:, OF THE UNIVERSITY OF CALIFORNIA. QIKT OK ORBIT OF PSYCHE A REVISED FORM OF A THESIS PRESENTED TO THE FACULTY OF THE UNIVERSITY OF MICHIGAN, IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF SCIENCE BY SIDNEY DEAN TOWNLEY OF THE UNIVERSITY OF SAN FRANCISCO C. A. MURDOCK & Co., PRINTERS 1905 TABLE OF CONTENTS. PAGE. HISTORICAL NOTE 1 COMPUTATION OF EPHEMERIS FOR OPPOSITION OF 1882 1 COMPUTATION OF EPHEMERIS FOR OPPOSITION OF 1885 1 DISCUSSION OF STAR PLACES 2 COLLECTION OF OBSERVATIONS 2 WEIGHTS OF OBSERVATIONS 3 COMPARISON OF OBSERVED AND COMPUTED COORDINATES 4 COMPUTATION OF DIFFERENTIAL COEFFICIENTS 5 LEAST SQUARE SOLUTION OF OBSERVATION EQUATIONS 5 COMPUTATION OF CHECK EPHEMERIS 7 COMPUTATION OF SPECIAL PERTURBATIONS BY JUPITER 1870-1876 7 COMPARISON WITH SCHUBERT'S PERTURBATIONS 7 DISCUSSION OF DIFFERENCES BETWEEN SCHUBERT'S AND' MY PERTURBATIONS. 8 SPECIAL PERTURBATIONS FROM 1900 TO 1906 11 THE OliBIT OF PSYCHE. By SIDNEY DEAN TOWNLEY. The planet Psyche was 1 discovered at Naples March 17, 1852, by Professor 'ANNIBEL DE GASPARis. 1 This discovery, as is the case in many others both greater and less, was made by two men, the second discovery being only one day later than the first. Mr. HIND, of London, charted this planet nearly two months before the discovery of DE GASPARIS and found it to be an asteroid on March 18th.- The planet was immediately observed at several ob- servatories, and soon afterwards the first elements were computed by RUMKER. :{ Orbits were also com- puted by VOGEL, SCHONFELD, SONTAG, and KLINKER- FUES. The last named was then an assistant of GAUSS, and he used GAUSS'S method of computing the orbit from four places, which, on account of the small in- clination of the plane of the orbit, gave a much better determination of the orbit than was obtained by the other computers, who used" only three observations.. 4 Ir; 1855 KLINKERFUES computed perturbations by Jupiter and corrected the elements by all the observa- tions made up to that time. 5 Afterwards AUWERS corrected KLINKERFUES 's elements by the observations of the oppositions of 1857 and 1858, computing per- turbations by both Jupiter and Saturn. About 1860 SCHUBERT commenced computations upon this planet and corrected KLINKERFUES 's elements from the ob- servations of eight oppositions. 7 Frequent observa- tions of the planet were made between 1860 and 1870, and SCHUBERT again corrected the elements and com- puted special perturbations by Jupiter from 1870 to 1900. 8 Since 1870 but few observations and no computa- tions have been made upon the planet. My first observations were secured at the opposition of 1894. A comparison of the observed and computed places showed the planet to be considerably out of the ephemeris place, and it was deemed advisable to cor- rect SCHUBERT'S elements. It was found that ob- servations had been made at the oppositions of 1879, 1880, 1882, 1883, and 1885. As there were no other observations until 1892 it was decided to divide the work into two parts, as follows: First To correct SCHUBERT'S elements from the observations of 1879, 1880, 1882, 1883, and 1885; with the corrected ele- ments, to recompute the special perturbations by Jupiter over enough of the period from 1870 to 1900 to enable one to determine empirical formulae from which to compute the corrections to SCHUBERT'S per- turbations at any epoch. Second To again correct the elements from the observations 1890 to 1900, and with these Anally corrected elements, to compute perturbations from 1900 on. Computation of Ephemeris. For each of the op- positions under consideration, those of 1879, 1880, 1882, 1883, and 1885, an ephemeris of Psyche, com- puted from SCHUBERT'S elements was published in the Berliner Astronomischcs Jahfbuch. At two of the oppositions, however, observations were made after the expiration of the ephemeris, and it became neces- sary to compute extensions for each. The results of the computations are as follows : Berlin M. T. a 5 log A Aberr. T. 1882 March 13d 12h 00m OOs 10h 9m 4s.23 4- 11 55' 50".8 0.358562 18m 57s 14 12 00 00 10 8 25 .66 4-12 5 .3 0.359788 19 15 12 00 00 10 7 47 .96 4-12 4 14 .1 0.361061 19 3 16 12 00 00 10 7 11 .18 4-12 8 16 .9 0.362379 19 6 17 12 00 00 10 6 35 .33 4-12 12 13 .7 0.363742 19 10 18 12 00 00 10 6 .47 4- 12 16 4 .3 0.365148 19 14 19 12 00 00 10 5 26 .61 4- 12 19 48 .5 0.366597 19 18 1882 March 20 12 00 00 10 4 53 .79 4- 12 23 26 .1 0.368087 19 22 1885 Dec. 1 12 1 37 4 15 1 .87 4- 16 19 40 .9 0.221533 13 49 4 11 47 6 4 12 22 .42 4-16 14 18 .0 0.223776 13 53 9 11 23 5 4 8 8 .61 4-10 6 31 .9 0.228924 14 3 23 10 17 43 3 58 32 .75 4- 15 54 56 .6 0.251592 14 48 1885 Dec. 30 9 46 31 3 55 24 .38 4-15 55 41 .0 0.266562 15 20 1 A. N. Bd. 34, 147, 5 Ibid. 38, 181. !( The plan of this work was adopted while the author was - Ibid. 34, 163. 8 ibid. 50, 375. a student at the University of Michigan. It is now recognized 3 Ibid. 34, 189. 7 Ibid. 57, 323. that the plan is open to criticism. In the first place, it would 4 Ibid. 35, 17. 