Asteroids IV

1. INTRODUCTION Determining asteroid orbits is a challenging problem that the scientific community has faced since 1801, when Piazzi discovered (1) Ceres, the first main-belt object. Shortly after discovery Ceres was heading for conjunction, thus making it impossible to keep collecting observations. To prevent Ceres from being lost, Gauss computed its orbit using the observations collected by Piazzi and successfully predicted where Ceres would be observable after emerging from the Sun. Interestingly, the orbit determination of Ceres represents the first remarkable application of the famous method of least squares, which since then has been one of the most used mathematical tools with applications in disciplines such as statistics, physics, biology, geodesy, economics, and indeed every physical science. One of the reasons why, after more than two centuries, orbit determination continues to be a fascinating branch of celestial mechanics is that it strikes an impressive fusion between a scientifically sound theory and several relevant applications. Celestial mechanics rigorously describes the motion of celestial bodies and the large number of observations of space bodies allows one to extensively test the validity of the theory. In particular, for asteroid orbit determination we have a rigorous mathematical theory to compute trajectories and more than 650,000 asteroids (as of September 2014) whose observations continuously serve as validation of the theoretical computations. Just to give an idea of the level of accuracy that can be reached, there are asteroids for which the available observational data reveal accelerations with a precision of a few fm s–2 (10–15 m s–2). Asteroid orbit determination has several important applications . By computing an orbit along with its uncertainty, we can inform astronomers on when and where they should observe the night sky so that they can collect additional observations. If a mission to an asteroid is planned, estimating trajectories is important to select a viable target, and reliable ephemerides are required to successfully navigate the spacecraft to the asteroid. When two asteroids get close enough we can measure the gravitational interaction between them and obtain an estimate of an asteroid’s mass. Perhaps the most interesting application is planetary defense. When an asteroid’s orbit is computed we can figure out where the object will be in the future and whether or not there are upcoming planetary encounters. In particular, it is possible to compute the probability of an Earth impact and contribute 815 Farnocchia D., Chesley S. R., Milani A., Gronchi G. F., and Chodas P. W. (2015) Orbits, long-term predictions, and impact monitoring. In Asteroids IV (P. Michel et al., eds.), pp. 815–834. Univ. of Arizona, Tucson, DOI: 10.2458/azu_uapress_9780816532131-ch041. Orbits, Long-Term Predictions, and Impact Monitoring Davide Farnocchia and Steven R. Chesley Jet Propulsion Laboratory/California Institute of Technology Andrea Milani and Giovanni F. Gronchi University of Pisa Paul W. Chodas Jet Propulsion Laboratory/California Institute of Technology In this chapter we review the methods currently in use to compute asteroid orbits and make ephemeris predictions. Despite a well-consolidated theory, the increasing number and ever higher quality of observational data, together with the goal of pushing forward the horizon for ephemeris predictions, pose new challenges in estimating asteroid trajectories. We discuss how to develop a realistic statistical error model for astrometric observations by removing star catalog systematic errors and suitably weighting astrometric data. Moreover, since the dynamical model has to be accurate enough to fit the observational data and reliably predict an asteroid’s position in the future, we analyze the relevance of the different components of the force model. In particular, we show that nongravitational forces can be relevant at a high level of precision. We also address the problem of estimating asteroid orbits when only a short arc of observations is available. Solving this problem has relevant implications for modern surveys, where the amount of observational data requires innovative methods to compute an orbit catalog avoiding an excessive computational load. Finally, we discuss the Earth impact hazard assessment. We describe the standard methods that have been in use for the last 15 years as well as newer techniques that allow long-term impact monitoring, including going beyond scattering planetary encounters and accounting for nongravitational perturbations. 816   Asteroids IV to the study of mitigation measures, such as deflection, in case they are deemed necessary. 2. BASIC THEORY 2.1. Orbits The equations of motion for celestial bodies are ordinary differential equations with a smooth righthand side. An orbit is defined by the evolution of the Cartesian heliocentric...