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Flight Navigation

Daytime Celestial Techniques

Filed Under: Special Celestial Techniques

Daytime fixing, using celestial techniques, is rather limited because often only one body, the sun, is visible. Ordinarily, three LOPs cannot be obtained for a fix from one body, because the LOPs plot nearly parallel to each other.

The Sun Heading Shot at High Noon

The azimuth of the sun changes very rapidly when the subpoint of the sun is directly over the longitude of the observer, which is called the time of transit. The LHA at transit time is 360°. This phenomenon is more pronounced at lower latitudes as the subpoint of the sun passes closer to the observer. This makes it extremely difficult to get an accurate celestial heading shot at the transit time. Therefore, if you need a heading shot near the time of transit, you must take extra precaution to get the heading observation exactly at the precomputed fix time. If the moon or Venus is available, consider using these bodies for an accurate celestial heading. If using the sun, you should weigh the increased possibility of an inaccurate heading shot. If the accuracy is questionable, get another heading shot as the sun’s rate of azimuth change slows enough to allow a more accurate shot.

 

Intercept Method

The intercept method is normally used in obtaining a noon day fix. If the sun passes close to the observer’s position, within about 4°, the subpoint method of plotting the fix may be used. This method differs from normal procedures in that three different precomps for three different times are computed. Because of the rapid change of the sun’s azimuth at or near transit, this variation is necessary. The procedure is:

  1. Determine the time of transit.
  2. Select the LHA before and after transit for which the change in azimuth is 30° or more. Since 1° of LHA is equal to 4 minutes of time, the difference in transit LHA and the new LHA can be converted to time in minutes. Thus, the time preceding and following transit can be determined.
  3. Plot the DR positions for times determined in 12.7.2. Select the appropriate assumed positions necessary for the computation and plotting of the LOPs. The assumed position for time of transit is also plotted.
  4. Determine the intercepts and azimuth for each LOP. Plot these data from the respective assumed positions.
  5. Resolve the LOPs to a common time, preferably that of the transit LOP.

NOTE: At 30° N latitude, the linear speed of the sun is approximately 780 knots. Thus, on westerly headings in highspeed aircraft, the DR distance involved before encountering a 30° change in azimuth is considerable.

Subpoint Method

When the observer is within approximately 4° of the subpoint of the body, the subpoint method of solution is normally used. This is because the radius of the circle of equal altitude is so small that a straight line does not approximate the arc and a straight line does not give an accurate LOP. The procedure is:

  1. Plot the subpoints of the body for the time of the observations (using GHA and/or Dec).
  2. Find the co-altitude of the shots and convert it to NM (90° – Alt × 60 NM).
  3. Advance the first subpoint and retard the third along the DR track, using best-known track and GS.
  4. Set the distance found from the co-altitude and strike it off from the resolved subpoints (with a compass or pair of dividers). Do this for each observation.

NOTE: The resulting intersection, or triangle, gives one ontime fix. If the LOPs form a triangle, the aircraft position is probably within the triangle.

The subpoint method is convenient because Pub. No. 249 is not used—only the Air Almanac. This method can also be used with a star near your assumed position and may be necessary if, for some reason, your Volume 1 is unavailable. The stars Dec and GHA are needed to determine if the observer is within 4° of the subpoint. The Air Almanac may be used to find the Dec and sidereal hour angle (SHA) of the star. The SHA of the star is added to the GHA of Aries to find the GHA of the star.

Eliminating Motions with the Bracket Technique

For sun observations, you can eliminate motion calculations by using a shooting schedule of 3 minutes early, on fix time, and 3-minutes late. With this schedule, the 3-minute early and 3-minute late shots have the same magnitude of motion but an opposite sign. Therefore, these motions cancel each other out and do not need to be computed. The on-time shot has no motions. Therefore, the three intercepts can be averaged for a single LOP. At night, shooting the same star 4 minutes early and late, with a different star shot on time, can employ a similar method. In this case, the intercepts for the same star’s 4-minute early or late shots can be averaged. This reduces workload, but only two LOPs are obtained.

DR Computer Modification

Rather than eliminating motions, your DR computer can be modified so both observer and body motions can be computed at one time, without entry into the Pub. No. 249. Make a GS and latitude scale. [Figure 12-8] After constructing these, the DR computer can be modified for quick and accurate computations of 1-minute motion adjustments.

Figure 12-8. MB-4 motions modification.
Figure 12-8. MB-4 motions modification.

Tape the GS scale (0 through 900) along the centerline of the grid scale. Match zero to zero, 300 to 50, and 600 to 100 as shown in Figure 12-8. Then, tape the latitude scale along the zero grid line so that 90° falls on the centerline and the scale extends to the left as shown. Check the accuracy of your placement: 30° latitude should fall 13 divisions left of centerline. Juggle the scale as necessary to provide the greatest accuracy between 30° and 45°.

