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(82.) 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.
Elements from Nautical Almanac.
Right ascension mean sun.
Nov. 25, 1853 .
16h 17m 22s
4 31 25
Right asc. of star.
22h 57m 26s
16 19 58
22 49 11
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.
Correction for parallax.
The place of a heavenly body as seen, or supposed to be seen, from the centre of the earth, is called its true, or geocentric place: the place of a heavenly body as seen from any point on the earth's surface is called its apparent place. Thus, let ▲ be the place of a spectator on the surface of the earth, p any heavenly body, as the moon. Through p draw the straight lines A pm, cpm from the surface and centre to the celestial concave ; then m is the true place, and m, the apparent place of the heavenly body p. The arc mm,
or angle A p c, is called the diurnal parallax.
It appears from the figure, that the effect of parallax is to depress bodies in a plane passing through the reduced zenith, which coincides nearly with a vertical plane; the diurnal
parallax a p c is therefore usually called the parallax in altitude. If I be the heavenly body in the horizon of the spectator, the angle AHC is called the horizontal parallax of p.
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 reduced 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 line A c subtends no measurable angle at a star: hence the fixed stars are considered to have no parallax.
Since the form of the earth is considered to be 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 observed from the poles of the earth. For let q be a spectator at the equator, and H 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 e earth, the horizontal parallax of the 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 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.
Correction or augmentation of the moon's horizontal
The moon's semidiameter given in the Nautical Almanac is the horizontal semidiameter. When the moon is above the horizon its diameter appears under a greater angle, since the body has approached nearer the
earth; for the distance of the moon at m from the centre of the earth being a little more than sixty times the radius of the earth, cm=60× c I. As the horizontal parallax, cm o, is about 1° only, the line m o is nearly equal to m c. Hence two observers placed, the one at o, the other at I, would see the moon, the first in his horizon, the other in his zenith: but o would see the heavenly body distant a little more, and I a little less, than sixty times the radius; the diameter in fact would appear to the former about 30" less than to the latter. At any intermediate point as at m, the moon's semidiameter would evidently appear to be greater than at o, and less than at I. The correction to be made to the moon's horizontal semidiameter on this account is called the augmentation. It has been computed for every degree of altitude, and may be found in the Nautical Tables.
Correction for refraction.
A ray of light passing obliquely from one medium to another of greater density, is found to deviate from its rectilineal course, and to bend towards a perpendicular to
the surface of the denser medium.
Hence to a spectator on
the earth's surface, a heavenly body seen through the atmosphere appears to be raised, and its true place, on this account, is below its apparent place. Observations show that refraction is greatest when the body is in the horizon (about 34'), and that it diminishes to zero in the zenith. A table of refractions for every altitude has been formed and inserted in the Nautical Tables.
The corrections for parallax and refraction are frequently combined, so that they form one correction, called the "correction in altitude.". The two tables of the correction in altitude for the sun and moon may also be found in most collections of nautical tables.
Correction for the contraction of the moon's semidiameter on account of refraction.
When the moon is near the horizon its disc assumes an elliptical form resembling H, A H, in consequence of the unequal effect of refraction at low altitudes, the lower limb being raised more than the centre, and the centre more than the upper limb. If, therefore, in a lunar observation a contact is made between a distant object s and some point a on the moon's limb, the
contracted semidiameter c A must be added to the arc A S to obtain the distance s c of the centres, and not cн, the moon's uncontracted semidiameter, which is evidently too great. This correction has been calculated, and may be found in the Nautical Tables.
Correction for dip.
The altitude of a heavenly body, observed from a place above the surface of the earth, as on the deck of a ship,
will evidently be greater than its altitude observed from the surface, since the observer brings the image of the body down to his horizon, which is lower than the horizon seen from the surface of the sea immediately below him. The difference of altitude from this cause expressed in minutes. and seconds, is called the dip of the sea horizon. Let a tangent at B, the point on the surface beneath the spectator supposed to be at T, meet the celestial concave at н, and through T draw the tangent т H, touching the earth at R; then, if м be the place of a heavenly body, the arc MH is its altitude observed at B, and м н, the altitude observed by the spectator at T: the arc н H, is the dip due to the height B T of the spectator above the surface of the sea, and is evidently subtractive, to get the true altitude. This correction is found in all collections of nautical tables.
The use of the preceding corrections and reductions will be best seen in the following examples.
Given, a star's observed altitude, to find its true altitude.
The stars are such a distance from the spectator that (excepting probably a few) the earth's orbit subtends no angle at the star: hence a star is considered to have no parallax: and the only corrections used for reducing the observed altitude to the true are the index correction (the correction of the quadrant or sextant used) the dip, and refraction. Hence this rule.
1. To the observed altitude apply the index correction with its proper sign.
2. Subtract the dip (taken from table of dip of horizon). 3. Subtract the refraction (taken from table of refraction). 4. The result is the true altitude of star.