Asteroid Orbit Computation

Bowell et al.: Asteroid Orbit Computation

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Asteroid Orbit Computation Edward Bowell Lowell Observatory

Jenni Virtanen University of Helsinki

Karri Muinonen University of Helsinki and Astronomical Observatory of Torino

Andrea Boattini Instituto di Astrofisica Spaziale

During the last decade, orbit computation has evolved from a deterministic pursuit to a statistical one. Its development has been spurred by the advent of the World Wide Web and by the greatly increased rate of asteroid astrometric observation. Several new solutions to the inverse problem of orbit computation have been devised, including linear, semilinear, and nonlinear methods. They have been applied to a number of problems related to asteroids’ computed skyplane or spatial uncertainty, such as recovery/precovery, identification, optimization of search strategies, and Earth collision probability. The future looks very bright. For example, the provision of milliarcsec-accuracy astrometry from spacecraft will allow the modeling of asteroid shape, spin, and surface-scattering properties. The development of deep, widefield surveys will mandate the almost complete automation of the orbit computation process, to the extent that one will be able to transform astrometric data into a desired output product, such as an ephemeris, without the intermediary of orbital elements.

1.

velopment and application of powerful new techniques of orbit computation have largely kept pace. The derivation of orbital elements from astrometric observations of asteroids is one of the oldest inversion problems in astronomy. Usually, the observational data comprise a set of right ascensions and declinations at given times, although other types of data, such as radar time delay and Doppler astrometry, may also be used. Six orbital elements, at a specified epoch, suffice to describe heliocentric motion, a common parametrization being by means of Keplerian elements a, e, i, Ω, ϖ, M (respectively, semimajor axis, eccentricity, inclination, longitude of the ascending node, argument of perihelion, and mean anomaly at a specified time). In the two-body case, only M changes with time. In the n-body case, in which account is taken of gravitational perturbations due to planets and perhaps other bodies, along with relativistic and nongravitational effects, all six elements may change with time. Most remarkably, the inversion method developed by Gauss in 1801 (e.g., see Gauss, 1809; Teets and Whitehead, 1999), in response to the loss of the first-discovered asteroid Ceres, has never really been supplanted (in fact, there exists a family of methods originating from the ideas of Gauss and elaborated by others). Gauss also introduced the method of least squares, which sowed the seeds for some of the probabilistic methods of orbit com-

INTRODUCTION

There has been a revolution in orbit computation over the past decade or so. It has been fueled in large part by observational efforts to find near-Earth asteroids (NEAs). For example, as Stokes et al. (2002) makes plain, the volume of observations has increased by almost 2 orders of magnitude, which has resulted in an even greater surge in the rate of orbit computation. In the book Asteroids II, Bowell et al. (1989) and Ostro (1989) seem to have been the only authors to refer to uncertainty in the positions of asteroids, the former in the context of recovering a newly discovered asteroid at a subsequent lunation, the latter in regard to the improvement of an asteroid’s orbit when radar observations have been incorporated. Indeed, Bowell et al. (1989) end their chapter uncertain about the future of then-infant CCD observations. There is no reference to the need for automated orbit computation or to the rapid dissemination of data worldwide, both of which we now take for granted, and both of which have been enabled by an enormous increase in computing power and a concomitant reduction in costs. One can argue that it was the development of the World Wide Web that created a synergy between observers and orbit computers, each driving advances in the other’s research, to the benefit of all. As we will describe, the de27

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putation that have been developed over the past decade. A description of these methods and their application will be our central preoccupation in this chapter. For more detailed summaries of the development of orbit computation, readers are referred to Danby (1988) and Virtanen et al. (2001). Hitherto, it has been customary to use the term orbit determination to cover the sequential processes of preliminary (initial) orbit fitting (normally using observations at three times) and differential correction (normally consisting of a least-squares fit using all the observations deemed accurate enough, planetary perturbations optionally being allowed for), independent of whether statistical or nonstatistical techniques are used. In this chapter, we largely ignore the terms orbit determination, determinacy, and indeterminacy because, in our experience, their use results in unnecessary confusion. The title of our chapter pertains to both the inverse problem of deriving the orbital element probability density from the astrometric observations and the prediction problem of applying the probability density. It seems of doubtful use to apply statistical techniques to asteroids having an arbitrarily small number of observations. When there are two observations only, the number of observations is smaller than the number of orbital elements to be estimated. Nevertheless, techniques that are best described as statistical, and that have been developed during the past decade for inverse problems involving small numbers of observations, are superior to earlier techniques and appear to capture the overall significance of the observational uncertainty. We advise that, when there are fewer than on the order of 10 observations, the resulting orbital-element probability density must be applied with particular caution. Note that, in this chapter, we are only peripherally concerned with advances in celestial mechanics; for example, work on integrators and solar-system dynamics is outside our purview. We concentrate on modeling the interrelations among the astrometric observations, orbital elements, and the predictions they allow. We cannot emphasize enough how much online software and databases have contributed to the explosion of observational activity in the post-Asteroids II era. They have provided observers powerful tools for planning observations. The Minor Planet Center (MPC) has been a pioneer in ephemeris and targeting services (http://cfa-www.harvard.edu/cfa/ ps/mpc.html). Lowell Observatory’s asteroid services (http:// asteroid.lowell.edu) have been most useful for main-belt asteroid observing; they are one of only two services that provide ephemeris uncertainties for non-near-Earth asteroids. The Jet Propulsion Laboratory’s Horizons ephemeris computation system (http://ssd.jpl.nasa.gov/horizons.html) is maintained by JPL’s Solar System Dynamics Group. At the University of Pisa, the Near Earth Objects–Dynamic Site (NEODyS; http://newton.dm.unipi.it/cgi-bin/neodys/neoibo) is a compilation of orbital and observational information on near-Earth asteroids. Its counterpart, ASTDyS (http:// hamilton.dm.unipi.it/cgi-bin/astdys/astibo), pertains to numbered and multiapparition asteroids. Both of the University of Pisa sites rely on an orbit-computation freeware pack-

