Knezevic et al.: Determination of Asteroid Proper Elements 603

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The Determination of Asteroid Proper Elements

Z. KnezevicAstronomical Observatory, Belgrade

A. LemaîtreFacultes Universitaires Notre-Dame de la Paix, Namur

A. MilaniUniversity of Pisa

Following a brief historical introduction, we first demonstrate that proper elements are quasi-integrals of motion and show how they are used to classify asteroids into families and studythe long-term dynamics of asteroids. Then, we give a complete overview of the analytical, semi-analytical, and synthetic theories for the determination of proper elements of asteroids, with aspecial emphasis on the comparative advantages/disadvantages of the methods, and on theaccuracy and availability of the computed proper elements. We also discuss special techniquesapplied in some particular cases (mean motion and secular resonant bodies). Finally, we drawconclusions and suggest directions for future work.

1. INTRODUCTION

The computation of asteroid proper elements is certainlyone of the fields of asteroid research that has undergone themost remarkable development in the last decade. The accu-racy and efficiency of the methods introduced in this periodimproved dramatically. Thus we were able to solve manyproblems that puzzled researchers in previous times. Wecould also recognize and investigate an entire spectrum ofnew problems, from novel classes of dynamical behavior todifferent phenomena that were previously either completelyunknown or impossible to investigate with available tools.

The history, definition, and applications of proper ele-ments are described in great detail in a number of reviews(e.g., Valsecchi et al., 1989; Shoemaker et al., 1989; Lemaître,1993; Knezevic and Milani, 1994; Knezevic, 1994). How-ever, for the sake of completeness, these topics are tackledin the following sections.

1.1. Historical Overview

A classical definition states that proper elements arequasi-integrals of motion, and that they are therefore nearlyconstant in time. Alternatively, one can say that they aretrue integrals, but of a conveniently simplified dynamicalsystem. In any case, proper elements are obtained as a re-sult of the elimination of short and long periodic perturba-tions from their instantaneous, osculating counterparts, andthus represent a kind of “average” characteristics of motion.

A concept of proper elements has been introduced byHirayama (1918) in his celebrated paper in which he an-nounced the discovery of asteroid families. Even if notusing the technical term “proper,” he employed Lagrange’sclassical linear theory of asteroid secular perturbations to

demonstrate that certain asteroids tend to cluster aroundspecial values of the orbital elements, which very closelycorrespond to the constants of integration of the solutionsof the equations of their motion, i.e., to a sort of averagedcharacteristic of their motion over very long timespans. Inhis later papers Hirayama (1923, 1928) explicitly computedjust the proper elements (proper semimajor axis, propereccentricity, and proper inclination) and used them for theclassification of asteroids into families.

The next important contribution came from Brouwer(1951). He computed asteroid proper elements again usinga linear theory of secular perturbations, but in combinationwith an improved theory of motion of the perturbing plan-ets (Brouwer and Van Woerkom, 1950). By including moreaccurate values of planetary masses, and the effect of the“great inequality” of Jupiter and Saturn, he was able to get amore realistic value for the precession rate of the perihelionof Saturn.

Williams (1969) developed a semianalytic theory of as-teroid secular perturbations that does not make use of atruncated development of the perturbing function, andwhich is therefore applicable to asteroids with arbitraryeccentricity and inclination. Williams’ proper eccentricityand proper inclination are defined as values acquired whenthe argument of perihelion ω = 0 (thus corresponding to theminimum of eccentricity and the maximum of inclinationover a cycle of ω). The theory is linearized in planetarymasses, eccentricities, and inclinations, so that the properelements computed by means of this theory (Williams,1979, 1989), even if much better than the previously avail-able ones, were still of limited accuracy.

Another achievement to be mentioned is that by Kozai(1979), who used his theory of secular perturbations forhigh-inclination asteroids (Kozai, 1962) to define a set of

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proper parameters to identify the families. The selectedparameters were semimajor axis, z component of the angu-lar momentum (integral of motion in a first-order theory,with perturbing planets moving on circular, planar orbits),and the minimum value of inclination over the cycle of theargument of perihelion (corresponding to ω = π/2).

