Orbital Evolution, Activity, and Mass Loss of Comet C/1995 O1 (Hale-Bopp). I. Close Encounter with Jupiter in Third Millennium BCE and Effects of Outgassing on the Comet's Motion and Physical Properties
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ORBITAL EVOLUTION, ACTIVITY, AND MASS LOSS OF COMET C/1995 O1 (HALE-BOPP).I. CLOSE ENCOUNTER WITH JUPITER IN THIRD MILLENNIUM BCE AND EFFECTSOF OUTGASSING ON THE COMET’S MOTION AND PHYSICAL PROPERTIES
Zdenek Sekanina & Rainer Kracht Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, U.S.A. Ostlandring 53, D-25335 Elmshorn, Schleswig-Holstein, Germany
Version September 12, 2017
ABSTRACTThis comprehensive study of comet C/1995 O1 focuses first on investigating its orbital motion overa period of 17.6 yr (1993–2010). The comet is suggested to have approached Jupiter to 0.005 AU on − ∼ ∼
12, a dust loading high enough to imply amajor role for heavy organic molecules of low volatility in accelerating the minuscule dust particles inthe expanding halos to terminal velocities as high as 0.7 km s − . In Part II, the comet’s nucleus will bemodeled as a compact cluster of massive fragments to conform to the integrated nongravitational effect. Subject headings: comets: individual (C/1995 O1) — methods: data analysis INTRODUCTION: MAKING CASE FOR A CLOSEENCOUNTER WITH JUPITER
The celebrated comet C/1995 O1 (Hale-Bopp) was ex-ceptional in a number of ways, one of which was its enor-mous intrinsic brightness. This trait rendered it possibleto discover and observe the comet at very large helio-centric distance and to collect astrometric observationsover a long period of time. Discovered visually on 1995July 23, more than 600 days before perihelion, as an ob-ject of mag 10.5–10.8 (Hale & Bopp 1995), the cometwas subsequently recognized at mag 18 on a plate takenon 1993 April 27 with the 124-cm UK Schmidt telescopeof the Siding Spring Observatory near Coonabarabran,N.S.W., Australia (McNaught 1995), 3.9 yr before per-ihelion and 13.0 AU from the Sun. With the latest as-trometrically measured images from 2010 December 4(when 30.7 AU from the Sun!), taken with the 220-cmf/8.0 Ritchey-Chr`etien reflector of the European South-ern Observatory’s Station at La Silla, Chile (Sarneczkyet al. 2011), the observed orbital arc covers a period of17.6 yr, with 13.7 yr of post-perihelion astrometry — anunprecedented feat for a single-apparition comet.No less astonishing is the orbital history of the comet(Marsden 1999). Soon after its discovery, the orbit couldsubstantially be improved thanks to the Siding Springpre-discovery observation. Although Marsden eventuallydetermined that the near-perihelion osculating orbitalperiod equalled about 2530 yr, he noticed that this wasdue in part to the perturbations by Jupiter during thecomet’s approach to within 0.8 AU of the planet about
Electronic address: [email protected]@t-online.de one year before the 1997 perihelion and that the orbit hadoriginally been more elongated, with a period of slightlylonger than 4200 yr. Obviously, the object did not arrivefrom the Oort Cloud; its previous perihelion passage hadoccurred in the course of the 23rd century BCE.As described by Marsden, this was only a minor partof the story. Because the comet’s ascending node placesit practically on Jupiter’s orbit, extremely close encoun-ters with the planet are possible, and Marsden (1999)argued that this was precisely what happened 42 cen-turies ago on the comet’s approach to perihelion. He evenventured to propose that the comet had been capturedby Jupiter’s gravity from its original Oort-Cloud-type,highly elongated orbit.In his investigation, Marsden (1999) was handicappedby a relatively short interval of time that the comet’sobservations then spanned. The last observation he em-ployed was dated 1998 February 8, so that the observedorbital arc was only 4.8 yr long, not much more than aquarter of the interval of time available nowadays.The fascinating problem of the long-term orbital evo-lution of C/1995 O1 and the related issues warrant a newinvestigation, given in particular that the much greaterspan of astrometric observations currently available al-lows us to conduct a considerably more profound exam-ination of the orbital scenario and its potential implica-tions than it was possible a dozen or more years ago [forsome later efforts similar to Marsden’s (1999), see, e.g.,Szutowicz et al. (2002) or Kr´olikowska (2004)]. We fur-ther propose to confront our approach and results withevidence from the comet’s physical observations made inthe course of the 1997 apparition and to draw conclusionsfrom this data interaction. Sekanina & Kracht DATA, TOOLS, AND METHOD OF APPROACH
We began by collecting 3581 astrometric observationsof comet C/1995 O1 from the database maintained bythe
Minor Planet Center . This is a complete list of thecomet’s currently available astrometric data in equinoxJ2000. Besides the single pre-discovery observation in1993 (McNaught 1995), the data set contains 830 obser-vations from the year 1995, 1458 from 1996, 653 from1997, 295 from 1998, 198 from 1999, 50 from 2000, 51from 2001, 9 from 2002, 14 from 2003, 14 from 2005(of which 6 from January–February and 8 from July–August), 5 observations from 2007 October, and 3 from2010 December 4. The primary objective of the first phase of our orbit-determination effort was to subject the collected set ofastrometric observations to tests in order to extract asubset of high-accuracy observations. To ensure that thishas indeed been achieved requires an introduction of acutoff or threshold of choice for removing from the col-lection all individual observations whose (observed mi-nus computed) residuals in right ascension, ( O − C ) RA ,or declination, ( O − C ) Decl , exceed the limit. In practice,the task is made difficult by the fact that a residual ex-ceeding the limit does so for one of two fundamentallydifferent reasons: either because the astrometric positionis of poor quality or because the comet’s motion is mod-eled inadequately. The aim of the procedure is to rejectthe data points with large residuals that are of the firstcategory but not of the second category; to discriminatebetween the two categories requires an incorporation ofan efficient filter. Typically, the category into which anobservation with a large residual belongs is revealed bycomparing, with one another, observations over a lim-ited orbital arc: if all or most of them exhibit systematicresiduals, the problem is in the comet’s modeled motion;if one or a few stand out in reference to the majority, itis the poor quality of the individual observations. Ob-viously, this filter fails when there is only one or a veryfew observations available widely scattered over a longorbital arc (Section 4).Since we did not define what we meant by inadequatemodeling of the comet’s motion at the start of the orbit-determination process, we used two tools to filter outthe observations of poor quality and at the same timeto ensure that no observation of required accuracy wasremoved. One tool was a stepwise approach, beginningwith a cutoff at 10 ′′ (Section 3) and tightening it up; theother tool was to divide the whole observed orbital arcinto a number of segments of approximately equal andshort enough length, fitting an osculating gravitationalsolution through each of the segments.In addition, throughout this work we consistently ap-plied a self-sustaining test of cutoff enforcement. Whena particular cutoff was introduced as a limit for reject-ing inaccurate observations, minor differences between See . On a website https://groups.yahoo.com/neo/groups/comets-ml/conversations/messages/19755/
D. Herald, Murrumbateman,Australia, reported a detection of C/1995 O1 on CCD images of atotal exposure time of 144 minutes taken on 2012 August 7 with a40-cm Schmidt-Cassegrain telescope. However, in response to ourmore recent inquiry, Mr. Herald stated in an e-mail message dated2014 August 30 that he had been unable to independently confirmthe detection. For additional information, see the Appendix A. two consecutive solutions — before and after eliminationof the poor-quality data — could cause that some datapoints with the residuals slightly exceeding the limit inthe first solution (and therefore removed from the set ofused observations) just satisfied the limit in the secondsolution (and were to be incorporated back into the set);while other data points with the residuals just satisfy-ing the limit in the first solution (and therefore kept inthe set) slightly exceeded the limit in the second solution(and were now to be removed from the set). In order tocomply with the cutoff rule, it was necessaary to iteratethe process until the number of astrometric observationsthat passed the test fully stabilized.As described in Section 3 below, the single observa-tion of 1993 is known to be accurate to about 1 ′′ , whilenearly a half of the observations from 2005 and all from2007-2010 were made with large telescopes; the more ad-vanced solutions (Sections 6.2–6.3) showed a posteriorithat these data points satisfied the tight limit and werenot subjected to the aforementioned test.With these basic rules in mind, we applied an orbit-determination code EXORB7 , written by A. Vitagliano,both in its gravitational and nongravitational modes. Us-ing the JPL DE421 ephemeris, the code accounts for theperturbations by the eight planets, by Pluto, and by thethree most massive asteroids, as well as for the relativis-tic effect. The nongravitational accelerations are incor-porated directly into the equations of motion (Section 4).The code employs a least-squares optimization methodto derive the resulting elements and other parameters. PAST WORK AND PURELY GRAVITATIONALSOLUTIONS
The comet’s pre-discovery image from 1993, measuredby McNaught (1995), is of major significance, because itextends the observed orbital arc by 27 months. Marsden(1999) commented on his (and other authors’) difficultieswith linking this data point to the post-discovery, preper-ihelion observations and he emphasized the magnitude ofthe problem at the time by pointing to the possibility of“a significant unmodeled perturbation [that] must havebeen operative over a relatively short time interval,” in areference to D. K. Yeomans. The situation was not allevi-ated by McNaught’s remeasurement of the comet’s 1993image, which showed that its astrometric position wasaccurate to within 1 ′′ . Also, McNaught’s positional ver-ification of a known asteriod on the same plate confirmedthe correct time. The problem was solved (temporarily)by assigning a greater weight to the 1993 data point. Newdifficulties arose in an effort to link the preperihelion ob-servations with the post-perihelion observations near theend of 1997, and the nongravitational terms had to be in-troduced into the equations of motion. At the time of hispaper’s submission (1998 early February), the 1993 ob-servation (assigned a unit weight) was noted by Marsden(1999) to leave a residual of 1 ′′ .8 in right ascension evenwith the nongravitational terms included.Interestingly, Marsden’s (2007) last published orbit was gravitational, but the 1993 pre-discovery positionwas left out and 2817 observations were linked from only1995 July 24 through 2007 October 22; the mean resid- This set of elements is also listed in Marsden & Williams’(2008) most recent edition of their catalog of cometary orbits. omet C/1995 O1: Orbit, Jovian Encounter, and Activity 3
Table 1
Our Purely Gravitational (PG) and Standard Nongravitational (SN) Orbital Solutions for Comet C/1995 O1 Compared withThree Previously Published Orbits ( ∗ Identifies Solutions with Stabilized Number of Used Observations)Purely Gravitational Solutions Standard Nongravitational SolutionsNo. of Used Mean Residual Time of Previous No. of Used Mean Residual Time of PreviousSolution Observations Residual Cutoff Perihelion (TT) Solution Observations Residual Cutoff Perihelion (TT)PG I 2735 ± ′′ .86 2 ′′ − ± ′′ .75 2 ′′ − ± − ± − ± − ∗ ± − ∗ ± − ± − a ± − c SN VI ∗ ± − b ± − c Marsden d ± − Notes. a Marsden (2007): derived from the orbital arc 1995 Jul 24 through 2007 Oct 22. b Williams (2011): derived from the orbital arc 1993 Apr 27 through 2010 Dec 4. c Estimated from the barycentric original semimajor axis; formal error of a few months. d Marsden (1999): derived from the orbital arc 1993 Apr 27 through 1998 Feb 8; parameter A = 0. ual equaled ± ′′ .9. On the other hand, the most recentlypublished orbit, by Williams (2011), is based on gravita-tionally linking 2737 observations from the entire period1993–2010, leaving a mean residual of ± ′′ .88; no detailsare available on the quality of fit.The lesson learned from the obstacles encountered byMarsden (and others) in trying to fit a 57-month-long arcof the comet’s orbit leaves no hope that a solution basedon the gravitational law alone could adequately representa 211-month-long arc, nearly four times as long. We alsosuspected that the small number of observations avail-able from the period of time after 2003 — not to men-tion the single 1993 pre-discovery observation — wouldeventually require their heavy weighting in comparisonwith the data from the interval of 1995–2003.Given this history of orbit determination of C/1995 O1and also in view of the need of a reference standard forour further computations, we first set to derive the bestachievable purely gravitational solution. To begin with,we assigned a unit weight to each of the 3581 observa-tions and computed an initial gravitational orbit. Thisstep was necessary because of the lack of a priori in-formation on the quality of the individual data points.This solution, whose mean (rms) residual came out to be ± ′′ .56, served to obtain a complete set of initial residu-als. We noticed the presence of a few residuals exceeding10 ′′ and a large number of residuals substantially exceed-ing 2 ′′ ; there were strong systematic trends in right as-cension, with the residuals exceeding 4 ′′ in 2010. The1993 pre-discovery observation left a residual exceeding3 ′′ in declination.Retaining an equal weight for all observations, these re-sults prompted us to set a residual cutoff at 2 ′′ in eithercoordinate and to iterate the gravitational least-squaresfitting to the observations until their employable numberstabilized. The left-hand side of Table 1 shows that thistask required four iterations to determine that the totalnumber of observations satisfying this condition settleddown — in the solution PG IV — on 2648 and the meanresidual on ± ′′ .81. This solution places the comet’s pre-vious passage through perihelion early in the year − O − C from the solution PG IV are plot-ted in Figure 1. The fit can by no means be consideredsatisfactory, so the predicted perihelion passage in − − ′′ in 2010. In declination, the pre-discovery observation leaves a residual greater in absolutevalue than 2 ′′ . Less prominent, short-term systematictrends are also seen at other times in both coordinates.Each residual that exceeds 2 ′′ is legitimate, as it does sobecause of an imperfect model. STANDARD NONGRAVITATIONAL SOLUTIONS
Having confirmed that gravitational solutions are un-acceptable, we next turned to solutions based on thestandard nongravitational formalism, proposed by Mars-den et al. (1973), in which the nongravitational termswere incorporated directly into the equations of motionand the magnitude of the nongravitational acceleration,as a function of heliocentric distance r , was expressed byan empirical formula g ice ( r ; r ), designed to mimick thelaw of momentum transfer driven by the sublimation ofwater ice, g ice ( r ; r ) = ψ Λ − m (1+Λ n ) − k , (1)where Λ = r/r , r is the scaling distance for water ice,exponents m , n , and k are constants, and ψ is a normal-ization coefficient, such that g ice (1 AU; r ) = 1. Marsdenet al.’s (1973) formalism employs a so-called isothermalmodel, which averages the Sun’s incident radiation overa spherical nucleus’ surface of constant temperature, as-suming an albedo of 0.1 in both the optical and ther-mal spectral passbands. For this model, the parametersare r = 2 .
808 AU, m = 2 . n = 5 . nk = 23 .
5, and ψ = 0 . ≀≀ ≀≀ ≀≀ ≀≀≀≀ ≀≀ ≀≀ ≀≀ +2 ′′ +1 ′′ ′′ − ′′ − ′′ − ′′ +2 ′′ +1 ′′ ′′ − ′′ − ′′ O − C − −
500 0 +500 +1000 +1500 +2000 +3000 +5000TIME FROM PERIHELION (days)RESIDUALS IN RIGHT ASCENSIONRESIDUALS IN DECLINATION
PG IV N obs = 2648 ✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉
Figure 1.
Temporal distribution of residuals O − C (observed minus computed) in right ascension (top) and declination (bottom) from thegravitational solution PG IV to 2648 accurate observations of comet C/1995 O1 between 1993 and 2010. The axis of abscissae is interruptedin the years 1994–1995, 2003, 2004, 2006–2007, and 2007–2009 over periods of time with no observations. A strong systematic trend isapparent in right ascension starting in mid-1998 and reaching 3 ′′ by 2010. The pre-discovery observation leaves a residual exceeding 2 ′′ indeclination. Less prominent systematic trends are apparent from 1995 to mid-1996 in right ascension and during much of 1997 and 1998through part of 1999 in declination. All observations were given a unit weight. Tested countless times in the past, this standardformalism stipulates a constant orientation of themomentum-transfer vector in an
RTN right-handed ro-tating Cartesian coordinate system. Its cardinal direc-tions, referred to the comet’s orbital plane, are the ra-dial R (away from the Sun), transverse T , and normal N components. The components j C ( t ) ( C = R , T , N ) ofthe nongravitational acceleration, to which the comet’snucleus is subjected, are, at time t (when the heliocentricdistance is r ), equal to j R ( t ) j T ( t ) j N ( t ) = A A A · g ice ( r ; r ) , (2)where A , A , and A are the magnitudes of the nongrav-itational acceleration’s components in, respectively, theradial, transverse, and normal directions at 1 AU fromthe Sun; they are the additonal parameters that are, to- gether with the orbital elements, determined from theemployed observations in the process of orbit determina-tion by applying a least-squares optimization procedure.Retaining again an equal weight for all observationsand a residual cutoff at 2 ′′ , we iterated the standardnongravitational least-squares fitting to the observationsuntil their number satisfying the conditions stabilized.Limited experimentation showed that the incorporationof the normal component of the nongravitational accel-eration in addition to the usually included radial andtransverse components improved the fit to a degree. Theright-hand side of Table 1 indicates that after three iter-ations the number of employable observations leveled offat 2900, with the mean residual of ± ′′ .75 in the solutionSN III. Thus, with somewhat higher accuracy than thesolution PG IV, it accommodates ∼
250 more observa-tions, suggesting that the incorporation of the nongravi-tational terms into the equations of motion definitely hadbeneficial effects.omet C/1995 O1: Orbit, Jovian Encounter, and Activity 5 ≀≀ ≀≀ ≀≀ ≀≀≀≀ ≀≀ ≀≀ ≀≀ +2 ′′ +1 ′′ ′′ − ′′ − ′′ +2 ′′ +1 ′′ ′′ − ′′ − ′′ O − C − −
500 0 +500 +1000 +1500 +2000 +3000 +5000TIME FROM PERIHELION (days)RESIDUALS IN RIGHT ASCENSIONRESIDUALS IN DECLINATION
SN III N obs = 2900 ✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉
Figure 2.
Temporal distribution of residuals O − C (observed minus computed) in right ascension (top) and declination (bottom) from thestandard nongravitational solution SN III to 2900 accurate observations of comet C/1995 O1 between 1993 and 2010. For more description,see the caption to Figure 1. The fit is now much better, but not quite acceptable. The pre-discovery observation leaves a residual of 2 ′′ inright ascension. A slight systematic trend in right ascension is seen between 1999 and 2010. Only in declination is the fit fairly satisfactory. The solution SN III puts the comet’s previous peri-helion passage late in the year − O − C from the solution SN III are plot-ted in Figure 2. The 1993 pre-discovery observation isnow fitted satisfactorily in declination, but it leaves aresidual of 2 ′′ in right ascension. The residuals from theyears 1999–2010 are also fairly satisfactory in declina-tion, but they have a tendency to being negative in rightascension. We conclude that the standard nongravita-tional law, although improving the match significantly,still fails to offer a perfect fit and leaves thus room forfurther improvement. In terms of the predicted perihe-lion time at the previous return to the Sun, the solutionSN III does not necessarily offer a better estimate thanthe purely gravitational solution PG IV. This questionalso raises doubt about whether the comet did indeedreturn to perihelion in the year − TIGHTENING THE CUTOFF FOR RESIDUALS
Before searching for a qualitatively superior approachin our quest for as perfect a distribution of residuals aspossible, we considered it appropriate to further tightenthe residual cutoff for the observations from the years1995–2003 and possibly even 2005. We felt that the cut-off should not be tighter than, or competitive with, thereported accuracy of < ∼ ′′ of the pre-discovery observa-tion (Marsden 1999), which we were determined to keepin the database under any circumstances. We eventuallyadopted a cutoff of 1 ′′ .5.To check the effect of the tightened cutoff on the samplesize, the predicted previous perihelion time, and the non-gravitational parameters under otherwise identical con-ditions, we continued the investigation by means of thestandard nongravitational law. We found that, with thenew cutoff, three iterations were needed to stabilize thenumber of employable observations; the runs were SN IV,SN V, and SN VI, the total numbers of used data pointswere, respectively, 2465, 2459, and 2456. The first andlast of these solutions are presented in Table 1. As ex-pected, the distribution of residuals from SN VI was,except for the drop in their range, very similar to thatfrom SN III and, therefore, not entirely satisfactory. Sekanina & Kracht Table 2
Parameters A , A , and A from Nongravitational SolutionsNongravitational Parameter (10 − AU day − )Solution A A A SN III +1.340 ± ± − ± ± ± − ± a +1.04 ± ± Note. a The nongravitational solution from Table 1 (Marsden 1999).
Table 1 shows that the number of employable observa-tions in the solution SN VI dropped by only about 15 per-cent compared with SN III, a result that is a tribute togood work by the astrometrists. The predicted perihelionpassage moved back in time to September of − A and A , just exceeded 3 σ of either run,whereas the transverse component’s parameters, A , ofthe two solutions agreed to within 1 σ . The discrepanciesbetween SN VI and Marsden’s (1999) orbit, computedwith an assumed A equal to zero, are rather startling,exceeding 7 σ in A and even more in A . MODIFIED NONGRAVITATIONAL SOLUTIONS
In one of the previous papers (Sekanina & Kracht 2015),we introduced a generic modified nongravitational law , g mod ( r ; r ), and applied it to much advantage in ourstudy of the dwarf Kreutz sungrazing comets, discoveredin large numbers with the coronagraphs onboard the So-lar and Heliospheric Observatory ( SOHO ; Brueckner etal. 1995) and, to a lesser degree, with the coronagraphsonboard the two spacecraft of the
Solar Terrestrial Re-lations Observatory ( STEREO ; Howard et al. 2008).The modified nongravitational (MN) law retains thevalues of the exponents m , n , and k of the standard law(1), but varies the scaling distance r and the constant ψ .This intervention is fully justified on account of Marsdenet al.’s (1973) finding that the shapes of the normalizedsublimation-rate curves against heliocentric distance fora variety of species are fairly similar in a log-log plot ex-cept for major horizontal shifts, which means that theyare relatively insensitive to the exponents m , n , and k ,but critically dependent on the scaling distance r . For agiven absorptivity and emissivity of the nuclear surface, r measures essentially an effective latent heat of subli-mation, L sub , of the observed mix of outgassing species,varying to a first approximation inversely as its square, r ≃ (cid:18) const L sub (cid:19) , (3)where a calibration by water ice gives for the constant avalue of 19 100 AU cal mol − in the case of an isothermalmodel but 27 000 AU cal mol − for outgassing from asubsolar region only, the two extreme values of the scalingdistance. In the following, I employ the first model for areason discussed briefly at the end of Section 9.4. For the dwarf comets of the Kreutz sungrazing systemthe application of this modified version of the nongrav-itational law was instrumental because the species thatwere sublimating profusely from their disintegrating nu-clei in close proximity of the Sun were much less volatilethan water ice (Sekanina & Kracht 2015); these refrac-tory materials have r ≪ . much more volatile than water ice; forthese ices, in particular for carbon monoxide and carbondioxide, r ≫ . Assigning Greater Weights to Critical Observations
A number of initial orbital runs based on the modifiednongravitational law consistently showed that the fit tothe observations could not be improved significantly andthe systematic trends in the residuals at the ends of theorbital arc removed, unless these first and last observa-tions — that we herefater call critical — were assignedsubstantially greater weights than the observations nearthe arc’s middle. We first allotted weight 40 to the pre-discovery observation in 1993, weight 20 to each of thethree observations on 2010 December 4, and weight 10to each of the five observations on 2007 October 20–22,so that we had at that point a total of 9 critical obser-vations. All other employed observations, from the years1995–2005, were allotted a unit weight.We searched for orbital solutions based on a set of theobservations weighted this way assuming several scalingdistances r and compared the mean residual left by allthe observations used with the mean residual left onlyby the critical observations. In the subsequent runs wechanged the weight assignment and included the 2005 ob-servations among the critical ones. Some of the solutionsobtained as we tried to stabilize the number of observa-tions satisfying the residual cutoff of 1 ′′ .5 (similarly to thesolutions PG I through PG IV and SN I through SN IIIin Table 1), are presented in Table 3. They show the as-signed weights and a dramatic impact of the stabilizationprocess on the number of retained observations.These results provide an answer to the stubborn prob-lem that Marsden (1999) was struggling with in his effortto accommodate the pre-discovery observation. The bot-tom line is that this could readily be done (with a residualof < ′′ .5) by weighting the observation heavily enough;however, the resulting orbital solutions demanded rejec-tion of almost one half of all observations that we startedwith ( ∼ ′′ . ± ′′ . − ′′ . ± ′′ . Table 3
Some Modified Nongravitational Solutions with Weighted Observations, and Offset of Pre-Discovery ObservationNumber of Scaling Mean Residual Offset fromAssigned Used/Critical Distance Pre-Discovery Time of PreviousSolution Weights a Observations r (AU) All Obs. Crit. Obs. Observation Perihelion (TT)MN VII 40; 1;10;20 2452/9 0.5 ± ′′ .77 ± ′′ .44 0 ′′ .86 − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − Note. a The four weights refer to all observations (called critical) used, respectively, from the years 1993, 2005, 2007, and 2010; all otherretained observations are assigned a unit weight. example, to +2 ′′ . ± ′′ . − ′′ . ± ′′ .
3, respectively,from the 88 rejected observations during 1997 October,and to +2 ′′ . ± ′′ . − ′′ . ± ′′ .
5, respectively, fromthe 40 rejected observations during 1998 February. Theseoffsets do not correlate with any cardinal direction, suchas the antisolar direction. From Table 3 we note the lackof correlation between the mean residual from all theused observations and the mean residual from the crit-ical observations. For example, comparison of the solu-tions MN IX and MN XI suggests that the fit with ascaling distance r of 0.5 AU is clearly better than thefit with r = 6 AU in terms of all the employed observa-tions, but slightly worse in terms of the critical observa-tions. The solutions MN XI through MN XV indicate,on the other hand, that for a given scaling distance thequality of fit does not change among the critical observa-tions, but it improves among all the used observations astheir number drops dramatically. The highly weighted1993 pre-discovery observation is nonetheless fitted thebetter the smaller is the mean residual from all the usedobservations. We further note from Table 3 that, as thenumber of retained observations drops, the previous per-ihelion passage moves systematically by many years toearlier times; the perihelion time predicted from the so-lution MN XV is within 11 years of the time predictedin Table 1 from the gravitational solution PG IV!The last four entries of Table 3, whose weights for thecritical data — 40 (1993), 5 (2005), 10 (2007), 20 (2010) —are hereafter called the Weight System I , are the mostimportant ones. The last observation of 2005 was nowrejected, leaving just 15 critical observations, with the to-tal number of observations eventually stabilized at 1540.MN XVIII and MN XIX, which fit the data points essen-tially equally well, are the basis for deriving the optimumsolution, the final step in our orbit-determination effort;it is dealt with in the next subsection.Before addressing that issue, we should comment onthe properties and the implications of the most importantsolutions in Table 3, MN XVI through MN XIX. Thesesolutions require, similarly to MN XV, that nearly all theobservations between mid-March 1997 and early March1998 be rejected. The nongravitational parameters fromMN XVI through MN XIX, listed in Table 4, reveal some interesting features. The most surprising is that by farthe highest acceleration is in the direction normal to theorbital plane, A . We will return to this result in thesection in which we confront orbital evidence with infor-mation on the nuclear rotation of the comet. Table 4 alsoshows that the ratio of | A | /A varies strongly with thescaling distance r , dropping from near 10 at r = 6 AUto a little over 2 at r = 3 AU, when the transverse com-ponent A is very small and poorly defined.Comparison with the solutions SN III and SN VI fromTable 2 also offers some rather unexpected conclusions.The total magnitude of the nongravitational accelera-tion, p A + A + A , is now lower by a factor of ∼
2. Forthe solutions MN XVI through MN XVIII this could beunderstood as an effect of the scaling distance (which ofcourse equaled 2.8 AU for the SN solutions), because agreater scaling distance means a greater contribution tothe integrated effect from larger heliocentric distances.However, this does not explain the factor of 2 betweenthe SN solutions and MN XIX, whose r was nearly thesame. The incompatibility of the SN and MN results isalso underscored by the opposite signs of the transversecomponent. It thus appears that assigning weights tothe critical observations and relaxing the scaling distancehad a major effect on the results. Weight System I: Final Optimization of Orbit
Table 3 shows that the uncertainty in the scaling dis-tance r implies an uncertainty of at least several yearsin the time of perihelion passage in the 23 century BCE.Because of this indeterminacy, it could never be proven Table 4
Parameters A , A , and A from Solutions MN XVI–MN XIXNongravitational Parameter (10 − AU day − )Solution A A A MN XVI +0.068 ± − ± − ± ± − ± − ± ± − ± − ± ± − ± − ± Sekanina & Kracht
Table 5
Orbital Elements and Encounter Parameters for Nominal Solutionof C/1995 O1 with Weight System I (Equinox J2000)Orbit at Apparition 1993–2010Epoch of osculation (1997 TT) Mar 13.0Time of perihelion (1997 TT) Apr 1.13747 ± ◦ .587433 ± ◦ .000071Longitude of ascending node 282 ◦ .469632 ± ◦ .000006Orbit inclination 89 ◦ .429219 ± ◦ .000016Perihelion distance (AU) 0.9141709 ± ± a ( osculating 2558.3 ± a ◦ .8034Latitude of perihelion +49 ◦ .4092Nongravitational parameters:Law applied modifiedScaling distance r (AU) 3.070Normalization constant Ψ 0.09105 A [radial] (10 − AU day − ) +0.266 ± A [transverse] (10 − AU day − ) − ± A [normal] (10 − AU day − ) − ± ± ′′ .66Time period covered 1993 Apr 27–2010 Dec 4Orbital arc (days) 6429.54Total observations used 1540Critical observations used 15 b Post-Encounter Orbit in 23rd Century BCEEpoch of osculation ( − − ◦ .4076Longitude of ascending node 282 ◦ .3700Orbit inclination 89 ◦ .0530Perihelion distance (AU) 0.907236Orbit eccentricity 0.996749osculating 4661Orbital period (yr) n future a ◦ .26Latitude of perihelion +49 ◦ .58Pre-Encounter Orbit in 23rd Century BCEEpoch of osculation ( − − ◦ .4131Longitude of ascending node 102 ◦ .3538Orbit inclination 92 ◦ .2032Perihelion distance (AU) 0.151986Orbit eccentricity 0.999986original a ∼ n osculating 1 090 000Longitude of perihelion 103 ◦ .14Latitude of perihelion − ◦ .57Encounter ParametersTime of perijove ( − R J ) c ◦ .0Jovicentric velocity at perijove (km s − ) 28.72 Notes. a Derived from the original (or future) semimajor axis referred to thebarycenter of the Solar System. b Of these, one in 1993, six in 2005, five in 2007, and three in 2010. c R J = 71 492 km is adopted equatorial radius of Jupiter. that C/1995 O1 did in fact experience a close encounterwith Jupiter. However, pursuing Marsden’s (1999) sug-gestion that “ it is not entirely improbable that [ the cometunderwent ] a recent dramatic approach ” to the planet, weexploited the tabulated dependence of the previous peri-helion time on the scaling distance. By slightly adjusting r , we were able to determine the time of perijove andthe corresponding Jovicentric distance that were com-patible with plausible pre-encounter orbital constraints.We found that the comet had to pass through perihelionin early December of − − r ≈ A <
0, in contradiction to the physicalmodel of nongravitational forces.On the other hand, a favorable configuration one Jo-vian revolution later would require that the comet’s per-ihelion occurred in mid-October of − − r < − t J , correlates tightly with both the Table 6
Correlations Between Time at Perijove, Jovicentric Distance,and Pre-Encounter Original Orbit (Weight System I)At Perijove on − t J Distance, a Semimajor Axis, b Period,(TT) ∆ J ( R J ) (1 /a ) orig (AU − ) P orig (yr)5:11:56 10.612 − − − − Notes. a R J = 71 492 km is adopted equatorial radius of Jupiter. b Referred to the barycenter of the Solar System. omet C/1995 O1: Orbit, Jovian Encounter, and Activity 9Jovicentric distance at the time, ∆ J ( t J ), and with thepre-encounter orbit’s original semimajor axis, (1 /a ) orig (referred to the barycenter of the Solar System), or theequivalent original orbital period, P orig . The interval of t J in Table 6, covering 12 hours on November 7 of theyear − r and shows that the minimum perijovedistance ∆ J was attained certainly before 7:00. Althoughwe will estimate the exact time below, it is outside therange of interest because the comet would then have ar-rived along a strongly hyperbolic orbit, an unacceptablescenario.Realistically, the comet should have passed throughperijove after 8:15:11, when its pre-encounter barycen-tric orbit was elliptical. If C/1995 O1 was a dynami-cally new comet arriving from the “core” region of theOort Cloud — for which we adopt with Marsden etal. (1978) a heliocentric distance of ∼
43 000 AU) — itwould have arrived at perijove just 10 seconds later, at8:15:21. The comet’s original orbital period would havethen amounted to some 3 200 000 yr, its Jovicentric dis-tance at perijove would have been near 10.8 R J , and itsJovicentric velocity about 29 km s − , as the nominal so-lution in Table 5 indicates.Of course, C/1995 O1 may have been a long-periodcomet before the encounter, but not dynamically new.If so, it would have arrived later still, perhaps as lateas 8:30 or even 9:00. Its perijove distance would thenhave been only slightly greater. For two related reasonsit is, however, less likely that the comet had moved in anorbit of a relatively short period before the encounter:(i) it would have had a high probability of approachingJupiter on a number of occasions at times before thisencounter and (ii) having made many revolutions aboutthe Sun with a very small perihelion distance, it wouldlong ago have gotten depleted of highly volatile species,in conflict with the observations (Section 9).The steep dependence of the original barycentric semi-major axis, (1 /a ) orig , on the time of perijove, t J , duringthe day of − t J than about 2 hr. A least-squares fit to the data points between ∼ ∼ /a ) orig = − . . t J − h ) , ± . ± . /a ) orig is in (AU) − and t J in hr.Similarly, the relationship between the time of perijoveand the perijove distance, ∆ J , is over the entire interval oftime from 5:10 to 11:30 approximated by a least-squarespolynomial∆ J =10 . . t J − h ) + 0 . t J − h ) , ± . ± . ± . J is expressed in units of the Jovian equatorialradius R J (1 R J = 71 492 km) and t J is again in hr. Asearch for a minimum provides (∆ J ) min = 10 . R J at( t J ) min = 5:35:25 or 2:39:56 before the perijove time forC/1995 O1 as an Oort Cloud comet. RECIPROCAL SEMIMAJOR AXISOF PRE-ENCOUNTER ORIGINALORBIT OF COMET C/1995 O1IN 23rd CENTURY BCE ASFUNCTION OF TIMEAT PERIJOVE ············································································································ ······························································································································ ································································ ······· ····································································································
OortCloud8:15:21
Oort Cloud(1 /a ) orig = − . . t J − h ) ❙❙♦ t J , ON − (1 /a ) orig (AU) − − − − ❤t ❤t ❤t❤t❤t ❤t ❤t ❤t ❤t Figure 3.
Plot of the reciprocal semimajor axis of the comet’s pre-encounter original barycentric orbit, (1 /a ) orig , against the time atperijove, t J , on − /a ) orig = +0 . − and described with a correspondingperijove time of 8:15:21 TT. The orbit in Table 5, referred to the osculation epochof 1997 Mar 13, is generally in accord with the publishedgravitational orbits. For example, it agrees with Mars-den’s (2007) orbit to better than 90 seconds in the peri-helion time, to better than 4 ′′ in the angular elements,to better than 1000 km in the perihelion distance, and tobetter than 20 yr in the osculating orbital period. How-ever, with the exception of the arument of perihelion, ourformal errors are smaller than the differences, in some ofthe elements by two orders of magnitude. The originalvalues of the orbital period derived by us and by Mars-den deviate by 20 yr, the future period by 8 yr. Ourresult predicts for the previous perihelion time the year − − ′′ .1 in the argument of perihelion,by 3 ′′ .6 in the longitude of the node, by 3 ′′ .0 in the in-clination, by 450 km in the perihelion distance, and by12 yr in the osculation orbital period. It predicts thatthe previous perihelion time occurred in the year − ≀≀ ≀≀ ≀≀ ≀≀≀≀ ≀≀ ≀≀ ≀≀ +1 ′′ ′′ − ′′ +1 ′′ ′′ − ′′ O − C − −
500 0 +500 +1000 +1500 +2000 +3000 +5000TIME FROM PERIHELION (days)RESIDUALS IN RIGHT ASCENSIONRESIDUALS IN DECLINATION
FINAL — WS I N obs = 1540 ✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉
Figure 4.
Temporal distribution of residuals O − C (observed minus computed) in right ascension (top) and declination (bottom) fromthe nominal nongravitational solution listed in Table 5, based on 1540 accurate observations of comet C/1995 O1 between 1993 and 2010,which left residuals not exceeding 1 ′′ .5 in either coordinate. The fit is judged perfect in declination, but many observations from 1998–2000still leave mostly positive residuals. ever been employed in orbital computations of comets —not only improves the fit, but turns out to be the mostprominent component, more than twice the magnitude ofthe radial component. On the other hand, the transversecomponent is very small and poorly defined. The overallmagnitude of the nongravitational acceleration at 1 AUfrom the Sun is 0.69 ± − , or about one halfthe acceleration that resulted from the standard nongrav-itational solutions (Table 2). This is noteworthy becausethe magnitude of the nongravitational acceleration de-rived earlier was judged to be too high to be compatiblewith the presumably large nucleus of the comet (Marsden1999, Sosa & Fern´andez 2011).The scaling distance r of the nongravitational law inthe nominal solution in Table 5 is only slightly greaterthan the scaling distance for water ice in Equation (1).This result can be interpreted to mean that the nongrav-itational effect was due largely to water ice, with onlyrelatively minor contributions from more volatile species.We return to this issue in Section 9.Somewhat surprising is the very small perihelion dis-tance, of only about 0.15 AU, of the pre-encounter or-bit. Two other prominent dynamically new comets ap-proached the Sun to a similar distance, C/1973 E1 (Ko-houtek) and C/2006 P1 (McNaught), but in terms of theinclination the pre-encounter orbit of C/1995 O1 is muchmore similar to the latter of the two comets. And whilethe orbital plane’s nodal line and inclination were af-fected very little by the Jovian encounter (by 1 ′ and a lit- tle more than 3 ◦ , respectively), the line of apsides movedby 69 ◦ ( sic! ), the nodes got interchanged, and the peri-helion distance increased by a factor of six. C/1995 O1passed through perihelion two months later than it wouldhave in its pre-encounter orbit.The distribution of residuals from the nominal solutionin Table 5 left by the 1540 retained observations is dis-played in Figure 4. The distribution is satisfactory indeclination, but there is a significant excess of positiveresiduals in right ascension in the course of 1998 and1999. This is a tail of the set of much more strongly posi-tive residuals left by the observations made between mid-March 1997 and early March 1998, all of which had to bediscarded from the nominal solution. There are two clus-ters of residuals in R.A. in Figure 4 in the 1998–1999 timeslot — an upper dense one and a lower sparse one — sep-arated by a tilted strait with no data points. The lower,lightly populated cluster contains observations that aremore consistent with the solution in Table 5.To investigate this peculiar double-cluster distributionin greater detail, we present in Table 7 a dozen obser-vations from either cluster, made in the period of timefrom early May 1998 through early May 1999. Each ob-servation with the residuals is identified by the observingsite code and the reference to a Minor Planet Circular (MPC), in which it was published. From the list of ob-serving sites that always precedes the list of reported ob-servations in the MPCs, we extracted the information oneach instrument’s aperture and the f ratio and convertedomet C/1995 O1: Orbit, Jovian Encounter, and Activity 11 Table 7
Systematic Residuals in Right Ascension, from Nominal Orbit with Weight System I, of Sample Observations from May 1998–May 1999Sparse Cluster (Residuals Not Exceeding 0 ′′ .7 in R.A.) Dense Cluster (Residuals Exceeding 1 ′′ .2 in R.A.)Time of Residual O − C Obs. Image Refer- Time of Residual O − C Obs. Image Refer-Observation Site Scale ence Observation Site Scale ence(UT) R.A. Decl. Code (mm − ) MPC (UT) R.A. Decl. Code (mm − ) MPC1998 May 8.96560 − ′′ .63 − ′′ .16 834 131 ′′
32 129 1998 May 6.96538 +1 ′′ .22 − ′′ .29 844 90 ′′
31 853July 4.82718 − − − − − − − − − − − − − − − − it into the image scale presented in column 5 of Table 7.The aim was to check whether there was any systematicdifference between the image scales (CCD pixel sizes wereunfortunately unavailable) of the instruments employedto make the seemingly more accurate observations on theleft and the image scales of the instruments used to makethe observations leaving systematic residuals in R.A. onthe right. As seen from the table, it turned out that thesame instrument sometimes provided acceptable residu-als, while at other times it did not. The average imagescale for the observations on the left is better than thaton the right, but only by a small margin of 124 ′′ mm − vs 151 ′′ mm − . This result means that the problem withthe residuals in R.A. either lies elsewhere or the cutoffof ± ′′ .5 is tighter than the level of accuracy achievableby the category of telescopes (with apertures of, gener-ally, 25–50 cm) used by most observers at the times thecomet was bright enough to be within their reach. Wenote that on the one hand, the systematic trend in R.A.is limited to the time interval of high activity (associatedwith a complex coma morphology), but it fails to corre-late with the comet’s orientation relative to the Sun, sothat the offsets cannot readily be attributed to sunwardor antisunward condensations. Introduction of New Weight System
The distribution of residuals might be sensitive to theweight assigned to the pre-discovery observation of 1993.It is recalled that it indeed was this observation’s resid-ual in R.A. that caused problems when attempts weremade in 1998 to link it with the post-discovery obser-vations (Marsden 1999). The weight system applied inSection 6.2, which assigned the 1993 observation weight40, resulted in an excellent fit to this data point by thenominal solution in Table 5, leaving a residual of − ′′ .23in R.A. and − ′′ .14 in decl. (Figure 4). Given the posi-tion’s uncertainty “within 1 arcsec” (Marsden 1999) —the 124 cm UK Schmidt telescope, with which the ob-servation was made, has an image scale of ∼ ′′ mm − — one should allow these residuals to become quite a bitgreater and still consistent with the stated astrometricerror. The implication is that the weight assigned to the 1993 observation was unnecessarily high and that a lowerweight might be more appropriate.The weight assignment to the critical observations, notonly to the pre-discovery observation, also has broaderramifications in terms of the orbital results and predic-tions of the comet’s motion in the previous return tothe Sun. It is therefore desirable to address this issue ingreater detail. To examine the impact, we chose a set ofdifferent weights, called hereafter Weight System II, asfollows: the observation of 1993 was assigned weight 20(instead of 40), the 2010 observations weight 15 (insteadof 20), while the weights of the 2005 and 2007 observa-tions were left unchanged at 5 and 10, respectively.We then proceeded in a way similar to that in Sec-tion 6.1, with the nominal solution in Table 5 used asa starting set of elements in the iterative procedure ofstabilizing the number of retained observations. How-ever, unlike before (Table 3), at each step of the pro-cess we optimized the scaling distance r of the modi-fied nongravitational law to fit the encounter time. Ac-cordingly, we were successively deriving the solutionsMN XXI to MN XXVII listed in Table 8.The results, which again allow a close approach onlyin the year − − − ± ′′ .5 before, would nowfit. Simultaneously the scaling distance increased dra-matically by almost 5 AU, while the offset of the 1993pre-discovery observation grew to about twice the off-set of the nominal solution in Table 5 (Figure 4), stillwell within the reported uncertainty. The first solutionwas far from stabilizing the number of retained observa-tions, and the trend continued: 121 additional observa-tions were accommodated by the second iteration, whichrequired a scaling distance greater by a yet another 2 AU,while the offset of the pre-discovery observation now in-creased to more than 0 ′′ .8 and was slowly approachingthe limit of the measured error. After five more itera-tions, the number of retained observations finally stabi-lized at 1950, an increase by more than 500 relative tothe solution in Table 5. Interestingly, in spite of these2 Sekanina & Kracht Table 8
Modified Nongravitational Solutions Using Weight System II, and Offset of Pre-Discovery ObservationNumber of Scaling Mean Residual Offset fromUsed/Critical Distance Pre-Discovery Time of PreviousSolution Observations r (AU) All Obs. Crit. Obs. Observation Perihelion (TT)MN XXI 1720/15 7.78 ± ′′ .629 ± ′′ .256 0 ′′ .61 − ± ± − ± ± − ± ± − ± ± − ± ± − ± ± − additional accommodated data points the mean residualdropped slightly in comparison with the nominal solutionin Table 5. However, the mean residual of the 15 criticalobservations increased a little, unquestinably owing tothe greatly increased offset of the pre-discovery observa-tion. This offset was now at the limit of uncertainty, withthe residuals, in the sense “observed minus computed”,amounting to − ′′ .86 in right ascension and − ′′ .57 indeclination.The nongravitaional parameters derived from the so-lutions MN XXI through MN XXVII likewise differ sig-nificantly from those for the nominal orbit in Table 5,as illustrated by comparing Table 9 with Table 4. Whilethe total magnitude is approximately the same, it is nowthe radial component of the nongravitational accelera-tion whose magnitude dominates that of the normal com-ponent, while the magnitude of the transverse componentis again insignificant and poorly defined.In summary, there is evidence that the strong sys-tematic trends in the residuals in R.A., which necessi-tated rejection of all observations made between mid-March 1997 and early March 1998 and showed ratherprominently in the residuals of the retained observationsthroughout May 1999, were caused by an excess weightassigned to the 1993 pre-discovery observation. Whenthe weight was reduced by a factor of two, the orbitalsolution accommodated more than 400 observations thathad to be rejected before, increased the scaling distanceof the modified nongravitational law by a factor of five,and decreased the magnitude of the normal componentof the nongravitational acceleration, so that the radialcomponent now dominates; all this at the expense of afit to the pre-discovery position, with the total residualincreasing from less than 0 ′′ .3 to ∼ ′′ , the estimated un-certainty of the astrometric measurement. Table 9
Parameters A , A , and A from Solutions MN XXI–MN XXVIINongravitational Parameter (10 − AU day − )Solution A A A MN XXI +0.423 ± − ± − ± ± − ± − ± ± − ± − ± ± − ± − ± ± − ± − ± ± − ± − ± ± − ± − ± Weight System II: Final Optimization of Orbit
To determine the comet’s nominal orbit and its ar-rival time from the Oort Cloud under the constraints ofWeight System II, we fine-tuned the solution MN XXVIIin Table 8. The results are listed in detail as a nomi-nal solution in Table 10, whose format is identical withthat of Table 5 to allow direct comparison with the nom-inal solution with Weight System I. The agreement be-tween the two sets of orbital elements is excellent, butthere are major differences in the nongravitational pa-rameters that will be addressed in Section 9. As for theencounter parameters, the time of perijove is now pre-dicted 24 minutes earlier and the Jovicentric distanceat perijove is merely 0.05 Jupiter’s equatorial radius, or3600 km, smaller. An earlier time for the encounter isconsistent with a greater scaling distance and a highervalue of A , both of which mean that under Weight Sys-tem II the orbital motion was subjected to higher inte-grated nongravitational effects, so the temporal gap be-tween the perijove in the year − /a ) orig (or the orbital period P orig ), and for the Jovicentric dis-tance at perijove, ∆ J , are similar to Equations (4) and(5), respectively. A dozen solutions, spanning more than10 hr in the perijove time t J and listed in Table 11, il-lustrate the relationships. A least-squares fit to the datapoints covering less than one hour of the relevant rangeof t J provides the following relation:(1 /a ) orig = +0 . . t J − h ) , ± . ± . /a ) orig and t J are again in (AU) − and hr, re-spectively. We do not show its plot, because it looks verymuch like Figure 3: the slope is only 2% steeper and thefitted straight line is shifted, as already pointed out, toan earlier time by 25 minutes.From a wider interval of t J , spanning about 7.5 hr, theperijove distance (in units of Jupiter’s equatorial radius)is given by an expression:∆ J = 10 . . t J − h ) + 0 . t J − h ) . ± . ± . ± . Table 10
Orbital Elements and Encounter Parameters for Nominal Solutionof C/1995 O1 with Weight System II (Equinox J2000)Orbit at Apparition 1993–2010Epoch of osculation (1997 TT) Mar 13.0Time of perihelion (1997 TT) Apr 1.13785 ± ◦ .587793 ± ◦ .000070Longitude of ascending node 282 ◦ .469751 ± ◦ .000010Orbit inclination 89 ◦ .428854 ± ◦ .000017Perihelion distance (AU) 0.9141677 ± ± a ( osculating 2542.4 ± a ◦ .8031Latitude of perihelion +49 ◦ .4089Nongravitational parameters:Law applied modifiedScaling distance r (AU) 15.363Normalization constant Ψ 0.002812 A [radial] (10 − AU day − ) +0.566 ± A [transverse] (10 − AU day − ) − ± A [normal] (10 − AU day − ) − ± ± ′′ .64Time period covered 1993 Apr 27–2010 Dec 4Orbital arc (days) 6429.54Total observations used 1950Critical observations used 15 b Post-Encounter Orbit in 23rd Century BCEEpoch of osculation ( − − ◦ .4081Longitude of ascending node 282 ◦ .3687Orbit inclination 89 ◦ .0539Perihelion distance (AU) 0.907234Orbit eccentricity 0.996727osculating 4615Orbital period (yr) n future a ◦ .26Latitude of perihelion +49 ◦ .58Pre-Encounter Orbit in 23rd Century BCEEpoch of osculation ( − − ◦ .1217Longitude of ascending node 102 ◦ .3535Orbit inclination 92 ◦ .1802Perihelion distance (AU) 0.156469Orbit eccentricity 0.999985original a ∼ n osculating 1 090 000Longitude of perihelion 103 ◦ .14Latitude of perihelion − ◦ .86Encounter ParametersTime of perijove ( − R J ) c ◦ .2Jovicentric velocity at perijove (km s − ) 28.75 Notes. a Derived from the original (or future) semimajor axis referred to thebarycenter of the Solar System. b Of these, one in 1993, six in 2005, five in 2007, and three in 2010. c R J = 71 492 km is adopted equatorial radius of Jupiter. The distribution of residuals from the nominal orbitin Table 10, left by the 1950 observations, is presentedin Figure 5. It depicts the ∼
400 reinstated data pointsbetween mid-March 1997 and early March 1998 that re-place the prominent gap in Figure 4. A residual gap of afew months centered on mid-June 1997 has to do with thecomet’s persistently small elongations from the Sun (witha minimum of 21 ◦ .5 on June 10), which was responsiblefor no astrometric data (with one exception) betweenMay 12 and July 14 and made the observations over afew adhering months difficult, less frequent, and of lowerquality. The tail of systematically positive residuals (inexcess of 1 ′′ ) in R.A. in 1998–1999 has likewise disap-peared. The strong systematic effect in R.A. in Figure 4may be slightly overcorrected in Figure 5, but there is notruly worrisome effect of the kind. A greatly improvedfit in R.A. to the observations from the denser 1997–1998cluster, which is apparent by comparing Table 12 withTable 7, corroborates a superior match to the observa-tions by the solution from Table 10. In declination, thedistribution of residuals is considered rather satisfactory,with no prominent long-term systematic trends.Concluding this section, we note that the nominal or-bit with Weight System II offers — in terms of orbitalquality — a solution superior to the nominal orbit withWeight System I: it accommodates 410 more observationsto within 1 ′′ .5, while still fitting the 1993 pre-discoveryobservation to ∼ ′′ . Both scenarios (Tables 5 and 10) al-low for an approach of C/1995 O1 to less than 11 equato-rial radii of Jupiter on − Table 11
Correlations Between Time at Perijove, Jovicentric Distance,and Pre-Encounter Original Orbit (Weight System II)At Perijove on − t J Distance, a Semimajor Axis, b Period,(TT) ∆ J ( R J ) (1 /a ) orig (AU − ) P orig (yr)4:29:23 10.561 − − − − − − Notes. a R J = 71 492 km is adopted equatorial radius of Jupiter. b Referred to the barycenter of the Solar System. ≀≀ ≀≀ ≀≀ ≀≀≀≀ ≀≀ ≀≀ ≀≀ +1 ′′ ′′ − ′′ +1 ′′ ′′ − ′′ O − C − −
500 0 +500 +1000 +1500 +2000 +3000 +5000TIME FROM PERIHELION (days)RESIDUALS IN RIGHT ASCENSIONRESIDUALS IN DECLINATION
FINAL — WS II N obs = 1950 ✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉✉✉✉✉ ✉✉✉✉✉✉
Figure 5.
Temporal distribution of residuals O − C (observed minus computed) in right ascension (top) and declination (bottom) fromthe nominal nongravitational solution listed in Table 10, based on 1950 accurate observations of comet C/1995 O1 between 1993 and2010, which left residuals not exceeding 1 ′′ .5 in either coordinate. The fit is overall judged to be quite satisfactory, implying for the 1993pre-discovery observation residuals that reach the limit of its estimated uncertainty. Table 12
Residuals, from Nominal Orbit with Weight System II, of Sample Observations from Table 7 (Sparse Cluster vs Dense Cluster)for Comparison with Residuals from Nominal Orbit with Weight System ITime of Residual O − C Obs. Image Refer- Time of Residual O − C Obs. Image Refer-Observation a Site Scale ence Observation Site Scale ence(UT) R.A. Decl. Code (mm − ) MPC (UT) R.A. Decl. Code (mm − ) MPC1998 May 8.96560 ∗ − ′′ .48 +0 ′′ .67 834 131 ′′
32 129 1998 May 6.96538 − ′′ .63 − ′′ .44 844 90 ′′
31 853July 4.82718 − − − ∗ − − − − − − − − − − ∗ − − − ∗ − ∗ − − ∗ − − Note. a Observations marked with asterisks were rejected because a residual in at least one coordinate exceeds the rejection cutoff of 1 ′′ .5. As illustrated in Figure 6, the comet was approachingJupiter from above the ecliptic plane and was diverted bythe planet’s gravity back above the ecliptic plane in a tra-jectory that was strongly hyperbolic relative to Jupiter,with an eccentricity of 4.0. The comet was launched to-ward the Sun along a new, very different orbit, whoseplane though agreed with the original one to 3 ◦ . At thetime of encounter the comet’s position was at close prox-imity of the descending node of the original orbit and the ascending node of the new orbit. After reaching perihe-lion about two months later and at a distance from theSun six times greater than it would have in the absenceof the encounter, the comet began to recede from theSun, once again approaching the ecliptical plane. Some35 days after perihelion it crossed the descending node(now on the other side of the Sun) for the second timeand continued its journey to aphelion, diving almost per-pendicularly to the ecliptic.omet C/1995 O1: Orbit, Jovian Encounter, and Activity 15 S ②r r rr rr rr ✵ SCALE (AU):0 1 2 3 ❢ J+C
MAJOR ORBITAL TRANSFORMATION OFCOMET C/1995 O1 IN 23rd CENTURY BCEOWING TO ENCOUNTER WITH JUPITER ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣
ΠJUPITER’SORBIT ❨✛ PRE-ENCOUNTERORBITPOST-ENCOUNTERORBIT rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣
Figure 6.