age called OrbFit (http://newton.dm.unipi.it/~asteroid/orbfit) (Milani, 1999), developed by a consortium of about a dozen astronomers. We begin our review with a description of the theoretical underpinnings of the principal new techniques of orbit estimation. All of them accommodate probabilistic treatment, by which we understand fitting orbits that satisfy the observations within their estimated uncertainty. In section 3, we describe the implications of the new work as it pertains, in a broad sense, to predicting asteroid positions; in section 4, we outline some areas of future research that are likely to become prominent. We conclude with some thoughts and speculations about what might engage the interest of orbit computers a decade hence. 2.

INVERSE PROBLEM

The inverse problem entails the derivation of an orbitalelement probability density from observed right ascensions and declinations. Gauss’ theory made use of the normal distribution of observational errors and the method of least squares. That single least-squares orbit solutions can be misleading was realized long ago. For one, “unphysical” orbital elements may result; for another, poor convergence of the differential correction process can be an indicator of large uncertainties in the orbital elements. Computer-based iterative orbit-estimation methods have made it possible to explore part of the parameter space of orbit solutions very rapidly. For example, Väisälä orbits, in which it is assumed that an asteroid is observed at perihelion (Väisälä, 1939), can provide a useful — though not foolproof — tool for discriminating between newly discovered NEAs and mainbelt asteroids (MBAs) and for estimating their ephemeris uncertainties. Regardless of an asteroid’s single orbit solution, a fundamental question to ask is, “How accurate are the orbital elements?” At the time of the Asteroids II conference, all star catalogs contained zonal errors, sometimes amounting to an arcsec or more. Moreover, their relatively low surface densities and lack of faint star positions introduced field mapping and magnitude-dependent positional errors in asteroid astrometry. Beginning in 1997, the HIPPARCOS reference frame was used for high-density catalogs such as GSC 1.2 and USNO-A2.0 that are practically free of zonal errors. Even better catalogs, such as UCAC, are in the pipeline (e.g., see Gauss, 1999). For asteroid astrometry at its current best accuracy of tens of milliarcseconds, these catalogs are for most purposes error free and therefore introduce no systematic error. The remaining practical problems are related to generally poor stellar magnitudes and color indices and, as time goes by, positional degradation arising from imperfectly known stellar proper motions. If one knows, or can infer, which reference catalog was used for asteroid astrometric reductions, it is possible, to an extent, to compensate for systematic zonal errors. Such a technique was used by Stone et al. (2000) in their prediction of stellar occultations by (10199) Chariklo. In principle, the majority of his-

Bowell et al.: Asteroid Orbit Computation

toric astrometric data could be so corrected, though the amount of labor might be large. A new format for the submission of astrometric data, to include reference-frame and S/N ratio information, will soon be implemented by the MPC. 2.1. Inversion Using Bayesian Probabilities In the inverse problem, the derivation of an orbital-element probability density is based on a linear relationship between observation, theory, and error: The observed position is the sum of the position computed from six orbital elements, a systematic error, and a random error. It is customary to assume that the systematic error has been corrected for or is sufficiently arbitrary to be accounted for by the random error. A probability density is assumed for the random error: Typically, based on statistics of the O-C residuals, the density is assumed to be Gaussian (accompanied by an error-covariance matrix). Using Bayes’ theorem, the orbital-element probability density then follows from the aforementioned linear relationship and is conditional upon (and thus compatible with) the observations. Because of the nonlinear relationship between the orbital elements and the positions computed from them, the rigorous orbital-element probability density is non-Gaussian. The orbital-element probability density may, if desired, be multiplied by an a priori probability density, such as the distribution of the orbital elements of known asteroids. Only during the past decade has it been understood that the assumption of a Gaussian distribution of errors is sometimes not applicable to asteroid and comet orbit computation. Particularly for observations made recently, accidental astrometric errors (due to observational noise) are small compared to reference-star positional errors, known to have regional biases (zonal errors), or imperfect field mapping. Thus the error distribution in O-C positional residuals appears to be non-Gaussian. Although Muinonen and Bowell (1993) studied non-Gaussian probability densities for the observational error — using methods based on so-called statistical inversion theory (Lehtinen, 1988; Menke, 1989; Press et al., 1994) — the Gaussian density remained the most attractive statistical model for the random error. In spite of criticism of Gaussian hypotheses, alternative statistical models — aside from the work by Muinonen and Bowell (1993) — have not been put forward. Gaussian hypotheses are further supported by the impossibility of discriminating between systematic and random errors; indeed, if there is a large number of sources of uncorrected systematic error, the total error is essentially random, and the central limit theorem elevates carefully constructed Gaussian hypotheses above all other statistical hypotheses. In treating orbits probabilistically, orbit computers concerned with astrodynamic applications have taken the lead over those working on the analysis of groundbased observations. For example, Cappellari et al. (1976) were pioneers in their analysis of spacecraft trajectories, although Brouwer and Clemence (1961) gave an introduction to orbital error analysis. The first application of statistical inversion theory

