Determination of the moon's parallax in altitude. (Art. 154.) 54′ 51′′ or 3291" log. = 3.517328 Cosine moon's altitude (38° 0′ 16′′) log. 9.896527 38 43 29.3 True altitude of the moon's centre The manner of obtaining the declination of the sun or moon for a given time and place, from the value of that element in the Nautical Almanac, has been explained (art. 316.); and, as an example, let it be required to find the sun's declination for a station whose longitude from Greenwich is 50 degrees or 3 ho. 20' (3.33 ho.) in time, westward, at 4 ho. 15' (= 4.25 ho.) P. M., apparent time at the station on the 4th day of May 1843. May 4. Declin., apparent noon, Greenw. (Naut. Alm.) = 15° 51′ 32′′.5 N. (Hourly variation 43.37, Naut. Alm.) Variat. declin. for 4.25 hours diff. of longitude +34.32 =2 24.57 = 15 57 1.39 N. If the variations in the Almanac diminish from one hour to the next, that which is taken must be considered as negative, and, in applying the variation for the difference of longitude, it must be observed that at a given physical instant, the hour at a place eastward of Greenwich is later in the day than the hour at Greenwich. Also, in applying the variation for a given number of hours after noon, that variation must be subtracted if the declination be decreasing; while, for a given number of hours before noon, the variation must be subtracted if the declination be increasing. PROB. I. 334. To find the latitude of a ship or station by means of an observed altitude of the sun when on the meridian. E S S P Let PZSH be part of the meridian in the heavens, C the centre of the earth, z the zenith of the station, P the pole of the equator, and E the intersection of the equator with the meridian. Then the correct altitude of the sun's centre being found as in Ex. 1. above, let it be represented by SH, its complement is zs the zenith distance. The sun's declination must be obtained from the Nautical Almanac, for the time of the observation, by an estimate of the distance of the ship, H or station, in longitude from Greenwich; let this be represented by SE (the sun being supposed to have north declination): then it is evident that the sum of the arcs zs and SE will give ZE, the required latitude. If the sun's declination had been south, as Es', it must have been subtracted from the zenith distance zs' in order to give the latitude. If the earth were a sphere, this would be the geocentric latitude; if a spheroid, it would express the angle between a normal at the station and the equator (art. 151.), and the correction, art. 152., may, if necessary, be applied in order to reduce it to the geocentric latitude. Ex. 1. At Sandhurst, July 28. 1843, by reflexion from mercury, the double altitude of the sun's upper limb was found to be 116° 5′ 15′′ Index error of the sextant (subtractive) 1 15 The corrections on account of refraction, the sun's parallax in altitude and his semidiameter, found as above-mentioned (art. 333.), amount to Correct altitude of the sun's centre Zenith distance of sun's centre To determine the sun's declination, the longitude of Sandhurst from Greenwich being about three minutes (in time) westward. Sun's declination at Greenwich apparent noon, from the Naut. Almanac is 19° 6' 12'.7 N., and the hourly variation is 35" hence the variation of declination for 3 minutes westward is Sun's declination at the time and place of observation (nearly) Latitude of the station - 1.75 In the above example, and also in those which follow, the correction on account of refraction is supposed to have been taken from a table of refractions: the sun's parallax in altitude may also be obtained from a table, or by multiplying the horizontal parallax (Naut. Alm., p. 266.) by the cosine of the observed altitude. By a process exactly similar to that which is employed for the sun may the latitude of a station or ship be computed from a meridian altitude of the moon, a planet, or a fixed star. A fixed star has no sensible parallax, and that of Jupiter scarcely exceeds two seconds when greatest, but the horizontal parallax of Mars may amount to about nineteen seconds, and that of the moon to above sixty one minutes; therefore, when the altitude of either of these last celestial bodies is employed in a problem of practical astronomy, the parallax in altitude, or the product of the horizontal parallax and the cosine of the altitude, is to be subtracted from the altitude in order to reduce it to that which would have been obtained from an observation at the centre of the earth. Ex. 2. To find the latitude of a station by a meridional altitude of a planet. 44° 11' 20" At Sandhurst, Nov. 27. 1843, at about 5 P. M., by reflexion from mercury, the double altitude of Mars was found to be Index error of the sextant (subtractive) 42 Ex. 3. To find the latitude of a station from an observed altitude of the moon when on the meridian. Nov. 29. 1843, at 6 ho. 36′ mean time at the station, the correct altitude of the moon's centre when on the meridian was (Ex. 2., art. 333.) 