Comets, moons and binaries: when a moving point is not a point

On 28 August 1993 the Galileo spacecraft, on its way to Jupiter, passed the main-belt asteroid 243 Ida at about 2,400 km. The images came back to Earth over the following months. On 17 February 1994 Ann Harch of the imaging team, working through the returned frames, noticed a second, much smaller body beside Ida. It was the first indisputable moon of an asteroid, later named Dactyl (NASA, 243 Ida). Six months separate the encounter from the recognition. The moon had been in the data the whole time.

Two and a half years later, on 14 July 1996, Eric Elst and Guido Pizarro photographed an object with the ESO 1-m telescope at about magnitude $V=18.3$ and saw that it had a tail. Its orbit was that of an ordinary outer-main-belt asteroid. That object, now 133P/Elst–Pizarro, became the prototype of the main-belt comets: bodies in asteroidal orbits that lose dust, apparently because they still hold ice (Snodgrass et al. 2017).

Both stories are about the same failure mode, and it is the subject of this chapter. Every pipeline in Part IV treats a moving object as a point: a PSF-shaped blob whose position and flux are fitted, linked into a tracklet and turned into an orbit (Chapter 10, Chapter 12). Some moving objects are not points. A comet adds a diffuse coma and a tail. A binary adds a second point, often closer to the first than the PSF is wide. Both are discoveries about a body, not of a body, and both are usually made after the body itself is known, sometimes decades after, in images that were already in an archive.

The two halves of the chapter have the same structure. We write down the image of an active or double body, derive what distinguishes it from a single point, and then follow the methods that have actually found such signals, with their sample sizes and their failures. The two problems share a mathematical core: once the orbit is known, the search is a small linear model fitted to a small cutout, and the hard part is calibrating the null hypothesis, not computing the fit.

Two ways not to be a point

It helps to fix vocabulary first, because the literature’s counts depend on it.

An active asteroid has an asteroid-like orbit and observable mass loss, usually dust. A main-belt comet (MBC) is an active asteroid whose activity is attributed to sublimating ice, either because it recurs near perihelion or because gas has been detected. An active Centaur orbits among the giant planets and loses mass at distances where water ice alone cannot sublimate. Impacts, rotational break-up and thermal fracture can all make a comet-like tail. A tail is evidence that material is escaping. It is not, by itself, evidence of ice (Jewitt et al. 2015).

For companions, a detached binary has two bodies on a mutual orbit. A contact binary is a single body with two lobes touching. An asteroid pair is two related bodies that are no longer bound. These categories nest: the Lucy spacecraft found that Selam, the moon of the small asteroid Dinkinesh, is itself a contact binary (Levison et al. 2024).

In both cases there are two discovery events: finding the body, and finding the property. They can happen in different surveys decades apart. When a paper quotes a “discovery distance” for a comet, it may mean the first image, the first linked detection, the announcement, the first recognised coma, or a modelled onset of activity. When a survey quotes a “binary fraction”, it means the fraction detected within that survey’s own window of separation, contrast and brightness. Keeping these apart is most of the work of reading this literature honestly.

The diagram below shows the two signals side by side. On the left, an active body: a compact nucleus, a symmetric coma and an asymmetric tail, with the apertures used to search for them. On the right, a pair of point sources closer than the PSF width, and what is left after a single-PSF fit is subtracted.

Comets and active asteroids: how the search grew

From fuzzy comets to active asteroids

The visual rule that comets are fuzzy and asteroids are stellar broke down early. Object Wilson–Harrington showed a tail in 1949 and was later identified with a near-Earth asteroid. The modern literature treats it as the first bridge between the categories (Chandler et al. 2024; Jewitt et al. 2015). 133P made the bridge a research programme, because its activity recurred and its orbit was firmly in the outer main belt.

Henry Hsieh’s Hawaii Trails Project turned the conjecture into an experiment. It deliberately targeted Themis-family asteroids and other plausible analogues of 133P, and made 657 observations of 599 asteroids. On 26 November 2005 two 120-s exposures with Gemini North showed a tail on the already-numbered asteroid 118401, now 176P/LINEAR. The combined image reached a $3\sigma$ surface-brightness limit of 26.8 mag arcsec$^{-2}$; the tail itself was about 25.3 mag arcsec$^{-2}$. A Lulin 1-m image a month earlier had shown nothing (Hsieh 2009). One detection in 599 targets is a low rate, but it was a measurement with a stated surface-brightness sensitivity, which most later searches lack. Michael Read had independently found 238P/Read on 24 October 2005 with the 0.9-m Spacewatch telescope. Three objects made a population.

Survey-scale searches

The next step was to stop choosing targets and mine every known asteroid that a survey happened to observe. Adam Waszczak and colleagues measured the extendedness of 2,013,279 detections of 221,402 known main-belt objects in Palomar Transient Factory data (Waszczak et al. 2013). Hsieh, Larry Denneau, Richard Wainscoat and colleagues analysed 760,475 Pan-STARRS1 observations of 333,026 main-belt objects between 20 May 2012 and 9 November 2013; PS1 discovered two MBCs and one disrupted asteroid in that interval (Hsieh et al. 2015). The PS1 paper is valuable for its candour: it records the comets it missed and explains why a simple tail finder failed on artefact-rich survey images.

Then came archives and volunteers. Colin Orion Chandler, Chadwick Trujillo and collaborators built a pipeline (SAFARI, then HARVEST) that cuts out every known minor planet from archival Dark Energy Camera images, and in August 2021 launched the Zooniverse project Active Asteroids. By 2024 about 8,300 volunteers had examined some 430,000 images in 6.7 million classifications. They found previously unknown activity on 15 asteroids and one Centaur. After dynamical analysis, only four of those 15 were active asteroids in the narrow sense: one MBC and three MBC candidates. The rest were Jupiter-family comets and quasi-Hildas on asteroid-like designations (Chandler et al. 2024). Sixteen activity discoveries are not sixteen new MBCs. The volunteer side of that story belongs to Chapter 28; its detection statistics appear below.

Spectroscopy finally made the ice hypothesis testable directly. JWST detected water from 238P/Read, then from 358P/PANSTARRS on 8–9 January 2024, with $Q(\mathrm{H_2O})=(5.0\pm0.2)\times10^{25}$ molecules s$^{-1}$ and CO$_2$ below $7.6\times10^{22}$ (Hsieh et al. 2024). In 2026 it measured water from 133P itself, $(1.9\pm0.6)\times10^{25}$ and $(1.4\pm0.4)\times10^{25}$ molecules s$^{-1}$ at two epochs (Hsieh et al. 2026). Yet the visibly dusty MBC 457P/Lemmon–PANSTARRS gave a $3\sigma$ water upper limit of $2\times10^{24}$ molecules s$^{-1}$ (Noonan et al. 2026). Appearance, recurrence and composition had become separately measurable, and they do not always agree.

