00 C\! r\i EXCHANGE DEFINITIVE ELEMENTS OF COMET 1898X, (BROOK'S) A THESIS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF PENNSYLVANIA BY JOHNATHAN T. RORER IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PHILADELPHIA 1910 DEFINITIVE ELEMENTS OF COMET 1898 X, (BKOOK'S) A THESIS PRESENTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF PENNSYLVANIA BY JOHNATHAN T. RORER IN PARTIAL FULFILMENT OF THE KEQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY PHILADELPHIA 1910 PRESS OF THE NEW ERA PRINTING COMPANY LANCASTER. PA. DEFINITIVE ELEMENTS OF COMET 1898 X, (BKOOK'S). Comet 1898 X was discovered October 20th, 1898, by Dr. W. K. Brooks of Geneva, N. Y. Observers generally recorded the brightness of its central condensation from 10 m to ll m . Mr. Perrine early called attention to the similarity of the elements of 1881 IV (Schaeberle) to those of this comet (A. J., XIX, 145). The provisional elements assumed' in this computation were those published by Hussey (A. J., XIX, 120), based on obser- vations of Oct. 21st, 23d and 25th. One hundred and eighty- four observations were collected from all available sources and compared with an ephemeris computed from the above elements. To perfect these and lessen the residuals, a least square solution regarding the orbit as elliptic was then made. The six normal equations thus formed lead to indeterminate results. It was at first suspected that this might be due to the perturbations of the Earth pad Venus ; the computation of these perturbations for the time covered by the observations showed them to be but small fractions of a second of arc and they were therefore disregarded. It was next determined to use the data at hand to obtain the most probable parabolic elements. Three new normal dates were selected, each being intermediate between successive pairs of former normal dates and the normal equations then used to obtain corrected positions for these dates. For a parabolic orbit from these, the following elements result : r= 1898 Nov. 23. 16356 B. M. T. = 123 31' 34".89 ,-,,... , , , ^ . fl = 9620'41".26 Ecliptic and MeanEqumox = 140 21' 5".05 log. g =9.8786106 The usual checks were applied to verify the work. 3 240959 From the above elements the positions of the comet were computed for ten selected dates of observation. The resulting residuals, while showing a marked improvement over those from Hussey's elements, were deemed capable of further improvement, and accordingly the residuals for the date of each observation were found by interpolation and correction of the previous resi- duals. Ten normal places were then formed from which correc- tions to the above elements were derived by a second least square solution. These corrections were : ATT = - 242". 80 Ai = + 51.36 AI 7 = - 0.00206 Ag = - 0.0000284, leading to the elements : T= 1898, Nov. 23. 16150 w = 123 29' 40".30 ft = 96 18' 33".05 i = 140 21' 56".41 log q = 9.8785943 After the completion of the above computation forty-five ad- ditional observations became available ; an effort was therefore made to still further reduce the residuals by the employment of these observations. Using the above elements a new ephemeris was computed at four-day intervals for the time covered by the observations and from this by interpolation the computed place was found for each of the two hundred and twenty-nine observa- tions. The residuals thus formed were plotted, and a smooth curve was traced representing the points as closely as possible. It was not deemed necessary to recompute the coefficients of the corrections to the elements used in the normal equations, and con- sequently the same normal dates were used. A least square solution reduced the twenty resulting equations to five normal equations of the usual form : [1] +38.083A7T 68.580Ai2 16.373A* 21.725A2\+8.323Ag 1 +2175.976=0 [2] 68.580 +124.173 +27.701 +38.568 15.216 3809.574=0 [3] 16.373 + 27.701 +47.936 +19.797 + 4.170 1983.742=0 [4] 21.725 + 38.568 +19.797 +15.131 - 2.766 1548.340=0 [5] + 8.323 - 15.216 + 4.170 2.766 + 3.548 + 255.329=0 From these there was obtained : AH = 31". 373 10 6 A^ = Aft = + 94 .901 ATT = +126.000 ( Ai = - 136.805 10 4 AT= A2; = + 559.540 A substitution of these values in the five normal equations leads to the residuals : -0.14 + 0.21 -0.85 -0.19 -0.05 Considering the size of the constants, the above residuals are thought to be satisfactory and the following elements resulting from the above corrections are adopted as the final values. FINAL ELEMENTS. Nov. 23. 21745 n AO or it fL Ecliptic and Mean Equinox o , V* of 1898.0 i = 140 19' 39". 60 log q = 9.8786488 The short time this comet was observed, 36 days, the flatness of the arc traversed, the slowness of the motion, and the difficul- ties arising from its physical appearance, all combine to produce indeterminate results. Much effort was expended to obtain a 6 greater refinement from the data, and the present publication has been long delayed in the hope that better elements might be ob- tained. Further investigation, however, warrants the conclusion that the above elements may be considered as final. Since the above calculations were completed S. Sharbe has pub- lished definitive elements of this comet (A. N., 164, p. 378). Both parabolic and elliptic elements are given, the latter being deemed definitive. These are here reproduced. A comparison with th e elements given above shows substantial agreement, the greatest differences being in the inclination of the orbit and the time of perihelion passage. Elements of the Author Sharbe's Parabolic Sharbe' s Elliptic T 1898 Nov. 23. 21745 Nov. 23. 189594 Nov. 23. 195124 w 123 32' 17". 67 123 31' 53". 96 123 32' 23".70 G 96 18' 1".68 96 18' 14".47 96 18 12".46 i 140 19' 39 // .60 140 2(X 57". 50 140 20' 51". 52 log q 9.8786488 9.8785281 9.8785038 e 1. 1. 0.9997421 The value of de given by Scharbe is 0.0002579 0.000- 2600. The limits of error are evidently so large that there is no sufficient reaaon for giving preference to elliptic elements. The probable errors of all the elements indicate the indeterminate character of the orbit, to which reference has already been made. MAKERS SYRACUSE, - NY,