61 60 But the angles NEM, NtM are inversely proportional to EM, tм nearly, and tM EM: 60: 61 nearly; therefore NtM = (P's+p), and this may be considered as the angular measure, at the station, of a semidiameter of the earth's shadow in the region of the moon. Again, OTR or QTN is the angle subtended by the sun's diameter; therefore it may be represented by 2s; and QTN may be considered as equal to QtN nearly; therefore the angle MtQ, or the angular semidiameter of the penumbra, is equal to + 2s, nearly. 61 (P′—s + p) 369. In order to determine the phenomena of an eclipse of the moon, there must be found from the Nautical Almanac, for the time above mentioned, the right ascension of the moon, in degrees, and the right ascension, in degrees, of the centre of the shadow; the latter being the sum or difference of the sun's right ascension and 180 degrees: also the declination of the moon, and of the centre of the shadow; the latter being the sun's declination with a contrary name, or on the opposite side of the equator. X P Now the portion of the orbit apparently described by the moon during an eclipse may be considered without sensible error as a straight line; therefore let XY be such portion, and let м be the moon's place in it at the time above mentioned: also let s be the place of the centre of the shadow at the same time, and PS a declination circle passing through it. S Y Let A, in seconds, represent the difference between the right ascension of the moon and of the shadow's centre, that is the angle SPM or the corresponding arc of the equator; and let D be the moon's declination: then a cos. D is equal (in seconds) to the arc Mq of a parallel of declination passing through M. Find, for the same assumed time, the hourly motions of the sun and moon, both in right ascension and declination (the former as well as the latter in seconds of a degree): let the difference between the hourly motions in right ascension, or the moon's relative hourly motion in right ascension, be represented by a, and the moon's relative hourly motion in declination by b. The term a cos. D will express the moon's relative hourly motion on Mq. Imagine the line mn to be drawn parallel to PS; then in the right angled plane triangle Mmn, Mm and mn may represent, respectively, the relative horary motions of the moon upon the parallel of declination and in declination; therefore these are known, and we have again, Mm mn: rad. tan.m Mn: cos. mMn rad. :: Mm: Mn; thus Mn is found, and its value is the moon's relative horary motion in her orbit. The arc sq represents the difference between the moon's declination and that of the centre of the shadow, that is the sum of the declinations of the sun and moon; and in the right angled triangle Msq, considered as plane, we have also sq Mq rad. tan. Msq; sin. Msq rad. :: Mq: MS: thus MS and the angle M8q are found. If sv be let fall perpendicularly on XY, the angle m Mn will be equal to PSV; therefore the angles Msq and PSv being found, we have the angle мsv; and MS cos. MS v = Sv. Let sx and SY be, each, equal to the sum of the semidiameters of the moon and of the earth's shadow; then x and Y will be the places of the moon's centre at the instants that the moon is in contact with the dark shadow of the earth; the former at the commencement, and the latter at the end of the eclipse: if sX, SY were, each, made equal to the sum of the semidiameters of the moon and of the penumbra; then x and y would be the places of the moon's centre at the instant that the moon is in contact with the penumbra. In the right angled triangle xsv we have SX sv rad. cos. XSv: therefore the angle xse is found, and subtracting from it Msv the angle XSM is obtained. Thus in the triangles XSM, YSM, the side SM and all the angles are known; and, by plane trigonometry, there may be found XM and YM: in the triangle MSQ there may be found MQ. Then, the moon's relative horary motion in her orbit being found as above; there may by proportion be found the times in which the moon would describe the lines XM, MQ. Mv and MY: these times being severally applied to the assumed time when the moon was at M, will give the instants of the commencement of the eclipse, the conjunction of the sun and moon in right ascension, the greatest phase or nearest ap proach of the centres of the moon and shadow, and the end of the eclipse. 370. In the Nautical Almanac, under the title "Phenomena," there is given, for an eclipse of the sun which is to take place, the Greenwich mean time of the true conjunction of the sun and moon in right ascension (that is, the conjunction which may be conceived to be observed by a spectator at the centre of the earth); and the first step in computing the times of the commencement, the greatest phase, and the end of an eclipse of the sun for any particular station on the earth's surface, is to ascertain, to the nearest minute, the horary angle of the sun and moon or the angle between the meridian of the station, at the instant of true conjunction in right ascension, and the hour circle passing through the sun and moon at that time. For this purpose there must be taken from the Almanac the sidereal time at Greenwich mean noon on the day of the eclipse, and to