8 Ibid. 75, 209. have been much better to have used all of the oppositions avail- 157974 Observations. Three different types of instru- ments were used in making observations upon this planet, the equatorial, the heliometer, and the meridian circle. Fifteen comparison stars were used in the equatorial and heliometer observations. Their places have been carefully investigated and the results are incorporated in the following tabulation. STRUVE'S precessional constants have been used in bringing up the catalogue places. Where possible reductions to the A. G. system have been applied. Proper motions have been applied in only one case star number eight, The values used are -f O s .0071 and 0".03, and were obtained from the Poulkova Catalogues. The positions of all the other stars were examined for evidence of proper motion, but none of a positive nature was found. In assigning weights to the various catalogue positions I have been guided largely by the table published by Professor NEWCOMB in his Cata- logue of Fundamental Stars and the one published by Dr. DAVIS in his Declinations and Proper Motions of Fifty-six Stars. The places of the stars from the Kam Catalogue were brought up from the original places published in the Astronomisclie Nachrichten. POSITIONS OF COMPARISON STARS. Xo. Epoch. a 1 1879.0 20h 12m 48s .57 2 " 20 7 3 .05 3 1880.0 4 28 41 .13 4 " 4 24 54 .27 5 " 4 19 46 .85 4 16 35 .23 4 28 8 .06 4 13 2 .70 6 .44 Wt. 12 ie 5 21 5 30 11 51 17 51' 55".5 -18 2 22 .2 + 16 49 28 .2 + 16 56 54 .6 + 16 28 26 .0 + 16 20 51 .2 + 16 56 58 .7 + 16 ^13 54 .2 15 37 55 .5 Wt. 12 16 5 17 27 30 11 45 17 10 1882.0 10 25 55 .72 20 + 10 1 43 .8 20 11 10 7 8 .49 15 + H 55 20 .5 15 12 10 7 19 .62 18 + H 56 11 .5 18 13 10 8 55 .30 3 + 12 5 41 .3 3 14 10 12 6 .46 21 .+ 11 25 37 .4 21 15 10 7 4 .97 + 11 51 25 .5 Authority. 1/12 [4 Arg. Weiss 16048 + 6 Kam. 3980 + 2 Mun.i 23872.] 1/16 L'Mun., 23483 +5 Mun., 9920 + 10 Cape (1890) 2511.] 5 Konig. Mer. 1/21, 1/17 [W. 514 + 10 A. G. C. Ber. A. 1207 + 10, 6 Konig. Mer.] 1/25, 1/27 [W. 391 + 6, 8 Yarn. 3 1952 + 10 A. G. C. Ber. A. 1178 + 8 Konig, Mer.] 1/30 [Lai. 8214 + W". 317 +4 Kam. 766 +4 Kam. 7(55 + 2, 5 Yarn. 3 1928 + 10 A. G. C. Ber. A. 1156 + 8, 5 Konig. Mer.] 1/11 [10 A. G. C. Ber. A. 1227 + Eiim. II, 2345.] 1/51, 1/45 [Eiim. II, 2345 + 3 Kam. 751 + 4 Kam. 752 + 10 Poul. 1855, 634 + 6, 5 Yarn. 3 1908 + 8, 3 Bad. II, 480 + 10 Poul. 1875, 980 + 10 A. G. C. Ber. A. 1138.] 1/22, 1/17 [2 Kam. 719 + 10, 5 Yarn. 3 1842 + 10 A. G. C. Ber. A. 1102.] 1/20 [5 Yarn.., 4476 + 5 Schj. 3856 + 10 A. G. C. Leip. I, 4048.] 1/15 [W. 76 + 2 A. N. 47, 135 + 10 Yarn. 3 4341 + 2 Kam. 1696.] 1/18 1 3 B. B. VI, 2169 + 6 Yarn. 3 4344 + 3 Kam. 1697 + 6 A. N. 74, 247.] 3 B. B. VI, 2178. 1/21 [W. 173 + 5 Yarn. 3 4380 + 5 Schj. 3772 + 10 A. G. C. Leip. I, 3989.] 1/6 [3 B. B. VI, 11 2187 + 3 Kam. 1694.] OBSERVATIONS. The following observations of Psyche were collected from the publications mentioned in the last column Date. a 5 Place. Inst. Star. Source. 1879 Aug. 4 20h 5m 3s .77 -18 8' 53". 3 Hamb. M. C. A. N. Bd. 97, 51. July 29 10 .00 -17 46 34 , ,2 Krems. A. N. Bd. 97, 351. Aug. 31 2 8 6 20 39 .08 .63 -18 53 1 58 , 38 .1 .4 Green. Green. Obs . 1879. 11 19 59 43 .72 33 33 .1 " " " " " 12 59 1 .51 36 56 , ,9 .< " " " July 30 20 9 6 .75 -17 50 41 .4 Leipz. E. 1 A. N. Bd. 96, 225. Aug. 1 7 27 .45 58 7 , 5, " " 2 " " " 3 5 51 .11 -18 5 19 .4 2 " able to correct SCHUBERT'S elements rather than to have used simply a block of five oppositions front 1879 to 1885. Second As is afterwards shown, it is not possible to determine empirical formulae with which to carry forward the corrections to SCHU- BERT'S perturbations. Third 'The perturbations of Psyche by Saturn are of considerable magnitude, and these should have been computed, because it will never be possible to get a satis- factory orbit of the planet by neglecting them. In the light of these facts it is deemed unwise to try to carry out the second part of the program outlined above. Instead of doing that, it is the author's intention to compute general perturba- tions of Psyche by both Jupiter and Saturn. 3 1880 Nov. 10 4 24 14 .()5 4 16 40 50 .7 Hamb. M. C. A. N. Bd. 100, 133. 20 20 43 .42 31 28 .2 " " ' " Dec. 23 o 18 1 .24 24 6 37 7 .7 .4 3 9 .06 4 13 .9 Green. " Green. Obs. 1880. 11 2 25 .29 + 15 52 7 .6 Nov. 27 14 21 .92 + 16 15 56 .1 Paris Comptes Een. Vol. 92. 20 20 30 .53 30 54 .3 Wash. T. C. Wash. Obs. Vol. 27. Dec. 1 10 34 .39 7 30 .5 6 6 15 .20 + 15 58 50 2 " " " " 9 4 3 48 .60 54 24 .8 13 46 .52 49 37 .5 Nov. 10 29 15 .30 + 16 55 25 .6 Konig. H. 3 A. N. Vol. 100, 243. 15 25 12 .62 43 30 .5 " " 4 " " " 21 24 19 17 50 6 .77 .80 29 22 11 23 .3 " " 5 .4 " " 6 " " " 8 30 45 .28 59 67 .3 Leipz. E. 7 A. N. Vol. 100, 357. 29 30 12 11 33 40 .76 .39 11 9 51 53 . -J O .7 " " 8 " " " Dec. 14 4 17 .38 + 15 48 56 .6 " " 9 " " " 1882 Feb. 18 10 26 25 .71 + 9 59 57 .6 Green. M. C. Green. Obs. 1882. Mar. 9 11 52 .15 + 11 37 28 .1 Paris " Comptes Een. Vol. 94. 