In a previous article in the Proceedings, the computation of the altitude and azimuth of a heavenly body by use of secant-cosecant formulas has been discussed in some detail. Dead Reckoning Altitude and Azimuth Table, H. O. 211, is now available for use in solving problems by those formulas.
In publishing H. 0. 211, it was not considered desirable to illustrate uses for the table other than for computation of altitude and azimuth. Formulas and the necessary special rules are given in the book for computing the time of the sun on the prime vertical, identification of unknown stars, and great circle course and distance, but no sample problems are included. The purpose of this article is to illustrate, in some detail, the aforementioned problems, and to present some additional uses of the table which may occasionally prove valuable to the practical navigator.
Time of the sun on the prime vertical.—When the declination of the sun is less than the latitude and the same name, the longitude may be very accurately determined by taking a sight when the sun is on the prime vertical. Refer to Fig. 1. The time the sun will be on the prime vertical may be computed by using the following formulas:
From these formulas, t (LHA) may be determined. Knowing the hour angle the LAT may be found. The error of the watch on LAT (previously computed) is applied to obtain the watch time when the sun will be on the prime vertical.
The following problem is based on the data of problem I in H. O. 211.
At 0730 on 31 December, 1931, the navigator of the U.S.S. Akron expected to be in D. R. position Lat. 42°-10'.5 S, Long. 68°-37'.8 W. Zone (+) 5. The declination of the sun is the same name and less than the latitude. Required: The time the sun will be on the prime vertical.
To find time on the prime vertical:
Reduction to the meridian.—When no more than five minutes are required to compute the azimuth and altitude difference ready for laying down on the chart, there is no particular advantage to be gained by computing the watch time of LAN. If it is desired to take a sight near LAN, merely add the LHA in hours, minutes, and seconds to the watch time of the a.m. sight, which will give a time sufficiently accurate to be within the limits of the method for reduction to the meridian here presented.
In order to use H. O. 211 for reduction of sights to the meridian, two conditions must be fulfilled.
(1) The LHA must be less than 20m time (5° arc).
(2) The angular distance between the sun and the zenith of the D. R. position (L~d) must be greater than 15° arc.
The following formulas are used:
Knowing the DR latitude, K and (K~L) may be combined to give the latitude at the time of sight. It will be noted that the first two of these formulas are the same as those used in computing the altitude and that the third is the reverse of the third altitude formula.
As long as the LHA is less than 5° and (L~d) is greater than 15°, the error will not usually be greater than O'.5 of latitude. Occasionally the 0'.4 possible error of the table and the O'.5 possible error of this approximation may combine to produce a maximum error of O'.9 in latitude.
In using this reduction to the meridian, it is not necessary to abandon the speed of precomputation. Refer to the problem illustrated above. When the ship is on a steady course and speed and the time it is desired to take a noon sight is known, K may be precomputed. After taking the sight at the predetermined time, correct Hs to obtain H0, compute (K~L) as shown above (the work of half a minute), and find the latitude at the time of sight by combining K and (K~L). In practice, it would be advisable to precompute several K’s at 5-minute intervals.
When (L~d) is less than 15°, the sun moves so rapidly near LAN that the error due to this approximation cannot be neglected. When the LHA is greater than 5°, the difference between the azimuth and the meridian may be so large as to introduce a significant error. For these reasons, these limitations should be observed in using the table for reduction to the meridian.
Identification of stars.—Here the problem is a reversal of the usual computation for altitude and azimuth. Instead of t, d, and L being the known values, Z, H0, and
the top of the table in accordance with Rule 4.
It is believed that this method of star identification may prove of value to air navigators, where larger, more unwieldy star finders cannot be used. For surface vessels, Rude’s star finder is recommended.
Great circle sailing.—While great circle charts afford the simplest means for solving this problem, the method here presented is believed to be simpler than other logarithmic methods for computing the great circle course and distance and plotting the great circle track on a Mercator chart.
