# Increasing HelioLinC’s TNO recovery potential and extending TNO discovery beyond 150 AU with multi-night tracklets

** Summary**: Multi-night tracklets inherently enable the recovery of isolated single detections on a night by forming tracklets accross different nights thereby increasing the number of objects that can be found compared to single-night tracklet formation. Additionally, very distant objects greater than ~150 AU appear functionally stationary over a single night thus no single-night tracklets can be created for them. Multi-night tracklets, however, allow sufficient separation for these slow movers which enables the recovery of extremely distant objects that single-night tracklets cannot see. Due to this slow on-sky motion, the radial search area for multi-night tracklet formation for TNOs is comparable to the search area for single-night MBA tracklet formation, thus a combinatorial explosion is avoided. Finally, multi-night tracklets provide a superior representation of an object’s orbit compared to single-night tracklets due to their reduced plane measurement error which produces better hypothesis orbits and enables tighter clustering tolerances. Using an arbitrary two week selection of DP0.3 data we count 27.7% more recoverable objects (those having at least 3 nights of detections) with multi-night tracklets than single-night tracklets (2,444 vs. 1,914) and HelioLinC recovers 2,443 (99.96%) of them.

** Caveats**: These are very preliminary results for one arbitrary week of data in the DP0.3 dataset. They are not peer-reviewed nor are they official Rubin estimates. These are just the results of initial tests with an experimental algorithm by an independent researcher.

## Tracklet formation

Traditionally, HelioLinC-style algorithms work by forming n=2 sized tracklets for all combinations of observations that are “reachable” from one another within a single night given some specified constraints on the minimum and maximum on-sky rate of motion and a time interval over which to search. Assuming an object has been observed at least twice on a single night, one of these combinations must be of the object with itself. Repeating this process for all nights of data, a set of observation pairs (tracklets) is produced. Each of these tracklets can then be projected to an asserted heliocentric range and propagated to two reference epochs to find tracklets that occupy the same position at the same time thus indicating that they likely belong to the same object. By testing varying heliocentric range hypotheses we can find objects at arbitrary heliocentric distances.

Unconstrained tracklet formation scales like O(N^{2}) but can be made tractable by the rate of motion and time interval constraints I mentioned above. However, since the on-sky rate of motion for TNOs is so slow, we can increase the time interval over which we search rather significantly. For MBAs we typically create tracklet pairs by searching for observations that could be moving up to 1.5 degrees/day at observation intervals no greater than 90 minutes (1.5 hours) from one another to maintain tractability. The search radius for MBA tracklet formation is show in Equation 1 below.

\[1.5^\circ\,\text{day}^{-1} \times (1.5\text{hr} / 24\text{hr}) = 0.09375^\circ\]

Equation 1. The maximum search radius for Main Belt Asteroids using single-night tracklets

For a TNO search, however, the maximum rate of motion on the sky is much less than 1.5 degrees/day. 100”/day (or 0.02777 degrees/day) is probably a reasonable maximum rate of motion on the sky. Thus, if we want to search a comparable radius to MBAs for TNO tracklet pairs we can calculate that time interval in Equation 2.

\[0.09375^\circ\,\mathbin{/} 0.02777^\circ\,\text{day}^{-1} = 3.375\,\text{days}\]

Equation 2. Solving for the TNO tracklet search duration that has the same search radius as MBAs

The TNO tracklet formation time interval can therefore span days. But in the end, the upper bound on the time interval for TNO tracklet formation is a question of how far you can push it and what you gain by pushing it. We know MBA searches are tractable with the above specified constraints and translating that search radius to TNOs, we get ~4 days in the calculation above. In experimenting with DP0.3 data, I found that using 7 days for the upper bound on the time interval allowed for recovery of *all* of the TNOs that were observed at least once a night on 3 separate nights, so that is what I chose for the upper bound.

Separately, I also claim that we don’t even want to use the tracklet pairs formed over the same night. I’ll justify this next, but limiting tracklet creation to observations on separate nights requires a minimum time interval separation of the observations of at least 0.5 days.

My final TNO tracklet formation criteria are shown in Equation 3:

** dts**: 0.5

*to*7 days

** dpds**: 0

*to*100/3600 °day

^{-1}

Equation 3. Tracklet formation search constraints. * dts* represents the time interval boundaries,

*represents the boundaries for rate of motion on the sky.*

**dpds**This search area exceeds the MBA search area I specified above because I've extended the search duration beyond 3.375 days, but it is still tractable and avoids a combinatorial explosion as I’ll show in the results below.

One final thing to point out is that forming tracklets across nights will only very rarely preclude the recovery of any object that single-night tracklets might find. As long as there are at least two detections of an object on two separate nights within 7 days, a multi-night tracklet will be formed for it. There were no instances where a tracklet could not be formed in my test dataset.

## Plane representation

Now I have to justify the decision to exclude single-night tracklets – the 0.5 lower bound on the *dt* time interval constraint. Again, my claim is not just that same night tracklets can be excluded for convenience sake, but that we do not even *want* to use them.