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(see references above) to asteroid orbit determination was explored by Muinonen and Bowell (1993), who studied nonGaussian observational noise using an exp(–|x|κ)-type probability density (κ = 2 is the Gaussian hypothesis). In certain cases, they observed noticeable differences between Gaussian and non-Gaussian modeling, but overall, the advantages of applying ad hoc non-Gaussian hypotheses remain questionable. Carpino et al. (2002) discussed a method for assessing statistically the performance of a number of observatories that have produced large amounts of asteroid astrometric data, the goal being to use statistical characterization to improve asteroid positional determination. In their analysis, which encompassed all the numbered asteroids, they computed best-fitting orbits and corresponding O-C astrometric residuals for all the observations used. They derived some empirical functions to model the bias from each contributing observatory — sometimes almost an arcsec — as well as some non-Gaussian characteristics. These functions could be used to estimate correlation coefficients among datasets. Bykov (1996) has pursued a similar goal, using Laplacean orbit computation, as did Hernius et al. (1997), who study large numbers of multiple-apparition asteroids. In a recent study, completing the work in Muinonen and Bowell (1993), Muinonen et al. (2001) found that nonlinear statistical techniques, in contrast to the linear approximations in section 2.2, are not automatically invariant in orbital element transformations. Interestingly, without proper regularization via a Bayesian a priori probability density, even using the same orbital elements at different epochs can affect the probabilistic interpretation, rendering the interpretation dependent on the epoch chosen. Muinonen et al. then put forward a simple a priori probability density — the square root of the determinant for the Fisher information matrix computed for given orbital elements — that guarantees invariance analogous to that of the linear approximation. There are several practical consequences of allowing for the invariance. First, the probabilistic interpretation no longer depends on which orbital elements — that is, Keplerian, equinoctial, or Cartesian — are being used. Second, the interpretation is specifically independent of the epoch of the orbital elements. Third, the definition of the linear approximation by Muinonen and Bowell (1993) is revised and now includes the determinant part of the a posteriori probability density. Finally, the invariance principle is highly pertinent to all inverse problems in which the parameter uncertainties are expected to be substantial. 2.2.

Linear Approximation

In the linear approximation (e.g., Muinonen and Bowell, 1993; Muinonen et al., 2001), the relationship between orbital elements and sky-plane positions at the observation dates is linearized, and the regularizing a priori probability density is assumed constant. The resulting orbital-element probability density is Gaussian, and can be sampled using standard Monte Carlo techniques. Typically, the lin-

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ear approximation works remarkably well for multiapparition MBAs, and even for single-apparition NEAs having observational arcs spanning a few months, whereas it can fail for multiapparition transneptunian objects (TNOs). Using the rigorous statistical ranging technique (section 2.4), the validity of the linear approximation can be studied thoroughly for different orbital elements. For shortarc orbits, there is a hint that Cartesian elements, in comparison to Keplerian and equinoctial elements, offer the best linear approximation. Intuitively, this can be understood in a straightforward way. First, because of the Gaussian hypothesis for the observational error, the two transverse positional elements at a given observation date tend to be Gaussian. Second, because the transverse velocity elements are related to the difference of two sky-plane positions having Gaussian errors, the two transverse velocity elements are also almost Gaussian. The line-of-sight position and velocity elements can, however, be remarkably non-Gaussian. Using the linear approximation, Muinonen et al. (1994) carried out orbital uncertainty analysis for the more than 10,000 single-apparition asteroids known at the time. Though of limited application, that analysis captured the dramatic increase of orbital uncertainties for short-arc orbits and led naturally to a study, via eigenvalues, of the covariance matrix by Muinonen (1996) and Muinonen et al. (1997). The latter found that there exists a bound, as a function of observational arc and number of observations, outside which the linear approximation can be applied, and inside which nonlinearity dominates the inversion. This notion is in agreement with the experience by B. Marsden (personal communications, early 1990s) that covariance matrices of the linear approximation can be misleading for short-arc orbits. Chodas and Yeomans (1996, 1999), relying on the linear approximation, described orbit computation techniques that can incorporate both optical and radar astrometry, including a Monte Carlo algorithm that uses a square-root information filter. Bielicki and Sitarski (1991) and Sitarski (1998) developed random orbit-selection methods, paying particular attention to multivariate Gaussian statistics, outlier rejection, and proper weighting of observations. Their approach is based on the linear approximation, where random linear deviations are added to least-squares standard orbital elements. Milani (1999) discussed the linear approximation using six-dimensional confidence boundaries. The ultimate simplification of the linear approximation is a one-dimensional line of variations along the principal eigenvector of the covariance matrix (Muinonen, 1996), a precursor to the more general one-dimensional curves of variation used in semilinear approximations (section 2.3). However, as pointed out by Muinonen et al. (1997) and Milani (1999), there are substantial risks in applying the line-of-variations method. 2.3. Semilinear Approximations The problem raised by nonlinear effects has been stated several times, starting from Muinonen and Bowell (1993), who defined the rigorous, non-Gaussian a posteriori prob-

