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TO CONVERT SIDEREAL TIME INTO MEAN SOLAR TIME, AND THE REVERSE.
THIS problem is of constant use wherever the periods of solar observations are noted by a clock regulated to sidereal time, or those of the stars by a chronometer showing mean time. A simple method of solution is given in the "explanation" at the end of the Nautical Almanac, which has the advantage of not requiring a reference to any other work, and also of all the quantities being additive.
The three tables used in this method are those of equivalents; the transit of the first point of Aries in the 22nd; and the sidereal time at mean noon, in the 2nd page of each month.
To convert sidereal into mean solar time:
To the mean time at the preceding sidereal noon, i. e. transit of the first point of Aries, in table 22, add the mean interval corresponding to the given sidereal time, taken from the table of equivalents.
To convert mean solar into sidereal time :
To the sidereal time at the preceding mean noon, found in table 2, add the sidereal interval corresponding to the given mean time also from the table of equivalents.
The mean right ascension of the meridian, or the sidereal time at mean noon given in the Nautical Almanac, is calculated for the meridian of Greenwich, and must, therefore, be corrected for the difference of longitudes between that place and the meridian of the observer.
One of Mr. Baily's formula for the solution of the same problem is
M = (SAR) — a
and SR + M + A
Where M represents the mean solar time at the place of observation, S the corresponding sidereal time, Æ the mean right ascension of the meridian at the preceding mean noon, found under the head of "sidereal time" in page 2 of each month; a, the acceleration of the fixed stars given in Baily's table 6 for the interval denoted by (S-); and A the acceleration shown in his 7th table for the time denoted by M.
Convert 8h 1m 10s sidereal time, March 6, 1838, longitude 2m 21.5s east, into mean solar time.
Mean time at preceding sidereal noon Green
wich, (table 22)
Correction for Longitude:
2 21.5 or 141.5 *.0027305
Table of Equivalents :
H. M. 8.
8 0 0
0 1 0
0 0 10
1 4 44-19
Mean time required
9 4 35.7487
Again, to convert 9h 4m 35.748s mean solar into sidereal time. O right ascension at mean noon Greenwich,
under head of "Sidereal Time," table 2. . 22h 55m 6.18.
Correction for Longitude E:
9h 4m 35.748 solar time, equivalent sidereal .
Sidereal time required
1 4 44.5763
7 59 51.1724
22 55 4.7926
* .0027305 is the change in time of sidereal noon in one second; and '0027379 is the charge in the sun's mean right ascension in one second of time, or 9.8565 in one hour.
TO DETERMINE THE AMOUNT OF THE CORRECTIONS TO BE APPLIED TO OBSERVATIONS FOR ALTITUDE, ON ACCOUNT OF THE EFFECTS OF ATMOSPHERIC REFRACTION, PARALLAX, SEMI-DIAMETER, DIP OF THE HORIZON, AND INDEX ERROR.
THE formula given by Bradley for computing the value of atmospheric refraction is r = a. tan (Z—br), where Z represents the zenith distance of the object, and a and b constants determined by observation; a, the average amount of refraction at an apparent zenith distance of 45°, being assumed = 57′′; and b = 3′′·2.
The formula of Laplace is
•99918827 × c tan Z-001105603 × c tan 3Z, where c is assumed =60"-66.
The tables constructed from these formulæ are of course not exactly similar, on account of the difference of the constants, which are also slightly varied in the tables of Bessel, Groom
The rate of the increase of refraction is evidently, from the above formula, nearly as the tangent of the apparent angular distance of the object from the zenith in moderate altitudes. In very low altitudes (which should always be avoided on this account) the refraction increases rapidly and irregularly, being at the horizon as much as 33'-more than the diameter of the sun or moon.
The next correction is for parallax, the explanation of which term has been given in page 165. The sine of its value in any altitude decreases as the cosine of that altitude; but the parallax in altitude may be obtained from the horizontal parallax without computation, by the aid of tables.
The parallax given in any ephemeris is the equatorial, which has been shown in page 166 to be always the greatest. The first correction, where great accuracy is required, is on account of the latitude of the place of observation, but this is seldom necessary except in altitudes of the moon. The mean horizontal parallax of the sun is assumed 8"-6; but as our distance from this luminary is always varying in different parts of the earth's orbit, this value must be corrected for the period of the year. In table 8, the sun's horizontal parallax is given for the first day of every month which will facilitate this reduction, the proportional parts being found for any intermediate day. In the Nautical Almanac, however, this quantity is given more correctly for every tenth day. The parallax in altitude, corresponding to this horizontal parallax, can also be ascertained by inspection, from the same general table.
The parallaxes of the planets are given for every fifth day, in the Nautical Almanac; but of those likely ever to be found useful in observation, Venus and Mars are the only planets to whose parallaxes any correction need be applied in observing with small instruments. The horizontal equatorial parallax of the moon
is to be found for mean noon and midnight of every day in the year, in the third page of each month, in the Nautical Almanac. The corrections for its reduction for the latitude of the place, and the moon's altitude, require, from their magnitude, more care than those of any other celestial body; but in observations at sea the former correction is generally neglected, and the latter is much facilitated by the use of tables giving the reduction for every 10' of the moon's altitude*. The example given in this case will explain the method of making these corrections.
The semidiameter † of the sun is given for mean noon on every day of the year, in the second page of every month of the Nautical Almanac; that of the moon in the third page of each month for both mean noon and midnight; and those of the planets (which are seldom required) in the same table as their parallaxes. The correction for semidiameter is obviously to be applied, additive or subtractive, wherever the lower or upper limb of any object has been observed, to obtain the apparent altitude of its centre;—the moon's semidiameter increasing with her altitude, from the observer being brought nearer to her as she approaches his meridian, must be corrected for altitude, which can be done by the aid of table 7+.
The dip of the horizon is a correction only to be applied at sea, and is necessary on account of the height of the eye above the
* See Table 8 of Lunar Tables, page 188 of Dr. Pearson's "Astronomy." Riddle's Table, page 154, includes the corrections both for Parallax and Refraction, and is useful for "clearing the lunar distance" to be hereafter explained.
+ All quantities in the Nautical Almanac are calculated for Greenwich time; allowance must therefore be made, where necessary, for difference of longitude, which is the same as difference of time.
The augmentation of the moon's semidiameter for every degree of altitude is given in Table 7 of Dr. Pearson's "Lunar Tables." Altitudes taken with an artificial horizon are obviously double those observed above the sensible horizon.