Major orbital transformation of C/1995 O1 on − ℧ . The first descending node, on the pre-encounter orbit, and the ascendingnode on the post-encounter orbit coincide on the scale of the plot with the comet’s and Jupiter’s position at the time of encounter. PREDICTED APPEARANCE OF COMET C/1995 O1IN THE 23RD CENTURY BCE AND SEARCHFOR POSSIBLE HISTORICAL RECORDS
Searching for historical records would be pointless, ifthe comet’s appearance was affected by unfavorable ob-serving geometry. While the viewing conditions for acomet moving in an orbit perpendicular to the plane ofthe ecliptic cannot remain inferior for long, we are pri-marily concerned with a short orbital arc near perihelion,when the comet is intrinsically the brightest. Given theperihelion distance near 0.9 AU and the argument of per-ihelion at 130 ◦ , the comet would at perihelion project tothe terrestrial observer at an angular distance of less than24 ◦ from the Sun (at a geocentric distance of more than1.7 AU), if the Earth should then be crossing the comet’sorbital plane on the side of the ascending node. On theother hand, at the descending node, at a heliocentricdistance of 1.1 AU, the comet would be, under the mostfavorable conditions, at opposition with the Sun and only0.1 AU from the Earth.An ephemeris, based on the post-encounter orbital el-ements from the 23rd century BCE in Table 10, showsthat the viewing circumstances in this configuration wererather favorable. In Figure 7 we present the comet’s pre-dicted path in the sky over a period of 4 months, from − − − ′ in diameter, practi-cally the angular size of the Moon. The predicted viewing geometry is described in a self-explanatory Table 13. A minimum geocentric distanceof 0.644 AU was found to have taken place on − ◦ .2 on − ◦ .8on − ◦ .5 on − − − −
770 (such as
Ch¯unqi¯u or Ch’un-ch’iu , the
Spring and Autumn Annals ; e.g., Liu et al. 2003). This prediction is based on 433 naked-eye estimates of the comadiameter of C/1995 O1 made in 1997 between 10 days before and10 days after perihelion, and reported to the
International CometQuarterly (Green 1997, 1998, 1999). The data average, a diameterof 1 150 000 km, was equivalent to 19 ′ .6 at the 1997 perihelion. AND AQR AQLARI CAPCET CYGDELERI HERLYRLAC OPHPEGPER PSC SGE SCTTAU TRI VULSymbol:Star mag: ✉ ≤ t s r q ♣ − ◦ ◦ +10 ◦ +20 ◦ +30 ◦ +40 ◦ +50 ◦ DECLINATION 5 h h h h h h h h h h h h h RIGHT ASCENSION ❢❢❢❢❢❢❢❢❢❢❢❢❢ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ ♣♣♣♣♣♣♣♣ ♣s♣r♣♣♣♣q♣♣♣♣♣♣♣♣♣♣qq♣q♣r♣♣♣♣♣♣♣q♣q♣♣♣♣♣♣qq♣q♣♣♣♣rqs♣♣♣♣♣♣♣♣♣♣♣q♣♣♣♣♣qqq♣♣♣q♣q♣q♣♣q♣r♣♣♣r♣♣♣♣qs♣♣♣s♣r♣♣♣♣♣q♣♣q♣♣r♣♣♣♣♣♣♣♣q♣♣♣♣♣♣♣q♣♣♣q♣♣rq♣q♣♣qq♣♣q♣♣q♣♣♣♣♣♣♣♣♣rr♣r♣♣sqq♣q♣♣♣♣♣♣♣♣♣♣q♣♣♣♣sq♣q♣q♣♣q♣♣qqqr♣qqq♣qq♣q♣qqr♣♣♣q♣♣♣r♣♣♣♣rqrr♣♣q♣♣♣q♣q♣♣q♣♣qq♣♣♣♣♣♣q♣q♣♣♣♣♣q♣♣♣♣q♣♣♣q♣q♣qq♣♣qqr♣♣♣♣♣♣♣♣qtqq♣♣♣q♣♣♣♣q♣♣♣♣q♣♣q♣rq♣♣q♣q♣qq♣♣♣rq♣♣qr♣♣q♣♣♣♣r♣♣♣♣r♣♣♣q♣♣qqq♣✉✉♣♣♣♣q♣♣♣♣q♣♣♣♣♣♣q♣r♣ss♣♣♣♣♣♣ q♣♣qr♣♣rq♣♣♣♣♣♣♣♣q♣♣r♣q♣♣♣♣♣♣r♣r♣r♣♣♣q♣♣♣♣♣♣♣q♣♣qrs♣♣qqrrq♣♣♣♣♣qrqsq♣♣♣qq♣♣♣♣q♣♣q♣qq♣♣♣♣♣♣♣♣rq♣♣♣♣♣q♣♣♣♣♣♣q♣♣♣♣✉♣♣♣♣♣♣qqq♣♣q♣r♣♣♣q♣♣♣♣q♣♣♣qr♣♣q♣♣♣♣rr♣♣♣♣♣♣♣q♣q♣♣♣q♣♣♣♣♣r♣♣♣q♣r♣q♣q♣♣♣♣♣q♣♣♣q♣♣q♣q♣♣♣♣rr♣q♣♣t♣q♣♣q♣♣♣q♣♣♣q♣♣r♣♣♣♣♣♣♣♣♣♣r♣♣♣♣♣q♣♣♣♣qq♣q♣♣♣♣r♣sq♣q♣♣q♣♣♣♣q♣q♣♣♣♣♣qtq♣♣q♣qsqq♣♣♣♣♣♣♣♣♣♣♣q♣♣♣♣♣q♣♣♣♣rqqqqq♣q♣♣♣♣♣♣♣♣rq♣♣♣♣♣♣♣♣♣♣s♣♣qq♣q♣♣♣♣♣♣♣rq♣qq♣q♣♣q♣qq♣♣q♣♣♣q♣♣♣♣q♣q♣♣qq♣♣♣qr♣♣r♣qqqr♣q♣♣♣♣♣♣q♣♣s♣s♣♣♣♣♣♣♣♣♣qq♣qq♣♣♣♣♣♣♣♣q♣q♣♣♣♣qq♣♣qqqq♣♣♣♣♣♣♣♣♣♣q♣q♣ ····································································································································· ······················································································································································································································································ ··············································································································································································································································································· ························································································································································· ·········································································· ······································ ······································ ··············· ······················· ······································ ····································································· ·········· ········
Figure 7.
Predicted path of comet C/1995 O1 over the sky during its arrival from the Oort Cloud in the year − − − − Equally or more uncertain are the time assignments forevents in other ancient civilizations. In partcular, thecurrently accepted chronology of the Egyptian history,introduced less than two decades ago (Shaw 2000), dif-fers in the 3rd millennium BCE from a previous chronol-ogy (Breasted 1906) by almost 300 yr. It thus comesas no surprise that most modern catalogs of historicalrecords of comets, summarizing information collectedfrom the ancient sources, do not reach the 3rd millen-nium BCE, and they especially do not list the orbital el-ements. Indeed, Williams (1871) begins in the year − − −
466 (presumably 1P/Halley),Porter (1961) and Marsden & Williams (2008) in − − liste g´en´erale and Pingr´e’s (1783) com`etographie (one ofBaldet’s sources) start, respectively, in the 24th and 23rdcenturies BCE. Pingr´e’s, Williams’, Baldet’s, and Ho’slists and many additional sources were incorporated byHasegawa (1980) into his catalogue of ancient and naked-eye comets, but with some notes and comments left out.A reference to C/1995 O1 in − − − − − − − − − − − − − − − − T’u shu chi ch’-ˆeng as the source of the earliest record of a bright object in the sky, but the yearis identified as − − − − − − − Table 13
Circumstances at Arrival of C/1995 O1 to Perihelion in − a Phase Elon- Vis.No. Date (TT) (days) Earth Sun Angle gation Mag1 − − ◦ .9 41 ◦ .7 1.82 Nov 2.0 − − − − − − − − − − − Note. a Time from perihelion: minus sign means before perihelion,plus signafter perihelion. omet C/1995 O1: Orbit, Jovian Encounter, and Activity 17has been a matter of controversy, because the time of hispresumed reign preceded the oldest known written docu-ments by about a millennium. Doubts have also persistedon the subject of an elaborate irrigation system that Yuas the ruler is believed to have overseen to control floodsthat for generations had inundated the vast plains of thecountry along the Yellow River. Pingr´e (1783) adoptedthat the reign of Yu began in − − − − − − which would mean that the datelisted by Pingr´e is too early by some 150 yr. Moreover,Pang & Yau (1996) argue that a statement in the
BambooAnnals that during Yu’s reign “ . . . the Sun disappearedby day and reappeared at night . . . ” is a reference to a“double sunset” eclipse of the Sun on − − − − − − − − − This date for the beginning of the Xia Dynasty was fixed by the
Xia-Shang-Zhou Chronology Project , a multi-disciplinary projectcommisioned by the People’s Republic of China in 1996 to deter-mine with accuracy the location and time frame of the Xia, Shang,and Zhou Dynasties. This Project provides dates that are generallylater than given by the numerous variations of the traditional histo-riography, based mainly on the work by Sima Qian (died − https://en.wikipedia.org/wiki/Xia Shang Zhou Chronology Project. To set a frame for dating records of comets, Pingr´e often refersto major historical events (e.g., the death of emperors), for whosetiming he usually relies on information from secondary sources,such as the works of the French Jesuits A. Gaubil (1689-1759) orJ.-A.-M. de Moyriac de Mailla (1669-1748), or the Belgian JesuitP. Couplet (1623-1693), all of whom were missionaries to China. PREDICTED MOTION OF C/1995 O1 OVER SEVERALFUTURE REVOLUTIONS ABOUT THE SUN
The constraint that C/1995 O1 had a close encounterwith Jupiter and was a dynamically new comet in theprevious return to the Sun fixes the comet’s motion overmore than one full revolution about the Sun. With thiswide enough base as an initial condition, it is plausibleto investigate the comet’s expected orbital evolution overa limited number of future returns to perihelion. Thislimit is determined by the rate of error propagation inthe orbital elements, which is in this case a function ofthe semimajor axis of the comet’s pre-encounter, originalbarycentric orbit. By prescribing an (unknown) error tothe value of (1 /a ) orig , we can estimate the propagatederrors in the elements, which, as shown below, increasein general with time (or with the number of returns)exponentially. The most sensitive element is by far theperihelion time, to which we pay particular attention.For the purpose of determining the rate of error prop-agation, we began the orbit integration forward in timefrom the osculating orbit in 1997 listed in Table 10, whichwas assumed to be independent of the error in (1 /a ) orig .[It should be the comet’s post-encounter orbit in − /a ) orig differed from the adoptedvalue of +0.000046 AU − (Section 6.2).] The results ofthe computations are shown in Figure 8, which suggests PROPAGATION OF ERROR INPERIHELION TIME WITHNUMBER OF RETURNS ǫ ( t π ) = 4 . × − e . N ret (1 − . N ret ) ❏❏❏❏❪ RETURN TO (YEAR OF)
PERIHELION, N ret ( N ret = 1 FOR RETURN IN 1997) − − − ǫ ( t π ) (days) ❤ ❤ ❤ ❤ ❤ ···························································································································································································································································································································································································································· ····················································································································· Figure 8.
Propagation of the error in the perihelion time, equiva-lent to an error of 0.000001 AU − , as a function of time, expressedin terms of a number returns to perihelion, N ret , since − Table 14
Predicted Long-Term Orbital Evolution of Comet C/1995 O1 with Weight System II (Equinox J2000)Orbital/Encounter Parameter Initial Arrival 1st Return 2nd Return 3rd Return 4th ReturnEpoch of osculation (TT) − − ◦ .4081 130 ◦ .5878 131 ◦ .0061 130 ◦ .8987 130 ◦ .8571Longitude of ascending node 282 ◦ .3687 282 ◦ .4698 282 ◦ .4963 282 ◦ .5688 282 ◦ .7443Orbit inclination 89 ◦ .0539 89 ◦ .4289 90 ◦ .4537 90 ◦ .2601 90 ◦ .3011Perihelion distance (AU) 0.907234 0.914168 0.915306 0.919657 0.918314Orbit eccentricity 0.996727 0.995092 0.991429 0.992210 0.992211Orbital period (yr) 4615 2542 1104 1283 1280Longitude of perihelion 101 ◦ .26 101 ◦ .80 103 ◦ .02 102 ◦ .87 103 ◦ .09Latitude of perihelion +49 ◦ .58 +49 ◦ .41 +48 ◦ .99 +49 ◦ .10 +49 ◦ .14Encounter Parameters:Time of perijove (TT) − a b Jovicentric distance at perijove (AU) 0.00513 0.771 0.428 4.54 3.85Jovicentric velocity at perijove (km s − ) 28.75 23.90 23.04 14.54 36.90 Notes. a There was another distant approach to Jupiter, to 4.84 AU, shortly after perihelion. b This was a post-perihelion approach; there was no pre-perihelion approach to Jupiter within 9 AU. that an error of 0.000001 AU − transforms to a negligi-bly small error of 0.05 second in the perihelion time atthe next return, to which we assign N ret = 2. However,at the following return ( N ret = 3), the propagated erroralready increases to 3.4 minutes and at the subsequentreturns to, respectively, 0.21 day ( N ret = 4), 14.4 days( N ret = 5), and 53.1 days ( N ret = 6). We terminated thecomputations at this point, because given that a realisticuncertainty of (1 /a ) orig of C/1995 O1, even if it was a dy-namically new comet, should be at least 0.000010 AU − and perhaps still greater, the corresponding uncertaintyof the perihelion time at the subsequent returns couldbecome comparable with Jupiter’s orbital period, ren-dering the comet’s encounters with the planet (and theresulting orbital transformations) unpredictable.The complete sets of orbital elements with Weight Sys-tem II, including the initial arrival and the current return(both from Table 10) are presented in Table 14, in whichthe future returns are terminated in the 7th millennium.To our surprise, the table shows that as long as the cometexperienced the very close approach to Jupiter in − three consecutive apparitions: a moderate approach in1996 will be followed by a yet another preperihelion en-counter at the next return, in the year 4392!For the scenario based on Weight System I, the resultsare rather similar for the expected return in 4393, theperihelion time taking place about 24 days earlier and theJovicentric distance at perijove amounting to 0.58 AU,but the predictions by the two scenarios of the next peri-helion time, in the 6th millennium, already differ by morethan 200 yr. Yet, the fundamental features of the orbitalevolution, especially the triple Jovian encounter at threeconsecutive returns, are common to both scenarios andthus independent of the weight system for the criticalobservations.In a recent paper (Sekanina & Kracht 2016), we calledattention to the fact that Jovian perturbations exertedon a comet in the course of recurring close or moderateencounters, especially when they take place at severalconsecutive returns (in a scenario that we described as a high-order orbital-cascade resonance), can provoke rapidinward drifting of the comet’s aphelion. In magnitude,these effects rival those triggered by a single extremelyclose encounter. When we wrote that 2016 paper, we didnot expect that C/1995 O1 — then the next object of ourinterest — would provide us with such a nice exampleof the process of orbital-cascade resonance, ending upwith an aphelion distance of ∼
200 AU after only threerevolutions about the Sun.Table 15 presents the details of this remarkable orbitalevolution: from the perihelion times in column 2 we de-termined the anomalistic orbital periods in column 3,which between − /a , of the semimajor axis, while the lastcolumn lists the differences between the neighboring val-ues of 1 /a , that is, the perturbations integrated over onerevolution about the Sun. The table shows that in merelythree revolutions about the Sun, by 5451, the comet willreduce its orbital period from some 3 million yr to onlyslightly more than 1000 yr. An equally surprising result Table 15
Rapid Inward Drifting of Aphelion of C/1995 O1 Driven byHigh-Order Orbital Resonance (Weight System II)Time of Anom. High- Reciprocal IntegratedPerihelion Orbital Order Semimajor Pertur-Re- Passage Period Reso- Devi- Axis bationturn (TT) (yr) nance ation (AU − ) (AU − ). . . (Oort Cloud) (3.2 × ) . . .. . . . . . . . . +0.0000460 -2250 Dec 8 +0.0037684246.313 1:358 − − − omet C/1995 O1: Orbit, Jovian Encounter, and Activity 19is that the peak integrated perturbation is not the oneassociated with the close approach of 0.005 AU to theplanet in − /a : between − − rev − ,being very systematic and always positive; whereas dur-ing the three revolutions after 5451 it equals, in absolutevalue, only 0.00046 AU − rev − and is random. CORRELATION AMONG NONGRAVITATIONAL LAW,ORBITAL HISTORY, AND ACTIVITYOF COMET C/1995 O1
In Section 6.4 we expressed our preference for the or-bital solution in Table 10 (Weight System II) over thesolution from Table 5 (Weight System I) solely on thegrounds of orbital quality (the distribution of residuals)and the number of accommodated observations. We no-ticed that by far the most striking distinction betweenthe parameters of the two solutions was the nongravita-tional law. The solution from Table 5 required a scalingdistance of r = 3 .
07 AU, implying the prevalence of wa-ter ice in the effects of the nongravitational acceleration,and, unexpectedly, the dominance of the acceleration’snormal component over the radial component. On theother hand, the preferred solution from Table 10 is inline with a scaling distance of more than 15 AU, sug-gesting that the contribution to the detected nongravi-tational acceleration by ices much more volatile than wa-ter ice was important. In addition, the radial componentof the nongravitational acceleration exceeded the normalcomponent, even though by a factor of ∼ Nongravitational Law and Major Contributors tothe Comet’s Post-Perihelion Activity
Our aim is now twofold: (i) to compare the two appliedversions of the nongravitational law (Weight Systems Iand II) with the sublimation curves of common ices re-leased from the nucleus and (ii) to examine the correla-tions between the laws and the observed production ratesof these species. To proceed with the first task, we de-scribe the sublimation curves of three major compounds:water ice, carbon dioxide, and carbon monoxide in theorder of increasing volatility.The isothermal approximation to the nongravitational(or sublimation) law for water ice, extensively tested onthe orbital motions of a large number of comets, is takenfrom Marsden et al.’s (1973) standard Style II nongrav-itational model, as already mentioned in Section 4. Forcarbon dioxide the critical data on the saturated vaporpressure and the latent heat of sublimation were takenfrom Azreg-A¨ınou (2005), whereas for carbon monoxidefrom an extensive compilation by Wylie (1958). The parameters of an empirical fit, of the type ex-pressed by Equation (1), to the normalized sublimation(or momentum-transfer) rates of the three species arelisted in Table 16, together with the latent heat of sub-limation and the ice temperature and sublimation rateat 1 AU from the Sun. These rates, per cm per s, are,respectively for carbon monoxide and carbon dioxide, 8.5times and 2.3 times greater than for water ice. Masswisethe ratios are still higher: 13.2 for carbon monoxide and5.6 for carbon dioxide. An approximation to the scal-ing distance given by Equation (3) provides the valuesof 114.5 AU for carbon monoxide and 8.9 AU for car-bon dioxide, in reasonable agreement with the tabulatednumbers obtained by fitting the sublimation rates de-rived directly from the relations for vapor pressure as afunction of temperature.To proceed with the second task, we next collected thedata on the production rates of water, carbon dioxide,and carbon monoxide from the nucleus of C/1995 O1 af-ter perihelion. For water ice the production rates weredetermined by Dello Ruso et al. (2000) directly from theground-based high-resolution 2–5 µ m infrared spectro-scopic observations and by Crovisier et al. (1999) fromobservations with two instruments on board the InfraredSpace Observatory (ISO); and, furthermore, by Combi etal. (2000) from the images of the hydrogen Lyman-alphacoma taken with the SWAN all-sky camera on board theSOHO spacecraft; by Weaver et al. (1999b) via OH pro-duction rates from ultraviolet observations with the Hub-ble Space Telescope (HST); by Stern et al. (1999) andHarris et al. (2002), both groups using in principle thesame technique to analyze their observations taken, re-spectively, with a mid-UV/visible imager on board theSpace Shuttle and with a wide-field ground-based instru-ment; and by Biver et al. (1999, 2002) and Colom et al.(1999), who employed a radio telescope to obtain mea-surements of the 18-cm emission of OH.Since the radiation from carbon dioxide cannot be ob-served from the ground (e.g., Crovisier 1999), the onlyexisting post-perihelion data on its production rate werederived from the ISO observations of the 4.25-micronband (Crovisier et al. 1999).
Table 16
Comparison of Isothermal Approximations to Sublimation Lawsfor Carbon Monoxide, Carbon Dioxide, and Water Ice a Parameter of Carbon Carbon WaterSublimation Law Monoxide Dioxide IceScaling distance r (AU) 107.60 10.10 2.808Exponent m n nk − . − . − . Latent heat of sublimation(cal mol − ) 1785 6400 b
11 400Temperature of ice at 1 AUfrom Sun (K) 39.4 111.1 194.7Sublimation rate at 1 AUfrom Sun (cm − s − ) 10 . . . Notes. a For assumed Bond albedo of 0.04 and unit emissivity. b Average over temperature range of 60 K to 160 K. − Minor Planet Center (see footnote 1 in Section2) by the observers from the selected sites; for the pe-riod from late 2007 on from the magnitudes published bySzab´o et al. (2008, 2011, 2012). Each resulting magni-tude ( H ∆ ) corr was obtained by normalizing the observedmagnitude to a unit geocentric distance, by subtracting Table 17
Normalized Magnitudes for Dust Coma of C/1995 O1 BetweenMid-1999 and 2011 ( r > . a orTime (UT) Sun (AU) Angle ( H ∆ ) corr Reference1999 June 28.3 8.750 6 ◦ .7 7.83 467Nov 10.5 9.763 5.8 8.39 47412.7 9.778 5.7 8.18 4742000 Mar 2.5 10.573 5.3 8.78 4747.4 10.607 5.3 8.74 474July 30.5 11.605 5.0 8.66 474Nov 23.6 12.376 4.5 9.19 428Dec 22.5 12.564 4.4 8.84 4282001 July 22.8 13.907 4.2 11.64 474Dec 17.5 14.806 3.7 10.10 4282002 Jan 3.6 14.908 3.6 11.31 413May 8.4 15.645 3.7 11.55 422Oct 30.5 16.651 3.4 12.00 4222003 Jan 27.1 17.148 3.2 12.94 809Dec 25.5 18.955 2.8 13.28 4222005 Jan 8.1 20.918 2.5 13.58 3042007 Oct 20.7 25.766 2.2 13.51 41321.2 25.769 2.2 13.73 Szab´o b >
19 Szab´o b , c >
17 Szab´o b >
18 Szab´o b Notes. a See . b Reference: Szab´o et al. (2012). c In Table 1 of Szab´o et al. (2012), the date 2009 Sep 08 for this entryshould read 2009 Mar 08. the contribution from the nucleus, and by referring theresidual brightness to a zero phase angle using the “com-pound” Henyey-Greenstein law, as modified by Marcus(2007). Table 17 indicates that in 2007 the comet, at25.8 AU from the Sun, was definitely active (Szab´o et al.2008), but by 2009, at 28 AU from the Sun, the nucleusbecame inert (Szab´o et al. 2012). This development isalso supported by the Spitzer Space Telescope thermal-infrared observations made in 2005 and 2008 by Krameret al. (2014). As the comet’s heliocentric distance be-tween the two dates increased from 21.6 AU to 27.2 AU,the flux dropped much more steeply than required byan inverse square power law, with the dust in the comahaving been substantially depleted over the period of thethree years.The collected data on the comet’s post-perihelion ac-tivity in Figure 9 show a number of important features.Comparison of the two modified nongravitational lawswith the sublimation curves for water ice, carbon diox-ide, and carbon monoxide — all normalized to 1 AU— suggests that, the law with Weight System I is, asexpected, very close to the water-ice sublimation curve,while the law with Weight System II extends to largerheliocentric distances than the carbon-dioxide sublima-tion curve. Thus, no mix of H O and CO could explainthe law with Weight System II; a contribution from COis necessary.The water-production rates are consistent with the the-oretical water-ice sublimation curve up to about 2.5 AUfrom the Sun. Beyond that point only two positive obser-vations exist, which suggest that the rate decrease pro-ceeded more slowly than predicted by the simple theory.Only two measurements exist in the post-perihelion pe-riod of time for the production rate of carbon dioxide(obtained with the ISO); they are fairly compatible withan expected ∼ r − sublimation curve between 3.5 AU and5 AU from the Sun. At 3.89 AU, the production rates ofH O and CO were comparable, about 3 × s − , butmasswise the rate of CO was more than twice greater.The production rate of carbon monoxide follows theapproximately inverse-square power law from perihelionall the way to 14 AU. Among more than 40 collected mea-surements only one is strongly out of line; it is the second(and last) post-perihelion measurement obtained withthe ISO on 1998 April 6 at 4.9 AU from the Sun. In Fig-ure 9 this anomalous rate exceeds the level of the nucleusoutgassing rate from 1998 March 18–19 (at 4.7 AU fromthe Sun), as determined by Gunnarsson et al. (2003), bya factor of 8. Crovisier et al. (1999) speculated on a pos-sible outburst, but related that no simultaneous flare-upwas reported in the light curve. We may add that noexcessive production of CO is seen in the plot to haveaccompanied the CO event either.The normalized magnitudes ( H ∆ ) corr of the dust comaare drawn to overlap the CO production rates in Fig-ure 9 in the range of heliocentric distances from 8.7 AUthrough ∼
14 AU. A satisfactory correlation between bothappears from the plot, suggesting that ( H ∆ ) corr = 10 isequivalent to Q CO ≃ . × s − , even though a cor-respondence between magnitudes and any other activityindex is only approximate because of the dust ejecta’sfinite residence time in the coma. The magnitudes beginto progressively deviate from the inverse-square poweromet C/1995 O1: Orbit, Jovian Encounter, and Activity 21 SUBLIMATION (NONGRAVITATIONAL) LAWS ANDPOST-PERIHELION ACTIVITY OF C/1995 O1 mod (WS II) ✟✟✟✟✙ mod (WS I) H O CO CO r = 20 AU ✡✡✡✢ ♠ H O rate ✷ CO rate ② CO rate ✐✉ mag (dust) ∗ thermal ( H ∆ ) corr ✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏ ✒✑✓✏ ❄ ✒✑✓✏ ✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏ ❄ Q H O (s- ) ✷ ✷ ⑥ ⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥ ⑥ HELIOCENTRIC DISTANCE, r (AU)10 − − − − − g x ( r ) Q CO ,Q CO (s- ) tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttsssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss ssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq⑥⑥ ⑥ ⑥ ⑥ ⑥⑥ ⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥⑥♠♠♠♠♠♠♠♠ ♠♠♠♠♠♠♠ ♠♠① ①①①① ①①① ①①①①① ① ① ①① ∗ ∗ rrrrrrrrrrr ✲ Cometinert rrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr
Figure 9.
