COMET
IN the present article it is proposed to exhibit formulae by means of which the orbital elements of a comet may be determined from three observations, assuming the comet to move in a parabola, an hypothesis upon which the apparent paths of the great majority of these bodies may be closely represented, appending thereto a fully worked example of the practical application of the formulae; also to put the reader in possession of methods now employed for calculating ephemerides of the apparent positions of a comet, to assist in observation. The limits within which we are confined will necessitate reference to other works for demonstration of our formulae, but care will be taken to name those authorities, which are not only most accessible, but by which the subject has been most clearly treated.
A list of comets of short or moderate period, so far as known at present, a class which offers particular interest to the student of this branch of astronomy, will likewise be included.
The method of calculating a parabolic orbit from three observations which we shall follow is the comparatively expeditious one proposed by Gibers, and demonstrated in his Abhandlung über die leichteste und bequemste Methode die Bahn eines Cometen zu berechnen, first published at Weimar in 1797, and since twice reprinted with considerable modifications and additions. The method is founded upon the principle that, if a c be the chord between the extreme positions of the comet in its orbit, and A C the similar chord of the earth’s path, the radii-vectores at the middle position cut a c and A C proportionally to the times occupied in describing the arcs, a supposition which, though not mathematically exact, is but little in error if the intervals between the observations are pretty nearly equal, and the arcs described small.
It may be convenient if the notation employed in the subsequent formulae be given here, at least as regards the principal quantities entering into our calculations. [table]
[table]
We suppose that the observations furnish three complete positions of the comet referred as usual to the equator, or expressed in right ascension and declination, with the mean times of observation at the respective places.
The first step will be to convert the observed right ascensions and declinations into longitudes (a) and latitudes (β), thus :— [table]
Thus we find α', a", a ", and β', β", β"', where the quantities with one accent apply to the first observation, those with two accents to the second place, and with three accents, to the last observation. This is to be understood throughout our formulae for the calculation of the orbit.
Now reduce the times of observation to the meridian of Greenwich by applying the longitude of the place of observation with its proper sign, and convert the times so reduced into decimals of a day; thus we have t ’ , t", t"'.
For each of these times interpolate from monthly page iii., in the Nautical Almanac, the sun’s longitude (A) and the logarithm of the earth’s radius-vector (R); the sun’s longitude in the Almanac being apparent, the amount of aberration (20" - 42 ÷ R), which is given in another part of the ephemeris, must be added to the apparent longitude, to obtain the true longitude required in the calculation. We have then A', A", N" and the logarithms of R', R", R'", and are ready to proceed with the application of Olbers’s method.
We commence by calculating Μ or , the ratio of the P
comet’s curtate distances from the earth at the first and third observations from [table]
The following equations must then be formed (fc is the chord of the comet-orbit between the extreme observations): [table]
If (f — i') be the .interval of time between the first and third observations we have, by Lambert’s theorem, [table]
With an assumed value for ρ' we calculate r, r ’ ", and ^k, and then t"' for comparison with the observed interval between the first and third observations, and vary p ’ in successive trials until the observed and calculated values agree. In this solution of the above equations by the method of trial and error, a first approximate value of ρ' may be inferred as follows :—
Writing the equation for V- [table]
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 ρ ’ 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 ρ' 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 :— [table]
The approximations to ρ' may be continued until z' - z" agrees with (½" - i'), within 2 or 3 in the fifth place of decimals, though if the computer has only rough observations at command, à larger error may be tolerated.
The comet’s curtate distance from the earth at the third observation is given by
ρ'"=M∕
With the final values of √, r"', ρ' and p"', the direct calculation of the elements of the orbit commences.
The heliocentric longitudes F, θ''', and latitudes λ', λ'", are obtained from [table]
in which equations the right-hand quantities are known.
The values of √ and r" ’, resulting from these equations, should agree with the preceding ones if the calculations have been correctly performed. This agreement forms the first verification of the work.
If θ"' is in advance of ff, the motion in the orbit is direct; if the contrary be the case, the motion is retrograde.