Finally, we refer readers to the work by Schubart (1982,1991), Bien and Schubart (1987), and Schubart and Bien(1987), who pioneered attempts to determine the proper pa-rameters for resonant groups, i.e., for Hildas and Trojans.Since the usual averaging methods do not apply in this case,they adopted slightly different definitions of the properparameters, the most important difference being the substitu-tion of a representative value measuring the libration of thecritical argument instead of the usual proper semimajor axis.

1.2. Proper Elements/Parameters

The notion of proper elements is based on the lineartheory of secular perturbations, which dates back to Lagrange.Linear theory neglects the short periodic perturbations, con-taining anomalies in the arguments; this results in a constantsemimajor axis that becomes the first proper element ap. Thelong-term evolution of the other variables is obtained byapproximating the “secular” equations of motion with a sys-tem of linear differential equations. Because of the linear-ity assumption, the solutions can be represented in the planes(k,h) = (e cos ϖ, e sin ϖ) as the sum of “proper modes,”one for each planet, plus one for the asteroid. Thus the solu-tion can be represented by epicyclic motion: For the aster-oid, the sum of the contributions from the planets representsthe “forced” term, while the additional circular motion is theso-called “free oscillation” and its amplitude is the propereccentricity ep. The same applies to the plane (q,p) = (sin icos Ω, sin i sin Ω), with amplitude of the free term givenby the (sine of) proper inclination sin ip. Figure 1 shows theoutput of a numerical integration of an asteroid’s orbit for20,000 yr plotted in the (k,h) plane, and an epicyclic modelfitting to the data. As is apparent from the figure, the ap-proximation of the linear secular perturbation theory is goodenough for a timespan of the order of the period of circula-tion for the longitude of perihelion ϖ. However, even oversuch a timespan the linear theory is only an approximation,and over a much longer timespan (e.g., millions of years)it would be a rather poor approximation in most cases.

Proper elements can also be obtained from the output ofa numerical integration of the full equations of motion: Thesimplest method is to take averages of the actionlike vari-ables a, e, i, over times much longer than the periods of cir-culation of the corresponding angular variables. However,this method provides proper elements of low reliability: Thedynamical state can change for unstable orbits and in suchcases the simple average wipes out this essential informa-tion. Thus, if the goal is to compute proper elements stableto 1% of their value or better, over timespans of millions ofyears, it is necessary to use much more complicated theories.This is in essence saying that the approximate dynamical

system, rather than a simple linear one, is by itself complex,even if it admits integrals that are used as proper elements.

Whatever the type of theory, on the other hand, if it isto be accurate enough to represent the dynamics in theframework of a realistic model, its full-detail descriptionrequires delving into a very long list of often cumbersometechnicalities. For this reason, in the present paper we onlygive a qualitative description of the computational proce-dures, and then proceed to discuss the quality of the results.

Several different sets of proper parameters have beenintroduced over time, but the most common set, usuallyreferred to as “proper elements,” includes proper semima-jor axis ap, proper eccentricity ep, (sine of) proper inclina-tion (sin) ip, proper longitude of perihelion ϖp, and properlongitude of node Ωp, the latter two angles being accompa-nied by their precession rates (fundamental frequencies gand s respectively).

The analytical theories and the previously mentionedtheory by Williams (1969) use a different definition ofproper eccentricity and inclination. Other authors introducedcompletely different parameters to replace the standardproper elements. Still, the common feature of all these pa-rameters is their supposed near constancy in time (or moreprecisely, stability over very long timespans), and one cansay that in this sense the term “proper” is practically a syn-onym for “invariable.”

As explained by Lemaître (1993), even if quite differentin terms of the proper parameters and the ways to compute

–0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08–0.06

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0

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h =

e s

in(ϖ

)

k = e cos(ϖ)

ep

Cf

O

e

Fig. 1. The orbit of the numbered asteroid 27633 over 20,000 yrprojected in the (k,h) plane (full line). Data have been digitallysmoothed to remove short periodic perturbations. Point C repre-sents average value of the forced term, f is forced eccentricity, andthe dashed circle of radius ep represents the best-fitting epicycle.The value of the eccentricity e, obtained as the length of the vec-torial sum of the forced and the free terms, is an approximationof the current value.

