Shadows on the Earth: occultations

On 3 June 2013 the Centaur (10199) Chariklo, a dark body about 250 km across orbiting between Saturn and Uranus, passed in front of a star of magnitude $R=12.4$. The event had been predicted, and observers across South America were waiting for a single dip of a few seconds that would give the body’s size. Three Chilean sites saw it. But seven sites also recorded thirteen additional brief interruptions, a few seconds before and after the main event, and at the Danish 1.54-m telescope at La Silla a camera reading out ten times a second resolved each extra interruption into two narrow components. When the positions of those dips from every station were projected onto the sky, they fell on two ellipses: two flat rings, with radii near 391 and 405 km and widths of about 7 and 3 km, around a body that nobody had imagined could hold rings at all (Braga-Ribas et al. 2014).

Nobody imaged those rings. They were found by the light they took away from a star. That is the whole idea of an occultation: the star supplies the photons, the foreground body supplies the shadow, and a telescope on Earth records where and when the shadow passes. Because the signal is a loss of starlight rather than light reflected by the body, it does not fade with the fourth power of distance the way reflected light does (Chapter 2). An opaque body at 40 au removes the same fraction of a star’s light whether its albedo is 4% or 90%.

This chapter covers two experiments that share this physics but differ in almost everything else. In a targeted occultation, the occulting body is already known, its orbit predicts where its shadow will fall, and observers deploy across the predicted path. The result is a measurement of a known system: its silhouette, its rings, its moons, its atmosphere, and a very precise position. In a serendipitous (blind) occultation survey, telescopes stare at thousands of stars at tens of frames per second and wait for an unknown body, typically a kilometre across, to cross one of them. The result is usually a statistical statement about how many such bodies exist, because the body that caused a sub-second dip is never seen again.

Two experiments with one shadow

The difference between the two modes is worth stating carefully, because it decides what the word “discovery” means here.

A targeted campaign’s main uncertainty is whether the shadow will pass over the observers at all. Its trial factor is modest because the time and approximate place are predicted. A blind survey’s main uncertainty is whether a dip is real, because it searches billions of samples and its candidate events lie in the extreme tail of a noise distribution nobody fully understands. Targeted occultations are, in effect, a high-resolution imaging technique for known bodies. Blind surveys are a counting experiment for bodies that are otherwise invisible.

The physics of a shadow

Geometry: shadow speed and chords

Take a body at distance $d$ from the observer. Its shadow is cast onto the plane perpendicular to the line of sight through the observer (the “fundamental plane”), and it sweeps across that plane at the relative transverse velocity $v_\perp$. Near opposition, a distant body on a prograde circular orbit moves slowly compared with Earth, so the shadow speed is mostly Earth’s own orbital velocity $v_\oplus \approx 29.8$ km/s minus the body’s heliocentric velocity, which scales as $d^{-1/2}$ for circular orbits:

At 40 au this gives $29.8 \times (1-0.158) \approx 25$ km/s (Bickerton et al. 2009). Far from opposition the speed falls, and near quadrature (90° from the Sun) the line of sight is nearly parallel to Earth’s velocity and the projected motion becomes slow. We will come back to that, because it is a design choice for blind surveys.

An observer whose straight track across the fundamental plane passes at impact parameter $b$ from the centre of a circular body of radius $R$ sees the star disappear for

Each station records a chord: an ingress time and an egress time, which, projected through the known geometry, become two points on the body’s limb. Several chords from stations spread across the shadow trace out the silhouette. A 250-km body at 25 km/s gives a central event of 10 s. A 1-km body would give 0.04 s if geometric optics held, but at that size it does not.

The Fresnel scale

Light from a distant star reaching an observer behind an obstacle is diffracted. The natural length scale of near-field (Fresnel) diffraction is

the convention used throughout the occultation literature (Sicardy et al. 2024). At $\lambda = 0.55\ \mu$m and $d = 40$ au, $F = \sqrt{0.55\times10^{-6}\ \text{m}\times 5.98\times10^{12}\ \text{m}/2} \approx 1.28$ km. A shadow at 25 km/s crosses one Fresnel scale in 0.051 s. These two numbers, a kilometre and a twentieth of a second, set the scale of the entire blind-survey field: a body of radius comparable to $F$ produces a pattern that is not a crisp shadow but a set of fringes, and it is over in a fraction of a second (Nir et al. 2023a).

Because $F \propto \sqrt d$, the diffraction width grows slowly with distance: about 0.45 km at 5 au, 2 km at 100 au and 5 km at 600 au in visible light. The instrument adds its own blur: during an exposure of length $t_e$ the shadow moves $v_\perp t_e$. A 0.1-s exposure at 25 km/s smears over 2.5 km, already twice the Fresnel scale at 40 au.

Diffraction by a small opaque disk

For a circular opaque disk of radius $a$ illuminated by a point source, the scalar Fresnel field at the observer, normalised to the unobstructed field, is one minus the contribution of the blocked aperture:

This is Babinet’s principle in Fresnel form. Using the circular symmetry and measuring radii in units of $F$, so that $\rho = a/F$ is the body radius and $\nu$ the observer’s distance from the shadow centre, the angular integral gives a Bessel function and the field becomes

the form used by the W-FAST team (Nir et al. 2023a). It is worth checking the limits, because conventions differ between papers and a sign error is easy. At the shadow centre, $\nu=0$, the integral is elementary and gives $U = e^{i\pi\rho^2/2}$, so $I=1$: the bright central Poisson spot has exactly the unobscured intensity. For $\rho\to\infty$ the integral tends to $(i/\pi)e^{-i\pi\nu^2/2}$ and $U\to 0$: a large body casts a dark shadow. Far from the body, $U\to 1$. Any implementation should pass those three tests before it is trusted (Nihei et al. 2007).

Two features of this pattern govern blind surveys. First, when $\rho \lesssim 1$ the width of the pattern is set by diffraction, not by the body, while the depth of the dip scales approximately with the body’s area. A body half the size gives a dip about four times shallower but no shorter. Second, the pattern is wider than the body. Nihei and colleagues give an empirical width that interpolates between the diffraction-limited diameter of the first ring, $2\sqrt3 F$, and the geometric diameter $2a$, with stellar size added:

where $\theta_\star$ is the star’s angular radius (Nihei et al. 2007). For a point star and $a = 0.5$ km at 40 au, $W \approx 2[(2.22)^{3/2}+(0.5)^{3/2}]^{2/3} \approx 4.8$ km, so the event lasts about 0.19 s at 25 km/s. That is a planning number, not a template: the real waveform has to be computed.