Then, if the motion be direct, the longitude of the ascending node ’ ( ß ) and inclination of the orbit to the ecliptic (i) will be found from [table]
and if the motion be retrograde from [table]
The distances of the comet from the ascending node reckoned upon the orbit, at the first and third observations (√, w"), are given in the case of direct motion by [table]
or, if the motion be retrograde, by [table]
The arc u" ’- u is equal to the difference of true anomalies, and the true anomaly at the first observation (√) will be obtained from [table]
or from [table]
The perihelion distance (<?) = /· cos. 2 ¾υ = r'. cos. 2 ?v .
The longitude of the perihelion, reckoned on the ecliptic to the node and thence on the orbit, is given by [table]
As a further verification of the calculations, we have— [table]
which should give the former value of l·.
We have now only to determine the time of perihelion passage (T), by finding the interval between the first observation and the perihelion (τ) from υ' and q, by means of the equation— [table]
Similarly we may find the interval from the third observation to perihelion by substituting υ"' 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 represented.
In the computation of a geocentric position from parabolic elements we may proceed thus :—
Find the interval from perihelion passage to the time for which we require to compute (t - T), in days and decimals. [table]
We have thus the true anomaly and radius-vector. Then, if the motion be direct,— [table]
or, if the motion be retrograde,— [table]
—equations which give the heliocentric longitude and latitude (θ, λ). The geocentric longitude and latitude (a, β) and the true distance from the earth (Δ) are then obtained from— [table]
If the position of the comet as referred to the equator is required,—. [table]
Then
_ . eos. (N + e)
tan. R. A. — - c03 y — .tan. α tan.. Decl. = tan. (N + e). sin. R. A
As an example of the calculation of the orbit of a comet by Olbers’s method, we will compute the elements of the comet discovered at the Observatory of Marseilles by Μ. Borrelly on the 6th of December 1874, employing three observations taken at that Observatory on December 7, 16, and 26, viz. :— [table]
Converting the right ascensions into arc, and the times into decimals of a day, after subtracting 21 m 35" from them, for reduction to the meridian of Greenwich, we have— [table]
The obliquity of the ecliptic, from the Nautical Almanac, was 23° 27' 28", and hence, by the formulae p. 183, we find the following positions referred to the plane of the ecliptic, and interpolating for the above times from the same ephemeris the corresponding longitudes of the sun and log. radii-vectores of the earth, correcting the sun’s longitudes for aberration, and reducing all to mean equinox of 1875 0. [table]
The right ascension and declination are thus converted into longitude and latitude for the first observation :— [table]
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 α' - A", a" - A", a"- A", <tc., 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,— [table]
We have— [table]
flf
The calculation of Μ = ^~τ , is as follows, by (II.) :— P [table]
Next, we form the equations (III.) for the determination of r' 2 , r"' 2 , and ∕c 2 by successive assumptions for the value of p',— [table]
Therefore the equations for r' 2 and r"' 2 are— [table]
So that the equation for F i is thus formed,— [table]
And thus substituting logarithms in the factors for p' and p' 2 , our equations stand thus, in the form for proceeding with the work— [table]
We have now to find the value of p' by trial and error. Here, if = F + G. o' + Hp' 2 , we have— [table]
And for a first approximate value of p', by (IV.),— [table]
In the earlier approximations we may use five-figure logarithms. With p' = 0'649, the work proceeds thus— [table]
If for a second approximation we take p' = 0'7139, and calculate z ’ — z" precisely as before, the error in the interval from the first to the third observation, or (i''' - i'), is found to be - 0 d '3606, which, compared with the error of the first assumption ( - 2 d '0488), shows a change of + l d, 6882 for an increase of 0O649 in p; or of 1 ⅛th part, and by mere proportion we have p = 0'72776, for a third approximation, giving the error in interval = + 0 d O222, 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 si 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 +0 d O0l67, or less than 2∣ 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 ρ'. 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 ρ'...0*7268994, with which the calculation is as follows :— [table] [table] [table]
which may be considered a perfect agreement.
