PROBLEMS.

PROBLEM I.

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 of all the quantities being additive.

The tables to be used by 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.

Rule.-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 taken from the table of equivalents.

To convert sidereal into mean solar time :

To the mean time at the preceding sidereal noon, i.e. the transit of the first point of Aries, in table 22, add the mean interval corresponding to the given sidereal time, also from the table of equivalents.

The mean right ascension of the meridian, or the sidereal time at mean noon, is given in the "Nautical Almanac" for the meridian of Greenwich; which must, therefore, be corrected for the difference of longitudes between this place and the meridian of the observer. One of Mr. Baily's formulæ for the solution of the same pro

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 (SAR),and A the acceleration (shown in his 7th table) for the time denoted by M.

Examples.

Convert 8h. 1m. 10s. sidereal time, for March 6th, 1838, in longitude 2m. 21.5. east, into mean solar time, by the method given in the "Nautical Almanac."

Mean time at preceding sidereal noon at Green

wich, table 22 for March

Correction for Longitude :

II. M.

S.

1 4 44.19

Again, to convert 9h. 4m. 35.748s. into sidereal time. O right ascension at mean noon Greenwich, under S head of "Sidereal Time," table 2

Correction for Longitude E:

22h 55m 6.18$

9h. 4m. 35.748. solar time, equivalent sidereal . 9 6 5.2112

* .0027305 is the change in time of sidereal noon in one second; and 0027379 is the change in the sun's mean right ascension in one second of time, or 9.8565 in one hour.

The same examples by Mr. Baily's formula :
:-

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° = 57, and b = 3.2.

The French formula of Laplace is

99918827 x c tan z - 001105603 x c tan 3Z where c is assumed = 60."66.

The tables constructed from these formula are, of course, not exactly similar, on account of the difference of the constants, which are slightly varied in the tables of Bessel, Groombridge, &c. They all suppose a mean temperature, and pressure of the atmo

sphere; and corrections have further to be made on account of the deviation of the thermometer and barometer from these assumed standards. These corrections are, however, rendered perfectly simple in operation, by the use of any of the numerous tables of refraction; that by Dr. Young, being given as table 4 in this volume.

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', or 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 98. 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, by the aid of tables.

=

The parallax given in any ephemeris is the equatorial, which has been shown in page 98 to be always the greatest. The first correction, where great accuracy is required, is to reduce this to the horizontal parallax in 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 (page 266), 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 latitude and altitude require, from their magnitude, more care than with any other celestial body; but in observations at sea the former 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 render the method of making these corrections obvious.

The semi-diameter of the sun is given for mean noon in 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 that of the planets (which is seldom required), in the same table as their parallaxes. The correction for semi-diameter 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 semi-diameter 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, on account of the height of the eye above the horizon, as on shore an artificial horizon is always used. A larger angle is evidently

""Riddle's

* See Table 8 of Lunar Tables, page 188 of Dr. Pearson's " Astronomy." Table," page 154, includes the corrections both for Parallax and Refraction, and is useful for "clearing the lunar distance" to be hereafter explained.

+ Allquantities 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 semi-diameter 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.