How bright is a hidden world?

In January 2016 Konstantin Batygin and Mike Brown argued that an unseen planet shapes the orbits of the most distant Kuiper belt objects (Chapter 8). Within three months, three groups had submitted papers asking what such a planet would look like. Esther Linder and Christoph Mordasini evolved the interior of a 10-Earth-mass ice giant over the age of the Solar System. At 700 au their nominal planet had a radius of 3.66 Earth radii and an effective temperature of 47 K, and it would appear at visual magnitude 21.7. Sunlight alone would have kept it at about 10 K. The rest of its warmth was heat left over from its formation and still leaking out (Linder & Mordasini 2016). Nicolas Cowan, Gil Holder and Nathan Kaib pointed out that a Neptune-sized planet at 40 K and 700 au would be a source of about 30 mJy at 1 mm wavelength. That is bright enough for cosmic microwave background (CMB) telescopes, which map large areas of sky at millimetre wavelengths repeatedly. Its annual parallax of about five arcminutes would separate it from stars and asteroids (Cowan et al. 2016). Jonathan Fortney and colleagues modelled the atmosphere and found that methane might freeze out of it. The planet could then shine far more brightly near 3–5 μm than its temperature alone suggests (Fortney et al. 2016).

Three predictions led to three kinds of search: an optical search through survey archives, an infrared search through WISE, and a millimetre search through ACT’s CMB maps. None found a planet. The searches were not redundant. Each covered a different part of the space of things the planet might be. This chapter explains why. An undiscovered planet has no single predicted magnitude. What can be predicted is a joint distribution over radius, albedo, distance, phase, internal heat, atmosphere and observing geometry. Turning that distribution into a decision about where to look is the subject of this chapter. The same physics governs every search for small bodies, from main-belt asteroids to a world beyond Neptune.

Three handles on a dark body

An unseen body announces itself in three independent ways. It reflects sunlight, roughly in proportion to $pR^2/r^4$, where $p$ is its albedo, $R$ its radius and $r$ its distance. It emits heat, in proportion to $R^2B_\nu(T)/r^2$, where $B_\nu$ is the Planck function at its temperature $T$. And it moves across the sky, mostly because Earth moves around the Sun, by an angle that scales as $1/r$. Combining these can tell a distant planet from a much smaller nearby body and can break some, but not all, of the degeneracies between size, albedo and temperature.

The most important distinction is between a passive body, whose temperature falls as it receives less sunlight, and an internally luminous planet, whose cooling interior keeps its temperature nearly independent of where it is now. Only the second has thermal flux falling as $r^{-2}$ at a fixed wavelength over a wide range of distance. A passive body’s total thermal output falls as $r^{-4}$, just like its reflected light, because it can only re-emit what it absorbs. Its flux at a particular wavelength depends also on where that wavelength sits on its cooling spectrum.

Reflected sunlight

Keep the two distances apart

Let $r$ be the Sun–body distance in au, $\Delta$ the observer–body distance in au, $R$ the body’s radius, $p_\lambda$ its geometric albedo at wavelength $\lambda$, and $\Phi_\lambda(\alpha)$ a phase function equal to one at phase angle $\alpha=0$ (the Sun directly behind the observer). Then, relative to the Sun’s flux at 1 au,

The factor $r^{-2}$ is the dilution of sunlight on its way out. The factor $\Delta^{-2}$ is the dilution of the scattered light on its way back. The geometric albedo already contains the normalization of disk-integrated scattering, so an extra factor of $\pi$ here would be a mistake.

In magnitudes,

where the absolute magnitude $H$ is the magnitude the body would have at $r=\Delta=1$ au and zero phase. For a distant body $\Delta\simeq r$, so $m\simeq H+10\log_{10}r$ and the flux falls as $r^{-4}$. The standard conversion from $H$ to diameter in the V band is

The coefficient depends slightly on the adopted magnitude of the Sun. Vilenius and colleagues, for example, write the same relation with $m_{\odot,V}=-26.76\pm0.02$ (Vilenius et al. 2018).