Distant activity

At the other end of the Solar System the question is how far out a comet becomes active. C/2017 K2 (PANSTARRS), discovered on 21 May 2017 at about 15.9 au, was found active in CFHT images from 2013 at 23.7 au; Hubble resolved a coma about $10^5$ km in radius and limited the nucleus radius to below 9 km for an albedo of 0.04 (Jewitt et al. 2017). Man-To Hui and colleagues pushed the record for C/2010 U3 (Boattini) to 25.8 au, in precovery images from November 2005 (Hui et al. 2019). Both of these are retrospective measurements, not real-time discoveries at those distances.

C/2014 UN271 (Bernardinelli–Bernstein) is the cleanest example of a comet found as a body. Pedro Bernardinelli and Gary Bernstein linked it out of about 108 million single-night transient detections in 80,000 Dark Energy Survey exposures, as a moving point, not because it looked fuzzy. It appeared in 42 DES images on 25 nights, giving 32 useful astrometric measurements on 21 nights, the first in 2014 near 29 au. A VISTA precovery extended the record to 2010 at 34.1 au. The announcement came on 19 June 2021, and only then was a diffuse coma found retrospectively in combined DES data (Bernardinelli et al. 2021). The 34.1-au figure is a body detection, not an activity detection. The linking itself is the subject of Chapter 12 and Chapter 6.

In 2025 ALMA detected carbon monoxide ($J=2$–1) from UN271 at 16.6 au, in observations from 8 and 17 March 2024, identifying the volatile that drives its distant activity (Roth et al. 2025). And in September 2026 Chandler’s group showed that activity survives in the shallow coadds of the Rubin Observatory’s second Early Data Preview: 59 active cutouts out of roughly 700 examined, spanning 32 objects, including 29 designated comets, the interstellar comet 3I/ATLAS (Chapter 14) and new activity on the asteroid-designated 2025 NC6 (Chandler et al. 2026).

The physics of a coma

The image model

Write the calibrated image of exposure $e$ as a sum of components:

Here $B$ is the smooth sky, $S$ the static scene of stars and galaxies, $F_N$ the flux of the unresolved nucleus, $C$ a normalised coma profile with flux $A$, $T$ a possibly asymmetric tail, and $n$ the noise. The position $\mathbf x_e$ comes from the orbit. $P^{\rm mov}$ is the PSF smeared by the body’s motion $\mathbf v_e$ during an exposure of length $t_e$:

This decomposition is a synthesis of the trailed-PSF and coma fits of Ferellec et al. (2022) and the scene modelling of Bernardinelli et al. (2021), not the notation of any one survey. One trap is worth naming at once. If the telescope tracked the comet, the stars are trailed and the comet is not, and adding another motion kernel double-counts the motion. Using a field star’s untrailed PSF as the reference for a trailed asteroid is worse: the asteroid looks extended because it moved, and a pipeline reports a coma that is not there.

Why a coma falls as $1/\rho$

Suppose dust leaves the nucleus steadily and isotropically at speed $v_d$, with $\dot N_d$ grains per second. Conservation of grains through a sphere of radius $r$ gives the space density

$n_d(r)=\frac{\dot N_d}{4\pi v_d r^2}.$

The surface brightness at projected distance $\rho$ from the nucleus is the line-of-sight integral of this density:

$I(\rho)\propto\int_{-\infty}^{\infty}\frac{dz}{\rho^2+z^2}=\frac{\pi}{\rho}.$

So a steady coma has $I\propto1/\rho$ outside the PSF core, and the flux inside an aperture of radius $\rho$ grows linearly with $\rho$, because $\int_0^\rho 2\pi\rho'\,\rho'^{-1}\,d\rho'=2\pi\rho$. This is why a $1/\rho$ profile convolved with the PSF is the standard coma template (Ferellec et al. 2022), and why the usual dust measure is divided by $\rho$:

$Af\rho=\frac{4r_h^2\Delta^2}{\rho}\,\frac{F_C}{F_\odot}.$

Here $r_h$ is the heliocentric distance in au, $\Delta$ the distance from the observer, $F_C$ the coma flux inside the aperture, $F_\odot$ the solar flux at 1 au, $A$ the grain albedo and $f$ the filling factor of grains in the aperture. For an ideal steady coma, $Af\rho$ does not depend on aperture. A bright nucleus inside the aperture breaks that property badly (Ye et al. 2023).

The $1/\rho$ law is a template, not a theorem about real comets. A finite age of activity, changing production, grain acceleration, jets and radiation pressure all bend it. The UN271 analysis considered logarithmic slopes of both $-1$ and $-1.5$ and measured stacked coma emission fainter than 30 mag arcsec$^{-2}$ (Bernardinelli et al. 2021). The template also needs an outer truncation radius to be normalised, and some prescription for its central singularity before convolution. A fitted “coma fraction” therefore means nothing without its truncation and background convention.

What switches activity on

A patch of exposed ice at distance $r_h$ balances absorbed sunlight against thermal radiation and the latent heat carried off by sublimation:

$\frac{(1-A_B)S_\odot}{r_h^2}=\chi\left[\epsilon\sigma T^4+L(T)\,\dot m(T)\right],$

with Bond albedo $A_B$, solar constant $S_\odot$, emissivity $\epsilon$, latent heat $L$, and a geometric factor $\chi$ between the subsolar case ($\chi=1$) and the isothermal one ($\chi=4$). The mass flux follows from kinetic theory,

$\dot m(T)=P_v(T)\sqrt{\frac{m}{2\pi k_BT}},$

where $P_v$ is the vapour pressure and $m$ the molecular mass (Hsieh et al. 2015). Because $P_v$ depends exponentially on temperature, activity switches off much faster than sunlight fades. The PS1 study’s thermal model gives subsolar water fluxes of $4.9\times10^{-5}$, $2.6\times10^{-5}$ and $1.2\times10^{-5}$ kg m$^{-2}$ s$^{-1}$ at 2.5, 3.1 and 3.8 au. Over this range the flux drops by a factor of 4.1 while insolation drops by $(3.8/2.5)^2=2.3$, and the gap widens farther out. A mantle of dust over the ice reduces production much further, so these are upper envelopes, not predictions of what a telescope will see.

Gas is not what an optical survey sees; it sees dust that the gas lifted. Grains must overcome cohesion and gravity, stay in the aperture long enough to be counted, and scatter light toward us. In a single-size approximation the dust mass in the aperture is $M_d=\tfrac43\rho_d a C_d$, for grain density $\rho_d$, radius $a$ and total scattering cross-section $C_d$, and the production rate is roughly $M_d$ divided by the residence time $\rho_{\rm ap}/v_d$ in an aperture of physical radius $\rho_{\rm ap}$. The grain size and speed usually dominate the uncertainty in the result.

Tails

Sunlight pushes grains outward with a force proportional to their cross-section, so a grain feels a reduced effective gravity:

$\ddot{\mathbf r}_d=-GM_\odot(1-\beta)\frac{\mathbf r_d}{|\mathbf r_d|^3},\qquad\beta=\frac{3L_\odot Q_{\rm pr}}{16\pi cGM_\odot\rho_d a},$

where $Q_{\rm pr}$ is the radiation-pressure efficiency. Small grains (large $\beta$) drift anti-sunward. Large grains stay near the orbit and form a trail. Curves of constant release time (synchrones) and of constant $\beta$ (syndynes), projected onto the sky, give physically motivated tail templates. Projection can produce an anti-tail pointing roughly sunward. A search that only looks in the anti-solar direction is therefore incomplete (Jewitt et al. 2015; Ye et al. 2023).

Forecasts use an empirical total magnitude $m=H_c+5\log_{10}\Delta+k\log_{10}r_h+\Phi(\alpha)$, with phase function $\Phi$ and activity index $k$; it is a fit, and extrapolating it past a change in activity moves forecast discovery dates substantially (Inno et al. 2024). Classification uses the Tisserand parameter with respect to Jupiter,

$T_J=\frac{a_J}{a}+2\cos i\sqrt{\frac{a}{a_J}(1-e^2)},$

for semimajor axis $a$, eccentricity $e$ and inclination $i$. Orbits with $T_J\lesssim3$ (the boundary is drawn at 3, 3.05 or 3.08) are comet-like; this is a dynamical label, not a compositional one.

Finding activity

A calibrated activity statistic

Once the orbit is known, activity detection is a matched-filter problem on a small cutout, with nuisance parameters. The research dossier behind this chapter proposes a clean formulation that connects the published PSF-plus-coma fits to the detection theory of Chapter 1. Whiten the cutout with its noise covariance, $\mathbf y=\mathbf C^{-1/2}\mathbf d$. Collect the whitened nuisance templates (nucleus PSF, background, centroid derivatives, PSF corrections) as the columns of $\mathbf X$. Project them out of the activity template $\mathbf t$:

$\mathbf R=\mathbf I-\mathbf X(\mathbf X^T\mathbf X)^{-1}\mathbf X^T,\qquad\tilde{\mathbf t}=\mathbf R\,\mathbf C^{-1/2}\mathbf t,\qquad Z=\frac{\tilde{\mathbf t}^T\mathbf y}{\sqrt{\tilde{\mathbf t}^T\tilde{\mathbf t}}}.$

Under Gaussian noise with a non-negative coma amplitude, $2\log\Lambda=\max(0,Z)^2$. The projection does the important work. It removes the part of the coma template that cannot be told apart from a slightly brighter nucleus or a slightly higher sky. If little norm survives, the observation contains little information about activity, however bright the source. This is the formal version of the familiar statement that an unresolved coma is degenerate with the nucleus.

The statistic is not yet a search. Maximising $Z$ over tail angles, widths, apertures, epochs and a million asteroids produces trials, and for $N_{\rm eff}$ independent Gaussian tests the tail is roughly $N_{\rm eff}[1-\Phi(z)]$. Real templates are correlated and real residuals are not Gaussian, so the threshold must come from inactive control objects and injected comae, not from that formula.

Catalogue extendedness

The first survey-scale methods never touched pixels. They compared catalogue shape measurements of each asteroid with those of stars.

Pan-STARRS1 flagged moving sources whose second moments exceeded the PSF’s (a pipeline change on 13 July 2013 shifted this statistic by $\sqrt2$, a reminder that definitions matter), then had experts review repeat detections and request deeper imaging. It produced hundreds to thousands of flags per night. From comets found by other surveys that PS1 had observed but not flagged, the authors put an upper bound of about 70% on discovery efficiency. That is a bound from missed opportunities, not an injection-based completeness (Hsieh et al. 2015).

Waszczak and colleagues used a concentration index in magnitudes: the difference between peak surface brightness and total magnitude for the asteroid, minus the median of the same difference for stars in that image. Positive values mean less concentrated light. The index also responds to brightness, motion, seeing, sky level and sub-pixel centring, so they built an empirical model of inactive asteroids in a five-dimensional space of those observables, ten bins per dimension, $10^5$ cells, and scored each detection by its residual from the cell mean. Screening selected 1,577 objects, about 0.7% of the sample, and retained 76 of 115 testable known comets, an efficiency of 66%. From the absence of new MBCs they set 95% upper limits of 33 or 22 active MBCs per million main-belt asteroids, under two efficiency assumptions. The search found four new comets, which is a different result from finding new MBCs (Waszczak et al. 2013).

The PTF approach costs $O(N)$ once catalogues exist and never re-reads an image. Its weakness is averaging: combining an object’s residuals over all its epochs dilutes a brief outburst, so epoch-level scores must be kept.

Motion-matched PSF plus coma

The pixel-level method, refined by Sarah Sonnett and colleagues and documented in detail by Ferellec et al. (2022), fits each cutout as a mixture of a bare nucleus and a nucleus with coma:

for each target and each exposure:
    bias/flat correct; mask defects
    pick nearby, slightly brighter, isolated stars; normalise; build empirical PSF
    trail the PSF by the asteroid's motion during this exposure
for the target:
    stack synthetic asteroid images with the same registration as the data
    convolve the synthetic asteroid with a 1/r coma profile
    fit (1 - f_c) * P_ast + f_c * P_coma to the data stack
    compare f_c with control stars; inspect residuals

Building frame-specific models before stacking keeps changing seeing from masquerading as dust. The Isaac Newton Telescope survey that used it took three to ten frames totalling 500 s per target at 0.333 arcsec per pixel, imaged 534 asteroids, and obtained valid tail results for 316 and coma fractions for 293 frame sets. It found one possible candidate, 279870 (2001 NL19), not a confirmed MBC. An optimiser problem left some coma fractions stuck near their initial value of 0.01 (Ferellec et al. 2022). For a fixed centroid and coma kernel the amplitudes are a linear least-squares problem and should be solved in closed form, with a background term. An iterative optimiser on a linear problem is both slower and, as this case shows, less reliable.

The cost is $O(EKP)$ for $E$ exposures, $K$ coma shapes and $P$ cutout pixels. On small stamps, direct convolution often beats FFTs. Template reuse matters more than clever transforms.