Knezevic et al.: Determination of Asteroid Proper Elements 605

them, all the theories follow several basic steps, which canbe summarized as (1) modeling of the asteroid motion (N-body, restricted three-body) and distinguishing the fast andslow angles (that is, separating the perturbations dependingon the mean longitudes from the rest); (2) removal of theshort periodic perturbations (analytical or numerical averag-ing, online filtering) and computation of the mean elements;(3) splitting of the resulting Hamiltonian into two parts, theintegrable (secular) part, and the perturbation (the long-periodic part depending on slow angles) — in the case ofsynthetic theories this step corresponds to the removal ofthe forced terms; (4) removal of the long-periodic terms andcomputation of the proper values — at this stage, analyticand semianalytic theories resort to averaging over the slowangles and iterative procedures to compute the inverse mapof canonical transformation, while in the case of synthetictheories this phase includes the Fourier analysis and extrac-tion of the principal harmonics from the time series of meanelements (freed from the forced terms); (5) output of theproper values, possibly accompanied with their errors (stan-dard deviations, maximum excursions), fundamental fre-quencies, quality, and resonant codes, chaotic behavior indi-cators, and other information.

1.3. Applications

For a long time proper elements have been used for thesole purpose of identifying the asteroid families. Even to-day this is one of their most common applications (Zappalàet al., 1990, 1995). Since family identification is the sub-ject of another chapter in this book (Bendjoya and Zappalà,2002), we are not going to discuss it here in further detail.

On the other hand, the proper elements are computed forvery different asteroids, from NEAs (Gronchi and Milani,2001) to Trojans (Milani, 1994; Beaugé and Roig, 2001).They are used to describe very different dynamical phenom-ena, like stable chaos (Milani et al., 1997), resonant dynam-ics (Morbidelli, 1993), or nongravitational effects (Farinellaand Vokrouhlický, 1999). The list of problem-solving appli-cations of proper elements is long, ranging from the deter-mination of locations of secular resonances in the asteroidbelt (Milani and Knezevic, 1994) and in a wider planetaryregion (Knezevic et al., 1991, Michel and Froeschlé, 1997),through physical and dynamical studies of asteroid fami-lies (Morbidelli et al., 1995) and the determination of theirage (Milani and Farinella, 1994; Knezevic, 1999), to such“exotic” uses as the study of an optimum strategy for theorbit maintenance of a low lunar polar orbiter (Knezevic andMilani, 1998), to mention but a few.

2. THEORIES

2.1. Analytical Theory

The main problem in the manufacturing of fully analyti-cal theories of proper elements is in the complete degen-eracy of the unperturbed dynamics, the two-body problem.

Degeneracy means that some of the fundamental frequen-cies are zero, and indeed in the two-body approximationboth the perihelia and the nodes do not precess at all. In theHamiltonian formalism, this is expressed by the statementthat the unperturbed Hamiltonian function H0 = H0(L) is afunction only of one variable L, which in turn is a functionof the semimajor axis. The perturbed problem with Hamilto-nian H0(L) + µH1(l, g, L, G) (the small parameter µ repre-senting the ratio of the mass of the planets to the mass of theSun) can be handled with different perturbative approaches,but they all have in common the use of a solution of theunperturbed two-body problem to be substituted into theperturbing function H1. Thus they also have in common theproblem that the angle variable l, the mean anomaly con-jugate to L, can be eliminated, but the angles g, conjugateto the other action variables not appearing in H0, cannot beremoved by averaging.

In practice, this implies that the procedure to computeproper elements must always be decomposed into two com-putational steps: the transformation from osculating orbitalelements to mean elements, free from the short periodicperturbations (with arguments containing the fast variablel), and the transformation of the mean elements into properelements. A fully analytical theory performs both steps bymeans of the computation of functions for which analyticalexpressions, in practice truncations of some series, are avail-able. Note that it is possible, sometimes even convenient,to mix two different methods: As an example, Lemaître andMorbidelli (1994) use mean elements computed analyticallyas a starting point for their semianalytic computation ofproper elements.

