For the time T' find from the Nautical Almanac the moon's right ascension and declination; and for the time T find the right ascension (art. 312.) of the meridian of the station: the difference between this right ascension and that of the moon being multiplied by 15 gives the moon's hour angle at the station for the instant of the observation. For the time T' find from the Nautical Almanac the right ascension and declination of the sun; also the true horary motions of the sun and moon, both in right ascension and declination. The geocentric latitude of the station is supposed to be known, or it may be found (art. 152.) from the geographical latitude. With the moon's declination, her hour angle, and the geocentric latitude of the station, find the moon's true zenith distance; and with this, find (art. 163.) the augmentation of the moon's semidiameter: the augmentation being added to the semidiameter which is in the Nautical Almanac will give her apparent semidiameter. Find in the Almanac the semidiameter of the sun; and let the sum or difference of these semidiameters be represented by s. For the time T' find from the Nautical Almanac the moon's equatorial horizontal parallax, and subtracting from it the horizontal parallax of the sun (N. A. p. 266.), the remainder is the relative equatorial horizontal parallax. Reduce this (art. 157.) to the relative horizontal parallax for the latitude of the station, and let it be represented, in seconds, by P'. With P', the geocentric latitude, the moon's hour angle, and her declination; find (art. 161. (II)) a, the relative parallax in right ascension, also find (art. 162. (v)) 8, the relative parallax in declination. Applying 8 by subtraction or addition to the moon's true declination, observing that d always increases the distance of the moon from the elevated pole, there will be obtained the moon's apparent declination; and adding to the moon's true hour angle found above the value of a, in time, the sum will be the moon's apparent hour angle. With P', the moon's apparent hour angle, her de clination, and the geocentric latitude of the station; find (arts. dA' dD' 371, 373.) the values of d and d dt dt the variations of the relative hourly motions in right ascension and declination. The differences between the true horary motions of the sun and moon in right ascension and declination, respectively, give the true relative horary motions; and the variations of the relative horary motions being subtracted from the true horary motions, the results will express the apparent relative horary motions in right ascension and declination: the former being mul tiplied by the cosine of the moon's declination gives the apparent relative horary motion on the moon's parallel of declination. Let the last horary motion be represented by p, and let the apparent horary motion in declination be represented by q. The difference between the true right ascension of the sun and moon, found as above for the time T', being taken, if to this be applied the value of a, by addition or subtraction; the result will be the apparent distance in right ascension between the centres of the sun and moon at the same time T', that is at the time of the observation being made at the station. In like manner the value of & being applied to the difference between the true declinations of the sun and moon at the time T will give the apparent distance in declination between the centres: let this last be represented, in seconds, by n. The apparent distance in right ascension being multiplied by the cosine of the moon's declination will give the apparent distance between the centres on the parallel of declination passing through the apparent place of the moon: let this be represented, in seconds of a degree, by m. P Let s be the apparent centre of the sun, and M that of the moon, at the instant of the observation, that is the instant at which the disks of the sun and moon are in contact (the commencement of the eclipse for example); then, if the estimated difference in longitude between Greenwich and the station were correct, it is evident that in the right-angled triangle MNS, the sides M MN and Ns might be represented, respectively, by m and n; and Ms being represented by s, we should have m2 + n2 = s2. But since there will, in general, be an error in that estimated longitude, lett (in decimals of an hour) denote that error; then m +pt being substituted for m, and n + qt for n in the equation, there may be from thence found the value of t. This value, according to its sign, being added to, or subtracted from the estimated difference of longitude, will give (in time) the required difference. Ex. May 15. 1826, at Sandhurst, the commencement of the solar eclipse was observed, by mean time, at 1 ho. 47′ 1′′ (T). Estimated longitude from Greenwich (in time), 3'. Greenwich mean time of the observation at Sandhurst, 1 ho. 50′ 1′′ (T′). With the elements above found, we have by spherical trigonometry, as in art. 358., the moon's true altitude at the station for the time T', 51° 18′ 23. Sin. moon's alt. 14" (art. 163.) 10".93 = Augmentation of moon's semi- Moon's augmented semidi. 15 0.53 diameter Sun's semidiameter 15 49.9 Sum of the semidiameters 30 50.43 or 1850".43 On putting in numbers, with the above elements, the formula for tan. a (art. 161.), we obtain, in angle, a= 17′ 2′′.5 ; and, in time, a=1'8".16. Again, on putting in numbers the formula for tan. 8 (art. 162.), neglecting cos.a, we have 8=30′15′′.2. Also, on putting in numbers the formulæ arts. 371. and 373., dD' d =78".35. In these operations for finding the pa dt rallaxes, logarithms with six decimal places will suffice. Now, putting t (in decimals of an hour) for the error in the estimated difference in longitude between Sandhurst and Greenwich, the equation m2 + n2 = s2 becomes = (1789.3 + 1115.29t)2 + (478.27 +485.85 t)2 = (1850.43)2, from which we obtain t (in hours) - 0.0014, or (in seconds of time)- 5".04; and therefore the difference between the longitudes is, in time, 2′ 55′′ nearly. This determination is, however, subject to an uncertainty which always exists in estimating the precise instant when an eclipse com mences. 383. When the moon passes between the earth and a planet or fixed star, if either the immersion or the emersion take place at the unenlightened edge of the moon's disk, the instant of its occurrence may be easily distinguished from certain parts of the earth's surface; and hence the longitude of the station may be determined with considerable precision. The process for determining the longitude of a station from the observed commencement or termination of the occultation of a fixed star or planet by the moon is very similar to that which has been just described, but is more simple since the parallax and the proper motion of a planet are very small, and those of a fixed star are insensible or may be disregarded. The instant (T) at which the immersion or emersion occurs may be expressed in mean time at the station, and there must be added to it, or subtracted from it, the estimated distance of the station in longitude (in time) from Greenwich: let the result be represented by T'. For the time T' find, from the Nautical Almanac, the moon's right ascension and declination; and, for the time T, compute the right ascension of the mid-heaven: the difference between these right ascensions being multiplied by 15 gives the moon's hour angle at the station for the instant of the observation. For the time T' there must be found from the Nautical Almanac the true horary motions of the moon in right ascension and declination; and also the right ascension and declination of the star. With the geocentric colatitude of the station, the moon's declination and her hour angle, find the moon's true zenith distance, and subsequently the augmentation of her semidiameter: let the augmented semidiameter of the moon be represented by s. For the time T' find, from the Nautical Almanac, the moon's equatorial horizontal parallax, and reduce it by art. 157. to the horizontal parallax (P') for the latitude of the station. With P', the geocentric latitude, the moon's hour angle and her declination, find (arts. 161. (11) and 162. (v)) the values of a and 8, which are the moon's parallaxes in right ascension and declination; and hence obtain the moon's apparent right ascension and declination. Find also the values dD' dt' of d dA' and d which are the variations of the moon's horary movements in right ascension and declination. |