14 8 31 .52 59 35 .0 " " " " " " 15 7 53 .90 + 12 3 43 .8 " " " 16 17 7 6 17 41 .30 .57 7 11 45 43 .3 " " " ' .7 " " " " " " 18 6 6 .52 15 31 .1 20 4 59 .90 22 55 .2 Feb. 10 09 24 85 + 9 18 34 .6 Wash. T. C. Wash. Obs. 1882. Mar. 2 16 50 .48 + 11 4 30 .6 4 15 19 .91 14 33 .2 14 8 23 .31 + 12 30 .0 Feb. 18 26 31 .54 + 9 59 19 .7 Wien E. 10 A. N. Vol. 102, 283. Mar. 13 9 13 .77 + 11 54 57 .7 " " 11 14 8 32 .50 59 28 .6 " " 12 " " " " 15 7 58 .13 + 12 3 15 .1 " 13 " ". " " 7 13 20 .36 11 27 46 .0 Dres. 14 A. N. Vol. 102, 187. 8 12 34 .35 32 49 .8 14 " " " " 12 9 54 .54 50 27 .3 " 15 1883 April 13 25 10 9 14 41 16 19 .24 .74 - 11 54 20 38 37 .0 15 " " " " .0 Wash. T. C. Wash. Obs. 1883. 27 39 46 .07 12 24 .1 30 37 23 .86 .5 May 2 35 48 .78 10 51 50 .9 3 35 1 .51 47 50 .7 " " " " " 9 30 19 .52 24 25 .6 2 35 59 .25 52 45 .4 Paris M. C. Comptes Ren. Vol. 97. 9 30 29 .70 " " " " " 12 28 12 .71 " 15 26 .08 3 33 .3 " " " " " " 16 25 16 .87 11 .6 " " " " " " 18 14 23 52 .51 9 53 37 .6 " " " " " " 1885 Dec. 1 4 15 6 .50 + 16 19 58 .8 Comptes Een. Vol. 102. 11 9 12 8 26 13 .98 .27 14 6 28 44 .9 23 3 58 37 .09 + 15 55 8 .5 30 55 28 .71 55 52 .2 " " " " " " Arranging these observations chronologically and have been determined injtt^Mlt);ipwanner : For comparing them with the ephemeris values, we have heliometer and microraetej', .pb^ierva'tions |he weights the following table of residuals, in which the weights (divided by six) of ttt compa^son staife (page 2) were taken as the weights of C. When clouds interfered or the' seeing was bad the weights were slightly reduced. Some reduction was also made in the weights of those observations dependent upon stars 6 and 8, because on account of the large number of catalogue places of these stars the weights became abnormally large. The mean of the weights of those observations dependent upon A. G. C. stars is 4. This weight, 4, was given to each of the meridian and transit circle observations. All weights were then divided by four and these Avere used in taking the weighted means. The residuals in declination for the Kremsmunster observations of 1879 July 29 and 31 differ about 10" from the residuals of the observations made at Hamburg, Paris, and Leipzig. Perhaps an error of 10" was made in reading the Kremsmunster circle, but no further evidence to support such a sup- position could be found. It is my judgment that more might be lost by including these residuals than by rejecting them ; hence zero w r eight has been as- signed to each. RESIDUALS. Date. C W O C W Remarks. 1879. July 29 4s .67 0.75 4 1".7 0.0 3Q 4 .73 .5 - 9 .2 .5 Clouds. 31 -4 .72 .75 4' 2 .1 .0 Aug. 1 -4 .94 .5 -13 .1 .5 Clouds. 2 -4 .98 1. - 9 .5 1. 3 5 .00 .5 -11 .9 .5 Clouds. 4 -5 .01 1. -11 .6 1. 11 -5 .13 1. - 9 .0 1. 12 5 .07 1. -10 .5 1. 1882. Feb. 10 44S.47 1.00 24".8 1.00 18 4 .60 .75 20 .4 .75 18 4 .32 1. 20 .4 1. Mar. 2 4 .53 1. 24 .3 1. 4 4 .53 1. 24 .0 1. 7 4 .48 1. 23 .3 1. 8 4 .24 1. 22 .3 1. 9 4 .52 1. 24 .8 1. 12 4 .18 .25 19 .7 .25 13 4 .05 .25 20 .9 .25 13 4 .56 .5 20 .9 .5 14 4 .38 1. 20 .5 3. 14 4 .42 .75 20 .8 .75 14 4 .54 1. 20 .7 1. 15 4 .06 .125 18 .7 .125 15 4 .37 L 19 .9 1. 16 4 .47 1. 20 .7 1. 17 4 .53 1. 18 .7 1. ' 18 4 .27 1. 21 .5 1. Mar. 20 44 .30 1. -18 .5 1. Mar. 9 4 4s .425 16.625* 21".55 16.625 1883. Apr. 25 4 Is .23 1. 3".4 1. 27 1 .33 1. 6 .9 1. 30 1 .13 1. 4 .1 1. May 2 1 .12 1. 3 .4 1. 2 1 .11 1. 4 .3 1. 3 1 .36 1. 5 .5 1. 9 1 .17 1. 8 .1 1. 9 1 .16 1. 12 1 .14 1. 15 1 .17 1. 4 .4 1. 16 1 .06 1. 5 .2 1. May 18 4 1 .00 1. 2 .5 1. Aug. 3 -4s .938 7.0 - 10".48 5.5 1880. Nov. 8 -)- 5s. 28 0.5 4 8".0 0.5 10 5 .30 .25 8 .4 .25 15 5 .52 .75 5 .7 .75 16 5 .26 1. 6 .5 1. 20 5 .34 1. 8 .9 1. 20 5 .43 1. 8 .5 1. 21 5 .33 1. 7 .2 1. 23 5 .50 1. 8 .8 1. 24 5 .39 1. 7 .5 1. 27 5 .43 1. 10 .8 1. 29 5 .32 1.25 10 .8 1.25 30 5 .31 1.25 10 .8 1.25 Dec. 1 5 .47 1. 10 .2 1. 2 11 .8 .5 Clouds. 3 5 .34 1. 10 .2 1. 6 5 .18 1. 10 .8 1. 9 5 .12 1. 10 .0 1. 11 5 .13 1. 11 .1 1. 13 5 .04 1. 11 .5 1. Dec. 14 4.4 .92 0.75 4 11 .5 0.75 Bad seeing. May 7 4 Is .165 12. 4".78 10. 1885. Dec. 1 44s.63 1. 4 17".9 1. 4 4 .56 1. 10 .3 1. 9 4 .66 1. 13 .0 1. 23 4 .34 1. 11 .9 1. Dec. 30 44 .33 1. 4 11 .2 1. Dec. 13 4 4s .504 5. 4 12".86 5. No smoothing out process has been applied to the residuals, and on account of the small number of observations in most of the series the computation of probable errors for the purpose of weighting the residuals has been avoided. It was thought that the method of procedure adopted would give just as reli- able results as a more elaborate process. Nov. 27 45s.298 17.75 4 9".53 38.25 5 DIFFERENTIAL COEFFICIENTS AND LEAST SQUARE SOLUTION. Transforming the ephemeris positions of the planet for the dates of the normals to the epoch of 1870, we have : a 5 log A 1879 August 5.5 300 56' 16" .5 -18 14' 5" .8 0.240293 1880 November 29.5 62 57 27 , ,3 + 16 9 58 , .3 0.218635 1882. March 7.5 153 8 4 . ,1 + 11 32 19 , o 0.352240 1883 May 7.5 217 49 26 , 2 10 29 16 .7 0.354937 1885 December 13.5 61 1 11 .7 4- 15 59 6 .4 0.234298 AVith these coordinates and SCHUBERT'S elements, differential coefficients were computed giving the follow- ing logarithmic observation equations : OBSERVATION EQUATIONS. 