The slowest movers: the outer Solar System, 2010–2026

In January 2018 a Subaru exposure caught a faint point of light that barely moved. Scott Sheppard, David Tholen and Chad Trujillo followed it with Gemini North and Magellan for three years, and in February 2021 they announced that 2018 AG37, nicknamed Farfarout, had been about 132 au from the Sun: the most distant Solar System body ever observed (NOIRLab 2021). On 16 May 2023 another team, using the same telescope, found 2023 KQ14 at only 71 au (Chen et al. 2025). By the usual measure of records Farfarout is the more remarkable object. By the measure that matters for the architecture of the Solar System it is the other way round. Farfarout’s early orbit brings it in to about 27 au, inside Neptune’s orbit, so Neptune could have thrown it out to where it was seen. 2023 KQ14, nicknamed Ammonite, never comes closer than 66 au. Neptune cannot have put it there.

These two bodies frame this chapter. Between 2010 and 2026 the search for the outer Solar System changed from finding bright objects to building samples that can be interpreted. Wide cameras, deliberate cadences, precise astrometry and new algorithms produced several different kinds of progress. The Outer Solar System Origins Survey (OSSOS) built a census whose biases are measured. The Sheppard–Trujillo programme hunted the rare extremes. The Dark Energy Survey (DES) turned five thousand square degrees of cosmology imaging into a second Solar System survey. DEEP and the New Horizons searches on Subaru made digital tracking (shift-and-stack along trial motions) routine over real survey fields. Rubin began delivering discoveries whose value depends on what happens a year later. None of these programmes found a new major planet. The connection to Planet Nine is real but indirect: the bodies found here are the test particles whose orbits make the case for and against a distant perturber, which is the subject of Chapter 8 and Chapter 9.

The chapter also treats the physics and strategy of the hardest targets, the slowest and most distant movers. The earlier history, from the first Kuiper belt object to Sedna and Eris, is in Chapter 5.

Two frontiers

A body’s “distance” can mean four different things, and much confusion in the popular record comes from mixing them. The heliocentric distance $r$ is where the body is now. The observer distance $\Delta$ is how far it is from the telescope, which near opposition is about $r-1$ au. The semimajor axis $a$ measures orbital energy. The perihelion $q=a(1-e)$ and aphelion $Q=a(1+e)$, with $e$ the eccentricity, are the closest and farthest points of the orbit.

A distant detection is therefore not necessarily a detached object. Farfarout was very far away when observed but has a Neptune-crossing orbit. 2012 VP113 has a perihelion near 80 au and stays remote throughout its orbit (Sheppard & Trujillo 2016). Population names follow the orbit, not the current position. “Sednoid” is used operationally for the strongly detached, large-orbit bodies, roughly $q>60$ au and $a>200$ au, following Chen et al. The boundary is a convention, not a discontinuity in the dynamics. “Inner Oort cloud” is even less fixed. The VP113 discovery paper used $q>50$ au and $150<a<1500$ au (Trujillo & Sheppard 2014), while the OSSOS paper on 2013 SY99 calls $a\sim1000$–2000 au the inner fringe of the cloud and reserves the name for larger orbits (Bannister et al. 2017). Any population count needs its cuts quoted alongside it.

The core high-perihelion sample is small: Sedna, 2012 VP113, Leleākūhonua (2015 TG387) and Ammonite. Nearby in discovery space but not in the same dynamical class are 2013 SY99, 2017 OF201, Farout and Farfarout. They link the detached population to orbits that Neptune still scatters or that slowly diffuse.

The physics of a distant mover

Three facts govern every search in this chapter: reflected light falls as the fourth power of distance, apparent motion is mostly a reflection of Earth’s own orbit, and an orbit’s energy is much harder to measure than its current distance. Each has a short derivation.

Light: the r⁻⁴ wall

For an unresolved body seen by reflected sunlight, the apparent magnitude in band $b$ is

with $r$ and $\Delta$ in au, $H_b$ the absolute magnitude, and $\Phi_b(\alpha)$ a phase correction that depends on the Sun–body–observer angle $\alpha$. Sunlight arriving at the body falls as $r^{-2}$, and the reflected light reaching us falls as $\Delta^{-2}$. Far from the Sun, near opposition, $\Delta\simeq r$ and $\alpha\simeq0$, so

where $F$ is flux, $D$ the diameter and $p$ the geometric albedo. A magnitude limit is therefore not a distance limit until a size and an albedo are assumed. Chapter 2 treats the brightness physics in full.

The consequence is brutal. Consider a hypothetical body with $H=2$, about as bright intrinsically as Sedna-class objects. It is magnitude 22 at 100 au, 25.0 at 200 au, 26.8 at 300 au, 29.0 at 500 au and 32 at 1000 au (estimates from the equation above). A survey that reaches $r=24.5$ sees it out to about 178 au. One that reaches 27.5 sees it to about 355 au. Each magnitude of extra depth extends the horizon by only $10^{0.1}=1.26$.

The cost in telescope time is worse still. For background-limited exposures with fixed seeing, signal grows as integration time $T$ and noise as $\sqrt T$, so

at fixed significance. Moving the same body twice as far away needs $2^8=256$ times the exposure. From 100 au to 1000 au the factor is $10^8$ (idealized estimates). For $N$ equal, independent, correctly registered exposures the gain in limiting magnitude is

so 100 exposures buy 2.5 magnitudes and a distance factor of $10^{0.25}=1.78$. Stacking recovers photons that are already in the data. It does not change how many are needed.

Motion: a reflection of our own orbit

Earth moves at 29.8 km/s. A body at distance $r$ on a circular orbit moves at $29.8/\sqrt r$ km/s, a tenth of Earth’s speed at 100 au and three percent at 1000 au. Near opposition both velocities are perpendicular to the line of sight and in the same direction, so the body appears to move backwards (retrograde) at an angular rate set almost entirely by Earth’s motion. To leading order

where $n_\oplus=2\pi$ per year is Earth’s mean motion, which works out to about 148 arcseconds per hour. The body’s own motion reduces this by a fraction of order $r^{-1/2}$, so the formula slightly overestimates the rate, by roughly ten percent at 100 au. Michael Brown and colleagues used the practical rule $r\simeq150/\mu$ in the Sedna discovery (Brown et al. 2004). Away from opposition the rule fails: the projected rate shrinks towards zero at the stationary points, so a small rate does not by itself imply great distance. The HSC-SSP search restricted itself to fields within 40° of opposition for exactly this reason (Chen et al. 2018).

