Orbit computation for transneptunian objects

Orbit computation for transneptunian objects

Jenni Virtanen,a,* Gonzalo Tancredi,b Karri Muinonen,c,1 and Edward Bowelld

a Observatory, University of Helsinki, P.O. Box 14, FIN-00014 Helsinki, Finlandb Departamento Astronomia, Facultad Ciencias, Tristan Narvaja 1674, 11200 Montevideo, Uruguay

c Astronomical Observatory of Torino, via Osservatorio 20, I-10025 Pino Torinese (TO), Italyd Lowell Observatory, 1400 West Mars Hill Road, Flagstaff, AZ 86001, USA

Received 14 November 2001; revised 16 September 2002

Abstract

Using statistical orbital ranging, we systematically study the orbit computation problem for transneptunian objects (TNOs). We haveautomated orbit computation for large numbers of objects, and, more importantly, we are able to obtain orbits even for the most sparselyobserved objects (observational arcs of a few days). For such objects, the resulting orbit distributions include a large number ofhigh-eccentricity orbits, in which TNOs can be perturbed by close encounters with Neptune. The stability of bodies on the computed orbitshas therefore been ascertained by performing a study of close encounters with the major planets. We classify TNO orbit distributionsstatistically, and we study the evolution of their ephemeris uncertainties. We find that the orbital element distributions for the most numeroussingle-apparition TNOs do not support the existence of a postulated sharp edge to the belt beyond 50 AU. The technique of statistical rangingprovides ephemeris predictions more generally than previously possible also for poorly observed TNOs.© 2003 Elsevier Science (USA). All rights reserved.

Keywords: Orbits; Transneptunian objects; Kuiper-belt objects

1. Introduction

Because of their faintness, transneptunian objects(TNOs) are a challenge to observers both for discovery andfollow-up. The known population of some 400 objects (as ofMay 2001) is only a fraction of the total number, Jewitt andLuu (2000) have estimated the number of TNOs larger than100 km to be tens of thousands. Moreover, only a few TNOshave accurate orbits—19 of them were numbered as of May2001—leaving a majority of the population poorly observedor lost. Most TNOs have likely undergone only minorchanges since their formation in the solar nebula. Therefore,dynamical and physical studies of these objects can givevital information on the formation and evolution of the outerSolar System. Initial orbit determination techniques play acrucial role in increasing the number of objects with rea-

sonably secured orbits which can then be studied for thephysical characteristics of the TNO population.

There are two major obstacles in the TNO follow-upprocess. First, due to their long orbital periods—about 250years or more—observations of even the best observedobjects cover only a small fraction of a revolution. The bestexamples of this are the first-ever transneptunian object,Pluto, observed since 1914 (observations now encompass-ing 35% of its orbit), and (20000) Varuna, for which ob-servations dating back to 1954 have been identified fromphotographic archives (16% of its revolution period cov-ered). However, half of the known TNOs have observationsspanning less than six months, corresponding to less than1% of their revolution periods. Thus, unlike some near-Earth or main-belt objects, where a few years’ observationsuffices for the determination of an accurate orbit, TNOswill have to be observed for decades. Second, the necessaryexpense and difficulty of using large telescopes to make theobservations implies the need for optimized observing strat-egies, which in turn mandates a firm understanding ofephemeris uncertainty.

* Corresponding author. Fax: �358-9-19122952.E-mail address: [email protected] (J. Virtanen).1 Current address: Observatory, University of Helsinki, P.O. Box 14,

FIN-00014 Helsinki, Finland.

R

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The good news is that the projection of the complicatedTNO orbital element probability density onto the sky planeresults in fairly simple distributions (in contrast to those ofsingle-apparition near-Earth asteroids, for example). Thismeans that many of the unnumbered TNOs actually haveminimal ephemeris uncertainty, making their recovery onsubsequent apparitions straightforward. However, this doesnot apply to the numerous objects with very short arcs of afew months or less.

Bernstein and Khushalani (2000) discuss TNO orbit de-termination and ephemeris prediction. They describe a lin-earized orbit-fitting procedure in which gravitational accel-erations are treated as perturbations to the inertial motion ofthe object. They also make use of Cartesian coordinates andtheir derivatives, which fix the orbit more stably than dotraditional orbital elements. Applied to map the ephemerisuncertainties of multiapparition TNOs, the method by Bern-stein and Khushalani is efficient for devising observingstrategies for orbit improvement. In their search for distantKuiper belt objects, Allen et al. (2001) used Bernstein andKhushalani’s software to derive orbital parameters for newlydiscovered objects. Although the method is applicable also topoorly observed objects, TNOs with very short observationalarcs (say, a few weeks or less) should be treated using morerigorous methods to derive the orbital uncertainties properly.

The Minor Planet Center (MPC) provides orbital ele-ments for all observed TNOs (in some cases taking intoaccount encounters with Neptune). A possible defect of theMPC’s technique of selecting short-arc orbits resemblingthose of known objects is that it causes a bias against objectsthat actually are on unusual orbits. Although the MPC doesnot provide estimates of the orbital uncertainties, it doesproduce ephemeris uncertainty plots for short-arc objects.Unfortunately, these objects are the ones with the largest un-certainties and the given coordinate offsets are often severalthousands of arcseconds. Thus, methods are called for to con-strain both orbital and ephemeris uncertainties in such cases.

The technique of statistical ranging has proved to be avery powerful orbit computation tool for sparsely observedobjects and/or short observational arcs (Virtanen et al.,2001, here VMB; Muinonen et al., 2003; Bowell et al.,2002), a criterion that the majority of the known TNOsmeet. In this paper, we carry out a statistical study of theorbits of the entire transneptunian population. Our purposeis to clarify the current understanding of the orbital structureof the transneptunian region and to prepare the ground formore detailed dynamical studies. We also present practicalmeans of treating ephemeris uncertainty for poorly observedTNOs. In Section 2.1, we briefly review the technique ofstatistical ranging and describe the upgrades implementedsince VMB. We describe the analysis of orbital stability inSection 2.2; and in Section 3, we present and discuss theresults of TNO ranging and ephemeris prediction (Section3.3). We hope to convince the reader that statistical ranging

affords a more general method of orbit treatment than hasbeen developed hitherto.

2. Orbit computation techniques

2.1. Statistical ranging

We examine the probability density function (PDF) ofTNO orbital elements using the method of statistical orbitalranging (VMB; Muinonen et al., 2001). The Monte Carloselection of orbits in orbital element space is based onchoosing angular deviations in right ascension (R.A.) anddeclination (Dec.) for two observation dates and assum-ing topocentric ranges (distances) corresponding to thesedates. The a posteriori PDF is mapped with a large set ofsample orbits, each associated with a weight proportional toexp(�1⁄2�2) based on the R.A. and Dec. residuals (see, forexample, Eq. 14 in VMB).

We have upgraded the ranging technique from that inVMB and Muinonen et al. Our objective has been to providefully automated software that can be used for systematicstudies of large numbers of objects. Thus, the steps needingmanual intervention (see VMB, e.g., Section 2.4) have beeneliminated. These are the iteration of the PDF value corre-sponding to the maximum likelihood orbit and, more im-portantly, the iteration of topocentric range intervals. Westart with joint initial values for the ranging parameters forall the TNOs, in particular with a wide interval of [30, 50]AU for the topocentric range covering more or less theentire transneptunian region. The first iterative step is thecentering of the Monte Carlo selection of orbits on theinterval compatible with the observations for each object.Additional refinements have been implemented as follows.

First, with a small number of sample orbits (say, 10), westudy the a posteriori PDF for the topocentric range anddetermine new lower and upper bounds for the range inter-val from the �3� cutoff values. By gradually increasing thenumber of sample orbits (10 3 200 3 2000), we improvethe topocentric range intervals and end up with an unbiasedphase space region of possible orbits. Second, in the firstiterative step for the range intervals, we now generate ran-dom topocentric ranges either from a uniform or a Gaussiandistribution, depending on which is more efficient. We startwith the assumption of uniform distribution over the initialrange interval. If the maximum number of trials has beencarried out and no sample orbits have been found, we adoptthe Gaussian distribution with a mean of 43.1 AU (standarddeviation of 7.4 AU), a mean distance found when the entireTNO orbit computation process was repeated a few timeswhile testing the implemented upgrades. This ensures thatorbits can also be found for those objects having well-defined distances (typical uncertainty of tenths of AU forobjects observed longer than a year) which are outside theadopted initial range interval of [30, 50] AU. In particular,for objects with observational arcs of several years, we

420 J. Virtanen et al. / Icarus 161 (2003) 419–430

make use of the topocentric ranges based on orbit solutionusing Gauss’s method, and we take them as the maximumlikelihood values for the Gaussian range distribution. How-ever, to compute, in the final iteration, the a posteriori PDFfor the orbital elements, we continue to apply the uniformdistribution to generate the ranges.