Different perturbation techniques can be used, with theLie series technique being the most convenient for theoriespushed to higher order (and therefore based upon series withmany terms). In essence, analytical perturbation techniquesexploit the approximation of the perturbing function H1 bymeans of a finite sum of terms, each with a simple expres-sion of the form

µ b(L) eh e'k i j i'm cos(pl + ql' + δ)

where h, k, j, m, p, and q are integers; the primed elementsrefer to some perturbing planet; b is a known function; andδ is some combination (with integer coefficients) of theangles included in g, the perihelia and nodes of the asteroidand the planet. The truncation of the series is mostly basedon the degree in the small parameters eccentricity and incli-nation (of both the asteroid and the planets), although trun-cation for large values of the integers p and q is also pos-sible. Thus we can describe the degree of completeness of atheory by means of the order (in the small parameter µ) andof the degree (in the eccentricities and inclinations). Fromthis arises the main limitation of the analytical method: Theaccuracy, and stability with time, of the proper elements de-creases as the asteroid eccentricity and inclination increase.There is a boundary between the region where the analyti-cal proper elements are most suitable and the region wheremore computationally intensive methods, such as the semi-

606 Asteroids III

analytic ones, need to be used; Knezevic et al. (1995) havemapped this boundary (see section 2.4). The simple analyti-cal form of the terms implies that it is possible to performboth derivatives and integrals analytically; thus the corre-sponding operations can be applied to the series term byterm. An analytical theory can be expressed by means ofderivatives, integrals, and arithmetic operations on theseseries; thus they can be explicitly computed by means of afinite, although large, number of elementary operations. Inpractice, the current analytical theories use series with sev-eral tens of thousands of terms.

The series used in the current theories are essentiallybased upon the expansions computed by Yuasa (1973) andcorrected and completed by Knezevic (1989, 1993). Theseare complete to degree 4 in eccentricities and inclinations;only a few special terms of degree 6 have been added later(Milani and Knezevic, 1994). Yuasa (1973) defined an al-gorithm enabling the computation of proper elements witha theory containing the main terms of order 2 in µ and com-plete to degree 4 in eccentricity and inclination, but oneessential step was missing. Milani and Knezevic (1990)found that, at orders >1 in µ, the formulas of perturbationtheory explicitly provide a map between proper and meanelements, in the opposite direction from the one we dealwith in practice. Thus the computation of proper elementsfrom mean elements requires the solution of an inversefunction problem, and this is possible only by an iterativeprocedure. Milani and Knezevic (1999) later applied thesame argument to the computation of mean elements fromosculating ones.

There are two reasons to use analytical proper elements,even when other methods (for example, synthetic ones) areavailable for the same objects.

First, the large number of terms and the iterative proce-dure notwithstanding, even with complications introducedin later versions (Milani and Knezevic, 1992, 1994), the com-putational complexity of the fully analytical computation ofproper elements is not important (with respect to the com-puting power available today). To recompute from scratchthe proper elements for more than 67,000 asteroids requirescurrently less than 2 CPU hours on a standard workstation.Thus the only limit to the size of proper element catalogs,when computed analytically, is the size of the catalogs ofosculating elements. Taking into account that asteroids ob-served only at a single opposition generally have osculatingelements affected by uncertainties larger than the inaccu-racies in the computation of proper elements, the AstDysinformation system (http://hamilton.dm.unipi.it/astdys/) pro-vides analytical proper elements for all numbered and multi-opposition asteroids. This catalog is updated every monthwith new discoveries, new numberings, and new recoveriesproviding multiopposition orbits. Since the asteroid familiesare statistical entities, the advantage of such a large set ofproper elements may largely compensate for an accuracythat is not as good as that for the synthetic proper elements.

The second main advantage of the analytical proper el-ements is that the computational algorithm automatically

provides information on all the resonances. Indeed, the it-erative procedures used in the computation can be diver-gent where a small divisor, resulting from the attempt tointegrate one of the trigonometric terms with very slowlyvarying arguments, occurs. On the other hand this is, bydefinition, a resonance. Difficulties in the computation ofmean elements (from the osculating ones) are due to mean-motion resonances, involving the anomalies of the asteroidand a planet. Difficulties in the convergence of the itera-tions for the computation of proper elements from meanelements indicate secular resonances, i.e., very small fre-quencies resulting from combinations of the frequencies gand s of the perihelia and nodes with the correspondingfrequencies for the perturbing planets. Thus the analyticalproper elements come with quality codes indicating not onlylower accuracy due to failure of convergence, but also theresonance responsible for the problem. By the same theo-ries it is possible to construct maps of secular resonances(Knezevic et al., 1991, Milani and Knezevic, 1994), includ-ing combinations of up to four secular frequencies (and insome special cases six or eight frequencies). In Fig. 2 wegive an example of such a map with the most importanttwo- and four-frequency secular resonances occurring in themain asteroid belt.

