During the night of the discovery of an asteroid, a large number of images spaced in time, that represent an arc too short to propagate an orbit, are obtained. Initially, it is necessary to recover the body in the celestial vault to have more observations to determine its orbit. The first step in this process is to establish the admissible region, defined as the region in space where the object can be found. In this paper we present the calculation of the Admissible Regions from data from a single night observation, considering the geocentric and topocentric versions and restrictions such as belonging to the Solar System, the object does not belong to the Earth-Moon gravitational system, and the body is at a minimum distance from Earth. This procedure was applied in the calculation of the admissible regions of 2003 BH84, 3122 Florence, 3200 Phaethon, 555 Norma, 1738 Oosterhoff and 2006 SO375. The respective admissible regions were generated in their geocentric and topocentric variant, and the respective metric changes were made to visualize their geometric characteristics. It was found that the topocentric version generates a simpler geometry than the geocentric version, decreasing the re-observation area. It was identified that the logarithmic metric is appropriate for the study of regions near the Earth (NEO’s).

This paper presents a methodology for Initial Orbit Determination (IOD) based on a modification of the Laplace’s geocentric method. The orbital elements for Near-Earth asteroids (1864) Daedalus, 2003 GW, 2019 JA8, a Hungaria-type asteroid (4690) Strasbourg, and the asteroids of the Main Belt (1738) Oosterhoff, (2717) Tellervo, (1568) Aisleen and (2235) Vittore were calculated. Input data observations from the Minor Planet Center MPC database and Astronomical Observatory of the Technological University of Pereira (OAUTP; MPC code W63) were used. These observations cover observation arcs of less than 22 days. The orbital errors, in terms of shape and orientation for the estimated orbits of the asteroids, were calculated. The shape error was less than 53 × 10–3 AU, except for the asteroid 2019 JA8. On the other hand, errors in orientation were less than 0.1 rad, except for (4690) Strasbourg. Additionally, we estimated ephemerides for all bodies for up to two months. When compared with actual ephemerides, the errors found allowed us to conclude that these bodies can be recovered in a field of vision of 95’ × 72’ (OAUTP field). This shows that Laplace’s method, though simple, may still be useful in the IOD study, especially for observatories that initiate programs of minor bodies observation.

The data set collected during the night of the discovery of a minor body constitutes a too-short arc (TSA), resulting in failure of the differential correction procedure. This makes it necessary to recover the object during subsequent nights to gather more observations that will allow a preliminary orbit to be calculated. In this work, we present a recovery technique based on sampling the admissible region (AdRe) by the constrained Delaunay triangulation. We construct the AdRe in its topocentric and geocentric variants, using logarithmic and exponential metrics, for the following near-Earth-asteroids: (3122) Florence, (3200) Phaethon, 2003 GW, (1864) Daedalus, 2003 BH84 and 1977 QQ5; and the main-belt asteroids: (1738) Oosterhoff, (4690) Strasbourg, (555) Norma, 2006 SO375, 2003 GE55 and (32811) Apisaon. Using our sampling technique, we established the ephemeris region for these objects, using intervals of observation from 25 minutes up to 2 hours, with propagation times from 1 up to 47 days. All these objects were recoverable in a field of vision of 95’ × 72’, except for (3122) Florence and (3200) Phaethon, since they were observed during their closest approach to the Earth. In the case of 2006 SO375, we performed an additional test with only two observations separated by 2 minutes, achieving a recovery of up to 28 days after its discovery, which demonstrates the potential of our technique.