ability density and developed a Monte Carlo method for it. Their work constitutes the backbone of many of the theoretical developments over the past decade. A cascade of one-dimensional semilinear approximations follows from the notion that the complete differential correction procedure for six orbital elements is replaced, after fixing a single orbital element [mapping parameter; for example, the semimajor axis or perihelion distance; cf. Bowell et al. (1993) and Muinonen et al. (1997)], by an incomplete procedure for five orbital elements. Varying the mapping parameter and repeating the algorithm allows one to obtain a one-dimensional, nonlinear curve of variation, along the ridge of the a posteriori probability density in the phase space of the orbital elements. While the incomplete differential correction procedure has been used by several researchers over the decades, only Milani (1999), in what he terms the multiple-solution technique, has systematically explored its practical implementation. In Milani’s technique, the mapping parameter is the step along the principal eigenvector of the covariance matrix computed in the linear approximation. However, the covariance matrix and the eigenvector are recomputed after each step, allowing efficient tracking of the probability-density ridge. Because of the nonlinearity of the inverse problem for short-arc asteroids, Milani’s multiple-solution technique has turned out to be particularly successful in many of the applications described in section 3 and in Milani et al. (2002). The nonlinear technique is very attractive and efficient because of its simplicity. 2.4. Statistical Ranging The linear and semilinear approximations, relying on differential correction procedures, run into severe convergence problems for short observational arcs and/or small numbers of observations. For such circumstances, the true six-dimensional orbital element probability density cannot generally be collapsed into a single dimension. Virtanen et al. (2001) devised a completely general approach to orbit estimation that is particularly applicable to short orbital arcs, where orbit computation is almost always nonlinear (see also Muinonen, 1999). They term it “statistical ranging,” and use Monte Carlo selection of orbits in orbital-element phase space. From two observations, angular deviations in right ascension and declination are chosen and topocentric ranges are assumed by randomly sampling the sky plane and line-of-sight positions. Then large numbers of sample orbits are computed from the pairs of heliocentric rectangular coordinates to map the a posteriori probability density of the orbital elements. In a follow-up paper, Muinonen et al. (2001) studied the invariance in transformations between different element sets and developed a Spearman rank correlation measure for the validity of the linear approximation. They were also able to accelerate the computational efficiency of the Monte Carlo method by up to two orders of magnitude. Computations for most short-arc objects — including, for example, all single-apparition TNOs — are carried out in seconds to minutes. For longer

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Bowell et al.: Asteroid Orbit Computation

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a (AU) Fig. 1. Evolution of nonlinearity in orbit computation for 1998 OX4. We mimicked the discovery circumstances and repeated orbital ranging for the asteroid as additional observations were obtained. In the lower panel, we illustrate the extent of the marginal probability densities for semimajor axis a and eccentricity e for the following six cases: (1) observational arc Tobs = 0.08 d (number of observations = 3; broad cloud of points); (2) 1.1 d (12) (diamonds); (3) 5.0 d (14); (4) 7.1 d (16); (5) 8.1 d (19); and (6) 9.1 d (21). Because the distributions for the four cases with the longest observational arcs (from 5.0 to 9.1 d) overlap, we have vertically offset them by 0.02 in e. The least-squares solution computed using the longest observational arc is marked with an asterisk and vertical line. The upper panel shows, for the six cases, the marginal probability density pP for the semimajor axis.

arclengths (say, several weeks for NEOs and MBOs), mapping the six-dimensional orbital-element phase-space region is tedious. In such cases, the ranging technique is more practical for detailed studies, such as exploring the validity of the linear approximation, than routine orbit computation. The statistical ranging technique can be applied to asteroids having only two observations, providing estimates on six orbital elements based on only four data points. The intriguing mathematical questions involved will be the subject of future study. Figures 1 and 2 show some of the results that can be obtained from statistical ranging. In Fig. 1, the evolution of possible orbits of the Apollo-type asteroid 1998 OX4 can be followed as the observational arc gradually lengthens.

0.15 1.20

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Fig. 2. Bimodal orbital solutions for 1999 XJ141 (format is similar to that of Fig. 1). The solutions occur in the regions around (a ~ 1.28 AU, e ~ 0.18), which contains the current orbit (cross) following identification with 1993 OM7 (see MPEC 2000-C34), and another region around a ~ 2.3 AU. Dashed lines separating Earth-crosser and Amor orbits (upper) and Amor and Mars-crosser orbits (lower) indicate that both solution regions imply an Amortype asteroid. The upper panel shows the marginal probability density pP for the semimajor axis.