2.2. Semianalytical Theory

The semianalytic calculation of proper elements was ini-tiated by Williams (1969) and then revisited by Lemaîtreand Morbidelli (1994). It is a classical perturbation method,where the two averaging processes (the first one on themean longitudes and the second on the pericenters andnodes) are performed numerically. This avoids the expan-sion in the eccentricity (e) and inclination (i) of the aster-oid, and makes the method particularly suitable for about20% of the asteroids with large values for these elements.

The latest version of the theory is written in a Hamilto-nian formalism and computed up to the second order in theperturbing masses (presently only Jupiter and Saturn) andup to the first degree in the eccentricities (e') and inclina-tions (i') of the perturbers.

Elimination of the short-periodic terms is performednumerically by the calculation of the double integrals (overthe two mean longitudes). After this averaging, the semi-major axis is constant and represents the first proper ele-ment. To compute the averaged Hamiltonian, a Fourierseries of the slow angles is used, with the coefficients evalu-ated on a three-dimensional grid (in a, e, and i) and stored;a triple linear interpolation is used each time the Hamilto-nian and its derivatives have to be evaluated.

The averaged Hamiltonian is split in two parts, based onthe smallness of the parameters e' and i'; the integrableproblem corresponds to circular and planar motion of theperturbing planets, while the perturbation part gathers allthe first-order contributions in e' and i'. The dynamics ofthe integrable problem has already been analyzed by Kozai(1962), and it reveals different behaviors for low and high

Knezevic et al.: Determination of Asteroid Proper Elements 607

inclinations; in the latter case, a critical curve separates thephase space into two regions, corresponding to librations(about 90° or 270°) and to circulations of the argument ofthe pericenter ω of the asteroid.

The removal of the long periodic terms is done by us-ing the action angle variables and is based on the semi-analytic method developed by Henrard (1990). The result-ing Hamiltonian K, after the second averaging process,depends only on two proper actions, J and Z, which are bothconstant. The result is a proper orbit of area proportionalto J, located in a plane identified by the value of Z. Eachpoint on the proper orbit is characterized by a value of thephase ψ, which can be considered as a proper argument ofperihelion. From the values of ψ, J, and Z, the correspond-ing values of e, i, and ω can be calculated. Any point canbe chosen as a “representative” of this orbit: For example,ψ = 0°, 180° corresponds to the minimum value of the ec-centricity and maximum of the inclination along the properorbit. On the other hand, ψ = 90°, 270° corresponds to themaximum of the eccentricity and the minimum of the in-clination; this option allows the definition of proper ele-

ments even for ω librators, since the libration center is ei-ther 90° or 270°.

The proper orbit is also characterized by the two basicfrequencies, g and s, calculated as the partial derivatives ofthe Hamiltonian K with respect to the actions J and Z. Theset a, g, and s is also a set of “proper elements” and is in-dependent of the choice of the representative point on theproper orbit.

The existence of two high inclined groups of asteroidswas clearly shown by this method: a Pallas family at about35° inclination (Lemaître and Morbidelli, 1994), and aHungaria family at 2 AU (Lemaître, 1994).

The precision is limited by the first-order developmentin e' and i' and could not be easily improved. However, thissemianalytic procedure has some advantages, such as thefact that it allows the calculation not only of the properfrequencies, but also of their derivatives with respect to theaction variables; to our knowledge, this last possibility hasnot been explored in specific applications.

The last calculation of a complete set of proper elementsby the semianalytic method was done in 1994, but only for

2 2.2 2.4 2.6 2.8 3 3.2 3.40

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g – g6 + s – s6

g + g5 – 2g6

g + g5 – 2g6

g – g5 + s – s6

g + 2g5 – 3g6

2g – 2g6 + s – s6

g – g6

Fig. 2. Map of a sample of secular resonances in the asteroid main belt, involving up to six fundamental frequencies. The locationsof the resonances are computed for ep = 0.1 and are marked with three lines corresponding to the zero value of the correspondingdivisor and to ±0.5 arcsec/yr (with the exception of g – g6 for which we plot wider boundaries corresponding to ±2 arcsec/yr). g ands are the frequencies of the asteroid perihelion and node, while g5, g6, and s6 are the corresponding values of Jupiter and Saturn.