Post-perihelion activity of C/1995 O1 and comparison of the isothermal approximations to the nongravitational and sublimationlaws, g mod ( r ; r ). The preferred nongravitational law has a scaling distance r = 15 .
36 AU (Weight System II or WS II), the other has r = 3 .
07 AU (Weight System I or WS I). Also shown are the sublimation laws for water ice, carbon dioxide, and carbon monoxide Inaddition, we plot the observed data collected from various sources (see the text): the production rates of water Q H O (large open circles;including two 3 σ upper limits — circles with arrows), carbon dioxide Q CO (squares), and carbon monoxide Q CO (solid circles; theanomalously high rate is an ISO data point, Crovisier et al. 1999); the total magnitudes of the dust coma at heliocentric distances greaterthan 8 AU (the CCD data, obtained by a selected set of observers and extracted from the Minor Planet Center ’s Observations Database,corrected for the contribution from the nucleus, and normalized to a unit geocentric distance and zero phase, ( H ∆ ) corr ; circled dots); and,in relative units, the dust coma’s thermal flux between 21.6 AU and 27.2 AU (connected asterisks), based on the Spitzer results by Krameret al. (2014). A crude fit to the dust magnitude data far from the Sun by a g mod ( r ; r ) law is depicted by a dashed line to show that theyare approximated by r ≈
20 AU. Szab´o et al. (2012) argue that the comet became inactive beyond a heliocentric distance of ∼
28 AU. law downward at about the time of termination of theCO production-rate data; the magnitudes follow, veryapproximately, a law similar to that with Weight Sys-tem II, but with a greater scaling distance r , estimatedat ∼
20 AU. The last three data points from Table 17 arenot plotted in Figure 9; they would be located way belowthe plot’s foot line, referring, in conformity with Szab´oet al.’s (2012) conclusion, to a vanishing dust coma.Two important inferences on the post-perihelion activ-ity of C/1995 O1 are (i) the production of carbon monox-ide did not apparently follow an inverse-square powerlaw beyond ∼
15 AU from the Sun, as it should (up to ∼
100 AU) if free CO were available on the nucleus sur-face, but a steep drop in the dust-coma brightness set inat ∼
15 AU and no dust could be detected in the comafrom 28 AU on; and (ii) the dynamically determined non- gravitational law of Weight System II approximates thecomet’s post-perihelion activity curve fairly successfullyup to almost 20 AU from the Sun, but to achieve thebest possible correspondence, the law’s scaling distanceneeds to be increased by some 30%.A peculiar post-perihelion evolution of the inner coma’sbrightness, observed extensively by Liller (2001), will beaddressed in Part II of this investigation. At this pointwe are interested in a potential perihelion asymmetry ofthe comet’s activity, for which purpose we now describethe developments along the incoming branch of the orbit.
The Comet’s Preperihelion Activity
The procedure we now followed was the same as for thepost-perihelion activity in Section 9.1. The data on thewater production rate were assembled, as before, from2 Sekanina & Kracht
Table 18
Pre-Discovery and Averaged Normalized Magnitudes for Dust Coma of C/1995 O1 Between Discovery and1996 March 20 ( r > H ∆ ) corr Type a of Observations Observer(s)1991 Sept 1 b ◦ .1 > c Sept 1 (1, phot) McNaught (?)1993 Apr 27.8 d d e Notes. a No color corrections applied. b Comet not found; estimate for the comet’s total brightness depends on the plate’s adopted limit of mag 21. c Conservative upper limit; the dust coma may have been fainter than mag 16 or 17, or absent altogether. d Pre-discovery observation. e Includes the discovery observation. the works by Dello Russo et al. (2000), Crovisier et al.(1999), Combi et al. (2000), Weaver et al. (1997, 1999b),Harris et al. (2002), Biver et al. (1997, 1999), and Colomet al. (1999); in addition, we used Weaver et al.’s (1999a)result derived from a ground-based observation of the4.65- µ m band.For carbon dioxide, the preperihelion list has six en-tries, two ISO measurements by Crovisier et al. (1999)and four HST upper limits by Weaver et al. (1999b). Onthe average, the CO production rate is about a factor oftwo or so lower than the CO production rate at the sameheliocentric distance (between 2.7 and 4.6 AU). Relativeto the production of water, the CO production is lowerby a factor of 4–5 at 2.9 AU, but they are comparable toone another at 4.6 AU.The preperihelion production of carbon monoxide wasmonitored at millimeter and submillimeter wavelengths,starting soon after discovery, by Biver et al. (1997, 1999).It was also measured by Jewitt et al. (1996) in 1995, byWomack et al. (1997) in 1995-1996, by Crovisier et al.(1999) in 1996, by DiSanti et al. (1999) in 1996–1997,and by Weaver et al. (1999a) in 1997.Normalized dust-coma magnitudes would provide lit-tle additional information on the comet’s preperihelionactivity, if there were no pre-discovery observations. Be-cause there were, these magnitudes do provide — theirlow accuracy notwithstanding — some fairly tight con-straints. The 1993 observation with the UK Schmidt(McNaught 1995), made by C. P. Cass and already re-ferred to, shows, when combined with the discovery ob-servation, that the rate of brightening between 13 AUand 7 AU was much steeper than r − . The activityis further constrained by the failed effort by McNaught (1995) to detect the comet on a plate obtained on 1991September 1. Assuming with McNaught that the lim-iting magnitude of the UK Schmidt plate was 21, thenondetection of the comet in 1991 may indicate that itsnucleus was nearly or completely inactive, because Szab´oet al.’s (2012) results imply that the nucleus should havehad a visual magnitude of about 22 at the time of the1991 observation, equivalent to a normalized magnitudeof about 16. The upper limit on the dust-coma magni-tude in Table 18 is therefore very conservative. On theother hand, the estimated normalized magnitude of thenucleus at the time of the 1993 detection is about 15,so that the dust coma was then much brighter than thenucleus. Even McNaught’s (1995) estimate of 19 for theso-called nuclear magnitude was at least 1.5 mag brighterthan the bare nucleus.In response to the request by the Central Bureau forAstronomical Telegrams for fortuitous pre-discovery ob-servations of the comet (Marsden 1995), another mes-sage — besides that by McNaught — was sent by George(1995), who reported an apparent image on a photographtaken with an 8.5-cm f/1.7 lens on a Kodak Royal Gold400 film by T. Dickinson on 1995 May 29.40 UT, whenthe comet was of mag 11.7. Although this observationwas made only some 8 weeks before discovery, it suggeststhat the comet was still brightening very rapidly.Once the comet was discovered, the number of visualobservations of its total brightness became overwhelm-ing, and in Table 18 we list averages of estimates by fourexperienced observers, all of whom used reflectors withapertures between 31 cm and 41 cm: J. Bortle, A. Hale,K. Hornoch, and A. F. Jones. To allow a sufficient overlapin heliocentric distance with the observations of carbonomet C/1995 O1: Orbit, Jovian Encounter, and Activity 23
SUBLIMATION (NONGRAVITATIONAL) LAWSAND PREPERIHELION ACTIVITYOF C/1995 O1 mod (WS II) ✡✡✢ mod (WS I) H O CO CO r = 12 AU ✟✟✟✟✯ ♠ H O rate ✷ CO rate ② CO rate ✐✉ mag (dust) ( H ∆ ) corr ✒✑✓✏✒✑✓✏✒✑✓✏ ✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏ ✒✑✓✏ ❄ ✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏ ✒✑✓✏✒✑✓✏ ✒✑✓✏ ❄ ✒✑✓✏ ❄ ✒✑✓✏ ❄ ✒✑✓✏ ❄ ✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏✒✑✓✏ Q H O (s- ) ✷✷ ✷ ❄ ✷ ❄ ✷ ❄ ✷ ❄ ⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥⑥ ⑥⑥ ⑥⑥ ⑥⑥⑥⑥⑥⑥⑥⑥ ⑥ ⑥⑥⑥⑥⑥⑥ ⑥ ⑥ ⑥ ⑥⑥ HELIOCENTRIC DISTANCE, r (AU)10 − − − − − g x ( r ) Q CO ,Q CO (s- ) tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq ♠①♠①♠①♠①♠①♠①♠①♠①♠①♠① ♠①♠①♠①♠① ♠① ♠① ♠① ❄ rrrrrrrr rrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr Figure 10.
Preperihelion activity of C/1995 O1 and comparison of the isothermal approximations to the nongravitational and sublimationlaws, g mod ( r ; r ), which are normalized the same way as in the plot of post-perihelion activity in Figure 9. Similarly, the scales for theobserved production rates of water, carbon monoxide, and carbon dioxide have not been changed. Note the increased scatter in the COdata below 3 AU from the Sun. The pre-discovery dust magnitude from 1993 is fitted by a g mod ( r ; r ) law with r ≈
12 AU. Comparisonwith Figure 9 suggests a striking asymmetry relative to perihelion. For a description of the symbols used, see the caption to Figure 9. monoxide, we covered a wide enough period of time fromdiscovery to 1996 March, during which the comet’s helio-centric distance dropped from 7.1 AU to 5 AU. These vi-sual brightness estimates are compared in Table 18 witha series of CCD magnitudes by A. Nakamura, who useda 60-cm f/6.0 Ritchey-Chr´etien reflector. Both the vi-sual and CCD data were taken from several issues of the
International Comet Quarterly (Green 1995, 1996).The collected information on the preperihelion activityof C/1995 O1 is presented in Figure 10. Although thereare similarities between this plot and Figure 9 in gen-eral, differences are readily perceived in detail. The wa-ter production rates are higher than the post-perihelionones farther from the Sun, but slightly lower near perihe-lion, so that the production of H O increases less steeplywith decreasing heliocentric distance before perihelionthan it decreases with increasing distance on the way out.The preperihelion variations in the water production ratecan be fitted with a modified nongravitational law (seethe beginning of Section 6) with a scaling distance of r ≃ r = 5 . ∼ by Prialnik (1997), andby Enzian et al. (1998).The normalized preperihelion magnitudes ( H ∆ ) corr arelinked with the CO production curve by assuming mag10 to be equivalent to 1 . × s − , which differs slightlyfrom the post-perihelion equivalent, apparently becauseof a somewhat different dust-to-CO mass ratio in thecoma (Section 9.5). When comparing Figures 9 and 10,the most striking feature is a major perihelion asymmetry at heliocentric distances larger than ∼ g mod ( r ; r ) law with r ≈
20 AU, the 1993pre-discovery magnitude and the 1991 upper limit areboth consistent with r ≈
12 AU. The dynamically de-termined scaling distance of ∼ correlation between thenongravitational acceleration exerted on the comet’snucleus and its observed activity . Nongravitational Laws IncorporatingPerihelion Asymmetry
The apparent detection of perihelion asymmetry in theactivity of C/1995 O1 leads up to the issue of an asym-metric nongravitational law, which, unfortunately, theemployed software package does not allow us to apply atpresent. This tool has over the past several decades beendebated in the literature fairly extensively, even thoughthe modern, sophisticated methods of accounting fora nongravitational acceleration, by incorporating it di-rectly in the equations of motion, focused from the begin-ning on the symmetrical models (Marsden 1969). Testedon 2P/Encke’s comet, the early orbital solutions basedon an asymmetric law failed (Marsden 1970) for reasonsthat have not been fully understood. Similarly, employ-ing an empirical nongravitational law that was consis-tent with 1P/Halley’s asymmetric water production-ratecurve, Yeomans (1984) was unable to improve upon theresults obtained with a symmetrical law. Rickman &Froeschl´e (1983) [see also Froeschl´e & Rickman (1986)and Rickmam & Froeschl´e (1986)] calculated the ex-pected nongravitational parameters for Halley’s cometfrom their thermal model of comets, but Landgraf (1986),applying them in his orbital solutions, concluded that, inspite of an improvement, a further refinement was desir-able. Subsequently, Sitarski (1990) developed a hybridmodel, employing the symmetrical nongravitational lawby Marsden et al. (1973), to derive precessional parame-ters — inherently requiring perihelion asymmetry — forshort-period comets.Returning to the problem of 2P/Encke and motivatedby a perception of major seasonal effects and nonran-dom distribution of active regions on the nucleus surfaceas well as by the observed statistical correlation betweenthe perihelion asymmetry of the light curves of comets An updated fit to the observations of C/1995 O1 by this modelappears in Capria et al. (2002). and the sense of the nongravitational effect, Sekanina(1988b) suggested that, at the risk of a potentially sig-nificant loss of generality, the form of an asymmetric lawcan be prescribed by applying the standard symmetriallaw (Marsden et al. 1973), g ice ( r ; r ) [see Equation (1)],in which the heliocentric distance r is taken not at thetime of observation, t , but at a time t − τ , hence r ( t − τ ),where τ is a constant “offset” time. The nongravitationalacceleration then peaks at a time t π + τ rather than atthe time of perihelion, t π ; that is, before perihelion if τ <
0, or after perihelion if τ > τ is to be determined by optimizing an orbitalsolution that fits the astrometric observations of an ex-amined comet. Yeomans & Chodas (1989) employed thisparadigm to develop an orbit-determination procedurethat allows one to establish the optimum offset time τ asthe value used in a solution with a minimum rms residual.Applying this approach to eight short-period comets (in-cluding 1P/Halley but not 2P/Encke), they found thatin a number of orbital runs this new law improved thefit to astrometric observations and indicated that the ra-dial, rather than transverse, component of the nongravi-tational acceleration accounted for practically the wholeeffect (the normal component having been ignored); inmost cases, the peak acceleration was offset from perihe-lion a few weeks in one direction or the other, both themagnitude and the sign of the offset time correlating withthose of the offset of the outgassing peak (usually repre-sented by the light curve) from perihelion.Sitarski (1994a, 1994b) went even further in his appli-cation of this type of asymmetric law. He started fromhis hybrid model, improving it markedly by incorporat-ing the offset time τ and its rate of variation directlyinto the least-squares procedure, which included all threecomponents of the nongravitational acceleration. Undercertain assumptions, he succeeded in fitting — in a sin-gle run — the orbital motion of comet 22/P Kopff over13 returns to perihelion between 1906 and 1990 with anrms residual of ± ′′ .56; and by rigorous extrapolation insatisfying, to better than ± ′′ , the discovery positions ofthe comet at its next apparition.While an asymmetric nongravitational law might im-prove a fit to the orbital motion of C/1995 O1 over the so-lution with Weight System II, the paradigm that is cus-tomized for applications to short-period comets shouldbe modified. A constant offset time τ , which — basedon Combi et al.’s (2000) extensive observations of near-perihelion water production rates — should equal about+19 days for this comet, could not explain the strik-ing difference between the heliocentric distances of thevery steep preperihelion increase of activity in Figure 10(from 13 AU down to 8 AU) and its equally steep post-perihelion drop in Figure 9 (from 12 AU up to 28 AU). Interms of the scaling distance r , the difference betweenthe post-perihelion and preperihelion activity curves is20 −
12 = 8 AU, whereas a time difference of 19 days isequivalent to merely ∼ g mod = 0 . r pre = 12 .
56 AU and r post = 19 .
37 AU and the equivalent times from perihelion t pre − t π = − t post − t π = +2538 days, so inomet C/1995 O1: Orbit, Jovian Encounter, and Activity 25absolute value their difference is 1178 days. For the g mod law to peak at t π +19 days and at the same time satisfy g mod (12 .
56 AU; 12 AU) = g mod (19 .
37 AU; 20 AU), it mustapply r ( t pre − τ ∗ ) = r ( t post − τ ∗ ) = r ( t ∗ ) and, accordingly, τ ∗ = 589 days, t ∗ = t π ∓ r ∗ = 16 .
14 AU.In order that g mod ( r ∗ ; r ) = 0 . r ought to equal r = ( − r n + m/k ∗ [ g mod ( r ∗ ; r )] /k r m/k ∗ [ g mod ( r ∗ ; r )] /k − ) /n = 16 .
10 AU . (8)We note that the scaling distance of the law for WeightSystem II differs from this value by only 0.74 AU.The offset times at peak outgassing and at ∼
16 AUfrom the Sun represent two critical check points of theasymmetric law g mod ( r ; 16 .
10 AU). Since the offset timeat ∼
16 AU is considerably greater — by a factor of 31 —than the offset time near perihelion, the law g mod ( r ; τ ; r )with a constant offset time τ in the argument of helio-centric distance, r ( t − τ ), could not obviously fit the ob-served perihelion asymmetry of C/1995 O1. It is legiti-mate to seek a generalized form of τ , called now τ ⋆ , asa product of the offset time at peak outgassing, τ , anda function h [ r ⋆ ] of heliocentric distance r ⋆ = r ( t − tau ⋆ ).This function should be increasing with r ⋆ , rather than r ( dh/dr ⋆ > h = 1, at thetime of peak outgassing, t peak [when t peak − τ = t π and r ⋆ = r ( t peak − τ ) = q ]: r ( t − τ ⋆ ) = r ⋆ = r ( t − τ h [ r ⋆ ]) . (9)We consider two general forms of h [ r ⋆ ]: one is a powerfunction, the other an exponential. In either case, it isnecessary to make sure that h [ r ] does not overcorrectthe asymmetry near aphelion. The argument of r shouldthen read either as r ⋆ = r ( t − min { τ [ r ⋆ /q ] ν , τ max } ) , (10)or as r ⋆ = r ( t − min { τ exp[ µ ( r ⋆ − q ) κ ] , τ max } ) , (11)where ν , µ , κ , and τ max are constants. In practice, ei-ther equation is readily solved by successive, rapidly con-verging iterations, starting with r ⋆ = r in the expressions( r ⋆ /q ) ν or exp[ µ ( r ⋆ − q ) κ ]. The two nongravitational lawsshould be referred to as, respectively, g mod ( r ; τ, ν ; r ) and g mod ( r ; τ, µ, κ ; r ), but when describing them, as well as g mod ( r ; τ ; r ), summarily, we use the same notation asfor the symmetrical law, g mod ( r ; r ). We note that anyof the asymmetric laws reduce to the symmetrical lawwhen τ = 0, regardless of the values of ν , µ , or κ . Simi-larly, the variable offset-time laws reduce to the constantoffset-time law when ν = 0 or µ = 0.For some comets, a better fit should be obtained withexpression (10), for others with expression (11). Themain difference between the functions (10) and (11) isthat the former allows the asymmetry to increase moregradually. The exponential, on the other hand, fits thecases where the perihelion asymmetry increases only in-significantly at small to moderate heliocentric distances,but rather dramatically farther from the Sun. As a prac-tical procedure, we suggest that the overall trend in theperihelion asymmetry of the gas production law betweenthe two end points be employed to choose the type of the function h [ r ⋆ ], whereas the magnitudes of the offset timeat peak outgassing and far from the Sun be used to fixthe function’s constants.Insight into the issue of selecting the function h [ r ⋆ ]is provided by determining the dependence of thedifference in log g mod ( r ; r ) between a preperihelion andpost-perihelion nongravitational effect (or a productionrate), ∆ asym log g mod ( r ; r ), on the offset time τ ⋆ as afunction of heliocentric distance. Differentiating the re-lation between the time from perihelion and heliocentricdistance for parabolic motion (thus obtaining a tightupper limit on the derivative dr/dt ) and equating thedifferential dt with τ ⋆ , we find∆ asym log g mod ( r ; r ) = 0 . √ r ⋆ − qr ⋆ (cid:18) m + nk Λ n⋆ n⋆ (cid:19) τ ⋆ , (12)where r ⋆ and q are in AU, τ ⋆ in days, and Λ ⋆ = r ⋆ /r .The difference ∆ asym log g mod ( r ; r ) has the same sign as τ ⋆ ; when positive, g mod ( r ; r ) is greater after perihelion,and vice versa.Comparing Figures 9 and 10 we note that the primaryfeatures of the perihelion asymmetry of C/1995 O1’s ac-tivity could be fitted much better by the offset timesthat follow an exponential law rather than a power law.The asymmetry is distinctly apparent in close proxim-ity of perihelion (particularly in the variations of theproduction rate of water) than it is between 3 AU and7 AU from the Sun (in the CO production). At stilllarger distances, the differences between preperihelionand post-perihelion rates of outgassing increase dramati-cally. When κ = 1 in Equation (11), the two check pointsrequire that µ = 0 . − , so that τ ⋆ = 75 days at r = 7 AU. When κ = 2, then one gets µ = 0 . − and τ ⋆ = 32 . r = 7 AU. And when κ = 3, it fol-lows that µ = 0 . − and τ ⋆ = 23 . ν = 1 .
20, so that τ ⋆ = 219 days at r = 7 AU.Inserting each of these values of τ ⋆ into Equation (12),the perihelion asymmetry in the nongravitational law g mod ( r ; r ) at r = 7 AU becomes ∆ asym log g mod = 0 . κ = 3 is especially satisfactory, as the asymmetry mim-icks day-to-day variations in the production rate of car-bon monoxide, while the power-law case is rather unac-ceptable.The superior quality of the exponential law is also ap-parent from the following figures, in which three asym-metric nongravitational laws are compared with a sym-metrical one within 200 days of perihelion (Figure 11)and on another time scale, spanning 18 years (Figure 12).The constant-offset law displays nicely an asymmetrynear perihelion, but gradually less so farther from theSun; it grossly underrates the asymmetry, seen in Fig-ures 9 and 10, at very large heliocentric distances, whereit is virtually equivalent to the symmetrical law. Thepower law fits the large perihelion asymmetry very farfrom the Sun, but underrates the nongravitational ef-fect before perihelion and overrates it after perihelion ona time scale of months from perihelion. A remarkablequality of the exponential law is that it mimicks the be-havior of the constant-offset law near perihelion, but ofthe power law far from the Sun.6 Sekanina & Kracht SYMMETRICAL AND ASYMMETRICNONGRAVITATIONAL LAWSFOR C/1995 O1 (NEAR PERIHELION) g mod ( r ; r ) (SYMMETRICAL) ❅❅❅❅❅❅❘ g mod ( r ; τ, µ, κ ; r ) (ASYMMETRIC) ❅❅❅❅❅❅❘ g mod ( r ; τ, ν ; r ) (ASYMMETRIC) ❅❅❅❅❅❅❘ g mod ( r ; τ ; r ) (ASYMMETRIC) ❅❅❘ r =16.1AU –200 –150 –100 –50 0 +50 +100 +150 +200TIME FROM PERIHELION, t − t π (days) g mod ( r ; r ) rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr rrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrrr 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r r r r r r r r r r r r r r r r rrrrrrrrrrrrrrrrrrrrrrrrrr r rrrrrrrrrrrrrrrrrrrrrrrrrrr r r r r r r r r r r r r r r r r Figure 11.
Comparsion, within 200 days of perihelion, of fourmodified nongravitational laws, g mod ( r ; r ), for comet C/1995 O1,all of them having the same scaling distance of r = 16 .
10 AU:(i) g mod ( r ; r ), a symmetrical law with respect to perihelion;(ii) g mod ( r ; τ ; r ), an asymmetric, law with a constant offset timeof τ = +19 days; (iii) g mod ( r ; τ, ν ; r ), an asymmetric law withthe offset time following a power function τ ⋆ = τ ( r ⋆ /q ) ν , where τ = +19 days and ν = 1 .
20; and (iv) g mod ( r ; τ, µ, κ ; r ), an asym-metric law with the offset time following an exponential function τ ⋆ = τ exp[ µ ( r ⋆ − q ) κ ], where τ = +19 days, µ = 0 . − ,and κ = 3). Note that, within about 0.5 yr of perihelion, the curvepresenting the exponential law — unlike that for the power law— nearly coincides with the curve with a constant offset time, andthey both gradually approach the curve of the nongravitational lawsymmetrical with respect to perihelion. The only purpose of τ max is to prevent the offset time τ ⋆ from reaching unphysically large values far from the Sun.This is generally not a problem with power laws (unless ν is very high), but is critically important for exponen-tial laws. For example, for C/1995 O1 the exponentiallaws reach τ ⋆ = 100 yr(!) at r ⋆ = 34 . κ = 1,at r ⋆ = 23 . κ = 2, and at r ⋆ = 20 . κ = 3, that is, at distances only slightly to moderatelylarger than the scaling distance r . This means that atdistances greater than 20–30 AU the integration of thecomet’s motion is carried with a constant offset time. Ofcourse, the computed nongravitational effect 30 AU fromthe Sun reaches merely a ∼ − th part of the effect atperihelion, so truncation of offset times far from the Sunhas no influence on the accuracy of the computed orbit.Because of a potential for improving the fitting of themotions of comets, it is advisable that the perihelion-asymmetry option in the expression for the nongravita-tional law discussed above be in the future incorporatedin the orbit-determination software. Loss of Mass by Outgassing IntegratedOver the Orbit. Trace Molecules
In Sections 9.1 and 9.2 we described the major contrib-utors to the activity of C/1995 O1 after and before peri-helion, respectively. Our account, based on an extensivelist of referenced work, covered the production of water,carbon monoxide, and carbon dioxide. Before we com-plete the inventory of the volatile species, we note thatin-depth investigations of bright comets lead, virtuallyuniversally, to two major conclusions: (i) the production
SYMMETRICAL AND ASYMMETRICNONGRAVITATIONAL LAWS FOR C/1995 O1 (FROM ONSET OF ACTIVITYTO ITS TERMINATION) g mod ( r ; r ) (SYMMETRICAL) ✓✓✓✓✓✓✴ g mod ( r ; τ, µ, κ ; r ) (ASYMMETRIC) ✓✓✓✓✓✓✓✓✓✴ g mod ( r ; τ, ν ; r ) (ASYMMETRIC) ✓✓✓✓✓✓✴ g mod ( r ; τ ; r ) (ASYMMETRIC) ✓✓✓✓✓✓✓✴ r =16.1AU –6 –4 –2 0 +2 +4 +6 +8 +10 +12TIME FROM PERIHELION, t − t π (years) g mod ( r ; r ) − − − 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♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣rrr rr r r r r r r r r r r r r r r r r rrrrrrrr rrrrrr r r r r r r r r r r r r r r r r rr rrrrrrrrrrr r r r r Figure 12.