Estimates near opposition from $\mu\simeq148/r$ ″/h, parallax $206265/r$ arcsec and $m\simeq H+10\log_{10}r$, ignoring the body’s own motion and phase effects.

Three numbers in that table drive survey design. In the 3.5-hour triplets that Trujillo and Sheppard used, a body at 300 au moves about 1.6 arcseconds, comparable to the seeing. In a half-hour pair a body at 1000 au moves 0.07 arcseconds, too little to measure reliably in faint ground-based images. Over a day the same body moves 3.5 arcseconds. “Too slow for a same-night tracklet” is not “stationary over a survey”. Over a year it traces a parallactic ellipse whose semimajor axis is $206265/r$ arcseconds, hundreds of times the seeing even at 1000 au.

The curvature of the path matters for stacking searches. For a circular Earth orbit the largest projected angular acceleration from parallax is of order $n_\oplus^2/r$, about 0.6 arcseconds per day squared at 100 au (estimate). Whether a straight-line track is good enough depends on the season, the length of the sequence and the tolerance, and the right test is to propagate real observer positions and measure the residual from the best straight line. Over one night every distant track is a straight line. Over a week it is not, which is why multi-night stacking needs the nonlinear methods of Chapter 20.

Distance is fast, orbits are slow

Parallax gives distance within days. Orbital energy needs the radial velocity as well, and a short arc barely constrains it. The specific orbital energy is

with $v_r$ and $v_t$ the radial and transverse velocities. Distant eccentric orbits are close to parabolic, so $|\epsilon|$ is small and a modest error in $\epsilon$ becomes a large error in $a$. Short-arc fits often have strongly non-Gaussian posteriors with tails that depend on the prior. Brown’s team saw this in 2003: a Bernstein–Khushalani fit and an unconstrained least-squares fit to Sedna’s first weeks of data predicted the same position on the sky but disagreed about $a$ and $e$ (Brown et al. 2004). Only precovery images reaching back to 2001 broke the degeneracy. They also showed restraint: a possible 1991 image was rejected because the chance of an accidental match was about 3%.

A second geometric fact explains why these bodies are rare in catalogues even if they are common in space. A body on an eccentric orbit spends nearly all its time far from perihelion. If $r_{\rm lim}$ is the distance out to which the body is bright enough to detect, and $q<r_{\rm lim}<Q$, then Kepler’s equation with a uniform mean anomaly gives the fraction of the orbit spent inside $r_{\rm lim}$:

where $E$ is the eccentric anomaly. For Leleākūhonua’s discovery orbit, $a=1170$ au and $q=65$ au ($e=0.944$), and an illustrative $r_{\rm lim}=100$ au, this gives $f_{\rm visible}\simeq0.0053$. The body spends about half a percent of its 40,000-year orbit where a survey like the one that found it could see it (estimates). The OF201 discoverers estimate a similar detectability fraction, about 0.5%, under their own assumptions (Cheng et al. 2026). Each such discovery therefore implies a much larger hidden population, with a multiplier that depends on the orbital distribution assumed.

The programmes

OSSOS: a survey built to be modelled

The problem the preceding era left behind was not a shortage of objects but the impossibility of interpreting them. Discovery catalogues mixed uneven sky coverage, varying depth, and recovery campaigns that succeeded preferentially for objects on familiar orbits. An unusual short-arc object could simply be lost because its assumed orbit put the next observation in the wrong place.

OSSOS set out to fix that. Michele Bannister, Brett Gladman, JJ Kavelaars, Jean-Marc Petit, Stephen Gwyn and collaborators designed a survey with the MegaCam imager on the Canada–France–Hawaii Telescope (CFHT), observing from 2013 to 2017. Eight blocks of sky near the invariable plane, at different longitudes, each received a discovery triplet of three exposures over about two hours. Rather than chasing each new object with a narrowly predicted ephemeris, the first-year programme re-imaged whole blocks repeatedly, with the pointings drifting at the mean TNO rate. Unusual orbits were captured without having to be guessed first. Better arcs then allowed targeted tracking in the second year. The final release supplied 20–60 astrometric measurements per object across two to five oppositions (Bannister et al. 2016; Bannister et al. 2018).

Detection ran on two independent branches, one based on SExtractor catalogues and one on wavelet-space point-source detection. Each removed stationary sources and linked the remaining detections into nearly linear motions. Only candidates found by both branches went forward, which suppressed artifacts unique to either. Two visual inspections followed, preferably by different people. The first blocks accepted rates of 0.4–15 arcseconds per hour, and the final characterization generally used a slow limit near 0.5 arcseconds per hour. These are operational cuts, not properties of the population.

The characterization is what made OSSOS different. The team planted PSF-matched artificial movers, about 10,000 per CCD in the final analysis, into copies of the images and measured the detection efficiency as a function of magnitude and rate. The efficiency was fitted as

where $\eta_0$ is the bright-end plateau, $c$ a slow decline with magnitude, $m_0$ the turnover and $\sigma$ its width. Human vetting lost about 0.75% of real sources, a loss folded into the efficiency. Tests on scrambled images found no surviving bright false positives in the first two blocks. The survey tracked 838 of its 840 characterized discoveries, 99.8%. Its two losses were nearby objects that fell at block corners on discovery night, not distant objects lost to an orbital prejudice. It made more than 8,000 images and about 37,000 individually inspected tracking measurements, and its 313 resonant objects, including 132 plutinos, turned discovery into a quantitative record of Neptune’s migration (Bannister et al. 2018).