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the asteroids for which the analytic method proved not to beapplicable. Catalogs of semianalytic elements for approxi-mately 2500 asteroids are available from http://www.fundp.ac.be/sciences/math/equadif.html.

2.3. Synthetic Theory

The latest contribution to the field of asteroid proper-element determination is the synthetic theory by Knezevicand Milani (2000). This is, in fact, a set of purely numeri-cal procedures by means of which one can derive classicalproper elements (ap, ep, sin ip, ϖp, Ωp) and fundamentalfrequencies (g, s). The theory employs the approach usedby Carpino et al. (1987) for the major planets, and consistsof the following steps: First, one numerically integrates theasteroid orbits for a sufficient timespan, together with theorbits of perturbing planets included in the model (the in-direct effects of the planets not included in the dynamicalmodel are accounted for by applying the so-called bari-centric correction to the initial conditions). The short peri-odic perturbations are removed by means of an online filter-ing of the osculating elements, performed in the course of theintegration itself; simultaneously, the maximum Lyapunovcharacteristic exponents are derived from the variationalequations to monitor the chaotic behaviors. The forcedoscillations are then removed from the output of the integra-tion and the resulting time series spectrally resolved underthe constraints set by the d’Alembert rules to extract prin-cipal harmonics (proper values) together with the associ-ated fundamental frequencies and error estimates (standarddeviations and maximum excursions).

Because it is purely numerical, this theory is compara-tively simple in principle. It furnishes results of a superioraccuracy with respect to the analytical theory and, more im-portantly, provides a straightforward way to estimate errorsof all the proper values and for each asteroid included inthe computation. We used a simple running box method,computing proper values over a number of shorter time in-tervals and then deriving the corresponding rms and maxi-mum deviation values. The length and shift (i.e., number)of boxes has been selected in such a way to best fit thedynamics, thus in a different way for the inner and the outerbelt, for shorter and longer integration timespans, etc. Thetheory applies equally well to the asteroids of any eccen-tricity and inclination, with the obvious exception of theplanet-crossing objects; since it is based on the averagingprinciple, it does not work for the resonant asteroids either.

As an example, in Fig. 3 the number frequency distri-butions of error estimates are given for the synthetic propereccentricities and inclinations of numbered asteroids. Theseresults are on average better by more than a factor of 3 thanthe typical accuracy of the proper elements derived fromthe analytical theory (Milani and Knezevic, 1994).

The numerical integration of tens of thousands of orbitsis, on the other hand, a time-consuming procedure, so thatthe synthetic theory is ostensibly less efficient in terms ofCPU time and more demanding in terms of necessary com-

puting power. This in turn leads to other potential draw-backs, such as getting results supposedly valid over verylong timespans from the integrations over comparativelyshort intervals; such short integrations might affect the de-termination of maximum LCEs for the objects subject to aslow chaos, etc.

These drawbacks, however, are more than amply com-pensated for by the accuracy achieved and by the otheradvantages mentioned above. The synthetic theory is at themoment certainly the best theory available on the market.

The synthetic proper elements are available from theAstDys service at http://hamilton.dm.unipi.it/astdys. Thereare three catalogs available for downloading; these catalogscontain the proper elements themselves, their standard de-viations, and their maximum excursions (plus some addi-tional information). The files are updated on a monthly basisfor the numbered asteroids between 2.0 and 4.0 AU, withthe exception of the planet-approaching asteroids; the cata-logs as of July 2001 contain proper elements for 24,910asteroids.

2.4. Comparison of Theories in Terms ofAccuracy and Efficiency

The only direct comparison between different theoriesto compute asteroid proper elements has been made byKnezevic et al. (1995). They integrated the orbits of fiveselected asteroids for 4.5 m.y., simultaneously computing

Fig. 3. Number frequency distributions of standard deviations(top) and maximum excursions (bottom) of the proper eccentric-ity (left) and proper sine of inclination (right). About 80% of theasteroids have standard deviation of proper eccentricity below0.001, while for the sine of proper inclination this percentage risesto more than 90%. Median rms is 0.00024 in proper eccentricityand 0.00011 in sine of proper inclination. Further improvementis still possible, depending on the available computing power.

