EXAMPLE. The moon's right ascension, July 12, 1836, was, by observation, 6h. 36m. 39s.35. Required the mean time of observation. Given the distance of the moon from a fixed star not marked in the Nautical Almanac, together with the altitudes of the objects, the mean time of observation, and the estimated longitude, to find the longitude of the place of observation. First solution, using the latitudes and longitudes of the moon and star. RULE. To the mean time of observation, by astronomical computation, add the estimated longitude in time if west, or subtract if east; the sum or difference will be the supposed mean time at Greenwich," corresponding to which, find the moon's latitude, by Problem I., also the longitude and latitude of the star, by Table XXXVII., and correct them for aberration, by Table XLI. With the apparent altitudes and distance of the objects, find the correct distance by the usual rules of working a lunar observation. To the correct distance, add the latitudes of the moon and star, and find the difference between the half-sum and the distance. Then to the log. secants of the latitudes of the moon and star, rejecting 10 in each index, add the log. cosines of the half-sum and difference, if the latitudes are of the same name, or the log. sines, if of a contrary name; half the sum of these four logarithms will be the log. cosine of half the difference of longitude, if the latitudes are of the same name, or its log. sine, if of a different name. The difference of longitude is to be added to the apparent longitude of the star, if the moon is east of the star, otherwise subtracted, (borrowing or rejecting 360° when neces sary;) the sum or difference will be the true longitude of the moon; whence the mean time at Greenwich may be found, by Problem XIV. The difference between this and the mean time at the ship, will be the longitude, which will be west, if the mean time at Greenwich be greater than the mean time at the ship, otherwise east. REMARK. This method, with a slight modification, can be used in finding the longitude from the observed distance of the moon from a planet, as Jupiter, Venus, Mars, or Saturn, in cases where they are not marked in the Nautical Almanac. The only difference in the rule, when a planet is used instead of a star, consists in finding from the Nautical Almanac, by Problem I., the geocentric longitude and latitude of the planet, which are to be used instead of the longitude and latitude of the star in the above rule. For the daily variation of the longitude and latitude of a planet is so small, that no error of moment can arise from calculating those quantities for the supposed instead of the true time at Greenwich; and the parallax and semi-diameter of the planet can be allowed for by the methods pointed out in working a lunar observation. The latitudes of the moon and the fixed star or planet, made use of in these observations, ought not to differ very much, on account of the decrease of the relative motion arising from this source. If the latitudes are of a different name, their sum, otherwise their difference, ought to be found, and if it does not exceed one third part of the difference of longitude of the two objects, they may in general be made use of. EXAMPLE. Suppose that, on the 7th of January, 1836, sea account, at 11m. 57s. past midnight, mean time, in the longitude of 127° 30' E., by account, the observed distance of the farthest limb of the moon from the star Aldebaran, was 68° 36' 0", the observed altitude of the star 32° 14', and the observed altitude of the moon's lower limb 34° 43'. Required the true longitude, without using the distances marked in the Nautical Almanac, upon the supposition that they are not given in it. This lunar observation has already been computed by the common methods, in page 232, where we have found that the supposed time at Greenwich is Jan. 6d. 3h. 41m. 57s., the moon's semi-diameter 15 15", the moon's horizontal parallax 55' 24", the star's apparent altitude 32° 10′, the moon's apparent altitude 34° 55', the apparent distance of the centres of * This time may also be obtained from the chronometer, if you have one which is pretty well regulated to astronomical time the moon and star 68° 20′ 45". With these we find the true distance of the centres of the moon and star, by the usual rules for working a lunar observation, to be 68° 3' 0", as in page 232. The moon's latitude, deduced from the Nautical Almanac, by Problem I., is 4° 59' 10 N. Then the star's longitude and latitude are found as below, by Tables XXXVII., XLI., making use of the sun's longitude, 285° 17', as given in the Nautical Almanac, these longitudes being counted from the mean equinox; with these elements the calculation is made in the following manner : Table XXXVII.....*'s longitude, Jan. 6, 1836.... 67° 29′ 47′′.1....... ..'s latitude....... 5° 28′ 394.0 8. Table XLI.......Aberration ........ +15.9. *'s apparent longitude.................. True distance.... D's latitude....... *'s latitude Sum........ Half-sum.... ..... 67 30 03 68° 03' 00" 4 59 10 N. 5 28 40 S... 78 30 50 ........Aberration......... +1.2 *'s apparent latitude 5 28 40 .0 8. ..Secant 0.00164 ......Secant 0 00199 39 15 25 Difference of half-sum and distance 28 47 35. 8h. 30m. 22s.=127° 35′ 30′′ E. from Greenwich, differing 5 15 from the calculation in page 232. The computed time at Greenwich, 3h. 41m. 35s., differs from the assumed time, 3h. 41m. 57s., only 22s.; and, during this interval, the moon's latitude varies so little, that it will not be necessary to repeat the operation on account of this variation; observing that an error of one minute in the moon's latitude affects the secant of the latitude about 0.00001, and this produces in the difference of the longitude an error of only 2" or 3" in the present example; and as the latitudes are always small, it will hardly ever be necessary to repeat the operation when this method is used. Second solution, using the right ascensions of the moon and star. RULE. To the mean time of observation, by astronomical calculation, add the estimated longitude in time if west, or subtract if east; the sun or difference will be the supposed mean time at Greenwich. This time may also be taken from the chronometer, if you have one which is pretty well regulated for mean time at Greenwich. With this time, enter the Nautical Almanac, and find from it the right ascension and declination of the star or planet, and the declination of the moon. With the apparent altitudes and distances of the objects, find the correct distance by the usual rules of working a lunar observation. To the correct distance add the declinations of the moon and star, and find the difference between the half-sum and the distance. Then to the log. secants of the declinations of the moon and star, rejecting 10 in each index, add the log. cosines of the half-sum and of the difference, if the declinations are of the same name, or the log. sines, if of a contrary name; half the sum of these four logarithms is to be sought for in the column of log. cosines, if the declinations are of the same name, or in the column of log. sines, if of different names; and the corresponding time in the column P. M. is the difference of the right ascensions of the moon and star. This difference of right ascension is to be added to the apparent right ascension of the star, if the moon is east of the star, otherwise subtracted, (borrowing or rejecting 24h. when necessary;) the sum or difference will be the true right ascension of the moon's limb. If the moon's true right ascension can be found exactly in the Nautical Almanac, the corresponding hour will be the mean time at Greenwich. If it cannot be found exactly, as will most commonly happen, take out the right ascensions for the hours immediately preceding and following, and note their difference, D, in seconds of time; take also the difference, d, in seconds of time, between the moon's true right ascension and that right ascension marked for the first hour in the Nautical Almanac. Then, to the constant log. 3.55630, add the arithmetical complement of the logarithm of D, and the logarithm of d; the sum, rejecting 10 in the index, will be the logarithm of a number of seconds, to be added to the hour first marked in the Nautical Almanac, to obtain the mean time of the observation at Greenwich. The difference between this and the mean time at the ship, will be the longitude, which will be west, if the mean time at Greenwich be greater than the mean time at the ship, otherwise east. We may observe, that we can, as in the first solution, use a planet instead of a star. We shall now calculate, by this method, the same example as in the first solution. In this case, for the supposed time at Greenwich, January 6d. 3h. 41m. 57s., we find, by means of the Nautical Almanac, Aldebaran's right ascension 4h. 26m. 31s.3, Aldebaran's declination 16° 10′ 29′′ N., and the moon's declination 21° 9′ 33′′ N. 1 Half-sum ...... 105 23 02 52 41 31... Difference of half-sum and distance 15 21 29. Difference of and D's right ascensions 4h. 48m. 56s.9.. Cosine 9.78254 2) 19.81460 Cosine 9.90730 Given the intervals of time between the passages of the moon's bright limb and a fixed star over two different meridians, to find the difference of longitude between the two meridians. This problem includes, also, the case where one of the observations is supposed to be made at Greenwich, considering the time of the transit of the moon's bright limb over that meridian, given in the Nautical Almanac, as an actual observation; the error arising from this supposition being very small, on account of the great degree of accuracy of the lunar tables used in the computation of the Nautical Almanac. We may, however, observe that, where good observations can be obtained at both meridians, it is always best to use them in preference to the computed transits in the Nautical Almanac. The principle upon which the longitude is found