We have thus for the direct calculation of the orbit Log. ρ'...9*8614743, and, since p'" = Mp', we find Log. p"'...9*5314543. Log. r'...0*0708694, and Log. ∕"... 0*1742799, as given by our final approximation.
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 :— [table]
so that our first verification is complete.
The comet’s heliocentric longitude at the third observation (0"') being less than that at the first observation (0'), the motion in the orbit is retrograde or contrary to the order of the signs, and we therefore proceed to determine the longitude of the ascending node (β) and the inclination sslbv the second set of e∩uations in (VITA ∙ thus— [table] [table]
Then for the arguments of latitude at first and third observations (u, u"') — [table]
.∙. √"-u'=10° 35'2"O, and i(u"'-√)-5 c 17'31" 0 .
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. ⅜z> in (IX.)— [table]
The perihelion distance (<7) = r'. cos. 2 ⅜√, or [table]
The longitude of the perihelion (ττ)— [table]
It remains only to determine the time of perihelion from (XII.), computing both from υ and √" = √ + (√"-√) = 109° 34'45"O, so as to have the final verification of the work— [table]
We have now the whole of the elements of the parabolic orbit, viz., as usually written and entered in catalogues,— [table]
As already remarked, it is always desirable to ascertain 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, iae., 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. [table]
Instead of using Barker’s table, we will compute the true anomaly directly by the formulae (XIII.); thus, [table]
We then obtain the comet’s heliocentric longitude on the ecliptic (0") and heliocentric latitude (λ"), the motion in the orbit being retrograde, from equations (XV.) :— [table]
[table]
So that the errors of elements for the second observation may be expressed in transcription thus :— [table]
These errors are not greater than may be looked for, in a computation upon the method we have adopted.
We have computed the true distance of the comet from the earth at the second observation Δ". If the true distances at the first and third observations are desired, we have Δ' = p , Δ'" = - p n ,,, , or, in the present case, cos. β cos. β
Δ' = 1’28437, Δ"'= 1 ’ 24574, so that the comet was slowly approaching the earth during the interval over which the observations extend.
If it be preferred to compare with the observed right ascension and declination, the formulae (XVII.) have yet 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. [table]
The equations are [table] [table]
The equations are [table] [table]
The equations are [table]
It will be remarked that the apparent motion of this comet was very slow during the interval we have taken; it afforded a case where the orbit could only be improved by increased length of observation.]
We may correct the elements thus obtained for the main effect of the error due to the assumption made on commencing our calculation, by the following process, also suggested by Olbers, and applicable to the same observations.
There are already found r', r"', v, v" t and p'.
τ , i P". sin. (√"-√') (i"'-A , R'". sin. (A ff '-A")
u p r' . sin. (√' -v')~(i" -t') an ^=R' . sin. (A" -A') sh t ' ~ t } m = . tan ∕,^ , as in the calculation of the (i"-i')∙ m sin. (a -A")
ratio of the curtate distances fa or Μ.
P
Then compute N from [table]
Multiply those terms in the equations for p'" 2 and λ 2 which contain Μ by H , and the term in the equation for p'" 2 which contains Μ 2 by H 2 ; the equation for p' 2 not containing Μ or M 2 remains unchanged. With this new system of equations we find corrected values of ρ' and of r' and r'", and the elements of the orbit therefrom as before, p'" is now Μ . H . f .
To apply the above formulae to our preceding example we have [table]
[table]
With this corrected value of log. Μ, we might recalculate the co-eflìcients of p' and p' 2 in the equations for r"' 2 and k 2 , and complete the calculation of the orbit, but as the method of procedure is precisely that already illustrated by an example, it is unnecessary to occupy space here by so doing.
In the majority of cases in practice, the first elements of a comet’s orbit are calculated from a much shorter interval of observation than has been taken in the preceding example,—not infrequently from observations on consecutive nights, and in such cases our elements may be open to considerable correction, though the natural desire of the astronomer to learn something of a new comet’s position in the system, its track in the heavens, or possible identity with a comet already calculated, induces as speedy a determination of the orbit, however approximate, as practicable.