To use the modified MB-4 computer for motion adjustments:

  • Set true north under the index. If computing for grid, set polar angle (PA) under the index. In the NW and SE hemisphere quadrants, PA equals convergence angle (CA). In the NE and SW quadrants, PA = 360 – CA. Next, place the grommet over the zero grid line. Mark a cross (+) at the assumed latitude. [Figure 12-9]

    Figure 12-9. Celestial motions–step one.
    Figure 12-9. Celestial motions–step one.
  • Set track (or grid track) under the index and position the slide so the GS is under the grommet. Place a dot on the zero point of the grid scale. [Figure 12-10]

    Figure 12-10. Celestial motions–step two.
    Figure 12-10. Celestial motions–step two.
  • Place the Zn (or grid Zn) of the body under the index. Position the slide so the cross or the dot, whichever is uppermost, is on the zero line of the grid. [Figure 12-11]

    Figure 12-11. Celestial motions–step three.
    Figure 12-11. Celestial motions–step three.
 

NOTE: The vertical distance between the zero line and the low mark is the combined 1-minute motion. Each line of the grid equals 1 minute of arc (1 mile). If the cross is on the zero line, the motion is positive. If the dot is on the zero line, the motion is negative. When solving for motions using grid, all directions must be grid directions.

EXAMPLE: Given the following information, find the combined 1-minute motion adjustment.

Assumed Latitude45° 10′ N
True Track270°
GS240 knots
True Zn171°
Answer+1′

Combinations of Sun, Moon, and Venus

The moon or Venus is often visible during daylight hours and can be used to obtain an LOP. Always consider fixing using these bodies during daylight celestial flights. When planning the flight, use the sky diagrams in the Air Almanac to determine the availability of the moon and Venus. If the bodies are available, they can be readily found by accurately precomputing their altitudes and azimuths.

When looking for Venus, take all the filters out of the sextant and point it at the precise location of the planet. A bright, small pinpoint of light is visible but hard to detect, unless sky conditions and separation from the sun are ideal. With practice, acquisition should become easier and you will be familiar with those conditions conducive to successfully making a Venus shot.

During the day when the sun is high, the moon or Venus, if they are available, can be used to obtain compass deviation checks. In polar regions during periods of continuous twilight, the moon and Venus are available if their Dec is the same name as the latitude.

Duration of Light

Sunrise and sunset at sea level and at altitude, moonrise and moonset and semiduration graphs will not be discussed in detail in this chapter. It is imperative; however, to preplan for any flight where twilight occurs during the course of the flight, especially at the higher latitudes where twilight extends over longer periods of time. An excellent discussion, with appropriate examples, is provided in the Air Almanac and should be sufficient for those missions requiring detailed planning.

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LHA Method of Fixing

Filed Under: Special Celestial Techniques

LHA Method of Obtaining Three-Star Fix

The LHA technique allows you to solve the motion problem for a three-star fix by applying a correction to the assumed position rather than computing a numerical solution on the precomp. This eliminates mathematical motion calculations, therefore reducing the chance of math errors on the precomp. To accomplish a three-LHA fix, you must plan 4 minutes between the midtime of each shot. [Figures 12-4 and 12-5] Because LHA changes 1 degree for every 4 minutes, the precomp has three successive LHAs, 1 degree apart. To correct for off-time motion, adjust the assumed position based on true course (TC) and groundspeed (GS). If a shot is planned earlier than fix time, the assumed position is advanced (down-track). For shots planned later than fix time, the assumed position is retarded (up-track).

Figure 12-4. Typical example of the three LHA method.
Figure 12-4. Typical example of the three LHA method. [click image to enlarge]
Figure 12-5. Plotting three LHA.
Figure 12-5. Plotting three LHA.

The example in Figures 12-4 and 12-5 shows the LHA method for a 12-8-4 early shooting schedule. This shooting schedule allows the fix and/or MPP to be resolved before the fix time. To adjust the assumed positions, plot the fix time assumed position and then advance it for 4 minutes of track and GS for each body. This satisfies motion of the observer. When shooting the selected bodies, take care to shoot them exactly on the prescribed times. This eliminates motion of the body.

A variation of advancing the assumed position is to use half motions. This enables you to plot all three LOPs from one assumed position. Table 1 from Pub. No. 249 lists corrections to position of the observer. Each correction is for 4 minutes of time. To use it, enter with your relative Zn (Zn-track) and GS. Now, look at the bottom of the table and note you can apply this correction to your tabulated altitude or observed altitude. It does not matter which you choose, but note that the sign changes dependent on where you apply it. Now, take the number and multiply it by the 4-minute increment of the shot. For example, Figure 12-6 shows the precomp for a 0300 fix using 3 LHAs and half motions. The 0248 shot, Alpheratz, relative Zn, and groundspeed were used to extract a +20 correction from Table 1. Because this shot is 12 minutes early, we need to multiply +20 by three before we apply it to the shot. Note the +60 correction was applied to the observed altitude and, therefore, kept its positive sign. The benefit of doing this is a reduction in plotting. See Figure 12-7 for the plotted LOPs. This technique can be applied to day celestial as well.

