The following equations must then be formed (k is the chord of the comet-orbit between the extreme observations): M2. p'2 R-2R'. cos. (a'— A'). p' + sec. 8'2.p'2 R2_2R". M. cos. (a"-A")p+sec.p2, = (r'2 + r'''2) — 2R'. R"". cos. (A"'—A') +2R"". cos. (a'— A''')p' + 2R'. M. cos. (a"- A′). p' - 2M. cos. (a"-a')p'2-2M. tan. B'. tan. ""'. p'2 (III.) The amount and direction of the error of interval between the extreme times of observation, resulting from this first value of p', will, after a little experience, guide the computer to another value nearer to the true one; and the error of the second assumption, compared with that of the first, again leads to a much closer value for the third approximation, and so on till the assumed value of p' produces an agreement between the calculated and observed intervals. In practice we have not found any great advantage on adopting one or other of the devices suggested for obtaining successive values of p' by use of tables or otherwise the simple method of continued approximation, by deducing a new value of the curtate distance proportional to the errors in the two preceding assumptions, will be found in the great majority of cases sufficiently expeditious and as little troublesome as any other. In working Lambert's equation, proceed as follows: tan. A'"'— tan. A'. cos. (e'''— e') } (VII.); sin. (e'"'—e') and if the motion be retrograde from tan. ""— tan. A'. cos. (0'— e'''), The distances of the comet from the ascending node reckoned upon the orbit, at the first and third observations (a', u'''), are given in the case of direct motion by tan. (0'-) tan. u'"'= tan. u' cos. i or, if the motion be retrograde, by tan. (―e') tan. u' cos. i. tan. (0"'"'— ~) } (VIII.); COS. tan. u'"= tan. (―0'''). cos. i The arc u'-u' is equal to the difference of true anomalies, and the true anomaly at the first observation (v') will be obtained from tan. ‡v'= cotan. ¿(u''' — u') — or from tan. '= sin. (u'"'—u1) √r'''. cos.2 } (u'''— u') — NF' √TM!''. sin. (u'''— u') v'''= v' + u'"'— u' (29) [8,5366114] constant [8-2355814] } (XII.) '+[9,5228787] tan.3 v. Similarly we may find the interval from the third observation to perihelion by substituting v for v'; the times thus separately determined should agree, and this agreement will afford a third check upon the accuracy of our work. Thus the whole of the elements of the parabolic orbit are found, and it is always desirable to ascertain how the geocentric place calculated from these elements for the time of the second observation agrees with the position observed; the first and third places are necessarily repre The comet's curtate distance from the earth at the third sented. observation is given by p""'- Mp' With the final values of r', r, p' and p', the direct calculation of the elements of the orbit commences. In the computation of a geocentric position from parabolic elements we may proceed thus: for which we require to compute (t-T), in days and Find the interval from perihelion passage to the time decimals. Put cotan. 2 v = · 3k (2q) — § (t − T) cotan. cotan. then tan.v= 2 cotan. 2r= } }· cos. A. cos. ( — 6): )= cos. (v + xcos. A. sin. (0 – 6) sin. (v +-). cos. i sin. (v +-). sin. i sin. A = or, if the motion be retrograde,— cos. A. cos. (8 − e) : = cos. (v− + n) cos. A. sin. ( — 0) = sin. (v. -+). cos. i sin. A sin. (v. -+). sin. i = (XIV.) (XV.) If e' is in advance of e', the motion in the orbit is direct; if the contrary be the case, the motion is retrograde.equations which give the heliocentric longitude and latitude (0, 2). The geocentric longitude and latitude (4, B) and the true distance from the earth (4) are then No. 6........ +0.2618976 +1.827669 0-3010300 9-9926916 +0.1033276 +1.268608 0.3010300 9.6699800 +9.9609124 No. 8........................... +0.913929 Log. 2 Log. M. 0-3010300 9-6699800 +0.5509163 +0.6852973 Log. tan. B' +0.1633710 Log. tan. 8"...... No. 9............. +4.845039 No. 8+No. 9.......+5.758968 Long. M. Eq. 1875-0..225 2 30 And so for the second and third positions. The interpolation of the sun's longitudes and the log. radii-vectores of the earth from monthly page iii. of the Nautical Almanac requires no illustration. We now form the angles a'-A", a'' - A'', a'"'-A", &c., and take out the sines and cosines required; and it is always convenient to have these functions and other of the principal quantities copied in plain figures on a paper separate from the calculations. Thus we have, Log. 2 Log. M........ Log. R'........ Log. cos. (a"-A')..+9.8654465 +9.8298155 No. 7................ .+0.675796 No. 6+No. 7......+1·944404 So that the equation for k2 is thus formed,— gu12 + go1118= 1.936783-2-126905.p' + 6·106359. - 1.827669 + 1·944404. p' — 5.758968./ 0.1091140-182501.p' + 0·347391.p' And thus substituting logarithms in the factors for p' and p', our equations stand thus, in the form for proceeding with the work Sine. 9-8088592 9.8365440 9.8988564 +9-9352968 +9.6696554 +9.8096060 +9.8654465 +9.9748170 + 9.9899024 ניייז = If for a second approximation we take p'-07189, and calculate z'z' precisely as before, the error in the interval from the first to the third observation, or (t'"-t'), is found to be -0.3606, which, compared with the error of the first assumption (-24-0488), shows a change of + 14.6882 for an increase of 0.0649 in p', or of th part, and by mere proportion we have p=0·72776, for a third approximation, giving