in this method is similar to that which is used in a common lunar observation, and depends on the observed motion of the moon; but, in the present problem, this motion is ascertained by observing the time when the moon's bright limb passes the meridian, instead of measuring the angular distance of the moon from the sun or a star. The variation of the moon's right ascension, corresponding to a change of 15° in the longitude, is given very accurately by the Nautical Almanac for every transit of the moon's limb at Greenwich. This variation is about 2m. in time for 1h. of longitude, and when the difference of the times of transit under different meridians has been found by observation, it is easy to get, by proportion, the corresponding longitude, as we shall see in the following examples. This method of computing the longitude is very much facilitated by the moon-culminating stars, inserted in pages 410-451 of the Nautical Almanac. construction of the table, we shall insert the following extracts from it, page 438 of the Nautical Almanac for 1836. new table of To show the contained in *Use sine if the declinations are of different names The stars whose right ascensions and declinations are inserted in this table, are called moon-culminating stars, because they have nearly the same declination as the moon, and do not differ much in right ascension, so that they are conveniently situated for observations of the differences of the times of the transit which are required in this problem. The first column of this table contains the date; the second, the name of the star or moon. If the bright limb of the moon be the first which passes the meridian, it is marked I.; but if it be the second limb, it is marked II. The upper culmination of the moon is marked v.c.; the lower culmination, l. c.; this last being of frequent use in high latitudes. The third column contains the magnitudes of the objects; that of the moon being denoted by her ge, expressed in days and tenths of a day. The fourth column contains the apparent right ascension of the moon's bright limb, at the time of the transit over the meridian of Greenwich; and the fifth column, its declination at that time: the same columns contain also the right ascensions and declinations of the moon-culminating stars at their upper culmination. The sixth column contains the variations in the right ascension of the moon's bright limb during the intervals of her transit over two meridians; one of these meridians being yo 30 W. from Greenwich, and the other 7° 30 E. from Greenwich; so that the distance of these two meridians is 15°, or 1h. in longitude. For convenience of reference, we shall call this variation the arc H, supposing it to be expressed in seconds of time, as in column 6. The arcs H, in the sixth column, are deduced from the right ascensions of the moon's bright limb, contained in the fourth column, so that they include the effect produced by the changes of the moon's semi-diameter. The seventh column contains the intervals of the transit of the moon's semi-diameter over the meridian expressed in sideral time; this time being generally used in making such observations, and for this purpose it is usual to note the times of transit by a clock regulated to sideral time. If the intervals are given in mean time, they may be reduced to sideral time by adding the correction in Table LI. corresponding to that time. Thus, if the interval is 6h. mean time, the tabular correction in column 1 of that table is 59s.1, making the interval 6h. Om. 59s.1, sideral time. If the interval be 6h. 58m. mean time, the corrections in Table LI., columns 1,2, are 59s.1+9s.5=1m. 8.6; consequently the interval in sideral time is 6h. 59m. 8s.6. The numbers in columns 4, 5, 6, 7, of the table of moon-culminating stars, correspond to the meridian of Greenwich, and may be reduced to any other meridian by the usual method of interpolation, as in Problem I., page 396. Thus, from the above extracts from this table, it appears that, at the time of the upper culmination, September 16, 1836, the right ascension of the moon's bright limb was 16h. 06m. 21s.71. At the following lower culmination, it was 16h. 37m. 12s.45, which may be considered as corresponding to the upper culmination, September 16, in a place 12h. in longitude west from Greenwich; and at the next upper culmination, the right ascension was 17h. 09m. 01s.65, which may be considered as appertaining to September 16, in a place 24h. west from Greenwich; according to the ancient method of counting the longitude, in a westerly direction completely round the globe. In like manner, in east longitude, we have, at the upper culmination at Greenwich, September 16, 1836, the right ascension of the moon's bright limb 16h. 06m. 21s.71, and we may suppose the preceding transit, 15h. 36m. 34s.71, to correspond to the longitude 12h. east, and so on. This being premised, we shall now proceed to show how to find, by interpolation, the moon's right ascension at the time of her transit over any meridian in east or west longitude from Greenwich. The process of calculation is very nearly the same as that in Problem I., page 396, but for convenience we have reduced it to the following form: RULE. To find the moon's right ascension at her transit over any meridian. 