Star Altitude and Azimuth Tables, or Tables of Computed Star Altitudes and Azimuths
By Captain Radler de Aquino, Brazilian Navy.—It is curious to observe that the first sets of tables giving altitude and azimuth alongside one another were published in the same year of 1903 in France and in Brazil.
Professor of Hydrography G. de Lannoy published in Paris, in 1903, his famous Tables Stellaires—Hauteurs cl Azimuls de 32 fijoiles—the first three argument Stellar Tables ever published. They were not general, as they only covered the latitudes north from 32° to 54° and were subdivided into 4 small and inexpensive volumes.
The tables look advantage of Professor Souillgouet’s beforehand assumed position, and, therefore, the latitude was rounded up to the nearest whole degree and the local hour angle to the nearest 4 minutes of time. (It was not customary then to give the hour angle in degrees.) The altitudes were given to the nearest minute of arc and the azimuths to the nearest degree. A small table of right ascensions from 1903 to 1915 was given also for each star, as well as the variations of the altitudes due to the change in declination of the stars from 1910 to 1915.
Although very useful these tables were never republished.
In the same year of 1903 the writer published his “A Navcgaqao sent Logarithms" (Navigation without Logarithms) in Rio de Janeiro, a simplified and improved edition of Sir William Thomson’s (Lord Kelvin’s) Tables for Facilitating Sumner’s Method at Sea, London, 1876 and 1886, with modified methods for finding, not the hour angle and the azimuth as he did, but the altitude and azimuth directly for use in Marcq St. Hilaire’s method, then very little used all over the world.
The writer felt deeply honored 35 years ago to be able to present his views on this subject in the Naval Institute Proceedings for December, 1908 in which he described for the first time his Altitude and Azimuth Tables, published one year later in London, in 1910, and favorably endorsed on February 1, 1909, by the U. S. Hydrographic Office before its publication.
Since then many tables and devices were made public for facilitating the work of navigators at sea, and later, from 1918, of the navigators in the air.
A Report No. 198 of the National Advisory Committee for Aeronautics, Astronomical Methods in Aerial Navigation, by Mr. K. Hilding Beij, published in Washington, Government Printing Office, 1924, contains an extensive bibliography from 1886 to 1924 and ends with the Brazilian Centenary Edition of the writer’s Altitude and Azimuth Tables, 1924.
He gives a short description of the various methods which have been proposed—logarithmic solutions, tabular, graphical, and mechanical solutions and solutions by means of nomograms and diagrams.
As Lieutenant C. H. Hutchings, U. S. Navy, expresses his surprise “that such Tables were not computed years ago” we might indicate 'their historical development and show some reasons why they were not computed. On pages 78-81, H. E. Wimperis, M.A., late Major R.A.F., in his little A Primer of Air Navigation, New York, 1920, gives a good description of his proposed “Two Star Altitude Tables,” in which he says:
If the altitudes of two different stars are measured simultaneously [in such case “simultaneously” means that the star observed first is observed again as a third reading and the mean of the first and third readings taken] it is possible by looking up a special book of tables to read off the latitude of the observer’s position and the sidereal time of the observation.
Mr. Beij explains on page 30 of his work referred to above why these Tables were never published. He says:
Tables may be constructed from which latitude and local sidereal time may be found directly from the altitudes of two stars. Such tables would eliminate all computations and even the use of the Nautical Almanac. Unfortunately, however, in order to reduce the Tables to a practical size, the altitudes cannot be tabulated at closer intervals than a degree, or half a degree at the most. Double interpolations are thus necessary, and the advantages of the tables are nullified to a great extent.
With these difficulties and inconveniences in mind, on page 34 he says:
The nomograms and diagrams which have thus far been described are perfectly general in application and may be used for observations of the sun and moon as well as the stars and planets. The changes in the right ascensions and declinations of the fixed stars are very small, and for the purposes of aerial navigation these stellar coordinates may be considered as constant for periods of several years. Thus a variety of diagrams is possible for the reduction of observations of the fixed stars. A few of these arc here described for illustration and because under certain conditions they may prove to be of value to the navigator.