–5 –4 –3 –2 –10

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log10(δ sin i)

Knezevic et al.: Determination of Asteroid Proper Elements 609

the proper elements and their instabilities by means of theanalytical (Milani and Knezevic, 1994) and semianalytical(Lemaître and Morbidelli, 1994) theories. Comparing theoutcomes (see Fig. 4 for an example) they concluded thatanalytic proper elements are more accurate for the aster-oids up to about 15° of inclination, while the semianalyticones are better above about 17°. In the intermediate transi-tion region both methods have roughly the same stability.

There is no direct comparison on a case-by-case basisfor the synthetic theory with respect to any of the othertheories. Thus, at the moment we can resort only to thegeneral estimate mentioned above of an improvement by afactor of more than 3, on average, of the synthetic properelements with respect to the results coming from the ana-lytical theory (section 2.3).

As for the efficiency in the computation, the analyticaltheory is by far the fastest and the synthetic theory is themost time consuming, with the semianalytical theories fall-ing somewhere in between these two. In practice, therefore,when choosing between the theories to be applied under agiven circumstance, one always has to find an optimumtradeoff between the conflicting demands of efficiency andaccuracy.

3. SPECIAL CASES

3.1. Resonant Proper Elements

The removal of the short- and long-periodic terms be-comes problematic in the neighborhood of a resonance — amean-motion resonance in the first average procedure, asecular resonance in the second one. If one of the frequen-cies is close to zero, then the usual averaging methods haveto be adapted to this particular situation. The nonresonantharmonics can be eliminated but the critical angle is stillpresent in the averaged problem.

Of course the first reaction is to treat the problem purelynumerically, integrating the orbits over millions of years andusing a frequency analysis to calculate the proper frequen-cies. This method seems very suitable, because the realcases of resonant asteroids are not so numerous and the timerequired by the integration is still reasonable. The Trojans,well-known 1:1 jovian resonant asteroids, or the Hildagroup were integrated following this idea (section 3.2).

However, the simulations concerning fictitious fragmentsof asteroids, for example, in order to statistically investi-gate the efficiency of a particular resonance in delivering

Fig. 4. Comparison of the proper eccentricities (top) and proper inclinations (bottom) for asteroid 387 Aquitania. Proper valuesdetermined by means of the semianalytic theory are defined as the values occurring for ω = 0° (ex, ix) and for ω = 90° (ey, iy), thus inthe former case corresponding to the minimum of eccentricity and maximum of inclination over a cycle of the argument of perihelion,and in the latter case corresponding to the maximum of e and minimum of i. Proper values obtained by means of the analytic theoryare in the middle (e, sin i). For this particular asteroid, having comparatively high orbital inclination, the unremoved instabilities ofproper eccentricity as obtained from the two theories are about the same, while for the proper inclination the semianalytical theoryalready gives better results.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

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610 Asteroids III

meteorites to the Earth, require the calculation of thousandsof orbits. This encourages the development of specific reso-nant seminumerical methods, allowing a picture of thephase space to be drawn and the computation of “resonantproper elements” in a much faster way.

Let us emphasize that these resonant proper elements arenot the same as in the nonresonant cases; they can be de-fined as any identifier of the trajectories obtained in themodel after the resonant averaging process. Their defini-tion is arbitrary (e.g., the values of the eccentricity or ofthe inclination when the critical angle equals some specificvalue, for example, zero) and cannot be directly connectedto the classical nonresonant proper eccentricity and inclina-tion. Thus it is not possible to mix resonant and nonresonantproper elements for purposes such as family classification.

In this approach, let us mention the calculation of secu-lar resonant proper elements by Morbidelli (1993), whoapplies the same semianalytic method as Lemaître andMorbidelli (1994) but adapted for the resonant case. Afterthe elimination of the short-periodic terms, instead of elimi-nating all secular perturbation terms in the second step, acritical angle (g – g6, for example) is still present in the in-tegrable part of the Hamiltonian. Typical cases such as582 Olympia, 945 Barcelona, or 739 Mandeville have beentreated by this method and show a good qualitative agree-ment with the numerical integrations. However, no catalogsor listings of secular resonant real asteroids with corre-sponding proper elements or identifiers are available; any-one interested in this calculation can contact the author ofthe method, A. Morbidelli (morby@obs-nice.fr).

