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Volume 6, 1873
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Art. XI.—On the Prediction of Occultations of Stars by the Moon.

[Read before the Auckland Institute, 10th November, 1873.]

All the methods in use for ascertaining the longitude, independently of chronometers, depend upon the observation of the moon's position at a certain instant of time at the place, then ascertaining from the tables of the moon in the nautical almanac, or other similar publications, the instant of time at Greenwich, or other standard position, at which she reaches that point. The difference between the two times so obtained is the difference of longitude between the two places.

The most complete, as well as the most simple, method of making this comparison, and the one almost invariably used for observatory purposes, is to note the exact time of the moon's crossing the meridian of the place by the transit instrument, taking, at the same time, her zenith distance, or not, according to the instrument employed.

But to effect this in at all a satisfactory measure requires an observatory and fixed instruments of an expensive character, and accurate observations kept up for a considerable period and elaborately reduced by computation. It is, therefore, inapplicable to the purposes of a traveller, either by sea or land. The method chiefly employed when an approximate result has to be obtained from a single set of observations, is by observing the moon's angular distance from the sun, a planet, or a fixed star, commonly called lunars.

The chief objections to this method depend on the circumstance that since the moon at fastest moves only about 1 second to 24 seconds of longitude, and ordinarily much less, every second of error in the angular measurement produces an error about thirty times as great in the longitude; and as the observations have often to be taken in very inconvenient postures, in which only light instruments held in the hand are available. On board ship accuracy cannot, as a rule, be expected from them, and in practice they are now but little used—far less frequently, as far as my observation goes, than they used to be forty years ago, though the trouble of computing them has been greatly lightened by special tables.

The only remaining method of importance is by occultations of fixed stars,

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by which a very accurate determination of longitude may be had from a single observation, for which no instrument is necessary but a hand telescope of moderate power.

This excellent method was scarcely available for travellers before binocular telescopes came into use, but they seem to me now to be very unduly neglected, especially by seamen. The causes of this neglect are not far to seek. The first is an irreparable one: they occur very rarely—there are not, on an average, more than twelve each month for each latitude, of which seldom one-half are fairly available for good observation on shore—and with the small-power telescopes, which it would be necessary to use on board ship, the really available cases would hardly occur oftener than two in a month. But another cause of the unpopularity of these observations is, that it is necessary to make a preliminary investigation for each star that seems likely to be available, the result of which, when made, very often is simply to show that it is useless; so that out of half-a-dozen stars predicted it is rarely that more than one proves altogether suitable, and even that one may be lost by a passing cloud.

It must be confessed that this is a discouraging circumstance at the best, and when the prediction required an elaborate calculation, involving the solution of three spherical, and at least two plane, triangles, it was fatal to its use by practical men. I shall proceed, however, to show that the trouble may be reduced to very small dimensions indeed.

The elements necessary for the prediction and computation of occultation are given in the nautical almanac, and more copious ones in the American nautical almanac; but they could only be given without great labour for belts of latitude, and a special investigation has to be made for each place. Various plans have from time to time been published for abridging the labour of these predictions. The first I am acquainted with is a pamphlet by Captain, now Admiral, Shadwell, which was published by the Admiralty in 1847. The principle on which it is based, is to use essentially the same processes as those required for the final computations, but to shorten them by using approximations instead of the accurate elements; by treating all the triangles as plane, and solving them by the use of the traverse-table, with which seamen are very familiar.

But the most practically useful method of approximate prediction is by the method of graphical projection. Drawing a diagram of the earth as it would appear to an eye situated in the star at the moment of conjunction in right ascension, showing the line on it upon which the spectator would be carried in given intervals of time by the earth's motion, then marking a point on the picture at which the moon's centre would be at the same moment, and a line to indicate the direction of her movement, with the points on it which she will reach in given intervals; then it is clear that if the figure of the

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moon covers any point on the earth at the time the spectator is there, the star will not be visible to him, or it will be occulted, and it is obvious that the moment when the moon's image on the drawing just touches the part where the spectator is at the moment, on either side, will be the time at which the disappearance or occultation and egress will occur.