The figure above shows what that computation gives. For a point source at a single wavelength, a 0.5-km body at 40 au produces a dip to about 64% of the stellar flux on either side of a central spot at full brightness, with bright fringes reaching 127%. A real star observed in a broad band behaves differently, for reasons covered next, and a camera integrating for 25 ms or 200 ms samples a smoothed and shallower version still. With 200-ms exposures, as in the original TAOS design, the central bin in this example dips to only about 90%.

Large bodies: the knife edge

For a body much larger than $F$, the limb is locally a straight edge. With $u = x/F$ the distance from the geometric edge (positive on the illuminated side), the intensity is the classical knife-edge result

where $C$ and $S$ are the Fresnel integrals. It falls to a quarter at the geometric edge, oscillates with fringes outside it, and goes smoothly to zero inside (Sicardy et al. 2024). For a narrow ring or a small body one must propagate the complex transmission of the whole aperture before squaring. Subtracting two knife-edge intensities, a tempting shortcut for a ring with two edges, throws away the interference between them.

The knife-edge width is not a resolution floor. If the edge profile is known and many photons sample it, the time of the edge can be estimated far more precisely than the Fresnel width. For photon arrival times with rate $n_\star f[(t-t_0)/\tau] + n_b$, where $f$ is the normalised profile of width $\tau$ and $n_b$ the background rate, the Fisher information on the edge time is

(Gomes & Bernstein 2025, Appendix B). The timing error falls as $n_\star^{-1/2}$, so a bright star can locate a kilometre-wide edge to a fraction of a kilometre. Resolving two separate features closer than $F$ is a different and harder problem.

Finite stars, bandwidth and exposure

A star is not a point. Its disk is incoherent, so the correct procedure averages the monochromatic intensity (not the complex amplitude) over the stellar surface, then over the photon-weighted passband, then over each exposure:

where the angle brackets denote the average over stellar surface positions $\boldsymbol\alpha$ weighted by surface brightness and over wavelength weighted by throughput and the stellar spectrum (Nir et al. 2023a). Each operation smooths the fringes. Stellar size matters more than intuition suggests: the projected stellar diameter at the occulter’s distance is approximately

(Sicardy et al. 2024). A solar-type star with $V=12$ and $V-K\approx1$, seen behind a body at 40 au, projects to about 0.9 km, comparable to the Fresnel scale. Because $F$ grows as $\sqrt d$ but the projected stellar radius grows as $d$, the stars that reveal fringes become rarer as the occulter recedes. Brightness alone is a poor criterion for choosing target stars: a bright giant can be so large that it smears the event more than diffraction does. The 2026 Ixion campaigns measured $\theta_\star = 0.670\pm0.010$ mas for their occulted giant star (Kilic et al. 2026). At Ixion’s geocentric distance of 39.26 au that star projects to a diameter of about 19 km, more than ten Fresnel scales.

A broad passband raises the photon count and lowers the noise but blurs the chromatic fringe structure. A narrow band keeps structure but loses photons. These are information trade-offs fixed by physics, not inefficiencies a faster code can remove.

Targeted occultations

From Uranus’s rings to Pluto’s air

The template for discovery by predicted occultation was set in 1977, when observations of a stellar occultation by Uranus revealed its rings. James Elliot, Edward Dunham and Douglas Mink led one team, with independent observations by Robert Millis and collaborators and by J. C. Bhattacharyya and M. K. Vainu Bappu. Neptune’s ring arcs followed from occultations in 1984. That story belongs to Chapter 4, and the lesson carries forward: an occultation aimed at a planet found something nobody was looking for (Sicardy et al. 2024).

Pluto supplied the next. Noah Brosch’s observation on 19 August 1985 suggested an atmosphere, and the event of 9 June 1988 established it. William Hubbard, Donald Hunten and their Tasmanian colleagues recorded it with the Hobart 1-m telescope and an EMI photomultiplier sampled at 1 kHz, averaged to one-second points. The star faded gradually over about 115 s rather than snapping off. Their first fits assumed methane; later work showed the atmosphere is dominated by nitrogen (Hubbard et al. 1988). Occultations then became a barometer. Events in 2002 showed that the pressure had risen since 1988, even though Pluto was receding from the Sun, and Meza and collaborators later reanalysed twelve events from 1988 to 2016 with a single consistent atmospheric model (Meza et al. 2019).

The figure shows that reanalysis. The pressure at radius 1215 km, which the light curves constrain directly, roughly tripled between 1988 and 2015. The surface pressure is an extrapolation through an assumed temperature profile, and Meza and colleagues warn that its quoted errors omit that systematic. A 2020 occultation observed from Devasthal in India gave a surface pressure of $12.23^{+0.65}_{-0.38}$ μbar, consistent with a plateau since 2015 (Sicardy et al. 2021). That paper explicitly disagrees with some reported rapid declines, which is a reminder that pressure trends are only as good as the consistency of the retrievals.

Beyond the Pluto system, the first trans-Neptunian object (TNO) occultation actually observed was by 2002 TX300 on 9 October 2009 (Elliot et al. 2010). The campaign put 21 telescopes at 18 stations across 5920 km. Seven stations lost to weather and nine saw nothing. Only two recorded the event: the 2-m Faulkes North telescope on Haleakala and a portable 0.36-m telescope at Mauna Kea’s mid-level station. Their chords gave a mean radius of $143\pm5$ km. The ratio of stations deployed to chords obtained is typical, and it is the defining constraint of the method. By 2019 a review counted 43 occultations by 22 TNOs and 17 by five Centaurs, excluding Pluto and Charon (Ortiz et al. 2019).

Rings become a population

Chariklo, described in the opening, was the first. The coherence of the dips across stations was what made the case: an isolated glitch at one site or an arbitrary set of jets could not place matching features on the same two ellipses as seen from different places (Braga-Ribas et al. 2014). The ring orientation also explained why Chariklo had dimmed and its water-ice absorption had weakened in earlier years, as the rings turned edge-on. Small shepherd satellites were proposed to confine the rings, but no such satellites have been found.