0.189186 ATT -f 9.155347 n V^Att -\- 9.044763 n At -f 0.485258 n 0.237386 0.194755 4- 7.938927 n 4- 9.467439 4- 0.409537 4- 0.280214 4- 0.880449 0.132406 4- 9.400944 n 4- 8.614869 4- 0.222787 4- 0.047791 4- 0.694169 0.138764 4- 8.976777 n 4- 9.606266 4- 0.068281 n 4- 0.036684 4- 0.723934 0.179010 4- 7.532970 4- 9.475303 4- 0.399199 4- 0.261723 4- 1.025690 9.400352 4- 9.870979 4- 9.835881 4- 9.691681 U 4- 9.451835 4- 9.995812 9.467262 4- 8.892532 n 4- 0.192931 n 4- 9.649330 4- 9.560284 4-0.157535 9.636191 n 4- 9.857489 n 4- 9.110204 4- 9.732451 n 4- 9.552548 n 4- 0.198388 n 9.629199 n 4- 9.371875 n 4- 0.114306 4- 9.524342 4- 9.524494 n 4- 0.209925 n 9.469337 4- 9.115697 n 4- 0.176690 n 4- 9.664063 4- 9.558994 4- 0.321169 The weights assigned are very approximately those of the weighted residuals. Each equation was multi- plied through by the square root of its weight, divided by 100, and the following substitutions were then made : [8.345270] ATT = x [8.008004] 1/10 Afl = y [8.343446] A = z [8.560052] A = u [8.430729] A. == v [9.030964] 1000 A/* = w The resulting weighted homogeneous observation equations are as follows : 9.843916 x 4- 9.147343 n y 4- 8.701317 n s -\- 9.925206 n u 4- 9.806657 r 4- 9.749106 4- 9.867266 = 0.000000 4- S.081438 n 4- 9.274508 4- o.oooooo 4- 0.000000 4- o.oooooo 4- 0.033196 n = 9.937651 4- 9.543455 n 4- 8.421938 4- 9.813250 4- 9.767577 4- 9.813720 4- 9.963652 n = 9.881539 4- 9.056818 n 4- 9.350865 4- 9.596274 n 4- 9.694000 4- 9.781015 4- 9.323144 n = 9.833740 4- 7.524966 4- 9.131857 4- 9.839147 4- 9.830994 4- 9.994726 4- 9.812562 U 9.055082 4- 9.862975 4- 9.492435 -f 9.131629 a 4- 9.021106 4- 8.964848 + 9.020361 = 9.272507 4- 9.035043 n 4- 0.000000 n 4- 9.239793 4- 9.280070 4- 9.277086 4-9.129608 n = 9.441436 n 4- 0.000000 n 4- 8.917273 4- 9.322914 n 4- 9.272334 n 4- 9.317939 U 4- 9.483962 = 9.371974 n 4-9.451916 n 4- 9.858905 4- 9.052335 4- 9.181810 n 4- 9.267006 n 4- 8.767473 = 9.124067 4- 9.107693 n 4- 9.833244 n 4- 9.104011 4- 9.128265 4- 9.290196 4-9.109241 n = Using the method of least squares, the normal equations are (numerical coefficients) : 4-3.4798 x 0.1090 y -\- 0.0036 z 4- 1.2117 w 4- 2.9460 f 4- 3.2623 w 2.0934 0.1090 4- 1.7953 4- 0.1056 0.0295 0.0919 0.1019 4- 0.0389 = 4- 0.0036 4- 0.1056 4- 2.1969 4- 0.0145 0.0006 4- 0.0022 0.0771 = 4- 1.2117 0.0295 4- 0.0145 4- 2.8858 4-1.1727 4- 1.4629 2.7746 = 4- 2.9460 0.0919 0.0006 4- 1.1727 4- 2.5804 4- 2.8479 -1.7881 = 4- 3.2623 0.1019 4- 0.0022 4- 1.4629 4- 2.8479 4- 3.2396 2.1488 =i The similarity of the coefficients in the first ar^d sixth of these equations shows that the value of the unknowns determined from them will be somewhat uncertain. The "process of solution Avas carried out, however, and the following elimination equations derived : 4- 3.4798 x 0.1090 y -f 0.0036 s -f 1.2117 u -f 2.9460 v -\- 3.2623 w 2.0934 4. 1.7919 4- 0.1057 + 0.0084 4- 0.0004 4- 0.0003 0.0267 4- 2.1907 4- 0.0127 0.0036 - 0.0012 0.0733 = 4- 2.4638 -j- 0.1469 4- 0.3270 2.0452 4- 0.0775 4- 0.0665 4-0.1060 = 4- 0.0808 0.0059 = whence w = 4- 0.0730 v 1.4303 u = 0.7351 The usual check quantities were carried through- out the formation and solution of the normal equa- tions. The smallness of the coefficients in the last two elimination equations makes the determination of v and w uncertain ; and the uncertainty of these also affects the determination of the other quantities, especially x. In order to determine the unknowns more accu- rately the first four of the elimination equations were chosen, and by successive substitutions u, z, y, and x were expressed as functions of v and w, giving the following equations : u 8.77542 n v 4- 9.12295 n w 4- 9.91913 s 7.29863 + 7.11965 + 8.45708 y 5.78568 n 4- 6.57640 4- 7.96937 x 9.91690 n 4- 9.95001 n 4- 9.49528 (A) Substituting these values of u, z, y, and x in the original homogeneous weighted observation equations, we have the following set of equations for the deter- mination of v and w: 9.05806 v 4- 8.70422 w 9.06032 4- 8.37561 n 9.22671 n 4-9.31658 n 9.04297 n 4-8.34031 n 8.86656 4- 9.46044 8.29842 4- 7.98687 8.37234 4- 7.31511 n 8.73056 4- 8.81948 8.57060 4- 8.03448 8.19166 4- 8.76841 Check [m' m'] = 0.2021, 4- 9.40401 = 4- 8.83731 = 4- 9.04634 n = 4- 9.46924 n 4- 9.14898 = 4- 8.63960 4- 8.58364 = 4- 8.56549 = 4- 8.98490 = 4- 7.34452 n [m m 4] = 0.2019 These equations were multiplied through by the number whose logarithm is 0.53076 and new un- knowns defined by the relations v' = [9.75747] v, w' = [9.99120] w were introduced. Solution by the method of least squares gave for the normals, 4- 2.7365v' + 1.3698-w;' + 2.1313 = 4- 1.3698v' 4- 1.6550w' + 1.0004 = 3 = -\- 0.0269 y = + 0.0102 x = -\- 1.4884 from which and hence v ' 0.8132 w' +0.0686 v = 1.4214 w = +0.0700 These values of v' and w' substituted in the obser- vation equations from which they were determined give [v' v'} = 0.6639. This divided by the square of the number whose logarithm is 0.53076 gives [v v] =: 0.0576, while [m m.6] 0.0565, which is, I think, considering the inherent uncertainty of the solution, a satisfactory agreement. Substituting the values of v and w in equations A, we have finally, + 1 4243 + 0.0094 + 0.0259 4- 0.9056 - 1.4214 4- 0.0700 x y z u V w Returning now to the relations between these quan- tities and the original unknowns (page 5) we have finally the following corrections to the elements : A7r 4-64". 