1. Take from the fourth column of the table of moon-culminating stars, the right ascensions of the same limb of the moon corresponding to four successive culminations," so that *Near the time of full moon, when the limb marked in the table changes from I. to II., there may be one or two of these quantities not marked in column 4th of the table for the limb which is wanted in the two may precede and two follow after the time of transit at the proposed place. Put these numbers below each other in their regular order; then find their first and second differences. Call the middle term of the first differences, the arc A; the mean of the 2nd differences, the arc B; and if the longitude be west from Greenwich, put T equal to that longitude in time; but if the longitude be east, put T equal to the difference between 12h. and that longitude. 2. To the constant logarithm 5.36452 add the logarithm of T in seconds of time, and the logarithm of A in seconds of time; the sum, rejecting 10 in the index, will be a proportional part, which is to be added to the second right ascension taken from the Nautical Almanac. 3. Enter Table XLV. with the arc B at the top, and the time T at the side; opposite to this will be the correction of second differences, to which prefix a different sign from that of the arc B, and place it under the second ascension and the proportional part above found. Connect these three quantities together, as in addition in algebra; the sum will be the sought right ascension of the moon at the time of her transit over the proposed meridian. The same process may be used for interpolating the numbers in columns 5, 6, 7, as we shall see in the following examples : EXAMPLE 1. Required the right ascension of the moon, September 16, 1836, astronomical account, at the time of the transit over the meridian of a place whose longitude is 3h. 48m. 29s. west from Greenwich; also, the value of the arc H, deduced from the numbers in column 6, for the time of this transit. Here we have T=3h. 48m. 29s., being the same as in Example I., page 396; this value being selected in order to show more readily the similarity of the present calculation with that in page 396. Hence it appears, that, on September 16, 1836, astronomical account, in a place 3h. 48m. 29s. west from Greenwich, the right ascension of the moon's bright limb at the time of passing the meridian, was 16h. 16m. 02s.42, and that the arc H, corresponding to that meridian, was 1538.30. This arc H represents the variation of the moon's right ascension between the times of the transit of her bright limb over the two meridians whose longitudes are T-30m. and T+30m., corresponding respectively to 3h. 18m. 29s. west, and 4h. 18m. 29s. west, from Greenwich. In the preceding example, the longitude of the place is given, to find the moon's right ascension at the time of the passage of her bright limb over the meridian of that place; but we may suppose that right ascension to be given, to find, by an inverse process, the longitude of the place of observation, or the time T. The solution of this problem is very similar to that of Problem XIV., page 426, changing longitude into right ascension, &c.; and it may be expressed as in the following rule : RULE. To find the longitude of any place from the moon's right ascension at her transit over the meridian of that place. 1. Take from column 4 of the table of moon-culminating stars, in the Nautical Almanac, the four right ascensions of the bright limb of the moon, as in the above example; and then compute, as in that example, the values of the arcs A, B, in seconds of time. 2. To the constant logarithm 4.63548 add the arithmetical complement of the logarithm of the arc A in seconds of time, and the logarithm of the difference in seconds of time between the given right ascension and the second right ascension taken from the Nautical adding or subtracting the sideral time of the moon's disk passing the meridian, deduced from column 7. Thus, at the upper culmination of the moon, September 16, the right ascension of limb I. is 16h. 06m. 21s.71, in column 4; and in column 7, the sideral time of the moon's semi-diameter passing the meridian is 72s.52 the double of this, or 145s.04=2m. 258.04, represents the time required for the whole disk to pass the meridian. Adding this to 16h. 06m. 21s.71, we get 16h. 08m. 46s.75, for the right ascension of the limb II., at the time of its transit. In like manner, if the quantity 2m. 25s.04 be subtracted from 16h. 08m. 46s.75, which corresponds to the limb II., we shall get the right ascension corresponding to the limb I., supposing the time of the transit of limb II. to be marked in the Nautical Almanac. |