Astron. Nachr. / AN 328, No. 2, 137 – 141 (2007) / DOI 10.1002/asna.200510665

Orbit of comet C/1860 M1 (Great Comet of 1860)

R.L. Branham, Jr.

IANIGLA, C.C. 330, Mendoza, Argentina

Received 2005 Nov 29 , accepted 2006 Sep 18Published online 2007 Feb 15

Key words celestial mechanics – stellar dynamics – comets: individual (C/1860 M1)

Comet C/1860 M1 (Great Comet of 1860) is one of a large number of comets with parabolic orbits. Given that thereare sufficient observations of the comet, 261 in right ascension and 251 in declination, it proves possible to calculate abetter orbit. The comet’s orbit is hyperbolic, and statistically different from a parabola. The comet, therefore, cannot beconsidered to be a Near Earth Oject.

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

1 Introduction

This paper continues a series on orbits of comets with cat-alogued parabolic orbits (Marsden and Williams 2003) butwhich nevertheless possess sufficient observations that wecan do better. The chief reason for studying these objectsresides in the possibility that a comet with a parabolic orbitmay be, depending on factors such as perihelion distance,a Near Earth Object (NEO). When a non-parabolic orbitis calculated we can know for sure. An additional consid-eration addresses craftsmanship: it is esthetically displeas-ing to leave an orbit in a preliminary state, preliminary be-cause many, perhaps most, parabolic orbits were calculatedby the method of Olbers as a computational convenience,when better can be done. And a better orbit implies betterstatistics for studying the origin of comets.

But why study the Great Comet of 1860 in particular?Because the comet was observed during four months, June21 to Oct. 18, and over 500 observations, in one coordi-nate or the other, of it are available. To leave the orbit asa parabola comes close to dereliction of one’s professionalduty. The reduction of the observations, moreover, provedchallenging because they represent uneven quality, and tocalculate an acceptable orbit required changing some of themethodology used previously.

2 Preliminary data reduction andephemerides

I conducted a literature search of the journals published inthe 19th century that include comet observations and alsoannual reports of some of the major observatories. Observa-tions of the Great Comet of 1860 were found in Astronomis-che Nachrichten, Comptes rendus de l’Academie des sci-ences, and Monthly Notices RAS. This yielded a total of 324

Corresponding author: rlb@lanet.com.ar

observations after a few duplicate observations had been re-moved. A small number of observations of insufficient pre-cision, such as a series made by a naval officer using a sex-tant at sea and a series at Williamsport, Australia, were ex-cluded. I have tried to use sextant observations of cometsbefore (Branham 2005), but without success. Many of theobservations were made in one coordinate only. The finalcount of observations is 261 in right ascension (α) and 251in declination (δ). Table A1 in the Appendix summarizesthe observations, and Fig. 1 graphs them.

The details of the treatment of the observations havebeen explained previously (Branham 2003). It suffices tosay that all of the observations were reduced to the formatof Terrestrial Time, α, and δ. Whenever a specific referencestar was given to which the comet observation had been re-ferred, its position was recalculated, with modern positionstaken from the Tycho-2 catalog (Høg et al. 2000), using thealgorithm in Kaplan et al. (1989). If differences in α and δfrom the reference star, ∆α and ∆δ, were given, they wereapplied, corrected for differential aberration, to the new po-sition. If ∆α and ∆δ were not given the differences in thepositions between the older catalog and Tycho-2 were ap-plied to the published positions of the comet. Because theobservations are 19th century, they were corrected for the E-terms of the aberration. Rectangular coordinates needed tocalculate (O −C)’s were generated along with numericallyintegrated partial derivatives to correct the comet’s orbit.

3 Errors or missing information in theobservations

When looking at large (O − C)’s it sometimes becomesevident that there is some sort of error with the publishedobservation. When the error is corrected, the (O − C) be-comes acceptable. What sorts of errors? Wrong identifica-tion of a reference star can happen. The wrong reference starcan sometimes be detected by taking an abnormally large

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

138 R.L. Branham, Jr.: Orbit of comet C/1860 M1

Fig. 1 The observations.

