THE STEREOGRAPHIC PROJECTION FOR STAR IDENTIFICATION AND STAR "SPOTTING"
By G.T. Rude, Hydrographic and Geodetic Engineer, U.S. Coast and Geodetic Survey
It is evident that in general problems which lend themselves to solution by means of tabular values may also be solved by the use of some graphic device. While this may in some cases entail a slight decrease in accuracy for theoretical problems, it is unquestionably sufficiently accurate for all practical purposes. A diagrammatic solution has the advantage however over tabular or computed solutions of requiring far less time and further that the results may be more easily and quickly grasped, the diagram tending to transform an abstract problem into concrete form.
With this idea in mind the writer, while navigator of a troop transport during the war, conceived the idea of employing a graphic means of star identification to take the place of star identification tables which at best are far from satisfactory. The stereographic projection was selected as the one best adapted to the purpose. The accuracy of the results obtainable and the fact that almucanter circles could be introduced, admitting of identification by altitudes only, justified the use of this projection. The introduction of the almucanter circles further made possible another application which may be called "Star Spotting" for lack of a better term. This permits the location of stars and planets very expeditiously and closely in azimuth and altitude before they become readily visible to the unaided eye, thus allowing the navigator to secure star observations while the horizon is clearest—immediately after sunset.
In the Proceedings for September, 1921, Lieut. Commander L.V. Keilhorn, U.S.C.G., in his article "Approximate Altitude and Azimuth" illustrated an interesting, practical use of the stereographic projection for securing altitudes and azimuths. He made use of the horizon projection, plotting the celestial bodies as occasion arose by means of hour and declination circles. The present writer employed two applications of the stereographic projection—the polar projection on which the celestial bodies are permanently plotted by means of their tabulated right ascension and declination, and the horizon projection on which are projected the zenith of the observer and azimuth and almucanter circles for the latitude of the observer.
By locating on the polar projection the zenith and meridian of the observer, and then superimposing upon these the zenith and meridian of the horizon projection the two projections are combined and altitudes and azimuths of celestial bodies for the position of the observer may be read by inspection from the horizon projection which is constructed on a transparent material.
Fig. 1 is a diagram of the northern celestial hemisphere on a polar stereographic projection, extending from the pole to declination 30° south. To obviate the confusion which the declination circles would cause when combining with the horizon projection, these are omitted from the drawing of the polar projection, and the necessary graduations for declination are supplied by means of a graduated revolving transparent arm (A, Fig. 1). The periphery is graduated for right ascension in hours, subdivided into 10- and 2-minute intervals. First magnitude, and some second and third magnitude, stars are permanently plotted on the projection by their declinations and right ascensions. The projection may also be used for the planets by plotting their positions occasionally. For the southern hemisphere a similar projection is used, extending from the south celestial pole to declination 30° north, thus allowing an overlap of 60° of declination on the two projections.
Fig. 2 is the stereographic horizon projection constructed on transparent celluloid for latitude 45°, N. or S. These may be called "Templates" for convenience. Results of sufficient accuracy for practical purposes can be secured three degrees either side of this parallel of construction; that is, from latitudes 42° to 48°, either north or south latitude. It is evident therefore that 11 such templates will cover the earth between latitudes 66° N. and 66° S—for practical purposes, the whole navigable world.
The cross near the center is the projection of the zenith of the observer; the starred straight line his meridian; the curved lines radiating from the central circle are azimuth circles graduated to each 5°, the heavier lines indicating the even tenth degree and the lighter the 5° intervals, and the nearly concentric circles numbered from 20° to 80° are almucanter or altitude circles for each 5°, the even tenth degree being accentuated by a heavy line as in the case of the azimuth circles.
When the celluloid horizon projection, or template, is used in connection with the polar projection (Fig. 3) on which the celestial bodies are plotted, the azimuth circles will indicate true bearings of these bodies from the observer for every 5° and the altitude circles true altitude, beginning with altitude 15°, which is the lowest circle constructed. The bearing and altitude, however, of any body falling between these lines may be readily estimated to the nearest degree. The figures at the ends of the azimuth lines indicate actual values of azimuth, the inner figures for the northern hemisphere and the outer for the southern.
With star diagrams and templates constructed on these projections as outlined above the navigator may determine at a glance for any dead reckoning position the following: The azimuths of celestial bodies with sufficient accuracy for plotting position lines by the Marcq St. Hilaire method; in advance the navigation stars and planets which will be visible at twilight or dawn; the approximate altitudes and azimuths for any given time of stars and planets ("Star Spotting"), and the identity of any star or planet, the altitude of which has been observed and its bearing estimated only, without actual compass bearing.
While the use of 11 templates, employing each template through 6° of latitude, makes the use of the projection practically universal, it is evident that all these templates will very seldom be required, depending of course, upon whether or not a vessel makes considerable change in latitude. On a vessel bound from New York to Liverpool only three templates would be necessary; from New York to the Mediterranean only one would be needed throughout the entire voyage.
When used on or near its parallel of construction altitudes and azimuths indicated by horizon projection template will agree very closely with observations or computed values. As departure is made from this parallel, altitudes and azimuths in general will differ not more than a degree from these values, even when 3° in latitude from the standard parallel. This is shown in the examples given below.
Since the projections may be used both for star identification and "star spotting," examples of both will probably best serve to illustrate the method of use and accuracy of results.
"Star Spotting."—At any convenient time the probable dead reckoning position of the vessel for the estimated time of observation is roughly determined. Bearings and altitudes of suitably located stars may then be obtained graphically for that estimated time. These values may be set on pelorus and sextant to bring the desired star within the field of the telescope. The example following will serve to illustrate the procedure
On December 25, 1921, in Latitude 44° 10' N., Longitude 25° 15' W., required to determine the names, approximate altitudes and bearings of navigation stars visible above 15° altitude at 15 minutes after sunset.
Solution:
? | h | m |
L.M.T. of sunset | 4 | 27 |
? | ? | 15 |
L.M.T. of observation | 4 | 42 |
R.A.M.S. December 25^{th} to nearest minute | 18 | 14 |
Red for G.M.T. | ? | 01 |
Local Sidereal Time | 22 | 57 |
?
By revolving the celluloid arm (A, Fig. 1) to the local sidereal time on graduated periphery of diagram and sliding the movable marker (B) to the degree of declination on the graduated arm corresponding to the dead reckoning latitude, it is evident that, since latitude equals declination of zenith and local sidereal time the right ascension of the observer's meridian, the center of the marker is the projection of the observer's zenith on the polar projection, and the center line of the arm the observer's meridian.
Superimposing the horizon projection on the polar projection (Fig. 3), pole toward pole, zenith over zenith and meridian over meridian, it will be seen that the two projections are conformable, and by reference to Fig. 3 (before photographing projections were oriented to the time and latitude of this example) the following stars on the polar projection will be seen through the transparent horizon projection, their bearings indicated by the radiating azimuth lines and their altitude by the nearly concentric almucanter circles. The first columns are the graphic values from the projection and the second columns are computed values, a comparison of which will furnish an idea of the degree of accuracy obtainable. The graphic values were taken by inspection from the writer’s projections, the polar one being 16 inches square which allows of a sufficiently large scale to take out the values with a fair degree of accuracy, and at the same time is a convenient size for practical use.
? | Altitudes | Azimuths | ||
? | Graphic | Computed | Graphic | Computed |
Capella | 27 ½° | 28° 01’ | 52° | 51° 45’ |
Aldebaran | 15° | 15° 48’ | 82 ½° | 82° 15’ |
Altair | 35° | 35° 45’ | 244° | 244° 00’ |
Deneb | 65° | 65° 35’ | 285° | 284° 30’ |
Vega | 41° | 41° 48’ | 287 ½° | 287° 15’ |
The results tabulated above were obtained for a latitude (44° 10’) very close to the parallel for which the template was constructed (45°). The values below, however, were obtained for a latitude 3° distant from the parallel of construction; that is, Latitude 48°. It will be seen that the graphic determinations, even under the extreme limits of difference in latitude under which a template is used, agree very closely with computed results and that the determination of azimuth is in this case sufficiently accurate for the plotting of Marcq. St. Hilaire position lines.
? | Altitudes | Azimuths | ||
? | Graphic | Computed | Graphic | Computed |
Capella | 30° | 30° 21’ | 53° | 53° 30’ |
Aldebaran | 16° | 16° 16’ | 84° | 83° 20’ |
Altair | 35° | 34° 00’ | 242° | 241° 30’ |
Deneb | 66° | 66° 14’ | 277° | 275° 40’ |
Vega | 42 ½° | 42° 51’ | 284° | 284° 10’ |
Star Identification.—For the identification of stars by means of these projections no compass bearing is necessary; the bearing of the body may be estimated only to the nearest point or so and the identification made absolute by means of the observed altitude and the almucanter circles of the horizon projection.
To illustrate, suppose that on the evening of December 25, after having previously determined the positions of the large stars as in our last example, none showed because of clouds, but that at G.M.T. 6^{h} 23^{m} 0^{s} through a break in the clouds an altitude of 46° 41’ 30" was observed of an unknown star of second magnitude, a little south of East. Required the name of the unknown star?
Solution:
? | h | m |
G.M.T. | 6 | 23 |
Long (25° 15’W) | 1 | 41 |
L.M.T. | 4 | 42 |
R.A.M.S. December 25, 1921 | 18 | 14 |
Red for G.M.T. | ? | 01 |
Local Sidereal Time | 22 | 57 |
Dead Reckoning Latitude | 44° | 10’ |
?
As in the case of the last example the zenith and meridian of the observer are located on the polar projection and the horizon projection superimposed and oriented.
Reference to Fig. 3, on which the template was placed for these examples before photographing, will show the star, altitude 46°, bearing 102 ½° (a little south of East) to be Hamal. This is the only second magnitude star in that general vicinity of the heavens at that altitude and it is therefore evident that only its approximate direction is necessary, the altitude serving to identify it.
To sum up, by the use of 2 star diagrams and 11 templates constructed on stereographic projections, the following data for all the navigable waters may be expeditiously obtained by the navigator:
- Determination of azimuths of heavenly bodies with sufficient accuracy for plotting position lines by the Marcq St. Hilaire method.
- The names of navigation stars which will be visible at twilight or dawn.
- The predetermined approximate altitudes and azimuths for any given time of navigation stars ("Star Spotting").
- The identity of any navigation star, the altitude of which has been observed and its bearing roughly estimated.