If the observations used in the first computations are near together, or the geocentric motion is slow, it will be preferable to wait for later positions, rather than occupy time in attempting a closer representation of the middle place. When later observations are available, the orbit ∕∕∕
may be re-calculated, Μ or v- , being determined by 01- bers’s formulae of correction, employing r', √", p ’, &c., as deduced from the first orbit. But the following general method of correcting approximate elements of a parabolic orbit, which has been widely used, will be found as satisfactory, though requiring great care in working. It is generally known as the method of variation of curtate distances.
We select three good observations at as wide intervals as practicable. These observations should be corrected for the effects of parallax aιid aberration by means of distances from the earth (Δ) calculated from approximate elements.
The aberration will be most conveniently taken into account by subtracting 497 β '8 × Δ from the time of observation, and interpolating the values of A and Log. R from the Nautical Almanac for the time thus reduced.
Then, introducing the observed longitude and latitude and the value of p, calculated from the approximate orbi⅛ find θ, λ, and r at the first and third observations from— [table]
From θ', λ', f, and θ'", λ"', r"', we compute the elements in the same manner as before, and thence the geocentric longitude for the time of the second observation, which call α 1 . (The geocentric latitude may be substituted for the longitude, if it be changing more rapidly.) Also find the time by these elements between the first and third observations, which call i 1 . Now vary p' by a small quantity, as 0 , 01 or 0'005 ( = m), and find 0', λ', r' again, and with these new values, combined with the previous ones for λ"', r"', coinpute the elements, and compare again with the second longitude, and call the difference from the longitude first computed r; also find the interval between the first and third observations, and call the difference in this case p. Next, with the first values of p', θ', λ', r', combine θ"', λ" ’ , r'" calculated from a similar slightly changed value of p"', (p"' + n), and, completing the elements, compare again with the longitude at the second observation obtained with the unvaried p', p"', and also with the corresponding interval between the extreme observations, and call the differences from the longitude and interval with unvaried curtate distances s and q. We have thus in the three calculations,— [table]
The three hypotheses and corrected values of ρ', p"' are then— [table]
And we have—∙ [table]
If the resulting corrections of the curtate distances are small, the true, or rather corrected, elements may be obtained by interpolation between the values obtained on the different hypotheses. When x and y are large, it is occasionally necessary to repeat the work, to have a close agreement between the middle longitude calculated from the corrected orbit and the longitude observed.
To make our article rather more complete, we may now refer to the calculation of ephemerides of the geocentric places of a comet from the parabolic elements, which are required during its visibility to facilitate observations. If a few places only are required, the right ascensions and declinations may be found in the manner already described; but if the comet is likely to continue visible any length of time, it is more convenient to work by rectangular equatorial co-ordinates, introducing the X, Y, Z, depending on the sun’s position, which are now given with much detail in the Nautical Almanac. Before proceeding further, however, it will be desirable to explain the use of Barker’s Table, to which reference has already been made, in the calculation of the true anomalies, as it dispenses with the longer computation introduced above, with the view to render our example independent of any other publication. The table had appeared from time to time, in one form or another, in various astronomical works; but in 1847 it was re-computed λvith extreme precision by Dr Luther, and printed in Encke’s edition of Olbers’s Abhandlung über diae leichteste und bequemste Methode die Bahn eines Cometen zu berechnen. It is much too extensive to be reproduced here.
The true anomaly in the parabola is related to the time from perihelion by the equation—
75 k (t,- r T}
—-——- — 75 tan. ⅛ -v + 25 tan. 3 ⅜ v <f- √2
where lc is the Gaussian constant [log. = 8’2355814], and q as before the perihelion distance. In the table—
II = 75 tan. ⅜ υ + 25.tan. 3 ⅛ v „ 75.*.(i-T)
0r m -
an equation, which, when q is known, allows either of (t — T) being found from Μ, and consequently from v, or when (t - T) is known, gives Μ, and then, by means of the table, the corresponding v. T5k
Put C——; C is therefore a constant and log. C = 9’9601277.
