« PreviousContinue »
2) 139 23 58
Sum 69 41 59
(MH+SK-MS) 26 8 54
M'H 36 27 54
S'K 7 41 31
the true lunar distance.
36 27 54
M'H+S'K = 44 9 25 nat. cosine
nat. cos M'S 95° 44′ 31′′
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 Mrs. Taylor's method, which requires the use of her "Tables," the true distance is obtained as follows:
+ 4 22
nat. cosine 0.617387
6 20 95 50 53
95 44 33
The apparent altitudes and distance are first obtained from those observed, by correcting them for semidiameter and dip if necessary. Then in Table 1 find the log of the corrections for the altitudes on account of the moon's parallax.
Trom Table 2 take the logs of the effect of the moon's horizontal parallax upon the distance.
Table 3 gives the minutes and seconds answering to these logarithms.
From Table 4, find the effect of the refractions of both objects on the observed distance.
And from Table 5, if the sun is one of the objects observed, the effect of his parallax.
These corrections, applied, with their proper signs, to the apparent distance, give the true distance as above. Mr. Airy makes the following remarks upon the effect of errors of observation in taking lunar distances and lunar transits. A certain error of time produces that same error in the deduced longitude; and an error in the measure of one second produces about two seconds of time in the longitude.
An error of one second of time in a lunar transit produces about 30 seconds error in the longitude.
An error of one second of time in a lunar zenith distance will produce at least 30 seconds of time error in longitude-sometimes considerably more. An error of one second in zenith distances produces at least two seconds of time in longitude. error of one second of time in an occultation produces one second of time in the longitude.
The same with eclipses of Jupiter's satellites.
Instead of measuring the distance between the moon and a star, for a comparison with the time at which the same distance is obtained by calculation for the meridian of Greenwich; altitudes may be taken simultaneously of the moon and a star, from the latter of which, its right ascension and declination being accurately known, the right ascension of the meridian can be computed. This right ascension applied to the moon's distance from the meridian (the angle P in the astronomical triangle) gives the right ascension of the moon, to be compared with the time at Greenwich at which it is identical, for the difference of longitudes.
Another method, applicable particularly to low latitudes *, is to select, when the moon is on or near the prime vertical, any star whose right ascension and declination are known; it being at the time within 8° or 10° of the zenith.
* Obtained from Mr. E. K. Horn.
Take the distance between this star and the moon; also the moon's altitude, and apply the moon's correction in altitude with a contrary sign as the correction in distance; then, with the corrected distance as a base, and the co-declinations as containing sides, the difference of right ascension, and consequently the moon's right ascension, and Greenwich time, are found.
If a star answering to the above conditions is not available, select any star having the same or nearly the same azimuth as the moon, and not less than 30° or 40° distant; the sum or difference of the corrections in altitude would then evidently be the correction in distance. If the star happened to be one of those given in the lunar distance, the Greenwich time is at once found; if not, with the corrected distance as a base, the problem is worked out as before.
The objection to both these methods is, that the moon's declination is required to be known accurately as an important part of the data, to compute which, it is necessary to know the longitude correctly (the very thing sought), except in cases where the moon's declination on either side of the equinoctial is nearly a maximum, and consequently for some time comparatively stationary. Under these circumstances a good result may be expected from the last method when the moon is on, or nearly on, the prime vertical.
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 of time, results a most
*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.
easy and accurate determination of the difference of meridians *; of which the following example is sufficiently explanatory.
At Chatham, March 9, 1838, the transit of a Leonis was observed by chronometer at 10h 52m 463, and of the moon's bright limb, at 10h 20m 7s; the gaining rate of chronometer being 15.
Eastern Meridian Chatham-observed transits.
On account of rate of chronometer .
Equivalent in sidereal time
Difference of sidereal time between the intervals Due to change in time of moon's semidiameter passing the meridian
Western Meridian Greenwich-apparent right ascensions.
9 59 46-18 10 27 16-76
10 52 46 11 20 7.5
0 27 21.5
-0 0 0·03
0 27 21:47
+0 0 ⚫01
Difference in D's right ascension
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 112.77s 1b:: 4.63: 147·80,2′ 27′′-8, 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
0 27 30.58
0 27 25.96
0 0 4.62
* For a more rigid method of computing the difference of meridians by lunar transits, see Baily's Formula and Problems, pp. 239 to 247.
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 :—
▷ Right ascension at Greenwich transit
▷ Right ascension at Greenwich transit
▷ Right ascension at the time, and sidereal
Greenwich mean time correspond-) H.
ing to the above right ascension.11 17 0.5
Or sidereal time at Greenwich 10 25 46.5
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, v. 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