OSSOS also shows the gap between search range and reach. Its motion search nominally extended to about 300 au, but its practical flux reach for $H_r=4$ bodies was about 100–130 au depending on the block, and its most distant discovery was at 82.5 au. Those facts are consistent: the small area (155.3 deg² searched) and the steep size distribution make large distant bodies rare in any one field. OSSOS released its selection functions with a survey simulator so that any model population can be passed through the same biases, a method treated in Chapter 16.

Sheppard and Trujillo: hunting the extremes

Chad Trujillo and Scott Sheppard ran the opposite kind of programme: wide, deep and tuned for the rarest, slowest objects. On 5 November 2012, in a 52-square-degree survey with the Dark Energy Camera (DECam) on the 4-m Blanco telescope, they found 2012 VP113. Their survey reached about 95% sensitivity at $m_r=24.3$ and 50% near 24.5. Each field received three visits separated by 1.5–2 hours, a baseline of about 3.5 hours chosen so that even the 1.6-arcsecond displacement of a body at 300 au would show. Machine candidates went to visual inspection. VP113 moved at 1.6 arcseconds per hour. Magellan observations in March, August and October 2013 and a CFHT precovery from 22 October 2011 secured the orbit. The paper, published in March 2014, gave $q\simeq80$ au and $a\simeq266$ au, a distance of about 83 au, and a diameter of about 450 km if the albedo is 0.15. The same survey found 90 objects between 30 and 50 au and only VP113 beyond 50 (Trujillo & Sheppard 2014). Sedna was no longer unique.

The 2016 survey report gives the strategy in detail (Sheppard & Trujillo 2016). By then the programme covered 1,080 deg²: 970 with DECam, 40 with Subaru, 60 with Magellan and 11 with the Kitt Peak 4-m. Exposures of about 330–700 seconds on the 4-m telescopes reached $r\sim24$ (or about 24.5 in a broad $VR$ filter), and Subaru and Magellan fields reached about 25.5. Most fields had three images over 3–5 hours. A moving-object finder searched for slow motions, generally below 5 arcseconds per hour, requiring a displacement of 3–4 pixels across the sequence. Objects beyond about 50 au were prioritized for recovery. About 90% of fields were sensitive to motion faster than 0.3 arcseconds per hour, corresponding to about 500 au near opposition, and about 14% had day-long baselines that could detect still slower motion. These are motion sensitivities, not flux sensitivities: a body at 500 au must still be bright enough. By the TG387 report the cumulative area had reached about 2,130 deg², with 300-second HSC exposures to about $r=25.5$ and 420-second DECam $VR$ exposures to about 25.

The same programme set the distance records. Farout, 2018 VG18, was first imaged with Subaru on 10 November 2018 and recovered with Magellan in December. The December 2018 announcement placed it at about 120 au, the first object observed beyond 100 au, with an albedo-dependent diameter near 500 km (Carnegie Science 2018). Farfarout had actually been imaged earlier, on 15–16 January 2018, at around magnitude 25.3, but it took until February 2021 to establish its orbit. The Minor Planet Center circular gives $r=132.83$ au on 10 February 2021 and an early orbit with $a=86.04$ au, $e=0.688$, $q=26.84$ au and an uncertainty parameter $U=9$, which marks a very poorly determined orbit (MPEC 2021-C187). The press release described a 27–175 au orbit. The two quoted aphelia should not be merged into one precise solution, but they agree on the robust point: extreme distance today does not imply permanent detachment from Neptune.

The bottleneck in this kind of programme is not computation. It is covering enough sky deeply, rejecting artifacts without discarding tiny motions, and securing recoveries over several years of telescope time. None of the discovery papers reports an end-to-end compute total.

DES: a cosmology survey searched for the outer Solar System

The Dark Energy Survey imaged about 5,000 deg² with roughly 80,000 DECam exposures between 2013 and 2019, typically 90 seconds in the $griz$ filters. DECam has about 520 million pixels, so the survey holds of order $4\times10^{13}$ pixel-exposures, or about 170 TB for a single uncompressed 32-bit image plane (estimate excluding weights, masks and templates). Its scheduler deliberately avoided returning to the same field on the same night. Requiring the conventional same-night tracklets of asteroid surveys would have thrown away the dataset’s main opportunity (Bernardinelli et al. 2022).

Pedro Bernardinelli, Gary Bernstein, Masao Sako and collaborators solved the problem by linking single detections from different nights directly, using the physics of distant orbits to limit the combinatorics. Their search of the first four years found 316 objects (Bernardinelli et al. 2020). Reprocessing all six years found 817 outer Solar System objects: 815 TNOs, one Centaur and one Oort-cloud comet, 461 of them first reported in that paper. Lowering the SExtractor detection threshold from 1.5 to 0.8 and switching from a boxcar to a Gaussian detection filter gained about 0.47 magnitudes per exposure in $r$. The survey’s fitted 50% completeness was $r_{50}=23.77$ over an effective area of 4,800 deg². A fixed archive became a better survey by changing software. Still lower thresholds were tested and produced too many noise transients to link, so the catalogue threshold was itself a computational limit.

The DES linker is described in the methods section below, and its full machinery belongs to Chapter 12. Its scale is the point here: about $1.6\times10^{10}$ single-epoch detections were cleaned to $1.08\times10^8$ transients, which generated at least $10^{13}$ pairs and about $10^{12}$ triplets before orbit fitting cut them to 31,064 candidates seen on at least seven nights, 9,081 after reliability cuts, 872 sent to visual inspection and 817 confirmed. The authors estimate 15–20 million CPU hours for the whole search. Of 3,749 fake objects that met the discoverability criteria, 3,710 were linked, about 99%. That is a conditional linking efficiency, not the fraction of all bodies in the footprint that were found.

The DES sample also produced a dynamical result. Its 16 extreme TNOs were consistent with an isotropic distribution of orbital orientations once the survey’s selection function was applied. That is not proof that no perturber exists. It shows why combining discoveries from different surveys without their selection functions is inadequate, a theme taken up in Chapter 8.

