Greenwich date, and then recalculate the R. A. mean sun for that date. It will be rarely necessary to repeat this method of approximation more than once; but the necessity for this repetition should be borne in mind in many of the subsequent rules when a wrong Greenwich date has been found to have been used. The following examples will show the effect of an error in the Greenwich date on the resulting ship mean time. EXAMPLES. 263. August 11, 1846, at 8h 50m P.M. mean time nearly, in long. 90° W., the hour-angle of Arcturus was 3h 56m 55 west of meridian: required correct mean time at the place. This result is slightly incorrect, arising from the estimated mean time, 8h 50m, being different from the true time. When great accuracy is required, the operation should be repeated, using mean time last found, namely 8h 45m, instead of the one used before; thus, 264. June 15, 1853, at 10h 10m P.M., supposed mean time nearly, in long. 10° 42′ W., the hour-angle of Arcturus was 2h 2m 30s EAST of meridian : required mean time at the place. Ans. 1st approximation, 6h 30m; 2d approx. 6h 30m 35. Elements from Nautical Almanac. Right ascension mean sun. 14h 8m 35 Right ascension star. 4 27 32 6 2 10 ...... 5 34 43 TO FIND SHIP MEAN TIME FROM THE HOUR-ANGLE OF THE SUN. If the heavenly body observed be the sun, its hour-angle will also be apparent time at the place if P.M. at the time of observation, and what it wants of 24 hours if A.M. Therefore, to find the corresponding mean time, we have only to apply apparent time thus found to the equation of time, with its proper sign, as pointed out in Rule 16, p. 90. Rule 22. To find at what time any heavenly body will pass the meridian. 1. Take out of the Nautical Almanac the right ascension of the heavenly body, and also the right ascension of the mean sun for noon of the given day. 2. From the right ascension of the heavenly body (increased if necessary by 24 hours) subtract the right ascension of the mean sun; the remainder is mean time at the ship nearly. 3. Apply the longitude in time, and thus get a Greenwich date; with this Greenwich date correct the right ascension of mean sun, and the right ascension of the heavenly body if necessary. 4. Then from the right ascension of the star subtract the right ascension of the mean sun thus corrected; the remainder is the mean time when the heavenly body is on the meridian. As in the last problem, the table of acceleration for correcting the R. A. of mean sun ought to have been entered with the correct mean time; but the error in this case is inappreciable. EXAMPLE. 265. At what time will Sirius pass the meridian of a place in long. 68° 30′ W. on Nov. 20, 1845? R. A. mean sun. Nov. 20..... 15h 57m 265 Star's R. A... 30h 38m 235 cor. 19h..... 3 7.3 R.A. mean sun 16 0 36 2.5.. ship M. T.. 14 37 47 15m..... R.A. mean sun 16 0 35.8 Therefore the transit of Sirius is at 14h 37m 47 on Nov. 20, or at 2h 37m 47 A.M. on Nov. 21. To find at what time it will pass the meridian on the morning of Nov. 20, we must evidently begin one day back, and take out the right ascension of the mean sun for Nov. 19. 266. At what time will a Pegasi pass the meridian of Portsmouth, long. 1° 6' W., on Nov. 25, 1853? Ans. Nov. 25, 6h 38m 58s. 267. At what time will the star Regulus (a Leonis) pass the meridian of Land's End, long. 5° 42′ W., on May 30, 1845? Ans. May 30, 5h 27m 458 P.M. 268. At what time will Antares pass the meridian of Portsmouth, long. Ans. Aug. 20, 6h 24m 11s. 1o 6' W., on Aug. 20, 1845? 269. At what time will a Leonis pass the meridian of Lisbon, long. 9° 8' W., on June 4, 1846? Ans. June 4, 5h 9m 4s. 270. At what time will the star Antares pass the meridian of Copenhagen, long. 12° 35′ E., on Aug. 20, 1846? Ans. Aug. 20, 6h 25m 213. 271. At what time will the star Fomalhaut pass the meridian of Calcutta, long. 88° 26' E., on Nov. 20, 1846? Ans. Nov. 20, 6h 52m 34s. To find the meridian zenith distance of a heavenly body, or how far it will pass north or south of zenith: 1. Take out the declination, and correct it, if necessary, for the Greenwich date. 2. Under the latitude of the place put the declination, with their proper names N. or S. 3. If the names are alike (both north or both south), take the difference and mark it with the common name of the latitude and declination, if the declination be greater than the latitude, otherwise on the contrary name. 4. If the names are unlike (one north and one south), take the sum and mark it with the name of the declination. 5. The result will be the meridian distance of the heavenly body from the zenith N. or S., according as the result was marked N. or S. EXAMPLE. 