Tails: sectors, wedges and polar maps

A thin tail can leave the central PSF almost untouched, so it needs its own search. The INT study split an annulus into 18 sectors. The inner radius followed the seeing, usually 10–20 pixels, and the annulus was 12 pixels (about 4 arcsec) wide, with a separate 2-arcsec outer annulus for the sky. The asteroid’s score is its brightest sector’s median flux above local sky, $F_{\rm ast}=\max_nF_n-F_{\rm local}$. Each comparison star gets the same maximum-sector treatment, and the asteroid is ranked against them:

$p_{\rm ast}=\frac{\#\{j:F_{\star,j}>F_{\rm ast}\}}{N_\star}.$

Taking the maximum over sectors for the stars too builds the look-elsewhere effect into the comparison. A permissive cut $p_{\rm ast}\le0.31$ sent the lowest-ranked fifth of targets to visual inspection. It was a triage threshold, not a false-alarm rate (Ferellec et al. 2022).

Annular tail finders have a mixed record, and the failures are instructive. PS1 intended to run one operationally and abandoned it because image artefacts produced too many false flags (Hsieh et al. 2015). Chandler’s wedge photometry, which sums annular wedges of several angular widths to measure tail significance and direction, was severely limited by background sources (Chandler et al. 2024). Galaxies and neighbouring stars are the dominant false positives. They do not average away, and they look like short tails.

The best-controlled recent example is a non-detection. David Jewitt, Max Mutchler and Yoonyoung Kim remapped a Hubble image of the asteroid Bennu into polar coordinates at 1-degree steps out to 100 pixels (4 arcsec), looked for excess brightness near the predicted trail angle, and calibrated visibility with artificial trails of 1-arcsec FWHM. No trail implied a dust production below $5.8\times10^{-3}$ kg s$^{-1}$ under their grain assumptions, while the near-nucleus profile gave only $<0.12$ kg s$^{-1}$ (Jewitt et al. 2026). Measuring away from the nucleus, where the PSF wings no longer compete, bought a factor of twenty. Bennu is also a reminder of physical limits: OSIRIS-REx saw particles leaving it in situ at rates far below what any remote image can detect.

Archives and volunteers at scale

HARVEST shows where the compute of an activity search actually goes (Chandler et al. 2024). It queries archive metadata, predicts which minor planets fall in each image through the SkyBot service, downloads and validates files, cuts out thumbnails, records provenance in a database, and sends the thumbnails to volunteers. It ran in Python and Bash under Slurm on a university cluster; no GPU use and no total CPU-hour figure are reported.

A depth cut does a great deal of the work. HARVEST dropped predicted objects less than one magnitude brighter than the image’s S/N = 10 depth, removing about 57% of SkyBot predictions, some 21 million. Later batches required a two-magnitude margin. The cut makes activity recognisable when it is there, but it also discards a faint nucleus whose outburst would have made it visible.

Each thumbnail received about 15 volunteer votes, combined as a weighted mean $L_i=\sum_u w_u y_{iu}/\sum_u w_u$ of binary votes $y_{iu}$, with a candidate threshold $L\ge0.473$. Each volunteer’s weight mixes, in the ratio 7:2:1, their accuracy on training images, their experience, and a penalty for saying “yes” too often on science images; the penalty matters because training images are all active, so a habitual “yes” would otherwise look perfect. Chapter 28 treats the scheme in detail and charts what each term bought. On a development sample of about 170,000 images the combined weighting cut the review load from 2,513 to 1,718 images, a third, not an order of magnitude. On a later sample of 406,082 images volunteers produced 4,171 candidates; experts flagged 526 for investigation, and separately found 138 candidates that the volunteer threshold had missed. The throughput numbers expose the real bottleneck. About 18 million vetted thumbnails waited, volunteers retired $805\pm678$ images a day, and some 5,000 new thumbnails arrived daily. At the mean rate the static backlog alone would take about 60 years (Chandler et al. 2024). That describes the historical workflow, not the project’s current state, but it shows that human attention, not pixel processing, set the rate of discovery.

Distant comets: scene modelling and spatial filtering

For UN271 the DES team jointly fitted the static sky scene and a moving point source across overlapping exposures, subtracted both, rotated the residuals into a common physical frame, and averaged them with inverse-variance weights. The point-source photometry showed little brightening between 34.1 and 23.7 au. Yet TESS, measured in a large aperture, recorded a brightening of 1.5 mag between 23.8 and 21.2 au, and at the earlier epoch was about $1.0\pm0.15$ mag brighter than contemporaneous DES point-source photometry (Bernardinelli et al. 2021). The two did not contradict each other. A point-source fit measures the nucleus plus whatever coma falls inside the PSF. A large aperture measures the coma.

Background subtraction can delete the signal it is meant to reveal. For P/2021 HS, 3,446 TESS images provided 574 hours of integration, and the authors compared static-sky removal, which can suppress extended features, with comet-centred, direction-aligned median stacks (Ye et al. 2023). That comet had been discovered as an asteroid by Pan-STARRS on 16 April 2021. Its coma became recognisable only near perihelion, when forward scattering of sunlight by dust at a phase angle of up to 139 degrees amplified extremely weak activity. Models suggested an active area of only about 700 m$^2$ on a nucleus 0.6–1.1 km across. Geometry, not depth, decided whether it looked like a comet. Long stacks along known orbits are the subject of Chapter 17. Here they must also allow the coma to change during the stack.

Photometric time series offer another route: a persistent excess over a modelled bare-nucleus light curve flags activity even when morphology is ambiguous. ATLAS, ZTF and Las Cumbres photometry show Chiron’s post-outburst excess decaying over about 1.4 years, though its rings also scatter light (Murtagh et al. 2026).

Forecasting Rubin

How many more comets will the Rubin Observatory’s Legacy Survey of Space and Time (LSST) find, and how early? Inno et al. (2024) asked a controlled counterfactual: if an idealised LSST-like survey had been running before each known long-period comet’s perihelion, when would it have seen it? With monthly ephemerides over a ten-year window, empirical total magnitudes and a faint limit of $r=24.5$, they found 868 observable comets, about 40% of them detectable at least five years before perihelion. Tightening the limit to $r=24$ delays about a third of the sample by about a year. The paper has inconsistencies in its object counts, and its “at least double” discovery rate is a conditional inference, not a cadence simulation. A real forecast needs population, activity physics, cadence, linking and an activity classifier, as Boe et al. (2019) built for 150 Pan-STARRS long-period comets (Chapter 16).

