The Lunar Apsides COLBERT. c Is i KniTiox, 100 COPIES. MOTION OF THE LUNAR A.VSLDKS. The motion of the lunar orbit lias long been a vexed question with the mathematicians. The following method of reconciling the theory with observation is novel, and perhaps will be accepted as conclusive : It may be mentioned, incidentally, that the motion of the lunar apsides has for the last two hundred years been a stumbling- block. NEWTON tried to account for it 011 the gravitation theory, but left it with the remark, "Apsis lumr est duplo velocior circiter" (the motion is about twice as great as this). CLAIRAUT showed, in 1750, how to account theoretically for the other half, but the attempt to reduce the equations to a numerical form still left a residual, and when LAPLACE attacked the problem he was only able to make the theory responsible for 444 parts out of 445. It has been attempted to bridge over the difficulty by adding 2 sin' 2 | -y . d Q> -f- d t to the motion of the perihelion ; which in the case of the moon is practically equal to 4 sin 4 ^y, because d Q -4- d D = -|- sin 2 y; =2 sin 2 \ y, nearly. It will be observed that this quantity is not needed in that shape, neither is the existence of a second moon required to account for the peri- geal motion. I may not be familiar with all the literature of the subject, but believe that the outstanding residual has not hitherto been eliminated by any investigator; and note^ a recent remark by G. W. HILL to the effect that it is not probable the perigeal motion will ever be accounted for by theory so closely as it can be obtained by a comparison of observations. If 1 c represent the motion of* the perigee divided by that of the moon, then c 2 and (1 c" 2 ) are the squares of two sides of the right-angled triangle the hypothenuse of which is unity; and v 1 c 2 is the perturbation of the radius vector. (This is not new.) The quantity 1 c- comprises a radial, which involves r 3 : a tangential, depending on the square of the velocity in the orbit, involving r 4 ; and one that originates in the displacement, being really a perturbation of the perturbation. The last is usually treated as a single quantity, namely as a function of r 4 . It is more philosophical to regard it as furnishing a multiple for each 736356 MOTION OF THE .of' * the : |)tliiiiv' f two ' instead of being a quantity simply additive. Also, for obtaining the mean motion of the apsides, it is sufficient to derive the constant portion of each function considered, being what we shall here call the "Average Value" of the quantity: With g the mean anomaly, e the eccentricity, r the radius vector, and a the semi -axis major, we have the following extension of a well-known equation: cos g 315 And an inversion of this series gives the following, which it is not necessary to carry out beyond the sixth power : - = ! + ( + - + j~- ) cos ,, y .. -- + 4- 120 LUNAH APSIDES. 3 liaising eacli of these expressions to the required powers, and omitting all that is periodical, we have the following as "average values": *> r i : i + f - ^ i+ J'* " '- : 1 4- 3e- -f- 3 <H + I,, e -Ii_ s 8 8~ 3-8 3 , } ,4 1 ?i 1 J Q '__ : 1 + oe* -f , K) e 4 4- _i_ c _ i__ e 8 4 8 9-8* 4-8* f *: 1 4- * 2 + ?*< + 15 e r 2 2 8 48 f l 3 : 14. ?2 4- 15.^4 + 7 r 3 284 4 o o 45 , 35 ^ : '8 e T For the solution of the problem we take the following as the most probable values of the quantities named. They are deduced from the figures given by NEWCOMB, in 1879, in his paper 011 the recurrence of solar eclipses. The epoch chosen is A. D. 1800. The processes of the subsequent computation are given, as they may be of use in verification; and all of the loga- rithms have been computed closely enough to secure accuracy in the last figure of the result as here presented. The mark ? following some of the numbers or logarithms indicates that the next succeeding figure would be nearly 5. The same mark inverted, thus, 4 shows that the given value is too great by half a unit, or nearly so, in the right hand place: ' Log&ri&ms. 's daily motion, 3548".1927904 3-5500072091 Sidereal year, days, 365 -2563647 2-56259 77924 D's synodical rev. days, 29 -53058844 1-4702721009' Sidereal rev. days, 27 -32166120 1-4365071016 ]> 's daily motion, 47434 X/ .890233 4-67609 78999' D 's Q's daily motion, 43886".697443 4-64233 29006 D 's TT, daily motion, 400".9187565 2-60305 63747 MOTION OF THK Half square ratio sidereal periods; = (l -i- 357-447). 1)7-44678 802:27 358-447 4- 357-447 '0-00121 32943 _ = 0-0027898145 O Nominal perturbation ; Then for e n the eccentricity of the by a preceding formula: 97-4455753284 earth's orbit, with e, = 0-01679228, we have by a ( a , 4- r,) 3 1-00042 31202 0-00018 37199 and taking an approximate value for J> 's c, with y = about 58 / 40".6 we obtain E, the perturbation of the perigeal motion due to the earth's elliptical figure, as follows : Constant of precession (JULIAN) = 54". 9 4625 Obliquity, (1800) = 28 27' 54".8 cos 50".40230 Daily soli-lunar 1-73993 805 9-96251 23' 1-70245 04 ]) 's m -- a 3 X D 's ( a -4- r) 3 1 _ (3 4. 2) sin' 2 y 's (a,-?- r,) 3 Sum of D and = 0".l 379940 1-00453806 2-154902 1-000423 9-1398602 0-33673 10 0-00196 64 9-99473 02 4 0-33342 75 0.49904 41 6-28988 37 94-87480 44 99-96251 23' 0-8601398 2-48638 44 99-12395 63 98-97424 36 98-0981999 93-42210 20 0-00001 36383 Our value of E is slightly larger than the one given by LAPLACE. The precession here used is greater than that observed ; the differ- ence being due to a planetary perturbation which causes the equinox to move forward a little more than 17" in a century. The number 306.468 is the earth's moment of inertia, divided by the momentum of the ring of matter that forms our equator- ial protuberance. 3-155325 (3 -J- 2) seconds sidereal arc in solar day, (Solar days in sidereal year)- a. c. Obliquity of ecliptic, cos Daily soli -lunar precession, 0". 13799 40 a. c. 306-468 Twice do X Moon's mass, = 7-51744 a. c. Lunar precession, 0". 09424 18 E 011 perigee, 0".01253 718 = d D (0-00000 02643 03) 2fi _i_ (1 _ C 2) = 1-00003 14038 5 = LCXAH APS1DKS. i> The, value of is, however, a direct function of the perturbation. We obtain it as follows: (1 <- 2 )'-r- 97-4479980098 Syn. .i. sid. period of D 0-03376 49993' = *- -4- / 2 97-48176 30091' Whence e = 0-0548997758 98.73957057 2 = -0030139854 97-47914/11416 e 4 = -0000090841 94-9582823 gC = -0000000274 92-4374 e 8 = -00000000008 89-92 t p 0-9969860146 9-9986890662 Then, for the averages on radius vector we have: (r 4- ) = 1-00904 53630 5' 0-00391 06910 (r 4- a) 4 = 1-01508 69601 8 0-00650 44721 7 Also for the inclination we have: 8 a _:_ 3 = _:_ 8 __ _|_ 339627 /' ' - - ^4- 0"804 sin 2 y' = 0-00804069057 97-90529335 sin 4 y ^ = . V 6 46^27 \ 9^105867 sin 8 y 42 91-6212 When n is % $n even power, the average value of sin" y is \ (n 1 ) . (n 2) . . .'. ($n + 1) . 2(1 . 2 ; 3 . 4 . . . !* ._ Jw) Giving ( 1 4- 2 ) for sin 2 ; (3^-8) for sin 4 ; (5 4- 16) for sin 6 ; (35 4- 128) for sin*; etc. Cos 2 ]> 's latitude = 1 sin 2 y sin 2 longitude. Hence we get the following values, not for the latitude at any particular point but the average cos, cos 2 , etc., of the D 's latitude: cos lat. =1 sin 2 y - sin 4 y sin 6 y ' sin y . 4 64 256 16384 cos 2 lat. = 1 J sin 2 y. 395 105 cos 3 lat. = 1 - sin 2 y 4- sin 4 y 4- sin 6 y 4- sin 8 y. 4 64 256 16384 cos 4 lat. = 1 sin 2 y -f- f sin 4 y. These relations give us: cos* lat. = 0-99397 85840' : 9-99737 70272 cos 4 lat. = 0-9919835542: 9-9965044722 MOTION OF THK And these multiplied into the average values of r a and r^ give the average third and fourth powers of the projection of r on the plane of the ecliptic: , It is important to note that the sum of the cube cosines for an inclination of 5 8' 40". 61 9 is equal to that for a medial value of 0".804 less; so that our computation gives us 5 8' 39". 81 5 5 f . This corresponds precisely to the HANSENIAN value of 58 / 39 // .9G corrected by the 0".15 which NEWCOMB deduced from a dis- cussion of the Greenwich and Washington observations from 18G2 to 1874. If be the )> 's distance divided by that of , and taking the parallaxes as equal to 3422". 75 and 8". 794, we have 2 _ 0-000006601803; and the value of 3m -f- 2 must be multi- 9 15 plied into (1 -j- _ 2 , etc.) and ( 1 + _ 2 , etc.) for the perturba- 8 8 tive series in the direction of r and perpendicular thereto. For the effect due to the "variation," let 1 + x an( l 1 x represent the semi -axes of the ellipse, the longer axis being in quadratures and the other in the syzigies. Let w be the mean angular distance from the direction of the minor -axis of this ellipse. Then if r denote the distance from the centre to any point in the circumference, we have, by comparison of the ellipse with its circumscribing circle: r 2 sin 2 ( w + dw ) . (1 + x ) 2 + cos 2 ( w + dw ) . ( 1 x )-; = 1 4~ a? 2 2x cos 2w -f- 4Q cc 2 sin 2 2w } if qx denote the maximum perturbation in longitude in the average orbit that which gives unequal areas in equal times. Now, 1 -jr x 2 = a 2 , if a be the radius of the circle of equal area that would have been described in the absence of compres- sion; because (I -\- x) . (I x) = 1 x 2 . Hence r o ~^~ a o 1 2cc cos 2w + 4Qx 2 sin 2 2w, if there were 110 change of area; and becomes + , . (1 _ 2x cos 2w + 4Qic 2 sin 2 2 w ) on account of solar perturbation on a Q . 'From this we have: r> = ( 1 + ) - ( 1 + | * 2 + 3 ^ + | Q^ 4 ) 4 1 + 2x2+ 4QX 2 + 6 Q2 X 4). LUNAR APSIDES. The numerical values are as follows: ( syn 4- sid ) 2 1 = = 4 x -4- ( 1 + * )- 0-1682344223 1-7750601269' 97-4791411416 9-2259148612 98-48011 6130 which is the square of the average eccentricity in the hypotheti- cal orbit described by the moon once in each synodical lunation. 1 + X I - X (1 4- x) 4- (1 - 1 + (* 2 -5- 2) -0076681608 00005 88006 9 = tan (45+ 1581".633) = 's daily motion 4- 402".857 average cos add 1581".633 = Variation; = 1984".490 Syn 4- sid 2144".934; And Q = Taking the logarithms, we have: For r 3 Function of e 0-00391 06910 y 9-99737 70273 x 0-00005 14323 0-0000032255 (log) (logs.) and the numbei-s are The logarithm of the sum (3 4- 2) m Solar; ( v 4- r,) 3 , Earth perturbation, PI anetary perturbation, 0-0013423761 97-88469 12105 0-00331 75364 9-9966569260 95-7693824210 0-0066606104 0-00001 27682 97-92511 926 4-68003 181 2-60515 107 3-29764 88 0-03376 50 3-33141 39 9-40603472' For r 4 0-00650 32488 9-99650 44722 0-0000941089' 53758 0-0031072057' 1-0030957170 1-0071802610 2-0102759780 -01683252457 0-3032556830' 97-9226965831' 0-0001837199 0-00001 36383 9-99999 96358 98-2261492602 THE LUNAR APSIDES. 1 c -00845198033 97-9:269584777 D 4-67609 78999* (1 c) = 400".91875926 4 2-6030563777, (The planetary perturbation is that adopted by HILL in his tables of Venus. It is what LAPLACE terms the "indirect" per- turbation; being that clue to the enlargement of the earth's radius vector by planetary action, which lessens the solar disturbing force. The direct planetary perturbation is neglected, being infmitessimal as between the earth and moon.) This result is identical with the value of the perigeal motion which NEWCOMB has obtained from a discussion of the eclipses of 2500 years preceding the present century. The difference between the two is less than one part in 100,000,000. Hence the problem is completely solved. The following is the resulting value of the daily motion of g, the mean anomaly: 47033".97147 4 4 NEWCOMB, 47033".97147 HASSEN; (Tables D ), 47033".97227 If any one should object to our deduction of the values of e and y from that of the quantity sought he is respectfully referred to the top of page 174 of Loomis' Practical Astronomy, with the fact that a comparison of the rates of change in the values of the quantities shows this to be a parallel case with that given by LOOMIS on page 173. It is not necessary to our result to carry out the logarithm of e 2 to ten places; but I think there needs be no doubt in the future in regard to the precise values of e, y, or x in the lunar orbit. Of course the numerical values of these quantities are slightly reduced since the beginning of the century by the decreasing eccentricity of the earth's orbit. There is still room for a possible very small correction to the assumed values of ' sidereal motion of the sun and moon. E. COLBERT, Formerly Superintendent Dearborn Observatory. CHICAGO, December 12, 1886. FERGUS PRINTING COMPANY, CHICAGO. Photomount Pamphlet Binder Gay lord Bros. Makers Stockton, Calif. PAT. JAN. 21. 1908 HOME USE CIRCULATION DEPARTMENT MAIN LIBRARY This book is due on the last date stamped below. 1 -month loans may be renewed by calling 642-3405. 6-month loans may be recharged by bringing books to Circulation Desk. Renewals and recharges may be made 4 days prior to due date. ALL BOOKS ARE SUBJECT TO RECALL 7 DAYS AFTER DATE CHECKED OUT. Z.W619 AON NQV OCT 1 MAR i 3 1982 It) LIBRARY USE MAR 3 LD21 A-40m-5,'74 (R8191L) General Library University of California Berkeley U.C. BERKELEY LIBRARIES