Numerical behavior of the Keplerian Integral methods for the initial orbit determination problem

We investigate the numerical behavior of the two methods for the computation of preliminary orbits presented previously in Linkage. We will reffer to these methods as link2 and link3 respectively. These polynomial methods allow a fast computation of orbits but their sensitivity to astrometric error has not been analyzed in previous works. Understanding the behavior of these methods is crucial to deal with large databases of VSAs such as the isolated tracklet file (ITF) available at the Minor Planet Center (MPC). For do this analysis we used different indicators as the $\chi_4$, $\Delta_\star$ and the $rms$ (see PAPER2021 and references there in for details).

The data

We generated our test data sets from real data submitted by Pan-STARRS1 to the MPC over the period from 2011-01-30 through 2019-07-28 inclusive. This technique ensures that we are using consistent data from a single survey with a real observation cadence within a night and across many years. We restricted the test data to main belt objects because they dominate the statistics of any asteroid survey and extracted triplets of random tracklets corresponding to the same object for 822 asteroids. We also required that each of the tracklets must have 3 or more detections. These observations are our matched real data which include all the vagaries of an actual operational survey including the requirements that the observations be acquired at night, when the telescope is operational, when the sky is clear, scheduling issues, etc. The average time between any pair of tracklets corresponding to the same object is about 2.6 years.

We then generated synthetic data sets for the same objects using the nominal orbit for each object as reported by the MPC and the actual times of observations from the real data. We did so using a simple 2-body calculation and a full $n$-body integration with all the planets and major asteroids as implemented in OpenOrb. In both the 2-body and $n$-body cases we generated synthetic data sets in which a random error was introduced into the astrometric observations by generating a random offset from the calculated positions according to a 2d Gaussian with standard deviations ($\sigma$) of 0.1’’, 0.2’’, 0.5’’, and 1.0’’.

The synthetic data set with no introduced error allows us to determine our algorithm’s performance on perfect data while the other data sets allowed us to characterize how the algorithms’s performance degrades with increasing astrometric error typical of other and historical asteroid surveys with our eventual goal of applying link2 and link3 to the MPC’s ITF.