32 Aft 4- 9 .26 Ai + 1 .18 A< 4. 24 .94 AM = 52 .72 A^ + .000652 Substituting these values in the original observa- tion equations, we have [pvv] = 576".32, which divid- ed by 10,000 gives an exact agreement with [vv] just found. On account of the small number of observa- tion equations the computation of probable errors of corrections to the elements has been carefully avoided. As a check upon the work an ephemeris was computed, with the corrected elements, for each of the normal dates giving the following coordinates, referred to the ecliptic and equinox of 1870.0 : 1879 August 5.5 1880 November 29.5 1882 March 7.5 1883 May 7.5 1885 December 13.5 A comparison of these coordinates with the observed coordinates gives the following residuals, those in a having been multiplied by cos 8 : 300 55' 10".6 62 58 36 .7 153 9 16 .0 217 49 41 .2 61 2 29 .6 18 14' 14".6 + 16 10 7 .6 + 11 31 55 .3 10 29 20 .6 + 15 59 18 .0 O C C in a in 8 -7". 8 1".9 + 6 .2 -0 .3 -7 .5 + 2 .7 + 11 .2 -3 .9 5 .3 + 2 .4 From these we find [pvv] = 513".3, which, consid- ering the element of indeterminateness inherent in the least square solution, is perhaps in satisfactory agreement with the value 576".3 found from the least square solution. The value of [pvv] before the least square solution was 31,003". 7. The above residuals are still rather large, but if we consider the fact, as is afterwards shown, that SCHUBERT'S perturbations are probably not strictly accurate, and also that perturbations by Saturn and other planets have not been computed, it seems useless to try and further reduce these residuals. The corrected elements are, Epoch and osculation 1870 Jan. 0. B. M. T. On account of the uncertainties of the least square solution, it is certain that these elements are not exact to-ihe tenths of a second, but these have been retained because it is customary to print the elements in that way in the Jahrbuch. Special Perturbation*. From the smallness of the corrections to SCHUBERT'S elements it is evident that practically the same perturbations should be obtained, no matter which set of elements is used in the com- putation. Nevertheless, in order to be perfectly sure about the matter, special perturbations by Jupiter were computed, with the corrected elements, from 1870 to 1876. These perturbations, however, showed much greater differences from SCHUBERT'S than was to be expected, amounting to 60" in dw and dM. The method of variation of constants was used, ( formulas from WATSON,) and the perturbations were computed for forty-day intervals upon the same dates as were used by SCHUBERT. To avoid the accumula- tion of perturbations of the second order, the elements were corrected for the perturbations at frequent in- tervals, the first date of a new computation being always the same as the last date of the preceding one. When Psyche was nearest Jupiter this interval was about six months, but when furthest from Jupiter it could be taken as great as one and one-half years. BESSEL 's mass of Jupiter was used and i' and O' were taken from the American Ephemeris, for these are probably the values used by SCHUBERT, as he was at M 330 59' 12". 8 that time a computer in the Nautical Almanac Office. 7T 15 51 33 .7 \ Ecliptic and The longitude and the radius vector of Jupiter were O 150 35 32 .9 i mean equinox taken from SCHUBERT'S tables published in A. N., i 3 3 59 .9) of 1870.0. vol. 75, p. 209. ^ 7 49 21 .2 !" 710" .7200 The following table, taken in the sense TOWNLEY log /* 2 .851699 SCHUBERT, exhibits the differences in the per- log a 0. 46554 turbations : Berlin M. T. Adi Adfi Ad(/ > Ad?r Ad M A/; a is a function of p, and as we have the increment of /*, A/t = -j- 0.00065 (see page 6), the increments of \/a and \/p may be computed by dif- ferentiation, as follows: = A V<* cos < \/a sin Carrying out the computations according to these formula 1 , we find, lo lo = 0.23277 log \/p = 0.22871 = 3 Jl820 n 10, log A \/p = 5.45806 U 10 And from a subtraction logarithm table it is easily seen that this value of log A\/P will produce a change of only seven units in the sixth place of log \/P- The increment of r is small, but not constant, on account of the increased eccentricity of the ellipse and of the changing value of Av. It is my purpose to consider at present only the former of these. Ar will then have a maximum value when the planet is at an extremity of the minor axis, and a minimum value at an extremity of the major axis. At this .]