Star Identification.

By Captain Harry S. Knapp, U. S. Navy.

Hydrographic Office publication No. 1560, listed in Portfolio No. 1 of every station chart catalogue under the title Diagram for the Graphical Solution of Triangles, though its real title is Graphical Method for Navigators, by Commander (now Rear-Admiral, retired) C. D. Sigsbee, U. S. Navy, has proved so useful to me for one particular purpose that I venture to invite the attention of the service to it. If any excuse is needed, it may be found in the fact that I have not yet run across anybody in the service who had used the diagram for the particular end in view until after I had spoken of it.

Every navigator who employs twilight sights appreciates any method by which a star or planet can be unhesitatingly identified, whatever the conditions of cloudiness or light. This is particularly the case at evening twilight, when the brightest stars may often be detected in broad daylight while the horizon is excellent, though the lesser stars of the constellations are still invisible. The identification can be made by waiting, if cloudiness does not intervene; or, with a loss of time, by other known methods, if the azimuth is made a part of the observation. A few minutes' work beforehand with the Sigsbee Diagram will enable the navigator to identify his star at once on sight; more than that, it will enable him to look for his star early in exactly the right place. I have myself frequently set my sextant and found a star by sweeping when the star was invisible in the sky, and I have seen others do the same thing.

The Sigsbee Diagram is a stereographic projection, on the plane of the meridian, of (a) the great circles traced by planes all passing through a common axis, which is also one axis of the meridian, and of (b) the one great circle and (c) the small circles traced by planes at right angles to this common axis. In Fig. 1, which is an incomplete diagram, the primitive, or bounding circle, M'BM"A is the celestial meridian, M'XM" is one of (a) the great circles passing through the common axis M'M", ACB is (b) the one great circle, and aXb is one of (c) the small circles traced by planes at right angles to the common axis M'M". In the complete diagram, the great circles through the axis and the small circles at right angles to it are all drawn at intervals of one degree where the diagram permits it without undue crowding of lines. If (1) the common axis passes through the poles, the first (a) are hour circles (meridians), the second (b) is the equator, and the third (c) are declination or latitude circles. Thus in Fig. 2, in which the full lines are exactly the same as in Fig. 1, but with different lettering, P is the elevated pole, S the depressed pole, PXS is an hour circle, EQ is the equator, and aXb is a circle of declination. If (2) the common axis passes through the zenith and nadir, the first (a) are azimuth circles, the second (b) is the horizon, and the third (c) are altitude circles. Thus in Fig. 3, Z is the zenith, N the nadir, and of the dotted lines HR is the horizon, ZX'N is an azimuth circle, and cX'd is an altitude circle. To use the diagram, there are only needed a sheet of tracing paper and a pencil; the rest of the work is arithmetical, or a matter of manipulation.

If the declination and hour angle of a heavenly body are known, its projection can be plotted on the diagram considered as made on assumption (1) above; if the altitude and azimuth of the same heavenly body are known its projection can equally be plotted on the diagram considered as made on assumption (2) above. If either set of co-ordinates is known, and the latitude as well, the other set of co-ordinates may be found corresponding to the same instant. The application of this general principle will be made in the solution of the particular problem in mind, and incidentally the method of using the diagram will be apparent.

The problem may be stated thus: At morning or evening twilights, what navigational stars or planets will be visible, if clear and what will be the approximate altitude and azimuth of each. The known data are: the latitude, the hour angle, and the tabulated declinations of the stars and planets.

The hour angle is found by applying the right ascension of the mean sun to the mean time of twilight, morning or evening (say 20 minutes after sunset and 20 minutes before sunrise), thus getting the sidereal time of the hour when observations can suitably be taken. It is not necessary to work closely for the purpose in view, and seconds of time may be ignored. Applying to the sidereal time of twilight the right ascension of any heavenly body gives its hour angle at twilight. With the sidereal time of twilight as a constant, the hour angles of all the navigational stars can be computed mentally by running down the table of mean places.

The solution of the problem is as follows: Put the tracing paper over the diagram, temporarily secured. Now considering the diagram as the projection in which the common axis of intersecting great circles (meridians) passes through the poles and the horizontal diameter represents the equator, mark on the tracing paper the center of the projection and the elevated pole, the latter on the circumference bounding the projection. Next on the tracing paper plot the projected position of any star or planet by its co-ordinates of hour angle and declination, seen through the paper from the diagram. This plotted position is correctly situated with reference to the center and the elevated pole, on the scale of the diagram, however the tracing be turned. Fig. 2 may be taken to represent the tracing paper, except that in actual work only the points C, P and X need be marked; all else is superfluous. The point X represents the projection of a star at twilight (west of the meridian, say) whose actual hour angle is 45° (3 hours), and whose actual altitude is 40°. It is found by marking on the tracing paper the intersection, seen through the paper, of the hour circle PXS, 45° from Q (QV = projected 45°), and the declination circle aXb, 40° from EQ.

Now forgetting the tracing for a moment, consider the diagram as a projection on which intersecting great circles have a common axis passing through the zenith and nadir, and the horizontal diameter represents the horizon (Fig. 3, dotted lines). Since the projection is on the plane of the meridian, and since all meridians pass through the poles, the elevated pole will be on the circumference of the projection, and its position thereon will be above the horizon line at the point where the circumference is cut by the circle of altitude that is equal to the latitude (latitude equals altitude of the elevated pole). Note this point, on the right side or the left, it makes no difference. In Fig. 3 it is P, taken 20° from H on the left side. The elevated pole will be considered north.

Reverting to the tracing paper, revolve it with its center mark over the center mark of the diagram until the elevated pole marked on the tracing is directly over the noted position of the elevated pole on the diagram. As said above, the plotted position of the heavenly body on the tracing paper is its correct projection with reference to the pole and the center, in whatever place on the bounding circumference the pole may be. In Fig. 3, the full lines represent the tracing paper, the pole has been revolved to P from its original position vertically above C, and X has been revolved until it occupies the position X'. X' is in exactly the same position relative to P and C in Fig. 3 that X was in Fig. 2.

Hence, since the pole on the tracing paper has been revolved to its proper position with reference to the zenith of the diagram. the point on the diagram immediately under the projected position of the heavenly body on the tracing is the proper projection of the body on the diagram with reference to the zenith and center, and its co-ordinates of azimuth and altitude can be read off from the diagram through the tracing paper. Thus in Fig. 3, where the dotted lines represent the diagram seen through the tracing paper, the point X' is seen to lie on the intersection of the azimuth circle ZX'N and the altitude circle cX'd. The azimuth is measured by the projected arc HY, and is N 52 ½° W in this case; the altitude is measured by the arc He (or Rd), and is 47° in this case. The full lines, though unnecessary and not traced in practice, have been left in Fig. 3 to assist in keeping in mind that X' on the tracing paper is correctly projected with reference to P and C by its co-ordinates of hour angle and declination.

What has been described for one star or planet can be done to all of them that are useful for navigation by one plotting on the tracing paper and one revolution of the paper for latitude, and thus by ten or fifteen minutes' work the navigator may prepare a table of the visible navigational stars by the aid of which he can instantly identify any one; and from the same table he can decide which will give the best cuts.

In practice, the table will not contain many stars and planets. In the first place, the number of stars ordinarily used for navigation is limited. Then again, some will be below the horizon, and others will not be above the horizon a sufficient distance, say 10° for reliable results. To eliminate the latter the procedure is as follows: After marking the pole and center on the tracing, revolve the paper until the pole corresponds with the latitude as explained above, and secure it temporarily. Then mark on the tracing paper the 10° circle of altitude that is above the horizontal line of the diagram, after which revolve the tracing paper back until the pole on it is again over the upper pole of the diagram, which is the proper position for plotting by hour angle and declination. Then in plotting the projection of stars or planets on the tracing paper by these co-ordinates, if any one is seen to fall below this 10 circle it is rejected. This is a simple matter of inspection in most cases. In Fig. 2, the 10° circle is shown by the broken dotted arc nm. Each star or planet that plots below (to the left of) this arc goes out at once.

There are two things that, unless understood clearly and kept in mind, will be a source of trouble in using this method. The first of these is that the hour angle must be measured from the extremity of the equator that is on the side from the center that is opposite to the one on which the pole will fall when the tracing is revolved. Thus, in Fig. 2, hour angle must be measured from Q, not from E, because in Fig. 3 the tracing has been revolved with the pole toward £. The reason is that the origin of hour angle is the meridian, zero hours occurring at the time of upper transit; moreover, for any heavenly body that can be observed at all, i.e., for any heavenly body whose upper transit can be seen above the horizon, upper transit takes place on the arc of the meridian that includes the zenith, and whose extremities are the elevated pole and the horizon. The intersection of this arc of the meridian with the horizon is necessarily on the opposite side of the zenith from the pole.

The second source of possible difficulty is to know whether any particular heavenly body will be east or west of the meridian. The diagram serves to represent equally well the projection of a body on either side of the meridian; but as two bodies of the same declination, symmetrically placed on opposite sides of the meridian, will plot superposed on the plane surface of the diagram, it is necessary to exercise care in determining whether the azimuth in any particular case is to be east or west. A thumb rule for determining the name of the azimuth is as follows:

If the sidereal time is greater than the heavenly body's right ascension, the azimuth will be westerly if the difference of the two is less than 12 hours, and easterly if that difference is greater than 12 hours. The contrary holds if the right ascension is greater than the sidereal time.

Better than any rule is a rough projection on the plane of the equator showing the relative traces of the meridians passing through the place, the star and the First Point of Aries.

Azimuth is, of course, reckoned as North so many degrees East (or West), or South so many degrees East (or West), according as it is measured on the diagram from the end of the horizon line nearer to or further from the position of the elevated pole (considered North here) in its revolved position on the tracing.

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