The geometry that explains this is quite simple. Figure 1 below shows two observations of an object with astrometric measurement error on two different nights viewed edge-on as if we’re looking out from a heliocentric position on the orbital plane. The solid gray line is the true plane. The dotted lines are the orbital planes for the same observations formed by single-night tracklets (top) and multi-night tracklets (bottom). The planes formed by multi-night tracklet are much closer to the true plane of the orbit. Propagating single-night tracklets will produce notably worse results than propagating the multi-night tracklets as plane errors create cumulative position errors when propagated in time.

**Figure 1.** Because there is measurement error in the observations, tracklets created on observations from the same night (top) deviate from the true plane more than tracklets created on observations on separate nights (bottom).

Now let's veryfy this numerically. Figure 2 below (images of numpy arrays) shows the observation times (first column) and heliocentric X,Y and Z positions (columns 2,3 and 4) for a random TNO in the Rubin DP0.3 data on the left. On the right are the sequentially calculated time deltas (first column) and normalized cross products of these position vectors – the unit normal vectors that define the plane formed by the two position vectors (columns 2,3 and 4). All of the unit normal vectors are similar and illustrate that all of the observations are on approximately the same plane, as we would expect for a solar system object, but there is some variation in the measured planes.

**Figure 2.** **Left**: (MJD,helioX,helioY,helioZ) The modified Julian date and heliocentric X,Y and Z positions in AU of a random TNO; **Right**: (dt,unitX,unitY,unitZ) The time difference between sequential observations in the left-side data along with their normalized sequential cross products representing the orbital plane the object is on.

Figure 3 breaks out these unit normal plane vectors from the right side of Figure 1 by duration. The unit normal vectors in the figure below on the left are the ones formed by sequential observations spanning less than 0.5 days (meaning the two observations are on the same night). The right unit normal vectors are the ones formed across sequential observations that span more than 0.5 days (meaning they’re across multiple nights).

**Figure 3.** **Left**: (dt<0.5,unitX,unitY,unitZ) The planes formed by observations with less than 0.5 days duration between them. **Right**: (dt>=0.5,unitX,unitY,unitZ) The planes formed by observations with more than 0.5 days between them. There is less variation in the calculated planes for sequential observations more than 0.5 days apart.

As you can see, the single-night planes (left of Figure 3) have more variation than the multi-night planes (right of Figure 3). There is no variation in the unit normals on the right to 3 decimal places. The takeaway here is that **tracklets with longer durations between observations produce better, more stable estimates of the orbital plane of the object**.

## HelioLinC Search Results & Discussion

Figure 4 below shows the high-level results for two different searches of DP0.3 data with HelioLinC using the RR clustering phase space variation. Both searches are of the first two weeks in the DP0.3 dataset. The left search uses single-night tracklets and the right search uses the multi-night tracklet formation outlined above. I want to stress again, __these are not official Rubin project TNO recovery estimates__. I'm just sharing my experiments with Rubin's preview data (DP0.3).

**Figure 4.** A comparison of HelioLinC recovered TNOs in the first two weeks of DP0.3 data using single-night tracklets (left) and multi-night tracklets (right).

HelioLinC recovers in excess of 99% of the recoverable objects using either single-night or multi-night tracklets, but multi-night tracklets enable a larger universe of recoverable objects than single-night tracklets do (see the second row in the lower half of the tables above). There are 1,914 recoverable objects with single-night tracklets and 2,444 recoverable objects with multi-night tracklets. I’m defining recoverability as 3 nights of linked detections here. There’s the separate question of orbit validation for 3-nighters, but for comparison sake this is the apples-to-apples comparison. This increased number of recoverable objects for multi-night tracklets stems from the ability to link nights with singular detections. I haven’t quantified it here, but even when single-night tracklets and multi-night tracklets recover the same object, multi-night tracklets have the potential to recover more nights of detections through the recovery of additional singular detections on a night for the linkage.

DP0.3 data doesn’t contain any objects more distant than 100AU, so the recoverable object comparison is actually even more favorable than the numbers suggest for multi-night tracklets because those are additional recoveries that multi-night tracklets would find that single-night tracklets wouldn’t are not even measureable in these results.

There are tradeoffs for this multi-night approach still. Multi-night tracklets are much more numerous than single-night tracklets (~47 million vs ~1 million respectively). And a multi-night tracklet HelioLinC search generates a larger amount of mixed linkages than a single-night search (52,705 vs. 42 respectively). This mixed linkage count is probably the biggest liability right now – not because 52,705 is so large, but because real telescope data might make it considerably larger. Reducing this mixed linkage number is probably where I will need to focus my efforts going forward.

## Acknowledgements

Thanks to Pedro Bernardinelli in particular for pointing out the potential blind spot to TNOs at greater than ~150 AU with single-night tracklets which led me to explore this. Additionally, thanks to Siegfried Eggl and Ari Heinze for feedback on this work. None of these people, however, have contributed any errors that may be present in this work. All errors in this work belong to Ben Engebreth alone.

## Credits

The DP0.3 data set was generated by members of the Rubin Solar System Pipelines and Commissioning teams, with help from the LSST Solar System Science Collaboration, in particular: Pedro Bernardinelli, Jake Kurlander, Joachim Moeyens, Samuel Cornwall, Ari Heinze, Steph Merritt, Lynne Jones, Siegfried Eggl, Meg Schwamb, Grigori Fedorets, and Mario Juric.

Published: 7/22/2024