“Celestial co-ordinator” is the general name applied to a variety of devices which provide graphic representation and approximate solution of problems involving the systems of spherical co-ordinates commonly used to designate locations of bodies on the celestial sphere. In teaching navigation, the important use of a device of this kind is to afford visualization and approximate solution of the various problems which arise in the study of nautical astronomy.
For this purpose, the two systems of co-ordinates employed are the horizon system of altitude and azimuth, and the equinoctial system of declination and hour angle (measured from the meridian of the observer). The co-ordinator consists of two concentrically pivoted discs, each representing one of these systems.
To provide a two-dimensional co-ordinator, the two systems must be projected on the same plane. The most convenient plane for this purpose is that of the meridian of the observer, which is both a vertical circle (in the horizon system), and an hour circle (in the equinoctial system). Since the observer’s meridan passes through the zenith, nadir, both celestial poles, and the north and south points of the horizon, it divides the celestial sphere into eastern and western hemispheres, each of which may be projected on the plane of the meridian.
The methods of projection most commonly employed are the orthographic and stereographic. Some co-ordinators use one, and some the other. Of course the same projection must be employed for both systems of co-ordinates. The orthographic projection of a hemisphere is illustrated in Fig. 1, and the stereographic projection in Fig. 2, with a parallel and secondary every 15 degrees.
It will be noted that the orthographic projection gives a good-sized scale near the center, but is crowded close to the meridian. In the stereographic projection, the scale in the center is only about half that of the orthographic, but gets larger near the meridian, where it is far superior.
as illustrated in Fig. 1. The ellipses which represent the projections of the secondaries may be determined as illustrated in Fig. 3. The semi-major axis is the radius of the circle representing the meridian. The semi-minor axis is the distance OD in the figure. That is, OD is the semi-minor axis of that ellipse which is the projection of the secondary “X” degrees from the meridian. The foci of this ellipse may be located by taking D as a center and the semi-major axis (OA) as a radius, and striking arcs which cut the vertical diameter at E and F, which are the foci. The ellipse itself may be drawn in several ways. One method is to locate a series of points in the manner illustrated in Fig. 3, and then to draw a smooth curve through the points so located. Another way is to put a tack at each focus, tie a loop of string around them reaching from one focus to the opposite end of the major axis, and by placing a pencil in the bight of the loop, draw the ellipse.
In the stereographic projection both the parallels and secondaries appear as arcs of circles. To construct the stereographic projection of a parallel, the center of the projected circle is located as illustrated in Fig. 4, by constructing a tangent to the meridian at the desired parallel. To construct the stereographic projection of the secondary “X” degrees from the meridian, the center of the projected circle is located as illustrated in Fig. 5, by projecting the point “2X” degrees from the pole of the secondary.
Since one system of co-ordinates must be superimposed over the other, it is necessary that the uppermost disc be transparent in order to read the co-ordinates of the system beneath. In some coordinators the horizon system is underneath, and in others the equinoctial system. In either case, the co-ordination of the two systems is accomplished by placing the elevated pole of the equinoctial system at an angle above the horizon equal to the latitude, or, which is the same thing, by placing the zenith at a declination equal to the latitude.
With the two systems oriented in this manner, a given altitude and azimuth can readily be converted into the corresponding declination and hour angle, or vice versa. The various problems arising in navigation and nautical astronomy which may be illustrated and solved by this device will be described in what follows. The orthographic projection will be used in the accompanying figures.
(1) The problem of determining altitude and true azimuth from latitude, hour angle, and declination. This is the problem
credits one John Blagrave of Reading (England) in 1585 with the first co-ordinator on the plane of a meridian. Rude’s “Star Finder” is a celestial co-ordinator employing a stereographic projection on the plane of the equinoctial. In this type, a series of templates is necessary in order to represent the horizon system at various latitudes. All other co-ordinators mentioned herein are on the plane of the meridian.
Astrographics contains a description of the construction and uses of a co-ordinator on the stereographic projection. The coordinator itself is attached to the inside of the back cover, and though it is but 4⅞ inches in diameter, all values may readily be determined to within one degree.
Another co-ordinator on the stereographic projection is George E. de Kausz’s “Navicard,” while those of Kellar and Commander H. P. Sampson employ the orthographic projection. In order to take advantage of the best features of each projection, this writer has developed a coordinator using the orthographic projection on one side of the card, and the stereographic on the other, being of the opinion that the orthographic projection is better for purposes of instruction and visualization. At the same time it provides a larger scale near the center, while the stereographic projection is unquestionably more accurate near the meridian.
It is not necessary so far as the solution of problems is concerned to have both discs inscribed with parallels and secondaries. The top disc can be perfectly plain, and the underneath one used for reading both sets of co-ordinates. There is provided herein a plate on the stereographic projection which can be detached (page 243) and turned into a co-ordinator if the reader will place a sheet of transparent paper over it and push a pin into the exact center of the diagram.
To solve the problem illustrated in Fig. 7 assume the diagram to represent the Horizon System, and the plain paper the Equinoctial System. Holding the paper fixed, put a mark at a point 30° above the south point of the horizon (S). This represents the location of the south celestial pole (Ps). Then indicate the location of the body by plotting a point on the transparent paper over the point whose altitude is 22.5° and whose azimuth is 315°. Now rotate the plain paper until the point representing the South Pole is at the Zenith. The equinoctial co-ordinates of the point may now be read from the underneath diagram, the new altitude of the point being its declination, and the new azimuth, its hour angle. These are found to be 22.°5 North, and 45° West as before.
It is necessary to observe that in measuring the hour angle, care must be taken to insure that it is reckoned from the upper branch of the meridian. To do this, when the zenith and pole are placed in coincidence, the hour angle must be measured from the left for south latitudes, and from the right for north latitudes. True azimuth is measured in accordance with the numbers, regardless of whether the latitude is north or south. If a body is to the eastward of the meridian, its true azimuth is less than 180°, while if it is to the westward of the meridian its true azimuth is greater than 180°.
To solve the problem of determining altitude and azimuth, from latitude, hour angle, and declination, proceed as follows, taking the problem illustrated by Fig. 6 as an example:
- Holding the plain paper firm, make a mark at the zenith, and label it Pn, and put a mark at the point whose declination (or altitude) is 36° and whose hour angle is 60° E., measured from the right, as previously explained.
- Now rotate the plain paper until the point you marked Pn is 40° above the north point of the horizon (N).
- Now read the altitude and true azimuth, which are found to be 43.°5 and 075°, respectively.
This method of using only one graduated disc works because at either pole the horizon system and the equinoctial system do in fact coincide.
It is believed by the writer that users of Weems’ Line of Position Book will be able to obtain their true azimuths with this device much more quickly and with less concern about rules than is possible with Rust’s Azimuth Diagram. As a matter of fact, this device shows clearly why Rust’s rules are necessary, since he works with angles only up to 90°. It will also be found easier to use than Wier’s Azimuth Diagram, and will give results more than sufficiently accurate for plotting lines of position and determining compass error.
The reader may enjoy working a few more problems. The following are given with answers, for anyone who may wish to try his hand.
- In the following, determine altitude and true azimuth: