ni 



PRECESSION AND NUTATION. 



KSSION AND NUTATION. 



KM uteslea of the earth, of which they really form a part ; that is to 

 ay, diurnal rotation, ptsesejlon, and nutation are the motions of th 

 earth about iu centre, independently of the motion of that centre 

 along iu orbit round the sun. It will asaist our comprehension o 

 the eubject to auppoee the earth's centre a fixed point, the relativ 

 motion* at the heavenly bodies being adjusted accordingly. [Mo 

 TIO.X.] 



The rotation of the earth round its axis is visible from hour t 

 hour by the change of plaee in the stars : the precession and nutation 

 are motion* of too alow a nature to be rendered visible, and indeet 

 could hardly be mad* so unlee* the ecliptic and equator were viaibl 

 circle*. If such were the cue, and if the motions were large enough 

 the equinoxes, or intersection* of the equator and ecli|>tic, wouli 

 appear to change place, the equator moving slowly round the eolipti 

 with a retrograde motion, that in, contrary to the annual course n 

 the sun. The vquinoxe* would appear to more with A variable 

 motion, eometime* a little faster and sometimes a little slower thai 

 the mean motion. At the same time the equator would appear tc 

 wing backwards and forwards to and from the ecliptic, turning upon 

 the equinox** as p. . "i these motions the average motion of the 

 equinoxes upon the ecliptic i the precession ; the alternate accelera- 

 tion and retardation is one part of the nutation ; and the alternate 

 Increase and diminution of the angle contained between the two is the 

 other part. It is however common to call the acceleration and re 

 tardation of the motion of the equinoxes by the name of the equation 

 of the equinoxes, and to reserve the name of nutation for the inotioi 

 of the pole which corresponds to it. 



I-et c be the centre of the earth, c r half its axis, r the north pole, 

 and ABB half of the equator. Let u y be part of the plane of the 

 ecliptic, and c(j a line perpendicular to it, pointing therefore to the 

 f the ecliptic in the heavens : and let the direction of the diurnal 

 rotation be that of the arrow marked on the equator. If then r were 

 carried uniformly round a circle perpendicular to CQ, so that CP 

 nhould describe a conical surface, the equinoxes B and A would be 

 carried round in a direction contrary to that of the diurnal motion, 

 and with them the equator B s A, the angle which the equator make* 

 with the ecliptic remaining unaltered This motion of B and A is the 

 precession. But suppose that instead of p being placed on the circle, 

 it is placed on the circumference of a small oval, which has it* centre 

 on the circle. While the centre of the oval moves forward on the 

 circle with the motion of precession, let the pole f move round the 

 oval with a motion much slower than that of the precession. It will 

 then trace out in space an undulating curve, as shown in the principal 

 diagram, and the effect will be an alternate retardation and accelera- 

 tion of the motion of the equinoxes along the plane of the ecliptic, 

 together with a vibration of the plane of the equator to and from the 

 ecliptic; which are the motions described as constituting the uu- 



The preceding is a description of the effect of any one of the 

 heavenly bodies, theoretically speaking, ii|>ou tin- axis of the earth. 

 The whole precession and nutation is the united effect of the nun, 

 moon, and planet*. The effect of the planet* however is insensible, 

 except in a slight annual alteration of the plane of the ecliptic, which 

 is mixed up with the precession, and makes it appear a very little 

 smaller than it would be if the system of the sun, moon, and earth 

 were undisturbed i.y the planetary attraction. The general reader 

 need only attend to the main phenomenon, namely, that the equinox 

 (the point of the heavens at which the sun is at the commence! 

 spring) mores slowly backward* along the ecliptic, at the rate of 

 SOJ second* per annum, or about 14 in one thousand yearn. Thi* 

 rate of motion i* subject to a very slow increase, which is not perhaps 

 sufficiently well determined to make it worth while to compute ex- 

 actly the tiin-- in which the equinox describes the whole heavens, a 

 period of between twenty-five and twenty six thousand years. 



A good notion of precession may be got from observing the spin- 

 ning of a top. As long as the axis of the top is not vertical, this axis 



11 will perkaiM k* en latalligible to imagine all the >tw> moring .lowly 

 forward* in parallel, to the ecliptic, the equinox remaining unchanged. 



itaelf revolves, but much more slowly than the top revolves round iu 

 ajd*. Let the top be supposed to remain with ita axis at one n. 

 the vertical, except only a slight balancing motion to and fro- 

 vertical, and let the conical motion of the axis be slightly accelerated 

 and retarded : thi* will give a complete notion of the phenomena of 

 precession and nut. 



