QB 3*75 M*.S58\ S3 UC-NRLF EXCHANGE 17 1916 COMPUTATION OF THE ORBIT OF PLANET (558) BY J. H. SCARBOROUGH SUBMITTED FQR THE DEGREE OF DOCTOR OF PHILOSOPHY (Pn.D.) TO VAKDERBILT UNIVERSITY, NASHVILLE, TENN. MAY, 1908 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER, PA. COMPUTATION OF THE ORBIT OF PLANET (558) BY J. H. SCARBOROUGH SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (PH.D.) TO VANDERBILT UNIVERSITY, NASHVILLE, TENN. MAY, 1908 PRESS OF THE NEW ERA PRINTING COMPANY S3 - COMPUTATION OF THE ORBIT OF PLANET (558). The small planet (558) was discovered by Wolf at Konig- stuhl-Heidelberg, February 9, 1905, and was reported in the Astronomische Nachrichten, Volume 167, page 208, as follows : " Photographische Avfnahmen von kleinen Planeten. 1905. Q,B, M. Z. Kgst. ll h 51.9**, Feb. 9, 1905. a 9 h 47.5 m B + 13 4'.0 Gr. 12.0 Tag. Bewegungen, 0.9 m + 4'.0. Q,B, und**** sind neue Planeten. M. Wolf." The value of Tag. Bewegungen as given in the above report is corrected on page 303 of the same volume, and should read, for declination, + 7' instead of + 4'. As reported by the same observer, page 349 of the same volume, another observation on March 13, 1905, gives: M. Z. Kgst. a B 9 h 16 0.5 m 25.5 m 24' The following observations made by Palisa at Wien (k. k. Sternwarte) are reported in Volume 168 of the Astr. Nach. : 1905 M. Z. Wien aapp. 1. par. A fiapp. 1. par. A h m s h m s Mar. 29 9 47 31 9 21 16.01 8.970 +17 20 12.9 0.656 " 31 9 37 20 9 21 9.24 8.955 +17 24 37.7 0.655 Apr. 4 10 17 35 9 21 13.23 9.265 +17 31 50.4 0.667 9 9 33 14 9 21 50.70 9.154 +17 37 34.3 0.659 23 10 37 14 9 26 38.49 9.491 +17 36 23.6 0.705 30 8 50 38 9 30 29.76 9.286 +17 27 2.5 0.670 May 10 9 27 18 9 37 34.88 9.465 +17 4 15.8 0.702 330514 The places for the two Heidelberg observations given on the preceding page are also given at another place in the Astr. Nach. (M. Z. Kgst.) : Feb. 9 Mar. 13 h m 11 51.5 11 0.5 h m s 9 47 51.19 9 25 35.30 +13 4 23^50 +16 25 34.90 I have examined the complete files of the Astr. Nach. from the date of the discovery of this planet up to December 8, 1905, a period of more than six months after the last observa- tion given in Vol. 168. So the above appears to be a com- plete list of all the observations made during the opposition period immediately following the discovery. This planet was numbered (558), September 26, 1905, by F. Bauschinger of the Astron. Recheninstitut, Berlin ; and this designation, (558), will be used in any future reference to it in this paper. The Provisional Elements of (558) have been computed by A. Berberich of the Astron. Recheninstitut, Berlin, and are published in Vol. 169, p. 285, of the Astr. Nach. y as follows Epoch, Feb. 9.5 (M. T. Berlin) 1905. M a> 41 IT 34".4 314 40 6 .0 144 15 43 .8 8 21 3.0 2 14 1 .0 /* 715".481 log a .463606. These elements are referred to the mean equinox of 1905.0. It is my purpose in this thesis : To construct, from the pro- visional elements just given, a provisional ephemeris for the opposition period from February 9 to May 10, 1905. To cor- rect the provisional elements by comparing the observed places, already given, with the computed places to be shown in the provisional ephemeris. I. COMPUTATION OF PROVISIONAL EPHEMEKIS. (a) Eccentric Anomaly. To determine the eccentric anom- aly, E, I used the transcendental equation M=E-e$\nE. (1) Assuming E = M, a first approximation was found ; using the first approximation and repeating the process, a second approxi- mation was determined ; the second approximation was corrected by means of the formula M-E'+e-smE' 1-e-cosE' and the results were found to satisfy the given equation (1). The results are given below : Date, Gr. M. T. 1" Approx. 