Figure 12-6. Half motions three LHA format.
Figure 12-6. Half motions three LHA format.

Figure 12-7. Plotting a half motions observation.
Figure 12-7. Plotting a half motions observation. [click image to enlarge]


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Introduction to Special Celestial Techniques

Filed Under: Special Celestial Techniques

Determining Availability of Celestial Bodies

By doing a quick comparison of Greenwich hour angle (GHA) to the observer’s position, it is easy to determine the availability of celestial bodies. For example, the observer anticipates being at 18°N 135° W at 0015Z on 28 September 1995. There are several bodies listed in the Air Almanac, but not all of them are available for observation. To determine availability, take the observer’s longitude and look 80° either side of it. Within this range, compare the GHA of a body. Looking at Figure 12-1, we see that the sun, moon, Venus, and Jupiter are within the 80° range and are therefore usable. Saturn is outside of the 80° range, so it is not usable. The declination (Dec) of a body is not normally a factor; however, at high latitudes a body may not be available when its subpoint is near the pole opposite the observer.

Figure 12-1. A quick check of body availability.
Figure 12-1. A quick check of body availability. [click image to enlarge]

Latitude by Polaris

Polaris is the polestar, or North Star. Because Polaris is approximately 1° from the North Pole, it makes a small diurnal circle and seemingly stays in about the same place all night. This fact makes Polaris very useful in navigation. With certain corrections, it serves as a reference point for direction and for latitude in the Northern Hemisphere. Latitude by Polaris is a quick method of obtaining a latitude line of position (LOP); only the tables given in the Air Almanac are needed.

Obtaining Latitude by Polaris

A latitude by Polaris LOP is obtained by applying the Q correction to the corrected observed altitude. [Figure 12-2] This adjusts the altitude of the pole, which is equal to the navigator’s latitude. The Q correction table is in the back of the Air Almanac. The entering argument for the table is exact local hour angle (LHA) of Aries. The effect of refraction is not included in Q correction, so the observed altitude must be fully corrected. When refraction is used for a latitude by Polaris LOP, it is applied to the observed altitude and the sign of the correction is negative. A Polaris LOP can also be plotted using the intercept method. In this case, the Hc is computed by reversing the sign of the Q correction and applying it to the assumed latitude (rounded off to the nearest degree). Refraction is positive when applied to get an Hc for the intercept method.

Figure 12-2. Polaris Q correction and azimuth tables from the Air Almanac.
Figure 12-2. Polaris Q correction and azimuth tables from the Air Almanac. [click image to enlarge]

Obtaining Azimuth of Polaris

For either method, the azimuth of Polaris is obtained from the Azimuth of Polaris table found in the Air Almanac or in the Pub. No. 249. [Figure 12-2] Whether plotted as an intercept or a latitude, the assumed position should be corrected for Coriolis, or rhumb line, and precession, or nutation. The resulting LOPs should fall in the same place for either method. To plot the LOP using the latitude method, choose the longitude line closest to the DR and plot perpendicular to the longitude line. For the intercept method, use the assumed latitude and plot the intercept normally using the azimuth of Polaris.

 

Latitude by Polaris Example

On 18 April 1995 for Greenwich mean time (GMT) 1600 at 23° 10′ N 120° W, with an observed altitude 23° –06′ at 31,000′. When doing a latitude by Polaris you must use the exact latitude and longitude. See Figure 12-3 for plotting.

GHA086° –18′
Longitude (West)–120° –00′
LHA326° –18′
True Course (TC)090°
Groundspeed (GS)400 knots
Coriolis/rhumb line7R
Corrected Observed Altitude23° –06′
Q (based on LHA 072-44)–15′
Refraction–01′
Latitude22° –50′
Azimuth (LHA 326° –18′, Latitude 23° N)000.8°
Figure 12-3. Plotting the Polaris LOP.
Figure 12-3. Plotting the Polaris LOP.

NOTE: If the Q correction table in Volume 1 is used, precession and nutation (P/N) and Coriolis, or rhumb line, must be used in plotting the LOP. This is because the Pub. No. 249 covers a 5-year period, and the further the years get from the Epoch year, the greater the error is when using the Polaris table. P/N compensates for this error.

Intercept Method Example

Refer to the previous problem and Figure 12-3 for plotting. NOTE: Applying 10A to assumed latitude gives 22° –50′ N, which is the same the answer in the latitude by Polaris example.