Having determined the initial course and distance, the navigator next wishes to know the latitude and longitude of the vertex of the great circle, in order to de- determine that the track does not run through dangerous waters. Refer to Fig. 4. In the terrestrial triangle PAB, P is the elevated pole, A the initial position, and B the final position. In addition to the nomenclature given in the previous section, the following parts are specially designated:
Formula le gives the latitude of the vertex. The longitude of the vertex is found by adding or subtracting tv from the longitude of the initial position. The use of dv will be explained later.
To find the latitude and longitude of any point X, refer again to Fig. 4. Here the initial course, the latitude and longitude of the initial position, and the distance to point X are known. Additional nomenclature follows:
While the method given in the previous section is accurate, there is a shorter method which is much simpler to use when a number of points along the track are to be plotted. In the November-December, 1931, issue of the Revista de Marina y Aviacion, published at Callao, Peru, Lieutenant Commander Cesar Rangel, Peruvian Navy, has published an article entitled “El Método Secante-Cosecante” descriptive of an article of the same name by the present writer which appeared in the October, 1931, issue of the Proceedings. Lieutenant Commander Rangel’s article is more than a translation in that he derived formulas for, and worked sample problems in, identification of stars and great circle sailing. His formulas for plotting points along the great circle track are shorter than those given above and may be placed in a form which greatly facilitates computation. The following additional nomenclature (Fig. 4) is given:
It will be observed that the first two formulas only are required to plot the great circle track, a large reduction in labor over that required with the formulas given in the preceding section. While Cx may be computed by adding two functions already used in solving the first two formulas, it is recommended that courses along the great circle track be obtained by following arcs of the great circle after it has been plotted on the Mercator chart.
The array of formulas given above may appear somewhat formidable, but the actual solution of problems in great circle sailing is quite simple. The essential data of the following problem are the same as those contained in example 7, page 55, Dutton’s Navigation and Nautical Astronomy.
Problem: The navigator of a ship in San Francisco, Calif., Lat. 37°-47'.5 N, Long. 122°-27'.7 W, wishes to follow the great circle track to Sydney, Australia, Lat. 33°-51'.7 S, Long. 151°-12'.7 E. His ship makes 15-knot speed. Required: (1) great circle course and distance; (2) lat. and long, of vertex and distance to vertex; (3) points along the great circle track one day’s run apart.
Great Circle Course and Distance
Note: See rules page 1032.
Note that between the fifth and sixth points Lx changes from north to south and that Tv_x is taken from the top of the table instead of the bottom. It will be observed that dv_xhas become less than 90° between the fifth and sixth points.
As has been previously stated, the vertex in this particular problem lies beyond the destination. When V lies between A and B (as in Fig. 4), there are two points on the great circle track which have the same latitude. In such cases, one computation obtains two points on the curve, the latitude being the same for both points, but Tv_x is east in one case and west in the other, two different longitudes being thus obtained by applying to Long.v.
The ramifications of great circle sailing are almost endless. The great circle part of composite sailing may be solved with H. O. 211, using formulas le and 2e as they are and 3e (reversed). For application of these formulas, refer to page 57, Dutton’s Navigation and Nautical Astronomy.
Precomputation of sights.—Mention has been made under “Reduction to the Meridian” of opportunity for precomputation of sights. Every navigator is familiar with the great advantage in precomputing the constant for a meridian altitude. It has been shown that this advantage need not be abandoned when computing the noon latitude by the method described in this article. It is possible and practical to extend precomputation further than to the meridian altitude.
When steaming on a steady course and speed for any length of time, so that the D.R. position may be projected into the future with a reasonable degree of accuracy, a very convenient use may be made of precomputation. When the cosine-haversine method, which is based upon the D.R. position, was more extensively used than at present, rather wide application was made of precomputation. The fact that H. 0. 211 is a D.R. position method makes it equally useful for this purpose.