The first real Rubin result is more modest and more concrete. Using predicted positions for 14,030 comets, depth maps and the exact list of visits contributing to each coadd, Chandler’s group selected about 700 cutouts in shallow regions of the Early Data Preview 2 coadds and found activity in 59. In one- and two-visit coadds the comet is present in only one visit, so its flux is diluted by the weights and can be clipped as an outlier. The authors therefore do not use those coadds for quantitative photometry, and the 8% active fraction applies only to those selection rules (Chandler et al. 2026). Coadds built for static science are hostile to moving sources; Chapter 39 explains why.

Moons and binaries: a short history

The first trans-Neptunian binary was found before the Kuiper belt was: James Christy and Robert Harrington’s 1978 discovery of Charon, whose first sign was an imperfect, distorted image of Pluto (Stephens & Noll 2006). The Pluto–Charon mutual events of 1985–1990, when each body passed in front of the other as seen from Earth, then measured both bodies’ sizes and surface brightness without resolving them (Proudfoot et al. 2026).

Asteroid moons had a less secure start. Several claims from anomalous light curves were never confirmed, which is why Dactyl in 1994 was decisive. In 1999 William Merline and collaborators reported a moon of (45) Eugenia from ground-based adaptive optics, and AO imaging went on to find satellites of Sylvia, Kleopatra, Minerva and other large asteroids. Petr Pravec and Gerhard Hahn reported 1994 AW1 as a photometric binary candidate in 1997, confirmed photometrically in 2008. The small moons of large main-belt asteroids and the close, similar-sized pairs among small asteroids appeared to trace different formation mechanisms (Margot et al. 2015).

Hubble then made binaries routine, and PSF fitting made them common. Denise Stephens and Keith Noll took 81 previously unclassified trans-Neptunian objects observed with NICMOS. Three binaries were visible by eye. Fitting shifted model PSFs found six more, for an observed rate of 11% (+5/−2 percentage points) within the survey’s detection domain, with cold classical objects contributing 6 of 27 and the dynamically excited sample 3 of 54 (Stephens & Noll 2006). Eighteen years later Simon Porter, Susan Benecchi, Anne Verbiscer, Will Grundy, Keith Noll and Alex Parker did the same with WFC3 images of 198 cold classical objects from the Solar System Origins Legacy Survey (SSOLS): 23 visually separated binaries and 19 more blended ones (Porter et al. 2024). In both cases fitting roughly doubled or tripled the yield from images already taken.

Archives and closer views continued the pattern. Alex Parker and colleagues found Makemake’s moon in WFC3 data from 27 April 2015: six 725-s exposures showed a co-moving companion 0.57 arcsec from the dwarf planet and $7.80\pm0.04$ mag fainter. It was gone in a second visit two days later, and earlier HST images had missed it too, probably because the orbit is seen nearly edge-on and the moon was often projected against the primary’s glare (Parker et al. 2016). Csaba Kiss and colleagues found the moon of 2007 OR10, now Gonggong, in archival HST images, prompted partly by the primary’s unusually slow rotation (Kiss et al. 2017). New Horizons’ LORRI camera, far closer to its targets than any telescope at Earth, resolved the near-equal pair 2011 JY31 with a mutual semimajor axis of $198.6\pm2.9$ km and flagged 2014 OS393 as a likely binary (Weaver et al. 2022). On 1 November 2023 the Lucy spacecraft passed Dinkinesh at 431 km and found its moon Selam, which turned out to be two lobes of about 210 and 230 m in contact, orbiting about 3.1 km from the 720-m primary every 52.67 hours (Levison et al. 2024).

Finally, binary searches became statistical surveys of astrometry. Luana Liberato and collaborators searched Gaia’s asteroid positions for periodic wobbles and reported 358 candidates from Data Release 3 in 2024, then 343 from the Focused Product Release in 2026, of which only 99 overlap (Liberato et al. 2024; Liberato et al. 2026).

The physics of a companion

Separation, contrast and resolution

At observer distance $\Delta$, a companion at projected physical separation $r_\perp$ appears at angle $\rho=r_\perp/\Delta$. Its flux ratio $f=F_2/F_1$ gives a contrast $\Delta m=-2.5\log_{10}f$. If the two bodies have equal albedo, $f\simeq(D_2/D_1)^2$. Projected separation is not the semimajor axis: phase, eccentricity and viewing angle all enter.

A telescope of aperture $D_{\rm tel}$ has its first Airy minimum at about $1.22\lambda/D_{\rm tel}$. For a 10-m telescope at 1.2 μm, the diffraction limit is about 30 mas, and main-belt satellites are commonly 5–10 mag fainter than their primaries (Margot et al. 2015). At 40 au, 30 mas is about 870 km. The Airy scale is not a hard cutoff, however. A well-known PSF fitted at high S/N can detect a pair closer than that, and the next subsection says how much closer.

The blended-binary signal

Model a cutout as a background plus $J$ point sources,

$m_p=B_p+\sum_{j=1}^{J}F_j\,P(p-\mathbf x_j),\qquad J\in\{1,2\}.$

For $J=2$, let the total flux be $F$, the flux fractions $w_1$ and $w_2$ (so $w_1+w_2=1$), and the separation vector $\mathbf s$. Put the origin at the photocentre, so that $\mathbf x_1=-w_2\mathbf s$ and $\mathbf x_2=w_1\mathbf s$. Expanding each PSF to second order in its offset,

The first-order terms cancel because $w_1\mathbf x_1+w_2\mathbf x_2=0$, which is what the photocentre means. The second-order coefficient is $\tfrac12(w_1w_2^2+w_2w_1^2)=\tfrac12w_1w_2$. This derivation is the dossier’s analytic assessment, not a published completeness law, but its consequences match the measurements.

The SSOLS injection tests agree in shape. Equal-brightness pairs at combined $V=21$ were recovered down to about 0.5 pixel, about 600 km at 40 au. At $V=22.5$ about one pixel was needed, and equal blended pairs effectively disappeared near $V=24.5$. At 5:1 contrast the limiting combined brightness was about $V=23$, and at 10:1 about $V=21.5$ (Porter et al. 2024).

The natural test statistic is $T=\chi^2_{J=1}-\chi^2_{J=2}$, maximised over separation and position angle. It does not follow a textbook $\chi^2$ distribution. Under the null the secondary’s flux is zero and its position undefined, the flux is constrained positive, and the search over positions adds trials. False-alarm rates must be calibrated by running the entire search, nuisance fits included, on simulated single sources.