Azimuth of Polaris359.5
Coriolis/rhumb line7R
Assumed Lat (rounded off)23° –00′ N
Q (reversed sign)+15′
Refraction+01′
Hc Polaris23° –16′
Ho Polaris23° –06′
Intercept10A

NOTE: In these examples, all information was taken from the Air Almanac. No P/N is required.

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Conversion of LOPs to a Common Time (Part Two) Planning the Fix

Filed Under: Plotting and Interpreting the Celestial Line of Position

Planning the Fix

In selecting bodies for observation, one should generally consider azimuth primarily and such factors as brightness, altitude, etc., secondarily. If all observations were precisely correct in every detail, the resulting LOPs would meet at a point. However, this is rarely the case. Three observations generally result in LOPs forming a triangle. If this triangle is not more than 2 or 3 miles on a side under good conditions and 5 to 10 miles under unfavorable conditions, there is normally no reason to suppose that a mistake has been made. Even a point fix, however, is not necessarily accurate. An uncorrected error in time, for instance, would require the entire fix to be moved eastward if observations were early and westward if observations were late, at the rate of 1 minute of longitude for each 4 seconds of time.

In a two-LOP fix, the ideal cut of the LOPs is 90°. In Figure 11-12, a 90° cut with a 5 NM error in one LOP causes a 5 NM error in the fix. If the acute angle between the LOPs is 30°, a 5 NM error in one LOP causes a 10 NM error in the fix. Thus, with a two-LOP fix, an error in one LOP causes at least an equal error in the fix; the smaller the acute angle between the LOPs, the greater the fix error caused by a given error in one LOP. Of course, if both LOPs are in error, the fix may be thrown off even more. In a three-LOP fix, the ideal cut of the LOPs is 60° (star azimuths 120° apart). With this cut, a 3 NM error in any one LOP causes a 2 NM error in the fix. With any other cut, a 3 NM error in any one LOP causes more than a 2 NM error in the fix. In a three-star fix, the cut will be 60° if the azimuths of the stars differ by 60° or if they differ by 120°. If there is any unknown constant error in the observations, all the Hos will be either too great or too small.

Figure 11-12. Effect of cut on accuracy of a fix.
Figure 11-12. Effect of cut on accuracy of a fix. [click image to enlarge]
Notice in Figure 11-13 that, if stars are selected whose azimuths differ by 120°, this constant error of the Hos causes a displacement of the three LOPs, either all toward the center or all away from the center of the triangle. In either case, the position of the center of the triangle is not affected. If you use any three stars with azimuths outside a 180° range, any constant error in observations tends to cancel out.

Figure 11-13. Effect of azimuth on accuracy of fix.
Figure 11-13. Effect of azimuth on accuracy of fix.

The three-star fix has two distinct advantages over the two-star fix. First, it is the average of three observations. Second, selecting the stars carefully can counteract the effect of constant errors of observation. There is also a third advantage. Each pair of two LOPs furnishes a rough check on the third. In resolving an observation into a LOP, you might possibly make a gross error; for example, obtaining an LHA that is in error by a whole degree. Such an error might not be immediately apparent. Neither would such a discrepancy come to immediate attention in a two-LOP fix. However, this third advantage does not apply when a single LHA is used in solving all LOPs, such as is done when precomputing and using motion corrections to resolve all LOPs to a common time. Because of these three advantages, it is evident that a three-star fix should be used, rather than a two-star fix, when possible.

Whatever the number of observations, common practice, backed by logic, is to take the center of the figure formed unless there is reason for deviating from this procedure. Center is meant as the point representing the least total error of all lines considered reliable. With three LOPs, the center is considered that point within the triangle equidistant from the three sides. It may be found by bisecting the angles, but is usually located by eye.

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Conversion of LOPs to a Common Time (Part One)

Filed Under: Plotting and Interpreting the Celestial Line of Position

Moving the LOP

One method of converting LOPs to a common time is to move the LOP along the best-known track for the number of minutes of GS necessary for the time conversions. This method is similar to that used in correcting for Coriolis or rhumb line and precession or nutation. For example, suppose the track is 110° and the GS is300 knots. LOPs are for 1500, 1504, and 1508, and a fix is desired at 1508. This means the 1500 LOP must be moved to the time of the fix using the track and 8 minutes of the best known GS. The 1504 LOP must be moved to the time of the fix using the track and 4 minutes of GS. The 1508 LOP is already at the fix time, so it requires no movement. Figure 11-9 shows the method of conversion as it is completed on the chart.

Figure 11-9. Conversion of lines of position to a common time.
Figure 11-9. Conversion of lines of position to a common time.

If, at any time, the LOP has to be retarded (moved back) to the time of the fix, use the following procedures. Using the reciprocal track and GS, obtain the correction in the regular manner for the number of minutes of difference. For example, suppose the fix is at 1800 and the last shot is at 1802.