it must be applied, by addition or subtraction, according as the place is eastward or westward of Greenwich, the longitude of the place from Greenwich in time; the result will be the sidereal time at the place or the right ascension of the midheaven of the place, at the instant of Greenwich mean noon. To this result there must be added the Greenwich sidereal time of the true conjunction in right ascension, which is obtained by reducing to sidereal time the mean time taken from the Nautical Almanac as above (either by adding the acceleration or using the table of time equivalents); the sum will express the right ascension of the midheaven of the place at the instant of true conjunction in right ascension. From this last result there must be subtracted the sun's right ascension taken from the Nautical Almanac, for the Greenwich mean time of true conjunction in right ascension; and there will remain (in time), for the instant of such conjunction, the angle contained between the meridian of the place and the hour circle passing through the sun and moon, or the apparent time at the place at the instant of true conjunction in right ascension. Let the angle thus found be represented by T. 371. With the value of 7 (in time) there must be found to the nearest minute the time of apparent conjunction in right ascension at the given place, or the time at which the conjunction in right ascension would be observed by a spectator at that place. Thus, let A' represent the true difference between the right ascensions of the sun and moon at any given instant, the moon being above the sun, and the positions of the luminaries such that by the effect of parallax their apparent distance becomes less than their true distance; then, if the sun and moon were in true conjunction in right ascension at a time denoted by 7, the diminution, which may be represented by da', would be the true relative parallax of the moon in right ascension (the true difference between the parallaxes of the sun and moon in right ascension) at the time 7, and it might be considered as the equivalent of a in the term sin. (T+a) (art. 161.), when the horary angle is represented by 7 in degrees. Now A-d A', the apparent difference between the right ascensions of the sun and moon, does not vary with the time; but during a short interval the differential of that difference may be considered as so varying; therefore d (A'-ds') being considered as the variation in a unit of time, as one hour, if t be any short interval expressed in hours, or in a fraction of an hour, reckoned from the given time 7, we should have dt d(A-d At for the apparent relative parallax in right as dt cension at the time expressed by T+t. As an approximation only is required at present, da' sin. 1" (d A' being expressed in seconds of a degree) may be considered as equal to tan. a (art. 161. (II)), and may be represented by the numerator in the equivalent of tan. a; thus we shall have where 7, in degrees, may be put for the horary angle ZPS' (fig. to art. 147.) at the Greenwich mean time of true conjunction in right ascension. From this equation we have, in arc, dr Here represents the diurnal motion of the moon in a unit dt of time, as one hour: now the mean length of a lunar day is about 24.85 solar hours; or, in that time the meridian of a place on the earth deviates 360 degrees from the moon; therefore dmay be represented by or 14° 29′ or 52140". dr 360° 24.85 This value being multiplied by sin. 1" becomes the length of an equivalent circular arc in terms of the radius. If t, in decimals of an hour, express the interval between the instants of true conjunction in right ascension and of becomes, at the latter instant, the apparent difference of right ascension, or the apparent relative parallax, expressed in arc. But this expression is equivalent to the value of sin. a (art. 161. (1)) when the horary angle ZPS' in degrees is equal to 7+15 t; and, in an approximation, for +15 t may be put 7; therefore the above expression may be considered sin. P' cos. l as equal to sin. T. Equating these expressions and COS. D putting P' sin. 1" for sin. P', we have, in decimals of an hour, d A' here (in seconds of a degree), the difference between the dt horary motions of the sun and moon in right ascension, may be taken from the Nautical Almanac for the Greenwich dτ mean time of true conjunction in right ascension: sin. 1" d t (a constant) is equal to 52140 sin. 1", or to 0.25278; and its logarithm 9.40274. The value of t thus found being added to the Greenwich mean time of true conjunction in right ascension will give, approximatively, the Greenwich mean time of apparent conjunction in right ascension: let the latter be represented by T. For the time T thus found there must now be obtained from the Nautical Almanac the following elements: The equatorial horizontal parallaxes of the sun and moon: the difference between these parallaxes is represented by P'. The horary motions of the sun and moon in right ascension: d A' the difference between them is represented by dt The horary motions of the sun and moon in declination: the d D' difference between them is dt The true right ascensions of the sun and moon: the difference between them is a'. |