Although the initial 2-h discovery arc leads to possible orbits extending to e ~ 0 and a ~ 180 AU, the probability that the asteroid is an Apollo type is as high as 74%. After being recovered one night later (second case, diamond symbols), Apollo classification is confirmed at very nearly 100% probability. One discordant observation on the second night (see also Fig. 7) is easily recognizable because of the existence of eight additional observations on the same night; it was thus omitted from the orbit computation. In the upper plot, the narrowing of the marginal probability density in a has become very pronounced after only 5 d of observation. The asteroid 1999 XJ141 was discovered near 90° solar elongation on December 13, 1999, followed till December 24, and then lost. It was only the realization that the observations of 1999 XJ141 led to a double solution — not uncommon for asteroids discovered near quadrature — that allowed the asteroid to be linked to independently discovered 2000 BN19 and then 1993 OM7, the last observed on only two nights. The statistical ranging technique very nicely

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resolves the orbital-elements ambiguity, as shown in Fig. 2. Applying statistical ranging to the observations made between December 13 and 24, 1999, and imposing a 1.1arcsec rms residual shows a bimodal distribution of possible solutions (lower panel). However, as shown in the upper panel, the probability mass is overwhelmingly concentrated in the region at smaller a, leaving vanishingly small probability for the larger-a, e solutions. A continuum of orbital solutions between the two regions can be generated only by allowing unexpectedly large rms residuals of several arcseconds. Virtanen et al. (2002) automated the statistical ranging technique for multiple asteroids, and applied statistical ranging to TNOs. Whereas TNO orbital element probability densities tend to be very complicated (in contrast to those of NEAs), the projected distributions of sky-plane uncertainties are remarkably unambiguous. Virtanen et al. found that dynamical classification of TNOs is reasonably secure for orbital arcs exceeding 1.5 yr and were able to use the known TNO population (as of February 2001) to make a statistical assessment of orbital types (see section 3.4). 2.5. Other Advances Approaches based on the variation of topocentric range and the angle to the line of sight have been developed by McNaught (1999) and Tholen and Whiteley (2002). The former technique is not readily amenable to uncertainty estimation, whereas the latter does sample the orbital-element probability density. In essence, both are methods that, instead of using two Cartesian positions, as does the statistical ranging technique, use Cartesian position and velocity. However, the statistical ranging technique has an advantage over the position-velocity techniques: For ranging, the two angular positions involved are well fitted by every trial solution without iteration at that stage; for position-velocity techniques, there is an a priori requirement that the two positions be timed so as to allow for an accurate estimation of motion. Although at opposition the apparent motion is usually a good indicator of an object’s topocentric distance (Bowell et al., 1990), the situation is quite different at smaller solar elongation, where confusion among the various asteroid classes is much greater and multiple orbital solutions may occur. Parallax is a very useful tool for short-arc orbital determination. For surveys of the opposition region it can be useful for improving the first guess of a Väisälä solution when observations from two nights are available. The improvement can be significant if, from at least one night, data are distributed over an arc of several hours. In a study on searching for NEOs at small solar elongation, Hills and Leonard (1995) suggested discriminating NEA candidates not only using the traditional method based on their large daily motion, but also by taking advantage of the parallax resulting from observations at different locations. For this purpose they considered a baseline of 1 R (Earth radius). Such a baseline could conveniently be implemented by ob-

serving at appropriate intervals using, for example, the Hubble Space Telescope (e.g., Evans et al., 1998), but it is not easily attainable by groundbased observers. Boattini and Carusi (1997) suggested that, even using a baseline of 0.1– 0.2 R between stations, good range estimates could be made up to distances of 0.5 AU if the observations are performed near simultaneously. Marsden (1991) modified the Gauss-Encke-Merton (GEM) method of three-observation orbit computation to deal with mathematically ill-posed cases (usually for short arcs). He extended the procedure to pertain to cases having two observations only, which has a useful application to newly discovered putative Earth-crossing asteroids, where Väisälä’s method yields only aphelic orbits. More recently, Marsden (1999) developed the concept of lateral orbits, in which the true anomaly is ±90° and the object is on the latus rectum. According to G. V. Williams (personal communication), these methods are used at the MPC to compute the ephemeris uncertainty for newly discovered NEAs, as published at http://cfa-www.harvard.edu/iau/NEO/TheNEOPage.html. L. K. Kristensen (in preparation, 2002) looked at the accuracy of follow-up ephemerides based on short-arc orbits and found that the geocentric distance and its time derivative are the essential parameters that determine the accuracy of predicted positions. Yeomans et al. (1987, 1992) assessed the improvement of NEA orbits when radar range and Doppler data are combined with optical astrometric observations. They found that long-arc orbits were improved only modestly, whereas shortarc orbits could be improved by several orders of magnitude, often to the extent that additional near-term optical observations would afford little further improvement. The relative weights of the optical and radar observations follow the Bayesian theory by Muinonen and Bowell (1993): Radar time-delay and Doppler astrometry are incorporated by defining the optical a posteriori probability density to be the radar a priori probability density. In practice, the inverse problem then culminates in the study of the joint χ2 of the optical and radar data. Regularization, such as that described in section 2.1, is not needed because the linear approximation is usually valid when radar data are incorporated. Both OrbFit and NEODyS have recently been upgraded to account for radar astrometry, too. Bernstein and Khushalani (2000) devised a linearized orbit-fitting procedure in which accelerations are treated as perturbations to the inertial motions of TNOs. The method produces ephemerides and uncertainty ellipses, even for short-arc orbits. They applied it to devising a strategy for computing accurate TNO orbits from a minimum number of observations, a matter of importance when one considers the expense of using large telescopes. 3. PREDICTION Here our fundamental question is, “What is an asteroid’s positional uncertainty in space or on the sky plane?” Answering this question allows observers to search for asteroids whose positions are inexactly known, and to know

