Modern wide-field astronomical surveys produce extensive catalogs of small-body tracklets. A fundamental challenge in astrodynamics is linking these tracklets to identify those which correspond to the same body. Tracklet linking has traditionally relied on orbit fitting methodologies, which, while effective, are extremely computationally intensive, scaling as $O({N}^{3})$, where $N$ is the number of tracklets. Holman et al. (2018) introduced an innovative algorithm called HelioLinC, which exploits the fact that tracklets intrinsically carry four pieces of orbital information, and thus require only two more parameters to describe a Keplerian orbit. By systematically scanning through the remaining two parameters, and synchronizing all tracklets to a common epoch, it is possible to identify groups of tracklets that correspond to the same body without the extreme combinatoric overhead of orbit fitting. HelioLinC's improved computational efficiency of $O(N\phantom{\rule{0.1667em}{0ex}}\mathrm{log}\phantom{\rule{0.1667em}{0ex}}N)$ will be critical for processing the enormous number of tracklets that will be produced by upcoming surveys such as LSST.

However, the original implementation of HelioLinC used local tangent plane coordinates, which limited its capability to link tracklets over extended time baselines. Heinze et al. (2022) presented a modification to the HelioLinC technique that does not require a tangent plane, but requires a systematic search through a grid of 3 parameters, and also breaks down over long time baselines.

Here we present a novel technique that substantially extends the time baseline for tracklet linking without sacrificing algorithmic efficiency. Rather than scanning through a grid of range and range rate, as previous efforts have done, we scan through a grid of two Keplerian invariants. The resulting orbital solutions are exact, meaning that our time baseline is limited only by the invariants' invariance. Furthermore, our approach does not require a reference epoch, meaning that the orbital uncertainties are much more well-behaved.