The accuracy of the numerical algorithm in reproducingthe two R.A. and Dec. input values is now typically on the10-�arcsec level, making the technique applicable to thehigh-precision astrometry likely to become available in thefuture (Muinonen and Virtanen, 2002).

As pointed out by Bowell et al. (2002), the obviousquestion of the underdeterminacy of the inverse problemusing only four measurements (that is, two astrometric ob-servations) remains to be discussed. The methods byJedicke (1996), McNaught (1999), and Tholen and White-ley (2003) also make use of four measurements (but, incontrast to the ranging technique, they use the Cartesianposition and velocity). Although these methods are bound tofind more or less the same orbital solutions, only that byTholen and Whiteley seems to be amenable to uncertaintyestimation.

JPL ephemerides DE-405 were used as a source for thepositions and velocities of all the planets (Standish 1990).Parts of the orbit-computation freeware package OrbFit 1.1(http://newton.dm.unipisa.it/�asteroid/orbfit) are adopted,namely, the routines by M. Carpino for inputting astromet-ric observations in the MPC format, for computing theposition and velocity of the observer, and for time conver-sion. Astrometric observations are from the MPC.

2.2 A priori constraints imposed by orbital stability

The perturbative effects of the four giant planets havebeen shown to play a major role in the dynamical evolutionof the transneptunian region (e.g., Malhotra et al., 2000). Inparticular, the high-eccentricity orbits that typically resultfor short observational arcs are sensitive to close encounterswith Neptune or even the other giant planets. Therefore, weconsider it important to study the stability of the sample oforbits resulting from statistical ranging; that is, we discardorbits in which TNOs experience close encounters with anymajor planet, because they are likely to be in unstable, andtherefore very unlikely, orbits.

We performed an analysis of the minimum orbital inter-section distances (MOIDs) of all the orbits computed withranging with respect to the four giant planets, namely,Jupiter, Saturn, Uranus, and Neptune. To compute theMOID we initially tested the widely used algorithm devel-oped by Sitarski (1968). We found many cases where thealgorithm did not converge, especially for high-eccentricityorbits. We then implemented a new algorithm based on thesimple idea of numerically searching for the minimum mu-tual distance between the orbits. We parametrized the orbitsusing true anomalies and searched for the minimum of a

two-variable function: the mutual distance computed fromthe Cartesian coordinates of points moving in the orbits. Themutual distance should have two minima, one smaller thanthe other. Choosing an arbitrary set of two initial anomalies,we look for the first minimum with the direction set methodin multidimension by Powell (Press et al., 1994). If thealgorithm did not converge after, say, 200 iterations, wechose a new set of initial values. After finding the firstminimum, we chose a new set of initial values by adding180° to each true anomaly solution, and we searched for thesecond minimum. If the new minimum turned out to be veryclose to the first one, we chose new sets of initial valuesuntil we found the second minimum different from the firstone. The MOID is then the smaller of the two.

We have defined a close planetary approach as MOIDbeing less than 3 Hill radii (rH) from any major planet(where rH is the radius of the “sphere” centered on theplanet within which the planetary gravitational attraction islarger than the Sun’s tidal attraction). The value of 3rH waschosen because it is generally considered that encounters atdistances closer than that can lead to sizable and rapidchanges in the dynamical evolution (see, for example, Tan-credi, 1998).

We also explore whether the orbit is currently protectedfrom close approach to Neptune by a mean-motion reso-nance. For this purpose, we have adopted a criterion forproximity to a resonance based on Fig. 1 of Morbidelli et al.(1995); if the semimajor axis and eccentricity of the orbitfall inside any region of possible libration with respect toNeptune, the orbit is selected for further detailed dynamicalanalysis. For each ranging orbit we follow the followingdecision tree:

Is the MOID with respect to Jupiter, Saturn, or Uranusless than 3rH?

1. YES 3 discard.2. NO: Is the MOID with respect to Neptune less than 3rH?

a. YES: Is the orbit close to a mean-motion resonancewith Neptune?i. NO 3 discard.

ii. YES: Integrate for 1 Myr; is there a close encoun-ter of less than rH with any planet?A. YES 3 discard.B. NO 3 accept.

b. NO 3 accept.

We used the mixed variable symplectic integrator byLevison and Duncan (1994) to carry out numerical integra-tions of the orbits, including the code swift_mvs, in whicha particle is discarded if it comes closer than rH to anyplanet.