He further adds that
Local sidereal time and latitude arc completely determined by the simultaneous altitudes of two stars. If latitudes be taken as abscissae and local sidereal times as ordinates, curves showing the simultaneous altitudes of two given stars may be constructed, as in Fig. 22. [A note indicates that he proposed the construction of these altitude curves in 1924.] From this diagram, and the Greenwich Sidereal Time, as indicated by his chronometer, the navigator can determine his position. No tables are required; even the Nautical Almanac is unnecessary. All computations, including the calculation of hour angles, are eliminated, save the simple addition and subtraction required to find the longitude from the local and Greenwich Sidereal Times.
This valuable suggestion was carried out very neatly in 1928 by Commander P. V. H. Weems, U. S. Navy, in his Star Altitude Curves and he added very properly the curves of altitudes of Polaris, facilitating the finding of the intersection of the other two altitude curves. A second edition for Latitude 30° to 40° North was published in 1938 positioned for Epoch 1 January 1945.
While this is ideal solution when the three stars are available, Weems declares on page VI:
The one obvious limitation of the Star Altitude Curves is that their use limits the observer to the three stars shown. The method supplements but docs not replace the longer methods of computing and laying down single curves of equal altitude, called lines of position, since the latter methods arc required for the sun, moon and planets and stars for which altitude curves arc not computed. They are only available in north latitude and within the limits assigned.
This brings us to the very interesting article on Tables of Computed Star Altitudes and True Azimuths published in the Naval Institute Proceedings for September, 1942, by Lieutenant Hutchings. Owing to the war, I was able to see it only a few weeks ago and I was very pleased to see the able way in which lie handled a very difficult situation.
I was especially interested when he indicated the value of the chronometer or watch showing immediately the G.S.T. in arc, or, as it is known now in all Air Almanacs: G.H.A.T, as I first suggested its use ten years ago in a radiogram to Captain J. F. Hellweg, U. S. Navy, then and now Superintendent of the U. S. Naval Observatory.
Answering, in part, the Questionnaire on a pink sheet attached to the Air Almanac for 1933 this radiogram was as follows:
July 2, 1933, Capt. J. F. Hellweg, U. S. Navy
Superintendent U. S. Naval Observatory,
Washington, D. C.
As all ships and aircraft should carry at least three modern second-setting chronometers or accurate watches: one for G.S.T., one for G.C.T., and one for G.A.T. minus 12 hours, all set once a day, Air Almanac only shows a slight advantage for moon. For sun, planets, and stars Air Almanac only saves conversion of time into arc. I find the turning of pages and the large corrections in arc with as many as six figures rather tedious. Nautical Almanac should give R and E as in British Nautical Almanac and their corrections are always very small and would be only used once a day to set watches which should show time in arc and Nautical Almanac could give all data in arc, as now in Air Almanac. As second-setting accurate watches must be kept on board all ships and aircraft they take care of the large corrections necessary with the Air Almanac as they go along. In example for Sun and Moon on page 200 of Nautical Almanac for 1933 the three second- setting watches would show times in arc respectively G.S.T. 67 deg. 42 min., G.C.T. 267 deg. 38.5 min., and G.C.T. minus E 84 deg. 34 min.; right ascension of moon would be 49 deg. 32 min. Thus in my opinion advantages of Air Almanac are more apparent than real and do not simplify matters as use of proper watches would. For last thirty years have advocated use of sidereal chronometer on board ship as you can see in my Altitude and Azimuth Tables for 1910. Will send by mail other suggestions. Please communicate these ideas to Admiral Gherardi and Commander Weems. Many thanks and best wishes.
Radler de Aquino,
Captain Brazilian Navy
The first watch indicating the exact G.S.T. in arc was constructed by Longines in 1938, the writer helping lo perfect it to answer the requirements of sea and air navigation and in the examples given by Lieutenant Hutchings its value is emphasized. However, with the British and American Air Almanacs giving all data now for every 10m of G.C.T. the value of a watch giving G.S.T. in arc or G.A.T. minus 180° (G.IL.A.) is not so great. My idea was to use this watch also to reduce the Air Almanacs to about one sixth, as I indicated in my article published in the Naval Institute Proceedings for October, 1937, under the title of “Sidereal or Mean Time Chronometers?” (presented to the U. S. Naval Institute in November, 1936).
