Multi-year stacking searches for solar system bodies

TL;DR

This work develops a differential-geometry framework to quantify the sensitivity and computational cost of fully coherent, nonlinear digital tracking over multi-year, all-sky surveys. By deriving a metric on orbital parameter space from the geometry of image-space detections, it defines local coordinates and a density of trial orbits that ensures complete coverage within a tolerance of maximal SNR. The approach is validated with ZTF data, including Sedna and blind recoveries, and is applied to linear motion as a baseline, then to six-parameter Keplerian orbits sampled over six years of ZTF observations. The findings indicate substantial depth gains for wide-area surveys but reveal prohibitive computational scales for full all-sky, multi-year searches, motivating hierarchical, segmented, or targeted strategies (e.g., Rubin LSST deep fields). Overall, the framework provides a principled blueprint for turning large surveys into deep digital-tracking searches capable of probing outer solar-system populations, including substantial portions of Planet 9 parameter space.

Abstract

Digital tracking detects faint solar system bodies by stacking many images along hypothesized orbits, revealing objects that are undetectable in every individual exposure. Previous searches have been restricted to small areas and short time baselines. We present a general framework to quantify both sensitivity and computational requirements for digital tracking of nonlinear motion across the full sky over multi-year baselines. We start from matched-filter stacking and derive how signal-to-noise ratio (SNR) degrades with trial orbit mismatch, which leads to a metric tensor on orbital parameter space. The metric defines local Euclidean coordinates in which SNR loss is isotropic, and a covariant density that specifies the exact number of trial orbits needed for a chosen SNR tolerance. We validate the approach with Zwicky Transient Facility (ZTF) data, recovering known objects in blind searches that stack thousands of images over six years along billions of trial orbits. We quantify ZTF's sensitivity to populations beyond 5 au and show that stacking reaches most of the remaining Planet 9 parameter space. The computational demands of all-sky, multi-year tracking are extreme, but we demonstrate that time segmentation and image blurring greatly reduce orbit density at modest sensitivity cost. Stacking effectively boosts medium-aperture surveys to the Rubin Observatory single-exposure depth across the northern sky. Digital tracking in dense Rubin observations of a 10 sq. deg field is tractable and could detect trans-Neptunian objects to 27th magnitude in a single night, with deep drilling fields reaching fainter still.

Figures (11)

• Figure 1: Illustration of the change in sky coordinates of an orbit under changes $\Delta\theta^\mu$ in the orbital parameters. Black curves show the path of Sedna during 2019. Points mark the intersections with ZTF, i.e. times when the object would be present in a survey image. The gray curves show how the orbital path shifts under a change in semi-major axis $a$ (top) or longitude of the ascending node $\Omega$ (bottom). The arrows illustrate the Jacobian of the transformation. The changes in orbital parameters are exaggerated for illustration. In practice the shifts used in digital tracking will be at the arcsecond level.
• Figure 2: Detection significance in stacked images as the semi-major axis of the trial orbit deviates from Sedna's true orbit. The black curve is for "perfect" images simulated without noise while white points correspond to actual ZTF data. The red dotted curve is the analytic form of expected significance (Eq. \ref{['eqn:SNRexact']}) and the dashed gray shows the metric approximation (Eq. \ref{['eqn:SNRmetricexpform']}). The predicted significance matches the simulation and observations for all trial orbits, while the metric approximation is valid near the true orbit.
• Figure 3: Illustration of local coordinates for orbital parameter space $\xi^\mu$ near the orbit of numbered minor planet 612911, as seen over six years of ZTF imaging. Each small dot and attached arrow correspond to a time when the object was present in a ZTF image. The arrow shows the shift in the orbital path corresponding to a change $\Delta \xi^\mu = 1$ holding the other local coordinates fixed (for visibility, arrows are magnified by a factor of 1000 and are colored according to time of year). The sets of arrows for the different $\xi^\mu$ form a basis for exploring the local region of orbital parameter space. The inset in each panel shows the combination of Kepler element shifts corresponding to a change $\Delta \xi^\mu=1$ in the local coordinate. The height of a bar is the logarithm of the absolute value of the Keplerian element shift $\Delta \theta^\nu$ in units of the Kepler element's natural length scale $\ell^\nu$ (see Eqs. \ref{['eqn:xi2theta']} and \ref{['eqn:elemscales']}). The horizontal black line corresponds to $\lvert \Delta\theta^\nu / \ell^\nu\rvert = 1$ and each gray line is a factor of 10.
• Figure 4: Recovery of (341520) Mors-Somnus in a blind search experiment. The large circles show the offsets in image coordinates between the trial orbit with the highest observed SNR and Mors-Somnus's true orbit. Each circle corresponds to one of the 780 intersections with ZTF (color shows the date). This trial orbit gave a detected at $\mathrm{SNR}=46$, whereas stacking along the true orbit gives $\mathrm{SNR}_\mathrm{max}=60$. The small sets of dots show the next three top trial orbits having $\mathrm{SNR}=31$ (blue), 28 (orange), and 24 (green).
• Figure 5: The fraction of orbital parameter space intersecting the ZTF survey in various numbers of images. The boundary of the gray shaded region shows the fraction of orbits that will be captured in at least one ZTF image (orbits are sampled as described in Sec. \ref{['sec:dataprep']} and binned in semi-major axis). The black curves show the distribution of the number of ZTF intersections for the orbits that have at least one intersection. Specifically, of all orbits that are present in ZTF, the curves shows the fraction that have greater than 100, 500, 800, etc intersections. For example, the median outer solar system orbit observed by ZTF will be present in a stack of about 1100 images.