The number of Solar System minor planet discoveries is growing rapidly thanks to the continuation of present day surveys such as Pan-STARRS and the Catalina Sky Survey, and upcoming surveys such as the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST) and NEOCam, recently renamed to NEO Surveyor. In about a decade, the number of known objects will grow from the currently known 1 million to about 6 million minor planets. Such an increase in discoveries will enable a higher resolution look into the dynamical evolution of our Solar System. However, identifying minor planets in survey images and linking their detections into orbits continues to be a challenging problem. First, linking asteroid detections across multiple nights is difficult due to the sheer number of possible linkages, made even more challenging by the presence of false positives. Second, the motion of the observer makes the linking problem non-linear as minor planets will exhibit higher order motion on the topocentric sky over the course of weeks. Finally, once potential linkages have been established, they need to be confirmed as possible orbits using computationally expensive orbit determination software. For example, the Vera C. Rubin Observatory estimates it will discover nearly six million Main Belt asteroids that will be observed hundreds of times over the course of its ten year survey. Naively attempting to link hundreds of millions of asteroid detections over a ten year period is not computationally feasible.
To make the linking problem more computationally tractable, surveys which aim to discover minor planets focus on constructing “tracklets”: two dimensional sky-plane motion vectors consisting of two or more detections spaced typically 20-90 minutes apart that constrain the direction and rate of motion of potential moving objects. Tracklets are constructed to reduce the number of possible linkages that could be formed by providing information on plausible direction and skyplane angular velocity. They are then linked into inter-night linkages known as “tracks”: sky-plane paths of motion containing several tracklets spanning up to ∼ 15 nights, typically modelled with low-order polynomials. In the case of the LSST, for a moving object to be discoverable it must be observed at least twice a night on at least three unique nights within the 15 day window to go through the trackletto-track creation process. In part due to the relative motion of the observer and the rate of motion of moving objects, both tracklets and tracks can exhibit high residuals relative to the fitted low-order polynomial, requiring relaxed fitting tolerances that can in turn lead to the creation of many spurious candidate linkages. Orbit determination (OD) algorithms are therefore required to run on each candidate track so spurious linkages can be identified and be removed.
The Zwicky Transient Facility (ZTF), an optical timedomain survey scanning the entire northern hemisphere of sky at a rate of more than 3700 deg² hr−¹ , can be seen as a precursor to the LSST. ZTF uses the ZTF Moving Object Discovery Engine (ZMODE) algorithm. Instead of linking tracklets directly into tracks, ZMODE first attempts to build a “stringlet”. A stringlet forms an intermediate step between tracklets and tracks which allows for the linking of pairs of detections across nights before tracks are built. This approach was designed to accommodate ZTF’s cadence during its main survey, where the cadence is frequently too sparse to form short intra-night tracklets.
Recently, Holman and colleagues has shown promising results by shifting the reference frame for linking detections to the heliocenter. By assuming a heliocentric distance and its rate of change, cleverly fitting tracklets for the remaining unknown parameters in inertial space, then propagating the resulting “arrows” to a common epoch, arrows corresponding to the same minor planets will form clusters. These clusters can then be extracted and subsequently validated by orbit fitting. As a testament to the effectiveness of HelioLinC, some 200,000 new minor planet orbits were recovered from the Minor Planet Center’s Isolated Tracklet File (ITF).
Common in all of these approaches is the requirement to build tracklets, which in turn requires a telescope to perform multiple revisits to the same field in a night, then more revisits a few nights later, and so on. For a survey that cannot cover the entire visible sky twice per night, this leads to up to a factor of two reduction of the nightly surveyed area. For a survey such as the LSST, which aims to balance four different science drivers, requiring such a cadence decreases the overall ease by which the other science drivers can be accommodated. It is therefore prudent to investigate whether linking algorithms that are cadence independent can be constructed and whether such algorithms can perform as good or better than the current methods. An algorithm that does not demand a high revisit cadence could increase the efficiency of future surveys, as well as help multi-science missions such as the LSST.