Comparison, from 6 yr preperihelion to 12 yr post-perihelion, of the four nongravitational laws plotted on a differentscale near perihelion in Figure 11. Note that only the exponentialand power laws fit the major perihelion asymmetry, whereby the ef-fects 6 yr before perihelion and 12 yr after perihelion are essentiallyequal. The offset times are limited to a maximum, τ max , of 3 yr. rate of water dominates and (ii) activity at large heliocen-tric distances is controled by carbon monoxide. Althoughseldom specified, the first of the two conclusions refers todistances near 1 AU from the Sun, which explains whythe two statements are not mutually contradictory. As arule, incomplete data allow to reliably determine neitherthe total mass lost by outgassing per orbit, nor the rela-tive contributions to the total provided by water, carbonmonoxide, and other species. However, because of thesheer amount of data collected for comet C/1995 O1, itstotal outgassed mass can be derived rather accurately forwater ice and carbon monoxide and at least estimated forcarbon dioxide and a number of other volatiles.Figures 9 and 10 suggest that the g mod ( r ; r ) law, withan appropriate choice of the scaling distance r , offers asatisfactory fit to the variations in the production rates ofcarbon monoxide (linked at large heliocentric distancesto the comet’s light curve) and water. If for a particu-lar species X the production rate at 1 AU from the Sunis Q ( X ) (in molecules per unit time), the loss of massby this species’ outgassing integrated over the orbit andmeasured from aphelion to next aphelion, is∆ M ( X ) = Q ( X ) µ mol ( X ) A Z ( P ) g mod ( r ; r ) dt, (13)where µ mol ( X ) is the species’ molar mass, A is the Avo-gadro number, P is the comet’s orbital period, and t istime. As described by Equation (1) with slight modifica-tions, the law g mod ( r ; r ) is a dimensionless function thatis symmetrical with respect to perihelion and normalizedto r = 1 AU. The scaling distance r depends on thespecies’ latent heat of sublimation, L sub ( X ), as shown byEquation (3). The dimension of the integral expressionin Equation (13) is time, defining ℑ ( X ) = ℑ ( L sub ( X )): ℑ ( X ) = Z ( P ) g mod ( r ; r ) dt = 2 k grav √ p Z π r g mod ( r ; r ) dυ, (14)omet C/1995 O1: Orbit, Jovian Encounter, and Activity 27where p = q (1+ e ), q is the comet’s perihelion distance, e its orbital eccentricity, k grav = 0 . day − the Gaussian gravitational constant, and υ the trueanomaly at time t . With r in AU, ℑ ( X ) equates the pro-duction of the species X integrated over the orbit withthe species’ constant production rate, equal to that at1 AU from the Sun integrated over a period of ℑ days,which we call an equivalent time. If the production ratein Equation (13) is expressed in molecules per unit time,the mass loss of the species integrated over the orbit be-comes ∆ M ( X ) = Q ( X ) µ mol ( X ) A ℑ ( X ) . (15)With a symmetrical law g mod ( r ; r ) one cannot accountfor the peak production’s offset from perihelion, but theasymmetry can be described approximately by choosingdifferent values of Q and r before and after perihelion,allowing a minor discontinuity at perihelion. Denotingthe respective production rates at 1 AU by ( Q ) pre and( Q ) post and the respective equivalent times by ℑ pre and ℑ post with the scaling distances of ( r ) pre and ( r ) post ,the total loss of mass of the species X per orbit is∆ M = ∆ M pre + ∆ M post (16)= µ mol A h ( Q ) pre ℑ pre + ( Q ) post ℑ post i . Addressing the perihelion asymmetry in some detail,we first fit the production rates of water, carbon monox-ide, and carbon dioxide in Figures 9 and 10. One tech-nique, applied to the production rates of water using the g mod ( r ; r ) law, aims at determining both r and Q .The other technique, applied to the production ratesof carbon monoxide, derives only Q , using the scalingdistances of 20 AU and 12 AU, already marked in thetwo figures. For the production rates of carbon dioxide,based on very limited data, we use two approaches: the g mod ( r ; r ) law with r = 8 . L sub inTable 16 and Equation (3), and the law intrinsic to CO ,whose parameters are in Table 16.The results in Table 19 show that the sublimation ofwater ice — while effectively confined to distances of lessthan 5 AU from the Sun — still prevails after integratingthe mass loss over the orbit. The abundance of carbonmonoxide (the component outgassed from the nucleus)makes up 14–20% of water in terms of the production rateat 1 AU from the Sun, but 22–32% of water in terms ofthe mass-loss rate at 1 AU, and it reaches 32% of the totalmass of water lost per orbit by the comet’s nucleus. Asimilar trend is shown by carbon dioxide. The summarymass of CO and CO lost per orbit by outgassing fromthe nucleus amounts to some the mass of lost water, soCO and CO are hardly minor constituents of the comet’svolatile reservoir.Some asymmetry relative to perihelion is apparent fromthe numbers in Table 19, but not beyond errors. Morewater and carbon dioxide was released after perihelionthan before perihelion, but nearly equal parts of car-bon monoxide. However, the fit in Table 19 ignores theanomalously high rate measured after perihelion by theISO. We already remarked in Section 9.2 on the incon-sistencies in the measurements of the preperihelion COproduction rates between 3 AU and perihelion, an effectthat complicates the mass-loss determination. Table 19
Loss of Mass of Water Ice, Carbon Monoxide, and CarbonDioxide Integrated Over Orbit of Comet C/1995 O1Water Carbon CarbonIce Monoxide a Dioxide b ( X = H O) ( X = CO) ( X = CO )Before PerihelionNumber of observations used 58 28 2Scaling distance ( r ) pre (AU) ± c ) 8.9 (fixed d )Production rate ( Q ) pre at1 AU from Sun (s − ) 10 . ± . . ± . . ± . Mass-loss rate ( ˙ M ) pre at1 AU from Sun (g s − ) 2 . × . × . × Integral ℑ pre (days) 82.47 98.09 94.22Mass lost over preperihelionhalf of orbit, ∆ M pre (g) 1 . × . × . × After PerihelionNumber of observations used 30 33 2Scaling distance ( r ) post (AU) ± c ) 8.9 (fixed d )Production rate ( Q ) post at1 AU from Sun (s − ) 10 . ± . . ± . . ± . Mass-loss rate ( ˙ M ) post at1 AU from Sun (g s − ) 2 . × . × . × Integral ℑ post (days) 80.19 103.08 94.22Mass lost over post-perihelionhalf of orbit, ∆ M post (g) 1 . × . × . × Mass lost per orbit, ∆ M (g) 3 . × . × . × Notes. a Outgassing from the nucleus only; we have made concerted effort tofilter out the contribution from the extended source (dust coma). b Using the g mod ( r ; r ) law, with a scaling distance ( r ) pre = ( r ) post ,derived from Equation (3) for L sub = 6400 cal mol − (Table 16). Analternative determination that employs the carbon-dioxide sublima-tion law (Table 16) results in the same integrated values of mass lostover the orbit and its two halves. c As derived in Figures 9 (post-perihelion) and 10 (preperihelion). d Scaling distance r assumed identical before and after perihelion be-cause of the absence of better information; discontinuity at perihelion. The level of activity measured by the integrated pro-duction of the three species is impressive, the total out-put per orbit amounting to 5 . × g or the mass ofa comet nearly 3 km in diameter at a bulk density of0.4 g cm − . This mass does not include dust and a largenumber of additional volatiles.The production-rate errors in Table 19 suggest thatthe results have an uncertainty of about ±
20% for wa-ter, around ± ±
40% for the poorly observed carbon dioxide. The scal-ing distance r for water ice is distinctly greater than2.8 AU, an effect that is clearly apparent from Figure 10.The discrepancy can be explained as a corollary to thepreferential sunward outgassing; the scaling distance of ∼ r , on its way to perihelion (Biver et al.1999), while the last one when the comet was 4.4 AU,or 1.02 r , on its way out (Biver et al. 2002). It maynot be completely a chance that the most distant de-tection of water takes place when the comet’s heliocen-tric distance approximately equals the scaling distance,because at these distances the g mod ( r ; r ) is very steep,always varying as r − . , so that a very minor changein r implies a dramatic change in the production rate.If this rule should apply generally, we predict that theleast volatile species that could be detected in a cometare those for which r = q ; in the orbit of C/1995 O1,an upper limit on the sublimation heat is nearly exactly20 000 cal mol − for the isothermal model, but slightlymore than ∼
28 000 cal mol − for the model with an ex-treme outgassing anisotropy (subsolar outgassing).Although the mass lost per orbit is the prime char-acteristic of each species’ outgassing, it is the observedproduction rate near perihelion, or near 1 AU from theSun, that is routinely tabulated by the observers. Forspecies other than water, it is fairly customary to presenttheir production rates in units of the water productionrate. Since an output ratio of a species relative to waterdepends on whether it is given in terms of the produc-tion rate or the mass-loss rate of the mass loss per orbit,we present the relationships among these three measuresthat the isothermal model provides. Let ℜ Q ( X ) be theratio of the production rate of a species X to the waterproduction rate (sometimes also called a relative abun-dance) at 1 AU from the Sun, ℜ Q ( X ) = Q ( X ) Q (H O) ; (17)let ℜ µ ( X ) be the ratio of the respective molar masses, ℜ µ ( X ) = µ mol ( X ) µ mol (H O) ; (18)let ℜ ℑ ( X ) be the nominal orbit-integrated production(i.e., equivalent time) ratio, ℜ ℑ ( X ) = ℑ ( X ) ℑ (H O) ; (19)and let ℜ ( X ) be the nominal orbit-integrated mass-lossratio, ℜ ( X ) = ℜ µ ( X ) ℜ ℑ ( X ) ; (20)we define ℜ M ( X ), the mass loss of the species X per orbitin units of the mass loss of water ice per orbit, ℜ M ( X ) = ∆ M ( X )∆ M (H O) , (21)as a product ℜ M ( X ) = ℜ Q ( X ) ℜ ( X ) = ℜ Q ( X ) ℜ µ ( X ) ℜ ℑ ( X ) . (22) Nominal here meaning for ℜ Q = 1. The observers publish ℜ Q ( X ), and it is ℜ ( X ) — andtherefore ℜ ℑ ( X ) — that we need to know in order todetermine the mass loss of X per orbit relative to water.Even though the actual function is unknown, an examplewith carbon monoxide below illustrates that applicationof Equation (14) with the scaling distance r tied to thesublimation heat L sub by Equation (3) is robust enoughto provide us with a reasonably informative estimate forthe factor that allows us to convert, according to Equa-tion (22), the reported production-rate ratio ℜ Q ( X ) intothe physically more meaningful ratio of the mass loss perorbit, ℜ M ( X ).The dependence of the equivalent time ℑ on the subli-mation heat L sub in the range of up to 30 000 cal mol − is for an isothermal model of the nucleus and a perihe-lion symmetry in outgassing displayed in Figure 13. Thegradual drop in ℑ on the left of the figure is understood:as L sub increases, the sublimation is effectively limitedto an ever shorter arc of the orbit around perihelion, sothat d ℑ /dL sub <
0. However, somewhat surprisingly, thecurve attains a minimum at L sub = 16 300 cal mol − andthen continues to climb, reaching the equivalent time ofwater [ ℑ (H O) ≃
135 days] at L sub ≈
20 000 cal mol − ,exceeding the equivalent time of hyper-volatile species,lim L sub → ℑ ≃
231 days, at L sub > ∼
25 000 cal mol − , andconverging to lim L sub →∞ ℑ ≃
261 days. This behaviorcan be explained by an increasing steepness of the sub-limation curve at close proximity to perihelion: mostmass of such volatiles is not lost until the comet getswithin 1 AU of the Sun and their abundances ℜ Q di-minish progressively with increasing L sub . This trend isconsistent with our independent conclusion above thatoutgassing of species with a sublimation heat exceed-ing 20 000 cal mol − should be increasingly rare inC/1995 O1 and that species with the sublimation heatnear or greater than ∼
28 000 cal mol − should not bedetected at all. The shape of the curve ℑ ( L sub ) dependsvery strongly on the perihelion distance (Appendix B).In an effort to assess a cumulative contribution to theorbit-integrated mass loss by various species, we list inTable 20 the parent molecules observed in C/1995 O1,as compiled by Bockel´ee-Morvan et al.’s (2004), andcompare them with a set of selected entries from Acree& Chickos’ (2016) compendium of organic and organo-metallic compounds, whose latent heat of sublimation(sublimation enthalpy) does not, in concert with the con-straints above, exceed 30 000 cal mol − .The table is ordered by the latent heat of sublima-tion, L sub , which is known to be only a weak function oftemperature. For most entries the listed values of L sub were compiled by Acree & Chickos (2016) in their com-pendium from an extensive set of sources. In the major-ity of cases the results from different sources for differenttemperature ranges agreed with each other quite well,often within a few percent or so. For some molecules,for which the sublimation heat was not listed by Acree& Chickos (2016), we were able to learn the information Note that r < .
37 AU for L sub >
16 300 cal mol − and that r < .
58 AU for L sub >
25 000 cal mol − ; thus, even perihelionof C/1995 O1 is in a thermal regime dominated by reradiation,with only a minor fraction of the solar energy spent on sublima-tion of species with such a high sublimation heat. Of course, the g mod ( r ; r ) law is then only an approximation to the genuine sub-limation law, which is exponential. omet C/1995 O1: Orbit, Jovian Encounter, and Activity 29 LAW g mod ( r ; r ) INTEGRATED OVER ORBITAL PERIODOF COMET C/1995 O1 H OCO CO CH OHH S HCNCH HCO HCOOHHCOOCH CH CONH C H NSC H COOHCH C H SO (COOH) CHOH(COOH) CH (COOH) L sub (X) (kcal mol − )100 25 10 5 3 2 1 0.7 0.5SCALING DISTANCE, r (X) (AU) ℑ (X) (days)100150200250 ••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••••• Figure 13.
Law g mod ( r ; r ), integrated over the orbit of C/1995 O1 as a function of the latent heat of sublimation L sub that determines thescaling distance r according to Equation (3), and expressed as an equivalent time ℑ , defined by Equation (14). Contrary to expectation,the equivalent time ℑ reaches a minimum near L sub = 16 300 cal mol − . The locations in the graph of 16 selected volatile species areidentified by their chemical formulas. All molecules to the left of the curve’s minimum and inclusive have been identified in C/1995 O1,with the exception of methyl hydrogen carbonate, CH HCO . None of the seven molecules to the right of the minimum has been detected. from the chemistry webbook of the National Institute ofStandards and Technology (NIST). For several tabu-lated species, mostly those observed in C/1995 O1, thesublimation heat apparently has not been determined; inthe majority of these instances its value could be approx-imated by the sum of values of the heat of vaporizationand the heat of fusion, if both known (though usually re-ferring to different temperatures) and is then parenthe-sized. For only three tabulated molecules we were ableto find only the heat of vaporization, which provides alower limit to the heat of sublimation; the heat of fusiontypically amounts to only a minor fraction of the heat ofsublimation.Highly relevant to the determination of the total massloss by outgassing are columns 5–7 of Table 20, in whichwe list, respectively, the water-normalized ratios ℜ µ , ℜ ℑ ,and ℜ , the last one being the factor that converts the pro-duction rate to the mass loss integrated over the orbit.For example, Bockel´ee-Morvan et al. (2004) state thatthe production rate of carbon monoxide from the nu-cleus was 12% of the water rate. Table 20 lists ℜ = 2 . For access, see http://webbook.nist.gov/chemistry/. bon dioxide. However, three of the detected moleculeswere more refractory than water; the most refractory one,ethylene glycol, has an estimated sublimation heat of ∼
18 000 cal mol − , somewhat below an upper limit onthe sublimation heat for an isothermal model.We note from Table 20 that the nominal mass-loss ratio ℜ is always greater than unity. Of the two factors thatcontribute to ℜ , ℜ ℑ has a generally minor effect, vary-ing between about 0.7 and 1.9. By contrast, ℜ µ rangesfrom 0.9 to nearly 17. The heaviest molecule detectedin C/1995 O1 was carbon disulfide with ℜ µ = 4 .
22, fol-lowed by sulfur dioxide ( ℜ µ = 3 .
56) and ethylene gly-col ( ℜ µ = 3 . ℜ are proportionallyhigher. Table 20 lists eight species whose molar mass ismore than six times as high as that of water ( ℜ µ > Its detection, not included in Bockel´ee-Morvan et al.’s (2004)compilation, was reported by Crovisier et al. (2004a). The three upper limits are meant to indicate that the probablevalues should be some 10–20% lower, still much greater than unity.
Table 20
Heat of Sublimation and Mass Loss for Observed and Potential Volatile Species in Comet C/1995 O1Latent Heat of Molar Molar Nominal Nominal Bullet IfSublimation, Mass, Mass Production Mass-Loss DetectedOutgassing Species, X Formula L sub (X) µ mol (X) Ratio, Ratio, Ratio, in Comet Notes a (cal mol − ) (g mol − ) ℜ µ (X) ℜ ℑ (X) ℜ (X) C/1995 O1Carbon monoxide CO 1785 28.01 1.55 1.65 2.56 • bMethane CH • Ethylene C H HCO CO > < < H • Acetylene C H • Hydrogen sulfide H S 5650 34.08 1.89 1.45 2.74 • Nitrous oxide N O 5820 44.01 2.44 1.43 3.49 cCarbonyl sulfide OCS (6000) 60.08 3.33 1.42 4.73 • eDimethyl ether (Methoxymethane) CH OCH (6300) 46.07 2.56 1.40 3.58 c,eCarbon dioxide CO • bAcetaldehyde CH CHO (6800) 44.05 2.45 1.37 3.36 • eMethyl formate (Methyl methanoate) HCOOCH > < < • dHydrogen cyanate (Isocyanic acid) HNCO > < < • dAmmonia NH • Methyl isothiocyanide C H NS 7530 73.12 4.06 1.32 5.36Formaldehyde H CO (7600) 30.03 1.67 1.31 2.19 • e,fCarbon disulfide CS (7800) 76.13 4.22 1.30 5.49 • eSulfur dioxide SO (7900) 64.07 3.56 1.29 4.59 • eCyanogen (CN) COCH • Ethyl cyanide (Propionitrile) CH CH CN (9900) 55.08 3.06 1.13 3.46 c,eMethyl cyanide (Acetonitrile) CH CN (10 000) 41.05 2.28 1.12 2.55 • eCyanoacetylene HC N 10 100 51.05 2.83 1.11 3.14 • Methanol CH OH 10 600 32.04 1.78 1.07 1.90 • Trimethylammonium cyanide (CH ) NHCN 10 800 86.14 4.78 1.05 5.02Trinitromethane CH(NO )
11 000 151.04 8.38 1.04 8.72Tetranitromethane C(NO )
11 300 196.04 10.88 1.01 10.99Water ice H O 11 400 18.02 1.00 1.00 1.00 • bPyrazine C H N
13 400 80.09 4.44 0.83 3.69Formic acid (Methanoic acid) HCOOH 14 600 46.03 2.55 0.74 1.89 • Acetic acid (Ethanoic acid) CH COOH 16 400 60.05 3.33 0.69 2.30 cHexanitroethane (NO ) C (NO )
16 900 300.05 16.65 0.70 11.66Formamide (Methanamide) NH CHO 17 200 45.04 2.50 0.71 1.78 • Methyl carbamate CH OCONH
17 700 75.07 4.17 0.73 3.04Cyanamide NH CN 18 000 42.04 2.33 0.75 1.75 cEthylene glycol (CH OH) (18 200) 62.07 3.44 0.77 2.65 • e,gMethylsulfonylmethane (CH ) SO
18 400 94.13 5.22 0.79 4.12Acetamide CH CONH
18 800 59.07 3.28 0.83 2.72Thioacetamide C H NS 19 800 75.13 4.17 0.98 4.09Benzoic acid C H COOH 21 300 122.12 6.78 1.24 8.41Methyl phenyl sulfone CH C H SO
22 000 156.20 8.67 1.36 11.79Oxalic acid (COOH)
22 300 90.03 5.00 1.41 7.05Dimethylglyoxime (CH ) C (NOH)
23 100 116.12 6.44 1.52 9.79Dithiooxamide (Rubeanic acid) (CH NS)
25 000 120.19 6.67 1.72 11.47Malonic acid CH (COOH)
26 000 104.06 5.77 1.78 10.27Tartonic acid CHOH(COOH)
27 800 120.06 6.66 1.85 12.32Thiosemicarbazide NH NHCSNH
30 000 91.14 5.06 1.90 9.61
Notes. a Sublimation-heat and molar-mass data taken mostly from Acree & Chickos’ (2016) updated compendium or from the National Institute ofStandards and Technology (NIST) webbook. Detection in C/1995 O1 as reported by Bockel´ee-Morvan et al. (2004), unless stated otherwise. b Latent heat of sublimation taken from Table 16; averages of the data listed in Acree & Chickos’ (2016) compendium are 1880 cal mol − and6250 cal mol − for CO and CO , respectively. c Listed by Crovisier et al. (2004b) and/or by Bockel´ee-Morvan et al. (2004) among species searched for but not detected in C/1995 O1. d Latent heat of sublimation unavailable; latent heat of vaporization used to estimate its lower limit. e Latent heat of sublimation unavailable, but approximated by a sum of latent heats of fusion and vaporization at available temperatures. f Released by polyoxymethylene (POM) decomposition from dust (e.g., Cottin & Fray 2008); production curve too steep for release from nucleus. g Detection in C/1995 O1 reported by Crovisier et al. (2004a). omet C/1995 O1: Orbit, Jovian Encounter, and Activity 31
Table 21
Trace Parent Molecules Detected in C/1995 O1 a Production Nominal Mass Loss Mass LossOutgassing Rate Mass-Loss Relative Per Orbit,Species, X Ratio, b Ratio, to H O, ∆ M (X) ℜ Q (X) ℜ (X) ℜ M (X) (10 g)Methane 0.015 1.45 0.022 0.074Ethane 0.002 c < < < < < < d Notes. a Not listed are formaldehyde (H CO), released from dust in thecoma (Table 20), and two unstable species that are not candidatesfor parent molecules: thioformaldehyde (H CS) and radical NS. b As compiled and published by Bockel´ee-Morvan et al. (2004) andby Crovisier et al. (2004a, 2004b). c Average taken of the range 0.001 to 0.003. d Including the contribution from hydrogen isocyanide. in connection with Table 19. If the production ratesfor the trace molecules, presented by Bockel´ee-Morvanet al. (2004), refer essentially to 1 AU from the Sun,the mass of each of the 18 parent species lost over theorbit of C/1995 O1 is listed in Table 21. By summingup the data, we show that while their total productionrate equaled merely 6.4% of the water production rate,their total mass lost per orbit amounted to 19% of thewater mass loss, thus averaging ∼
1% per trace species.Together with carbon monoxide and carbon dioxide (Ta-ble 19), this makes up 75% of the water mass loss.Additional molecules that were searched for, but notdetected, in C/1995 O1 (Crovisier et al. 2004b, Bockel´ee-Morvan et al. 2004) are listed in Table 22, which againshows that an upper limit on the production rate is muchtighter than the limit on the orbit-integrated loss of massby outgassing. The latter is 3 times higher than theformer in Table 21 and 2.8 times higher in Table 22.Since the parent molecules detected in C/1995 O1 (aswell as in other comets) are primarily hydrocarbons, of-ten oxygen, nitrogen, and/or sulfur-bearing, one wonders— given the enormous variety of compounds into whichthese elements can combine — how many more similar,fairly volatile species are there still waiting to be dis- Excluding formaldehyde, probably released from dust grainsin the comet’s atmosphere (e.g., Cottin & Fray 2008), as well asthioformaldehyde and a radical NS (unstable compounds that arenot candidates for a parent molecule), deuterated species, and ions. covered in comets and, in particular, what is their totalmass relative to the mass of the water-ice reservoir. Thisquestion prompted Crovisier et al. (2004b) to inspect thedistribution of the relative abundances of more than 20detected species, that is, their cumulative number, N , asa function of their relative production rate, ℜ Q . Theirplot shows the dataset to have a tendency to vary as apower law, N ∝ ℜ − . , and suggests that the dataset isincreasingly incomplete at ℜ Q < − . Crovisier et al.(2004b) pointed out that this effect appeared to tie inwith the well-known existence of numerous unidentifiedfeatures in diverse parts of cometary spectra.As we are primarily interested in estimating a totalmass lost by outgassing per orbit of C/1995 O1, weprepared in Figure 14 a plot of the distribution of theorbit-integrated mass loss, ∆ M , of the species presentedin Tables 19 and 21. The dependence of a cumulativenumber N of the species on the ratio ℜ M expresses themass loss per orbit in units of the total mass loss of waterice. Whereas ℜ M has up to now been a function of thespecies X , we now treat the cumulative distribution as acontinuous function of ℜ M and note that the mass lossin an interval between N and N + d N is ℜ M ( N ) d N .We assume the function N to vary as a power of ℜ M (with N = 1 for ℜ M = 1), N = ℜ − y M , (23)so that d N = − y ℜ − ( y +1) M d ℜ M , and for the total mass CUMULATIVE DISTRIBUTION OFMASS LOSS PER ORBIT BYPARENT MOLECULESIN C/1995 O1 (cid:0)(cid:0)(cid:0)✒ y = 0 . (cid:0)(cid:0)(cid:0)✠ y = 0 . (cid:0)(cid:0)(cid:0)✒ y = 0 . ❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅❅ ♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣♣ qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq N = ℜ − y M ℜ M ,MEASURED IN UNITS OF H O MASS LOSS110 N ①①①①①①①①①①①①①①①①①①①①① Figure 14.
Cumulative distribution of the orbit-integrated massloss by the outgassing parent molecules detected in C/1995 O1.The number of species, N , is plotted against the mass lost by themover the orbit, ℜ M , measured in units of the mass lost by thesublimation of water ice, (0 . ± . × g. The total masslost per orbit by all volatile species is ℜ ∞ = y (1 − y ) − , where − y is the slope of the cumulative distribution. In order that ℜ ∞ befinite, it must be y <
1. The thick line, y = 0 .
72, is a fit through thethree most abundant species (water, carbon monoxide, and carbondioxide); the thin line, y = 0 . y = 0 .
5, isa constraint required by an extreme case, when water outgassingaccounts for all the lost mass, ℜ ∞ = 1. Table 22
Upper Limits on Volatile Species Searched for Though Undetected in C/1995 O1 a Production Latent Heat of Molar Molar Nominal Nominal Mass Loss Mass LossOutgassing Rate Sublimation, b Mass, Mass Production Mass-Loss Relative Per Orbit,Species, X Formula Ratio, L sub (X) µ mol (X) Ratio, Ratio, Ratio, to H O, ∆ M (X) ℜ Q (cal mol − ) (g mol − ) ℜ µ (X) ℜ ℑ (X) ℜ (X) ℜ M (10 g)Phosphine PH < < < CO < > < < < < CCH < > < < < < O < < < OCH < < < H O < < < SH < < < CH CN < < < CH OH < < < O < >
11 600 34.01 1.89 < < < < OH < < < COOH < < < OHCHO < >
16 700 60.05 3.33 > ∼ > ∼ < < CN < < < < c < < CH COOH < < < NH < ∼ ? ) ∼ ? ) < ? ) < ? )Sodium hydroxide NaOH < ∼ ? ) ∼ ? ) < ? ) < ? )Fulminic acid HCNO < ∼ ? ) ∼ ? ) < ? ) < ? )Thioformaldehyde H CS < ∼ ? ) ∼ ? ) < ? ) < ? )Cyanodiacetylene HC N < ∼ ? ) ∼ ? ) < ? ) < ? )Total (rounded off) . . . . . . . . . < < < Notes. a The 3 σ upper limits as published by Crovisier et al. (2004b) and by Mockel´ee-Morvan et al. (2004). Sublimation-heat and molar-mass datataken mostly from Acree & Chickos’ (2016) updated compendium or from the National Institute of Standards and Technology (NIST) webbook. b When unavailable, L sub was approximated by a sum of latent heats of fusion and vaporization (parenthesized values); when the fusion heatwas unavailable, a lower limit to L sub was equated with vaporization heat; when the vaporization heat was unavailable (last five entries), weapproximated the ratio ℜ ℑ by unity (values with a question mark). c The value determined from the Antoine equation. loss per orbit experienced by all outgassing species, ℜ ∞ ,we get ℜ ∞ = Z ∞ ℜ M d N = − y Z ℜ − y M d ℜ M = y − y , (24)where 0 . < y <
1. The limits on y are dictated by re-quiring that ℜ ∞ be finite, positive, and greater thanunity. The infinite upper limit in the first integral impliesmerely a very large set under consideration that includesspecies with the abundances many orders of magnitudelower than the abundance of water, so that N → ∞ when ℜ M →
0, in compliance with Equation (23).A tighter condition for y follows from our above find-ing that the mass lost per orbit by all known outgassingspecies, including water, was 175% of the mass loss ofwater, which implies y = 0 .
