Celest Mech Dyn Astr (2010) 106:157–182 DOI 10.1007/s10569-009-9245-y ORIGINAL ARTICLE Circular and zero-inclination solutions for optical observations of Earth-orbiting objects K. Fujimoto · J. M. Maruskin · D. J. Scheeres Received: 24 July 2009 / Revised: 16 October 2009 / Accepted: 17 November 2009 / Published online: 28 January 2010 © Springer Science+Business Media B.V. 2010 Abstract Situational awareness of Earth-orbiting particles is important for human extra- terrestrial activities. Given an optical observation, an admissible region can be defined over the topocentric range/range-rate space, with each point representing a possible orbit for the object. However, based on our understanding of Earth orbiting objects, we expect that certain orbits in that distribution, such as circular or zero-inclination orbits, would be more likely than others. In this research, we present an analytical approach for describing the existence of such special orbits for a given observation pass, and investigate topological features of the range/range-rate space by means of singularities in orbital elements. Keywords Situational awareness · Orbit determination · Space debris · Admissible region 1 Introduction Situational awareness of Earth-orbiting particles such as active satellites and space debris is highly important for future human activities in space. Presently, over 300,000 particles have been estimated to exist, and over 80,000 observations are made per day (Rossi 2005). Obser- vations are made either by radar or optical sensors. For optical observations, which are usually made for objects in medium Earth orbit (MEO) and geostationary orbit (GEO), only the angles and angular rates of the track can be determined. That is, the range and range-rate remain largely unknown. Rossi conducted a detailed study on the dynamics and evolution of objects in MEO (Rossi 2008). Milani et al. have proposed a method for heliocentric orbits where each track is expressed in a four dimensional quantity called the attributable vector, and by placing a few physical constraints they restrict the range and range-rate to a region called the admis- sible region (Milani et al. 2004; Milani and Kneževi´ c 2005). Tommei et al. expanded this K. Fujimoto (B ) · D. J. Scheeres Department of Aerospace Engineering Sciences, The University of Colorado-Boulder, Boulder, CO, USA e-mail: kohei.fujimoto@colorado.edu J. M. Maruskin Department of Mathematics, The San José State University, San José, CA, USA 123