Retarding the LOP 2 minutes of GS on a track of 70° would be the same as advancing it 2 minutes of GS on a track of 250°.

Motion of Observer Tables

A second method of conversion of LOPs to a common time is with a Motion of the Observer table such as the one in Pub. No. 249. This table gives a correction to be applied to the Ho or Hc so that the LOP plots in its converted position. The correction obtained from Table 1 in all volumes of Pub. No. 249 is for 4 minutes of time. An additional table allows you to get the correction for the number of minutes needed. For example, suppose the LOP needs to be advanced for 11 minutes and the Ho of the body is 33° 29′ and Zn is 080°.

The track of the aircraft is 020o and the GS is 240 knots. In Table 1, Correction for Motion of the Observer for 4 minutes of Time [Figure 11-10], the entering arguments is Rel Zn and GS. Rel Zn is azimuth relative to course (Zn minus track or track minus Zn). Subtract the smaller angle from the larger and enter the table with the answer. In this case, Zn – track = 080° – 020° = 060° (Rel Zn) and GS is 240 knots. Entering this table with these arguments, the correction listed is +08′ for 4 minutes of time.

Figure 11-10. Entering arguments are relative true azimuth and groundspeed.
Figure 11-10. Entering arguments are relative true azimuth and groundspeed. [click image to enlarge]
Figure 11-10. Entering arguments are relative true azimuth and groundspeed (continued). [click image to enlarge]
Use the whiz wheel to calculate the total motion for 11 minutes. In this case, the 11-minute correction totals 22′. By applying any other correction (refraction, sextant correction, etc.), a total adjustment is derived. By changing the sign, this total may be applied to the Hc. To apply the correction to the Ho, the sign of the adjustment would remain the same. Apply the adjustment to the intercept as the rules state in Table 1. In each case, the resultant intercept would be the same.

Suppose the Hc was 33° 57′. Applying the correction –22 yields 33° 35′. Comparing this with our Ho 33°29′ results in an intercept of 6 NM away. If you decide to apply the correction to the Ho, 33° 29′ + 22′ yields 33° 5l’. Comparing this to the Hc 33° 57′ yields the same result, 6 NM away. When using the Motion of the Observer table and when the fix time is earlier than the observation (LOP to be retarded), the rule for the sign of the correction is also printed below Table 1.

 

Moving the Assumed Position

Another method of converting LOPs to a common time is to move the assumed position. This method is recommended for shots 4 minutes apart computed to give all three bodies a single assumed position. However, it is not limited to that type of computation. The assumed position is moved along the best-known track at the best-known GS. For example, again suppose the track is 330° and the GS 300 knots. LOPs are for 1500, 1504, and 1508 and a fix is desired at 1508. [Figure 11-11] Since the first LOP would have to be advanced 40 NM (8 minutes at 300 knots), the same result is realized by advancing the assumed position 40 NM parallel to the best-known track. The 1504 LOP must be advanced 20 NM; therefore, the assumed position is advanced 20 NM miles parallel to the best-known track. The third shot requires no movement, and it is plotted from the original assumed position.

Figure 11-11. Moving assumed positions.
Figure 11-11. Moving assumed positions.

It should be noted that the first shot is always plotted from the assumed position, which is closest to destination. In this method, if observations are precomputed and the assumed position is moved prior to shooting, the following procedure is used when shooting is off schedule. For every minute of time that the shot is taken early, move the assumed position 15 minutes of longitude to the east. For every minute of time that the shot is taken late, move the assumed position 15 minutes of longitude to the west. In addition, the affected LOP must be moved along the best-known track for the number of minutes of GS the observation was early or late. If the shot was early, advance the LOP; if the shot was late, retard the LOP.

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Interpretation of an LOP

Filed Under: Plotting and Interpreting the Celestial Line of Position

Navigation has two aspects—the mechanical and the interpretive. The mechanical aspect includes operation and reading of instruments, simple arithmetical calculations, plotting, and log keeping. The interpretive aspect is the analysis of the data that have been gathered mechanically. These data are variable and subject to error. You must convert them into probabilities as to the position, track, and GS of the aircraft and the direction and speed of the wind. The more these data are subject to error, the more careful the interpretations must be and the less mechanical the work can be. LOPs and fixes especially require careful interpretation. It is convenient to think of a fix as the true position of the aircraft and of the LOP as a line passing through this position, but these definitions are optimistic. It is almost impossible to make a perfect observation and plot a perfect LOP. Therefore, a LOP passes some place near this position, but not necessarily through it, and a fix determined by the intersection of LOPs is simply the best estimate of this position on the basis of one set of observations. In reality, a fix is a most probable position (MPP) and a LOP is a line of MPP.