(O − C) and seeing if there is a relatively bright star nearthe published reference star that would reduce the (O − C)to something reasonable. Sometimes a differential observa-tion gives the wrong sign for a ∆α or ∆δ. Simply changingthe sign produces a good (O − C). Sometimes a clericalerror occurs, such as writing a 3 for a 5 or 5 for 8. 19thcentury typesetting of observations was based on a writtenmanuscript, where it was easy to confuse these numbers. If,for example, a declination is published ending with some-thing like 54.′′6 and the (O − C) is close to 20′′, it is rea-sonable to assume that the declination really should havebeen 34.′′6. If, one the other hand, the (O − C) were some-thing like 15′′, then the assumption of confused numberswould be dubious, and the observation should be left alone.An error in both the (O − C) in α and in δ indicates ei-ther a wrong reference star or an error in the time. The lattercondition can be detected by seeing if a change in the timereduces both (O − C)’s to something reasonable.

Table A2 exhibits the errors in the observations thatcould be corrected, by the actions indicated, for the benefitof anyone who wishes to further study and perhaps improvethe orbit of this comet.

4 Treatment of the observations

The treatment of the observations proved more difficult thanin previous studies of 19th century comet orbits becausethree observatories, Rio, Santiago, and Sydney, produced33% of the 512 observations, but with (O − C)’s higherthan normal. The median of the absolute values of the pre-fit (O − C)’s for these observatories is 13.′′12, with a meanabsolute deviation (MAD, the sum of the absolute values ofthe residuals divided by the degrees of freedom) of 26.′′95,versus 3.′′16, with a MAD of 9.′′56, for the other observa-tories. The Sydney observer, W. Scott, admits to problemswith the observations: “The observations are very unsatis-factory; owing partly to the faint and diffused nature of the

Fig. 2 Distribution of weights.

Comets [sic] light, which rendered it unpossible [sic] to fixon any particular point as a nucleus, and partly to the verydefective instrumental means at my command.” LikewiseM. Moesta at Santiago writes, “. . . the instrument couldnot be mounted equatoreally [sic], and could only be placedprovisionally on loose ground ... I consider the enclosed ob-servations of no great value.” The Rio observer, M. Emm.Liais, makes no comment, but one must assume that similarcircumstances were operative.

To weight the observations by the impersonal weight-ing scheme used previously, the biweight, would assignzero weight to 19% of the observations. I feel uncomfort-able with such extreme trimming. The observations may be,to use Moesta’s words, “of no great value”, but they areof some value. I decided, therefore, to use in lieu of thebiweight another impersonal, but less extreme, weightingscheme, the Welsch weighting (Branham 1990, p. 117). Onescales the post-fit residual ri by the median of absolute val-ues of the residuals and assigns a weight wt as

wt = exp(−r2i /2.9852), |ri| < ∞ . (1)

The robust L1 criterion (Branham 1990, Ch. 6) calculatesthe first approximation. Because the first approximation isgood, it becomes unnecessary to iterate the solutions. Forthe Great Comet the median weight was 0.89. Figure 2shows the distribution of the weights. 72.3% of the obser-vations received weights greater than 0.5, 49.0% weightsbetween 0.9 and 1, and 11.5 % of the weights were less than0.001. With the Welsch weighting the median of the abso-lute values of the post-fit residuals decreases from 5.′′06 to2.′′91.

5 The solution

Table 1 shows the final solution for the rectangular coor-dinates, x0, y0, z0, and velocities, x0, y0, z0, along withtheir mean errors for epoch JD 2400800.5 and the mean er-ror of unit weight, σ(1). Also shown is a solution based on

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org

Astron. Nachr. / AN (2007) 139

Table 1 Solution for rectangular coordinates and velocities forepoch JD 2400800.5 and equinox J2000.

L1 solution Least squares

x0 0.3295446062 0.3295824692±0.0000153977y0 –1.4582796028 –1.4582481422±0.0000219332z0 –3.4645214441 –3.4645113145±0.0000219090x0 0.0016596382 0.0016597550±0.0000000558y0 –0.0014750776 –0.0014749166±0.0000001200z0 –0.0123642601 –0.0123641662±0.0000001402σ(1) 3.′′25

the robust L1 norm. The two solution differ little, the dif-ference in their norms is only 1.8×10−5, showing that theWelsch weighting function has done a good job of handlingthe large errors present in some of the observations.

Table 2 shows the covariances and the correlations.The correlations are moderate, aside from three correlationsabove 90%, and the condition number of the matrix of theequations of condition, 7.9×103, is relatively low. The lin-ear system, therefore, seems well-conditioned and shouldresult in a reliable solution.

Table 2 Covariance (upper triangle) and correlation (lower tri-angle) matrices.

