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The Determination of Asteroid Proper Elements

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The Determination of Asteroid Proper Elements Knezevic et al.: Determination of Asteroid Proper Elements 603 The Determination of Asteroid Proper Elements Z. Knezevic Astronomical Observatory, Belgrade A. Lemaître Facultes Universitaires Notre-Dame de la Paix, Namur A. Milani University 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 study the 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 a special emphasis on the comparative advantages/disadvantages of the methods, and on the accuracy and availability of the computed proper elements. We also discuss special techniques applied in some particular cases (mean motion and secular resonant bodies). Finally, we draw conclusions and suggest directions for future work. 1. INTRODUCTION demonstrate that certain asteroids tend to cluster around special values of the orbital elements, which very closely The computation of asteroid proper elements is certainly correspond to the constants of integration of the solutions one of the fields of asteroid research that has undergone the of the equations of their motion, i.e., to a sort of averaged most remarkable development in the last decade. The accu- characteristic of their motion over very long timespans. In racy and efficiency of the methods introduced in this period his later papers Hirayama (1923, 1928) explicitly computed improved dramatically. Thus we were able to solve many just the proper elements (proper semimajor axis, proper problems that puzzled researchers in previous times. We eccentricity, and proper inclination) and used them for the could also recognize and investigate an entire spectrum of classification of asteroids into families. new problems, from novel classes of dynamical behavior to The next important contribution came from Brouwer different phenomena that were previously either completely (1951). He computed asteroid proper elements again using unknown or impossible to investigate with available tools. a linear theory of secular perturbations, but in combination The history, definition, and applications of proper ele- with an improved theory of motion of the perturbing plan- ments are described in great detail in a number of reviews ets (Brouwer and Van Woerkom, 1950). By including more (e.g., Valsecchi et al., 1989; Shoemaker et al., 1989; Lemaître, accurate values of planetary masses, and the effect of the 1993; Knezevic and Milani, 1994; Knezevic, 1994). How- “great inequality” of Jupiter and Saturn, he was able to get a ever, for the sake of completeness, these topics are tackled more realistic value for the precession rate of the perihelion in the following sections. of Saturn. Williams (1969) developed a semianalytic theory of as- 1.1. Historical Overview teroid secular perturbations that does not make use of a truncated development of the perturbing function, and A classical definition states that proper elements are which is therefore applicable to asteroids with arbitrary quasi-integrals of motion, and that they are therefore nearly eccentricity and inclination. Williams’ proper eccentricity constant in time. Alternatively, one can say that they are and proper inclination are defined as values acquired when true integrals, but of a conveniently simplified dynamical the argument of perihelion ω = 0 (thus corresponding to the system. In any case, proper elements are obtained as a re- minimum of eccentricity and the maximum of inclination sult of the elimination of short and long periodic perturba- over a cycle of ω). The theory is linearized in planetary tions from their instantaneous, osculating counterparts, and masses, eccentricities, and inclinations, so that the proper thus represent a kind of “average” characteristics of motion. elements computed by means of this theory (Williams, A concept of proper elements has been introduced by 1979, 1989), even if much better than the previously avail- Hirayama (1918) in his celebrated paper in which he an- able ones, were still of limited accuracy. nounced the discovery of asteroid families. Even if not Another achievement to be mentioned is that by Kozai using the technical term “proper,” he employed Lagrange’s (1979), who used his theory of secular perturbations for classical linear theory of asteroid secular perturbations to high-inclination asteroids (Kozai, 1962) to define a set of 603 604 Asteroids III proper parameters to identify the families. The selected 0.08 parameters were semimajor axis, z component of the angu- lar momentum (integral of motion in a first-order theory, 0.06 with perturbing planets moving on circular, planar orbits), and the minimum value of inclination over the cycle of the 0.04 argument of perihelion (corresponding to ω = π/2). ) e Finally, we refer readers to the work by Schubart (1982, e p ϖ 0.02 1991), Bien and Schubart (1987), and Schubart and Bien (1987), who pioneered attempts to determine the proper pa- f C 0 rameters for resonant groups, i.e., for Hildas and Trojans. h = e sin( O Since the usual averaging methods do not apply in this case, they adopted slightly different definitions of the proper –0.02 parameters, the most important difference being the substitu- tion of a representative value measuring the libration of the –0.04 critical argument instead of the usual proper semimajor axis. –0.06 1.2. Proper Elements/Parameters –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 k = e cos(ϖ) The notion of proper elements is based on the linear theory of secular perturbations, which dates back to Lagrange. Fig. 1. The orbit of the numbered asteroid 27633 over 20,000 yr Linear theory neglects the short periodic perturbations, con- projected in the (k,h) plane (full line). Data have been digitally taining anomalies in the arguments; this results in a constant smoothed to remove short periodic perturbations. Point C repre- sents average value of the forced term, f is forced eccentricity, and semimajor axis that becomes the first proper element ap. The long-term evolution of the other variables is obtained by the dashed circle of radius ep represents the best-fitting epicycle. approximating the “secular” equations of motion with a sys- The value of the eccentricity e, obtained as the length of the vec- torial sum of the forced and the free terms, is an approximation tem of linear differential equations. Because of the linear- of the current value. 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- system, rather than a simple linear one, is by itself complex, oid, the sum of the contributions from the planets represents even if it admits integrals that are used as proper elements. the “forced” term, while the additional circular motion is the Whatever the type of theory, on the other hand, if it is so-called “free oscillation” and its amplitude is the proper to be accurate enough to represent the dynamics in the eccentricity ep. The same applies to the plane (q,p) = (sin i framework of a realistic model, its full-detail description cos Ω, sin i sin Ω), with amplitude of the free term given requires delving into a very long list of often cumbersome by the (sine of) proper inclination sin ip. Figure 1 shows the technicalities. For this reason, in the present paper we only output of a numerical integration of an asteroid’s orbit for give a qualitative description of the computational proce- 20,000 yr plotted in the (k,h) plane, and an epicyclic model dures, and then proceed to discuss the quality of the results. fitting to the data. As is apparent from the figure, the ap- Several different sets of proper parameters have been proximation of the linear secular perturbation theory is good introduced over time, but the most common set, usually enough for a timespan of the order of the period of circula- referred to as “proper elements,” includes proper semima- ϖ tion for the longitude of perihelion . However, even over jor axis ap, proper eccentricity ep, (sine of) proper inclina- ϖ such a timespan the linear theory is only an approximation, tion (sin) ip, proper longitude of perihelion p, and proper Ω and over a much longer timespan (e.g., millions of years) longitude of node p, the latter two angles being accompa- it would be a rather poor approximation in most cases. nied by their precession rates (fundamental frequencies g Proper elements can also be obtained from the output of and s respectively). a numerical integration of the full equations of motion: The The analytical theories and the previously mentioned simplest method is to take averages of the actionlike vari- theory by Williams (1969) use a different definition of ables a, e, i, over times much longer than the periods of cir- proper eccentricity and inclination. Other authors introduced culation of the corresponding angular variables. However, completely different parameters to replace the standard this method provides proper elements of low reliability: The proper elements. Still, the common feature of all these pa- dynamical state can change for unstable orbits and in such rameters is their supposed near constancy in time (or more cases the simple average wipes out this essential informa- precisely, stability over very long timespans), and one can tion. Thus, if the goal is to compute proper elements stable say that in this sense the term “proper” is practically a syn- to 1% of their value or better, over timespans of millions of onym for “invariable.” years, it is necessary to use much more complicated theories.
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