The best interpretation of LOPs and fixes means they are used, to the best advantage, with DR. But good interpretation cannot compensate for poor LOPs, nor can good LOPs compensate for careless DR. To get good results, every precaution must be taken to ensure the accuracy of LOPs and exact DR calculations.

Intelligent interpretation requires fine judgment, which can be acquired from experience. You can be guided, however, by certain well-established, though flexible, rules. The following discussion pertains especially to celestial LOPs and fixes. It also applies to LOPs and fixes established by radio and, to some extent, to those obtained by map reading.

Single LOP

Previous discussions dealt with the basic plotting of a LOP and errors in LOPs, but they did not show the actual mechanics of the plotted corrections that must be applied. The LOP must be corrected for Coriolis or rhumb line correction and also for precession and/or nutation correction if it is based on a Volume 1 star shot. Coriolis or rhumb line correction becomes a very significant correction at higher speeds and latitudes. For example, suppose the correction determined from the Coriolis or rhumb line correction table is 9 NM right (of the track). The LOP must be moved a distance of 9 NM to the right of track. This can be done either by moving the assumed position prior to plotting or by moving the LOP itself after it is plotted. (Remember, the assumed position is not used in the plotting of the LOP obtained from a Polaris observation.) Consider Figure 11-5, which shows a track of 90°. Notice that, in both methods, the corrected LOP is in the same place with respect to the original assumed position and that the intercept value is the same. The resultant LOP is the same regardless of the method used.

Figure 11-5. Two methods of coriolis/rhumb line correction.
Figure 11-5. Two methods of coriolis/rhumb line correction. [click image to enlarge]
If, in addition to the Coriolis or rhumb line correction, a precession and/or nutation correction of 3 NM in the direction of 60° is required, it would have been further applied as shown in Figure 11-6. Again, the corrected LOP is the same, using either method, because the intercept and resultant position of the corrected LOP to the original assumed position are the same. The corrected LOP alone gives very little information; hence, a position must be arrived at only after considering the LOP and the DR position for the same time.

Figure 11-6. Two methods of coriolis/rhumb line and precession/nutation correction.
Figure 11-6. Two methods of coriolis/rhumb line and precession/nutation correction. [click image to enlarge]

Most Probable Position (MPP) by C-Plot

The MPP is just what the name implies. It is not a fix; however, since it is the best information available, it is treated as such. Notice in Figure 11-7A that the DR position and celestial LOP (for the same time) do not coincide.

Figure 11-7. Most probable position by C-plot.
Figure 11-7. Most probable position by C-plot. [click image to enlarge]
Obviously, the DR information, or celestial information, or both is in error. Notice that the prior fix has no time on it. Suppose this prior fix had been for the time of 1010. It would then be very likely that most of the error is in the celestial information and the probable position is closer to the DR position than to the celestial LOP. On the other hand, suppose the prior fix had been for the time of 0900. Since the accuracy of the celestial information is unaffected by the time from the last fix, it would, in this case, be most likely that the actual position is closer to the LOP than to the DR position.

 

A formula has been devised to position the observer along the perpendicular to the LOP according to the time factor. The formula is:

where t is time in minutes, p is the perpendicular distance between the DR position and the LOP, and d is the distance from the DR position for the time of the MPP measured along the perpendicular to the LOP. Look at Figure 11-7B and C and see how the formula works for the two problems cited above if the perpendicular is 20 NM in length. In Figure 11-7B, t is 15 minutes and p is 20 NM, so the MPP would be located along the perpendicular about 8½ NM from the DR position.

Now, consider Figure 11-7C where t is 1 hour 25 minutes or 85 minutes, p is 20 NM and, in this case, the MPP would be over 16 NM away from the DR position along the perpendicular to the LOP.

If you prefer not to use the formula, a simple table can be easily constructed to solve for d with entering arguments of t and p. [Figure 11-8] The table could easily be enlarged to handle larger values of t and p. In most fixes, the DR position is so close to the LOP that the midpoint between these two can be considered the MPP. A good rule to use is to take the midpoint of the perpendicular if the total distance between the DR position and the LOP is 10 NM or less. If the value of p is greater than 10 NM, use a table or the formula to determine the MPP. Up to this point, determination of the MPP has been rather mechanical. Experienced navigators frequently further adjust the position of the MPP for other factors not yet considered. For example, if the LOP is carefully obtained under good conditions or if it is the average of several LOPs, you may further weight the MPP in the direction of the LOP by an amount that judgment dictates. However, the reverse may be true if the LOP is obtained under adverse conditions of rough air. In the latter case, you might move the MPP closer to the DR position by some amount determined by sound judgment.

Figure 11-8. To solve for distance.
Figure 11-8. To solve for distance.