0.953 0.711 0.198 0.003 0.004 0.0020.524 1.933 0.852 0.002 0.010 0.0070.146 0.441 2.181 0.006 0.006 0.0120.957 0.325 0.138 0.000 0.000 0.0000.486 0.990 0.554 0.295 0.000 0.0000.197 0.530 0.994 0.168 0.634 0.000

Table 3 gives the orbital elements corresponding withthe rectangular coordinates of Table 2: the mean anomaly ofperihelion passage, M0; the eccentricity, e; the semi-majoraxis, a; perihelion distance, q; the inclination, i; the node,Ω; and the argument of perihelion, ω. Rice’s (1902) proce-dure was used to calculate the mean errors for the ellipti-cal elements. To express Rice’s procedure in modern nota-tion let C be the covariance matrix for the least squares so-lution for the rectangular coordinates and velocities. Iden-tify the errors in a quantity such as the node Ω with thedifferential of the quantity, dΩ. Let V be the vector ofthe partial derivatives

(∂Ω/∂x0 ∂Ω/∂y0 · · · ∂Ω/∂z0

).

Then the error can be found from

(dΩ)2 = σ2(1)V ·C · V T . (2)

The partial derivatives in Eq. (2) are calculated from thewell known expressions linking elliptical orbital elementswith their rectangular counterparts. The solution shows ahyperbolic orbit, and the mean error indicates that the hy-perbola is statistically distinguishable from a parabola. TheGreat Comet of 1860, therefore, can be reliably ruled out asa possible NEO.

Table 3 Elliptic orbital elements and mean errors for epoch JD2400800.5 and equinox J2000.

Value Mean Error

M0 222.4169120988 0.1184494260a –324.7165187113 1.4908710233e 1.0009022247 0.0000041614q 0.2929672638 0.0003153304Ω 82.6125705552 0.0526656748i 81.5555646468 0.0197720653ω 100.6078294903 0.0409526918

Fig. 3 Final residuals.

Figure 3 shows the residuals weighted by Eq. (1). Theresiduals are random: a runs test on the 501 residuals greaterthan the machine epsilon shows 234 runs out of an expected250 with standard deviation of 11.2. One must thus acceptat an 85% confidence level that the residuals are random.Random, but not normal. The residuals are skewed, factorof skewness = −0.019, platykurtic, kurtosis = −0.343, andlighter tailed than a normal distribution, Hogg’s Q factor of0.233 versus 2.580 for a normal distribution. But given thatthey are random one can consider the solution acceptable.

Although all of the observations are post-perihelion, itwould be interesting to see if there is any indication of sig-nificant non-gravitational motion such as would be causedby mass loss during perihelion passage. An easy way tocheck for this is to perform runs tests on subsets of the resid-uals. A statistically significant deviation from randomnessin a subset gives evidence for an effect not contemplated inthe data reduction model. I divided the residuals into twogroups, as Table 4 shows.

From this table one deduces that the first set of resid-uals is highly random, 47% chance of being random, thesecond slightly less so, 22% chance of being random. Thisis insufficient evidence to assert a significant difference inthe behaviour of the two groups of residuals. We thus assertthat there is no reason to suspect non-gravitational forces atwork on this comet.

www.an-journal.org c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

140 R.L. Branham, Jr.: Orbit of comet C/1860 M1

Table 4 Runs tests on subsets of residuals.

Time span Resid. > eps Runs Expected

2400583.5–2400642.9

124 58 62

2400642.9–2400702.5

376 176 188

6 Conclusions

An orbit for Comet C/1860 M1 (Great Comet of 1860),based on available observations, 261 in α and 251 in δ, isgiven. The orbit is hyperbolic and statistically different froma parabola. The comet can, therefore, in no way be consid-ered an NEO.

References

Branham, R.L. Jr.: 1990, Scientific Data Analysis, Springer, NewYork

Branham, R.L. Jr.: 2003, PASA 20, 1Branham, R.L. Jr.: 2005, P&SS 53, 1437Høg, E., Fabricius, C., Markarov, V.V., Urban, S., Corbin, T.,

Wycoff, G., Bastian, U., Schwenkendlek, P, Wicenec, A.:2000, The Tycho-2 Catalogue of the 2.5 Million BrightestStars, CD-Rom Version, U.S. Naval Observatory, Washington

Kaplan, G.H., Hughes, J.A., Seidelmann, P.K., Smith, C.A.: 1989,AJ 97, 1197

Marsden, B.G., Williams, G.V.: 2003, Catalog of Cometary Orbits2003, Smithsonian Astrophysical Obs., Cambridge, Mass.

Rice, H.L.: 1902, AJ 22, 149

c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.an-journal.org

Astron. Nachr. / AN (2007) 141

A Summary of the observations and errors and missing information in the observations

Table A1 Observations of the Great Comet of 1860.