636 for the lower limit to y ,as depicted in Figure 14. Since most data points in thefigure congregate along y ≃ .
5, their distribution is in-consistent with the required law and suggests that the setis substantially incomplete. A fit through the three mostabundant species — water, carbon monoxide, and car-bon dioxide — predicts that y = 0 . ± .
12. Combiningthe constraints, a probable range for the relative loss ofmass by outgasssing per orbit of C/1995 O1 amounts to2 < ∼ ℜ ∞ < ∼
5, so that water ice may account for as littleas ∼
20% of the orbit-integrated outgassed mass. If the cumulative distribution of sublimating species does fol-low a power law, their number with mass loss per orbitgreater than 10% of the water mass loss is predicted tobe between four and seven (compared to three known);those with ℜ M greater than 1%, between 21 and 46 (com-pared to 9 known, an incompleteness of more than 50%);those with ℜ M greater than 0.1%, between 100 and ∼ ℜ M greaterthan 0.01%, between ∼
460 and ∼ ℜ M limit, the number of species to outgass fromthe nucleus equals ∼ y = 0 . ℜ µ ≫ ℜ ℑ > ℜ Q . In addition, one cannotrule out that C/1995 O1 carried many hundreds of parentmolecules with a mass-loss ratio ℜ M > − and perhapsmore than a thousand with ℜ M > − .omet C/1995 O1: Orbit, Jovian Encounter, and Activity 33We next compare this predicted variety of species inC/1995 O1 with the number of diverse molecular formu-las generated as a sum of all possible combinations ofspecific elements and subject to prescribed assumptions.Disregarding for a moment the chemical issues and ex-cluding the homonuclear molecules, the number of math-ematically possible compounds is a function of (i) a num-ber N of the elements, E ( i ) ( i = 1 , , . . . , N ), that make upthe molecules and (ii) maximum numbers K i of atoms, E ( i ) k i ( k i = 0 , , . . . , K i ), that each of the elements contrib-utes to the molecules. A general chemical formula of suchcompounds is E (1) k E (2) k . . . E ( N ) k N ; ( i = 1 , , . . . , N ; k i = 0 , , . . . , K i ) , (25)where k i = 0 means that an i -th element does not con-tribute to the particular molecule. However, the exclu-sion of the homonuclear molecules requires that therealways be at least two elements ( N > i and j = i , inany molecule for which k i > k j >
0. The completeset of compounds is symbolically denoted as h E (1) K E (2) K . . . E ( N ) K N i (26)and the number N of mathematically possible molecularcompounds is N = N Y i =1 ( K i +1) − N X i =1 K i ! . (27)The molecules consisting of N − N −
2, . . . , 2 elementsthat make up subsets of the set (26) are included in N . Note that for K = K = . . . = K N = K Equation (27) issimplified to N = ( K +1) N − (1+ KN ). We next illus-trate this exercise, including the chemical properties ofthe resulting compounds, on an example of a molecu-lar set consisting of three elements, each of which con-tributes up to two atoms per molecule, [C H O ] ( N = 3; K C = K H = K O = 2), in Table 23; we show that, of the to-tal number of compounds, N = (2+1) − (1+2 ×
3) = 20,fully 16 (80%) have been detected outside the Solar Sys-tem, whereas 8 (40%) have been observed in C/1995 O1,five (25%) of them being parent molecules and the restdissociation products. Two molecules seen in interstel-lar or circumstellar space have not been detected in thecomet in spite of attempts to do so. Among the four com-pounds not observed in either environment is glyoxal; two of the three remaining ones — ethylenedione andethynediolide (together with its isomer glyoxalide) — arederivatives of glyoxal (Dixon et al. 2016). The spectrumof solid glyoxal was found to have fundamental bandsbetween 3.5 µ m and 18.2 µ m (Durig & Hannum 1971).The last undetected molecule is the hydrocarboxyl func-tional group COOH, of fleeting existence as a separatecompound; it is contained in all carboxylic acids, one ofwhich (formic acid) is listed in Table 23, while another(glyoxylic acid) is related to glyoxal; the oxalic acid hasthe formula of the group’s dimer (Table 20). If homonuclear molecules should be included among the math-ematically possible compounds, then N =1 is allowed and Equa-tion (27) changes to N = Π Ni =1 ( K i +1) − ( N +1). From our standpoint, glyoxal is the most interesting amongthree C H O isomers; the other two are acetylenediol HOCCOHand acetolactone H COCO.
Table 23
Molecular Species Made Up of Carbon, Hydrogen, and OxygenWith Up to Two Atoms of Each Element a ( N =3 , K =2)Molecule Detected b AtomsPer Out- in C/1995 O1Ele- Molecule’s Name Formula side c ment Solar Any ParentC H O System Status OnlySet [C H O ]1 1 1 Formyl radical HCO t CO t t t t t t d HOCCO2 2 1 Ketene CH CO t ❞ ❞ H ]1 1 0 Methylidyne radical CH t t t H t H t t t Subset [C O ]1 0 1 Carbon monoxide CO t t t t t t O t d OCCOSubset [H O ]0 1 1 Hydroxyl radical OH t t t O t t t t ❞ ❞ Notes. a Homonuclear molecules excluded. b Open circle in comet columns means the molecule was searched forbut undetected. c In interstellar and/or circumstellar space; for original discovery re-ports, see https://en.wikipedia.org/wiki/List of interstellar andcircumstellar molecules . d Derivative of glyoxal; glyoxalide OHCCO, an isomer of ethynedi-olide, is yet another derivative (Dixon et al. 2016).
A total number of molecules detected in the interstellarmedium up to now is about 200 — nearly 10 times thenumber of parent molecules observed in C/1995 O1 —with a peak molar mass well over 100 g mol − , eitheras gas or as ice. Some are radicals that could not ex-ist in a cometary nucleus without being bound chem-ically to other, more stable ices. A majority of inter-stellar molecules are made up of up to five elements:carbon, hydrogen, oxygen, nitrogen, and sulfur. Thenumber of hydrogen and carbon atoms seldom exceedssix each per molecule; oxygen rarely contributes morethan two to three atoms per molecule, while the otherelements mostly not more than one atom per molecule(see footnote c to Table 23). Under these conditions, amathematically complete set, [C H O − NS], consists of
N ≈ N and K i . In fact,isomers exist even among the compounds presented inTable 23. In addition to the isomers of glyoxal, referredto above, there is hydroxymethylene (HCOH), an isomerof formaldehyde; ethynol (HC OH), an isomer of ketene;and dioxyrane (H CO ), an isomer of the formic acid.If one expects a degree of analogy between the presenceof complex molecules in the interstellar medium and incomets, and detection incompleteness by a factor of threeto four in the interstellar medium on the assumption thata great majority of compounds complies with a symbol-ical formula [C H O − NS] in Equation (26), then a de-gree of detection incompleteness of parent molecules in acomet like C/1995 O1 is estimated at a factor of ∼ ∼
760 volatile species in C/1995 O1 whose mass loss isgreater than 0.01% of the water mass loss, based on thedata in Figure 14. In any case, the present inventory ofparent molecules in C/1995 O1 appears to be extremelyincomplete, by a factor much greater than 10, and themass loss of water is quite probably less than 50% of thetotal outgassed mass lost per orbit. The conclusion onthe existence of hundreds of very complex molecules thatcontribute to the process of outgassing from comets isalso broadly consistent with very recent results from theRosetta Mission, which suggest that carbon in dust parti-cles from comet 67P/Churyumov-Gerasimenko is boundin organic matter of high molecular weight (Fray et al.2016) and which support the presence of polyoxymethy-lene (POM) and other polymers in the dust grains asa plausible source of organic matter in gaseous phase,such as formaldehyde (Wright et al. 2015). A potentialpresence of polycyclic aromatic hydrocarbons (PAHs),such as phenanthrene, anthracene, etc., as discussed byBockel´ee-Morvan et al. (2004) and others, would onlystrengthen this general conclusion.As a caveat, the degree of approximation to the law ofoutgassing by the isothermal model is open to question,as the nominal orbit-integrated production ratio ℜ ℑ de-pends on the distribution of the sublimation rate over thenucleus. For a distribution other than that conforming tothe isothermal model, the constant in Equation (3) ex-ceeds 19 100 AU cal mol − , a maximum of 27 000 AU cal mol − being reached when the sublimation proceedsfrom a subsolar area only. The scaling distances for thewater sublimation curves in Table 19 would be fitted bestby choosing for the constant the values of 24 800 AU calmol − before and 23 600 AU cal mol − after perihelion.Nonetheless, use of the isothermal model is warranted byits successful application in orbital studies and by expec-tation that the scaling distances for other molecules arelikely to be affected similarly, resulting in Table 21 in anet ℜ ℑ effect close to nil except for the species whose L sub exceeds ∼
16 000 cal mol − , i.e., formamide and ethyleneglycol, whose contributions would moderately drop. Production of Dust and Its Mass Loading ofthe Gas Flow
Further insight into the issue of the mass lost over theorbit by outgassing is provided by investigating the mass-loss rates of dust and by examining the terminal veloci-ties of dust particles. Since it is well known that, owing to a fairly flat particle-size distribution, the large-sized dust(submillimeter and larger grains) provides a dominantcontribution to the total particulate mass lost, we focuson the investigations of C/1995 O1 based on imagingobservations at submillimeter and longer wavelengths.Jewitt & Matthews (1999) observed the comet at 850 µ m on 16 dates in 1997 between February 9 (1.27 AU pre-perihelion) and October 24 (3.17 AU post-perihelion),deriving a dust mass-loss rate of (1 . ± . × g s − at 1 AU before perihelion and (1 . ± . × g s − at 1 AU after perihelion, and an orbit-integrated massloss of 3 × g, equivalent to the mass of a cometmore than 5 km in diameter, at an assumed density of0.4 g cm − . They employed a blackbody approximationand a model of an optically thin, spherically symmetrical,steady-state coma, by-passing the particle-size distribu-tion function by adopting a dust opacity of 0.55 cm g − at 850 µ m. They equated the mass production rate witha ratio of the particles’ mass to their residence time inthe beam. However, the residence time depends not onlyon the ejection speed of the particles (and therefore ontheir size), but also on the direction of their motion rel-ative to the observer. In particular, particles moving indirections close to the line of sight have a residence timemuch longer than particles moving perpendicular to theline of sight, a scenario considered by Jewitt & Matthews(1999). Sekanina & Kracht (2014) developed a techniquethat properly accounts for the effect of residence timein a spherically symmetrical, steady-state coma environ-ment, also employing a more realistic approximation forthe particle velocity. This technique relates a total massproduction rate of dust at time t ,˙ M d ( t ) ≡ ˙ M d ( s , s ∞ ) = ˙ N d ( t ) Z s ∞ s πρ d s f d ( s ) ds, (28)and a geometric cross-sectional area of all dust particles(of assumed sphericity) ejected from the nucleus that attime t reside in a beam whose radius at the comet is r b , X d , r b ≡ X d , r b ( s ,s ∞ ) = Z s ∞ s πs f d ( s ) ds Z t −∞ ˙ N d ( t )Φ( s,t ; r b ) dt, (29)where f d ( s ) ds is a normalized distribution function ofparticle radii s whose lower and upper limits are, respec-tively, s and s ∞ , ρ d is a particle bulk density, ˙ N d ( t ) isthe number of particles released at t per unit time, and0 ≤ Φ( s,t ; r b ) ≤ ε d ≤ λ , the information provided by themeasured thermal flux density F λ in the beam is F λ ∼ Z s ∞ s ε d ( s ) πs f d ( s ) ds Z t −∞ ˙ N d ( t )Φ( s, t ; r b ) dt, (30)but since a variation with particle size of the emissivity at λ is unknown, the flux density is interpreted in terms ofa blackbody’s cross-sectional area, that is, it is assumedomet C/1995 O1: Orbit, Jovian Encounter, and Activity 35that ε d ( s ) = 1 for any s between s and s ∞ . And sinceparticles whose radii s satisfy a condition x = 2 πsλ ≪ s → ε d ( s ) = 0, the truegeometric cross-sectional area of the particles in thebeam should in fact be much greater than offered bythe flux-density measurement. To account for the effectof emissivity, we note that there must exist particle sizes s min > s and s max ≤ s ∞ such that Z s ∞ s ε d ( s ) πs φ d ( s ) ds = Z s max s min πs φ d ( s ) ds = X d ( s min , s max ) , (32)where φ d ( s ) equals f d ( s ) R t −∞ . . . dt from Equation (30)and X d is the measured blackbody cross-sectional area.This formalism approximates the unknown variations in ε d ( s ) by introducing discontinuities at s min and s max ,with ε d ( s ) = 0 at s ≤ s < s min and s max ≤ s ≤ s ∞ , but with ε d ( s ) = 1 at s min ≤ s ≤ s max . In the case examined hereone can safely adopt s max = s ∞ . If the variations in theemissivity can, for example, plausibly be approximatedby ε d simx for x < x and by ε d ≃ x ≥ x , their totaleffect is accounted for by taking 2 πs min /λ ≃ x , thatis, s min ≃ λx / π . As x is near unity, for submillimeterwavelengths this condition implies that s min ≫ s , whichgreatly affects the results, as seen below.Based on the work by Sekanina & Kracht (2014), therelationship between the total mass production rate ofdust and the cross-sectional area X d ( s min , s max ) is then˙ M d = 8 ρ d πr b X d ( s min , s max ) Z s ∞ s s f d ( s ) ds Z s max s min v d ( s ) s f d ( s ) ds , (33)where v d ( s ) is a size-dependent ejection velocity of thedust particles that generally is a function of heliocentricdistance. We adopt a power law for the size distribution, f d ( s ) ds = C (cid:16) s s (cid:17) α , Z s ∞ s f d ( s ) ds = 1 , (34)where α is the distribution’s index or power and C is anormalization constant; and make use of an expressionpreviously employed for the ejection velocity (Sekanina &Kracht 2014), v d ( s ) = v χ √ s (35)with v and χ being the parameters, v potentially vary-ing with heliocentric distance. The total mass productionrate is then as follows˙ M d = 8 v ρ d s ∞ πr b (3 − α )( − α )4 − α (cid:18) s ∞ s max (cid:19) − α × (1 − ǫ − α ∞ ) X d ( s min , s max ) (36) × h ( − α )(1 − ǫ − α )+(3 − α )(1 − ǫ − α ) χ √ s max i − , where ǫ ∞ = s /s ∞ , ǫ = s min /s max , and α = 3, , and 4;otherwise the powers of s ∞ , s max , ǫ ∞ , and/or ǫ should bereplaced with the respective logarithms. Before we apply Equation (36) to Jewitt & Matthews’(1999) set of the cross-sectional data, we carefully selectthe model parameters. We first focus on the constantsthat define the size distribution function, to which the re-sults are most sensitive. The index α was for C/1995 O1determined numerous times and in different ways; a use-ful compilation, published by Lasue et al. (2009), showsthat 3 . ≤ α ≤ .
7. The entries most relevant to theimportant submillimeter and larger particles were pro-vided by the investigations based, at least in part, onthe ISO observations, whose spectral reach extended tonearly 200 µ m. Min et al. (2005) derived α = 3 .
48 fromthe ISO observations made in September 1996, about sixmonths before perihelion, while Harker et al. (2002) de-duced from the observations covering nine months that α = 3 . α = 3 . α = 3 . We adopt α = 3 .
55 asan optimum mean value; we estimate its error at ± s ,has a negligible effect on the mass production rate; weemploy s = 0 . µ m, since Min et al. (2005) showed thatincorporating grains of amorphous olivine and pyroxeneof this minute size improves a fit to the comet’s observedspectral energy distribution.An upper boundary to the size distribution, s ∞ , influ-ences the total mass production rate much more signif-icantly than the lower boundary. We assume that thelargest particles must reach an escape velocity from thenucleus at the time they decouple from the gas flow,which according to Probstein (1969) is at a distance of ∼
20 nucleus’ radii. The size of the nucleus is amongthe subjects of Part II of this study; here we only men-tion that Szab´o et al.’s (2012) results suggest that atthe relevant distance the escape velocity amounts to v esc ≃ − . From Equation (35) the particle radiusat the upper boundary of the size distribution functionis then equal to s ∞ = (cid:18) v v esc − (cid:19) χ − , (37)so that this task is reduced to finding the particle velocityparameters v and χ .Jewitt & Matthews (1999) calculated that the velocityof particles 1 mm in radius ejected from C/1995 O1 was80 m s − at 1 AU from the Sun (although they eventuallyused a value three times lower ), assuming that the diam-eter of the nucleus equasled 40 km. With its dimensionsnearly twice as large (Szab´o et al.’s 2012), the ejectionvelocity of the millimeter-sized particles was more likelyto amount to at least 110 m s − . In fact, Vasundhara &Chakraborty’s (1999) direct fit to dust-coma features inthe comet’s images taken between 1997 February 18 andMay 2 imply for these particles ejection velocities thatare still higher, 123–145 m s − . Both Gr¨un et al. (2001) and Harker et al. (2002) used a sizedistribution law that was introduced by Sekanina & Farrell (1982)and subsequently employed extensively by Hanner (e.g., Hanner1983, 1984). The Sekanina-Farrell law differs from a power law forsubmicron- and micron-sized grains, but in terms of α both laws arepractically identical for submillimeter-sized and larger particles. Table 24
Total Mass Production Rates of Dust, ˙ M d , for C/1995O1 FromSubmillimeter Observations by Jewitt & Matthews (1999)Time of Distance Blackbody Dust Rate ˙ M d (10 g s − )Observation to Sun Cross-Section1997 (UT) (AU) (10 cm ) a Nominal s min = s J&MFeb. 9.05 1.271 2.6 15.8 0.63 1016.98 1.186 3.1 21.2 0.84 13Mar. 9.92 0.997 2.7 24.7 0.98 1623.00 0.932 2.6 25.8 1.02 1630.83 0.915 2.6 25.5 1.01 17Apr. 6.92 0.920 2.9 27.1 1.07 1826.82 1.023 3.8 27.8 1.10 18May 2.23 1.062 3.4 23.3 0.92 156.21 1.103 3.3 21.3 0.84 1412.22 1.161 4.3 25.7 1.02 16June 14.92 1.558 2.8 11.5 0.46 7July 6.92 1.836 3.1 10.7 0.43 7Sept. 9.92 2.644 3.5 8.9 0.35 613.92 2.692 2.5 6.3 0.25 4Oct. 13.71 3.044 0.88 2.0 0.08 1.324.68 3.170 0.81 1.8 0.07 1.2
Note. a Relative errors from ±
8% to ± At the other end of the dust-particle size spectrum, theejection velocities of submicron-sized grains that madeup the leading boundaries of recurring expanding haloswere determined by Braunstein et al. (1999) from imagestaken over a period of 61 days and approximately cen-tered on perihelion. Corrected for the projection effects,the ejection velocities averaged 670 ±
70 m s − , rangingfrom 603 m s − to 775 m s − , with no systematic trendsbetween 0.91 AU and 1.10 AU from the Sun. Vasundhara& Chakraborty (1999) derived the ejection velocities ofup to 650 m s − for submicron-sized particles near 1 AU,with a rate of decrease with heliocentric distance r onlyslightly steeper than r − (to be adopted here). To rec-oncile these high ejection velocities of microscopic grainswith the velocities of 130–140 m s − for millimeter-sizedparticles, the parameters in Equation (35) should equal,after rounding off, v = 700 m s − at 1 AU from the Sunand χ = 13 cm − .With these values and an escape velocity of ∼ − inserted into Equation (37), the upper boundary of theparticle size distribution is at s ∞ = 180 cm. Referring toour previous arguments, we adopt this same number for s max , while with a rather conservative value of x ≈ . x is expected to be near unity), we find s min = 80 µ m.The remaining physical parameter in Equation (36) —a particle bulk density — is rather uncertain, becauseavailable information is indirect, based on research ofcomets other than C/1995 O1. Exposure of Stardust’ssample collector to impacts of dust grains ejected from81P/Wild 2 points to a very broad range of bulk densi-ties, from compact particles with ρ d ∼ − to highlyporous aggregates for which ρ d is as low as ∼ − (e.g., H¨orz et al. 2006). A recent investigation of the dustdetected by the Grain Impact Analyzer and Dust Accu-mulator (GIADA) on board the Rosetta spacecraft ledto a conclusion that for 67P/Churyumov-Gerasimenko amean dust-particle bulk density equals 0 . +0 . − . g cm − (Fulle et al. 2016). We henceforth adopt ρ d = 0 . − . With the known angular beam radius of 7 ′′ .65, thelinear radius becomes r b = 5550∆ km, where ∆ is thecomet’s geocentric distance in AU. This completes theprerequisites for deriving the total mass production ratesof dust, which are in Table 24 listed in a column marked“Nominal” and compared with the numbers by Jewitt &Matthews (1999), in the “J&M” column, and with therates derived in a case of the neglected emissivity effect(in a column marked “ s min = s ”).Table 24 suggests an unexpectedly good correspon-dence between our nominal mass production rates andthe rates determined by Jewitt & Matthews (1999); oursolution offers rates that, on the average, are only 1.55times higher than are theirs. We are convinced that thisagreement is fortuitous, in part because the enormousdifferences between both approaches are demonstratedby the employed ejection velocity: our value of 137 m s − for grains 1 mm in radius, is a factor of 5.5 times higherthan the value used by Jewitt & Matthews. According totheir Equation (6), our nominal rates should have beenhigher by the same factor. In addition, the maximumradius of the particles escaping from the nucleus in theirmodel was ∼ − the wavelengthimplies the dust production rates that are too low by afactor of about 25.As for the uncertainties involved in our nominal massproduction rates in Table 24, we note that they are mostsensitive to the size distribution index α . The error of ± α = 3 . α = 3 .
4. The un-certainties in the other parameters are of lesser impact.A change by a factor of two in a particle size at the upperboundary of the distribution, s ∞ , results in a range from0.77 to 1.31 the nominal rate. The same change in a par-ticle size at the distribution’s lower boundary, s , has anentirely negligible effect on the production rates, but achange in the lower limit of the size of the particles thatcontribute to the measured thermal flux density, s min ,does affect the rates moderately; a change by a factor oftwo causes the rates to vary from 0.87 to 1.15 the nominalrate. The column s min = s in Table 24 shows a changein the rates after the nominal value of s min was reducedby a factor of 8000. Equation (36) implies that changesin the particle density and ejection velocity project lin-early as changes in the production rate (disregarding aminor effect from the parameter χ ).The next tasks were to test whether the mass produc-tion variations of dust with time could be fitted by thesame g mod -type law as the gas production variations; toestimate a total loss of dust mass from C/1995 O1, de-rived by integrating this production rate ˙ M d over theorbit from aphelion to next aphelion; and to compareomet C/1995 O1: Orbit, Jovian Encounter, and Activity 37 Table 25
Loss of Dust Mass and Dust-to-Water Mass-Loss Ratiofor Comet C/1995 O1Parameter ValueParticle size distribution index, α s ( µ m) 0.01Particle radius at upper end, s ∞ (m) 1.80Particles contributing to measured thermal signal:Minimum particle radius, s min ( µ m) 80Maximum particle radius, s max (m) 1.80Particle ejection velocity:Parameter v at 1 AU (km s − ) 0.70Parameter χ ( µ m − ) 0.13Power γ of heliocentric-distance variation, r − γ ρ d (g cm − ) 0.80Before Perihelion After PerihelionNumber of observations used 7 a b Distances from Sun (AU) 4.58–0.92 0.92–3.17Scaling distance ( r ) d (AU) 6.28 ± ± . AU from Sun (10 g s − ) 24.0 ± ± ℑ pre or ℑ post (days) 88.5 85.1Dust mass lost per half orbit (g) 1 . × . × Dust loading of water flowat 1 AU from Sun, Ψ (0)
H2O . × Dust loading of water flowper orbit, Ψ ∗ H2O
Notes. a Based on 5 nominal dust mass production rates from Table 24 com-bined with 2 adjusted data points by Gr¨un et al. (2001); orbital arccovered by Jewitt & Matthews’ (1999) observations alone too shortto determine scaling distance. b Based on nominal dust mass production rates from Table 24 exceptfor anomalously high ones from September 9 and 13 (outburst?),which had to be eliminated; fit used to adjust Gr¨un et al.’s rates. it to the mass loss of water presented in Table 19. Webegan by ascertaining that the more extensive set of thepost-perihelion nominal data points from Table 24 is sat-isfied by a law g mod [ r ; ( r ) d , post ] and by determining thetabulated scaling distance ( r ) d , post . We then turned tothe preperihelion data in Table 24, which cover a shorterrange of heliocentric distances. We noticed that, fortu-nately, Gr¨un et al. (2001) reported the dust productionrates at three larger heliocentric distances in 1996 and1997, derived from the ISO observations. The chronologi-cally last of these data points, from 1997 December 30,when the comet was 3.90 AU from the Sun, was used byus to adjust Gr¨un et al.’s scale of dust production to agreewith ours. It turned out that when their production rateat 3.9 AU was multiplied by a factor of 1.3, it fitted thevalue predicted for this time by the post-perihelion mod-ified law. We then adjusted Gr¨un et al.’s (2001) preperi-helion data points, referring to 1996 April 27 (the cometat 4.58 AU from the Sun) and to 1996 October 7 (2.82 AUfrom the Sun) by the same factor, and linked them withthe five preperihelion nominal production rates from Ta-ble 24; we obtained a rather satisfactory fit by apply- ing another modified law, g mod [ r ; ( r ) d , pre ]. Togetherwith a summary of the adopted parameters, the dust-production results are listed in Table 25, separately foreither orbital branch as well as for the whole orbit.We found that the orbit-integrated mass loss of dustby C/1995 O1 equaled 4 × g, close to — but slightlyhigher than — the total by Jewitt & Matthews (1999).The dust production appears to be only marginally, andwithin 1 σ errors, higher after perihelion than before per-ihelion. Figure 15 illustrates that the data points are fit-ted by the two g mod -type laws quite satisfactorily. How-ever, we would not expect the laws to be applicable toheliocentric distances larger than approximately 4.5 AU.The production rate of carbon monoxide catches up withthe water production rate at 3.6–3.7 AU, thus graduallytaking control over the process of dust release, includ-ing non-sublimating water-ice grains, at larger distancesfrom the Sun.The mass ratio of the production of dust to the pro-duction of water is called in Table 25 and hereafter a mass loading of the water flow by dust or shortly a dustloading of the water flow . We distinguish a normalizeddust loading , Ψ (0) H2O , given asΨ (0)
H2O = mass production rate of dust at 1 AU from Sunmass production rate of water at 1 AU from Sun , (38)whose values are different before and after perihelion;and a dust loading integrated over the orbit, or simplya dust loading per orbit or integrated dust loading , Ψ ∗ H2O ,defined asΨ ∗ H2O = total mass of dust lost by ejection per orbittotal mass of sublimated water ice per orbit , (39)which consists of the preperihelion and post-perihelioncontributions to the total. When data are limited to aheliocentric distance r different from 1 AU, one may onlybe able to determine a nominal dust loading , Ψ H2O ( r ), atthat particular distance, which — assuming the validityof a g mod -type law for both water and dust — is relatedto the normalized dust loading byΨ H2O ( r ) = Ψ (0) H2O " ( r ) n d +1( r ) n H2O +1 ( r ) n H2O + r n ( r ) n d + r n k , (40)where n and k are the exponents of the modified law,as defined below Equation (1), while ( r ) d and ( r ) H2O are, respectively, the scaling distances for the dust (fromTable 25) and water (from Table 19), which both applyto either the preperihelion or post-perihelion branch ofthe orbit, as do the values of the nominal and normalizeddust loading of the water flow. Additional observations of the comet in a spectral range (nearor beyond 1 mm) that should warrant a proper account of the con-tributions from massive grains were reported by Senay et al. (1997)at three wavelengths between 1.1 mm and 2.1 mm in late February1997 and by de Pater et al. (1998) at wavelengths 2.6–3.5 mm and7.0–13.3 mm during March and April 1997. We do not use theseobservations in our computations because they cover the same pe-riod of time as the nominal data points in Table 24, but note thatSenay et al. derived a fairly low production rate of 3.2 × g s − five weeks before perihelion, while de Pater et al. deduced a produc-tion rate on the order of 10 g s − within four weeks of perihelion,in conformity with the data we employ. TEMPORAL VARIATIONS INPRODUCTION RATE OFDUST FROM C/1995 O1 t π ① DATA BY JEWITT & MATTHEWS (1999) ❤t ADJUSTED RESULTS BY GR ¨UN ET AL. (2001)
OUTBURST? ❅❅❅■ g mod [ r ; ( r ) d , pre ] LAW ( r ) d , pre = 6 . AU (cid:0)(cid:0)(cid:0)✒ g mod [ r ; ( r ) d , post ] LAW ( r ) d , post = 5 . AU CALIBRATIONDATA POINT –320 –240 -160 –80 0 +80 +160 +240TIME FROM PERIHELION, t − t π (days)10 ˙ M d (g s − ) ❤t ❤t ①① ① ①①① ①①①① ① ① ①① ①① ❤t ·································································································································································································································································································································································································································································································································································································································································································································································································································································································································································································· 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· Figure 15.