From orbit to mass

A companion’s real value is the mass it measures. For a relative orbit of semimajor axis $a$ and period $P$, Kepler’s third law gives the system mass

$M_1+M_2=\frac{4\pi^2a^3}{GP^2}.$

Two scales bound where companions can live. The Hill radius $r_H\simeq a_\odot(1-e_\odot)[(M_1+M_2)/3M_\odot]^{1/3}$, evaluated near the system’s heliocentric perihelion, marks where solar tides compete with the pair’s mutual gravity. It is a scale, not a strict stability boundary. The DES binary 2013 RJ124 has a mutual semimajor axis of about 34.9 thousand km and a Hill radius of about 490 thousand km, so it orbits at about 7% of its Hill radius (Bernardinelli et al. 2023). At the other end, a fluid Roche limit does not exclude cohesive or contact systems, as Selam shows. Close pairs also precess under the primary’s oblateness, at a leading-order rate

$\dot\Omega\simeq-\frac32nJ_2\left(\frac{R}{a}\right)^2\frac{\cos i}{(1-e^2)^2},$

for mean motion $n=2\pi/P$, primary radius $R$ and oblateness coefficient $J_2$. For Huya, adding such non-Keplerian terms improves the orbit fit dramatically (Rommel et al. 2025), and a Keplerian ellipse fitted blindly can miss real structure.

Mutual events and occultations

An unresolved pair’s light curve combines both bodies’ rotation with events in which one hides the other:

$F(t)=F_1^{\rm rot}(t)+F_2^{\rm rot}(t)-\Delta F_{\rm occult}(t)-\Delta F_{\rm eclipse}(t).$

In an occultation one body blocks our view of the other; in an eclipse one body’s shadow falls on the other. Away from opposition they happen at different times. If the secondary disappears completely, the fractional drop is $\delta=F_2/(F_1+F_2)$, or $-2.5\log_{10}(1-\delta)$ in magnitudes. A 0.03-mag attenuation floor corresponds roughly to a diameter ratio of 0.17, below which photometric detection becomes difficult (Margot et al. 2015). An elongated single body seen equator-on varies by about $2.5\log_{10}(a/b)$ for axis ratio $a/b$, so a large amplitude alone cannot distinguish a contact binary from an elongated single body.

A stellar occultation works differently, because it measures the star’s light, not the companion’s. A moon too faint to see can still cast a shadow. For shadow speed $v_\perp$, a spherical moon of radius $R_2$ crossed at impact parameter $b$ blocks the star for a time $\tau$ with $v_\perp\tau=2\sqrt{R_2^2-b^2}$, so a single chord gives a lower bound on diameter. At 20 km s$^{-1}$ a 100-km moon hides the star for at most 5 seconds. The sharp edge is softened by the exposure time, the star’s angular size, and Fresnel diffraction on the scale $\ell_F\sim\sqrt{\lambda\Delta/2}$, about 1.3 km for visible light at 40 au. Chapter 25 treats occultation physics in full.

Photocentre wobble

Gaia does not resolve asteroid binaries. It measures the light-weighted centre, which moves around the centre of mass as the pair orbits. With mass ratio $q=M_2/M_1$ and flux ratio $f$, the photocentre $\mathbf x_{\rm ph}=(\mathbf x_1+f\mathbf x_2)/(1+f)$ and centre of mass $\mathbf x_{\rm CM}=(\mathbf x_1+q\mathbf x_2)/(1+q)$ differ by

$\mathbf x_{\rm ph}-\mathbf x_{\rm CM}=\left[\frac{f}{1+f}-\frac{q}{1+q}\right]\frac{\mathbf r_\perp}{\Delta}.$

If the bodies share density and albedo, mass scales as $D^3$ and flux as $D^2$, so $f=q^{2/3}$. The bracket then vanishes both for a negligible moon and for identical twins, and peaks near $q=0.154$, a diameter ratio of 0.536 (Liberato et al. 2024). Gaia only sees the projection of this vector onto its scan direction, which changes from transit to transit.

The figure shows a physical blind spot, not a computational one. The pairs that are easiest to resolve in images are exactly the ones astrometry cannot see, and no faster periodogram can recover a displacement that is zero.

Finding companions

The table summarises what each method measures and what it misses. The subsections follow it in order.

Hubble: separated and blended companions

The HST procedure starts from a known orbit: register the dithered exposures, fit one and two PSFs, check that the second source keeps its offset across exposures and filters, and confirm with more epochs, since a background star separates from a moving target while a companion shares its motion.

The NICMOS search is unusually reproducible. With the NIC2 camera’s 0.075-arcsec pixels and typical exposures of 256 s in F110W and 512 s in F160W, Stephens and Noll generated PSFs with the Tiny Tim model, shifted on a 0.1-pixel grid, giving 100 sub-pixel positions per pixel over a search box 4–5 pixels on a side. The algorithm tried pairs of positions and adjusted fluxes at fixed total. A detection needed consistent separation (to 0.01 arcsec), position angle (to 10 degrees) and flux fractions (to 10%) across the dithers, and more than 7,000 synthetic images mapped the sensitivity (Stephens & Noll 2006). A literal position-pair search costs $O(EG^2K)$ for $E$ exposures, $G$ candidate positions and $K$ pixels per stamp. Precomputing the image–PSF inner products and the PSF–PSF overlaps reduces each pair to a $2\times2$ linear solve.

SSOLS used four F606W and four F814W exposures per object, at distances of 35.98–53.43 au where one WFC3 pixel spans 1,044–1,550 km. The team masked defects, made target-centred stacks upsampled by two, and fitted Tiny Tim single and double PSFs separately in the two filters. Their score turns the $\chi^2$ improvement into $L_r=\exp[(\chi^2_{\rm single}-\chi^2_{\rm double})/2]$ and $r_1=L_r/(1+L_r)$, then penalises disagreement between the filters beyond 0.25 pixel in separation or 1 mag in contrast. Nineteen blended systems scored above 95% (Porter et al. 2024). That number is not a calibrated posterior probability: no evidence integral or parameter-volume penalty enters it, so “95%” cannot be read as a 5% false-discovery rate across the catalogue. The injection tests, with real backgrounds, 0.1-pixel jitter and flux ratios of 1:1, 2:1, 5:1 and 10:1, are what give the result its weight.

Adaptive optics: subtracting the halo

From the ground, AO images pose a harder inference. A bright main-belt primary may be resolved, the PSF changes with the atmosphere, and the moon sits in a structured halo. The Euphrosyne team removed the halo with running medians in boxes about 50 pixels in azimuth and 40 pixels in radius, inserted 100 synthetic point sources into each science image to measure how much flux the processing removed, fitted positions with 2-D Gaussians, and fitted the orbit with the GENOID code. The moon appeared at five of seven epochs; the orbit fitted five positions over 26 days with an RMS of 1.52 mas (Yang et al. 2020). The lesson carries over directly to exoplanet imaging (Chapter 33): halo subtraction removes part of the companion too, so injections must go in before subtraction and recovery must be mapped in radius and angle. An azimuthally averaged contrast curve can hide blind sectors.