In his proposed tables Lieutenant. Hutchings has very properly taken as arguments the latitude and the L.S.T. or our old friend the R.A.M. (the Right Ascension of the Meridian) now called L.H.A.T or simply IL.A.T. As five stars are available to which may be added Polaris, and as the Air Almanac gives the single correction lo be applied to the correct sextant reading to find the latitude with L.H.A.T, as the argument, the tables will be most useful and easier to use than any others. The same considerations that we mentioned above in which Beij showed the advantages of his proposed Star Altitude Curves apply equally well to the tables proposed by Lieutenant Hutchings. The curves have the advantage of giving right away the latitude and local sidereal time, while the tables require the ordinary plotting of the intercepts or altitude-differences along their respective azimuths in order to find the fix.
I have read with pleasure the Discussions by Commander A. A. Ageton, U. S. Navy, and Commander P. V. H. Weems, U. S. Navy, in the Naval Institute Proceedings and fully agree with them in their constructive remarks. Only time will tell whether they will be successful, but I am quite sure they will be most useful in the navigation of aircraft above the clouds during twilight and at night in the Northern Hemisphere, where Polaris and the five tabulated stars will always be visible; also in the Southern Hemisphere, if the tables provide for southern latitudes.
Five years ago H. M. Nautical Almanac Office on behalf of the Air Ministry began the tabulation of a new set of star tables for 22 bright stars. In a discussion on the Air Almanac Dr. D. H. Sadler, then and now the Superintendent, gave a good description of these tables saying:
The Nautical Almanac Office has recently undertaken, on behalf of the Air Ministry, to compute and publish such Tables giving the computed altitude and azimuth with intervals of 1° in declination, latitude and hour angle. The accuracy of the tabulations will be 1' in altitude and 1° in azimuth and I hope it will be sufficient for all navigators.
First of all, the declination of the stars remains constant, or practically constant, for many years, and consequently we shall be able to give separate tables for the stars, for which no interpolation for declination will be necessary for several years. We will also provide a method by which these tables can be corrected for periods when the star has moved away from its present position. Therefore, the general arrangement of the tables will give tabulations for declinations from 0° to 29° and also for certain individual selected stars.
This statement is made on the assumption that the position chosen is such that the latitude and the local hour angle are both integral degrees. That involves slight difficulties of plotting on the chart, but it certainly makes the tables easier to use.
In completing his remarks on this subject Dr. Sadler said:
I should like to say a few words about solutions based upon short methods. These short methods are dependent upon the fact that the astronomical spherical triangle can be divided into two rightangled triangles in several ways. In my opinion, some of these methods are ideal for the solution of the problem, but I am used to working in an easy chair at a desk, and 1 am quite sure that the air navigator would not be able to make as full use of these methods as he would of straightforward altitude and azimuth tables. The altitude and azimuth tables do take up far more room. They require a large number of small volumes or a small number of large volumes, whereas the short solutions can be put into a book of a hundred pages or thereabouts. Although this is a great advantage, these solutions do involve some addition and subtraction at an intermediate stage, and possible confusion may result. For practical work, I think it is definitely preferable to have a large number of volumes than to have a short table in which it is possible to make mistakes.
In speaking for the short table makers Dr. L. J. Comrie, late Superintendent of H. M. Nautical Almanac Office, said: “My main contention therefore is based on size. Shall we have a small table and spend a very little extra time on reduction or shall we have a large set of tables and avoid doing that extra work?”
Sea and air navigators will have to answer this question themselves in accordance with their personal idiosyncrasies.
The excellent tables referred to by Dr. Sadler have been published under the title of Astronomical Navigation Tables (A.N.T.) in 14 small volumes and represent, in the writer’s opinion, the best solution of the problem so far, if size and cost arc of no consequence to the navigator.
The excellent tables H.O. No. 214 published in 8 large octavo volumes, comprising 2,080 pages, by the U. S. Hydrographic Office give the altitude to the nearest tenth of a minute of arc and require a final correction of the altitude for the odd minutes of declination for all bodies, besides a glance at two adjacent values of the altitude to determine the sign of the declination coefficient.