Now, Moeyens and colleagues presented one such cadence- and observer-independent linking algorithm: “Tracklet-less Heliocentric Orbit Recovery” (THOR). Rather than shifting the origin to the heliocenter like Holman and colleagues, they choose to shift the linking frame of reference to a series of dynamically selected heliocentric “test orbits”. The main insight is that transforming detections into the frame of a test orbit linearizes the motion of all objects in a relatively thick bundle of orbits near the test orbit (in phase space), which can then be picked out with line-detection algorithms such as the Hough transform. This provides a path to scanning an otherwise voluminous 6D phase space with a finite number of test orbits and at feasible computational cost.
“By sparsely covering regions of interest in the phase space with “test orbits”, transforming nearby observations over a few nights into the co-rotating frame of the test orbit at each epoch, and then performing a generalized Hough transform on the transformed detections followed by orbit determination (OD) filtering, candidate clusters of observations belonging to the same objects can be recovered at moderate computational cost and little to no constraints on cadence.”
They validated the effectiveness of this approach by running on simulations as well as on real data from the Zwicky Transient Facility (ZTF). They have shown that, THOR can link Main Belt asteroids and more distant populations at high completeness and at moderate computational cost. On two weeks of simulated data, THOR recovered 91.3% of the 18,332 that were ideally findable, whereas a tracklet-based linking algorithm would have recovered none.
On two weeks of ZTF data, THOR linked 97.2% of the 21,542 objects with at least five detections (a factor of ∼ 2 recovery increase over MOPS and a factor of ∼ 1.5 increase over ZMODE). THOR recovered orbits for 97.4% of objects beyond 1.7 au, with 98.4% of objects recovered beyond 2.5 au.
Furthermore, by comparing the 2018 sample to the catalog of orbits as presently known (April 2021), they showed that the lower limit on purity of THOR sub-missions to the MPC would be 97.7% and – assuming all candidates shown in Figure 1 above, are confirmed as real – possibly as high as 100%. This, in combination with its capability to discover objects regardless of cadence or observer, renders it immediately useful for Main Belt and Kuiper Belt Objects (KBO) searches on survey data and archival datasets.
According to authors, while THOR can be applied to running surveys, application to archival datasets – such as ZTF, CSS, or PanSTARRS – is interesting as well. The most straightforward way to do this would be to run a sliding ∼two week window over the dataset, from beginning to the end, running THOR at each window instance. For each window run, orbits could be computed for the discovered objects and projected to the past (and future) for discovery of additional observations (and improvement of orbital solutions). This is the approach they themselves plan to take with the ZTF archival data.
Assuming the 11 objects discovered here are representative of remaining undiscovered objects in ZTF, this search would likely yield on order of 1, 000 asteroids. The discovery potential with deeper archival datasets (e.g., DES data or the DECam archive) is likely to be significantly larger.
The THOR package and demo Jupyter notebooks are open source and available at this https URL.
Featured image: The observations of the 11 discovered candidates are plotted in red, with the sky-plane motion of their best-fit orbits plotted as lines. 10 of the objects show MBA-like best fit orbit solutions. The remaining object has a hyperbolic orbit solution and corresponds to recovery observations of the hyperbolic comet C/2018 U1. The “wiggles” apparent in some of the best-fit orbit lines are due to the motion of the observer (topocentric motion). This figure was generated using plots ztf.ipynb. © Moeyens et al.
Joachim Moeyens, Mario Juric, Jes Ford, Dino Bektesevic, Andrew J. Connolly, Siegfried Eggl, Željko Ivezić, R. Lynne Jones, J. Bryce Kalmbach, Hayden Smotherman, “THOR: An Algorithm for Cadence-Independent Asteroid Discovery”, Arxiv, pp. 1-22, 2021. https://arxiv.org/abs/2105.01056
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