A few rules follow directly. Doubling a body’s diameter brightens it by 1.505 mag. A tenfold higher albedo brightens it by 2.5 mag. Doubling the distance of a distant body dims it by 3.01 mag. At a fixed magnitude limit, the farthest distance at which a body can be seen scales as $R^{1/2}p^{1/4}$. A large uncertainty in albedo therefore matters less for how far a survey reaches than for how big an object it implies.

Albedo, the hidden variable

The same brightness can come from a large dark surface or a small bright one, so optical magnitude alone gives a size only through an assumed albedo. That assumption is often poor. Vilenius and colleagues measured the thermal emission of bodies in the Haumea collisional family with the Herschel and Spitzer space telescopes. Family members, coated in fresh water ice, have a median geometric albedo of $0.48^{+0.28}_{-0.18}$. Interlopers on similar orbits have $0.08^{+0.07}_{-0.05}$ (Vilenius et al. 2018). A single universal albedo for the outer Solar System would mistake a difference in surface composition for a difference in size by a factor of about 2.5.

Planets have their own range. Fortney and colleagues found that if methane condenses out of a cold planet’s upper atmosphere, scattering by hydrogen molecules could push the optical geometric albedo toward the pure-Rayleigh limit of about 0.75 (Fortney et al. 2016). In 2025 David Russell and Terry White took a different route. They used the measured masses and radii of cool exoplanets to argue that a planet of Planet Nine’s inferred mass is most likely a mini-Neptune of 2.0 to 2.6 Earth radii. With albedos from 0.333 to 0.465, that puts it at apparent magnitude 21.9 to 22.7 at an assumed aphelion of 630 au (Russell & White 2025). That is an extrapolation from exoplanets, not a measurement. Their sample was much more strongly irradiated than anything at hundreds of au, so a search should keep broader physical possibilities open.

Phase, rotation and colour

For an airless sphere that scatters like a Lambert surface, the phase function is

At quadrature ($\alpha=90^\circ$), $\Phi_L=1/\pi$, a loss of 1.24 mag. That matters for near-Earth asteroids and for directly imaged exoplanets. It hardly matters in the outer Solar System. From Earth, the phase angle of a body at distance $r$ can never exceed about $\arcsin(1/r)$: $1.43^\circ$ at 40 au and $0.082^\circ$ at 700 au. Planet Nine would always be essentially full.

Real surfaces are not Lambertian. Shadow hiding and coherent backscattering make many airless bodies brighten sharply near zero phase. Vilenius and colleagues discuss an enhancement of about 0.1 mag below roughly $0.2^\circ$ and measured phase slopes between about 0.01 and 0.1 mag per degree for several Haumea-family bodies (Vilenius et al. 2018). These are measurements of particular bodies, not a universal law, and an asteroid phase law should not be applied to a gas atmosphere.

Rotation changes the projected area and surface brightness, so a search whose exposures span several rotations sees a varying flux. It should model that variation or marginalize over it. The Dark Energy Survey’s joint analysis of trans-Neptunian objects treats absolute magnitude, colours and light-curve amplitude together for this reason (Bernardinelli et al. 2025). Colour also decides the best filter. The band with the most solar photons is not necessarily the band with the best signal-to-noise ratio, once the body’s spectrum, sky brightness, detector efficiency and seeing are included. Fortney’s methane-depleted planets might look blue, but that is a model possibility, not a guaranteed colour to search for.

The numbers

The table gives V magnitudes at opposition for five hypothetical bodies, with the maximum angular speed caused by Earth’s motion. All values are estimates from the equations above, with $\Delta=r$ and $\alpha=0$.

Two things stand out. An Earth-sized planet with albedo 0.3 stays brighter than $V=21.5$, the 50% depth of the Pan-STARRS1 Planet Nine search, only out to about 320 au, and it reaches $V\approx24.8$ at 700 au. A 1,000 km body with albedo 0.1 is already at magnitude 28 at 300 au and passes magnitude 30 near 500 au. Deep stacking can reach some of that territory. Single-exposure surveys cannot.

Heat

Equilibrium temperature

A body that absorbs sunlight must radiate the same power. For a sphere at a single uniform temperature, with Bond albedo $A_B$ and thermal emissivity $\epsilon$,

The Bond albedo is the fraction of all incident power reflected in all directions. It is not the geometric albedo. The two are related by $A_B=qp$, after weighting over the solar spectrum, where $q=2\int_0^\pi\Phi(\alpha)\sin\alpha\,d\alpha$ is the phase integral. Using one number for both is a further physical assumption. At 700 au with Linder and Mordasini’s Bond albedo of 0.31, the formula gives under 10 K.

Asteroids: radiometry with a fudge factor

Airless asteroids are not isothermal. They are hottest at the subsolar point and cold on the night side. The Near-Earth Asteroid Thermal Model (NEATM), introduced by Alan Harris in 1998, approximates the dayside as

with no emission from the night side. Here $\zeta$ is the angle from the subsolar point and $\eta$ is an empirical “beaming parameter” that absorbs the effects of thermal inertia, spin and roughness. With $\eta=1$ the subsolar temperature is $\sqrt2$ times the isothermal one, which matters a great deal on the short-wavelength side of the spectrum (Masiero et al. 2017).

Combining optical and thermal measurements gives two energy budgets for two unknowns. Reflected light measures roughly $pD^2$. Thermal emission measures roughly $(1-A_B)D^2$, which for a dark body is nearly $D^2$ alone. Together they give both diameter and albedo. This is how the Wide-field Infrared Survey Explorer (WISE) sized asteroids by the thousand. During its primary mission in 2010 it detected more than 150,000 asteroids and comets and discovered more than 30,000 new objects. In its third reactivated year as NEOWISE it gave NEATM diameters and albedos for 170 near-Earth objects and 6,110 main-belt asteroids, with near-Earth object diameters agreeing with independent measurements to within 26% at one sigma (Masiero et al. 2017). For warm, dark asteroids the thermal infrared is the better discovery band. Their optical albedo is low, and their thermal emission is strong.

Planets with their own heat

A giant planet carries heat from its formation, released as its interior cools and contracts. Its internal luminosity and effective temperature are related by

Far from the Sun $T_{\rm eq}$ is negligible and the planet’s temperature is set by its interior. How fast that interior cools depends on its composition and on how heat moves through it. Linder and Mordasini’s nominal planet reaches 47 K today. A variant with inhibited cooling reaches 33 K, and one with low initial luminosity 27 K. The V magnitudes of these variants change only a little, because V measures reflected light. Their mid-infrared magnitudes change enormously (Linder & Mordasini 2016). Fortney and colleagues add a warning. Standard adiabatic cooling models reproduce Neptune’s heat output but not Uranus’s, which is far lower. So their predicted temperatures are upper limits within their framework, and a cooling track that fits Neptune does not set a lower bound on the luminosity of another planet (Fortney et al. 2016).

Where in the spectrum to look

For an unresolved, uniform blackbody disk,

with $R$ and $\Delta$ in the same units. The peak of $B_\lambda$ falls at $\lambda_{\max}T\simeq2898$ μm K, so 40 K and 47 K bodies peak near 72 and 62 μm, in the far infrared. Figures should say whether they show the spectrum per unit wavelength or per unit frequency, because the two peak at different wavelengths.

How flux scales with distance depends on which side of the peak the observing band sits. In the Rayleigh–Jeans limit $h\nu\ll kT$, $F_\nu\propto R^2T\nu^2\Delta^{-2}$. For an internally warm planet with fixed $T$ this gives the gentle $r^{-2}$ law. For a passive body, $T\propto r^{-1/2}$, so its Rayleigh–Jeans flux falls as $r^{-5/2}$. On the Wien side, $F_\nu\propto r^{-2}\exp[-h\nu/kT(r)]$ falls much faster. There is no thermal advantage that holds independent of wavelength.

At 1 mm, $h\nu/k\approx14.4$ K, so even a 40 K spectrum is only roughly in the Rayleigh–Jeans regime. An exact blackbody calculation for Linder and Mordasini’s nominal planet at 700 au gives 17.3 mJy at 1 mm and 4.7 mJy at 2 mm (estimates), lower than Cowan’s 30 mJy, which scaled from Neptune’s spectrum for a Neptune-sized planet. The ACT collaboration made its own band-integrated forecast. A planet of 5 Earth masses, 2.94 Earth radii and 42.2 K at 500 au would give 2.3, 5.3 and 11 mJy in its 98, 150 and 229 GHz bands (Naess et al. 2021). Differences of a factor of two between such forecasts come from different assumptions and passbands. They are not errors.

At very low temperatures what a telescope measures is the planet’s emission minus the background it blocks. For an opaque blackbody in front of the CMB the contrast is $\pi(R/\Delta)^2[B_\nu(T)-B_\nu(2.725\,\mathrm K)]$. For ACT’s nominal planet this correction is 2.6%, 1.4% and 0.6% in its three bands. For a passive body only a few kelvin above the CMB it can dominate the signal.

The largest departures from a blackbody come from atmospheres. If methane condenses out, photons in the 3–5 μm window escape from deeper, hotter layers. In Fortney’s models the emission there exceeds a blackbody at the effective temperature by about twenty orders of magnitude. The planet’s total luminosity does not change. Its brightest W1 case is $W1\approx16.1$ for a 50 K, strongly methane-depleted model, and clouds can close the window (Fortney et al. 2016). The window is physically motivated. How bright it is remains uncertain, because the relevant low-temperature opacities are poorly known.

The table compares warm planets with a passive 1,000 km body. All values are blackbody estimates with unit emissivity. The passive body has Bond albedo 0.1 and is heated by the Sun and the CMB, and its 1 mm flux is the contrast against the CMB. The 40 K temperature of the compact model is illustrative, not derived from its radius.

At small distances the fixed-temperature entries are only normalizations, since sunlight would add heat there. At hundreds of au that correction is small for these warm models. At thousands of au the passive model is incomplete, because interstellar radiation and any residual internal heat would dominate the trickle of sunlight.

Motion: distance written on the sky

A distant body’s path across the sky is, to first order, a reflection of Earth’s orbit scaled by the inverse of its distance. Its angular position is

where $\boldsymbol\mu_{\rm hel}$ is the slow drift from the body’s own orbital motion and $\mathbf P_\perp$ is the observer’s position around the Sun, in au, projected onto the sky. The parallax term traces an ellipse once a year, with a semi-major axis of $206{,}265/r$ arcseconds. Its shape depends on the body’s ecliptic latitude. Earth’s orbital speed moves the line of sight at up to about $147.8/r$ arcseconds per hour. At 700 au that is a 295 arcsecond ellipse and at most 0.21 arcsecond per hour (estimates). Cowan and colleagues’ figure of about 0.1 arcsecond per hour is a half-year average, not the peak rate at opposition (Cowan et al. 2016).

Including the body’s own orbital motion, a body on a circular prograde orbit seen at exact opposition moves at

about 3.19 arcseconds per hour at 40 au and 0.203 at 700 au (estimates). Eccentricity and projection change this. Near the stationary points of the yearly loop the instantaneous motion says almost nothing about distance. The curvature over months is what pins it down.

Motion sets the longest useful exposure too. To keep the trail within, say, 0.7 arcsecond, an exposure at opposition can last about 13 minutes at 40 au and about 3.45 hours at 700 au (estimates). Slow motion allows long exposures, but it can defeat pipelines that need to see a body move within one night. In Pan-STARRS1 the usual within-night baseline is only 20 to 60 minutes, during which a very distant body moves about as much as its astrometric uncertainty (Holman et al. 2025).

The parallax law also has a quiet consequence for search design. The displacement is linear in $\gamma=1/r$, not in $r$. A bank of trial distances spaced evenly in $r$ wastes most of its templates on far distances, where neighbouring trajectories are almost identical. The tolerance argument of Chapter 1 gives the right spacing: $|\delta\gamma|\max|\mathbf P_\perp|\lesssim\delta\theta$ for an allowed positional error $\delta\theta$ in radians, and $\delta\mu\lesssim\delta\theta/T_{\rm span}$ for velocity over a time span $T_{\rm span}$. ACT already spaced its distance grid non-uniformly for this reason (Naess et al. 2021).