38° 43′ 29.3. Therefore the zenith distance = Determination of the moon's declination at 6 ho. 39′, the Greenwich mean time of the observation. 51° 16′ 30."7 Moon's declin. at 6ho. P. M. (Naut. Alm.)=—0°3′ 43′′.2 (S.) + 7 48.1 335. Since a method of determining the latitude of a ship without taking the altitude of a celestial body may have some use when great accuracy is not required, it will be proper to mention here that an approximation may be made to a knowledge of that element by observing with a watch the time in which the diameter of the sun ascends above, or descends below the horizon. Let this be done: and, in the annexed The \ diagram, let a and B represent the places of the sun's centre when its upper edge and lower edge touch the horizon, as at a and b, in descending, for example; and let c be the place of the sun's centre when in the horizon. lines Aa, Bb, are semi-diameters of the sun, passing through the points of contact; therefore perpendicular to the horizon, and passing through z, the zenith of the observer, if pro B C a A Z E duced. The right angled triangles AaC, Bbc, are equal to each other, and on the parallel AB of declination, the arcs AC, Ad (the latter equal to Aa) have to one another the same ratio as the angles APC, APd (P being the pole of the equator); that is the same as half the time in which the diameter descends has to half the time in which the diameter would pass, by the diurnal movement, over the meridian, or over any horary circle. Now, in the triangle aac considered as plane, AC Aa: rad. : sin. aCA; that is, the time of the semidiameter descending, is to the time of its transit over the meridian (Naut. Alm., p. I. of the month), as radius is to sin. aCA. But, in the spherical triangle CEt, formed by the equator Et, the horizon CE and the hour circle PC, in which triangle ct denotes the sun's declination, and the angle ECt is the complement of a CA, we have (art. 60. (ƒ)) Rad. cos. CEt cos. Ct sin. ECt. = Thus there may be found the angle CEt, which is measured by PZ, the colatitude of the ship: the required latitude is therefore found. In this problem no attention has been paid to the effects of refraction. B 336. In the arctic or antarctic regions, when the sun, the moon, or a star has considerable declination towards the elevated pole, it is visible at its culmination below as well as above the pole; and in this case the latitude of a station or ship can be obtained by the meridional altitudes in both situations. Thus the benefit of having two observations is gained, and the knowledge of the declination can be dispensed with. Let the primitive circle QR be the horizon, z the zenith, and P the pole of the equator; also let A and B be the places of any celestial body at the times of culmina R tion above and below the pole, the observed altitudes having been corrected on account of parallax and refraction: then, the polar distances PA and PB being equal, we have AZ + ZP=ZB-ZP; whence 2 Z PZ B—ZA: that is, the colatitude ZP is equal to half the difference between the two zenith distances, or between the two altitudes. On land, in the tropical regions, the chief difficulty attending the determination of the latitude by meridional observations with an artificial horizon, arises from the sun, moon, and planets being, at the time, very near the zenith; on which account the ordinary reflecting sextants or circles, from the great obliquity of the mirrors to each other, cannot be used to take angles equal to twice the altitude of the celestial body above the horizon: also the sun, moon, and stars then change their altitudes very slowly, so that it is difficult to ascertain the moment when they are in the meridian. In this case, recourse must be had to observations taken at times considerably before or after the time of culmination. Observations taken at such times are, in fact, most generally employed in all climates, as many circumstances, particularly a cloudy sky, may prevent the celestial body from being observed on the meridian: but before the formulæ relating to such observations are investigated, it will be proper to give the problems for determining at any instant the hour of the day or night at the station or ship. PROB. II. 337. To find the hour of the day by an observed altitude of the sun; the latitude of the station, and the sun's declination being known. N. B. It is to be understood that the time of the day should be previously estimated within an hour of the truth, and also that the longitude of the station is known or can be ascertained within a few degrees, in order that the sun's declination may be found for the time, from the Nautical Almanac, with sufficient precision. Let the diagram represent a projection of the sphere on the plane of the horizon of the station; then z, the centre, will be the zenith, and a diameter as PZH will represent the meridian. The altitude of the upper or lower limb of the sun being observed, and the corrections made by which the altitude of the sun's centre is obtained, the complement of that altitude may be expressed by the arc zs in the sphere. E D A H S O |