It, then, there be calculated for any comet the quantity—
C we shall have—
M = m(i-T) = 75 tan. ⅛ v + 25.tan. 3 ⅛ v.
To afford the reader a clearer idea of the great assistance which a table of this kind renders in cometary calculations, we will apply Luther’s table in the two cases where we have used direct formulae in our example,—
(1.) To obtain (i' - T) from √ = 98° 59' 43"O. [table]
The table, of which the argument is the true anomaly (fi), with interval 100", furnishes these values of log. Μ, near the above value of v— [table]
Wherefore, by simple interpolation, we find 98° 59' 43'' - 0 corresponds to log. Μ...2’1069519; from this value of log. Μ subtract log. m, as found above, and we have l , 6908245 for the logarithm of the time (in days and decimals) from perihelion, corresponding to 49 d 07096, as before.
(2.) In the reverse process, the determination of v, when (t - T) is given, we have in our example, as referring to the second observation with which the elements were compared— [table]
Near this value of Μ the table gives us— [table]
Whence, again by simple proportion, we find log. M = 2’1832053, which corresponds to 104° 52' 25’9", differing only 0" ’ l from the value found in the example.
The student should procure the last edition of the werk above-named for the sake of this table, and for the extensive catalogue of orbits of comets, coming down to 1864, and by far the most complete and reliable yet published. . It has been remarked that, where an ephemeris of geocentric positions (right ascension and declination) for any length of time is required, it is convenient to calculate with rectangular equatorial co-ordinates, instead of by the process we have followed in comparing the orbit with the middle observation.
For this purpose we compute what may be termed coordinate constants, from the elements τr, a, and i, and obliquity of ecliptic (∈), by the following formulae :— [table]
The angle ψ is to be taken in the first quadrant with its proper sign, and when the comet’s motion is retrograde, i must be used with a negative sign. The heliocentric co-ordinates of the comet (x 1 y, z) will then be obtained from— [table]
X being measured in the direction of the first point of Aries, from the sun as origin of co-ordinates, y towards 90° of Right Ascension, and z from the plane of the equator, positive to the north.
Similar co-ordinates of the sun, X, Y, Z, with the earth as origin, are found in the Nautical Almanac for Greenwich noon and midnight.
The Right Ascension and Declination are given by [table]
The true distance of the comet from the earth (Δ) ≡ Z + z sin. δ ’
As an example of this calculation we will find the coordinate constants applying to the elements of Borrelly’s comet in this article.
Here a = 282° 12' 48"Ι τ> - a =344° 36' 45"’4.
i = 80° 56' 27"'5 (the motion being retrograde).
The mean obliquity of ecliptic, 1875’0 = 23° 27' 19" - 9. [table]
[table]
We thus have, as the expressions are usually written, [table]
lhe true anomaly reckoned from 0° to 180°, in a direct orbit, is to be applied to A', B ’, and C', with a negative sign if the time for which we are calculating be before the perihelion passage, and with a positive sign if the time be subsequent thereto. In a retrograde orbit the contrary rule is to be observed.
In our comparison of the elements of Borrelly’s comet with the second observation we found v= 104° 52'26"O, and log. r— 0’1258616; the calculation of the heliocentric co-ordinates and geocentric place with these values stands thus :— [table]
For the demonstration of the method of determining a parabolic orbit, which has been here adopted, the reader may consult Olbers’s work, already mentioned, and for various modifications and refinements he is referred to Encke’s treatise, U eher die Olbers ’ -sche Methode zur Bestimmung der Cometenbahnen, in the appendix to the Berliner Astronomisches Jahrbuch for 1833; he will obtain much additional information from the treatise on Theoretical Astronomy, by Prof. Watson of Ann Arbor, U.S., and from Prof. Oppölzer’s work, Bahnbestimmung der Kometen und Planeten, Leipsic, 1870.