From brightness to a detection

A brightness is not yet a detection. The first algorithm in any search is a forward model. For each epoch it propagates the body and the observer, computes $r$, $\Delta$, phase and motion, evaluates the reflected and thermal spectra, and integrates them through the instrument. A photon-counting detector then expects

source electrons in exposure time $t$ through a telescope of area $A_{\rm tel}$, where the throughput $\mathcal T$ includes all losses and the detector’s quantum efficiency. With background $b$ and dark current $d$ in electrons per pixel, read noise $\sigma_{\rm read}$ and an effective PSF noise area $n_{\rm eff}$, a useful estimate is

Real difference images and millimetre maps have correlated noise and subtraction residuals, and the detection statistic should carry that covariance rather than this Poisson formula. Magnitude systems also need care. Linder and Mordasini’s Johnson and WISE magnitudes are Vega-based, and comparing them numerically with an AB magnitude without converting zero points is an error (Linder & Mordasini 2016).

To turn this into a selection function, as Chapter 16 describes, inject the predicted signal into the real data and run the whole pipeline:

for physical_orbit_sample in population_prior:
    for each real exposure (time, band):
        geometry = propagate(body, observer, time)
        spectrum = reflected_model(geometry, radius, albedo, phase)
                 + thermal_model(geometry, interior, atmosphere)
        counts   = integrate_throughput(spectrum, band, exposure)
        inject_trailed_PSF(counts, predicted_position, real_data)
    recovered = run_entire_discovery_and_confirmation_pipeline()
    record(recovered, physical_parameters, observing_conditions)

With $N_{\rm phys}$ physical samples, $E$ epochs and $L$ wavelength samples, the photometry costs about $O(N_{\rm phys}EL)$ once spectra exist. Solving an atmosphere or evolving an interior is a separate and often dominant cost. Cached spectra and vectorized band integration usually make this stage cheap compared with searching unknown trajectories.

The rewards of deeper optical integration are modest in distance. Stacking 100 equal exposures gains $1.25\log_{10}100=2.5$ mag (Chapter 1). Under the $r^{-4}$ law that extends the distance reach for a given body by only $100^{1/8}\approx1.78$. Under the $r^{-2}$ law of a warm planet’s thermal emission the same 2.5 mag gain would extend it by $10^{2.5/5}\approx3.2$. Deep stacking is still enormously valuable, because at a fixed distance it reveals much smaller bodies. But the comparison explains why thermal emission becomes attractive for a warm planet far out.

Brightness completeness is not orbit completeness. Chip gaps, masks, poor subtractions, cadence, minimum-detection rules and failed linkages can lose a body that is bright enough. The Dark Energy Survey analysis of Planet Nine simulated 100,000 reference planets. Of them, 11,709 crossed the wide survey footprint and 10,187 (87.0%) would have been recovered, ruling out about 5% more of the parameter space beyond what ZTF had excluded. It used measured exposure efficiencies but did not reprocess raw images with injected planets (Belyakov et al. 2022). A 2025 Pan-STARRS1 search by Matthew Holman and colleagues injected synthetic detections into the catalogue and modelled detection and linkage. It found 692 real objects, including 642 trans-Neptunian objects, but no new large planet. It predicted that 75,769 of 100,000 reference planets would have been recoverable (Holman et al. 2025). Catalogue injection measures what the catalogue pipeline preserves. It cannot speak for sources that never made it into the catalogue.

Predictions meet surveys

Each prediction of 2016 led to a search, and each search tested one part of the space of possible planets.

The WISE search came first. Meisner and colleagues built time-resolved coadds from about six million individual 3.4 μm (W1) exposures, 129,483 coadds in all, covering more than three quarters of the sky. They detected moving transients and tried to link them across survey passes. The bookkeeping was heavy. A prescreen enumerated more than 92 billion triplets and 275 million quadruplets of transients and called an orbit fitter on more than eight million possible linkages among about five million filtered transients. They found no valid linkage of five or more detections. At high Galactic latitude they were 90% complete at $W1=16.70$ and 50% at 16.86 (Meisner et al. 2017). That is close to the brightest W1 that Fortney’s methane-depleted models predicted. It was a test of one spectral possibility, not a limit on a planet of given mass. And most of the difficulty was combinatorial association, not photometry.