Curve of the dust production rate of C/1995 O1 as a function of time. It was derived using the technique described in thetext, applied to the dust cross-sectional data at a submillimeter wavelength (Jewitt & Matthews 1999), which were linked to the adjustedISO-based production rates by Gr¨un et al. (2001), and fitted by a modified law, g mod [ r ; ( r ) d ], separately before and after perihelion (thickcurve). Of the 11 Jewitt-Matthews post-perihelion data points, 9 were successfully matched by a g mod law, the two discarded ones possiblysuggesting the presence of a minor outburst some 160 days after perihelion. The data point by Gr¨un et al. (2001) at 3.90 AU from theSun (+273 days from perihelion) was employed to calibrate their other observations; it required a multiplication factor of 1.3 to fit the g mod curve. The preperihelion data by Gr¨un et al. at 4.58 AU and 2.82 AU from the Sun were then adjusted by this factor, linked to theother five preperihelion data points, and all subsequently fitted by a g mod law, as depicted in the figure. At perihelion the two laws aredisconnected, the post-perihelion production rate being nominally 23% higher than the preperihelion rate, as illustrated by the dots. Thisdiscrepancy is bridged by an empirical dashed curve, peaking shortly after perihelion. A logarithmic differentiation of Equation (40), d ln Ψ H2O ( r ) d ln r = nkr n h ( r ) n d − ( r ) n H2O ih ( r ) n d + r n ih ( r ) n H2O + r n i , (41)implies that it is the difference between the scaling dis-tances of the g mod laws for the production of dust andwater, respectively, that exclusively determines whetherthe dust loading increases or decreases with heliocentricdistance: when the scaling distance for the dust produc-tion is greater, as is the case for C/1995 O1 both beforeand after perihelion, the dust loading increases with in-creasing distance from the Sun. However, Ψ H2O does notdiverge, as lim r →∞ d Ψ H2O /dr = 0. The limit, equalinglim r →∞ Ψ H2O ( r ) = Ψ (0) H2O " ( r ) n d + 1( r ) n H2O + 1 k ≃ Ψ (0) H2O " ( r ) d ( r ) H2O nk , (42)is to be employed in practice with caution because of theaforementioned transformation of the dust ejection pro-cess from a water dominated mode to a carbon-monoxidedominated mode near 4 AU from the Sun.Table 25 suggests that our results for both the normal-ized and integrated dust loading of the water flow rangebetween about 11 and 12 and that they are at most onlymarginally higher prior to perihelion. The parameters from Tables 19 and 25 allow one to predict the mass load-ing by dust of the water flow, Ψ H2O ( r ), as well as thatof the combined flow of water and carbon monoxide,Ψ H2O+CO ( r ), and of water, carbon monoxide, and carbondioxide, Ψ H2O+CO+CO2 ( r ), for a broad variety of heliocen-tric distances r . The results of these computations, inTable 26, show that the dust loading increases system-atically, both before and after perihelion, only for theflow of water. Once we consider a more realistic case,a combined flow of water with other species, the dustloading peaks and then may drop at larger heliocentricdistances. This behavior is a corollary of Equation (41)written for a flow or more than one species, becausethe scaling distance of dust is smaller than the scalingdistances of carbon monoxide and carbon dioxide bothbefore and after perihelion. We note that in the caseof a combined flow of water, carbon monoxide, andcarbon dioxide, which in Table 26 approximates the trueconditions in the comet’s atmosphere most closely, thedust loading never exceeds 11 before perihelion and 10after perihelion. The table also allows one to derive adust loading of a flow of carbon monoxide, Ψ CO , and ofcarbon dioxide, Ψ CO2 :Ψ CO ( r ) = h Ψ − H2O+CO ( r ) − Ψ − H2O ( r ) i − (43)Ψ CO2 ( r ) = h Ψ − H2O+CO+CO2 ( r ) − Ψ − H2O+CO ( r ) i − . omet C/1995 O1: Orbit, Jovian Encounter, and Activity 39 Table 26
Nominal Mass Loading by Dust of Gas Flows in C/1995 O1Dust Loading of Gas Flow Pre- and Post-PerihelionDistance Ψ
H2O ( r ) Ψ H2O+CO ( r ) Ψ H2O+CO+CO2 ( r )from Sun, r (AU) pre post pre post pre post1.0 11.3 10.9 8.6 8.9 7.6 7.51.5 11.4 11.0 8.7 9.0 7.6 7.62.0 11.8 11.5 8.9 9.3 7.7 7.82.5 13.0 13.0 9.4 9.5 8.1 8.23.0 15.8 16.8 10.6 11.8 8.9 9.03.5 22.4 25.8 12.6 13.8 10.1 9.44.0 38.0 46.3 14.5 13.3 10.7 7.84.5 75.5 90.1 13.9 8.7 9.5 4.7 Before comparing the numbers in Table 26 with theresults reported elsewhere, a caveat first. Whereas care isusually taken, as it should, to clearly distinguish betweena dust loading of the water flow, Ψ
H2O , on the one handand a dust loading of the gas flow (that is, of the totalflow of a number of volatile species), Ψ gas (Ψ gas < Ψ H2O ),on the other hand, the two quantities are not alwaysdifferentiated (e.g., A’Hearn et al. 1995; Schleicher et al.1997), apparently with a tacit, but — as seen from Table26 — rather questionable, premise that water dominatesthe sublimation process.A high mass loading by dust in C/1995 O1 was com-mented on by numerous researchers. With one exception,the reported values of the dust loading based on the ob-servations at submillimeter and millimeter wavelengthsare in good to excellent agreement with the numbers inTable 26. The exception is an observation at wavelengthsbetween 1.1 mm and 2.1 mm in late February 1997, fiveweeks before perihelion, by Senay et al. (1997), who re-ported for Ψ
H2O a fairly low value of 3.5. On the otherhand, Jewitt & Matthews (1999), whose data we employextensively, estimated a lower limit to Ψ
H2O+CO at 5 closeto perihelion; having accounted for the contribution fromcarbon monoxide as a fifth of the contribution from wa-ter, they effectively estimated a lower limit to Ψ
H2O at 6.For greater heliocentric distances, the dust loading wasreported by Gr¨un et al. (2001) to equal 9 at 4.58 AUbefore perihelion, 6 at 2.82 AU before perihelion, and 10at 3.90 AU after perihelion. For the gas production theyemployed the combined rate of water and carbon monox-ide. At 4.58 AU before perihelion we find from Table 19a gas production rate of 3.0 × g s − (15% H O, 85%CO), so that Ψ
H2O+CO ≃
10; at 2.82 AU before perihelion,2.4 × g s − (70% H O, 30% CO), so that Ψ
H2O+CO ≃ × g s − (33%H O, 67% CO), so that Ψ
H2O+CO ≃
10, in excellent agree-ment with Gr¨un et al. (2001). Since we adjusted theirdust production rates by a factor of 1.3 in Figure 15,the dust loading of the H O+CO flow with these num-bers becomes 13, 8, and 13, respectively, still in goodagreement, given that the uncertainty in Gr¨un et al.’sdust production rates was estimated by the authors at afactor of two in the least.Three investigations employed a dynamical approach,and in this case it was a dust loading of the total gas flow,Ψ gas , that was determined. Fitting the dust morpholog-ical features in the head of C/1995 O1, Vasundhara & Chakraborty (1999) determined a lower limit of 4.8 ± ∼ CO must have been at least 5; theydeduced a dust production rate of 8 × g s − some320 days before perihelion, about a factor of 5 belowthe curve in Figure 15. Table 19 suggests that the COproduction rate was in fact near 2.7 × g s − , leadingto a dust loading of only 3 with Fulle et al.’s (1998) dustproduction rate. From Table 26 we find Ψ CO ≃
17 at adistance of 4.5 AU. Fulle et al.’s dust rate estimate of5 × g s − in 1993, at a heliocentric distance of 13 AUimplies Ψ CO ∼ r − between 13 AU and 6.7 AU and as approximately r − between 6.7 AU and 4.6 AU before perihelion.Lisse et al. (1999) determined a dust production rateof ∼ × g s − and a dust loading of the gas flowof ∼ µ m spectral energy distribution on1996 October 31, when the comet was 2.54 AU from theSun. With this dust rate we find a loading of 5.9 forwater and 4.3 for water plus carbon monoxide. FromTable 25, we compute a dust production rate about 2.2times higher. Similarly, their estimate of 1 × g for anorbit-integrated loss of dust is a factor of 3–4 lower thanJewitt & Matthews’ (1999) and ours.Finally, several investigations determining the massloading by dust were based on photometry or spectro-photometry of scattered sunlight at optical wavelengths;with no exception, the mass of dust ejecta (and thus theloading) was in these studies underestimated usually by awide margin. It is well known that millimeter-sized par-ticles dominate the mass distribution, while micron-sizedparticles the cross-sectional distribution of cometary dustejecta. Optical observations always sample primarilythe smaller grains and fail to adequately account for thelarger ones, as remarked by Jewitt & Matthews (1999).This effect manifests itself in the dust production ratesreported by Weaver et al. (1997, 1999b) for heliocentricdistances exceeding 2.4 AU both before and after perihe-lion, derived via a proxy parameter Af ρ (A’Hearn et al.1995) from their HST and IUE observations. The dustproduction rates are much too low by about one orderof magnitude, even though the computed dust loading ofthe water flow exceeds unity for all entries in Table I ofWeaver et al. (1999b). This effect is nicely illustrated by McDonnell et al.’s (1991)Figure 13 for the dust ejecta from 1P/Halley detected during theGiotto encounter. The values of Af ρ , derived from the spectra of the Interna-tional Ultraviolet Explorer (IUE), were based of course on scatteredsunlight at UV rather than optical wavelengths; one of two Echellespectrographs worked in a spectral range of up to 3300 ˚A at most. Af ρ , which they did not convert to the dustproduction rates. They did, however, point out that thegas production (including OH) was in C/1995 O1 about20 times higher than in 1P/Halley, while the dust pro-duction was more than 100-fold greater at comparableheliocentric distances. Since the best estimate of thedust production rate of 1P is 5 × g s − (McDonnellet al. 1991) at the encounter time of the Giotto space-craft (0.90 AU from the Sun post-perihelion), one shouldexpect a rate of at least 4 × g s − for C/1995 O1 at1 AU from the Sun after perihelion, which is by morethan 2 σ higher than predicted by the model in Table 25— still a fair agreement, especially considering that thiscomparison involves a premise of comparable perihelionasymmetry for the two objects. However, when one em-ploys a conversion factor between Af ρ and the dust pro-duction rate recommended by A’Hearn et al. (1995), oneobtains ∼ × g s − at 1 AU before perihelion,a factor of ∼ Af ρ system-atically underrates the dust production; the data thatSchleicher et al. (1999) referred to for 1P are representedby Af ρ = 10 . cm at 1 AU preperihelion (A’Hearn et al.1995) and therefore by a dust rate of only 8 × g s − .On the one hand, this rate indeed is more than 100 timeslower than the rate for C/1995 O1, but on the other handit is also more than 6 times lower than McDonnell et al.’s(1991) post-perihelion rate for 1P normalized to 1 AUfrom the Sun. The lesson appears to be that compari-son of two comets in terms of Af ρ provides a correct dustproduction rate for the comet of interest, if a correct dustproduction rate is available for the other comet.Monitoring the dust distribution between 4.6 AU and2.9 AU before perihelion in a red passband, Rauer et al.(1997) derived production rates much too low by a factorof 2 to 4 relative to the curve in Figure 15; their values ofthe dust loading of the flow consisting of water and car-bon monoxide, 2.3 through 6.0, then become 9 through13, in good agreement with the numbers in Table 26.A more extensive investigation by Weiler et al. (2003),covering a range of heliocentric distances from 4.6 AUto 2.9 AU preperihelion (using in part the observationsexamined by Rauer et al. 1997) and from 2.8 AU to12.8 AU post-perihelion, also began by determining Af ρ ,which was converted to a dust production rate on cer-tain assumptions concerning the dimensions and activityof the nucleus and regarding the size distribution, termi-nal velocity, bulk density, geometric albedo, and phasefunction of dust particles. The fit was relatively insensi-tive to variations in some of the tested parameters andparameteric functions with an exception of the activefraction of the surface: 19% of a nucleus 30 km in ra-dius provided the best match; this indicated an activearea of about 2150 km . Unfortunately, Weiler et al.’sresulting dust production rates within 4.6 AU of the Sunwere up to almost a factor of 10 lower than the ratesby Gr¨un et al. (2001) or those implied by the curve inFigure 15. At 7 AU post-perihelion, Weiler et al.’s dustrate was close to 1 × g s − , again at least one order ofmagnitude lower than needed to satisfy Sekanina’s (1996)dust-loading condition at ∼ ∼ × g s − was a factor of 2–3 lower than Fulle et al.’s (1998) dustrate at 13 AU preperihelion. Weiler et al.’s major resultof a nearly constant dust loading of Ψ H2O+CO ≃ . ± . × g of dust was lost by the comet over its orbitabout the Sun, an amount of mass that was by more thanfive orders of magnitude (sic!) lower than our, Jewitt &Matthews’ (1999), and Lisse et al.’s (1999) estimates.To summarize, we are convinced that only investiga-tions based on thermal flux observations at wavelengthsgreater than about 100 µ m and/or on carefully executeddynamical analysis of the dust ejecta provide reliable in-formation on the total losses of dust from C/1995 O1.These results suggest that the comet’s production of dustexceeded a rate of 10 g s − near perihelion and that thetotal loss of dust per orbit was on the order of 10 g.The dust loading of the gas flow was more than a fac-tor of 10 for water (even in the proximity of perihelion)and close to 10 for water plus carbon monoxide. Such ahigh dust loading is bound to affect downward the dust-particle velocities, an issue that is addressed next. Effects of Dust Loading on Terminal Velocities ofMicroscopic Grains, and Expansion of Dust Halos
A mass loading by dust of the gas flow has a significanteffect on the drag acceleration that the gas imparts to thedust particles during their liftoff from the nucleus. Theterminal velocity v d ( s ) that a particle of radius s attainsat the time its motion decouples from the gas flow is acomplicated function of the interaction (Probstein 1969),but the relationship between v d and Ψ gas is simplifiedand can be written in closed form for a particle of trivialdimensions; for its terminal velocity, ( v d ) lim , Probsteinprovides an expression:( v d ) lim = lim s → v d ( s ) = (cid:18) T gas gas (cid:19) n ( c p ) gas (44) × (cid:2) ( γ gas − M (cid:3) + c dust Ψ gas o where T gas , M gas , ( c p ) gas , and γ gas are, respectively, thetemperature, the initial Mach number, the specific heatcapacity at constant pressure, and the heat capacity ratio(Poisson constant) for the gas that drives the dust awayfrom the nucleus, and c dust is the specific heat capacity ofthe dust particle. We note that ( v d ) lim is identical withthe parameter v from Equation (35); by its substitutioninto Equation (44), we obtain for a dust loading of thewater flowΨ H2O = 2( c p ) H2O T H2O h ( γ H2O − M H2O i − v v − c dust T H2O . (45)Since γ H2O = 1 .
33 and M H2O < H2O >
0, the secondterm in the square brackets is much smaller than unity.This equation is solved by computing a first-guess valueof Ψ
H2O with this term ignored; by finding an initial Machnumber from an approximate formula, M H2O = 1 − Ψ . ; (46)omet C/1995 O1: Orbit, Jovian Encounter, and Activity 41and by iterating Equations (45) and (46) until they con-verge; Equation (46) provides a reasonable approxima-tion to M H2O for all values of Ψ
H2O .Our objective is now to apply Equation (45) to ap-propriate observations in order to test whether the veryhigh dust loading of the water flow that resulted fromcomparison of the production rates of water and dust isindependently confirmed in this fashion. The best ap-proximation to a particle of trivial dimensions, which— according to Equation (44) — is accelerated to thehighest terminal velocity, is obviously provided by thesmallest ejected grains. We already remarked that Minet al. (2005) advocated the presence of grains as smallas 0.01 µ m in radius in order to fit the comet’s spectralenergy distribution. An overabundance in C/1995 O1 ofunusually small submicron-sized dust grains — equal toor smaller than 0.1 µ m in radius, especially near per-ihelion — was reported by numerous researchers (e.g.,Williams et al. 1997, Harker et al. 1999, Lisse et al. 1999,Hayward et al. 2000). These grains, with the highest ter-minal velocities near 700 m s − , populating the leadingboundaries of a succession of expanding dust halos, wereobserved extensively over a period of about two monthsaround perihelion, as already noted in Section 9.5. Be-sides their velocities, application of Equation (45) re-quires the knowledge of the mineralogical compositionof the submicron-sized grains, because the heat capacity c dust is strongly temperature dependent.The composition of microscopic dust from C/1995 O1was examined many times with use of a variety of tech-niques (e.g., Hanner et al. 1999, Wooden et al. 1999,2000, Hayward et al. 2000, Gr¨un et al. 2001, Harker et al.2002, Moreno et al. 2003, Min et al. 2005). Below we com-pare the results of two very different investigations, oneby Harker et al. (2002) and the other by Hayward etal. (2000); both dealt with the grain populations over thecritical time near perihelion. Harker et al. considered fivecategories of dust: amorphous carbon, amorphous andcrystalline olivine, and amorphous and crystalline py-roxene (orthopyroxene). Their synthetic spectral energydistribution over a range from 1 µ m to 50 µ m, designedto model the comet at 0.93 AU from the Sun (10 daysafter perihelion), was fairly consistent with the contribu-tions, by mass, of fully 64% of crystalline olivine with aradiative equilibrium temperature of 220 K; 9% of crys-talline orthopyroxene with a temperature of 320 K; 12%of amorphous pyroxene (closest to Mg . Fe . SiO ) with435 K (for grains 0.1 µ m in radius); and even smallercontributions from amorphous olivine and carbon.Hayward et al. (2000) combined their investigation ofthe thermophysical properties of the microscopic dust inC/1995 O1 with analysis of particle dynamics and themorphology of the dust halos. They concluded that thethermal emission of the halos arose from submicron-sizedparticles subjected to radiation pressure accelerationssmaller than the Sun’s gravitational acceleration, whichexplains a fairly uniform spacing of the halos, controledby a constant ejection velocity of the dust that populatedtheir leading boundaries. These relatively low accelera-tions are consistent with silicate, but not carbonaceous,material. Whereas carbon grains accounted largely for Harker et al.’s (2002) results were amended in an importanterratum that was published two years later; see the reference. the continuum near 8 µ m and almost exclusively forthe 3–5 µ m continuum, silicates dominated the 10 µ mregion of the thermal spectrum. Unlike Harker et al.(2002), Hayward et al. concluded that amorphous py-roxene was the most abundant silicate (at least 40% bymass) and that the pyroxenes contributed almost two-thirds of the silicate grain population. Hayward et al.also pointed out that even though cometary silicates tendto be magnesium rich (whose temperatures are typicallybelow the blackbody temperature), a strong contamina-tion by absorbing material raises their temperature abovethe blackbody temperature regardless of their intrinsiccomposition. Hayward et al. determined that the con-tinuum color temperature was elevated in the halos ofC/1995 O1; measured from the spectra taken over a pe-riod of 1997 March 24–28 it averaged 395 K at 0.92 AUfrom the Sun, implying a superheat of 1.36.Even though carbon grains with temperatures in excessof 500 K were present in C/1995 O1, we dismiss their rolein populating the leading boundaries of the expandinghalos because they were subjected to much higher radi-ation pressure accelerations (greatly exceeding the Sun’sgravitational acceleration) than were silicate grains, thusfalling increasingly behind in the course of the halos’ ex-pansion. We compute the specific heat capacity c dust fortwo effective dust-particle temperatures; after normaliza-tion to 1 AU from the Sun these are 210 K [correspond-ing to the dominant particles in Harker et al.’s (2002)model] and 380 K [consistent with Hayward et al.’s (2000)model]. Although radiative equilibrium temperatures donot necessarily vary as r − , we will use this power law asan admissible approximation in a narrow range of helio-centric distances near 1 AU.The variations in the specific heat capacity of solidswith the temperature T are known to folow the Debyelaw at very low T and to converge to a constant at veryhigh T . For our application of Equation (45) it will sufficeto approximate c dust ( T ) by c dust ( T ) = ξ T ξ T + ξ T + ξ T , (47)which is readily seen to satisfy either of the two condi-tions, as c dust ≃ ξ T when T → c dust → ξ /ξ when T → ∞ . The constants ξ , . . . , ξ are determined by fit-ting appropriate data. Combining the Debye law for lowtemperatures with the Dulong-Petit law for high temper-atures, we obtain from Equation (47) an estimate for theDebye temperature, T D , T D = 2 π r π ξ . (48)In practice, we employed the heat capacities that werecomputed by Yomogida & Matsui (1983) for a numberof meteorites — as comet-dust analogs — in a temper-ature range from 100 K to 500 K (at a 50 K step) fromtheir mineral compositions and specific-heat data com-piled by Touloukian (1970a, 1970b). The numbers areknown to be rather compatible with the more recent re-sults by Consolmagno et al. (2013). To streamline the cu-bic fit, the averaged expressions of T /c dust for the ninestandard temperatures between 100 K and 500 K werelinked with a synthetic data point at T = 20 K that wasvaried until a condition was satisfied that required, in2 Sekanina & Krachtaccordance with a Debye-temperature normalized heat-capacity curve, that at T D the heat capacity reach about95% of its limiting value, which in the notation of Equa-tion (47) equals ξ /ξ . This constraint was equivalent toassigning c dust ( T ) = 0 . − K − at T = 20 K, whichresulted in the following formula for a representative heatcapacity of the meteorites: c dust ( T ) = 2 . T . T +0 . T +2 . T , (49)where T is expressed in units of 100 K and c dust ( T ) comesout in J g − K − . For the Debye temperature this relationoffers T D = 838 K and c dust ( T D ) = 0 .
98 J g − K − , while( c dust ) lim = lim T →∞ c dust ( T ) = 1 .
03 J g − K − , so that, in-deed, c dust ( T D ) / ( c dust ) lim = 0 .
95. For olivine and pyrox-ene the Dulong-Petit law indicates that ( c dust ) lim equals,respectively, 1.01 and 1.07 J g − K − , in good agreementwith the result from Equation (49), thus suggesting thatin terms of the specific heat capacity the meteorites areacceptable analogs for the dust in C/1995 O1. For thecritical temperatures between 210 K and 380 K at 1 AUfrom the Sun, c dust varies from 0.60 to 0.84 J g − K − ,in a range that we employ below in the applications ofEquation (45).The two quantities for water vapor in Equation (45)that still need to be addressed are its temperature, T H2O ,and specific heat capacity at constant pressure, ( c p ) H2O .The temperature of water vapor is lower than the tem-perature of water ice on the nucleus, T ice , from which itsublimates. The relationship between the two is a func-tion of the initial Mach number, M H2O , and the heat ca-pacity ratio, γ H2O ; the gas dynamics approach providesthe following expression (e.g., Cercignani 1981): T H2O = T ice h π +1)Ω − π Ω(1+Ω ) i = T ice h (1+Ω ) − π Ω i (50) ≃ T ice (cid:16) − π Ω (cid:17) for Ω ≪ , whereΩ = M H2O γ H2O − γ H2O +1 (cid:16) γ H2O (cid:17) = 0 . M H2O . (51)Since in the presence of dust M H2O <
1, Ω is much smallerthan unity, making the approximation in the last line ofEquation (50) reasonably accurate.The temperature of water ice was, as a function of he-liocentric distance, determined by solving the energy bal-ance on the comet’s nucleus, using the isothermal model(Section 4). Within a few tenths of AU of a unit heliocen-tric distance, it can be approximated with high accuracy(to better than ± T ice = 194 . − . r −
1) + 5 .