2' ' Approx. A* E Feb. 9.5 42 46 30.81 42 49' 4.62 4.53 42 49 9.15 13.5 43 35 36.09 43 38 10.30 4.50 43 38 14.80 ti 17.5 44 24 40.31 44 27 14.84 4.42 44 27 19.26 11 21.5 45 13 43.49 45 16 18.18 4.39 45 16 22.57 14 25.5 46 2 45.59 46 5 20.35 4.32 46 5 24.67 Mar. 1.5 46 51 46.61 46 54 21.29 4.24 46 54 25.53 < 5.5 47 40 46.56 47 43 21.02 4.17 47 43 25.19 M 9.5 48 29 45.36 48 32 19.52 4.09 48 32 23.61 |4 13.5 49 18 43.04 49 21 16.75 4.02 49 21 20.77 II 17.5 50 7 39.56 50 10 12.73 3.92 50 10 16.65 (( 21.5 50 56 34.95 50 59 7.40 3.86 50 59 11.26 {( 25.5 51 45 29.13 51 48 0.82 3.75 51 48 4.57 M 29.5 52 34 22.15 52 36 52.90 3.68 52 36 56.58 Apr. 2.5 53 23 13.95 53 25 43.69 3.56 53 25 47.25 14 6.5 54 12 4.51 54 14 33.13 3.47 54 14 36.60 II 10.5 55 00 53.93 55 3 21.23 3.37 55 3 24.60 14 14.5 55 49 42.07 55 52 7.96 3.27 55 52 11.23 14 18.5 56 38 28.93 56 40 53.33 3.18 56 40 56.51 (4 22.5 57 27 14.54 57 29 37.35 3.06 57 29 40.41 (r 26.5 58 15 58.86 58 18 19.94 2.94 58 18 22.88 II 30.5 59 4 41.91 59 7 1.13 2.84 59 7 3.97 May 4.5 59 53 23.66 59 55 40.93 2.73 59 55 43.66 it 8.5 60 42 4.06 60 44 19.29 2.63 60 44 21.92 " 12.5 61 30 43.18 61 32 56.23 2.54 61 32 58.77 All the values of E given in the above table have been veri- fied by substituting in the given equation (1). (6) True Anomaly, Radius Vector, and Longitude. The true anomaly, V, the radius vector, r, and the longitude of the planet in its orbit, u, were determined from the following : V sin - = sin (45 E sin V E cos -^ = T/2a cos (45 + }) cos -~, TT = a) + a = 458 55' 49".8. The results are shown in the following table : Date, Gr M. T. V u logr o / // o // Feb. 9.5 44 21 34.00 359 i 40.00 0.451 010 13.5 45 12 3.66 359 52 9.66 0.451 180 u 17.5 46 2 30.68 42 36.68 0.451 350 ft 21.5 46 52 55.18 1 33 1.18 0.451 530 25.5 47 43 17.54 2 23 23.54 0.451 706 Mar. 1.5 48 33 37.14 3 13 43.14 0.451 886 u 5.5 49 23 54.54 4 4 0.54 0.452 066 9.5 50 14 9.10 4 54 15.10 0.452 250 it 13.5 51 4 20.74 5 44 26.74 0.452 438 i ( 17.5 51 54 30.56 6 34 36.56 0.452 626 n 21.5 52 44 36.98 7 24 42.98 0.452 818 (i 25.5 53 34 41.14 8 14 47.14 0.453 010 ii 29.5 54 24 42.74 9 4 48.74 0.453 204 Apr. 2.5 55 14 41.56 9 54 47.56 0.453 400 da ,, * da da da A cos o -r- ATT -f cos o -77^1! -f- cos o -yv A^ + cos o -jj Ad> dir dl di d dM dp 13 In order to obtain the six equations between the variations of the six elements, I took the combined results of the vari- ations of the coordinates given for the normal places ; the three changes in right ascension giving me three equations, and the three changes in declination giving the other three. I might have taken each observation separately, and from the seven observations obtained fourteen equations, and then com- bined these into six equations. By either method of obtaining the six equations, all the observations available are made use of. The equations of condition for finding the values of the vari- ations of the elements were thus found to be : + 1.275012 Air 0.045347 AQ+0.035849 Ai+1. 895022 A^-f 1.347753 AJf 1.752088 Aju= 3".8650 0.196062 0.187308 +0.213273 -0.250601 -0.209490 +13.394215 = 4 .110 + 1.161339 0.041779 +0.044644 +1.778783 +1.225739 + 5.436958 = 8 .5754 0.175604 0.171191 +0.262144 -0.229251 -0.187798 +13.018983 = 2 .270 + 1.025810 0.038474 +0.057220 +1.654673 +1.078990 +20.764357 =15 .469 -0.164980 0.149625 +0.313806 -0.234041 -0.175896 + 9.383532 = 3 .820 The second member of each of the odd-numbered equations above is cos 8 Aa, while the second member of each of the others is A8. The second member in each is expressed in sec- onds of arc. To verify the computations of the coefficients in the above equations, I assumed a small arbitrary change in each element, namely : Let ATT = - 20" -10 Ai = -f 10 A< = + 10 AJf = + 10 A/A = + 00.91 Applying these changes to the provisional elements, and using the elements thus corrected, I computed the right ascension of the planet for April 14.5 and compared it with 14 the right ascension already computed from the elements before the changes were applied. I found the difference to be, Act = 8.48". I also found the value of Act directly from the third equa- tion in the list, using the same date, and the agreement was close enough to indicate that the computation of the coefficients in this equation is correct. As the computation of the places in this verification