Further, consider the validity of the DR position in relation to factors other than time. A DR position at the end of 40 minutes would be more reliable with Doppler drift and GS versus one based on metro information. These factors may also adjust the original MPP closer to or farther away from the DR position, along the perpendicular. However, these last mentioned factors are judgment values that come only with experience. In fact, with experience you may mentally calculate all the factors involved and arrive at the final position of the MPP without recourse to a formula or table.

Finding a Celestial Fix Point

Up to this point, only the single celestial LOP and what to do with it have been considered. Now, the celestial fix should be considered. To establish a fix, two or more LOPs must be obtained. Since, in most cases, two or more LOPs cannot be obtained simultaneously, they must be converted to a common time. For example, a LOP obtained at 1010 must be converted to the LOP obtained at the fix time of 1014. There are several methods for making this conversion that are discussed in this chapter. Consideration is also given to the planning of the fix and the final interpretation of the fix itself.

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Methods of Plotting and Interpreting the Celestial Line of Position

Filed Under: Plotting and Interpreting the Celestial Line of Position

Subpoint Method

A detailed explanation of the theory concerning the subpoint method is in the Computing Altitude and True Azimuth category, and in the Celestial Precomputation category. [Figure 11-1] Following is a summary of the steps involved:

  1. Positively identify the body and measure the altitude using a sextant.
  2. Because no tabulated information for azimuth or elevation is required for this method, corrections for refraction, parallax, semidiameter, wander error, and sextant correction are applied directly to the Ho.
  3. The resultant measurement is subtracted from 90° to obtain the co-altitude (co-alt). To convert to NM (1°= 60 NM), multiply the number of degrees times 60. Any fractional portion of degrees is added to the previous value.

Figure 11-1. The subpoint method.
Figure 11-1. The subpoint method. [click image to enlarge]

Example: Vega is observed at an altitude (Ho) of 88° 23′. Sextant correction is –03′.

88° 23′ – 03′ = 88° 20′
90° – 88° 20′ = 1° 40′
1° 40′ = 60′ + 40′ = 100 NM

In this example, 100 NM represents the distance from the observer’s position to the subpoint of the body. The coordinates of the body are its corresponding declination (Dec) and Greenwich hour angle (GHA). For this example, Vega’s Dec is N38° 46′. The GHA is obtained by applying the sidereal hour angle (SHA) of Vega to the GHA of Aries.

Example:

SHA = 080° 59′
GHA Aries = 039° 18′
GHA Vega = 120° 17′

Subpoint of Vega is located at 38° 46′ N l20° 17′ W. The observer is now ready to apply the information:

  1. Plot the subpoint on an appropriate chart.
  2. With dividers or compass, span the co-alt distance; in this case 100 NM.
  3. Use the body’s subpoint (38° 46′ N l20° 17′ W) as the center and 100 NM (co-alt) as the radius. The circle is called the circle of equal altitude and the observer is located on that portion of the circle nearest the dead reckoning (DR) position. There are definite advantages to this method. It requires no precomputation values and plotting is very simple if the observer and body are reasonably close together. When the observer and body are separated by great distances, some disadvantages appear.
  4. If a body is observed at 20° above the horizon, the observer is 4,200 NM from its subpoint. To swing a LOP from this subpoint, the subpoint and the arc must be plotted on the same chart. To permit plotting of any LOP, the chart must cover an area extending more than 4,000 miles in every direction from the DR position. This means that the chart must be either of such large size that it cannot be spread out on a table in the aircraft, or of such small scale that plotting on it is inaccurate. To cover an area 8,000 miles across, a chart 4 feet square must be drawn to a scale of about 1:10,000,000. Furthermore, measuring would be difficult because of distortion.
  5. Since a celestial LOP cannot always be drawn by the subpoint method, the intercept method, based on the same principles, is often used.
 

Intercept Method

You can eliminate the need for plotting the body’s subpoint and still draw the arc representing the circle of equal altitude. [Figure 11-2] By using the following formula, you can calculate the altitude and azimuth of the body for the DR position:

Hc = SIN-1 [SIN (DEC’) SIN (LDr) +COS (DEC’)COS (LDr) COS (LHA)]
Z = COS (Z) = [SIN (DEC’) – SIN (LDR) SIN (HC)]/[COS (Hc COS (LDr)]
Zn = Z if SIN (LHA) < 0
Zn = 360 – Z if SIN (LHA) > 0

Figure 11-2. Line of position computed by intercept method.
Figure 11-2. Line of position computed by intercept method. [click image to enlarge]
The calculations may be performed quickly using a programmable calculator, or they may be extracted from the appropriate volume of the National Imagery and Mapping Agency’s Sight Reduction Tables for Air Navigation in a publication referred to as Pub. No. 249. This method enables the observer to use any of the navigational bodies available at the appropriate fix time. Here is a brief review:

  • Compute a DR for the time of the position, using preflight or inflight data.
  • Determine the necessary entering values for the Pub. 249 volume being used (Lat, LHA, Dec contrary, or same) and extract all the necessary values of computed altitude (Hc) and azimuth angle (Z).
  • After making all the necessary conversions and corrections (Chapter 10), compare the Ho and corrected Hc. This difference is the intercept. If the Ho equals the corrected Hc, then the circle of equal altitude passed through the plotting position. If the Ho is greater than the Hc, the difference is plotted in the direction of the true azimuth (Zn). The Zn represents the azimuth from the observer’s position to the subpoint. If the Ho is less than the Hc, plot the difference 180° from the Zn.
  • NOTE: If HO is MOre, plot TOward the subpoint (HO MO TO)

Example: The assumed position is 38° N, 121° 30′ W for a shot taken at 1015Z on Aldebaran. The Ho is 32° 14′. The Hc is determined to be 32° 29′ and the Zn is 120°. A comparison of Ho and Hc determines the intercept to be 15 NM away (15A).

 

Plotting LOP Using Zn Method

  1. Plot the assumed position and set the intercept distance on the dividers. [Figure 11-3]
  2. Draw a dashed line through the assumed position toward the subpoint.
  3. Span intercept distance along dashed Zn line.
  4. Place plotter perpendicular to Zn.
  5. Draw LOP along plotter as shown in Figure 11-3.
Figure 11-3. Celestial line of position using true azimuth method.
Figure 11-3. Celestial line of position using true azimuth method. [click image to enlarge]

Plotting LOP Using Flip-Flop Method

  1. Plot the assumed position and set the intercept distance on the dividers. [Figure 11-4]
  2. Measure 120° of the Zn with point A of the dividers on the assumed position and place point B of the dividers down. In this case, away from 120° or in the direction of 300° from the assumed position. Slide the plotter along the dividers until the center grommet and the 100/200-mile mark are lined up directly over point B of the dividers marking the intercept point.
  3. Remove point A of the dividers from the assumed position, keeping point B in place. Flip point A (that was on the assumed position) across the plotter, at the same time expanding the dividers so that point A can be placed on the chart at the 90°/270° mark of the plotter.
  4. Flop the plotter around and place the straight edge against the perpendicular, which is established by the dividers.
  5. Draw LOP along the plotter as shown in Figure 11-4.

Figure 11-4. Plotting celestial line of position using flip-flop method.
Figure 11-4. Plotting celestial line of position using flip-flop method. [click image to enlarge]
When using the intercept method, remember:

  • For some assumed position near the DR position, find the Hc and Zn of this body for the time of the observation. This is done with the aid of celestial tables, such as Pub. No. 249 or a programmable calculator.
  • Obtain needed corrections, sextant correction, refraction, etc., and apply these to the Hc by reversing the sign. Remember, we are striving to derive a precomputed value to ensure the correct body is shot. Measure the altitude (Ho) of the celestial body with the sextant and record the midtime of the observation.
  • Find the intercept, which is the difference between Ho and Hc. Intercept is toward the subpoint if Ho is greater than Hc, and away from the subpoint if Ho is smaller than Hc.
  • From the assumed position, measure the intercept toward or away from the subpoint (in the direction of Zn or its reciprocal) and locate a point on the LOP. Through this point, draw the LOP perpendicular to the Zn.
 

Additional Plotting Techniques

The preceding techniques involve the basic plotting procedures used on most stars and the bodies of the solar system. However, there are certain techniques of plotting that are peculiar to their own celestial methods; for example, the plotting of LOPs obtained by using Polaris, which is discussed later. Also, certain precomputation techniques lend themselves more readily to other plotting techniques, such as preplotting the true azimuths or plotting the fix on the DR computer.

These last plotting techniques are discussed in Pub. No. 249 in the section on precomputation. Other special techniques are discussed in the section on curves, in which the celestial observation is plotted on a graph rather than on the chart.

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Preplotting True Azimuth (Zn)

Filed Under: Celestial Precomputation

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.
Figure 10-5. The fix can be plotted quickly. [click image to enlarge]
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.

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Precomputation Techniques (Part Two)

Filed Under: Celestial Precomputation

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.
Figure 10-2. Typical celestial precomputation format. [click image to enlarge]

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

Hs31° 05
REFR–01
PERS/SEXT–05
Ho30° 59
Hc30° 40
INT19T

Corrections Applied to Hc

Hc30° 40
REFR+01
PERS/SEXT+05
ADJ Hc30° 46
Hs31° 05
INT19T

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 10-3. Mathematical solution. [click image to enlarge]
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.
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.

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Precomputation Techniques (Part One)

Filed Under: Celestial Precomputation

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.
Figure 10-1. Correction for motion of the body. [click image to enlarge]
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.

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