Subaru: HSC-SSP, FOSSIL and Ammonite

Subaru’s 8.2-m aperture and wide field made it the deepest wide survey instrument of the period, but its programmes followed several distinct paths. They were not one interchangeable “Subaru survey”.

Ying-Tung Chen and collaborators mined the HSC Subaru Strategic Program (HSC-SSP), a galaxy survey, for moving objects (Chen et al. 2018). They reorganized catalogue detections into HEALPix regions, removed stationary sources, classified the remainder with a real–bogus classifier using 49 catalogue parameters, searched for nightly linear alignments within 2 arcseconds of trial lines, and linked tracklets across nights. The first search covered March 2014 to November 2015, and about 221 deg² of the 333.5 deg² footprint met the 40°-from-opposition cut, with individual $r$ images reaching about 25.0–25.5. A source was called stationary if it repeated within 0.5 arcseconds over more than 20 minutes, which the authors recognized put objects beyond about 100 au near opposition at risk. More than 500 candidates were inspected and 231 accepted by at least two team members. The striking result was geometric: requiring detections in six exposures, with single-exposure fill factors of 85–90%, kept only 38–53% of objects even when bright. The measured bright-end efficiency was about 0.38. A large aperture does not guarantee a complete sample if the dither pattern and multiplicity cuts are hostile to moving targets.

FOSSIL II, an extension of the Formation of the Outer Solar System: An Icy Legacy project whose observing proposal was led by Fumi Yoshida, designed its cadence for moving objects: 16 closely packed HSC pointings covering about 25 deg², intra-night triplets about 80 minutes apart, and separations of months between epochs. Weather and technical failures then took over. Of the planned May 2023 discovery run, only one triplet survived, on 16 May, reaching $r\approx25.2$. Technical problems removed the June recovery time (Chen et al. 2025). That night’s data contained 2023 KQ14 at $r\simeq24.6$ and about 71 au.

The orbit was assembled from what followed, partly backwards in time. August 2023 Subaru observations still left the perihelion uncertain. CFHT director’s discretionary time in July 2024 tightened the orbit enough to search archives with the Solar System Object Image Search (SSOIS). That found a 2021 DECam image in which the object was visible in the raw exposure but had been destroyed in the public calibrated product by resampling around a bright-star streak. Adding the 2021 measurement made the 2014 precoveries findable. The final 10.16-year arc gave $a=251.9\pm0.3$ au, $q=65.9\pm0.2$ au and $i=10.98°$ with a mean residual of 0.1 arcseconds. The diameter estimate of 220–380 km spans assumed albedos of 0.15 to 0.05 and is not a measurement. Ammonite’s perihelion points in a different direction from those of the three earlier sednoids, and it fills a previously sparse range of perihelion distance. Its existence constrains some Planet Nine models but does not rule them all out.

2017 OF201: discovery without new photons

In 2025 Sihao Cheng, Jiaxuan Li and Eritas Yang reported a large body found entirely in archival data (Cheng et al. 2026). They searched the DECam Legacy Survey (DECaLS) with a method they describe as similar to Bernardinelli’s DES approach but optimized for DECaLS’s sparse epochs. Ten DECam detections in $g$, $r$ and $z$ from 2014–2018 gave a tentative orbit. That ephemeris led to nine CFHT images from 2011–2012. Nineteen measurements over 2,618 days, about seven years, then established $a=838.3\pm6.8$ au, $q=44.91$ au and $Q=1632\pm14$ au. The body was about 90.5 au away at announcement and about 85 au in the archival images, at $r\sim22.6$, bright enough that its estimated diameter is about 700 km for an albedo of 0.15.

OF201’s perihelion of 45 au places it among the scattering bodies, not the sednoids. Its importance for method is that the orbit was settled without a new observing campaign. DECam and CFHT astrometric scatters of about 0.13 and 0.03 arcseconds, combined with a parallactic signature that shrank by about 5% over the seven-year arc as the body receded, pinned the distance and its rate of change. The paper explicitly defers the search algorithm to future work, so it should not be described as a multi-year pixel-stacking discovery, and no limiting magnitude, trial count or runtime is published.

DEEP: digital tracking at survey scale

The DECam Ecliptic Exploration Project (DEEP), led by David Trilling, David Gerdes, Mario Jurić, Chad Trujillo and collaborators, was designed for digital tracking from the start (Trilling et al. 2023). Its 46.5-night allocation produced long stares of 52 to 130 consecutive 120-second exposures in a broad $VR$ filter. The design goals of $VR\sim27$ and “thousands of TNOs” are goals. The published results come from two complementary analyses of the first fields.

The orbit analysis by Hayden Smotherman, Pedro Bernardinelli and collaborators ran KBMOD, the GPU shift-and-stack code described in Chapter 17, on difference images built with the LSST Science Pipelines (Smotherman et al. 2023). The median sequence had 96 exposures. The search tried every starting pixel with 50 speeds from 1.4 to 4.5 arcseconds per hour and 30 directions within 45° of the direction of decreasing ecliptic longitude, 1,500 velocity hypotheses in all. Median processing was 576 seconds per CCD on an NVIDIA A40, including loading, searching, filtering and postage stamps, and about 282 GPU-hours overall. Searches in the reverse direction supplied an empirical background of artifacts. The likelihood threshold of 7 was raised to 10 in some slow or crowded cases. A convolutional neural network (CNN) and blinded visual inspection followed. Each nightly detection became a tracklet with fitted start and end positions and a full covariance, and the DES linker connected tracklets across nights and years. Requiring at least four nights and an arc longer than 0.8 years yielded 110 objects with multi-year orbits, four of them previously known, reaching about $m_{VR}=26.2$.