is rather long, I did not apply it to all the equations. However, it is very probable that if an error should occur in the computation of the coefficients of any one of the equations, it would affect the coefficients in the others also. The solution of the six equations given above, involving the variations of the six elements, gives AJf = + 3' 19".49 A< = + V 46".15 Ai = + 4'.80 AH= + 8".33 ATT = - 6' 5".99 A/i = - 0".001386. I found Act from the formula, A =ct f - , the result being, Aa = 0.00000376. Applying the corrections just found to the provisional ele- ments, we have the following : M= 41 20' 53".89 7r= 98 49 43 .81 ft = 144 15 52 .13 i= 8 21 8 .80 < = 2 15 47 .15 H= 715 .4796 Note that TT, used above, is equal to fl -f &> (used on page 3). 15 The epoch is the same as on page 3, and the elements are referred to the mean equinox of 1905.0. These are the corrected elements sought. The approximate time of the second opposition is May 17, 1906, the time of first opposition being February 4, 1905. From the mean daily motion as corrected, the sidereal period is found to be 1811.37 days ; the synodic period can be readily found by comparing with the period of the earth it is nearly 457 days. The perturbations of Jupiter and Saturn have not been taken into account. This topic would of itself form an inter- esting one for a thesis. Conclusion. I cannot hope that the corrected elements here given are free from error. The best results of computations based upon the observations of one opposition only, cannot in any case give us results altogether free from error. The short period over which the available observations extend, is such as to prevent results altogether accurate. Some other difficulties have already been pointed out. I selected this planet from a large list of newly discovered asteroids after examining all the available observations of each one. This afforded me the best material at hand for the work I desired to accomplish, for the observations were more numerous and extended over a longer period than those of any other one in the list. The fact that all computation and checking of results had to be done by one person has made the task much longer than I had anticipated. However, I hope to be able at some future time to carry the work forward to other oppositions, as they are observed, and thus make still further corrections of the elements. We cannot hope for results of extreme accuracy, without carrying the work forward through several oppositions. In the prosecution of the work, I cannot claim originality as to the theory and methods used. In fact, the theory of the 16 computation of orbits of bodies moving about the sun as given by Gauss in his " Theoria Motus " a century ago, is the basis of all the work of the computation of orbits by the astronomers of the present day. I have consulted freely, and used formulas, from the following : " Theoria Motus," by Gauss, translation by Davis. " Lecons sur la Determination des Orbits," par Tisserand, arranged by Perchot. " Les Orbits des Cometes et Planetes," by D'Oppolzer, French translation. "Tafeln zur Theoretischen Astronomic," by Bauschinger. " Theoretical Astronomy," by Watson. Astronomische Nachrichten, complete files to December, 1905. I am under special obligation to Wm. J. Vaughn, LL.D., head of the Department of Mathematics and Astronomy, Vanderbilt University, for his many valuable suggestions, and also for the free use of his excellent private library. I am also under special obligation to Professor D. T. Wilson, of the Case School of Applied Science (Cleveland, O.), who has kindly verified most of the computations in this thesis. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW AN INITIAL FINE OF 25 CENTS WILL BE ASSESSED FOR FAILURE TO RETURN THIS BOOK ON THE DATE DUE. THE PENALTY WILL INCREASE TO SO CENTS ON THE FOURTH DAY AND TO $I.OO ON THE SEVENTH DAY OVERDUE. J LD 21-100TO-12, '43 (8796s) Gay lord Bros. Makers Syracuse, N. Y. PAT. JAN. 21, 1908 3 3 t- UNIVERSITY OF CALIFORNIA LIBRARY