On the solution of Lambert’s equation, we may refer him to a paper by Mr Marth in Astronomische Nachrichten, vol. lxv., Nos. 1557-60, which he will find accompanied by elaborately constructed tables. To obviate extending this article to inconvenient length, the introduction of tables has been avoided throughout; and should it fall under the notice of any one practised in such calculations, we must beg him to attribute any deviation from general rules to the wish to make the article complete in itself, so that the student may compute parabolic elements of any new comet, and its apparent track in the heavens therefrom, without extraneous assistance.
For the method of calculating elliptical orbits, when decided deviation from the parabola is indicated, the reader is referred to Gauss’s classical work, Theoria Motus Corporum Coelestium, originally published in 1809, a translation of which, by Commander Davis, U.S.N., was printed at Boston in 1857; in this volume he will also find the demonstration of the formulae employed in calculating geocentric places from the elements of the orbit.
In presenting elements of comets of short period we shall include in Group A the comets for which periods of less than fifteen years are either established or have been assigned with greater or less degree of probability, and in Group B comets of longer periods, but not exceeding eighty years. We take the comets in order of length of period.
Group A.
I. Ended s Comet. [table]
T is expressed in Greenwich mean time.
The revolution of this comet in about 3⅜ years was discovered by Encke on its appearance in 1818-19, when it was detected by Pons. Encke, having calculated the effect of perturbation by the planet Jupiter, showed that the comet had been previously observed in 1786, 1795, and 1805, though missed at the intervening returns. It has been observed, with more or less success, at every appearance since 1819. Encke’s investigations soon led him to infer that the comet’s period had slightly diminished since 1795, and that this diminution might be owing to the effect of a resisting ìnedium. The late researches of Dr von Asten of Pulkowa indicate that it is only in certain revolutions that an effect of this nature - can be suspected, so that great doubt is thrown upon the validity of Encke’s theory.
II. Βlanpain's Comet. 1819, IV. [table]
These elements were calculated by Encke, who had ascertained that the motion of the comet could not be represented by a parabolic orbit. Clausen thought the comet was identical with one observed in 1743, which also exhibited a deviation from the parabola, and Gibers favoured his view, but no elliptical, elements have yet been deduced directly from the observations of 1743. Elements which appear in some of the catalogues of comet-orbits with Clausen’s name attached, were merely inferred from an assumed semi-axis major of 3’10, founded upon the hypothesis of identity with Blau pain’s comet of 1819. The latter has not been obsen ed since that appearance. [table]
Though there is great probability of the identity of this comet with Winnecke’s comet (No. VII.), it is preferable at present to treat of these bodies separately. The comet of 1766 was undoubtedly moving in an elliptical orbit of no long period, as was established by Burckhardt after a great amount of computation, but the observations were not sufficiently precise to allow of the length of revolution being found with exactness. [table]
Discovered by Tempel at Milan, 1873, July 3. The marked deviation from a parabolic orbit was soon independently noticed in England and on the Continent. It does not appear that there is any record of this comet having been observed prior to the year 1873. [table]
Discovered by De Vico at Eome, 1844, August 22. Notwithstanding the elaborate calculation of the orbit by Professor Brünnow, the comet has not been detected since 1844. The researches of Μ. Leverrier make it highly probable that this body is identical with the comet observed by Lahire at Paris in 1678, though it was not detected at any one of the many returns to perihelion which must have taken place in the interim. [table]
Discovered by Brorsen at Kiel, 1846, February 26, and since observed at several returns, including the last in 1873, to which year, it will be seen, the above elements apply. The comet appears to have been detected at its first passage of the perihelion in the present orbit, the form of which is evidently due to a very near approach to the planet Jupiter in May 1842. A first approximation to the elements previous to this encounter with Jupiter has been given by the late Professor D’Arrest. [table]
A comet discovered by Professor Winnecke, 1858, March 8, was soon found to be moving in an elliptical orbit of short period, and to be identical with the third comet of 1819, to which Encke had assigned a revolution of 5’62 years. The two appearances of 1819 and 1858 have since been connected, from calculation of the perturbations produced by the planet Jupiter, by Clausen, so that there is no possible doubt of the identity of the comets of these years. According to Clausen’s computation, the intermediate perihelion passages took place, 1825, February 5; 1830, August 21; 1836, March 3; 1841, September 13; 1847, March 29; and 1852, October 11, the comet being missed at every return. Our elements are for the last appearance in 1875. (See No. III.) [table]
Certainly a comet which, at the time it was detected by Pigott (of variable-star celebrity) at York, 1783, November 19, was moving in an orbit with short period of revolution. This was first established by Burckhardt about the year 1819, but the most accurate orbit will be that of Professor Peters of Clinton Observatory, New York, who has used improved solar places. His orbit is given above. The eomet has not been observed since 1783, but Peters has pointed out that with a major axis differing very little from that he had found, the comet might have encountered the planet Jupiter at some one of the aphelion passages subsequent to 1783, and thus have undergone an entire change of orbital elements. [table]
Discovered by Tempel in April 1867; the periodicity established at the same appearance. We have here given orbits both for that year and for the next return in 1873, on account of the heavy perturbations which the elements suffered from the attraction of Jupiter, near to which the comet passed about the aphelion passage. It will be seen that the node was thereby thrown back more than 22°, the inclination increased 3⅜°, and the revolution lengthened upwards of three months. The least distance between the planet and comet was about 0 32, the earth’s mean distance being taken as unity.
X. Biela's Comet.
The periodicity of this comet was ascertained on its discovery in February 1826 by Biela, at Josephstadt, Bohemia, and independently by Gambart at Marseilles about ten days later. Both discoverers remarked the similarity of elements to those of the comets of 1772 imperfectly observed by Montaigne at Limoges, and the first comet of 1806. Clausen’s calculations showed that the period of revolution between 1772 and 1826 had been about 6§ years, the comet having escaped observation at the intermediate returns, excepting that of 1806, when it was detected by Pons. Professor Santini of Padua has principally occupied himself with the investigation of the motions of this comet, and more recently the late Professor Hubbard of Washington. The comet was observed at its re-appearance in 1832, but missed in 1839, from proximity to the sun’s place. At the next return in 1845-6 it was again found, and at this appearance a remarkable separation into two distinct nebulosities took place under the eyes of astronomers. In 1852 both components were re-observed, and were found to have considerably widened their distance from one another. At the return to perihelion in 1859, there was no possibility of observations from unfavourable position in the heavens; but in 1866, according to calculation, which involved accurate determination of the planetary perturbations, its track seemed likely to favour observation, yet, notwithstanding long and minute search, the comet was not found, nor was anything seen of it as a comet in 1872. In this year, however, there occurred an extraordinary shower of meteors at the end of November, which beyond doubt were moving in the orbit almost identical with that of Biela when last observed. Subjoined are the elements of the two nebulosities forming the comet at the last appearance in 1852 :— [table]
Discovered by D’Arrest at Leipsic, 1851, June 27; he suspected a marked deviation of the orbit from a parabola as early as July 8, and elliptical elements were very fairly determined at this first observed visit. MM. Villarceau and Leveau of Paris have principally occupied themselves with the motions of this body, which, like others of the group, approaches very near to the orbit of Jupiter, thereby at times undergoing considerable perturbation. This was the case in 1861, when the distance between the two bodies was only 0’36, so that the elements we have given above for 1877 are materially different from what they were in 1851; the period of revolution is now about 100 days longer than when the comet was discovered. It was observed in 1857-58 and in 1870, but missed at the intermediate return. [table]
Discovered by Μ. Faye at Paris, 1843, Nov. 22, and period determined at this appearance. Μ. Leverrier has made very extensive researches respecting the previous motions of this comet, which he considers to have been revolving in an orbit of the above limited dimensions since the year 1747, when it may have passed so near to the planet Jupiter as to have its orbit completely changed. The calculations relating to this comet have for many years been in the hands of Dr Axel-Möller of Lund, the results of whose masterly computation of the perturbations will be found in various volumes of the Astronomische Nachrichten. Observations have been made at every return to perihelion since 1843. [table]