On 21 January 2017 twelve telescopes at ten observatories caught the dwarf planet Haumea. Ortiz, Santos-Sanz, Sicardy and colleagues found a projected limb of $1704\pm4$ by $1138\pm26$ km, larger than expected, and additional dips compatible with a ring of radius $2287^{+75}_{-45}$ km if circular, about 70 km wide, lying in Haumea’s equator and aligned with the orbit of its moon Hiʻiaka (Ortiz et al. 2017). Combined with rotational photometry, the silhouette revised Haumea’s shape, albedo and density.

Quaoar was a retrospective discovery. An Australian observer’s ring detection in 2021, with a ring independently suspected in 2019, led Bruno Morgado and colleagues to find secondary events in observations from 2018 to 2020, including one from the CHEOPS space telescope. Four events fixed a ring near 4100 km, about 7.4 Quaoar radii, far outside the classical Roche limit of about 1780 km for 400 kg m⁻³ material (Morgado et al. 2023). The Roche limit is where a fluid satellite is torn apart by tides, which is not the same as the distance beyond which colliding ring particles must accrete. Restitution in collisions, velocity dispersion and resonances all matter, and a ring outside that boundary is a puzzle for ring dynamics, not evidence against gravity. A 9 August 2022 occultation observed with Gemini North and CFHT gave nine body chords and a second ring, Q2R, at $2520\pm20$ km if circular and coplanar with the first (Pereira et al. 2023).

Converting a ring’s dip into a physical optical depth needs care. Let $p'$ be the observed fractional loss of stellar flux and $B$ the ring opening angle. The apparent optical depth is $\tau' = -\ln(1-p')$, and the naive normal optical depth would be $|\sin B|\,\tau'$. But for particulate rings observed from far away, diffraction by individual particles throws light out of the beam and roughly doubles the effective extinction. In the regime adopted by Morgado and colleagues the normal optical depth is $\tau_N = \tfrac12|\sin B|\,\tau'$ (Morgado et al. 2023). Even then, the equivalent width $\int \tau_N\,dr$ does not give a ring mass without a particle model.

JWST has since observed occultations without atmospheric scintillation. Its Chariklo event of 18 October 2022 found a mean normal opacity of $0.431\pm0.012$ for the inner ring, against $0.303\pm0.028$ from the historical ground-based sample, while the outer ring was weak in the infrared (Santos-Sanz et al. 2025). Either the ring changed or its opacity depends on wavelength, and the data cannot yet decide. A JWST Quaoar occultation, dated 28 August 2024 in the preprint’s text, placed a 3σ upper limit of 1 nbar on a global methane atmosphere (Proudfoot et al. 2025). (That preprint’s Table 1 labels its timestamps 2025, an internal inconsistency to check against the journal version.)

Moons, double stars and atmospheres

An unexplained short dip near a body is tempting evidence for a satellite. It may instead be a double star. On 7 March 2017 an occultation by Vanth, the large moon of Orcus, produced events from Vanth crossing two different stellar components. Speckle imaging later resolved the star as a double, and two chords plus a nearby negative gave Vanth a diameter of $443\pm10$ km if spherical (Sickafoose et al. 2019). Satellite inference usually combines occultations with imaging. Rommel and colleagues combined three single-chord observations of Huya’s moon with HST and Keck astrometry to give a minimum satellite diameter of 165 km and a system density of $1073\pm66$ kg m⁻³ that depends on assumed volumes (Rommel et al. 2025a). On 16 March 2025 the IRTF caught both Haumea and its moon Namaka, giving an $83\pm2$ km chord across Namaka and therefore a lower bound on its diameter (Rommel et al. 2025b). Satellites as a discovery problem are the subject of Chapter 27.

Atmospheres are detected by refraction rather than absorption. A radial gradient in refractivity bends rays by an angle $\alpha(r)$ that depends on the density profile, spreading neighbouring rays apart and dimming the star gradually. With refractivity $\nu(r) = K(\lambda)\,n(r)$ for number density $n$,

maps a ray at closest approach $r$ to the shadow plane at radius $z$, and the flux in a ray branch is $\phi = |r/z|\,|1-\Delta\,d\alpha/dr|^{-1}$ (Sicardy et al. 2024). Near the shadow centre the limb’s curvature focuses rays into a central flash. The bending profile gives density by Abel inversion or forward ray tracing, and hydrostatic balance plus the ideal gas law give pressure and temperature, but only after one assumes a composition and an upper boundary temperature.

In 2026 Ko Arimatsu and collaborators reported the first atmosphere on a TNO other than Pluto (Arimatsu et al. 2026). On 10 January 2024, 2002 XV93 occulted a star observed by the Tomo-e Gozen camera on the 1.05-m Kiso Schmidt in Japan, and the star faded over about 1.5 s. Diffraction would give a transition of about 0.05 s and the stellar disk about 0.004 s, so neither explains a 1.5-s fade. Adding an atmosphere improved the best-fit $\chi^2$ from 89.2 for 30 degrees of freedom to 28.7 or 28.5 for 29. The surface pressure is $124^{+15}_{-12}$ nbar for pure methane or $177^{+18}_{-19}$ nbar for nitrogen. The composition is unknown, and whether the gas is a transient from an impact or supplied from inside is open. A pressure time series would tell.

Predicting the shadow

Everything in a targeted occultation depends on knowing where the shadow will fall. In the tangent plane, the shadow coordinate is approximately

for observer-body distance $\Delta$, star and body angular positions $\boldsymbol\theta_\star$ and $\boldsymbol\theta_b$, and the observer’s projected position (Gomes-Júnior et al. 2022). An angular error becomes a distance on Earth:

At 40 au, 1 mas of error moves the predicted shadow by 29 km. A 100-km body casts a 100-km shadow, so to place observers usefully one needs both the star and the body at the few-mas level, and for a 30-km body like Arrokoth, at about a milliarcsecond. Gaia changed this. Its star positions reach about 10 μas, and proper motions about 20 μas per year, for bright stars at the catalogue epoch (Gomes-Júnior et al. 2022). Since then the dominant error is usually the body’s orbit. The star’s full covariance, including proper motion and parallax, still has to be propagated to the event epoch, and the covariance on the path combines

under the simplifying assumption that star and body errors are independent. They are not, if the orbit was fitted against the same stars.

Orbits for occultation work come from dedicated fits. NIMA (Numerical Integration of the Motion of an Asteroid), used by the Lucky Star collaboration led by Bruno Sicardy, refits orbits as new astrometry arrives, and a last-minute update can decide whether a campaign succeeds (Desmars et al. 2015). There is a virtuous loop: each successful occultation yields a position accurate to milliarcseconds, which improves the orbit and the next prediction. Rommel and colleagues derived such positions for four Centaurs and nineteen TNOs and showed how they tighten later ephemerides (Rommel et al. 2020). Orbit determination itself is treated in Chapter 13.

The prediction search is a spatial cross-match between body ephemerides and a star catalogue:

fit orbit and covariance from calibrated body astrometry
for each ephemeris segment:
    query Gaia cells touched by Earth + system + uncertainty tube
    for each candidate star:
        propagate stellar position and covariance to event epoch
        minimise apparent separation with light-time-consistent ephemeris
        if an Earth (or spacecraft) intersection remains possible:
            project shadow and covariance onto candidate stations
            save probability, duration, magnitude drop, provenance

The search radius must be the projected radius of Earth plus the system’s radius (including rings and moons) plus the uncertainty allowance, not the body’s angular radius. With $N_b$ bodies, $N_t$ ephemeris samples each and $N_\star$ stars, a naive all-pairs search costs $O(N_bN_tN_\star)$. A spatial index reduces this to roughly $O[N_bN_t(\log N_\star+k)]$, with $k$ the local candidates per query, plus refinement of the close passages (an estimate made for this book, not a measured scaling).

Deploying observers

Where should the telescopes go? If the cross-track position of the shadow centre is Gaussian, $Y\sim N(\mu,\sigma_y^2)$, a station at cross-track coordinate $y$ sees a circular body of radius $R$ with probability

This is the single-station view, and it is misleading on its own. All stations depend on the same random $Y$, so their outcomes are correlated: if the shadow shifts north, every southern station misses together. The useful objective is the probability of several well-separated chords, plus coverage of where rings and moons might be, given weather and access. Monte Carlo draws from the orbit posterior make that a practical calculation (Sicardy et al. 2024). A plan that only maximises the chance of one body chord will undersample satellites and broad rings.

Reducing the light curves and fitting the limb

Light-curve extraction for targeted events is ordinary high-cadence aperture or PSF photometry, divided by comparison stars, with careful time-stamping. The timing is the hard part on cheap equipment: exposure-start conventions, dead time, dropped frames and the latency between a GPS pulse and the detector all have to be known, because a 0.1-s timing error at 25 km/s is a 2.5-km limb error. The community uses a heterogeneous toolchain. Tangra and PyMovie extract light curves from video, PyOTE and ROTE fit ingress and egress times, and SORA (Stellar Occultation Reduction and Analysis) predicts events, models diffraction, star size and exposure, projects chords and fits limbs (Gomes-Júnior et al. 2022).

SORA’s search for the event in a light curve starts with a box-fitting scan over position, width and depth, and then fits a physical diffraction model. With prefix sums, all start times and $W$ widths can be scored in $O(NW)$ for $N$ samples. The full model is a likelihood

usually simplified to a diagonal $\chi^2$. A body that is itself bright dilutes the drop: if the star contributes rate $S_\star$ and the body $S_b$ to the unresolved flux, a total occultation only falls to $S_b/(S_\star+S_b)$ of the baseline, and subtracting the body’s mean flux does not remove its photon noise.

The chord endpoints are then projected into the fundamental plane and fitted with a limb, usually an ellipse with five parameters: centre (two), semimajor axis, axis ratio and position angle. The objective weights radial residuals $d_j$ by the projected timing error and an extra term for real topography:

The 2022 SORA paper explored this with a default of ten million random ellipses. That is a parameter-fitting trial count, not ten million attempts at a detection, and the current SORA code defaults to least squares and offers differential evolution as well, with CPU multiprocessing and no GPU backend in the inspected source (SORA repository). Negative chords matter too: a station that was observing and saw nothing excludes the silhouette from its track, but only if its sensitivity and timing coverage are recorded. With enough campaigns, the limb becomes a three-dimensional shape. The 2026 Ixion analysis combined eight campaigns, 51 observations and 30 positive chords into an area-equivalent diameter of $696.78^{+10.75}_{-8.87}$ km (Kilic et al. 2026).

Machine learning has entered at the light-curve stage. ODNet, a convolutional network classifying paired target and reference-star image snippets, was evaluated on 66 real observations (24 positive, 42 negative). It achieved 91.3% precision and 87.5% recall, against the legacy method’s 91.7% recall and 64.7% precision, and cut total analysis time from about twenty minutes to two to four (Cazeneuve et al. 2022). Those are results on one small dataset, not calibrated false-alarm rates.

Occultations and Planet Nine

Could an occultation find a distant planet? There are two very different routes. The direct one is to catch the planet’s own shadow. An illustrative planet of three Earth radii at 600 au subtends about 44 mas, and a central chord at 30 km/s lasts about 21 minutes. Diffraction is negligible at that size, and the body blocks light regardless of its albedo. But the shadow is still only some 40,000 km wide, the planet’s sky position is uncertain by many degrees, and the swept area, roughly $2\theta_R\ell$ for angular radius $\theta_R$ and track length $\ell$, contains few stars. Without a localised orbit this becomes a blind survey, and a single occultation does not yield an orbit. No such detection exists (Chapter 9).

The indirect route uses occultations as an astrometric instrument. Gomes and Bernstein propose a meridian array that would time millions of occultations of Gaia stars by known main-belt asteroids and Trojans. A distant planet of mass $M_X$ at distance $R_X$ in direction $\hat{\mathbf R}_X$ imposes, after removing the common acceleration of the Sun and planets, a tidal acceleration

on a body at heliocentric position $\mathbf r$. Fitting this quadrupole jointly with asteroid orbits, nongravitational forces and the Kuiper belt’s distributed mass is a test for an unseen perturber of the kind discussed in Chapter 8. Their simulated survey, 200 telescopes of 40 cm spaced 5 km apart over ten years at a nominal capital cost near $15 million, would reach 5σ sensitivity to a 5 Earth-mass planet at 800 au over more than 90% of the sky, with about 1800 TNO occultations as a by-product (Gomes & Bernstein 2025). This is a forecast. It assumes Gaia performance, Rubin-quality orbits, sub-microsecond systematic timing and centimetre station positions, and it finds that adding telescopes eventually stops helping because of degeneracy with the unknown masses of large TNOs.

