choosing the precomp assumed position.

Figure 10-4. Graphical solution.

Limitations

Precomputational methods lose accuracy when the assumed position and the actual position differ by large distances. Another limiting factor is the difference in time between the scheduled and actual observation time. The motion of the body correction is intended to correct for this difference. The rate of change of the correction for motion of the body changes very slowly within 40° of 090° and 270° Zn, and the observation may be advanced or retarded for a limited period of time with little or no error. When the body is near the observer’s meridian, however, the correction for motion of the body changes rapidly due in part to the fast azimuth change and it is inadvisable to adjust such observations for long (over 6 minutes) periods of time.

NOTE: Errors in altitude and azimuth creep into the solution if adjustments are made for too long an interval of time. Because of these errors, the navigator should attempt to keep observation time as close as possible to computation time.

Preplotting True Azimuth (Zn)

To speed up fix resolution, some navigators preplot the Zn of the bodies. This technique works best when used on a constant scale chart and using a technique of precomputation that gives one assumed position. Before making any observations, plot the assumed position, correct it for Coriolis and precession and/or nutation (if required), and draw the Zn of the bodies through this point. Label each Zn as the 1st, 2d, or 3d as shown in Figure 10-5, or use the name of the bodies. Use arrowheads to identify the direction of the body. Suppose the corrected assumed position is 30°40′ N, 117° 10′ W and the following Zn were computed for the bodies:

1st shot ZN 020°
2d shot ZN 135°
3d shot ZN 270°

Figure 10-5. The fix can be plotted quickly.

The original assumed position of 31° N; 117° 08′ W has been corrected for precession and/or nutation and for Coriolis or rhumb line error to obtain the plotting position. When the first intercept is found to be 10A, second intercept 40A, and the third intercept 50T, the fix may be plotted quickly by constructing perpendicular lines at the correct point on the respective Zn line. This greatly reduces the time necessary to plot the fix.