To construct such a diagram as this does not require any considerable calculations, but it is tedious, and, in practice, a pretty expert computer and draftsman would hardly compute it in less than an hour, which is a good deal of labour to expend on a mere preliminary, which may have to be many times repeated before one is come to which proves to be available for observation. A rather large book by Mr. F. C. Penrose, containing an elaborate method of shortening the labour of this graphical process, was published in England three or four years ago. In it there are diagrams ready made, upon which the elements of an occultation, as taken from the nautical almanac, may be laid down, the reduction necessary being made by means of a slide rule. I hardly think that this method will be much used. It is possible that I may be so wedded to methods to which I am accustomed that I do not readily take to other ones; but, to me, it appears quite as troublesome as the ordinary plan as given in Loomis and many other astronomical books, to which, after giving every attention to Mr Penrose's method, I have found it most convenient to adhere; but, in practice, I have adopted some mechanical aids which, without in any degree altering or modifying the plan, assist so largely in carrying it out that I now find that by their use I can predict at least four in the time it used to take me to project one, and without any sensible diminution in accuracy, the result being, that when an occultation occurs while the moon is within an hour or two of the meridian, the prediction is true within a limit of about two minutes, the possible error increasing to about double that quantity when the moon is four hours from the meridian.

In laying down a diagram by any process it is necessary to lay down in their true relative values the magnitude of the earth, or, at least, of the ellipse into which the observer's latitude-parallel is projected; of the different hour spaces upon it; and of the moon and her position and motions. The ready way of doing this is either to take the moon's horizontal parallax in seconds, and to adopt that on a suitable scale as the earth's radius, in which case the values of the hour intervals and of the observer's distance from where the star is vertical must all be reduced to that radius; or an arbitrary radius, as 1,000, may be used, and the value of the hour intervals in the observer's latitude may then be laid off one for all, the same diagrams being used for an indefinite number of predictions; but then all the other quantities must be reduced to that radius, and in either case the ellipses into which the observer's latitude-parallel is projected must be set out for each occultation or eclipse predicted,

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and the values of the hour-spaces marked on it, and this is the most troublesome part of the operation. Penrose's method gives several ellipses already drawn, from which one may be selected which corresponds most nearly with the circumstances of the eclipse to be predicted.

Now the method I employ is the first and simplest one. I take out roughly, by inspection, the moon's horizontal parallax at conjunction, and adopt that as a radius; the position of the observer and of his antipodes are then got out by taking the sum and difference of the declination and the sine of the observer's latitude into the adopted radius. These can then be laid off by scale on a vertical line drawn on any sheet of paper from a fixed point at its upper end, through which a line is drawn perpendicular to it. The moon's place at conjunction can then be marked on the same line by scaling off the distance south in seconds, as given in the nautical almanac, and the moon's hourly motion in right ascension, reduced to an arc of a great circle, is measured on the horizontal line on top, and her motion north or south on the perpendicular line. A line parallel to the diagonal of those co-ordinates drawn through the moon's place at conjunction will give her course, and the distance she travels in parts of an hour may be marked off on it, the times of which should be marked on them. All this is just as would have to be done on the ordinary method, but then, instead of constructing the ellipses representing the parallels of latitude and computing the hour divisions, I keep a set of ellipses, cut out of cardboard, for every 30″ of horizontal parallax, and for every 100′ of semi-minor axis, on which the hour divisions are permanently marked. I see at a glance which ellipse suits the conditions best—that is, the one drawn to the same horizontal parallax, and of which the minor axis corresponds with the distance in seconds of the observer and his antipodes—and at once rule in the curve from the card, and also mark the hour divisions from it. I have then only to take off the moon's semi-diameter, which, bearing a fixed proportion to the horizontal parallax, may be marked off on each cardboard ellipse, and it is the work of a minute to see the moment at which the ingress and egress occurs, and the point on the moon's perimeter at which the star enters and emerges. It is obvious that the same process is equally applicable to solar eclipses, taking of course the differences of their parallaxes and motions and the sum of their semi-diameters.

Now that binocular telescopes are so largely used, and are made to powers so considerable as 7 or 8 diameters, the observation of a star of fourth magnitude entering on the dark limb of the moon—that is before the full—may be perfectly well observed on board ships, and, I believe, in clear weather fifth magnitude stars could be seen; and, as one observation will give a longitude thoroughly trustworthy within very narrow limits, it seems a pity that they should so seldom be used.

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I have here the computations of five occultations which I have observed with a portable telescope at different times without any special care. Of these four agree with one another within a maximum of seven seconds; the fifth is apparently affected by some blunder, as it varies upwards of twenty seconds from the others, but, as even this discrepancy is only about four and a-half statute miles, it might be considered accurate in comparison with the approximations generally available on board ship.

The calculation of longitude from the observations is, it will be observed, not by any means formidable for its length, and it presents no intricacies which an ordinary navigator may not easily master.