Observatory Obsns. in α Obsns. in δ Reference1

Sydney, Australia 54 54 AN, 1860, Vol. 54, pp. 259/60Kremsmuenster, Austria 2 2 AN, 1860, Vol. 53, pp. 349/50Vienna, Austria 8 8 AN, 1860, Vol. 53, pp. 317/18Rio de Janeiro, Brazil 11 11 CR, 1860, Vol. 51, pp. 503/04Santiago, Chile 17 20 MN, 1861, Vol. 21, p. 186

MN, 1865, Vol. 26, pp. 59/60Altona, Germany 4 4 AN, 1860, Vol. 53, pp. 287/88

AN, 1860, Vol. 53, pp. 325/26Berlin, Germany 3 3 AN, 1860, Vol. 53, pp. 177/78Konigsberg, Germany 6 6 AN, 1860, Vol. 53, pp. 71/72Athens, Greece 9 9 AN, 1860, Vol. 53, pp. 317/18

AN, 1860, Vol. 53, pp. 345/46Florence, Italy 6 6 AN, 1860, Vol. 53, pp. 299/300

AN, 1860, Vol. 53, pp. 319/20Naples, Italy 6 6 AN, 1860, Vol. 53, pp. 323/24Padua, Italy 5 5 AN, 1860, Vol. 53, pp. 325/26Rome, Italy 13 13 AN, 1860, Vol. 54, pp. 43/44Leiden, Netherlands 1 1 AN, 1860, Vol. 53, pp. 299/300Utrecht, Netherlands 1 1 AN, 1860, Vol. 53, pp. 299/300Capetown, South Africa 82 71 CR, 1861, Vol. 53, pp. 29/39Cambridge, USA 22 20 AN, 1861, Vol. 54, pp. 251/52(Old) U.S. Naval, USA 11 11 AN, 1861, Vol. 54, pp. 9/10

Total 261 251

1AN: Astron. Nach.; MN: Mon. Not. Roy. Astron. Soc.; CR: Comptes Rendus

Table A2 Errors and missing information in the observations of the Great Comet.

Reference Date Error or missing data

AN, 1860, Vol. 53, pp. 287/88 June 23, 11h α probably 6h45m03s

AN, 1860, Vol. 53, pp. 317/18 June 22, 9h α probably 6h35m26.s91

AN, 1860, Vol. 53, pp. 317/18 July 2, 10h δ probably 3503′07.′′6AN, 1860, Vol. 53, pp. 319/20 June 8, 8h ∆δ probably −15′13.9AN, 1861, Vol. 54, pp. 9/10 July 9 Star *3 is Tycho 1419-01260-1AN, 1861, Vol. 54, pp. 9/10 July 11, 8h ∆δ probably +6′54.85AN, 1861, Vol. 54, pp. 9/10 July 13, 8h ∆δ probably +20′13.21AN, 1861, Vol. 54, pp. 43/44 July 2 Star f is Tycho 286-00457-1AN, 1861, Vol. 54, pp. 43/44 July 3 Sign of ∆δ is +AN, 1861, Vol. 54, pp. 43/44 July 4 Sign of ∆δ is +AN, 1861, Vol. 54, pp. 251/52 July 14, 9h ∆α probably −1m49.s03AN, 1861, Vol. 54, pp. 259/60 July 27 Star a Tycho 6696-01101-1, b Tycho 6696-00492-1AN, 1861, Vol. 54, pp. 259/60 Aug. 16, 21h ∆δ probably −10′26′′

AN, 1861, Vol. 54, pp. 259/60 Aug. 17 Unidentified star is Tycho 7811-00685-1CR, 1860, Vol. 51, p. 503 Upper part of page Positions are apparentCR, 1860, Vol. 51, p. 503 July 13, 7h ∆δ probably +3′10.′′0CR, 1860, Vol. 51, p. 503 July 20, 7h Time probably 7h58m07.s88CR, 1860, Vol. 51, p. 504 All Equinox of stars (a)–(h) 1860.0; Star a Tycho 0262-00930-1” ” b Tycho 4925-01147-1; c Tycho 5509-00767-1” ” d Tycho 5512-00428-1; e Tycho 5516-00401-1” ” f Tycho 6093-00622-1; g Tycho 6106-00172-1” ” h Tycho 6097-01648-1MN, 1865, Vol. 26, p. 59 Aug. 23, 9h ∆δ probably +9′14.′′3

www.an-journal.org c© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim