Art. LIV.—A New and Simple Graphic Method of projecting Occultations and Solar Eclipses not hitherto published.

Transactions and Proceedings of the Royal Society of New Zealand, Volume 26, 1893, Page 477

Art. LIV.—A New and Simple Graphic Method of projecting Occultations and Solar Eclipses not hitherto published. By T. B. Harding. Communicated by R. C. Harding. [Read before the Wellington Philosophical Society, 6th September, 1893.] Plate LIII. If any apology be necessary for bringing the following paper before the Society it will be found in the fact that there will be a good occultation of the planet Venus on the evening of the 13th instant. Conjunction of ♀ with the Moon takes place at 4h. 49m. p.m., New Zealand standard time, and the planet's hour-angle is then 2h. 45m.; so that, while those bodies are sufficiently high in the heavens for observation, it will be late enough to see them well defined. To proceed: When we consider the abstruse and difficult methods usually employed for the computation of the circumstances of an occultation or solar eclipse for any particular station on the Earth's surface it will be conceded that a ready and easy process of doing this is a desideratum to many who, while they wish to perform the work, are deterred by the labour of the methods usually employed. In this now proposed, and used by myself for several years, we have such a method, and one in which the mathematical work so much dreaded by the ordinary individual is almost entirely eliminated, and by which any person of only ordinary intelligence and skill in the use of rule and compasses may very readily perform the work. For this purpose we require—(1) An ordinary diagonal scale, by which we can get three places of figures; (2) a sector, or line of chords (which is to be found on any good scale), for measuring and setting off angles; (3) the “Nautical Almanac” for the year. To understand the details of the scheme, we have to suppose ourselves, pro tempore, at the star's (or sun's) centre, from which we are able to see the Earth and Moon as circular discs, revolving in their orbits, and during an occultation we observe that of the Moon passing between us and the Earth, and covering a certain zone of her disc. We note the parts so hidden from us, and the times at which they are covered and reappear. We find all the data for this in the pages of the “Nautical Almanac,” where the calculations are made for the Earth's centre; but from our position we are able to note places on

the Earth's surface projected to a plane passing through that centre. The data for the forthcoming occultation of ♀ required are the following:—  1. The equatorial horizontal parallax, corrected for the latitude of the place and diminished by that of ♀ (P.)  2. The geocentric latitude of the station (l).  3. The declination of ♀ (δ).  4. The difference of δ of ♀ and ☽ (Δ).  5. Time of ♂ in R.A.  6. Hour angle of ♀ at that time.  7. Moon's semi-diameter (μ).  8. Moon's hourly motion E.  9. Moon's hourly motion S. Unless great accuracy be required, the corrections for parallax and latitude need not be made, but the figures taken as they are given in the “Nautical Almanac.” They will be sufficient to tell us when to watch for the phenomena. The corrections being made, great accuracy can be obtained. P. The corrections for parallax have been calculated for all latitudes, and may be found in books of nautical tables. It is O” at the equator, and increases towards the poles, where it amounts to 12″, or ⅕ of a minute of arc. In the present case P. is 56; the correction is 5″, to which we add that of ♀ =7″, making together 12″ to be subtracted: 56′—12″ = 55′ 48″ = 55·76′. Take then from the diagonal scale in the compasses 55·8 for Earth's radius, and describe a semicircle, to represent the S. half of Earth's disc. The centre ⊕ represents Earth's centre. From ⊕ erect a perpendicular, which is the axis of projection, through which passes the plane of projection (at present perpendicular to the paper). The planet's declination (δ) is 9° 57′ S. We suppose ourselves in the plane of the paper towards the left hand. Our declination being S., the S. pole of the Earth must be turned towards us by the amount of our δ. Mark off the arc of 9° 57′ (10°) towards the left of axis of projection, through this draw the axis of Earth. We next want the parallel of Wellington. Its latitude is = 41° 17′ S., its co-latitude 48° 43′. Adding 6′ for correction, we have 48° 49′ (say, 50°) which set off from the polar axis in the same direction and mark the point l. Set off the same (50°) to the right of polar axis, mark it l1, and through these points draw the parallel of latitude, whose centre is on the polar axis. Now, looking from the left, we see these three points l, c, and l1 projected on the axis of projection. We have, however, only to do with two of them, l and c, l being our place when ♀ is in transit, and c our place at her 6-hour angle. We next imagine the whole diagram turned a quarter round, so that from being at the left in plane of the paper