64 ( r − , (52)where r is in AU. At heliocentric distances 0.9 AU to1.1 AU the temperature of ice varies only by 3.2 K; eventhe extreme temperature at the subsolar point is merely ∼
10 K higher.The specific heat capacity of water vapor at constantpressure, ( c p ) H2O , is nearly independent of temperaturebetween at least 50 K and 250 K; to two decimal places,
Table 27
Upper Limit to Mass Loading by Dust of Water Vapor Flow inC/1995 O1 from Halo Expansion RatesHalo Upper Limit to Dust LoadingDate Distance Expansion of Water Vapor Flow a , Ψ H2O r (AU) v exp (km s- ) T =210 r - T =380 r - Feb. 27 1.10 0.718 0.57 0.83Mar.12 0.98 0.660 1.3 2.313 0.97 0.623 2.0 5.223 0.93 0.686 1.0 1.624 0.93 0.775 0.25 0.3328 0.92 0.691 0.91 1.5Apr. 2 0.91 0.660 1.3 2.58 0.92 0.628 2.0 5.015 0.95 0.610 2.4 8.319 0.97 0.664 1.2 2.221 0.98 0.610 2.3 7.722 0.99 0.743 0.43 0.5927 1.02 0.603 2.5 9.1Average 0.97 0.667 1.2 2.1Va&Ch b Notes. a The first of the two columns refers to the dominant population ofcrystalline olivine grains in Harker et al.’s (2002) model, the secondcolumn to the dominant population of amorphous pyroxene grainsin Hayward et al.’s (2000) model. b Based on a value of v determined by Vasundhara & Chakraborty(1999) from analysis of an April 10 image, thus providing true load-ing rate (not an upper limit); estimated error of v is ± − . it is approximated by 1.85 J g − K − in this entire range(Freedman & Haar 1954, Wagner & Pruss 2002, Murphy& Koop 2005).We are now ready to calculate the mass loading bydust of the water flow near perihelion to check whetherthe high loading rates, exceeding 10, established fromcomparison of the production rates in Table 26, are cor-roborated. As mentioned in Section 9.5, Braunstein etal. (1999) systematically investigated the expansion rateof the concentric dust halos of C/1995 O1 on 13 daysbetween 25 February 1997 (33 days before perihelion)and 27 April 1997 (26 days after perihelion). Derivedfrom the spacing of the leading boundaries of successivehalos and an accurately determined rotation period, theexpansion velocity was an average of the terminal ve-locities of the smallest ejected grains, whose radii wereestimated at 0.1 µ m at the most. As such, this halo ex-pansion velocity, v exp , should be by at least several tensof meters per second lower than the limiting velocity v .Nonetheless, substitution of the halo expansion velocity v exp for the limiting velocity v in Equation (45) still isa very good approximation that should provide us witha fairly tight upper limit on the mass loading by dust ofthe water vapor flow near perihelion of C/1995 O1. Surprisingly, Table 27 demonstrates that when appliedto Braunstein et al.’s (1999) halo expansion velocities,Probstein’s (1969) model provides us with the dust load- The effect of projection onto the plane of the sky could haveslightly been overcompensated by Braunstein et al. (1999) in somecases in which the halos emanated from sources that were ratherfar from the subsolar latitude; still, the deprojected velocities area much better measure of v than the uncorrected velocities. omet C/1995 O1: Orbit, Jovian Encounter, and Activity 43ing rates Ψ H2O that are, on the average, at least one orderof magnitude lower than are the expected numbers (Ta-ble 26). A dust loading of ∼
11 would require at 1 AUfrom the Sun that the expansion velocities be lower than0.53 km s − for the dust temperature of 210 K and lowerthan 0.60 km s − for 380 K. This effect could not possi-bly be caused by systematic errors in Braunstein et al.’s(1999) expansion-velocity measurements, because a sim-ilar result was independently obtained by Vasundhara& Chakraborty’s (1999), who investigated the dynamicsof the dust features. Unfortunately, the April 10 imagewas the only image that Vasundhara & Chakraborty an-alyzed from the period of time between late Februaryand late April of 1997, when the straightforwardly inter-pretable concentric halos were observed (Braunstein etal. 1999). The result by Vasundhara & Chakraborty islisted in Table 27 for comparison; in this case the pro-jection issue was mute, because one of their parameterswas directly related to the tabulated value of v . Ofinterest is the statement by these authors that a lower limit on the dust loading of the gas flow was equal to3.4 ± < ∼
11. Indeed, a nomi-nal lower limit on the sublimation rate that Vasundhara& Chakraborty (1999) offer for the April 10 image is2.6 × − g cm − s − , while an average sublimation rateof water ice at the same heliocentric distance is, accord-ing to the isothermal model, 1.1 × − g cm − s − , or 0.4the nominal lower limit. Unfortunaely, the derived subli-mation rate is burdened by a very high error, so that thissublimation-rate (but not the dust-loading) argument israther weak. The net result of this discussion is that the sublimation of water ice alone could not explain thehigh expansion velocities of the prominent dust halosand that outgassing by other parent molecules shouldhave contributed to the effect . This conclusion is rem-iniscent of, and supports, the conclusion we made in Sec-tion 9.4 that water ice did not account for more than the total mass outgassed from C/1995 O1 and may have— in an extreme case — accounted for as little as .We are now in a position to infer, in general terms,possible properties of the missing parent molecules ; wesearch for a type of volatiles, each of which is, on theone hand, very highly loaded with dust, Ψ gas → ∞ , yet,on the other hand, can — near 1 AU from the Sun —accelerate submicron-sized silicate grains to terminal ve-locities of nearly 0.7 km s − . Returning to Equation (44),we find a limit for infinitely high dust loading,lim Ψ gas →∞ ( v d ) lim = v = (2 c dust T gas ) . (53)This is a condition for the temperature of a heavily loadedgas as a function of the velocity and specific heat capacityof the smallest (silicate) grains in the expanding halos,which implicitly involves the dust-particle temperature.However, the gas temperature is essentially identical withthe temperature of the sublimating solid (i.e., a nonwaterice), T gas → T solid because for a very high mass loading bydust the initial Mach number of the gas flow approacheszero ( M gas →
0) and Ω → T solid = v c dust . (54) Next we find a solution to Equation (54) for a heliocen-tric distance of Hayward et al.’s (2000) thermal infraredspectra, r = 0 .
92 AU. From Equation (49) it follows thatthe specific heat capacity c dust = 0 .
62 J g − K − for thecrystalline olivine grains that Harker et al. (2002) ad-vocated ( T = 220 K), but c dust = 0 .
85 J g − K − for theamorphous pyroxene grains preferred by Hayward et al.( T = 395 K). With v = 0 .
667 km s − Equation (54) gives: T solid = 359 K (Harker) or
262 K (Hayward) . (55)The relatively cool grains of Harker et al. can be ruledout, because no ice can have a temperature of ∼
360 Kon the surface of a comet at a heliocentric distance of0.92 AU, at which the blackbody temperature is 290 K.However, the hot grains suggested by Hayward et al. dofit the proposed hypothesis, which thus implies that low-volatility ices that sublimate just below the blackbodytemperature could upon release accelerate hot grainsto terminal velocities near 0.70 km s − . This conclu-sion is not meant to contest the participation of waterin the process of dust ejection, but to emphasize thatthe low-volatility ices (owing to their higher temperatureupon sublimation) are critical for imparting the grainstheir high velocites, otherwise unattainable. Still hottergrains, up to 440 K (e.g., Hanner et al. 1999), imply evenlower T solid , down to 253 K. And since Ψ gas , though high,is finite, these values of T solid tend to be upper limits.In an effort to further learn about these low-volatilityices, in Figure 16 we present a plot of sublimation tem-peratures, T sub , at a heliocentric distance of 0.92 AU for ·············································································································································································································································································· ································· ································ EXAMPLES OF LOW-VOLATILITY ICES
BLACKBODYTEMPERATURE EQUILIBRIUMTEMPERATURE ❙❙❙♦ T sub ✓✓✓✼ f sub r = 0 .
92 AU
Adopted rangeof upper limits (cid:8) on T solid
14 16 18 20 22 24LATENT HEAT OF SUBLIMATION, L sub (kcal mol − )180 K210 K240 K270 K300 K T sub f sub H COOCH COOH3 . . . . . .C H COOH4 . . . . . .CH CONH H (CH) H (CO) O7 . . . . . C H (NH ) H NH NO H (OH)
10 . . . . C H CHOOH ①❤ ①❤ ①❤ ①❤ ①❤ ①❤ ①❤ ①❤ ①❤ ①❤ ····················································································································································································································································································································································································································································································································································································································································································································································································································································· ···························································· ························································································································································································································································································································································································································································································································································································································································································ ··················································································· ······································································
Figure 16.
Sublimation temperature, T sub (solid circles), and afraction of the impinging solar radiation that is spent on sublima-tion, f sub (open circles), for ten organic low-volatility ices at a he-liocentric distance of 0.92 AU. In the order of increasing heat of sub-limation, L sub , they are: methyl methacrylate (1), acetic acid (2),propanoic acid (3), acetamide (4), pyrocatechol (5), glutaric anhy-dride (6), m-phenylenediamine (7), nitranilin or 3-nitroaniline (8),hydroquinone (9), and parahydroxybenzaldehyde (10). The black-body temperature at this distance from the Sun is 290 K, while theequilibrium temperature of a nonsublimating body with a Bondalbedo of 4% and a unit emissivity is 287 K, a limit that no ice atthat albedo and emissivity can exceed. Seven of the species have T sub near or exceeding the critical temparature T solid . The corre-lation between the sublimation heat and sublimation temperatureis high, with scatter not exceeding a few K at most. In the dis-played range of the heat of sublimation (14,000–25,000 cal mol − ),the fraction of the solar energy that is spent on the sublimation (asopposed on the thermal reradiation) spans about one order of mag-nitude. A range of upper limits on the ice temperature T solid thatsupports a dust-grain velocity of 0.667 km s − is also depicted. Table 28
Sublimation Rate of and Drag on Dust Grains by Low-Volatility Ices Near Perihelion of C/1995 O1 (Two Grain Temperatures Assumed)Latent Sublimation Rate Per Grains Accelerated to VelocityLow-Volatility Molar Heat of Unit Area (Water = 1) v (km s − ) Near PerihelionOrganic Molecule Formula Mass Sublimation(g mol − ) (cal mol − ) by number by mass T gr =395 K T gr =440 KAcetamide CH CONH H (CH) H (CO) O 114.10 20 550 0.167 1.06 0.676 0.687m-Phenylenediamine C H (NH ) H NH NO H (OH) H CHOOH 122.12 24 300 0.053 0.36 0.691 0.703 ten organic molecules whose sublimation heat, L sub , ex-ceeds 14,000 cal mol − , which is about 20% higher thanthe sublimation heat of water ice. The molecules wereselected out of a very limited pool of organic compoundsfor which — in addition to L sub — the saturated pressurewas available as a function of temperature from the NISTwebbook (see footnote 10) in terms of the three constantsof the Antoine equation, even though sometimes not, un-fortunately, covering the needed temperature range, thusrequiring an extrapolation. Figure 16 shows a high de-gree of correlation between the two quantities. Of partic-ular interest are the critical values of T sub and the corre-sponding values of L sub that are close to and higher than T solid . For our suggested range of 253 K ≤ T solid ≤
262 Kone finds that T sub = T solid for the ices with a sublima-tion heat between about 19 000 and 20 000 cal mol − . Allices with a higher sublimation heat accelerate minusculedust particles to terminal velocities that are higher than0.667 km s − , while the ices with only a moderately lowersublimation heat still should support velocities of aboutthis magnitude.Not all species in the range of Figure 16 should neces-sarily fit the curve of T sub ( L sub ) as closely as do the tenmolecules plotted, and the relationship should not be ex-trapolated to L sub <
14 000 cal mol − ; indeed, water icehas on the given assumptions (isothermal model and theadopted albedo and emissivity) at 0.92 AU a tempera-ture of 196 K and would leave an offset of more than 15 Kfrom the curve. In fact, the sublimation temperature ofwater is a bit higher than those of the two most volatilemolecules plotted in Figure 16.At the other end of the sublimation-heat range, thecurve can confidently be extrapolated (with an error ofa few K), because of the blackbody temperature limit. Astronger constraint is provided by the equilibrium tem-perature of a nonsublimating body, if the albedo is higherthan the emissivity drop from unity. For 0.92 AU fromthe Sun, the equilibrium temperature of our isothermalobject is 287 K. The temperature of the least volatile iceamong the ten is only 6 K below this limit.Also plotted in Figure 16 are the variations with L sub in the fraction of the incident solar radiation that is be-ing spent on the ice’s sublimation, f sub . When f sub → f sub →
1, the Sun’s radiation is spent entirely on sublimation, the sur-face temperature is nearly constant, and the sublimationrate varies as an inverse square of heliocentric distance.The first three most volatile species in Figure 16 use at0.92 AU a greater fraction of the energy on sublimation,whereas the opposite is the case with the seven leastvolatile molecules. The fraction f sub spans about one or-der of magnitude between the most and the least volatileices in the figure and the sublimation and reradiationbreak even for a sublimation heat of ∼
18 800 cal mol − ,when f sub = .Sublimation temperatures close to or above the criticaltemperature T solid that are necessary to accelerate thehot submicron-sized silicate grains to terminal velocitiesnear 0.7 km s − are reached by only seven among themolecules plotted in Figure 16, one of them marginally.These are merely examples among a sheer number of ex-isting organic compounds, for most of which needed dataare unavailable. Listed in Table 28, these seven moleculesillustrate what types of species, whose existence was im-plied in Section 9.4, are believed to have amply contrib-uted to the activity of C/1995 O1. They are, first of all, low-volatility ices, with a minimum sublimation heatnear 19 000 cal mol − ; in addition, they are mostly veryheavy molecules, several times heavier than water, withthe molar mass typically exceeding 100; as a corollary,their sublimation rates per unit surface area are, bymass, comparable to that of water ice , even thoughthey are much lower in terms of number of molecules be-cause of their lower volatility. Given the large numberof more simple organic compounds already detected inC/1995 O1, their enormous overall number and varietymake their summary major contribution to activity compelling, whereas a sublimation heat from 19 000 to25 000 cal mol − for many of them makes, near 1 AUfrom the Sun, this population of organic compounds tobe both sufficiently volatile to outgas vigorously and to sublimate at temperatures high enough to acceleratehot submicron-sized grains of dust to high terminalvelocities to explain observational evidence that is incon-sistent with water ice-based activity. While many maynever be detected individually, we propose that, in theirsum, these ices are the molecular compounds that wereresponsible for the incompleteness of the integratedmass-loss distribution in Section 9.4 (Figure 14).Large organic molecules whose sublimation heat ex-ceeds ∼
25 000 cal mol − sublimate too insignificantly atheliocentric distances near 1 AU, with their role becom-omet C/1995 O1: Orbit, Jovian Encounter, and Activity 45ing progressively marginalized. We do not attend to thiscategory of material, although it becomes increasinglyimportant for comets with perihelion distances ≪ ∼
18 200 cal mol − , near the lower boundaryof the category of organic ices that is proposed to drivehot microscopic dust into the atmosphere — is known(Crovisier et al. 2004a), as already pointed out. RAMIFICATIONS, AND FUTURE WORK
The proposed population of heavy organic molecules oflow volatility has implications for the dynamical behaviorof the nucleus of C/1995 O1. Although individually theabundances of these compounds represent only a minorfraction of the water abundance, their sheer number anda variety of their composition may make them to sum-marily contribute a mass of gas that is comparable to —if not greater than — the mass of water vapor. Accord-ingly, the heavy organic low-volatility ices could upontheir release add substantially to the total momentumthat the nucleus is exposed to while revolving about theSun and thus augment the nongravitational effect that ismeasured as part of the orbit determination.Part II of this investigation will focus on the issues re-lated to the nucleus of C/1995 O1. Because of an enor-mous size of the nucleus (see Szab´o et al. 2011, 2012 forupdates), the sublimation-driven nongravitational effectswere expected to be trivial and their eventual incorpora-tion into the equations of motion was accompanied bymuch reluctance (e.g., Marsden 1999). Nonetheless, theirintroduction turned out to be essential to improving theorbital solution; although our preferred Weight System II(Table 10) offers a total nongravitational acceleration of A = p A + A + A = (0 . ± . × − AU day − at1 AU from the Sun, only ∼ the acceleration derived byMarsden (1999) and that determined by Szutowicz etal. (2002) and by Kr´olikowska (2004) from much shorterorbital arcs, the effect on the orbital velocity integratedover one revolution about the Sun still reaches as muchas 2 . ± .
14 m s − . In fact, Sosa & Fern´andez (2011)pointed out that while they employed the nongravita-tional effects to successfully predict masses and dimen-sions for a number of long-period comets, the applicationto C/1995 O1 failed utterly, an exception they admittedwas a puzzle: the predicted diameter of the nucleus wasa factor of seven much too small and the fraction of thesurface that was active came out to be more than seventimes the predicted surface area! Given that the nongrav-itational acceleration is unquestionably genuine and welldetermined, the qualitative explanation offered by Sosa& Fern´andez is the comet’s extreme hyperactivity and,parenthetically, a smaller nucleus. If so, its enormouscross-sectional area of 4300 km (Szab´o et al. 2011, 2012)is still unexplained. The nature of the orbital motion ofC/1995 O1 thus remains an enigma to this day.With all available evidence to be scrutinized in Part II,there appears to be no escape from a conclusion that thenucleus was an unresolved and rather compact cluster offragments (some possibly still in contact) with dimen-sions of up to at least 10 km, the detected nongravita-tional acceleration being that imparted to the most mas-sive object. Even though the nongravitational accelera- tions on the active fragments that were less massive wereaccordingly higher, they triggered perturbations relativeto the central mass that remained undetected. CONCLUSIONS
The prime conclusions from this first part of a compre-hensive investigation of C/1995 O1 are as follows:(1) With thousands of astrometric observations cover-ing an orbital arc of 17.6 yr, C/1995 O1 offered us an op-portunity to derive a set of orbital elements of exception-ally high quality. As a result of an extensive examinationof the data and in-depth dynamical analysis, we presenttwo nominal nongravitational sets of orbital elements,which are distinguished from each other by different sys-tems of weighting the critical observations at both ends ofthe orbital arc. The superior solution, based on WeightSystem II (the single pre-discovery position of weight 20,the last three positions of weight 15 each), fits 1950 obser-vations from 1993–2010 with an rms of ± ′′ .64; 1631 ob-servations with a residual exceeding 1 ′′ .5 in either coordi-nate were eliminated in the process of orbit improvement.(2) Our orbit-determination method fully accommo-dates Marsden’s (1999) proposal that the comet under-went a preperihelion close encounter with Jupiter in theprevious return to the Sun, in the 23rd century BCE, andthat it was then captured from an Oort-Cloud-type orbit.Although we establish unequivocally that the encounterand capture occurred in − − , with the radial component pos-itive and dominant, the normal component negative andequal to about the radial component, and the trans-verse component two orders of magnitude smaller andpoorly determined. The scaling distance of the modifiednongravitational law applied equals 15.4 AU, ∼ − −
2. Wepainstakingly searched for a historical record that mightrefer to C/1995 O1, but the time-frame uncertainties inthe 23rd century BCE prevented us from making anydefinite identification.(5) The orbital motion of C/1995 O1 was extrapolatedforward in time, with a prediction for the next return toperihelion in the year 4393 and afterwards in 5451. It isnoted not only that the actual orbital period is gettingshorter with time, from 4246 yr between the years − × g for water, 1.1 × g for car-bon monoxide, and 0.8 × g for carbon dioxide. Intheir sum, they are equivalent to the mass of a sphere1.7 km in diameter at a density of 0.4 g cm − , with wa-ter accounting for less than of the total mass.(9) Outgassing by a number of additional, mostly or-ganic, parent molecules identified in C/1995 O1 (as com-piled by Bockel´ee-Morvan et al. 2004) was examined inthe same fashion. For each of 18 trace compounds, themass lost per orbit was computed from the nongravita-tional law with a scaling distance determined from thelatent heat of sublimation, and their sum found to addanother ∼ × g to the total production of gas; thetrace molecules searched for but not detected were notcounted. Because of their high molar mass, the organicmolecules contribute disproportionately more masswise,compared to their abundances. Their mass brings the to-tal mass lost per orbit by outgassing to 6 × g in theleast and the share of water to not more than 57%.(10) On account of a sheer number and variety of com-pounds into which the elements C, H, O, N, S (possiblyothers) could combine, and because of a large tally of asyet unidentified bands in the spectrum, particularly inthe near-infrared, submillimeter, and microwave regions,one can expect the presence in the nucleus of numerousadditional, mostly organic, ices currently unknown. Wefirst examined the number of observed species as a func-tion of their orbit-integrated mass loss relative to wa-ter. The data for C/1995 O1 (by far the most extensivefor any comet) suggested that the cumulative numberof species followed a power law of the normalized massloss per orbit with a slope near − ≥
1% of the mass loss of water ice, bya factor of up to 19 for the molecules whose total massloss was ≥ ≥ × g (in fair agreement with the estimate by Je-witt & Matthews 1999) and comparing it with the orbit-integrated production of parent molecules, we concludedthat the mass loading by dust of the water flow (thedust-to-water mass-loss ratio) integrated over the orbitabout the Sun reached 11.8, while the dust loading ofthe combined flow of water, carbon monoxide, and car-bon dioxide equaled 7.5. The dust loading of the totalgas flow from C/1995 O1 may have been as low as 3.(14) Another piece of evidence for the existence of alarge number of parent molecular compounds in the nu-cleus of C/1995 O1 is provided by the terminal veloc-ities of submicron-sized silicate grains that populated thecomet’s expanding halos, observed for about two monthsnear perihelion. The halos expanded with velocities closeto 0.7 km s − , which — as implied by a hydrodynamicmodel — is too high to be accounted for by a drag ac-celeration exerted by sublimating water molecules, giventhe extremely high mass loading of the water flow by thedust. Probstein’s (1969) theory was used to compute anupper limit on the expected mass loading by dust fromthe observed halo-expansion velocities that was typicallyone order of magnitude lower than the dust loading de-rived directly from the production data. This contradic-tion provides strong evidence that the terminal velocitiesof submicton-sized silicate particles were supported notonly by the expanding flow of water vapor but by addi-tional sublimating matter as well.(15) Further application of the hydrodynamic modelsuggests that a likely critical driver of the hot microscopicsilicate dust in the halos of C/1995 O1 was a mixture ofheavy organic molecules of relatively low volatility (witha range of sublimation heat of ∼
19 000–25 000 cal mol − )with sublimation temperatures just below the blackbodytemperature at heliocentric distances near 1 AU.omet C/1995 O1: Orbit, Jovian Encounter, and Activity 47(16) These heavy organic molecules of low volatilityappear to be identical with the missing compounds thatwe found to be responsible for a major incompletenessof the observed distribution of sublimating ices sortedby the relative orbit-integrated mass loss. In Part II wewill argue that the presence of these as yet undetectedspecies should also slightly alleviate the problem of thesizable nongravitational effects detected in the orbitalmotion of C/1995 O1, although the primary source ofthe disproportion of the momentum transferred to thenucleus by the mass lost by outgassing appears to be thenature of the nucleus as a compact cluster of massivefragments in near-contact with one another, a boldhypothesis — to be addessed in Part II — that, giventhe strong evidence, is extremely difficult to avoid.This project, whose orbital part was inspired by theinsightful paper written by the late B. G. Marsden in1998, was initiated by the first author (ZS) years ago incollaboration with P. W. Chodas, whose contribution ishereby acknowledged and greatly appreciated. The or-bital results presented in this paper were achieved thanksprimarily to the second author (RK). This research wascarried out in part at the Jet Propulsion Laboratory,California Institute of Technology, under contract withthe National Aeronautics and Space Administration.APPENDIX AMOST RECENT EFFORTS TO OBSERVECOMET C/1995 O1The 2012 attempt by D. Herald to observe this comet(Section 2) was not the only effort of this kind, followingthe 2010 astrometric observations made with the 220-cmf/8.0 Ritchey-Chr´etien telescope at the European South-ern Observatory (ESO) (S´arneczky et al. 2011; Szab´oet al. 2011). The comet was in fact observed with the820-cm Antu unit of the ESO’s Very Large Telescope(VLT) as a stellar object of an average V magnitude of24.20 and R magnitude 23.72 on 2011 Oct 5, 23, and25 (Szab´o et al. 2012), but no astrometry is available.Further information is provided in a blog by N. Howes,G. Sostero, and E. Guido, who employed remotely the200-cm f/10.0 Siding Spring-Faulkes South telescope tosearch for the comet on five occasions between 2012 Sept25 and Oct 9; it was detected on none of 45 to 75-minuteimages with a limiting magnitude from 21.5 to 23.5. Allthese efforts point consistently to a conclusion that theobject of magnitude 21.8, exposed by Herald with his40-cm reflector on 2012 Aug 7, could not be the comet,in agreement with the observer’s own final statement.APPENDIX BSHAPE OF THE CURVE ℑ ( X ) AS FUNCTION OFA COMET’S PERIHELION DISTANCEWhile, in general, the g mod ( r ; r ) function is integratednumerically, approximate values of ℑ ( X ) can be foundin closed form in the hypothetical cases of an infinitelyvolatile species, when L sub → r → ∞ ), and an infinite-ly refractory species, when L sub → ∞ ( r → ψ in terms of r and the http://remanzacco.blogspot.it/2012/10/deep-south.html. exponents m , n , and k , g mod ( r ; r ) = r − ( m + nk ) (cid:18) r − n +1 r − n + r − n (cid:19) k = r − m (cid:18) r n + 1 r n + r n (cid:19) k , (56)we find lim r →∞ g mod ( r ; r ) = r − m for L sub → , (57)lim r → g mod ( r ; r ) = r − ( m + nk ) for L sub → ∞ . Thus, the integral ℑ ( L sub ) from Equation (14) may ineither case be written as ℑ = 2 k √ p Z π (cid:18) e cos vp (cid:19) ζ dv (58)= 2 ζ +2 k p − ( ζ + ) Z π cos ζ u (cid:2) − ∆ e (1 − sec u ) (cid:3) ζ du, where u = v , ∆ e = 1 − e and ζ = m − L sub → ζ = m + nk − L sub → ∞ . For C/1995 O1 as a nearlyparabolic comet, ∆ e → e has thenature of a minor correction to the cos ζ u term. Let usreplace the variable u with an appropriate constant valueof h u i , so that sec h u i = η and the integration is readilyexecutable. We obtain ℑ = √ πk ζ +1 p − ( ζ + ) Γ( ζ + )Γ( ζ +1) (cid:2) − ζ (1 − η )∆ e (59)+ ζ ( ζ − − η ) (∆ e ) − . . . (cid:3) , where Γ is the Gamma function.Let us now insert the exponents m , n , and k for ζ andcalculate the ratio of the two limiting equivalent times;we obtain ℑ ∞ ℑ = lim L sub →∞ ℑ ( L sub )lim L sub → ℑ ( L sub )= (cid:18) p (cid:19) nk Γ( m + nk − )Γ( m + nk −
1) Γ( m − m − ) (60) × n − (cid:2) ( m − η − η ∞ )+ nk (1 − η ∞ ) (cid:3) ∆ e + . . . o , where η is the value of η for L sub → η ∞ its valuefor L sub → ∞ . By numerical integration we determinethat ℑ = 231 .
07 days, requiring η = 19 .
36 and there-fore h u i = 76 ◦ .
9. On the other hand, ℑ ∞ = 260 .
98 days,so that η ∞ = 1 .
02 and h u ∞ i = 8 ◦ . ℑ ∞ / ℑ is a verysteep function of the orbit parameter p = q (1+ e ), vary-ing as p − . . For parabolic orbits the ∆ e correction termdisappears and p = 2 q , so that ℑ ∞ / ℑ = 0 . q − . .This formula implies that ℑ ∞ = ℑ for a parabolic or-bit with q = 0 .
919 AU, only very slightly larger thanthe perihelion distance of C/1995 O1. On the otherhand, ℑ ∞ = 10 ℑ for q = 0 .
833 AU and ℑ ∞ = 100 ℑ for q = 0 .
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