Serendipitous occultations: counting the invisible

Origins and an X-ray warning

M. E. Bailey asked in 1976 whether “invisible” bodies in the Solar System could be found by monitoring stellar occultations. Françoise Roques, Michel Moncuquet and Bruno Sicardy worked out the diffraction theory in 1987, and Roques and Moncuquet assessed small-body detectability in 2000 (as summarised in Nihei et al. 2007). Early ground-based searches showed the central difficulty immediately. A William Herschel Telescope campaign reported three candidates, but none could be securely placed in the classical Kuiper belt. Later work by Bickerton, Kavelaars and Welch turned the problem into survey design: diffraction templates, realistic scintillation spectra, sampling and explicit false-positive calculations. They recommended about 40 Hz near opposition, which captures roughly 95% of the modelled pattern’s power at classical-belt distances (Bickerton et al. 2009).

Rates and what a null result means

Let $n(a,d)$ be the number of bodies per square degree per unit radius and per unit distance. A star watched for time $T$ sweeps a strip of sky whose width is the effective cross-section of the diffraction pattern. With unit efficiency within impact parameter $b_{\max}$, the event rate from a population of surface density $\Sigma$ at distance $d$ is

where the factor converts the physical strip, divided by $d^2$, into square degrees (Pass et al. 2018). Real detection efficiency $\epsilon$ is not a top hat in $b$. It depends on body size, distance, impact parameter, star size, speed and exposure phase, and it is sometimes not even monotonic in $b$. So the proper quantity is an effective sky coverage for each star,

measured by injecting synthetic events into the real data, and the expected number of events is $\mu=\sum_s\int da\,dd\;n\,\Omega_s$ (Zhang et al. 2013).

With zero accepted events and negligible background, the Poisson probability of seeing none is $e^{-\mu}$, so the 95% upper limit is $\mu_{95}=-\ln 0.05\approx3$. Every null survey’s headline number is that 3 divided by its effective coverage. For a power-law size distribution $dn/da\propto a^{-q}$, the cumulative number is $N(>a)\propto a^{1-q}$, and a limit on $q$ requires a normalisation at large sizes, usually set by the reflected-light surveys of Chapter 5. With candidates and an uncertain background, a point-process likelihood $\mathcal L=e^{-(\mu_s+\mu_b)}\prod_j[\lambda_s(x_j)+\lambda_b(x_j)]$ is the principled formulation, though the historical surveys did not all use it. A frequentist false-alarm probability is not the probability that a candidate is false.

The detection statistic itself is a matched filter (Chapter 1). For detrended residuals $x$, a template $h=m-1$ and noise covariance $C$,

and for stationary noise with power spectrum $P(f)$ the numerator becomes $\sum_f \widetilde h^*(f)\widetilde x(f)/P(f)$. The trouble is that each survey scans this statistic over every star, every time sample and every template, so the threshold must control the maximum over $10^{10}$ or more trials, in noise that is neither Gaussian nor stationary. W-FAST’s own illustrative arithmetic makes the point: 1,000 hours on 1,000 stars at 25 Hz is about $10^{11}$ samples, which under independent Gaussian noise would give about 0.1 false detections at $z=7$ and 0.003 at 7.5 (Nir et al. 2023a). Correlated templates, red noise and rare atmospheric events all push the real tail outward.

TAOS: coincidence built into the instrument

The Taiwanese-American Occultation Survey, led by Charles Alcock, Matthew Lehner, Shiang-Yu Wang, Zhi-Wei Zhang and colleagues, began operating at Lulin Observatory in Taiwan in February 2005 with what became four 50-cm f/1.9 telescopes watching the same field (the fourth joined in August 2008). A CCD “zipper mode” read strips of the chip repeatedly with the shutter open, giving 5-Hz photometry of many stars at once. The defence against noise was coincidence: an event had to appear in at least three telescopes (Zhang et al. 2013).

TAOS did not assume a parametric noise distribution. After subtracting a rolling mean (33 points) and dividing by a rolling standard deviation (151 points), it ranked each telescope’s samples, with the lowest flux given rank 1, and combined $K$ telescopes as

for $M$ samples. If ranks were independent and continuous, $Z$ would follow a Gamma distribution with shape $K$. TAOS computed the exact discrete tail combinatorially, removed runs whose cross-telescope rank correlations failed a test at $p<0.01$, and set the threshold so that each tuple’s tail probability was $0.25/N_T$, with $N_T$ the number of simultaneous flux tuples searched. That allows about a quarter of a false event in the whole survey under the stated null, split across the window widths searched. To catch events longer than one sample, it also ranked moving averages over 3 samples. One instrumental trap was subtle: in zipper readout a satellite or aircraft crossing the column used for background estimation raised the background, and over-subtraction produced a negative flux excursion in the star, simultaneously in several telescopes. TAOS stored and analysed the background columns to veto these.

The seven-year analysis, from 7 February 2005 to 8 September 2011, covered 1,159,651 star-hours, of which 292,514 had all four telescopes, and found no events. Its limit on the size distribution slope was $q<3.34$ to 3.82 at 95% confidence, depending on the normalisation (Zhang et al. 2013). The successor TAOS II, at San Pedro Mártir in Mexico, is designed with three 1.3-m telescopes to monitor more than 10,000 stars at 20 Hz. Its real-time pipeline is projected to handle about 20 billion photometric measurements per night (Huang et al. 2021). These are design figures. No TAOS II population result had been verified as of October 2026.

HST: guiding telemetry as a survey

Hilke Schlichting, Eran Ofek, Re’em Sari and collaborators noticed that the Hubble Space Telescope already ran a fast photometer on thousands of stars: its Fine Guidance Sensors (FGS), which count photons from guide stars at 40 Hz to keep the telescope pointed, free of atmospheric scintillation. In about 12,000 star-hours at low ecliptic latitude, accumulated over four and a half years, they found one candidate at latitude about 14°, fitted by a body of radius $520\pm60$ m at $45^{+5}_{-4}$ au assuming a circular orbit. Bootstrap resampling gave about a 2% chance that random noise would produce a comparable event somewhere in the dataset (Schlichting et al. 2009).