The population analysis by Kevin Napier and collaborators instead built explicit weighted stacks over about 60 deg² of field-visits with a mean limiting magnitude near $r=26.2$ (Napier et al. 2023). They generated the hull of angular rates allowed by bound orbits, filled it with k-means centroids (about 20% fewer grid points than a rectangular grid), ran several CNNs and human voting, and calibrated the votes with reverse stacks and implants. About 2,300 candidates above signal-to-noise 6.5 then entered the population inference probabilistically, without each needing a confirmed orbit. One detail matters for this chapter: the analysis discarded sources slower than 0.79 arcseconds per hour, roughly beyond 150 au for its geometry, because subtraction artifacts piled up there. A grid that nominally spans great distances did not produce an equally broad science sample.

New Horizons: searching for a reachable target

The search for a post-Pluto target for New Horizons solved a different optimization: find a body that an already flying spacecraft can reach. Marc Buie, John Spencer, Simon Porter, Alex Parker and colleagues combined ground searches with a Hubble Space Telescope (HST) campaign. Across all efforts they report 89 Kuiper belt discoveries, including Arrokoth and 11 objects later observed from the spacecraft at a distance (Buie et al. 2024).

The 2014 HST search used WFC3/UVIS with the broad F350LP filter, five exposures of about 370 seconds per visit, with HST tracking a representative reachable orbit. Instead of a rectangular grid of constant velocities, the shift-and-stack templates came from simulated reachable orbits and the exact motion seen from HST, chosen so that the frame-to-frame mismatch stayed within about one WFC3 pixel, 0.039 arcseconds. About 20 guide orbits sufficed for consecutive visits and as many as 80 for longer gaps. Of 83 fields, five were unusable, and the search found five objects. Arrokoth was discovered in images from 26 June 2014 at about magnitude 26.8. This bank of physically motivated templates is an early operational example of using a population prior to shrink a search at equal sensitivity.

After the Arrokoth flyby the search moved to Subaru, looking for bodies downstream of the spacecraft in half-night sequences of 90-second HSC exposures in very crowded fields near the Galactic plane. Wesley Fraser and collaborators used KBMOD from 2021 on, with 35 rates from 0.5 to 4 arcseconds per hour and 15 directions within 20° of the ecliptic (Fraser et al. 2024). The grid allowed the flux to smear over as much as about 1.4 arcseconds, about twice the typical seeing, as an explicit trade between depth and processing. A likelihood threshold of 5 produced 3,000–5,000 candidates per CCD. An ensemble of neural classifiers cut the bogus candidates by a factor of 500–1,000 at a cost of about 0.2 magnitudes in depth. Nightly efficiency turnovers reached about magnitude 27.0 for sequences of about 120 exposures. The report lists 239 multi-night detections, eleven with best-fit distances beyond 70 au. Their arcs were at most 77 days, and the authors state that false-detection and false-linkage rates were not characterized. The suggested excess of distant objects is a candidate result.

A separate analysis by Fumi Yoshida, Toshifumi Yanagisawa and colleagues applied a field-programmable gate array (FPGA) method adapted from space-debris detection to the same programme (Yoshida et al. 2024). It selected 32 equally spaced images, removed stationary backgrounds, binarized the pixels near sources and tested moving alignments before human inspection. It found 84 candidates in the 2020–2021 data and linked seven objects, two with designations. The binary representation is fast but discards flux information, so its sensitivity should be compared to matched filtering through recovery curves, not throughput.

Pan-STARRS: a characterized search beyond 80 au

The clearest quantitative search aimed specifically at distant planets is Matthew Holman, Kevin Napier, Matthew Payne and Jacob Kurlander’s search of Pan-STARRS1 catalogues (Holman et al. 2025). It used 708,554 exposures and injected 10,000 synthetic orbits into the source catalogues, carrying them through every cut. The pipeline built its own catalogue of sources stationary over years and rejected detections within 1 arcsecond of them, losing an estimated 0.5% of moving-object detections. It linked tracklets in 20 inverse-distance slices corresponding to 80–1600 au, required at least six tracklets per object, and ran for several days on 100–200 CPU cores. The final list of 3,807 candidates contained 3,115 implants and 692 real objects, 109 of them not yet in the Minor Planet Center database. The search recovered Sedna, Eris and Gonggong and found no new object beyond 60 au. The range of distances searched is not the range of distances discovered.

Rubin’s first year

The Vera C. Rubin Observatory combines an 8.4-m telescope, a 3.2-gigapixel camera with a 9.6-deg² field, and a nominal single-visit depth near $r=24.5$ (Jurić et al. 2025). Its first imagery was released on 23 June 2025. The widely reported 2,104 new asteroids were an all-asteroid count, not a TNO count (NOIRLab 2025). A later photometric study analysed eight First Look TNOs in about 24 deg² observed on nine nights, five of them new: 2025 ML58, 2025 MM66, 2025 MP35, 2025 MV13 and 2025 MW47 (Colazo & Alvarez-Candal 2026). Rubin’s pipeline and alert system are the subject of Chapter 15.

The first lesson of Rubin’s outer Solar System was about orbits, not detection. Chantal Hemmann, Brett Gladman, Justine Obidowski, Jean-Marc Petit and JJ Kavelaars selected three potentially large-orbit objects from the 2025 data, whose moving-target depth they estimate at about $r=23.5$, and recovered them with CFHT in 2026 (Hemmann et al. 2026). 2025 LS2 remained a genuinely large orbit, with barycentric $a\simeq510\pm110$ au. The other two did not. The preliminary Minor Planet Center semimajor axes of 2025 ME278 and 2025 MX348 were 134 and 515 au. After recovery both were near 39 au, likely plutinos in the 3:2 resonance with Neptune. MX348 was also identified with 2015 KF176, extending its arc further. These are three object histories, not an error rate for Rubin’s discovery stream, but the moral is clear.