Thus, we are left with orbits for which the major-planetMOIDs are greater than 3rH and resonant orbits with respectto Neptune having MOIDs that are less than 3rH but stableover 1 Myr.

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http://newton.dm.unipisa.it/~asteroid/orbfit

3. Results and discussion

3.1. Orbital uncertainties

We carried out an orbital uncertainty analysis for thepopulation of TNOs; as of February 2001, there were �400objects classified as TNOs, of which we considered 396. Inparticular, we studied the evolution of orbital uncertainty asa function of observational arclength.

We selected TNOs from the MPC’s distant object list-ings dated February 2001. To start with, all 367 TNOs wereincluded. From the list of Centaurs and scattered disk ob-jects, we chose the 29 objects that had perihelion distances(q) greater than 30 AU. Because of the enormous orbitaluncertainties for short observational arcs, it would be usefulalso to include the most poorly observed apparent Centaursin a future study. Of the known objects, 19 had been num-bered, whereas 66 objects (17% of the total number) hadbeen observed on only two nights.

We computed orbits for 396 (367 � 29) objects andapplied statistical ranging to the 362 unnumbered objects.Because of the relatively short intervals of observation, atwo-body dynamical model is accurate enough for ourstudy. The epochs of the orbital elements were taken to bethe midtime of each set of observations. We fixed bothmaximum deviations and residual cutoffs for R.A. and Dec.at 2.0 arcsec and assumed 1.0-arcsec observational noise,which is conservative considering the current astrometricaccuracy, but we choose to be overpessimistic rather thanoveroptimistic in this matter. Two-thousand orbits werecomputed for each object—a compromise between datahandling problems and requirements for statistical sig-nificance. On a 500-MHz workstation, orbit computationfor the 362 TNOs took about a day. We checked the influ-ence of the regularization presented in Muinonen et al.(2001) and found that the use of the determinant ofthe information matrix as a priori PDF has no notableeffect on the final probability densities. Thus, the regular-izing a priori PDF, while guaranteeing invariance, didnot play a predominant role in the total orbital ele-ment PDF.

For 15 objects, satisfactory orbit could not be found in areasonable time using ranging. All the missing objects aremultiapparition TNOs having observational arcs longer than2 years. As pointed out in Bowell et al. (2002), in such casesthe ranging technique is more practical for detailed studiesthan for routine orbit computation. For these objects, as wellas for the 19 numbered TNOs, we computed least-squaresorbits and covariance matrices. However, slippage factors(see Muinonen and Bowell, 1993) were typically near 2, andtests for the validity of the linear approximation (Muinonenet al., 2001) imply that the approximation almost alwaysbreaks down, with the possible exception of Pluto and(20000) Varuna, which have the longest observational arcsamong TNOs (�86 and 46 years, respectively). The orbitaluncertainties based on the least-squares covariances should

thus be treated with caution. The semilinear approach byMilani (1999) applies multiple solutions along the line ofvariations of the element covariance matrix to map thenonlinear confidence region. Generalization of this curve-of-variations method from a one- to six-dimensional ap-proach would make it suitable for objects for which the useof least-squares covariances fails.

To illustrate the extent of the computed orbital elementdistributions in a simple way, we computed quality metricsfor the orbital elements (see Muinonen and Bowell, 1993,and Muinonen et al., 1994), which also serve as a measureof orbital goodness. We chose the semimajor axis and theinclination for the analysis; the latter is typically the bestdetermined of the orbital elements, while short observa-tional arcs can lead to a large variety of values for theformer. For the metric, we used the standard deviations ofthe corresponding rigorous marginal PDFs. For the 15 mul-tiapparition and 19 numbered objects, we incorporated theslippage-corrected 1� uncertainties based on the least-squares covariance matrices. In Fig. 1, we plot the standarddeviations for the 396 TNOs versus the length of the ob-servational arc (Tobs). From the upper plot, there is anapparent division of objects into three categories: the firstcategory consists of 1-apparition TNOs (Tobs � 0.5 year),

Fig. 1. Standard deviation of the rigorous PDF for semimajor axis andinclination as a function of observational arclength (396 objects).


the middle category of TNOs observed at two apparitions(0.5 � Tobs � 1.5 years), and the third category of �3-apparition TNOs (Tobs � 1.5 years). Here we use the terms1-apparition, 2-apparition, and �3-apparition to mean ob-jects observed in single, two, and more than two apparitions.In what follows, we focus our attention on these threecategories; the numbers of objects in each category are 204,99, and 93, respectively. For the 1-apparition objects, thestandard deviation of the semimajor axis is on the order oftens of AUs, whereas in some cases the inclination is accu-rate to a tenth of a degree after only 30 days of observation.Among the 2-apparition objects, there is large dispersion inthe accuracy of the semimajor axis solutions, even thoughthe inclinations are determined reasonably well.

All 362 � 2000 sample orbits that represent the orbitalelement distributions were analyzed for their stability. Afterapplying criteria 1 and 2.a.i, we discarded 41% of theshort-arc orbits, 7% of the medium-arc ones, and 3% of thelong-arc orbits. For the remaining orbits, we applied crite-rion 2.a.ii, and we were left with almost 50,000 orbits tointegrate for 1 Myr. Let us note that only 5% of the short-arcorbits have a Neptune MOID less than 3rH and fall close toa resonance. This number is close to the value obtained ifone generated random orbits in the range a � [30, 100] AUand e � [0, 1] and checked whether they fell inside aresonance (see the area covered by the mean motion reso-nances in Fig. 1 of Morbidelli et al., 1995). On the otherhand, most of the �3-apparition orbits with low NeptuneMOID were close to a resonance. The integrations tookalmost 10 days of CPU time in a Pentium II 400-MHzcomputer under Linux. Of the integrated orbits, most of theshort-arc orbits (75%) were removed because of close en-counters with a giant planet (criterion 2.a.ii.A), whereasonly half of the medium-arc orbits and a very small fractionof the �3-apparition orbits were removed.

The remaining non-planet-encountering orbits (criterion2.a.ii.B) as well as those that do not have a MOID less than3rH to any planet (criterion 2.b) were left for the populationanalysis, i.e., 55% of the original short-arc orbits, 89% ofthe medium-arc orbits, and 96% of the �3-apparition orbits.In what follows, we refer to these orbits as filtered orbits,and to the original sample orbits as unfiltered orbits.

Previous analyses of the dynamical characteristics of theTNO population have concerned the extent of the transnep-tunian region and the population of the mean-motion reso-nances (for some of the most recent analyses, see Allen etal., 2001, and Malhotra et al., 2000). To give insight into thedistribution of TNO orbits and also into the previous ques-tions, we computed orbital element number densities acrossthe transneptunian region, i.e., the average numbers of ob-jects with orbital elements in given intervals. The weightsfor individual orbits have been incorporated and the PDF foreach TNO has been normalized so that each TNO has unitweight.

In Figs. 2 and 3, we have plotted the histograms forvarious orbital parameters for our three TNO categories. As

there were no significant differences in the numbers offiltered and unfiltered orbits for the 2-apparition and �3-apparition objects, we only give the histograms for thefiltered orbits in Fig. 3. However, Fig. 2 shows the compar-ison between the short-arc orbit histograms before and afterthe MOID study. For both cases, there are long tails in thesemimajor axis and eccentricity distributions. The unfilteredeccentricity histogram peaks at e � 0.8 but these high-eorbits are the most unstable ones, as can be seen from thefiltered histogram, which suggests a nearly uniform distri-bution for the eccentricities. Although the number of short-arc orbits is greatly reduced with the stability study, it has asurprisingly small effect on the overall distribution of theorbital elements. Moreover, since well-observed objectsonly produce long-term stable orbits, it is likely that eventhe short-arc objects turn out to be on long-term stableorbits.

For objects observed in more than one apparition (Fig. 3,dashed line for the 2-apparition and solid line for the �3-apparition objects), the large-a tail quickly dies down buteccen-tricities as high as 0.7 are found—most likely corre-sponding to objects in the scattered belt. While retrogradeorbits cannot be excluded for the shortest arcs, a significantnumber of �3-apparition objects have inclinations up to30°. A clear observational bias toward perihelion discover-ies can be seen for the short-arc objects and is also presentfor the longer arclengths. Also, a bias against observationsnear the galactic plane at � � 90° and � �270° is evidentfrom the longitude histograms. It is noteworthy that theheliocentric distances compatible with observations span-ning half a year or less extend to 80 AU while there appearsto be an upper bound of 50 AU for the �3-apparitionobjects. Comparing the distance histograms for the pre-1999 and post-1998 discoveries shows that both the maxi-mum and the tail of the distance distribution have clearlybeen moving outward. This is because the recent surveyswith fainter limiting magnitudes have found fainter and alsomore distant objects. For pre-1999 discoveries the averagediscovery magnitude was 22.9 and the faintest object had amagnitude of 25 while for post-1998 discoveries the corre-sponding magnitudes were 23.5 and 26.5.

Levison and Stern (2001) analyzed the orbital elementsfrom the MPC for the nonresonant TNOs, which they termthe main Kuiper belt. They find a correlation between theinclinations and absolute magnitudes for these objects, in-dicating the existence of a superposition of two dynamicallydistinct populations. We find independent confirmation oftheir result.

3.2. Classification

The classification of orbital type is a straightforwardapplication of the orbital element PDFs. Following thegrouping scheme described by Gladman et al. (2001) andintroducing some modifications, we classify the TNO orbitsas follows:


1. The classical belt: a � 40–48 AU and q � 35 AU.2. The inner belt: a � 36–39 AU and q � 35 AU.3. The outer belt: a � 48 AU and q � 36 AU.4. The scattered disk: a � 49 AU and q � 36 AU.5. The low-order resonant orbits: 4:3, 3:2, 5:3, 2:1, 5:2,

and 3:1 (as in Section 2.2, we use the definitions byMorbidelli et al. (1995) for the widths of the regionsof possible libration).

One should note, however, that there are no firm defini-tions for different dynamical classes, except for the resonantpopulations.

The phase-space structure of the transneptunian region inthe (a, e) and (a, i) planes is illustrated in Figs. 4 and 5. Thecontour plots in Fig. 4 show the concentration of the prob-ability mass to the low-e and low-i orbits in the classicalbelt. The inner belt is lightly populated in all the contourplots. No orbits are found in the colder part (very low e andi) of outer belts for the 2- and �3-apparition plots. Eventhough in the (a, e) contour plot of short-arc filtered orbitsthere is a small peak at a � 55 AU and e � 0, there is onlya probability of 8 � 10�6 of finding an orbit having a � 52AU and e � 0.1. On the other hand there is a probability of0.07 of finding a short-arc orbit having large eccentricity (e� 0.9). From the resonance populations, the Plutinos in the3:2 resonance are the most abundant in the 2- and �3-

apparition plots. Nevertheless, there is clear evidence thatthe other low-order resonances have some member candi-dates. More detailed dynamical studies are needed to con-firm the libration of the resonant argument.

A more detailed view of the unfiltered orbits in the (a, e)planes vs the arclength in Fig. 5 reinforces the evident lackof low-e resonant orbits, as well as the lack of near-circularnonresonant orbits with a � 50 AU. No fine structure can beseen for 1-apparition objects, whereas the strong correlationin the middle plot (strip-like structure) is due to the reason-ably well determined topocentric distance for the 2-appari-tion objects.

In Table 1 we give the probabilities of finding an orbit inthe dynamical classes previously defined for the three TNOcategories. The probabilities were computed by adding theweights for the individual orbits belonging to each dynam-ical class and normalizing the TNO PDFs so that each TNOcontributes with a weight 1/(total number of TNOs for thegiven arclength). For the 2- and �3-apparition objects, wenote that �50% of the orbits belong to the classical belt and�20% of the orbits are resonant (most of them in the 3:2resonance). The probabilities for the scattered objects are16% and 5%, respectively. The outer belt is populated by�5% of the orbits. However, as defined here, these orbitsare not in the cold part of very small e and i which is

Fig. 2. Marginal PDFs before and after MOID filtering for various orbital parameters (�, ecliptic longitude) including objects having Tobs � 183 days (2041-apparition objects, each with unit weight). Unfiltered orbits (dashed line) and filtered orbits (solid line).


predicted to be stable by Duncan et al. (1995). A betterclassification for this region should be made in the future.

The distribution of orbits for the 1-apparition objects isremarkably different: 43% of the orbits belong to the scat-tered disk and 26% to the outer belt, which is unrealistic incomparison with corresponding probabilities for the 2- and�3-apparition objects. Also, 10% of the orbits do not fallwithin any of the classes discussed. Using a stricter criterionfor the scattered disk objects, based on orbital integrationsby Duncan and Levison (1997), shows that very few of theobjects originally classified as scattered disk objects have ahigh probability of belonging to this class a posteriori.Although our results show that the dynamical classificationof TNOs is very uncertain for observational arclengthsshorter than half a year, this also leads us back to theproblem of the lack of proper definitions for the TNOclasses. In the future, we aim to improve the current TNOclassification scheme in light of the results presented here.

3.3. Ephemeris prediction and linkage

For new discoveries, we are often less interested in theactual orbital element distribution than in knowing what theuncertainties in its current and future positions are. In fact,as concluded in Bowell et al. (2002), the orbits are nowa-

days often only an intermediate state between the astromet-ric observations and the applications of the orbital elementprobability density, in this case the ephemerides. Anothermatter of great interest is the linkage of independent sets ofobservations to the same object; with the increasing numberof discoveries, the possibility that a new object has alreadybeen discovered is also increasing.

Using statistical ranging, we can rigorously project theorbital element PDF into the sky plane, resulting in theephemeris uncertainty distribution at a specified epoch. Wecomputed ephemerides for 362 TNOs and studied the evo-lution of ephemeris uncertainty vs the time elapsed from thelast observation. The size of the uncertainty region wasevaluated for the following epochs: 30 days, 1 year, and 5years after the date of last observation. As expected, theuncertainty grows nearly linearly with time (Fig. 6). Toroutinely recover an object 1 year later (i.e., the uncertaintyis on the order of arcminutes at the maximum) observationsfrom multiple apparitions are needed (note that the uncer-tainties for �3-apparition objects are likely to be overesti-mated by a factor of 2 or 3 due to the 1.0-arcsec assumptionfor the observational noise). It has in practice turned out thatmultiapparition orbits result in reliable ephemeris predic-tions, although the orbital uncertainties would be open todoubt, because of, e.g., the degeneracy of the least-squares

Fig. 3. Marginal PDFs after MOID filtering for various orbital parameters, including objects observed at more than one apparition; 183 � Tobs � 547 days(dashed line, 99 2-apparition objects) and Tobs � 547 days (solid line, 93 � 3-apparition objects).


covariance matrix. Nevertheless, based on orbital uncertain-ties from the standard orbit computation methods, objectswith 2- to 3-month arcs are in general considered lost ifobservations on the following apparition fail. Moreover, foreven shorter arclengths, long-term ephemeris uncertainties

cannot be evaluated at all (for example, because eccentric-ities have to be assumed in the orbit computation process).Using orbital uncertainties from statistical ranging, ephem-eris uncertainties—although often large—can be computedfrom the discovery night onward.

Fig. 4. Contour plots in the (a, e) and (a, i) planes for the three TNO classes. From top to bottom: unfiltered and filtered 1-apparition orbits; filtered2-apparition orbits; filtered � 3-apparition orbits. The three curves of constant perihelion distance, q � (20, 30, 33) AU, are also shown.


Unfortunately, those objects most in need of additionalobservations also have the widest distributions of currentand future ephemeris uncertainty. To overcome the difficul-ties in the follow-up process, we provide ways of shrinkingthe cloud of positional uncertainty surrounding a TNO’snominal position. We note that the study of the MOIDs is astrong a priori constraint for short-arc objects. From theephemeris probability density, we can compute various con-fidence regions by choosing appropriate confidence levels.Conventional choices are the 1-, 2-, and 3-� levels as

defined for the Gaussian distribution. In the Bayesian for-mulation of statistical ranging, we can optionally incorpo-rate a priori information into the final probability density,which results in improved ephemeris prediction. For TNOs,the a priori distribution based on the orbital element distri-bution of known objects (see VMB) is impractical becauseof the small number of numbered and multiapparition ob-jects. Another choice of a priori knowledge would be theuse of dynamical groupings as discussed in the previoussection. Instead, we suggest here the use of semimajor axis,eccentricity, and inclination filtering to focus search efforts.We illustrate the use of this technique with some examples.

1998 WW31 was followed up for two months after itsdiscovery in November 1998 and was in danger of beinglost when it was recovered in December 2000. It thenbecame apparent that 1998 WW31 is actually the first binaryTNO discovered after Pluto and Charon. Fig. 7 shows thesky-plane uncertainties for 1998 WW31 at the recovery date.Eccentricity filtering would have proved useful in this case;by fixing a cutoff value of 0.1, the extent of the R.A.uncertainty can be reduced to �40% of the original width.Although the search region is still �15 arcmin wide, anobserver using wide-field imaging could apply such a strat-egy when planning recovery observations, starting at e3 0and working outward.

Another example of efficient filtering is 2000 CM114, aTNO that was observed twice on two nights, 54 days apart.As can be seen from Fig. 8, the current ephemeris uncer-tainty (1.5 years after the last observation) has spread overseveral degrees in R.A. However, first applying an inclina-tion filter to exclude retrograde orbits, and then using 0.1 or0.2 cutoff values for eccentricity, shrinks the sky-planeregion that needs to be searched down to �70 or 30 arcminin R.A., respectively.

Application of the ranging technique to TNOs shows thatthe linkage problem in this case coincides with the inverseproblem. By making use of the first and last observation inthe computation of sample orbital elements, we actuallycarry out simultaneous linkage. For �3-apparition objects,the time elapsed between the two observations can be sev-eral years. Gauss’s method can also be used to combineTNO observations from different apparitions; likewise, oc-casionally the least-squares fitting can be used. The advan-tage of the ranging technique is that the linkage takes placeroutinely and is accompanied by rigorous uncertainty esti-mates.

Fig. 5. Evolution of structuring in the (a, e) plane as a function ofobservational arclength; orbits contributing 68.3% (top panel) and 95.4%(two lower panels) of the probability mass are shown. From top to bottom:1-, 2-, and � 3-apparition orbits. In the bottom panel, pluses indicatenumbered and � 3-apparition TNOs with least-squares orbits. The curvesmarking q � 30 and 35 AU are shown.

Table 1Probabilities of finding an orbit belonging to the different dynamical classes for the three TNO categories (1-, 2-, and �3-apparition objects)

Arclength(appar.)

Nonresonant Resonant

Class. Inner Outer Scattered Total 4:3 3:2 5:3 2:1 5:2 1:3 Total

1 0.11 0.0060 0.26 0.43 0.80 0.0088 0.017 0.012 0.028 0.014 0.014 0.0932 0.51 0.014 0.047 0.16 0.73 0.0084 0.11 0.020 0.047 0.016 0.0072 0.21�3 0.59 0.034 0.068 0.054 0.76 0.017 0.14 0.0004 0.045 0.031 0.0001 0.24


4. Conclusion

We have studied the orbital distribution of the transnep-tunian population. In particular, we have applied the tech-

nique of statistical ranging to the most sparsely observedTNOs having observational arcs of only a few days. Al-though the resulting distributions for the semimajor axis andeccentricity are broad, and the dynamical classification

Fig. 6. Evolution of sky-plane uncertainty in R.A. vs time elapsed from the last observation (362 objects). Triangles, dots, and diamonds indicate the extentof the uncertainty region 30 days, 1 year, and 5 years after the last observation, respectively.

Fig. 7. Ephemeris uncertainty and the application of (a, e, i) filtering for 1998 WW31 (Tobs � 57 days) at the recovery epoch, December 22, 2000. Imposingan a priori constraint by assuming the eccentricity to be small (e � 0.1) reduces the extent of the uncertainty region to �15 arcmin, successfully containingthe recovery observation (�).


based on such short arclengths is very uncertain, the rangingtechnique gives us the means to make rigorous ephemerispredictions in a straightforward way. The computed elementprobability densities are the groundwork for more detaileddynamical and physical studies of the transneptunian region.

Although our population analysis does not find supportfor near-circular orbits beyond 50 AU, the existence ofobjects with low-to-moderate eccentricities is not ruled out.The reasons for the nondetection—as discussed by Allen etal. (2001)—could be selection effects in discovering newobjects or the fact that this region has actually been emptiedby some dynamical process (like excitation). These ques-tions are, however, beyond the scope of this paper. We planto return to the question of the edge of the transneptunianregion after more detailed dynamical studies have beenmade.

Because of their long orbital periods and the need to uselarge telescopes for follow-up observations, understandingthe evolution of orbital uncertainties is fundamental forpreventing newly discovered TNOs from being lost and forensuring their eventual numbering. (a, e, i) filtering can beused to improve ephemeris prediction. Filtering can alsoprove particularly useful for planning recovery observationsof TNOs whose sky-plane uncertainties are on the order ofdegrees, which in turn requires an optimized search effort.

One objective of our ongoing work on the TNO ephemerisprediction problem is to provide observing strategies formaximum orbital improvement that require a minimumnumber of observations. Another aim is to offer a Web-based TNO ephemeris prediction service to help observersto plan follow-up and recovery observation for short-arcTNOs (Virtanen et al., 2003).

The successful application of the optimized ranging tech-nique for TNOs has several implications for future studies.First, it would be valuable to apply the ranging technique tothe case of poorly observed comets. Second, the automatedroutine can be used, for example, to study single-apparitionmain-belt objects for their classification in asteroid families.

Acknowledgments

J. Virtanen wishes to thank the University of HelsinkiScience Foundation for financial support. This research issupported, in part, by the Academy of Finland.

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Orbit computation for transneptunian objects