Discovered at Naples by Dr Peters, now director of the Observatory of Clinton, U.S., 1846, June 26, and not observed elsewhere, except at Rome on July 2. The period of revolution is uncertain to ±1’6 years, according to Peters’s last discussion of the observations of 1846, since which year the comet has not been found, though there appears to be no doubt of the great deviation of the orbit from a parabola. [table]
Periodicity discovered in 1858, when the comet was detected by Mr Tuttle at Cambridge, U.S. (January 4). It was soon found to present a great similarity of elements to those of the second comet of 1790, which was found by Mechain at Paris on January 9, and further investigation established the identity of the comets, five revolutions having been performed between 1790 and 1858. The two appearances have been connected by the calculation of the perturbations in the interval by Clausen and Tischler. [table]
This body is widely known as the “comet of the November meteors,” which have been found to move in an orbit that is almost identical. It was discovered by Tempel, 1865, Dec. 19; the best determination of the elliptical elements is due to Prof. Oppölzer. The ensuing return to perihelion will be looked for in 1899. in which year a repetition of the grand meteoric display of 1866 may also be expected. [table]
Discovered by Stephan at Marseilles, 1867, Jan. 27. The elliptical orbit which represents the whole course of observation very closely is due to Mr Searle of Cambridge, U.S. This comet appears to make a very close approach to the orbits of Mars and Uranus, and it is likely the actual form of orbit may be owing to an encounter with the latter planet near the aphelion. [table]
Discovered by Westphal at Göttingen, 1852, July 24. The elliptical character of the orbit was first established by Marth in the same year. Elements, in the calculation of which the effect of planetary attraction during the period of observation is included, have been published both by the discoverer and by Dr Axel Moller, whose elaborate investigations relating to Faye's comet have been already mentioned. [table]
Detected by Pons at Marseilles, 1812, July 20, and independently at a later date by Wisniewsky at Novo-Tcherkask. To Encke is due the discovery of the periodicity, but it is doubtful if the length of revolution can be inferred from the observations, within several months, so that although another perihelion passage is approaching, no reliable prediction of the track in the heavens is at present practicable. [table]
Discovered by De Vico at Rome, 1846, Feb. 20, andin a few weeks recognized as a periodical comet., ellipses having been calculated by Breen, Hind, and Peirce. The most reliable orbit, which is given above, is by Van Galen. This comet makes a near approach to the orbit of the planet Venus. [table]
Discovered by Olbers at Bremen 1815, March 6. The elements transcribed were calculated by Bessel upon the whole course of observation; very similar ellipses were also found by Gauss, Nicolai, and Nicollet. Bessel computed the effect of planetary attraction upon the motion of the comet in the actual revolution, and assigned the next perihelion passage to 1887, Feb. 9; but unfortunately this date is not to be relied upon within anything like narrow limits. [table]
Discovered by Brorsen at Altona, 1847, July 20. While there appears to be no doubt of the ellipticity, the period is yet open to considerable uncertainty, Dr Gould (the present director of the Observatory of Cordova) having inferred a revolution of 81 ’ 05 years. It is one of those comets which yet require a more minute calculation.
VIII. Halley's Comet.
In the case of this celebrated body we shall content ourselves with reproducing the elements for the appearance of 1835-36, as elaborately worked out by Westphalen, and the elements assigned for the next return in the year 1910 by the late Count de Γor.t⅛∙ coulant. [table]
It would be useless to attempt to present a history of Halley’s comet within the space to which this article must be limited. With regard to its history previous to the year 1456, the earliest visit known to Halley, the reader may refer to a paper on “The Past History of the Comet of Halley,” Hind in Monthly Notices of the Royal Astronomical Society, vol. x., in which the appearances of the comet are traced back with a greater or less degree of probability to the year b.c. 12, chiefly by aid of the details preserved to us in the Chinese annals; also to a remarkable memoir by the late Dr Angström of Upsala, Sur deux inégalités d'une grandeur remarquable dans les apparitions de la comète de Halley, Upsala, 1862, which is corroborative of the conclusions in the first-mentione I paper.