Optical archive searches tightened the optical branch. In 2024 Mike Brown, Matthew Holman and Konstantin Batygin searched Pan-STARRS1 to a 50% completeness of $V=21.5$. Together with the earlier ZTF and DES searches, this excluded 78% of their adopted reference population of Planet Nine orbits and masses (Brown et al. 2024). That percentage belongs to a specific physical and orbital prior. The Pan-STARRS1 analyses of Brown and of Holman use much of the same data and must not be added together. Holman and colleagues explicitly defer combining them because their efficiency models differ and their data overlap (Holman et al. 2025).

The far infrared, near a cold planet’s thermal peak, came last. In 2025 Terry Long Phan and colleagues paired sources from IRAS and AKARI, two all-sky far-infrared surveys taken 23 years apart. They used adopted detection floors of 0.2 Jy at 60 μm and 0.21 Jy at 90 μm and required fluxes, colours and separations plausible for a distant planet. Thirteen pairs passed the first selections and one survived image inspection, with fluxes of about 0.24 and 0.27 Jy (Phan et al. 2025). Amos Chen and colleagues instead used AKARI alone. They required repeat detections over hours and disappearance in a later pass, and excluded known stationary sources, cirrus and cosmic rays. Two candidates survived, with paired 90 μm fluxes of 0.61 and 1.62 Jy and of 1.27 and 0.51 Jy. The large changes within each pair and the sparse sampling mean no orbit can be determined (Chen et al. 2025). These are candidates needing follow-up, not detections.

Shadows and pulls

Two channels avoid reflected and emitted light altogether.

An occultation watches a background star for a brief dimming as a body passes in front of it. The body’s albedo does not enter. For a body much larger than the star’s projected size, the event lasts about $D/v_\perp$. A 2.3-Earth-radius planet crossing at an illustrative 30 km/s would cause a central event of about 16 minutes. At 700 au it would subtend only 0.058 arcsecond (estimates). Small bodies are governed by diffraction, with the Fresnel scale

setting the size of the shadow’s structure: about 1.2 km at 40 au and 5.1 km at 700 au for $\lambda=0.5$ μm (estimates). Bodies smaller than $F$ produce shallow diffraction patterns rather than clean dips (Sicardy et al. 2024). Occultations can find kilometre-scale bodies far too faint to image. They are not an efficient blind search for one planet, because a single planet covers a tiny solid angle and the events are rare (Chapter 25).

Gravity responds to mass, whatever the luminosity. A distant mass accelerates the Sun and the planets almost equally, so what can be measured is the tidal difference across the planetary system,

where $\mathbf x$ is a known body’s position relative to the Sun and $\mathbf n$ points toward the perturber. For five Earth masses at 700 au, $GM/r^3\approx1.7\times10^{-27}\,\mathrm s^{-2}$. Across 1 au that is a differential acceleration of about $2.6\times10^{-16}$ m s⁻² transverse to the perturber’s direction, twice that along it (estimates). The inverse-cube law is steep. A dynamical null result cannot be drawn on a size–distance plot without a mass–radius model and a full ephemeris fit, which Chapter 7 and Chapter 26 treat.

The regime map

The figure below draws, for each distance, the smallest body that a given channel could detect under stated assumptions. The optical lines assume an albedo of 0.1. For a fixed limiting magnitude the minimum diameter is $D_{\min}=1329\,r^2\,10^{-m_{\lim}/5}/\sqrt p$ km, rising as $r^2$. The thermal lines assume a body held at a fixed temperature, so they rise only as $r$. The 6 mJy limit near 150 GHz lies within ACT’s published range. The 0.21 Jy line is the AKARI catalogue floor that Phan and colleagues used. $V=21.5$ is Pan-STARRS1’s 50% depth, and 24.5 and 27 are assumptions for deeper surveys and stacks.