For instance, suppose that at 9:00 p.m. the fleet has settled on a steady course and speed for the night. The navigator has determined, by examination of sunrise tables, the time of morning twilight. From long experience, or from previous morning twilights, or by use of Rude’s star finder, he knows that he can locate certain stars which are so well spaced around the horizon as to give him a good fix. Experience has taught him what time will be favorable for the best combination of clear-cut horizon and good visibility of stars. In his charthouse, then, at 9:00 p.m., he runs forward his D.R. position to the time he wishes to take his morning star sights. Using this D.R. position, the navigator may compute the altitudes and azimuths of several stars. With the computed altitudes, he may pick out the corrections for his observed altitudes.
When morning arrives, the navigator observes the stars at the times used for computing, and their altitude differences are quickly obtained. Within five minutes from the time of the last sight, practical application of this method has shown that a 3-point fix may be laid down on the plotting sheet. Furthermore, the leisurely computation the previous evening has materially reduced the chance for error in computation.
Should the ship not have made good her course and speed during the night, the work of the previous evening is not wasted. When lines of position are plotted from the D.R. position used in computation, the sights automatically correct for any error in the D.R. position. The chief requirement is that the sights be taken at the times used in computation. Even should it be overcast at morning twilight, the only loss is the short time spent computing the previous evening. Should the sky be partly overcast with occasional stars shining through, often the stars observed will turn out to be some of those for which computations have been made.
When the ship remains on a steady course and speed during the morning, it is no great extension of this principle to consider precomputing sun sights. The advantage is not so great as with stars, but precomputation will remove the pressure which usually attends working at high speed. With the sun, it is a much simpler matter to take a sight at a predetermined time than it is with a star.
A further extension seems quite logical. Lieutenant Commander P. V. H. Weems, U. S. Navy, has long advocated use of precomputed altitude curves for air navigation. For sea navigation, there is no such demand for speed as there is in the air, but construction of such a curve is so simple that its use might well be a fine thing for marine navigators.
Here, too, the chief requirement is that the ship shall maintain a predicted course and speed. The D.R. track is projected into the future and positions picked off at hourly or half-hourly intervals. Using watch times indicated and the D.R. position corresponding, the altitudes and azimuths are computed. Typical curves are shown in Fig. 5.
Curves should be plotted to large scale on cross-section paper. To use the curves, a sight is taken at any time and the observed altitude plotted on the sheet. With dividers, pick off the distance to the altitude curve vertically and measure it on the altitude scale. This is the altitude difference. The azimuth for the time of sight is picked from the azimuth curve. Air navigators will recognize the usefulness of H.O. 211 for computing data for such curves.
Recent developments in computing local hour angle.—The Hydrographic Office and the Naval Observatory have under consideration at this time development of an air almanac, which may well be a most significant advance in the art of navigation. This air almanac is designed to simplify greatly the problem of determining the local hour angle. Lieutenant John E. Gingrich, U. S. Navy, has submitted to the Hydrographic Office a new method for tabulating data on navigational stars which was described in the Proceedings, February, 1932.
The air navigation research desk under Lieutenant Commander Weems is planning to extend the principle suggested by Lieutenant Gingrich so that the sun, moon, and planets will have the Greenwich hour angle and the declination tabulated for each hour of GCT during the year. Questionnaires have been sent out by the Hydrographic Office to over 3,000 navigators. If the reaction is favorable, this data will be issued as an air almanac, but there is every indication that this new almanac will be as popular with marine navigators as with aviators.
With the proposed almanac, it will be necessary to enter the book only once to pick out the GHA in arc, its correction, and the declination of the body. It is believed that the difficulty so often experienced with sidereal time will be eliminated and a simple, arithmetical method for computing the LHA substituted.
Mr. Elmer B. Collins, the principal scientist at the Hydrographic Office, has made a suggestion which, if carried out in the proposed air almanac, will be of great assistance to the navigator using H. 0. 211. Reference to a problem solved by H. O. 211 will show that functions of the declination from both A and B columns of the table are used. Mr. Collins suggested that these functions be tabulated with the declination at the head of the star pages for each month.
Since the declination is nowhere else used in computation, the functions may be entered in the work form direct from the air almanac, with a notation that the declination is north or south. This would save one entry in H. O. 211 when working star sights.