Rubin’s value for the most distant movers depends on linking across nights more than on single-visit depth. A half-hour pair of visits is a weak motion measurement for a body beyond a few hundred au, and an ordinary sidereal coadd is not a moving-object survey, because such a body’s annual parallactic ellipse is orders of magnitude larger than the PSF (LSST Solar System Science Collaboration 2020; Geringer-Sameth et al. 2025). Reaching fainter needs dedicated stares. A 2025 proposal by Kavelaars and collaborators asks for 30 hours, as six five-hour visits over a year to a field along the New Horizons direction, each visit made of 540 thirty-second exposures. It targets $r\simeq27.5$ and forecasts up to 730 Kuiper belt objects under its population model: a forecast, and not 730 sednoids (Kavelaars et al. 2025). Geringer-Sameth et al. show how such depths slip in practice. An eight-hour stare of about 800 thirty-second exposures with a 0.7-arcsecond PSF and single-image depth 24.6 has an ideal matched-filter 10σ depth of $r=27.5$. Their illustrative coarse template bank loses up to a factor $e^{-1}$ in signal-to-noise between templates, which brings the completeness down to about $r=26.4$ at the same threshold. Recovering the ideal depth needs a finer bank and more compute.

Methods for the slowest movers

Three approaches recur. Dedicated surveys make their intra-night baseline long enough to see slow motion in detections that are individually significant. Catalogue searches exploit parallax and multi-year sampling without needing same-night tracklets. Pixel-domain searches add up flux below the single-image threshold along trial trajectories. The last goes deepest but faces an immense search space. The best practical pipelines combine all three with prompt follow-up and archival precovery.

Triplets, recovery and precovery

A dedicated survey begins with a cadence chosen for the slowest target, not the most common nearby population. After calibration, background estimation, PSF measurement and astrometry, it extracts point sources or difference-image detections, searches for consistent motion across the discovery images with rates well below asteroid defaults, ranks the candidates, inspects them and schedules recovery. The discovery baseline should give a displacement comfortably larger than the uncertainty of a position difference, about $\sqrt{\sigma_1^2+\sigma_2^2}$ per coordinate for positional errors $\sigma_1$ and $\sigma_2$. VP113’s 3.5-hour triplets are the example. FOSSIL II’s multi-season design supplied the lever arm for the orbit.

Naively trying all triples of sources across three images costs $O(n^3)$ for $n$ sources per image. Spatial indexing with a bound on the admissible displacement reduces it to $O(n\log n)$ for tree construction plus the number of candidate neighbours actually generated, though confusion and artifacts can still dominate. Fitting an orbit to a surviving candidate is cheap. The expensive problem is generating candidates.

Precovery should be iterative: propagate the orbit’s uncertainty, search the most recent useful archive epoch, accept a detection only after assessing the chance of a coincidence with an unrelated source, refit, and step farther back. Sedna’s rejected 1991 image shows the first safeguard. Ammonite’s raw-image recovery shows a second: a source that is statistically bright can be algorithmically absent from a standard data product. Attribution and precovery in general are covered in Chapter 13.

Linking in inverse distance

Catalogue linking across nights uses the coordinates introduced by Bernstein and Khushalani. In a tangent-plane frame whose $z$ axis points at the field, with the body at $(x_0,y_0,z_0)$ at reference time $t_0$, define

The dotted quantities are scaled Cartesian velocities, not time derivatives of $\alpha$, $\beta$ and $\gamma$. With $\tau=t-t_0$, observer position $\mathbf x_E(t)$ and gravitational departure from straight-line motion $\mathbf g(t)$, the predicted sky position is

and similarly for $\theta_y$ (Bernardinelli et al. 2020). The $-\gamma x_E$ term is the parallax, the reflex of the observer’s motion, scaled by inverse distance. Fix $\gamma$ (a distance hypothesis) and set $\dot\gamma$ to zero, subtract the parallax from every detection, and what remains is nearly straight-line motion. This is the mathematical basis of efficient distant-object linking.

Physics then bounds the search. A bound orbit requires, to leading order,

so after removing parallax a second detection at time separation $\Delta t$ must lie within a radius $r_2=\sqrt{2GM_\odot\gamma^3}\,|\Delta t|$ of the first. A third must lie in a small parallelogram set by the remaining uncertainty in distance and radial velocity. The number of chance pairs scales as the square of the transient density times $\gamma^3$, and the leading number of triplets as $n^3\gamma^{7/2}$. Searches for closer objects are dramatically more expensive, and searches for distant ones correspondingly cheap per unit of sky.

The DES pipeline, in outline:

for each exposure:
    detect sources; calibrate positions and fluxes
    compare with deep stationary-source catalog; keep transients; mask artifacts
for each sky patch and inverse-distance bin gamma:
    remove nominal parallax from transient coordinates
    enumerate physically allowed pairs
    test allowed third-detection parallelograms
    fit and grow surviving orbits through other exposures
merge duplicates; apply night-count, arc and orbit-quality cuts
confirm with flux stacked from independent images without detections
measure completeness with blinded synthetic orbits

Two engineering details carry lessons. In the six-year search the third-detection test changed from several kd-tree ball queries covering the parallelogram to a linear transform: multiply each displacement by the inverse of the matrix of the parallelogram’s two spanning vectors and test whether both coordinates lie in the unit interval. With parallel Numba code this sped up the triplet stage about a hundredfold, a stage benchmark, not a hundredfold speed-up of the survey (Bernardinelli et al. 2022). Second, the final confirmation used information excluded from discovery. At each candidate’s predicted positions, DES stacked the exposures in which it had not been detected, excluding the nights used for linking, and computed a sub-threshold significance (STS). Candidates seen on at least nine distinct nights were accepted outright, and others had to pass $\mathrm{STS}>1.2\sqrt{N_{\rm images}}$ after review. The threshold was calibrated against visual assessment and is not a universal Gaussian criterion, but the pattern of proposing with some data and confirming with independent data is general.

The Pan-STARRS search used the same transformation with a different index: an assumed distance converts each detection’s direction into a barycentric position, a three-dimensional kd-tree of unit vectors finds neighbours within the bound-orbit radius, and tracklets with residuals above 0.4 arcseconds or magnitude differences above one are rejected. A practical cost model is $O(DN\log N+L)$ per sky partition, with $D$ distance slices, $N$ detections and $L$ the candidate-generation work, which can become combinatorial in artifact-dense fields (Holman et al. 2025).