the error in interval + 04.0222, so that we are now approaching the true value, and with p=0·7269562, obtained from the errors of the second and third trial in the same way that the third value of p' was inferred, we may substitute seven-figure logarithms and work more closely; it will thus be found that the error in interval corresponding to the fourth assumption for p' is reduced to +04-00167, or less than 24 minutes, and if we are only seeking an approximate knowledge of the orbit, the direct calculation of the elements might proceed with this fourth value of p'. However, to make the computation in this example a little more complete, we work out two further hypotheses, and finally adopt for the correct value of p'...0.7268994, with which the calculation is as follows: Log. p'...9.62448 + 0.49443 R's. 0.96988 Log. p'2 9-62448 No. (2.)... + 1·31494 (2.)... +0.11891 + 2.28482 No. (1.)...1.10135 + 0.47485 Log. p' 9-62448 (4.)... +0.09933 No.(1.) 2-619482 -1.233560 1-385922 +9.54082 +'?.. 1.18347 0.07316 Log. r'?.. 0.03658 Log. r.... (3.) 9.4948313 9.2612653 (4.)...+0.1978022 Log.r'2 +9.5408186 Log. r' 0.1417388 0-0708694 Log. p'.... 9.81224 Log. p' 9.62448 = Log. p'... 9-8614713 Log.p 9-7229486 (5.)........-9.1227396 (6.)...+9.2637672 12. 1.94487 Log. 2. 0.28890 Log. r'".. +2.22387 +0.25543 No. (3.)... — 0.27900 2-543796 No. (5.) +0-11844 No. (3.)....... 0-312487 0.14445 1.08788 0.400011 0.200006 1-177251 With the aid of the formula (VI.) the heliocentric longitudes and latitudes of the comet at the first and third observations are found as follows: Log... Log. sin. (0'+0'')... - 9.8243819 +90517209 -0.7726610 Log. tan. A'... +0.3134658 Log. tan. (-0')... 2-e'. 9.5408048 160° 50' 37".7 Add '... 121° 22′ 10′′:4 282° 12′ 48′′-1 ...282° 12′ 48"-1 e""...114 54 5.8 -e""...167° 18′ 42′′-3 Then for the arguments of latitude at first and third observations (u', u''')— Log. tan. (U—e')..-9.5408048 Log.tan. (-0'')..-9.3524609 Log. cos. i.. 9-1971480 Log. cos. i.. 9-1971480 Log. tan. u'..- 0.3436568 Log. tan. ""..- 0.1553129 """..124° 57' 59''-8 ... u'''— u'— 10° 35′ 2′′-0, and 1(u'"'— u′) — 5° 17′ 31′′.0. u'..114° 22' 57"-6 = We have now to calculate the true anomaly at the first observation from the radii-vectores r', r'"' and the included angle u'-u', which is =v''' — v', and for this purpose will employ both expressions for tan. v in (IX.)— By the second formula. Log. Nr............ 0.0871400 Log. cos. (u — u'). ......+9·9681449 No. 1............... Log. Nr. .......... 0.0852849 +1.2169840 0.0354347 Log. sin. (u'"'-u'r'"'). Log. tan. '. No. 2................................................. +1.0850125 No. 1 No. 2............ . +0.1319715 Log............+ 9.1204801 +9.0520150 ............+ 0.0684651 tv'.......+49° 29′ 51′′-5 how the comet's geocentric position, calculated from the elements thus obtained, agrees with the observed position. A close agreement where good observations have been employed, of course, indicates that the real path of the comet in space does not much differ from a parabola, while a considerable difference, i. e., one exceeding the probable error of the observation, may be due to the ellipticity of the orbit, and the comet may prove to be one of no long period. We will, therefore, proceed to compute the longitude and latitude from the above elements for the time of the second observation. Perihelion passage (T), October 19.19240 Date of second observation (t"), December 16-68190 t''T.........+58.48950 Instead of using Barker's table, we will compute the true anomaly directly by the formulæ (XIII.); thus, Log. (3k).... 8.7127027 Log. cotan. v.... Log. (t"-T).... +1-7670779 Log. cotan. +0-4797806 Log. Cotan.} Log. (29).... 9-9955453 Log. cotan. 2v.... +0.4842353 0-7964958 Log. sin. (v"+-)..+9.9363812 0.2654986 ....28° 29' 7"-6 9.8130004 Log. cos. (v''+-π)..-9.7023823 ....282° 12' 48"-1 2v.... 18° 9' 18"-8 Log. cotan. 2.... .... 9° 4' 39"-4 Log. 2.... 19-4167 29 30-0 These errors are not greater than may be looked for, in a The equations are computation upon the method we have adopted. We have computed the true distance of the comet from the earth at the second observation A". If the true distances at the first and third observations are desired, we have A'= p' Alll= cos. B cos. B/7, or, in the present case, A-1.28437, A-1.24574, so that the comet was slowly approaching the earth during the interval over which the observations extend. 5 19-2 163 7-8 0.00325 +9 22-3 176 43-8 0-00165 p = 1·01911 + [9·82521] p' + [0·01721] April 12-60769 320 30-8 If it be preferred to compare with the observed right ascension and declination, the formula (XVII.) have yet The equations are to be applied, the calculation, as will be seen, being very similar to that in the conversion of right ascension and declination into longitude and latitude. [The formation of the equations for the determination of p', p'', and k will perhaps be found the most slippery part of the computation by the beginner, and we add therefore two or three sets of data from observation and the ephemeris, with the resulting equations, which may be verified for the sake of obtaining a better acquaintance with this part of the work. 21.00666 [9.96708] p' + [0·01060]p' gl/21.01916 + [9·57247] p' + [9.48948]p' k 0.16416 [9·7247C] p' + [9·83437]p” Comet 1874. The Great Comet of Coggia. β " +46 35 3 28-37122 92 42 7+45 52 May 9.39543 92 55 24 +45 28 23 |