272. In latitude 25° Ń. find how far north or south of the zenith the following heavenly bodies will pass the meridian, their declinations being 10° N., 30° N., 10° S., and 50° S. respectively: (1.) ... (2.) ... (3.) lat. ...25° N. lat. ...25° N. lat. ...25° N. decl....10 N. ... decl....30 N. ... decl....10 S. sum ...75 S. 273. At what time will a Columbæ pass the meridian of a place in lat. 42° 20′ S. and long. 54° 40′ W. on May 10, 1856, and at what distance N. or S. of the zenith? Ans. 2h 19m; 8° 11' N. of zenith. 274. At what time will Sirius pass the meridian of a place in latitude 61° N. and long. 10° W. on March 16, 1860, and at what distance N. or S. of the zenith? Ans. 7h 0m; 77° 32′ S. of zenith, We will conclude this chapter by giving brief explanations of some of the principal corrections required for reducing the observations used for finding the latitude, longitude, time at the ship, and variation of the compass the subjects of the next chapter. CORRECTIONS OF THE OBSERVED ALTITUDE OF A HEAVENLY BODY. (79.) The altitude observed at sea by means of the sextant is called the observed or apparent altitude. To obtain the true altitude, or that defined in p. 68, we must apply to the observed altitude (in addition to the index error of the instrument itself) several corrections, the principal of which are the parallax in altitude, refraction, and dip. CORRECTION FOR PARALLAX IN ALTITUDE. (80.) Let A be the place of the spectator on the surface of the earth, c the center, z the zenith, x a heavenly body, and zmr the celestial concave. Through x draw the two straight lines AXMI and cxm to the celestial concave. Then m1 is the observed or apparent place, and m the true place of the heavenly body x. Draw Ar, a tangent to the earth's surface, at A; draw also CR through the center parallel to Ar; then con Z m R sidering the infinite distance of the points and R from the earth, the earth's semidiameter AC will subtend no angle at r or R, and Ar may be conceived to coincide with CR, and therefore the arc rR=0. The observed altitude of x (without reckoning at present refraction) is measured by the arc m1r, and its true altitude by mR=mr. The difference mm, between the true and apparent altitudes, or the angle AXC, is called the parallax in altitude. It appears from the figure that the effect of parallax is to depress bodies, so that the true altitude mR is greater on this account than the apparent altitude m、R, and that the true altitude may be obtained by adding the parallax in altitude to the observed altitude. If x be the same body when in the horizon, the angle Axc is called its horizontal parallax. (81.) It is also evident from the figure that the parallax of a heavenly body is greatest when in the horizon, and that it diminishes to zero in the zenith; that the parallax for different bodies will differ, depending on their distance from the spectator; that the nearer the body is to the earth, the greater will be its parallax: thus the moon's parallax is the greatest of any of the heavenly bodies: the fixed stars, with perhaps a few exceptions, are at such an immense distance, that the earth dwindles to a point so indefinitely small that the radius of the earth AC subtends no measurable angle at a star; hence the fixed stars are considered to have no parallax. Since the form of the earth is an oblate spheroid, the equatorial diameter being about 26 miles longer than the polar diameter or axis, the horizontal parallax of a heavenly body, as observed from some place on the equator, will be greater than the horizontal parallax of the same heavenly body if P' H E observed from the poles of the earth. For let o be a spectator at the equator, and н a heavenly body in his horizon, then the angle H is the equatorial horizontal parallax of the body at H. Similarly to a spectator at P, the pole of the earth, the horizontal parallax of the H' same body would be H', which is evidently less than H, since it is subtended by a smaller radius of the earth; thus it appears from the figure that the horizontal parallax is greatest at the equator, and that it diminishes as the latitude increases. The moon's horizontal parallax put down in the Nautical Almanac is the equatorial horizontal parallax. To find the horizontal parallax for any other place a correction (see Nav. Part II. p. 123) must be applied, which is evidently subtractive: this correction is seldom made in the common problems of Navigation: in finding the longitude by occultations or solar eclipses, it ought not to be omitted. It is inserted in most collections of Nautical Tables. |