we (♀) are brought in front, the points l and c to the axis of projection, the polar axis coincident with that of projection, and we are prepared to plot the path of the station along the plane of projection through a revolution of the Earth, as seen from the planet Venus. If there were no declination (if ♀ was at the equator) all the circles of latitude would appear as straight lines across the Earth's disc; at 90° (the poles) they would appear as concentric circles; between these extremes they appear as ellipses more open as our declination increases; at our declination of 10°, as an ellipse whose semi-major axis equals the radius of the circle of latitude (not being shortened by projection), and whose semi-minor axis equals the distance between l and c on the axis of projection. We have only to do with one quarter of this ellipse—that from Venus's meridian passage to her 6-hour angle. To draw this quadrant is not a matter of difficulty, as the following will show:— 1. Draw a line across the diagram through c perpendicular to the axis of projection. Measure from c towards the right at quantity equal to radius of circle of latitude. Mark that point 6. 2. With distance c-6 in the compasses and centre c describe the large quadrant from 6 downwards till it meets the axis of projection; and 3. With distance c-l and same centre describe the small quadrant, concentric with the other. 4. Divide both by lines radiating from centre (c) into six equal parts (of 15°) (hours). 5. Through all the five points of division on the small quadrant draw horizontal lines, and through those of the larger one perpendicular lines. The intersection of these lines marks at the same time the curve of the elliptical quadrant, and the points of the hours from 1 to 5. These are the positions of ♀ at her hour angles, as projected. Next we want the place of the Moon at conjunction, and her path in orbit during her course between ♀ and the Earth. The difference of declination (Δ =29·75′) must be measured from the same scale as before; and, as it is S., it must be measured upwards from the Earth's centre on the axis of projection. Mark this point; also mark it the time of ♂, which is when the Moon's centre is there. This in ♀ hour-angle time is 2h. 45m.; from this point and time she is moving eastward and southward. We get from the “Nautical Almanac” her motion in both directions, and from these plot her motion in orbit. 1. Her motion E. (R.A.) is given in time for 10m. as 19s. Six of these go to an hour, and 4sec. of time = 1′ of arc. We

therefore call seconds of time minutes of arc. Add one-half the quantity and we have her hourly motion E., 28·5′. 2. Her motion S. (δ) is as easily got: for 10m. it is 141″—that is, 14·1″ for 1m., or 14·1′ for the hour. To avoid confusion we set these quantities off to the left of the diagram, and draw the diagonal through the point of ♂ This diagonal, being measured on a narrow slip of paper, is divided into 12 parts of 5m. each, and these divisions, set off on the Moon's path in orbit, continued to the right as far as necessary. We have the position of the Moon's centre at these times. The elliptical quadrant being also so divided where it is found necessary, we have the relative position of the two bodies. Now, in the compasses (or on a slip of paper) measure the semidiam. of the moon; add that of ♀ (15·20 + 11 = 15·31′), and, laying it between the two paths, we shall find two points where it will just reach the same time on both. The first of these, 3h. 30m., is the time of first contact; the second, 3h. 59m., that of last contact. To these times, if we add that of ♀ meridian passage, 2h. 4·5m. (2h. 5m.), we get 5h. 35m. and 6h. 1m. as the New Zealand mean time of these phases. The south point of the Moon is that at its apex in the diagram (Plate LIII.). Its vertex is on a line drawn through its centre parallel with one joining Earth's centre and star. The angles are measured in the usual way. In a solar eclipse the Sun's hour-angle is the same as apparent time. Explanation of Plate LIII. The semicircle represents the southern half of the Earth's disc, as seen from the planet Venus, when ⊕ is Earth's centre, and the horizontal line passing through ⊕ the origin of co-ordinates. Any convenient scale of equal parts may be used, but the larger the better, as enlarging the time divisions. The divisions of the scale are taken as minutes of arc. P. is Earth's radius as seen from Moon's centre, μ is Moon's semidiameter seen from Earth's centre, and the two bodies bear the same relative proportion when seen from Venus. We take the parallax of Venus from that of the Earth so as to ascribe her slight motion to the Earth, and leave her in one position during the occultation. Also, we use the co-latitude of Wellington, because we measure from the pole and not from the equator. At conjunction the centres of the three bodies are in one plane, and while the Moon is moving eastward and southward in the line of her orbit Wellington is travelling along the curve of the ellipse, the places of each being indicated by the time marked on the respective lines. Fig. A.—Elements; ♀. Hour-angle time ♂, 2h. 45m. P. Relative parallax, 5·5′. Declination of ♀ S., 10°. Δ. Diff. of declination of Moon, 29·75′ S. l. Wellington reduced co-lat., 48° 49′. +'s hourly motion E., 28·5′. +'s hourly motion S., 14·1. μ. Moon's semidiam., 15·2′. ♀ 's semidiam., 6·5″. μ + ♀, 15·31′. Results: First contact, 3·30 = 5h. 35m. N.Z. mean t.; last contact, 3·59 = 6h. 1m. First contact, 120° E. of south point and 80° E. of vertex; last contact, 173° E. of south point and 126° E. of vertex. Fig. B.—Apparent path of ♀ behind Moon during the occultation on the evening of the 13th September, 1893.

Occultation of Venus