Stacking along trajectories

When a body is below the single-image threshold, the search must add flux along trial trajectories. Write each calibrated difference image as

where $i$ labels exposures, $p$ pixels, $P_i$ is the point-spread function (PSF), $a_i$ converts a reference flux $F$ into that exposure’s units, $X_i(\boldsymbol\theta)$ is the position predicted by trajectory parameters $\boldsymbol\theta$, and $n_{ip}$ is noise of variance $\sigma_{ip}^2$. For a fixed trajectory define

Then the maximum-likelihood flux is $\widehat F=\Psi/\Phi$ with variance $1/\Phi$, and the detection statistic is

Under a Gaussian null with a fixed template, $\nu$ is standard normal. In practice $\Psi$ and $\Phi$ are precomputed as images (the PSF-weighted data and the PSF-weighted inverse variance), and a trajectory’s score is a sum of pixel lookups along its path. This is the likelihood behind KBMOD and the DEEP coadds (Smotherman et al. 2023). Derivations and the history of the idea are in Chapter 17 and Chapter 18.

For $N_p$ starting pixels, $N_v$ velocities and $N_t$ exposures the direct cost is $O(N_pN_vN_t)$ lookups. DEEP’s 1,500 velocities over about 96 exposures took a median of 576 seconds per CCD on one A40. A GPU changes throughput enormously but not the scaling. The velocity grid is itself a sensitivity choice: the New Horizons Subaru search accepted up to about 1.4 arcseconds of smearing, and its lower rate limit of 0.5 arcseconds per hour means an opposition search is not automatically sensitive to bodies at 500–1000 au.

Linear tracks fail once parallax curves the path. Heliostack, by Napier, Holman and collaborators, instead chooses a distance, radial and tangential velocities and an orbital branch, and maps each image onto a barycentric “canvas” on which a body with those parameters stays fixed, while the image supplies the remaining two coordinates (Napier et al. 2026). In the April 2026 demonstration on 1,118 archival HST images spanning 15 days in 2003, covering about 0.02 deg², it searched cold classical orbits at 38–51 au with 1,148,817 initial conditions. Each condition yields a full canvas, so the search tested about $2\times10^{14}$ trajectories. A 7σ first pass returned 109,383 peaks, trajectory matching cut these to 533, and pixel-level tuning left 145 above an empirically chosen boundary at 8.8σ: 141 implants and four real objects, two known and two new. The fitted 50% completeness was 29.21 ± 0.03 in HST’s ST magnitude system. The whole run took a few hundred NVIDIA H200 GPU-hours and a few thousand CPU-hours. This demonstrates nonlinear stacking at ordinary Kuiper belt distances in exceptionally clean space data. It is not yet a sednoid search, and the authors note that the bank would grow by about a factor of 300 for a wider Kuiper belt region.

The look-elsewhere effect sets the threshold. For $M$ independent standard-normal trials, the expected number of false exceedances of $z$ is $M\,\bar\Phi(z)$, where $\bar\Phi$ is the Gaussian upper-tail probability. With $M=2\times10^{14}$ and $z=7$ this is about 256. Heliostack’s first pass found 368 non-implanted candidates, broadly consistent with that scale. Real trials overlap, so the effective number is smaller than the raw count, but real residuals are not Gaussian, which pushes the other way. Thresholds are therefore set empirically, from reverse searches and implants, and do not transfer between datasets.

For sparse multi-year data, Alex Geringer-Sameth, Nathan Golovich and Keita Iwabuchi derived a metric on orbit space (Geringer-Sameth et al. 2025). For Gaussian PSFs of width $s_i$, the fractional loss of signal for a small parameter error $\delta\boldsymbol\theta$ is, to second order,

where $J_i$ maps parameter perturbations to image displacements in exposure $i$ and the weights $w_i$ sum to one. Template density follows from $\sqrt{\det g}$, exactly as in gravitational-wave template banks (Chapter 38). The method’s showcase is Sedna in Zwicky Transient Facility (ZTF) data: 971 unmasked image intersections with median single-image signal-to-noise 3 stack to 127. The same study estimates about $10^{19}$ trial orbits for a six-year northern-sky search beyond 100 au at its tolerance, with a density scaling roughly as $r^{-7}$ and growing with time baseline as about $T^3$ for distant orbits. Those are empirical scalings for that setup, not theorems, but they show why brute force does not reach the 100–1000 au frontier.

Measuring what you missed

Every result above becomes science only through its selection function. With $f(\mathbf z)$ the intrinsic distribution of orbits, absolute magnitudes, colours and light curves, the expected number of discoveries is

where the factors are the probabilities of detection in images, linking and tracking to a secure orbit. They need not be independent. OSSOS measured and released the first and nearly eliminated orbit-dependent tracking loss. DES measured catalogue selection and linking with injected orbits. DEEP injected sources into the images before detection. Injecting into a final catalogue cannot reveal losses that happened earlier, for instance when a slow source was masked as stationary or absorbed into a template. The Pan-STARRS team went a step further and fitted a probabilistic model of linking success from each object’s sequence of detections, with about 90% validation accuracy (Holman et al. 2025).

Landmark results

The sample of extreme bodies also feeds directly into the Planet Nine debate. In April 2026 Amir Siraj, Christopher Chyba and Scott Tremaine applied a conditional-likelihood method, designed to cancel survey footprint effects under stated assumptions, to 25 stable or metastable distant TNOs, including OF201 and Ammonite. They report 1.9σ apsidal clustering with their three-dimensional method, compared with 2.7σ for their earlier 21-object sample (Siraj et al. 2026). These are model-specific significances, and Chapter 8 weighs them against other analyses.

Limitations and open problems

The 100–1000 au gap is mostly a luminosity gap, compounded by computation. A large eccentric body can visit 1000 au and be discovered near 80 au. That is not a survey of the population living at 1000 au. A non-detection should be stated in terms of absolute magnitude and orbit with the full efficiency, not as an empty shell of space, and unknown size and albedo distributions prevent a unique conversion to mass.

