Topocentric orbit determination: Algorithms for the next generation surveys

Abstract

The process of calculating a good orbit from astrometric observations of the same object involves three main steps: preliminary orbit determination, least squares orbit fitting, and quality control assessing the orbit's uncertainty and reliability. For the next generation sky surveys, with much larger number density of observations, new algorithms, or at least substantial revisions of the classical ones, are needed. The classical theory of preliminary orbit algorithms was incomplete in that the consequences of the topocentric correction had not been fully studied. We show that it is possible to rigorously account for topocentric observations and that this correction may increase the number of alternate preliminary orbits without impairing the overall performance. We have developed modified least squares algorithms including the capability of fitting the orbit to a reduced number of parameters. The restricted fitting techniques can be used to improve the reliability of the orbit computing procedure when the observed arcs have small curvature.False identification (where observations of different objects are incorrectly linked together) can be discarded with a quality control on the residuals and a 'normalization' procedure removing duplications and contradictions. We have tested our algorithms on two simulations based on the expected performance of Pan-STARRS -one of the next generation all-sky surveys. The results confirm that large sets of discoveries can be handled very efficiently resulting in good quality orbits. In these test we lost only 0.6 to 1.3% of the possible objects with a false identification rate in the range 0.02 to 0.06%.

LSST Science Book, Version 2.0

“…LSST will discover on the order of 10,000 comets, with 50 observations or more of each of them (Solontoi et al 2009). This will dwarf the current roster, providing answers to many questions regarding the physical properties of today's cometary population.…”

Multiple solutions in preliminary orbit determination from three observations

“…Examples of occurrence of multiple solutions with real data of asteroids are discussed in [9]. It is however difficult to find a real case with three solutions such that the one closest to the Earth is the best starting point for differential corrections: in fact the reliability of the heliocentric two-body approximation is questionable for values of the topocentric distance ρ < 0.01AU .…”

“…Following [9] we show how to obtain the dynamical equation in Laplace's and Gauss' methods. For a review of these methods see also [2].…”

“…observations made from a point on the surface of the rotating Earth, is quite natural in Gauss' method (see Appendix 7.2), and also Laplace's method can be modified to consider this effect (see [9]). In both cases from the two-body dynamics we obtain an equation like (2) (see for example (47)), that has the same algebraic structure as (32) but depends on the additional parameter γ, and reduces to equation (32) for γ = 1.…”

Corrección topocéntrica de parámetros orbitales obtenidos mediante las integrales de Kepler para asteroides MBA y NEO

“…Sin embargo, existen otras familias de asteroides, como los NEO o los TNO (transNeptunian objects), para los cuales esta suposición produce errores importantes. Esto se debe a que los primeros son cuerpos tan cercanos a la Tierra que la posición relativa del observador sobre la superficie y la rotación del planeta se vuelven relevantes, y los segundos se mueven muy lentamente sobre la esfera celeste, lo que implica que las observaciones generalmente no son suficientes para obtener la convergencia de los métodos, (Milani, et al, 2008). La segunda razón consiste en que los métodos clásicos poseen una convergencia muy débil si la información utilizada representa una sección muy pequeña de la órbita (Milani & Gronchi, 2010) y, en general, esto es lo que ocurre normalmente: se detecta el movimiento del cuerpo usando una serie de imágenes de la región (usualmente entre tres y seis) en el lapso de una hora de observación y se espera poder utilizar toda la información proveniente de tales imágenes para obtener la órbita, lo cual, en principio, no puede hacerse directamente con los métodos clásicos.…”