/ = M + tp. M= 2e sin M 2M sin 3M - latter point Ar = Art = - i, which, how- ever, is not sufficient to produce any change in the sixth place of log a. The change in r would be greater than this at perihelion and less at aphelion. Since & a cos , the increment of b due to the in- crements of a and <, may be computed from the rela- tion A& = Act cos a sin : A sin I". The result is log A& = 5.6819 n , which would produce a change of eight in the sixth place of log &. At rr + 1".2, which makes A log sin i, 47 units in the sixth place. The difference sin U T sin s , in which the sub- scripts stand forTowNLEY and SCHUBERT, can be found from the relation u = v -j- TT O, provided Av can be found. This can be done from the known rela- tion between v and M, AM being already known. We have : 3sinM) + ...... hence differentiating AM = AM -f /A/* Ai> = AM j 1 -+- 2c cos M -f 5 / 2 e 2 cos 2M j + Ae ( e 2 } -. e sin M+ 5 / 2 e sin 2M-f ~j(13 sin 3M 3 sinM)-f . . Y Omitting for the present the term A/*, and calling the corresponding values of Ai> by Av , we have, taking e = 0.136, e- = 0.0185, r '/ 2 e 2 = 0.0462, AM = 52". 7, Ae = + 24".7 M 45 90 135 180 225 270 315 A^ 70" 18" 3" 15" 41" 70" 98" 107" Since ATT AC = + 55", the values of Aw will lie between -f- 52" and 52". A log sin u will vary greatly, depending upon the value of u. The period of u is five years, the sidereal period of Psyche. The effect of A?t is indicated in the following tabulation. The small term /A/A has been allowed for in the values of Av. Date. M u Av AM A log sin u A AA 1869 Dec. 11 327 182 28' - 107" 52' -0.00254 + 0" .01 0" .0000 1870 Apr. 10 351 212 57 - 84 29 10 2 .34 + . 0006 Aug. 8 34 244 20 48 + 7 + 1 13 .96 . 0003 Dec. 6 38 274 27 23 + 32 1 39 .03 + . 0009 1871 Apr. 5 62 301 52 - 10 + 45 . Q 53 .12 + . 0075 Aug. 3 85 326 23 3 + 52 17 30 .53 + . 0121 Dec. 1 109 348 19 7 + 48 49 6 .73 + . 0077 1872 Mar. 30 133 8 18 - 13 + 42 + 60 + 2 .49 . +0 . 0034 July 28 157 27 5 27 + 28 + 12 + 3 .69 + 0. 0010 Nov. 25 180 45 15 41 + 14 + 3 + 2 .33 + . 0002 1873 Mar. 25 204 63 27 O / 2 + o .80 0000 July 23 228- 82 15 72 -17 1 .07 + . 0000 Nov. 20 252 102 18 - 89 -34 + 1 .21 .0000 1874 Mar. 20 275 124 18 - 99 44 + 6 + o .08 + . 0000 July 18 299 148 54 -105 50 + 18 + o .32 + . 0001 Nov. 15 323 176 33 -107 52 + 0.00181 + .07 + . 0003 Taking up now the investigation of Z, we have : Z = m' k 2 h r' sin ft. No statement is made by SCHUBERT concerning the value of the mass of Jupiter, or of k used by him. For the present I shall assume that we have used the same values. The increment of h depends upon that of p, which latter might be found by differentiating the expression p- = r' 2 + r 2 2rr' cos ft cos (w 1 w). It will, however, be suf- ficient for our present purpose to use approximate val- 1 1 ues of A/o. h =1 , hence A/i 3p' 4 Ap = 3p-* Ar for a maximum value. Taking now Ar = V2 (Aa + A&), a mean value, neglecting the incre- ment of v, we have the effect of A/i illustrated in the following table: 10 Date. lo g V P 4 log A7i A log li A AA 1869 Dec. 11 7.1913 3 .0493 0.00024 -f 0" .01 0".0000 1870 Apr. 10 7.5154 3 .3733 2 , 2 .34 1 Aug. 8 7.9098 3 .7678 1 -13 .96 3 Dee. 6 8.2984 4 ,1564 1 39 .03 9 1871 Apr. 5 8.5052 4 ,3632 2 53 .12 25 Aug. 3 8.4223 4 .2803 2 30 .53 14 Dec. 1 8.1709 4 .0289 1 6 .73 2 1872. Mar. 30 7.8931 3 .7511 1 i 2 .49 1 July 28 7.6397 3.4977 1 + 3 .69 1 Nov. 25 7.4182 3 .2762 2 i 2 .33 1 1873 Mar. 25 7.2224 3 .0804 3 + .80 1 July 23 7.0468 2 .9048 15 - .07 Nov. 20 6.8870 2 .7450 2 .21 1874 Mar. 20 6.7420 9 .6000 + .08 July 18 6.6139 2 .4719 + .32 / Nov. 15 6.5083 2 .3663 0.00000 + .07 .0000 The signs of these corrections have not been con- sidered. ft' is the heliocentric latitude of the disturbing planet with respect to the fundamental plane the instantaneous orbit of the disturbed planet in this case. The maximum possible value of A/3' is therefore A*. The values of A sin ft' could be computed from its known relations, but so many auxiliary quantities enter that the process might become quite compli- cated. The maximum value of ft' is 2 28'. If A/2' have its maximum value when ft' is a maximum, then A log sin ft' would amount to six units in the fifth place of decimals. The following tabulation shows the effect of a possible maximum A/3' during the interval under consideration: Max. A Date. /3' log sin ft' A AA 1869 Dec. 11 2 0' 0.00007 + 0" .01 O^'.OOOO 1870 Apr. 10 2 14 6 2 .34 3 Aug. 8 2 23 6 -13 .96 19 Dec. 6 2 28 6 39 .03 55 1871 Apr. 5 2 27 6 53 .12 75 Aug. 3 2 22 6 30 .53 43 Dec. 1 2 13 6 - 6 .73 10 1872 Mar. 30 2 7 + 2 .49 4 July 28 1 44 8 + 3 .69 7 Nov. 25 1 25 10 + 2 .33 6 1873 Mar. 25 1 4 13 -f .80 2 July 23 41 21 .07 ,0 Nov. 20 18 47 .21 2 1874 Mar. 20 6 134 + .08 2 July 18 29 30 + .32 2 Nov. 15 51 0.00017 + o .07 .0000 To sum up : A Vp and Ar are each very small, the former, on the average, somewhat larger than the latter. We can with safety neglect further consid- eration of these factors. A sin i is constant and of sufficient magnitude to affect A in the hundredth^ of second for several (six) dates about the time of perijovian distance. There remain in the numerator three periodic terms, namely, sinu, sin ft' and h, the periods of which are, respectively, the sidereal period of Psyche (5.00 years), the sidereal period of Jupiter (11.86 years), and the synodic period of Psyche with respect to Jupiter (8.64 years). The course of analysis just employed shows that none of these has a controlling influence in determining the differences between SCHUBERT'S and my perturbations in n. By following out exactly similar lines of investi- gation for d ics and ~j- , which it seems hardly Civ necessary to print, it became certain that the differ- ences between SCHUBERT'S elements and my own were quite insufficient to produce the large differences between his and my values of the perturbations. Much larger increments to the elements would be necessary to produce such large differences in the perturbations. It was then thought that the differences might have been produced by the use of different values of the mass of Jupiter. No statement was made by SCHU- BERT of the value of the mass of Jupiter employed by him, but an examination of his earlier publications makes it very probable that he used a mass factor quite different from the one employed by me BESSEL'S. In' publishing general perturbations of Harmonia by Jupiter, A. N. 66, p. 27, December, 1865, SCHUBERT makes the following statements : " In all my calculations of the special perturba- tions of asteroids, and also for the general pertur- bations of Melpomene and Eunomia, the mass of Jupiter by BESSEL 1A ._ QQ has been used ; but as I have 1U4: ( .OO 11 been led to think the mass by NICOLAI nearer the truth, the above perturbations of Harmonia have been computed with it. On former occasions I have remarked that I felt certain BESSEL 's mass of Jupiter needed a correction, and that it would be possible to determine it by means of the perturbations of Leu- kothca. . . . My special reasons for preferring the mass by NICOLAI to that by BESSEL are the following : In the great many definite determinations of orbits of asteroids by means of the special perturbations, it would appear to me as if the normals could be repre- sented better with a smaller mass of Jupiter, and, therefore, for the sake of a mere trial, once in the case of Thalia, I introduced the correction as the sev- 1 v onth unknown quantity and found for it 57^. The oUUU nine normals upon which my tables of Eunomia are based are represented better with NICOLAI 's mass than with BESSEL 's. The same result has been arrived at by Professor BRUNNOW in the case of 7m, for which he has computed the general perturbations by Jupiter (first and second order), Saturn, and Mars." Assuming, then, that SCHUBERT used NICOLAI 's value, I proceeded to compute the increments to r- and - - but even this was not sufficient to ex- df dt plain away the large differences between his and my perturbations. In fact, if these increments be plotted, an inspection of the curves shows that the differences between the perturbations increase with the time, and are independent, at least to a certain extent, of the magnitude of the perturbation. This attempt to find an explanation of the differences be- tween the two sets of perturbations has been made especially difficult by the fact that SCHUBERT pub- lished no details whatever in regard to his work, there being even no reference to the formulae and constants used, and inquiry at the Nautical Almanac office disclosed the fact that the original computation can no longer be found. There remain but two explanations of the differ- ences: First, that SCHUBERT did not correct his ele- ments from time to time, as is usually done in the computation of special perturbations in order to avoid the accumulation of perturbations of the second order. Second, that either one or both of the computations is affected by numerical errors of a constant or pro- gressive nature. As to the first of these, it seems hardly possible that such an experienced computer as SCHUBERT should have done this, yet an inspection of my com- putations shows that if this had been done, the result- ing perturbations would differ from those obtained, in the same way that SCHUBERT'S do. As to the second hypothesis, it may be said that SCHUBERT had had long experience as a computer, especially in the computation of general perturbations, but I believe Psyche was the first planet for which he computed special perturbations. I have every confidence in the- correctness of my own computations. Not that I never make mistakes, alas! I have learned from bitter experience that the opposite is true, but that every figure of the computation has been carefully checked, and parts of it have been completely and independently recomputed after an interval of two or three years. It becomes necessary therefore to abandon the attempt to explain the differences between SCHUBERT'S and my perturbations, and it is of course useless to try to construct empirical formulas by which to carry the corrections forward over the balance of the thirty- year period. Additional Perturbations. Instead of carrying out the second part of this plan of computation, as stated on page 1, I expect now to compute general perturbations for Psyche by HANSEN'S method. While that is being done, however, it is . desirable to keep track of the planet, and I have therefore computed special perturbations by Jupiter from 1900 to 1906, which are given in the following table. They have been computed from my corrected elements brought up to 1900 by means of SCHUBERT'S perturbations, and they are as follows : Epoch 1900 Jan. 0. Berlin M. T. M 332 1' 19" TT 16 34 55 \ Ecliptic and O 150 31 44 \ mean equinox 3 4 31 ) of 1900.0. 7 51 3 /* 710" .4390 The constants, elements, and coordinates of Jupiter have been taken from the Berliner Jahrbuch. The elements were corrected for the accumulated pertur- bations whenever it was necessary to do so. 12 PERTURBATIONS BY JUPITER. Berlin M. T. /-1900 di di r 4* /fc dM 1900 January 0.0 0".0 0".0 0".0 0" .0 0" .0000 0".0 0".0 rBRAjV>x, February 9.0 40 -0 .1 .6 4- 6 .6 21 .1 .0425 .8 4-22 .2 OF THE March 21.0 80 .3 1 .7 13 .7 40 .1 .0868 3 .4 41 .0 IVER3!TY April 30.0 120 .4 3 .4 21 .0 56 .0 .1314 7 .8 55 .3 OF /' June 9.0 160 .5 5 .7 28 .4 68 9 .1749 13 .9 64 .5 J^QfiliW^ July 19.0 200 .6 8 .6 35 .5 76 .9 .2157 21 .7 69 .1 August 28.0 240 .7 11 .9 42 .2 82 .8 .2528 31 .1 69 .8 October 7.0 280 .7 15 .7 48 .5 87 .1 .2851 41 .9 67 .8 November 16.0 320 .7 19 .7 54 .2 91 .1 .3119 53 .9 64 .6 1900 December 26.0 360 .7 23 .9 59 .5 95 .8 .3328 66 .8 61 .5 1901 February 4.0 400 .6 28 .1 64 .4 102 .4 .3478 80 .4 59 .6 March 16.0 440 .5 32 .2 69 .0 111 .6 .3567 94 .5 59 .8 April 25.0 480 .4 36 .1 73 .6 123 .7 .3598 108 .8 62 .7 June 4.0 520 .2 39 .7 78 .1 138 .9 .3573 123 .2 68 .4 July 14.0 560 .0 42 .8 82 .9 156 .9 .3495 137 .3 76 .7 August 23.0 600 4-0 .2 45 .4 88 .0 177 .1 .3368 151 .1 87 .3 October 2.0 640 .4 47 .4 93 .6 198 .8 .3195 164 .2 99 .5 November 11.0 680 .7 48 .9 99 .6 221 .4 .2981 176 .5 112 .7 1901 December 21.0 720 .9 49 .8 106 .2 243 .7 .2730 188 .0 126 .0 1902 January 30.0 760 1 .2 50 .0 113 .5 264 .9 .2443 198 .3 138 .5 March 11.0 800 1 .5 49 .7 121 .3 284 9 .2124 207 .4 149 .5 April 20.0 840 1 .7 49 .0 129 .7 300 .7 .1776 215 .2 158 .1 May 30.0 880 1 .9 47 .7 138 .6 313 .5 0' .1403 221 .6 163 .7 July 9.0 920 2 .2 46 .1 148 .0 322 .0 .1006 226 .4 165 .5 August 18.0 960 2 .3 44 .2 157 .7 325 .5 .0588 229 .6 162 .9 Septemb'r 27.0 1000 2 .5 42 .0 167 .8 323 .5 .0153 231 .0 155 .6 November 6.0 1040 2 .7 39 .8 178 .0 315 .7 4- o .0300 230 .7 143 .2 1902 December 16.0 1080 2 .8 37 .5 188 .4 301 .8 .0767 228 .6 125 .4 1903 January 25.0 1120 2 .9 35 .2 198 .7 281 .8 .1245 224 .6 102 .4 March 6.0 1160 2 .9 33 .2 208 .8 255 .6 .1732 218 .6 74 .1 April 15.0 1200 3 .0 31 .3 218 .6 223 .6 9994 210 .8 40 .7 May . 25.0 1240 3 .0 29 .7 228 .1 186 .0 .2719 200 .9 4- 2 .9 July 4.0 1280 3 .0 28 .4 237 .1 143 .5 .3212 188 .9 38 .9 August 13.0 1320 3 .1 27 .4 245 .4 97 .0 .3699 175 .1 83 .8 Septemb'r 22.0 1360 3 .1 26 .8 253 .1 47 .3 .4174 159 .3 130 .8 November 1.0 1400 3 .1 26 .5 260 .1 + 4 .3 .4631 141 .7 178 .6 1903 December 11.0 1440 3 .1 26 .5 266 .3 56 .0 .5061 122 .3 225 .5 1904 January 20.0 1480 3 .1 26 .7 271 .8 106 .6 .5454 101 .2 270 .2 February 29.0 1520 3 .1 27 .1 276 .6 154 .1 .5799 78 .7 310 .8 April 9.0 1560 3 .1 27 .5 281 .0 196 .4 .6080 54 .9 345 .3 May 19.0 1600 3 .1 27 .8 1'85 .1 231 .6 .6280 30 .1 371 .7 June 28.0 1640 3 .1 . 28 .1 289 .2 257 .4 .6377 4 .8 388 .1 August 7.0 1680 3 .1 28 .1 294 .0 272 .0 .6345 4- 20 .7 392 .8 Septemb'r 16.0 1720 3 .1 28 .0 300 .0 273 .9 .6156 45 .8 384 .7 October 26.0 1760 3 .1 27 .8 307 .8 262 9 .5775 69 .7 363 .:; 1904 December 5.0 1800 2 .9 27 .7 318 .5 237 .1 .5 1 ().l 91 .7 329 .5 1905 January 14.0 1840 2 .7 28 .3 332 .7 199 .9 .4292 110 .7 285 .5 February 23.0 1880 2 .4 30 .3 351 .4 153 .3 .3129 125 .7 235 .3 ' April 4.0 1920 2 .0 35 .0 375 .1 100 .6 + .1670 135 .3 183 .9 .May 14.0 1960 1 .4 43 .9 403 .5 4- 45 .3 -0 .0050 138 .7 137 .1 June 23.0 2000 4- .7 59 .3 435 .6 - 12 .0 .1929 134 .7 97 .5 August 2.0 2040 .0 83 .0 469 .0 75 .7 .3775 123 .2 62 .8 Septemb'r 11.0 2080 .7 116 .4 500 .6 157 .8 .5314 104 .9 - 21 .4 October 21.0 2120 1 .0 158 .5 526 .9 275 9 (1 .62;"(l SI .(i 4-45 .5 1905 November 30.0 2160 1 .0 206 .1 545 .8 442 .1 .6383 56 .1 157 .2 1906 January 9.0 2200 .5 254 .3 557 .7 661 .0 .5700 31 .7 322 .5 February 18.0 2240 4-0 .3 298 .4 564 .6 920 9 .4376 4- 1 1 .3 534 .9 1906 March 30.0 2280 4-1 .4 -335 .5 4-569 .0 -1199 .3 .2674 2 .8 4- 776 .7 . 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