A 2012 follow-up searched 19,500 more star-hours and found a second candidate of radius $530\pm70$ m at $35\pm9$ au, again assuming a circular orbit (Schlichting et al. 2012). The search fitted point-source templates, integrated over 400 to 700 nm with impact parameters from 0 to 1 Fresnel scale in steps of 0.2, as $m_i=m[1+A(l_i-1)]$, with event time and duration scanned over values allowed by HST’s, Earth’s and the body’s motions. The candidate threshold was $\Delta\chi^2>63$, expected to give about 0.5 random triggers. The candidate scored $\Delta\chi^2=71.9$. A Keck spectrum of the background star estimated its angular size.

The expensive part was calibrating that tail. The authors removed the event, resampled the data and reran the search on 3.1 million simulated star-hours, about 161 times the real low-latitude exposure. Eight simulated events reached $\Delta\chi^2\ge71.9$, giving about a 5% survey-wide false-alarm probability. That calculation took about 2,300 CPU-days (Schlichting et al. 2012), the most concrete compute figure in this field. The vetting was equally instructive. Telemetry anomalies and pointing excursions generated many triggers, and a high-latitude control sample contained an unexplained occultation-like event at latitude 81.5°, which the authors argued was not a KBO. Combining the two candidates gave $N(a>250\ \text{m})=1.1^{+1.5}_{-0.7}\times10^7$ deg⁻² near the ecliptic. If small bodies are instead spread uniformly over latitudes ±20°, the same two events give $4.4^{+5.8}_{-2.8}\times10^6$ deg⁻², a reminder of how much the population model matters.

OASES: two amateur telescopes

Ko Arimatsu and colleagues built OASES (Organized Autotelescopes for Serendipitous Event Survey) from commercial parts: two 279-mm telescopes with CMOS cameras on Miyako Island, Japan, recording a common field at 15.4 Hz, separated by about 39 m (Arimatsu et al. 2017). On 28 June 2016 at about 12:56:05 UT, both recorded a dip in a $G=12.1$ star at ecliptic latitude +0.2° (Arimatsu et al. 2019).

The search selected about 60,500 simultaneous star-hours, or 3.35 billion measurements per system, from about 130,000 star-hours collected. A rank statistic on smoothed fluxes, followed by flat-light-curve tests, produced 2,431 candidates. Re-measuring with larger apertures removed PSF-induced false dips, a per-system significance cut at $p<3.3\times10^{-4}$ left ten, and image inspection and agreement between the two systems left one. Its per-system probabilities under a flat model were $5.2\times10^{-5}$ and $1.4\times10^{-5}$, combining to $1.6\times10^{-8}$. With duration and coincidence requirements, the chance-coincidence probability per tested opportunity was about $9\times10^{-12}$, but multiplied over all the opportunities searched, the expected number of such false events was about 0.03. The survey-wide false-alarm probability is therefore about 3%, not $10^{-11}$. An empirical check formed about $4.8\times10^{14}$ pairs of streams from measurements within two hours of the event and recovered a similar probability. The fitted body has radius $1.3^{+0.9}_{-0.1}$ km at $33^{+17}_{-3}$ au (conditional on a circular orbit and the estimated star size), implying $N(a>1.2\ \text{km})=5.5\times10^5$ deg⁻² for the adopted size distribution. The authors acknowledge that a sufficiently fast atmospheric event could evade their false-positive calculation, because 39 m of separation removes ordinary scintillation but does not prove that the rarest excursions are independent.

OASES was upgraded in 2024. A commissioning test with 144-ms exposures at about 6.95 frames per second monitored about 9,300 stars above light-curve signal-to-noise 5, 2,300 above 10 and 400 above 15 (Arimatsu 2024). Those are star counts, not event rates: at 144 ms the shadow of a sub-kilometre body is not resolved.

Colibri: a cheap trigger and an expensive check

Emily Pass, Stanimir Metchev, Peter Brown and Steven Beauchemin built Colibri’s pipeline around a hierarchy (Pass et al. 2018). It calibrates image cubes, extracts light curves, convolves each with a Ricker (Mexican-hat) wavelet, picks the strongest dip, applies a 3.75σ threshold, compares surviving windows with physical diffraction kernels, and requires timing agreement between cameras. The early trials used two 5.8-cm lenses at about 17 Hz. In about 48,000 paired five-minute series, about 4,000 simultaneous star-hours at high ecliptic latitude, one candidate survived and was judged likely false because its fitted parameters disagreed between the cameras.

The injection results show where a hierarchy loses events, and they are not what intuition predicts.

Large bodies survive every stage. A 1.5-km-diameter body crossing centrally reaches the dual-detection stage only 4.4% of the time, mostly because the wavelet stage fails to pick the right dip, while the same body at 750 m impact parameter survives 22% of the time. Diffraction puts a bright central spot in the middle of a small body’s shadow, and an offset chord can produce a cleaner single dip. A first-stage filter tuned to one shape loses others. If $Q$ of the $N$ samples per star survive the wavelet stage, the search costs about $O(NL_w+QBL)$ for wavelet support $L_w$, $B$ physical kernels of length $L$, against $O(NBL)$ for exhaustive matching. The saving is real when $Q\ll N$, and so is the risk.

The final Colibri array at Elginfield, Ontario, has three 50-cm telescopes 110 to 160 m apart, with CMOS cameras at 40 frames per second covering about 1.43°×1.43° each, reaching about $G=12.1$ at temporal signal-to-noise 5, and streaming 1.5 GB/s (Mazur et al. 2022). The same paper reports about 20 ms per image just to convert the camera’s native format to FITS, nearly the 25-ms frame interval. At 1.5 GB/s an uninterrupted eight-hour night is about 43 TB (an estimate from the published rate, not a measured night).

W-FAST: whitening, stochastic banks and blind injections

The most complete and reproducible modern pipeline is W-FAST’s, by Guy Nir, Eran Ofek, Barak Zackay, David Polishook and Sagi Ben-Ami, and it borrows openly from gravitational-wave search methods (Chapter 37). The Weizmann Fast Astronomical Survey Telescope is a 55-cm telescope with a 7-deg² field, producing 10 to 25-Hz photometry from an image stream of about 6 Gbit/s. It keeps only 15×15-pixel cutouts around 1,000 to 5,000 bright stars (Nir et al. 2023b).