Slow motion is lost before any search begins. A template made from epochs too close in time partly subtracts a very slow mover from itself. Catalogue persistence filters can reject a body near its stationary point. Bright-star masks and blends remove real targets along parts of their paths, as Ammonite’s 2021 image shows. HSC-SSP’s stationary-source rule and DEEP’s 0.79-arcsecond-per-hour cut make the loss explicit. These losses depend on distance and cadence and must appear in injection tests.

Depth is a distribution. Weather, chip gaps, masks, trailing, colour and crowding make a single “limiting magnitude” misleading. The faintest discovery gives no completeness guarantee.

Short arcs and priors distort the frontier. Precise current distance does not imply a precise orbit. The positions at the two ends of a nightly digital-tracking fit are correlated, and DEEP carries that covariance into its orbit fits. The New Horizons distant-candidate analysis suppressed $e>0.9$ solutions for arcs shorter than two months to tame implausible posterior tails, which may be sensible for some samples. A sednoid search cannot treat high eccentricity as an artifact, though: Leleākūhonua and OF201 both have discovery-fit eccentricities near 0.945. Search assumptions must be kept separate from measured constraints.

False alarms are structured. Residual stars, shared templates, interpolation, detector features and overlapping trajectories create correlated backgrounds. Reverse searches are strong controls only if their artifact conditions match the forward search.

Recovery is part of the survey. Ammonite needed targeted CFHT time. Bodies several magnitudes fainter would need comparable or larger telescopes and more stacking. An algorithm that produces many unrecoverable candidates can raise the apparent yield while shrinking the usable dynamical sample.

Compute is rarely reported. DES gives an end-to-end CPU estimate and DEEP and heliostack give GPU budgets. Most discovery papers report neither wall time nor hardware, and data volumes in bytes are inconsistent. That prevents any honest ranking of compute efficiency across the period (Chapter 41).

Where the room is

The physical limits here are hard: shot noise, aperture, the size distribution of the population, and the missing radial velocity in a short arc. A tenfold faster pipeline cannot extend a fixed dataset’s photon limit. If it instead makes ten times as many useful exposures searchable, the ideal gain is 1.25 magnitudes and a distance factor of only $10^{0.125}=1.33$. No credible argument turns routine Rubin imaging into a census of ordinary sednoids at 1000 au. The engineering and algorithmic limits are different, and they are large. What follows is assessment, not demonstrated improvement.

Accelerate the combinatorics, not the orbit integrator. DES’s pairs and triplets make materialization and data movement the obvious targets. The published hundredfold triplet gain came from changing a geometric query. A GPU version could batch the inverse-distance transforms, sort detections into cells, stream pair blocks without writing rejected pairs to disk, and fuse the parallelogram test with early orbit-fit bounds. Amdahl’s law limits the payoff: if a stage takes a fraction $f$ of the runtime and is sped up by $s$, the total speed-up is $1/[(1-f)+f/s]$. No measured $f$ is published for DES, so neither the historical stage gain nor a kernel benchmark predicts an end-to-end factor.

Exploit inverse distance first, and place templates by the metric. Beyond 80 au Earth’s known motion supplies nearly all the curvature, so a bank in inverse distance plus small intrinsic drift is more economical than a generic acceleration bank. Pan-STARRS and heliostack already do this, so novelty must come from implementation or from a proven sensitivity-preserving reduction. The Geringer-Sameth metric says which directions need dense sampling. A short arc barely constrains the radial direction and should not be sampled as densely as sky position. DEEP’s 20% grid reduction and the HST reachable-orbit bank are real precedents, but neither is an order-of-magnitude gain.

Make the $10^{19}$ honest. As a rough scale only, heliostack processed about $2\times10^{14}$ trajectories in a few hundred GPU-hours, of order $10^8$ trajectories per GPU-second. At that rate $10^{19}$ trials would need of order $10^7$ GPU-hours (an order-of-magnitude estimate across different trial definitions and data). The room is in reducing the number of trials, not in summing faster. One proposal, made by Geringer-Sameth et al. themselves, is hierarchical: build permissive short-segment tables of the signed sums $\Psi$ and $\Phi$, combine only dynamically compatible segments, and refine survivors on the original pixels. Rejecting every segment that falls below a conventional threshold would lose exactly the sources stacking is meant to find, so pruning needs bounds on the remaining evidence or calibrated probabilistic gates. These ideas parallel hierarchical searches for continuous gravitational waves (Chapter 37) and the tree methods of Chapter 21.

Optimize geometry and data movement, not only summation. In Golovich et al.’s benchmark of a distributed nonlinear tracking code, computing orbit–image intersections took 44.33 seconds against 9.31 seconds for pixel retrieval on one node, so intersection was about 83% of that subtotal (Golovich et al. 2025). Eliminating retrieval entirely would give only about a 1.2-fold gain. Preindexed footprints, grouped orbit cells, cached ephemerides and data locality may matter more than a faster addition kernel (Chapter 40).

Make the stationary background target-aware. Jointly fitting persistent stars and slow-moving templates, excluding predicted path pixels from reference images, or using season-separated templates could recover sources discarded upstream, especially near the Galactic plane where catalogue searches lose completeness. The cost is extra nuisance parameters and a possibly higher false-alarm threshold.

Move artifact rejection earlier, with a sensitivity ledger. Transient density enters linking at high powers, so a conservative classifier upstream reduces downstream work disproportionately. The New Horizons survey’s 0.2-magnitude loss after neural filtering is the warning: suppressing candidates by a factor of a thousand can carry a scientific cost that must be measured near stars, for unusual PSFs and at the slowest rates (Chapter 22).

Treat observing time as part of the algorithm. One well-timed recovery can shrink an orbit’s uncertainty far more than many extra discovery photons. Choosing recovery times by the expected reduction in posterior volume, weather and access, and optimizing securely classified objects per telescope-hour and compute-hour together, may yield more science than any single kernel. OSSOS’s drifting blocks, DEEP’s widening fields, HST’s consecutive orbits and the CFHT recoveries of Rubin objects are four successful answers to that joint problem. The broader view of these opportunities is in Chapter 42.

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