Seeing-limited surveys: scene modelling

A wide-field survey with 1-arcsec seeing can still find wide binaries if it models the whole scene. For DES, Bernardinelli, Bernstein, Jindal and colleagues fitted the static background as a 20×20 grid at 0.35-arcsec spacing (7×7 arcsec) jointly with the moving source, mapping it into each exposure through the known WCS and PSF. For a binary they added a second PSF and solved the fluxes linearly at each trial geometry. From about 25,000 measurements, two systems showed a companion at S/N ≥ 5 in multiple images: the known 2014 LQ28 and the new 2013 RJ124, with mutual semimajor axes of about 32.9 and 34.9 thousand km, periods of about 1,470 and 1,850 days, and system masses of about $1.4\times10^{18}$ and $1.0\times10^{18}$ kg (Bernardinelli et al. 2023).

Light curves and mutual events

Photometry finds companions in two ways. Detached binaries show mutual eclipses and occultations on top of rotation; contact binaries show light curves that no single convex body can produce. The front end is calibrated differential photometry, light-time correction, removal of geometric trends and a period search. Two similar minima can put the strongest periodogram peak at half the true period, so rotation and events must be fitted jointly, not one after the other.

Lucy’s Dinkinesh campaign is the worked example. From 267 images, an iterative two-component Fourier model separated the primary (amplitude about 0.25 mag, spin period about 3.7387 h) from Selam (about 0.82 mag). Eighteen discrepant points were excluded, and the remaining dips were identified as mutual eclipses using the encounter geometry. The spacecraft photometry alone gave Selam’s period as $52.44\pm0.14$ h, consistent with the ground-based 52.67 h (Levison et al. 2024). Resolved images and photometry together made the case. Photometry alone would have been an interpretation.

For contact-binary candidates, the method compares ellipsoids, Roche-shaped binaries and general shapes under one photometric likelihood. Audrey Thirouin and colleagues measured a 17.43-h period and a $0.70\pm0.07$-mag amplitude for Logos, which already has a resolved moon, Zoe. Their rendering models favour a close or contact pair inside Logos, about 143 km apart, making the system a likely triple (Thirouin et al. 2025). For 2001 XR254, ten nights of 4.3-m Lowell Discovery Telescope photometry between 2021 and 2026 gave an 11.17-h period, a 0.42-mag amplitude and a sharp minimum suggesting a possible inner pair (Thirouin et al. 2026). Both are inferred architectures, not resolved images. Forward modelling a mesh of $V$ facets at $T$ epochs for $Q$ trial shapes costs at least $O(QTV)$, a natural fit for GPU ray tracing. But no amount of rendering supplies viewing angles that were never observed.

Mutual events can also be predicted: Benjamin Proudfoot, Will Grundy and Darin Ragozzine propagate orbit posteriors (500 realisations per event) to forecast event seasons for Altjira, Logos–Zoe, Huya and others, often with large timing uncertainties (Proudfoot et al. 2026).

Stellar occultations

A satellite’s shadow can pass well away from the primary’s predicted track, and earlier or later than the primary event. Campaigns therefore need a “fence” of stations across the shadow path, spaced at most about half the target’s diameter, and recordings long enough to cover plausible moon positions. For known TNO binary orbits and an assumed 20 km s$^{-1}$ shadow speed, observing for 15 minutes covers both sides of the orbit for 65% of systems, 30 minutes for 73%, and one hour for 92% (Braga-Ribas et al. 2025). Those fractions describe the known-orbit sample. And recording longer does not help if no station lies under the moon’s path.

The close binary 2014 WC510 was revealed by a December 2018 occultation. Its components are 349 km apart, about 13 mas on the sky, less than a third of a WFC3 pixel (Leiva et al. 2020, as summarised by Porter et al. 2024). For Huya, three single-chord satellite detections in 2021 and 2023 gave precise positions and a lower limit of 165 km on the moon’s diameter, and a system density of $1073\pm66$ kg m$^{-3}$ under the adopted shape assumptions (Rommel et al. 2025).

The event search itself is cheap: prefix sums test $D$ trial durations over $N$ samples in $O(ND)$. What limits these campaigns is geometry: the star, the prediction, and where the stations stand.

Gaia wobble searches

The Gaia search is the largest statistical workload in the subject. First fit each asteroid’s heliocentric orbit and form along-scan residuals. Because the scan angle changes, a fixed projected sinusoid is a reasonable model only over short windows; Liberato et al. use windows of at least ten transits within at most ten days. A transit lasts about 40 s and holds up to nine CCD measurements, combined by inverse-variance weighting into one residual $y_k$ with uncertainty $\sigma_k$. A systematic error common to the whole transit does not average down with the number of CCDs, and the 2026 analysis models it separately (Liberato et al. 2026).

In each window, the test compares a constant against a constant plus sinusoid,

$H_0:\ y_k=\mu+\epsilon_k,\qquad H_1:\ y_k=\mu+a\cos\omega t_k+b\sin\omega t_k+\epsilon_k,$

with a generalised Lomb–Scargle periodogram (computed with the nifty-ls library). The significance of the maximum over the frequency grid is calibrated with 10,000 Monte Carlo noise realisations per window, using the actual timestamps and uncertainties. The eligible data were 47,896 objects in 60,152 windows over the 66-month FPR interval. For objects with several windows, the p-values are combined by Fisher’s method and by the minimum-p method, and Benjamini–Hochberg selection is applied at a deliberately permissive 75% target false-discovery rate. That yields 1,448 statistically selected objects, of which the authors expected roughly a quarter, about 362, to be real. Checks on implied density and separation, consistency across windows and rejection of special cases leave 343 candidates in 410 windows, including nine known or strongly suspected binaries. In noise-only control simulations the typical number of detections was 88% lower than in the real data. Another 45 objects show repeated trends that could indicate wide companions (Liberato et al. 2026).

Two cautions apply. FDR control on each selection does not carry over to their union, as the paper notes. And physical plausibility does not prove a moon: Lutetia’s detected wobble period of about 8.12 h matches its rotation, and the Rosetta flyby excluded a satellite large enough to cause it, so the wobble comes from the body’s shape. The 343 are candidates, with no demonstrated contamination rate. The Monte Carlo alone amounts to $60{,}152\times10^4\approx6\times10^8$ simulated window series. No hardware or run time is reported. This workload returns in the last section.