The pipeline buffers 2,000 measurements per star, processes overlapping 200-frame segments and searches only their central 100 frames. It estimates the noise power spectrum by Welch’s method with 100-frame windows, updated every 100 frames, and weights the data by its inverse before filtering, which suppresses the red noise from scintillation and tracking (Nir et al. 2023a). The template bank is built stochastically: random draws over stellar radius (0 to 3 Fresnel units), body radius (0.3 to 2), impact parameter (0 to 2) and speed (5 to 30 Fresnel units per second) are kept if no existing template overlaps them enough, stopping after 20,000 consecutive rejections. At 25 Hz this gives a coarse bank of 32 templates with minimum overlap 0.7 and a full bank of 307 at overlap 0.95. At 10 Hz the counts are 18 and 507, so a slower cadence does not automatically mean fewer distinguishable templates. A coarse-bank score above 5 triggers the full bank, and a full-bank score above 7.5 makes a candidate. Template placement is the subject of Chapter 38.

for each calibrated batch:
    extract flux, centroid, PSF width, background, detector diagnostics
    update rolling buffers and empirical power spectra
    compute quality mask and usable exposure per star and time
    x = detrend(segment); xw = inverse_PSD_weight(x)
    coarse = correlate(xw, coarse_bank)
    for windows where coarse > 5:
        full = correlate(window, full_bank)
        keep candidates with full > 7.5
    inject hidden synthetic events into selected real light curves
    run the same recovery, quality cuts and blinded human vetting
    accumulate efficiency and effective coverage
jointly fit and diagnose surviving real candidates

Quality cuts on image motion, focus, bad pixels, timestamps, background and correlations between stars kept 1,070,837 of 1,756,333 star-hours (61%). Injections were blinded: synthetic events were multiplied into real light curves, and human vetters did not know which were real. In a demonstration with 6,220 injections, high-amplitude events were recovered at only about 30% of their theoretical white-noise signal-to-noise, because the correlated noise lives at the same frequencies as the signal. Whitening weights the data optimally under its model, but it cannot recover information the noise has buried (Nir et al. 2023a).

The 2020 to 2021 data, about $10^6$ star-hours in total, produced no occultations. Detection efficiency was 10% at radius 1.1 km and 50% at 2.0 km. The 95% limit is $N(a>1\ \text{km})\lesssim10^6$ deg⁻², and for a power law normalised to large (about 45 km) KBOs, $q<3.93$ (Nir et al. 2023b). The authors also tested the OASES result. The density implied by the OASES event, run through W-FAST’s selection function, predicts 7.8 detections. Seeing zero has Poisson probability $4\times10^{-4}$. That is a test of one population model against another survey’s coverage, not a false-alarm probability for the OASES dip. The paper also catalogues seven rejected occultation-like events and a larger set of flare-like events, some still unexplained.

CHIMERA: proving what was not there

CHIMERA, on the 5.1-m Hale Telescope at Palomar, took a different statistical route (Zhang et al. 2023). It observed the central 5′×5′ of the globular cluster M22, whose stars are distant and angularly small, for 63.1 hours over 24 nights, at about 33 Hz simultaneously in $g'$ and $i'$. Crowding made photometry the limiting problem. Even with image-specific PSFs and models of neighbouring stars, the light curves kept a correlation length of about four frames, 0.1 s, exactly the occultation timescale.

Instead of trying to prove that a rare dip was celestial, CHIMERA asked which occultations could be excluded. For each of 144 grid templates (four distances from 35 to 50 au, nine body diameters from 0.5 to 2.0 km, four impact parameters from 0 to 1.5 km), it computed the Mahalanobis distance $(x-m)^{T}C^{-1}(x-m)$ between template and data, calibrated against injections with a false-negative fraction of about 15.9%, and stepped through light curves in order of sensitivity until a limiting event prevented further exclusion. This gives a conservative rate limit without requiring a fully understood false-alarm tail: $N(D>1\ \text{km})\lesssim10^7$ deg⁻² at 95% confidence, where $D$ is diameter. The limiting events in the red channel look like photometric artefacts in the blue channel and the images. The paper estimates that about 5,000 stars should reach photon-limited signal-to-noise above 10, but only about 800 do in practice.

Archives and the quadrature trick

Space photometry avoids scintillation but integrates for a long time. Gaudi forecast in 2004 that a proposed Kepler mission might detect roughly 1 to 20 occultations over four years under its assumed integrations and scattered-disk model (Gaudi 2004), and Ofek and Nakar found Kepler’s yield of Oort-cloud events could range from zero to about 100 depending on the model (Ofek & Nakar 2010). Both were forecasts. If an event of duration $\tau$ lies inside an exposure $T\gg\tau$, its depth is diluted to roughly its equivalent width divided by $T$, its signal-to-noise falls as $T^{-1/2}$ once unresolved, and all the diffraction information that would give distance and size is gone.

TESS adds an onboard problem. The original instrument handbook describes grouping ten short pixel readouts, discarding the highest and lowest, and summing the rest to reject cosmic rays (Vanderspek et al. 2018). A genuine short dip can be thrown away as the lowest readout before any light curve exists. Measuring that loss requires injecting events at the pixel level and applying the onboard algorithm, not injecting dips into downloaded light curves. No verified Kepler or TESS KBO occultation count or limit was found for this chapter. Archival space photometry for moving bodies is covered in Chapter 23.

The opposite move is to observe where events are slow. Ofek, Spitzer and Nir point out that near quadrature the projected relative velocity is small, so events last longer, longer exposures suffice and fainter, angularly smaller stars become usable. They estimate one to three orders of magnitude more suitable stars for sub-metre telescopes, offset by a lower event rate per star, and discuss second-long integrations for Oort-cloud searches (Ofek et al. 2024). This is a design calculation, not a detection.

Landmark results

Sources: as cited in the text above.

The two halves of the table look very different. Targeted occultations have a steady record of secure, reproducible discoveries about known bodies, and the record is accelerating as Gaia improves predictions. Blind surveys have produced three published candidates, each with a survey-wide random-noise probability of a few percent, and several null results that bracket them. The serendipitous field has not yet produced a reproducible detection with an independently measured background.

