740
ASTRONOMY
h being a numerical constant. It follows from this form that the ratio of error in 7r0 would he multiplied three times in the resulting value of p. As a matter of fact we can determine p with greater proportional precision than the parallax, so that the logical course is to determine 7r0 from p rather than the reverse. The fundamental gravitational relations on which the motions of precession and nutation depend may be stated as Precession follows :—We put P, P' = the portions of the lunisolar and precession in a Julian year, due to the action of the nutation, moon and sun respectively, N = the constant of nutation, p. = the ratio of the mass of the moon to that of the earth, p! — ^ , A, C = the equatorial and polar moments ’ 1+M —A a quantity which may be called of inertia of the earth, <7 = C—^—, the mechanical ellipticity of the earth, and 6 = the obliquity of the ecliptic. The theory of the moon’s motion and of its action on the earth give the following equations :— N =[5'40289] p! q cos e ] P = [5•975052] p' q cos e Ml) P'= [372509] q cos e J In these expressions q and p' are absolute constants to be determined, while e varies slowly, but is known. The values of N and of P + P' are given by observation. At a conference of the directors of four national astronomical ephemerides, held at Paris in 1896, it was decided to adopt N = 9,210". By a discussion undertaken at the request of the same conference it was found that, for the epoch 1900, P + P, = 50-3722" (2) For the same epoch the obliquity is e=23° 27' 8-26". With these numerical values the equations (1) and (2) are four in number, which suffice to determine p', q, P, and P' for 1900. The solution gives the following results :— Julian Tear Solar Year. Lunar precession for 1900 34-3877" 34-3870" Solar „ „ „ 15-9842" 15-9838" Lunisolar „
„
,, 50-3719" 50-3708" C-A 9'=-^—= 0-0032813 Mass of moon-1-mass of earth = 1-^81-65. Recent research enables us to compute the obliquity of the at: ,ast Obliauitv i and future epochs with an error not nf -..tL exceeding 1" per table century p ‘ in the following
- — elapsed. The result is shown
Year. Year. Obliquity. Obliquity. B.C. A.D. // tr 3000 24 1 16-5 0 23 41 42-4 2900 24 0 44-1 100 23 40 57-9 2800 24 0 11-1 200 23 40 132700 23 59 37-5 300 23 39 281 2600 23 59 3-5 400 23 38 42-8 2500 23 58 28-9 500 23 37 57-4 2400 23 57 53-7 600 23 37 11-8 2300 23 57 18-1 700 23 36 26-1 2200 23 56 41-9 800 23 35 402100 23 56 5-3 900 23 34 54 T 2000 23 55 28-2 1000 ; 23 34 71900 23 54 50-6 1100 23 33 21-6 1800 23 54 12-5 1200 ! 23 32 35-2 1700 23 53 34-0 1300 I 23 31 48-7 1600 23 52 55-1 1400 23 31 2-1 1500 23 52 15-8 1500 23 30 15-42 1400 23 51 36-0 1600 23 29 28-69 1300 23 50 55-9 1700 ! 23 28 411200 23 50 15-3 1800 23 27 55-10 1100 23 49 34-4 1900 23 27 81000 23 48 53-1 2000 23 26 21-41 900 23 48 11-4 2100 23 25 34-56 800 23 47 29-4 2200 23 24 47-73 700 23 46 47T 2300 23 24 0-91 600 23 46 4-4 2400 23 23 14500 23 45 21-5 2500 23 22 27-40 400 23 44 38-2 2600 23 21 40-73 300 23 43 54-7 2700 23 20 54-13 200 23 43 10-9 2800 23 20 7-61 100 23 42 26-8 2900 23 19 21-19 0 23 41 42-4 3000 23 18 34-87
The obliquity was at a maximum about 7200 B.c., or 9100 yearsago, when its value was 24° 13'. It will reach a minimum about 9600 years hence, when its value will probably be between 22° 30' and 22° 40', but cannot be more exactly stated.
The Solar Parallax. The problem of the distance of the sun has always been regarded as the fundamental one of celestial measurement. The difficulties in the way of solving it are very great, and up to the present time the best authorities are not agreed as to the result, the effect of half a century of research having been merely to reduce the uncertainty within continually narrower limits. The mutations of opinion on the subject during the last fifty years have been remarkable. Up to about the middle of the 19th century it was supposed that transits of Yenus across the disc of the sun afforded the most trustworthy method of making the determination in question; and when Encke in 1824 published his classic discussion of the transits of 1761 and 1769, it was supposed that we must wait until the transits of 1874 and 1882 had been observed and discussed before any further light would be thrown on the subject. The parallax 8-5776" found by Encke was therefore accepted without question for several decades* Doubt was first thrown on the accuracy of this number by an announcement from Hansen in 1862 that the observed parallactic inequality of the moon was irreconcilable with the accepted value of the solar parallax, and indicated the much larger value 8-97". This result was soon apparently confirmed by several other researches founded both on theory and observation, and so strong did the evidence appear to be that the value 8-95" was adopted in the Nautical Almanac for a number of years. The most remarkable feature of the discussion since 1862 is that the successive examinations of the subject have led to a continually diminishing value, so that at the present time it seems possible that the actual parallax of the sun is almost as near to the old value of Encke as to that which first replaced it. Five fundamentally different methods of determining the distance of the sun have been worked out and applied. They are as follows :— I. From measures of the parallax of either ^fe^frs Yenus or Mars the parallax of the sun can be mination. immediately derived, because the ratios of distances in the solar system are known with the last degree of precision. Transits of Yenus and observations of various 1 sorts on Mars are all to be included in this class. II. The second method is in principle extremely simple, consisting merely in multiplying the observed velocity of light by the time which it takes light to travel from the 2 sun to the earth. The difficulty is to determine the time in question. 9 III. The third method is through the determination of the mass of the earth relative to that of the sun. In astronomical practice the masses of the planets are commonly expressed as fractions of the mass of the sun, the latter being taken as unity. When we know the mass of the earth in gravitational measure, its product by the 9 denominator of the fraction just mentioned gives the mass 2 of the sun in gravitational measure. From this the distance of the sun can be at once determined by the fundamental equation of planetary motion. IY. The fourth method is through the parallactic 1 inequality in the moon’s motion. This method was described in the ninth edition of the Ency. Brit. V. The fifth method consists in observing the displacement in the direction of the sun, or of one of the nearer planets, due to the motion of the. earth round the common centre of gravity of the earth and moon. It requires a