Bowell et al.: Asteroid Orbit Computation

when it is useful or necessary (from the standpoint of orbit improvement) to secure additional observations. It allows orbit computers to identify images of asteroids in archival photographic or CCD media, forms the basis of Earth-impact probability studies, and guides spaceflight planners in pointing their instruments and in making course corrections. This type of analysis belongs to the prediction (or projection) problem of applying the orbital-element probability density derived in section 2. 3.1. Ephemeris Uncertainty, Precovery, and Recovery In the linear approach to finding a lost asteroid, an asteroid is sought on a short segment of a curve, known as the line of variations, which is defined as the projection of the asteroid’s orbit on the sky plane. It is generally computed by varying the asteroid’s mean anomaly M. The extent of the line of variations that needs to be searched is readily computed using Muinonen and Bowell’s (1993) linear approximation and forms the basis of a number of URLs developed by Bowell and colleagues at Lowell Observatory (see http://asteroid.lowell.edu). For example, the asteroid orbit file astorb.dat contains current and future ephemeris uncertainty information for more than 127,000 asteroids, and the Web utility obs may be used to build a plot indicating when an asteroid is observable, based in part on sky location, brightness, and maximum acceptable ephemeris uncertainty. Although a good approximation when the sky-plane uncertainty is small, the linear approach is usually inappropriate for short observational arcs or for completely lost asteroids. Milani’s (1999) semilinear theory identified sources of nonlinearity. The observation function, or the mapping of given orbital elements onto the sky plane, is also nonlinear. Such effects cannot be neglected if one or more close approaches to a planet occur in the interval between the two. Because the observation function is nonlinear, the image of the confidence ellipsoid will appear as an ellipse on the sky plane only if the ellipse is quite small. Milani (1999) and Chesley and Milani (1999) considered semilinear methods of ephemeris prediction. They explored the validity of the linear approximation and introduced methods for propagating the orbital uncertainty using semilinear approximations. In uncertain cases, they recommended that the linear approximation be used only as a reference. The computation of the observation function is relatively straightforward as the solution of differential equations is not involved. Thus it is possible to overcome the difficulty of nonlinearity on the sky plane by computing it at many points. A burden of this approach is that delineating the sixdimensional confidence region uniformly requires a large number of samples, many of which may result in predictions close to each other when mapped onto the two-dimensional space of the observations. Milani (1999) developed the concept of the semilinear confidence boundary, in which an algorithm is used to identify points, in the confidence

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ellipsoid, that map close to the boundary of the confidence region. This approach has been used for the recovery and precovery of asteroids. Following the pioneering efforts of the Anglo-Australian Near-Earth Asteroid Survey (AANEAS) program from 1990 to 1996 (Steel et al., 1997), newly discovered NEOs are now routinely sought in photographic and CCD archives. Indeed, several groups of “precoverers” are currently active. For example, the DLR-Archenhold Near Earth Objects Precovery Survey (DANEOPS) group in Germany (http://earn.dlr.de/ daneops) and the team of E. Helin and K. Lawrence at the Jet Propulsion Laboratory have made significant contributions, focusing on NEOs. DANEOPS was able to precover 1999 AN10, the first asteroid known to have a nonzero collision probability before its precovery (Milani et al., 1999), and Helin and Lawrence precovered 1997 XF11. Boattini et al. (2001a) started an NEA precovery program called the Arcetri NEO Precovery Program (ANEOPP) in 1999. Although most of their identifications have been made using good initial orbits, ANEOPP has successfully made use of the mathematical tools developed for OrbFit to guide precovery attempts of NEOs having poor orbits. The multiplesolution approach has proved to be a most effective search method. Other groups such as that at Lowell Observatory and amateur astronomers such as A. Lowe have conducted extensive precovery programs on asteroids in general. Precovery activity has spawned other scientifically interesting results in the past decade. For example, the identification of the Amor-type asteroid (4015) 1979 VA with periodic Comet Wilson-Harrington (1949 III) on a plate from 1949 (Bowell and Marsden, 1992) showed clear evidence that the distinction between asteroids and comets is not straightforward. Recovering completely lost asteroids (i.e., asteroids having similar probability of being located at any orbital longitude) requires different techniques. Using a semilinear approach, Bowell et al. (1993) attempted to recover thenlost (719) Albert by eliminating, through searches of archival photographic media, parts of the perihelion distance space where the asteroid was not seen (“negative observations”). (They would have succeeded had sufficient resources been available for the search.) Several groups are searching for 1937 UB (Hermes), observed (poorly) for less than 5 d. For example, ANEOPP is using the multiple-solution technique to examine regions of Palomar and U.K. Schmidt plates. Currently, ~10–15% of the orbital element phase space has been examined. L. D. Schmadel and J. Schubart, reporting on their work on Hermes at http://www.rzuser. uni-heidelberg.de/~s24/hermes.htm, indicate that they used a method similar to that devised by Bowell et al. (1993). For asteroids having very poor orbit solutions, though not completely lost, the semilinear confidence boundary is very effective for planning a search, in particular because the recovery region is usually very elongated; even when its length is tens of degrees, the width may only be a few arcseconds. One can also compute a sequence of solutions along the major axis of the confidence region. This multiplesolution approach has recently been used to recover some

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make use of the orbital-elements distribution of known TNOs to limit the search region even more, though such an approach could bias against successfully following up TNOs having unusual orbits.

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Fig. 3. Precovery of 1998 KM3. Forty-one continuous curves, each corresponding to one of 41 orbits generated using the multiple-solution method out to ±5σ (actually, 201 orbits were used to sample the confidence region to be searched, but for clarity only one in five of them is shown). The angular distance from the nominal solution y = 0 is plotted against time. The precovery location of 1998 KM3 is indicated by the dot.

particularly difficult targets. For example, Fig. 3 shows how recovery was achieved of 1998 KM3, an Apollo asteroid whose sky-plane uncertainty was several tens of degrees at the time of the observations (see Boattini et al., 2001b). The plot shows the angular distance from the nominal solution given by y = 0, which is orbit #21, and the confidence region is populated by 20 orbits on each side of the nominal solution (actually, 201 orbits were considered in the original computation). Note that the asymmetric shape of the confidence region is an indicator of nonlinearity. 1998 KM3 was recovered on December 30, 2000, the asteroid being located between orbits #8 and #9. The challenges of recovering TNOs center on accurate ephemeris prediction. Although projection of orbital uncertainties onto the sky plane results in almost linear distributions, the need to use large telescopes for follow-up observations mandates a firm understanding of possible TNO orbits. An efficient method of ephemeris uncertainty ellipse computation was presented in Bernstein and Khushalani (2000). However, to map the ephemeris uncertainties for very shortarc TNOs (a few days to weeks of observations) over intervals of a year or more, rigorous methods, such as statistical ranging, are appropriate. Even for longer arcs (two apparitions, say), TNO orbits may be very imperfectly known. Figure 4 shows the results of applying statistical ranging to 1998 WU31, a TNO discovered in the fall of 1998 and recovered in the fall of 1999. The sky-plane uncertainty distributions are based on computations that assume 1.0-arcsec observational noise (probably an overestimate by a factor of 3–4). The correlations between ephemeris and orbitalelement uncertainties — especially between right ascension and semimajor axis or eccentricity — provide a good way of reducing the sky-plane search region. One could also

A great deal of effort has recently been expended to establish identifications among asteroid observations, that is, trying to link observations of two independently discovered asteroids and hence showing that they are the same object. Orbit computation plays a key part in the asteroid identification process, as Milani et al. (1996, 2000a, 2001), Sansaturio et al. (1996, 1999) show in developing identification algorithms (see also http://copernico.dm.unipi.it/ identifications). Such algorithms can be classified into two types, depending on the number and time distribution of the two available sets of astrometric observations. The first type is based on a comparison of the orbital elements, as computed by a least-squares fit to the observations making up each arc, and provides so-called orbit identification methods. These methods use a cascade of filters based on identification metrics and take into account the difference in the orbits weighted by the uncertainty of the solution, as represented by the covariance matrix. Orbit pairs that pass all the filters are subjected to accurate computation by differential correction to compute an orbit that fits all or most of the observations. In the second type of algorithm, an orbit computed for one of the arcs is used to assess the observations that make up the other arc. These constitute so-called observation attribution methods. The key to such methods lies in finding a suitable single observation, termed an attributable, that fits the observations of the second arc and at the same time contains information on both the asteroid’s position on the sky and its apparent motion. As in the orbit-identification algorithm, the attribution method uses a cascade of filters that allow comparison in the space of the observations and in that of the apparent motion, and point to a selection of the pairs to be subjected to least-squares fits to all the observations. A. Doppler and A. Gnädig (Berlin), starting in 1997, developed an automated search technique. After generating ephemerides, candidate matches are identified in surrounding regions whose sizes are dependent on the properties of the initial orbit. The candidate matches are then subjected to differential orbit correction, and, for apparently successful fits, additional identifications are sought. Doppler and Gnädig have to date made about 10,000 identifications. One of them involved 2001 KX76 (now numbered as 28976), which turned out to be the largest known small body after Pluto. Applying statistical ranging to TNOs, Virtanen et al. (2002) showed that the identification problem can sometimes coincide with the inverse problem. They applied statistical ranging routinely to most of the multiapparition TNOs; that is, they succeed in deriving sample orbital elements linking observations at different apparitions without any preparatory work. The first and last observations used

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Bowell et al.: Asteroid Orbit Computation

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in the computation are sometimes separated by several years. 3.3. Collision Probability Because this topic is the subject of another chapter in this book (see Milani et al., 2002), we comment only on some aspects of orbit computation. The collision probability was mathematically defined by Muinonen and Bowell (1993) in their Bayesian probabilistic work on the inverse problem. A first indicator of Earth or other planetary close approach is afforded by the value of the minimum orbital intersection distance (MOID) (Bowell and Muinonen, 1993; Milani et al., 2000b), the closest distance between an asteroid’s orbit and that of a planet. An asteroid’s Earth MOID can usually be determined surprisingly accurately, even for

short observational arcs. In the absence of close planetary approaches or strong mean-motion resonance, MOIDs do not change by more than 0.02 or 0.03 AU per century. Therefore, if it can be established that an asteroid’s planetary MOID is larger than, say, 0.05 AU one can be sure that no planetary collision is imminent. If it is less, further calculations are necessary. As a next step, Bonanno (2000) derived an analytical formulation for the MOID, together with its uncertainty, which allows one to identify cases of asteroids that might have nonnegligible Earth impact probability even though the nominal values of their Earth MOIDs are not small. To assess the possibility of an Earth impact in 2028 by asteroid 1997 XF11, Muinonen (1999) developed the concept of the maximum likelihood collision orbit: Given the time window for the collisional study and the astrometric observations, it is possible to derive the collision orbit, or

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“virtual impactor” in the terminology of Milani et al. (2000c), that is statistically closest to the maximum likelihood orbit. A safe upper bound for the collision probability follows by assigning all the probability outside the confidence boundary of the maximum likelihood collision orbit to the collision. For 1997 XF11, the rms values of the maximum likelihood collision orbit were 3.81 arcsec in right ascension and 4.06 arcsec in declination, rendering the collision probability negligible (Muinonen, 1998). The collision orbits derived in that study in spring 1998 were the first of their kind. Soon thereafter, collision orbits became a subject of routine computation for essentially all research groups assessing collision probability.

Whereas maximum likelihood collision orbits can be very useful for assessing primary close approaches, they do not yield the actual collision probability. Based on the technique of statistical ranging, and motivated by the approximate one-dimensional semilinear techniques of Milani et al. (2000c), Muinonen et al. (2001) developed a general sixdimensional technique to compute the collision probability for short-arc asteroids. In that technique, every sample orbital element set from ranging is accompanied by a onedimensional line of acceptable orbital solutions. Along these lines, collision segments are derived, allowing estimation of the general collision probability (as well as the phase space of collision orbits).

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Fig. 5. Classification using statistical ranging of 1990 RM18, observed over a 3-d arc. Five thousand orbits were computed assuming 1.0-arcsec observational noise. The upper panels show the extent of the a posteriori probability in the (a, e) and (a, i) planes. In the lower panels, the a, e, i probability density of known multiapparition asteroids has been incorporated as a priori information. Accurate orbital elements of 1990 RM18 are indicated by a triangle.

Bowell et al.: Asteroid Orbit Computation

3.4. Classification One of the motivations for developing methods of initial orbit computation is the classification of asteroid orbital type. The question is particularly relevant for Earth-approaching asteroids, which need to be recognized from the very moment of discovery to ensure follow-up. Because of the small numbers of observations, we call for special caution in the discussion of probabilities. In his study of detection probabilities for NEAs, Jedicke (1996) derived probabilities that a given asteroid is an NEA based solely on its rate of motion. His approach, entirely analytical, requires modeling the orbital element and absolute magnitude distribution of the asteroid population. The era of statistical orbit inversion has given us the means of addressing the classification problem probabilistically. The orbital element probability density can be mapped rigorously using the method of statistical ranging (Virtanen et al., 2001), which leads in a straightforward way to an assessment of probabilities for different orbital types. Figure 5 illustrates the case of 1990 RM18 (now numbered as 10343). Using no a priori information, 1990 RM18 had a 28% probability of being an Apollo asteroid and also had nonzero (0.2%) probability of being an Aten. Incorporating the probability density of known multiapparition asteroids as a priori information secures its classification as an MBO with 99.9% certainty, with peaks near the widely separated Flora and Themis family regions (the asteroid’s orbital ele-

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Fig. 6. Hierarchical Observing Protocol (HOP), requiring very few observations, shows how 2001 LS8 could be numbered in 2007. Vertical bars show windows of observability from Lowell Observatory, assuming a V-band limiting magnitude of 19.0, a minimum solar elongation of 60°, a minimum galactic latitude |b| = 20°, and a southern declination limit of –30°. The continuous curve shows the time evolution of the ephemeris uncertainty, assuming linear error propagation as described by Muinonen and Bowell (1993), vertical drops being when additional ±1-arcsecaccuracy observations are called for. Dashed curves indicate subsequent ephemeris uncertainty evolution in the absence of followup observations.

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ments are now known to be consistent with Themis family membership). Dynamical classification of TNOs has been attempted using statistical ranging by Virtanen et al. (2002), as described in section 2.4. Although lacking a clear definition of the various dynamical classes, their results show that classification is very uncertain for observational arcs shorter than six months. Their population analysis does not support the existence of near-circular orbits beyond 50 AU, although the existence of objects having low to moderate eccentricities is not ruled out. They call for more detailed dynamical studies to address the question of the edge of the transneptunian region. 3.5.

Optimizing Observational Strategy

Bowell et al. (1997) devised a method they term Hierarchical Observing Protocol (HOP) to choose optimum times to observe asteroids so their orbits would be improved as much as possible. The method combines some of the results from Muinonen and Bowell (1993) and Muinonen et al. (1994). Of course, the result of a given optimization strategy depends on the choice of metric. HOP is predicated on minimizing the so-called leak metric, a metric based on longitude, eccentricity, and angular momentum (k) (Muinonen and Bowell, 1993). HOP results from four semiempirical rules are: (1) To be numberable, an asteroid’s ephemeris uncertainty should be