3.2. Trojans and Hildas

The Trojan asteroids are a special case, because they arelocked in the 1:1 resonance with Jupiter. Because this issuch a strong resonance, librations with very large ampli-tude are possible: The critical argument λ – λ' can havesmall librations around 60° and also oscillate between =30°and =100°. Moreover, the inclination can be very large, upto 33°. As a result, the development of an analytical theory isvery difficult, and indeed it has been achieved only within asimplified three-body model (Érdi, 1988). A semianalyticaltheory is conceivable, but also difficult to compute within arealistic model.

The synthetic approach has been pioneered by Bien andSchubart (see Bien and Schubart, 1987; Schubart and Bien,1987; and references therein). They use as proper elementsa proper eccentricity and proper inclination defined essen-tially in the same way as in the synthetic theories for main-belt asteroids and an amplitude of libration of λ – λ'. Milani(1993, 1994) used a synthetic approach adapted to Érdi’s(1988) theory, and computed proper elements for 188 num-bered, multiopposition, and well-observed single-oppositionTrojans. Although the results suggested the possible exist-ence of some Trojan families, the number of asteroids inthe catalog was too small for a firm conclusion. Milani(1994) also mapped the secular resonances involving the

node by fitting the results of the synthetic computations ofthe frequencies, especially s, with a polynomial in the ac-tion variables.

More recently there has been interesting semianalyticwork by Beaugé and Roig (2001), who produced a catalog(available from the authors upon request) of proper ele-ments for 533 Trojans, also proposing a number of poten-tial dynamical families. However, these elements are oflower accuracy than those produced by means of a syntheticapproach. As of July 2001 there are 679 Trojans with goodquality orbits (either numbered or multiopposition), thus thecomputation of a new catalog of accurate proper elementsshould allow statistically significant conclusions about theexistence of Trojan families.

In regard to Hildas, even if important results on theirdynamics have been obtained in recent years (e.g., Ferraz-Mello et al., 1998), very little work has been done abouttheir proper elements. The computation of “three character-istic parameters” of orbits of Hilda-type asteroids initiatedby Schubart (1982) has been later extended to newly dis-covered objects by the same author (Schubart, 1991). Evenat present the only source of data for this group of bodiescan be found on the Web site maintained by J. Schubart(http://www.rzuser.uni-heidelberg.de/~s24/hilda.htm).

4. CONCLUSIONS AND FUTURE WORK

Activity in the field of determination of asteroid properelements has been very intense in the previous decade, andone can safely claim that more work has been done andmore improvement achieved in this period than in all thepreceding time.

The proper elements are now computable for essentiallyall asteroids for which there are accurate osculating orbits;thus we currently have proper elements determined forabout 70,000 objects. The three principal theories intendedfor production of large catalogs of proper elements for themain-belt objects, plus a number of specially adapted onesfor specific groups of objects, have been developed and suc-cessfully applied during this period.

The accuracy, long-term stability, and reliability of thecomputed proper elements have all remarkably improved(by orders of magnitude). This enabled a lot of importantproblems to be tackled and provided explanations for manyinteresting properties of individual asteroids and of theirfamilies, which, in turn, decisively contributed to a betterunderstanding of the dynamical and collisional evolutionof the asteroid belt as a whole.

Apart from providing a reliable basis for the identifica-tion of families, the studies related to the determination ofproper elements led to the development of complete theo-ries of asteroid motion, revealing at the same time manynovel dynamical features and concepts (nonlinear secularresonances, stable chaos, dynamical mass loss, etc.)

All the main theories to compute asteroid proper ele-ments have now reached a high level of “maturity,” beingdeveloped almost to their corresponding theoretical limits

Knezevic et al.: Determination of Asteroid Proper Elements 611

(Milani and Knezevic, 1994; see also section 2.2), while onthe other hand being amply and thoroughly tested throughyears of application. Therefore, it is plausible to conjecturethat in the near future we can expect more improvementin terms of their practical application than in the theoriesthemselves.

There are two principal directions in which the develop-ment can reasonably take place: improvement of the accu-racy and reliability of determination of proper elements(e.g., by extending the timespan over which the averagingis done in the synthetic theory), and computation of properelements for an ever-increasing number of asteroids withdifferent dynamical properties (for the purpose of enlarg-ing and completing the database); in this latter class onecan include computation of proper elements for objects suchas Centaurs and TNOs for which this has not yet been done.

Acknowledgments. We thank referees F. Namouni and C.Murray for useful suggestions, and A. Morbidelli for providinginput on the resonant case.

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