Limitations and open problems

The rarest atmospheric events are not understood. A power spectrum describes the second-order statistics of scintillation, not the incidence of intermittent turbulence. Two telescopes tens of metres apart may be nearly independent most of the time yet share their rarest excursions. W-FAST’s catalogue of unexplained occultation-like and flare-like events is the best evidence that this tail exists (Nir et al. 2023b).

A short single-site dip does not give an orbit. Size, distance, impact parameter, star size, speed and exposure phase trade against one another. The published distances for the HST and OASES candidates assume circular orbits. Independent velocity information from long baselines, resolved chromatic fringes or multiple stations can help, but no catalogue-quality orbit follows from a blind event.

Completeness is a function, not a number. W-FAST’s efficiency rises from 10% at 1.1 km to 50% at 2.0 km. “Sensitive to kilometre bodies” hides that slope, and errors in star size, unresolved binary stars, exposure phase, quality masks and human vetting all belong in it.

Different methods weight the population differently. Occultation rates integrate along sightlines, crater counts weight impact probability and geological survival, and reflected-light catalogues weight albedo. Shannon and collaborators show that a more distant population could reconcile occultation and cratering inferences under an explicit dynamical model, which is a possible explanation, not proof that the candidates are real (Shannon et al. 2021).

In targeted work, alignment is the selection function. A perfect detector sees nothing if the shadow misses, if clouds cover the site, or if no observable star lies on the path. Positive-only catalogues distort ring frequency and binary rates. A silhouette is not a volume or a centre of mass, and in precision astrometry the photocentre, figure centre and barycentre must be tracked separately. An atmospheric fit at a reference radius is a measurement, while a surface pressure is an extrapolation. Ring evolution and wavelength dependence are confounded until the same ring is seen at several wavelengths at nearly the same time.

Where the room is

Both halves of the field are data-limited in ways no algorithm can fix: finite photons, finite stellar disks, bandwidth and exposure averaging, subframes discarded on board, and the rarity of the alignments themselves. Weather and the number of observers on the ground are the binding constraints for targeted campaigns. What computation can change is how much of the collected information survives into the statistic, and how cheaply the null hypothesis can be calibrated. Chapter 42 collects these opportunities across the book.

Photometry first. CHIMERA’s gap between about 5,000 potentially photon-limited stars and about 800 that achieve that quality in practice is the clearest target (Zhang et al. 2023). Batched GPU scene modelling of crowded tiles, fitting a spatially varying PSF and common motion and solving for all stellar fluxes as a sparse linear problem, attacks the actual loss. It does not guarantee a proportional gain in discoveries, since the recovered stars may still be scintillation-limited. Colibri’s 20-ms format conversion at 40 frames per second is a simpler engineering target: ingest the native stream, fuse unpacking, calibration and extraction, and keep a rolling buffer of cutouts with enough full-frame context around triggers to diagnose PSF and neighbouring-star anomalies (Mazur et al. 2022).

Template banks in the right metric. W-FAST’s 32 coarse and 307 full templates already form a hierarchy, and if triggers are rare the full bank is evaluated about ten times less often than an exhaustive search would require (an arithmetic ratio, not a measured end-to-end speed-up). Three ideas could go further. Banks can be placed in the noise-weighted metric $\langle h_i,h_j\rangle_P=\sum_f\widetilde h_i^*\widetilde h_j/P(f)$, shared across stars with similar spectra and sizes, rather than in raw Euclidean distance. A low-rank factorisation $H\simeq UV^{T}$ replaces $B$ convolutions with $r$ convolutions plus a matrix product that maps onto tensor cores. The rank needed at fixed recovery is unknown. And a conservative bound replaces the heuristic first stage: if $h=\widehat h+e$, then $|\langle x,h\rangle|\le|\langle x,\widehat h\rangle|+\|x\|\|e\|$, so a branch whose bound cannot reach threshold can be dropped without loss. Colibri’s injection funnel shows why this matters: a heuristic first stage loses central small-body events preferentially. Templates here last a few to a few tens of frames, so direct convolution, overlap-save FFTs and batched matrix products should all be benchmarked rather than assumed (Chapter 39, Chapter 40). No occultation equivalent of the fast dedispersion transform is known, because radius, impact parameter, star size and bandpass change the waveform shape, not just its position and stretch.

What faster code does not buy. In the small-body regime the dip amplitude scales roughly as $a^2$ at fixed width. A factor $g$ improvement in effective signal-to-noise lowers the radius threshold by about $g^{-1/2}$, and if $N(>a)\propto a^{1-q}$ the idealised yield rises as $g^{(q-1)/2}$. For $q=4$ and $g=2$ that is a factor of about 2.8 (an estimate from this simple model, which fails when finite stars or structured completeness dominate). Evaluating the same statistic faster does not change $g$ at all. Sensitivity comes from more observing time processed, a better noise and waveform model, better extraction, or a justified lower threshold at the same false-alarm rate. Any claim of a tenfold gain must hold the false-alarm rate and the injection-derived coverage fixed and report the whole reduction-plus-search time.

Geometry and scheduling. Observing strategy can outgain kernels. The quadrature argument trades event rate for one to three orders of magnitude more usable stars (Ofek et al. 2024). Multi-site designs need care too: a 100-m baseline at 25 km/s gives an along-track delay of at most about 4 ms, shorter than a 25-ms exposure, so compact arrays mainly supply coincidence, not velocity. Longer baselines measure velocity but reduce the fraction of events seen by every station. The objective to optimise is joint detection efficiency at fixed false-alarm rate, with a correlated-noise alternative in the likelihood.

Targeted occultations. The prediction cross-match between Rubin-era catalogues of millions of bodies (Chapter 15) and Gaia is a batched spatial-query problem well suited to GPUs, and its metric should be correctly predicted observable events at fixed miss probability, with calibrated covariances. The paper-era ten-million-ellipse fit is embarrassingly parallel, but gradients and sensible parameterisation shrink it more than hardware does. Atmospheric ray tracing at fixed refractivity basis is a reusable linear operator, the most natural tensor-core application in the subject, provided reduced precision cannot bias a pressure or invent a weak refractive edge. The largest return may be campaign planning that optimises expected information, allocating mobile stations between body, rings and possible moons under orbit and weather uncertainty. None of the targeted-occultation papers reviewed here reports an end-to-end GPU benchmark, so the order-of-magnitude case remains to be made.

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