Before proceeding to the mathematical and physical description of 

 we shall show the manner in whirl, they m:iy h.ivo 

 an historical and chronological importance. Let the reader take a 

 globe or a map of the stars, he will see that the ecliptic crosses the 

 equator under the tail of one of the Pishes, so that the vernal e , 

 is nearly in a lin.> with the stars a Andromeda; and 7 Pegasi (Alphe- 

 rat and AlgeniM. These stars then are invisible at the 

 the spring, l>. -in : in tin- region of the heavens nearer* Let 



from twelve to thirteen thousand years elapse, and the slow pro- 

 cessional motion of the equinoctial point* will reverse the i 

 the equinoxes, so that the above-named stir* will ! ii.-ar the me- 

 ridian at midnight at the commencement of spring, as the stars in the 

 head of Virgo are now. An Aratus of our day would , 

 brightness of Virgo in the nights of spring, while one of thirteen 

 thousand years hence must choose Pisces for that purpose. The 

 seasons of the year at which different stirs begin to be distant enough 

 from the sun to shine Brightly i* undergoing a gradual alteration. 

 Heriod, for instance, says that in hie time and country Aretiirun 

 rose at sunset in sixty days after the winter solatic, 

 1 it. 1 that this took place about B.C. 870, whieh i* one of 

 for pupi>osiug that the poet was alive about that time. 'I'll 

 noiiifii.m however in itaelf rather vague, and Hesiod may be su] 



t in IUN description. H is genenilh I that 



'i attached too much importance to such data in 

 system of chronology. 



About two thousand years ago the equinox was twenty-eight degrees 

 more advanced among the signs, and was near the i 

 constellation Aries. It was about this time that the precession . 

 equinoxes was discovered by Hipparchus [AsTROJJOM v ; 1 1 

 in Bioo. Div.], and since that time the venial equinox has pn 

 the title which it properly held at the time when its 

 discovered, namely, the first point of Aries. Certain astron, 

 fictitious constellation* l,a\ v i H . en made to move with the equin 

 least until lately; counting from the equinox, the first thirty <! 

 of the ecliptic have been always called Aries, the weond thirty 

 Taurus, and so on. Thus when an astronomer of the middle ages 

 inserts the longitude of a star to be ^ 18 22', he means th.it it is in 

 18 22' of the astronomical Libra, a constellation supposed to begin at 

 6 x 80 or 180 distance from the vernal equinox. An astronomer of 

 our day would say the star's longitude was 198 22'. 



Taking the beginning of the year 1750 as the starting point (as is 

 generally done since the publication of the ' Mecanique Celeste ' 

 calling it the number of years elapsed, the whole motion of the 

 equinoxes from precession, including the effect of the planetary action 

 on the ecliptic, is 



S0"-176068 t + 0"-0001221483 t s , 

 while the precession in one year ia 



50"-1 76068 *-0"-0002442966 t. 

 M. Bessel substitutes 50"-21129 for 50"-176068. 



The obliquity of the ecliptic, assumed at 23 28' 18" in 1750, is 



23 28' 18' - 0"-48368 t- 0"-00000272295 t ! ; 



ts yearly diminution, arising from the planetary action, independently 

 if nutation, being 



0"-48868 + 0"-0000054469 t. 



Y%?^rx ir '' 63loT ih * ****** irso ' - d 



The nutation affects both the place of the equinoxes and the ob- 

 iquiiy of the ecliptic. Let AA denote the nutation in longitude, Ac 



ot upon the obliquity of the ecliptic. Then, according t 

 cssor Peters (' Xumcrus Const-ins Niitat i 



A\=-17"-2491 sin 8 + 0"-2(i;;; .si,, u IJ,-0"'2041 sin 2<[ + 



A = + 9"-2235 cos ft - 0"'0897 cos 2 & + 0"-0886 cos 2 <? + 0"-5508 x' 



cos 2 O + 0"-0093 cos (Q +*), 



n which expressions M denotes the longitude of the moon's asc, 

 node, C the longitude of the moon, the longitude of the sun, T 

 ho longitude of the p.-riheiion of the earth's orbit, aud ', th, 



i the moon's perigee. 

 It thus appears that by far the largest |rt of the effect of the 



mtition arises fn.m the m and depends not upon the place of the 



i "t upon that of iu node. This node [M mis a 



omnlete regressive i. volution in about eigb.tr, 



ime the main effect of nutation goes : I iu change*. 



of the nutation, found out and assigned the 

 its largest term ; tKe remaining on 



It took him twenty years of observation 

 ban the whole ] ii,,d ,,i the change, to detect t> 



whose existence he became sensible of immediately after hi* discovery 

 f aberration had cleared away the largest part of the then 

 motions of the stars. The discovery was completed and published at 



