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The same example, by Mr. Riddle's first method, which will be found in his "Navigation," gives 95° 44 29" for the corrected lunar distance.
BY THE METHOD OF MOON CULMINATING STARS.
The proper motion of the moon causing a difference in the interval of time between her transit and that of any star, over different meridians, affords another method of determining the longitude.* The times of transit (or apparent right ascension) of the moon's enlightened edge, and that of certain stars varying but little from her in declination, are calculated for Greenwich mean time, and given among the last tables in the "Nautical Almanac." The transits of the moon's limb, and of one or more of these stars are observed at the place whose longitude is required, and from the comparison of the differences of the intervals
* The time of the moon's transit compared with that observed at, or calculated for, another meridian, would be sufficient data for ascertaining differences of longitude; but by making a fixed star the point of comparison, we obviate any error in the position of the instrument, and also of the clock.
of time, results a most easy and accurate determination of the dif ference of meridians; of which the following example is sufficiently explanatory.
At Chatham, March 9th, 1838, the transit of a Leonis was observed by chronometer at 10h 52m 46s, and of the moon's bright limb, at 10h 20m 75; the gaining rate of chronometer being 1.5.
Difference of sidereal time between the intervals
Due to change in time of semidiameter passing meridian
Difference in D's right ascension
10 27 16.76
0 27 30.58
0 27 25.96
0 0 4.62
0 0 .01
0 0 4.63
The variation of D's right ascension in 1 hour of terrestrial longitude is, by the "Nautical Almanac," 112.77 seconds. Therefore as 112.77s1h: 4.638: 147.80, 2′ 27′′ .80, the difference of longitude.
But when the difference of longitude is considerable, instead of using the figures given in the list of moon-culminating stars for the variation of the moon's right ascension in one hour of longitude, the right ascension of her centre at the time of observation should
be found, by adding to, or subtracting from the right ascension of her bright limb at the time of Greenwich transit, the observed change of interval, and the sidereal time in which her semidiameter
passes the meridian. The Greenwich mean time corresponding to such right ascension being then taken from the "Nautical Almanac," and converted into sidereal time, will give, by its difference from the observed right ascension, the difference of longitude required. For instance, in the above example :—
D Right ascension at Greenwich transit
dian of place.
▷ Right ascension at Greenwich transit Observed difference
> Right ascension at the time, and sidereal
time at the place, of observation
Greenwich mean time correspond
H. M. S.
ing to the above right ascension. 11 17 0.5
Or sidereal time at Greenwich 10 25 46.5
Difference of longitude
0 2 27.9
BY OCCULTATIONS OF FIXED STARS BY THE MOON.
The rigidly accurate mode of finding the longitude from the occultation of a fixed star by the moon, involves a long and intricate calculation, an example of which will be found in the 37th chapter of "Woodhouse's Astronomy"; and the different methods of calculating occultations, are analyzed at length by Dr. Pearson in his "Practical Astronomy," commencing at page 600, vol. ii.
The following rule, however, taken from " Riddle's Navigation," will give the longitude very nearly without entering into so long a computation :
Find the Greenwich mean time from knowing the local time and the approximate longitude, and for that time take with the greatest exactness from the "Nautical Almanac" the sun's right ascension, and the moon's polar distance, semidiameter, and parallax, applying all corrections.
To the apparent time, add the sun's right ascension, and the difference between this sum, and the star's right ascension, will be the meridian distance of the latter. Call this distance P; the star's polar distance p; its right ascension R; the reduced colatitude 7; the moon's polar distance m; her reduced horizontal parallax H; and her semidiameter s.
sum, rejecting twenty, will be the tangent of arc a, of the same affec
sum rejecting twenty, will be the tan. of arc b, (always acute). When l is greater than p, a +b=arc c; and when l is less than p, barc c.
Add together tan c, cosec l, cosec P, and prop. log. H, and the sum rejecting tens, is prop. log. of arc. d. When arc c is obtuse, p+darc and when c is acute, p d
Add together cosec. l, cosec. P, prop. log H, and with the sum S, and p, take the correction from the subjoined table, and applying it with its proper sign to e, call the sum or the remainder e'. difference of m and e' is arc f.
To S add sin e', and the sum, rejecting tens, is the prop. log of
To the prop. logs of s+f, and s--f, add twice the sine of arc e, and half the sum, rejecting the tens, is the prop. log. of arc h.
Then the moon's right ascension = R± g±h, where g is additive west of the meridian, and subtractive east; and h is additive at an emersion, and subtractive at an immersion.
Having found the moon's right ascension, the corresponding Greenwich time is to be found from the " Nautical Almanac," the comparison of which with the local time gives the longitude of the place of observation.
TO DETERMINE THE DIRECTION OF A MERIDIAN LINE AND
THE VARIATION OF THE COMPASS.
In the spherical triangle ZPS already alluded to, as the astronomical triangle, and in which the co-latitude ZP, and the time represented by the angle P, were ascertained by the method of absolute altitudes in pages 154 and 158, the azimuth of any celestial body S is measured by the angle Z, which is found from knowing either the time, or the latitude, in addition to the observed altitude. This calculated azi_ muth compared with the magnetic bearing of the object observed at the same instant, and determined with reference to some well defined terrestrial
mark, affords the means of laying down a meridian line, and gives the variation of the compass.