Thermal excess

A final method needs no image of the companion at all. Two unresolved bodies with similar surfaces emit like one body of diameter $D_{\rm th}$ with $D_{\rm th}^2\simeq D_1^2+D_2^2$. If an occultation measures $D_1$ directly, a thermal diameter larger than $D_1$ implies a hidden companion. Gómez-Limón and colleagues combined Herschel and Spitzer data with occultation sizes for ten objects, four of them known binaries for validation, in a Bayesian model assuming equal albedo and beaming for both components (128 emcee walkers, 500 burn-in steps). They inferred a nonzero companion size for three of the known binaries and for three unconfirmed systems: 2002 TC302, 2002 KX14 and Máni, with satellite diameters of $272^{+114}_{-92}$, $251^{+65}_{-57}$ and $534^{+58}_{-54}$ km (Gómez-Limón et al. 2026). A posterior mode away from zero is not a Bayes factor, and unequal surfaces, shape or rings can mimic the excess. These are follow-up targets, not moons.

Landmark results

Limitations and open problems

Confusion, not photon noise, sets the false-positive rate. A faint galaxy looks like a short tail. A neighbouring star changes sector ranks. Tracking jitter, PSF ellipticity and detector effects mimic blended companions. Rotation can masquerade as eclipses, and shape can produce astrometric wobble. Null controls must match the target in motion, brightness, crowding, seeing, detector position and processing history (Ferellec et al. 2022; Hsieh et al. 2015).

No single completeness number exists. Activity detectability depends on nucleus flux, coma fraction, tail length, width and angle, phase angle, background and epoch. Binary detectability depends on primary brightness, flux ratio, projected separation, PSF state, orbital phase, inclination and cadence. A non-detection, such as Makemake’s second visit, constrains one region of that space at one epoch. Hsieh’s 2009 comparison of his surface-brightness limit with a known weak tail remains rare: later surveys have more objects and less sensitivity calibration.

The physics of activity is uncertain. Recurrent perihelion activity supports sublimation but does not exclude rotational help; (6478) Gault’s recurrent activity was not tied to perihelion, so a perihelion-prioritised search selects against other mechanisms. Water-only models miss supervolatile activity, while extrapolated power laws overpredict distant brightness. Optical $Af\rho$ cannot be converted to water production in general, as 457P shows (Noonan et al. 2026).

The close-binary population is barely measured. SSOLS’s recovery limits leave room for many faint, compact pairs that New Horizons and occultations show exist. Wladimir Lyra points to a dearth of measured binary masses around $10^{19}$–$10^{20}$ kg, near absolute magnitude $H\approx4$–5. That could reflect formation history, undetected companions, or the difficulty of measuring mutual orbits; it is not proof of an empty mass interval (Lyra 2025).

Statistics near threshold are fragile. The Gaia study tested its own confidence intervals. Amplitude intervals approach 95% coverage only for wobbles above about 1 mas, and period intervals reach about 90% coverage only above about 2 mas, while most selected amplitudes are 0.3–1 mas (Liberato et al. 2026). Scores reported as percentages (SSOLS’s 95%, a volunteer’s 0.473) are rankings until calibrated against independent positives, negatives and injections.

Where the room is

Both problems share a structure that favours the algorithm designer. Once the orbit is known, each test is a small linear model on a small cutout, and the expensive parts are repetition (millions of objects, thousands of null simulations) and I/O. Nothing here is asymptotically hard. The gains come from not redoing work, from statistics that preserve information, and from moving the human out of the inner loop. These are proposals drawn from the research dossiers. None has a published, controlled speedup at equal sensitivity. Chapter 42 puts them alongside the book’s other open problems.

Read each exposure once. An archive activity search like HARVEST touches each image for many asteroids. The efficient design reads an exposure once, builds its PSF, background and source model, and scores every asteroid cutout in it as one batch. A compact coma and tail basis evaluated with batched matrix products, and amplitudes solved in closed form, turn the science kernel into dense linear algebra. The data volume is the real constraint. As an illustrative estimate, $10^6$ cutouts of $64\times64$ float32 pixels occupy 16.4 GB, about 49 GB with variance and one working plane. At that scale, decompression and transfer can outweigh the scoring, and caching at the detector level matters more than a faster kernel. The same reuse applies to binary searches. For fixed templates the fit needs only $P_i^TC^{-1}d$ and $P_i^TC^{-1}P_j$, which can be cached across a grid of separations instead of recomputed per pair, with PSF-derivative templates as a cheap screen for the quadrupole signal.

Gate, then infer. If a cheap gate costing $c_0$ passes a fraction $f$ of cutouts on to an expensive fit costing $c_1$, the speedup is

$\frac{c_1}{c_0+fc_1},$

which reaches ten only if $f<0.1$ and $c_0\ll c_1/10$. The gate must be shown to preserve weak-activity completeness, measured as a function of morphology, with an audit sample of rejected cutouts. The Active Asteroids weighting cut candidates by a third. That is useful ranking, but nowhere near a validated order-of-magnitude gate.

Sum likelihoods along orbits, not detections. When a comet’s activity or a companion is marginal at each epoch, requiring a significant detection at each epoch throws information away. For activity with a known time profile $g_e$, accumulate signed matched-filter numerators and their normalisations along the orbit, and test change points or a small basis of production histories so that a brief outburst is not diluted by a long steady stack. For binaries, sum the pixel likelihood along a trial mutual orbit around the known primary. This is the relative-orbit version of Chapter 20 and of orbit-stacking in exoplanet imaging (Chapter 33). The template space grows, and so does the trials penalty, which must be calibrated the same way. Sum normalised likelihoods from native exposures, not residual images from which subtraction may already have removed the secondary.

Move humans to the hard cases. The Active Asteroids backlog (18 million thumbnails, 5,000 new per day, 800 retired per day) is a throughput mismatch that no faster kernel fixes. Machine scores can retire the obvious cases, active learning can send ambiguous ones to volunteers, and sequential vote aggregation can stop early on confident subjects. A randomised audit stream must be kept, because classifiers that share a systematic error do not confirm one another. Locally propagated ephemerides joined against footprints, as in the Rubin Ponder workflow, can replace repeated remote queries (Chandler et al. 2026).

Build the benchmark. The most valuable missing product is a survey-matched set of weak comae, tails and close binaries, with realistic inactive and single controls, end-to-end injections, pixel likelihoods for both hypotheses, and follow-up outcomes. Without it, a rise in activity rate or binary fraction can come from a deeper fit or a looser false-positive policy, and a tenfold faster pipeline at a changed threshold is not a tenfold better one.

Know which limits are physical. Photon noise, angular resolution, unfavourable scattering geometry, the degeneracy of an unresolved coma with its nucleus, the zero wobble of twin binaries, the absence of mutual events at bad orientations, and an occultation shadow that missed every station are limits of information. More computation cannot recover what the photons do not contain; Bennu’s in-situ particle activity lies far below any remote limit. PSF errors, lost diffuse emission, unnecessary resampling, repeated downloads, uncalibrated thresholds and follow-up latency are engineering and statistical limits. That second list is where the room is.

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