The crossings are where the methods trade places, and they move with the assumptions. For the compact 2.3-Earth-radius, 40 K planet with albedo 0.4, a uniform $V=24.5$ survey would reach about 1,050 au and a $V=27$ stack about 1,870 au. A 6 mJy limit at about 2 mm reaches only 356 au for the same planet and 618 au for Linder and Mordasini’s warmer, larger nominal model. The 0.21 Jy floor at 90 μm reaches about 383 and 826 au (all estimates). The gentler distance law of thermal emission does not by itself make thermal surveys the winner. At several hundred au a deep enough optical search beats current thermal archives for a compact, fairly bright planet. Beyond a few thousand au a warm planet is much brighter thermally than optically, but both signals may be below any wide-area survey. A passive 1,000 km body is already far beyond both at several hundred au: $V\approx31.6$ and under a microjansky at 1 mm at 700 au. There, occultations probe populations of small bodies and gravity can reveal enough mass, if there is enough.

So the map has branches rather than borders.

• Small, distant, inactive bodies favour optical imaging and stacking until reflected light becomes prohibitively faint. Beyond that, occultations can probe bodies too small and faint to image, at the cost of sparse events and many stars monitored.
• Warm asteroids favour thermal-infrared discovery and radiometric sizes, especially when their optical albedo is low.
• A cold mini-Neptune at several hundred au may be accessible in optical light, possibly in atmospheric windows, and in millimetre maps. Which wins depends on survey area, cadence, confusion, and the planet’s actual radius and heat.
• Farther out, a warm planet’s thermal emission falls more gently than reflected light, but wide-area thermal surveys are still shallow. For a genuinely cold compact body neither channel guarantees detection, and gravitational constraints become valuable but depend strongly on geometry.

Where the room is

Some of these limits are physical. Photon statistics, dilution, finite angular resolution and a body’s energy budget do not yield to a faster kernel. In a background-limited search, signal-to-noise grows only as the square root of exposure time. For reflected light, doubling the distance needs 16 times less flux to be detected. At fixed background and instrument that means about 256 times the exposure. For a warm planet’s thermal emission at fixed temperature, the factors are 4 in flux and 16 in exposure (estimates). These follow from the signal laws, not from any particular pipeline.

An order-of-magnitude saving in compute does not change the photons in an archive. If it allows ten times as many existing equal-quality exposures to be stacked coherently, the limiting magnitude improves by 1.25 mag. The reflected-light reach then grows by $10^{1/8}\approx1.33$ and the fixed-temperature thermal reach by $10^{1/4}\approx1.78$. If there are no extra exposures to use, the saving buys more sky, a larger orbit bank, less mismatch or better vetting. A claim of improvement should say which of these it bought. With that framing, several opportunities stand out.

Compress the physics before searching the pixels. For each plausible spectrum, integration through the bandpasses gives a short vector of relative fluxes. A low-rank spectral basis or a small vetted template library can replace recomputing atmospheres during the trajectory search, so the expensive trajectory geometry is shared across all physical models. Validate it on the cold, cloudy and methane-depleted extremes, and do not force a single optical-to-infrared colour.

Search in inverse distance. Parallax is linear in $1/r$, so banks should be spaced in $1/r$ and in velocity according to the metric of Chapter 1, with the real cadence and PSF weights. ACT already did this and reused shifts across smooth blocks, so neither idea is new alone. Combined with a proper metric for real survey cadences, they still matter (Chapter 38).

Keep signed sufficient statistics in hierarchical searches. Short segments can store their summed numerator and information and be combined along increasingly precise parallax and orbit hypotheses. Per-night maxima or powers would lose sensitivity. Masks, PSF normalization, the transfer function of sky subtraction and a calibrated false-alarm rate must all be kept (Chapter 21).

Model slow movers and the static sky together. Subtracting a static reference loses flux from a body that moves less than a beam, as ACT’s most distant limits show. A joint fit of moving source and static sky, or a projector that accounts for the effect of the subtraction on the planet template, can recover the correct likelihood. It cannot separate a perfectly stationary source from an unconstrained static background. The test that matters is the recovery of injected slow sources against real residual backgrounds.

Spend speed on calibration. Large injection campaigns across radius, albedo, temperature, sky density and motion would show where a search is limited by flux, by association, by background subtraction, or by an over-narrow physical prior. GPUs make such campaigns affordable. They must use the same masks and selection rules as the discovery run. The open problems this raises are collected in Chapter 42.

References

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