Heat: finding planets by their own glow

On 11 October 1983 the Infrared Astronomical Satellite (IRAS) recorded the same faint source five times in half a day. It was not where a star should be. Between the first and last detections it moved about one and a half degrees across the northern sky, near right ascension 17h 30m and declination $+59^\circ$. S. Green of the University of Leicester reported the fast-moving object, and the next night C. Kowal found it with a telescope at Palomar, looking like an ordinary asteroid. IAU Circular 3878, dated 14 October, gave it the designation 1983 TB, listing five IRAS positions and two from Palomar (IAUC 3878). Today it is (3200) Phaethon, and JPL’s small-body database credits its discovery to IRAS on 11 October 1983 (JPL SBDB).

Phaethon was found by its warmth. A cooled satellite saw the mid- and far-infrared glow of a Sun-heated rock, and the discovery chain that followed is the template for this chapter: repeated detections, motion inconsistent with a fixed star, rapid reporting, then confirmation. A thermal colour alone proved nothing. The motion did.

Two populations make heat worth chasing. Sun-warmed asteroids radiate strongly in the mid-infrared, so infrared surveys measure their size almost independently of how dark or bright their surfaces are. And a distant giant planet may radiate mostly its own stored internal heat. If its temperature does not depend on its distance from the Sun, its received thermal flux falls only as the square of distance, while reflected sunlight falls roughly as the fourth power. That simple contrast motivated a series of searches for a hidden planet in IRAS, WISE and AKARI data and, more recently, in the maps of telescopes built to study the cosmic microwave background (CMB). None has found a planet. They have produced the cleanest null results and the most carefully calibrated selection functions in the field, and one of them, the Atacama Cosmology Telescope’s 2021 search, is a model of how to do a weak-signal search over a huge template bank.

The chapter covers the thermal infrared first, then the millimetre. The dynamical case for a distant planet is in Chapter 8, the optical searches in Chapter 9, and reflected-light brightness physics in Chapter 2.

Why heat: the physics of thermal signals

Emission from a sphere

For an unresolved, uniformly bright spherical blackbody of radius $R$ at distance $\Delta$ from the observer, the flux density is

This is a starting model, not a universal planetary spectrum. A real asteroid has a temperature that varies across its surface, so the flux is an integral of $\epsilon_\nu B_\nu[T(\mathbf s)]$ over the visible surface weighted by the projected area, with $\epsilon_\nu$ the emissivity. And a detector measures a band integral, not $F_\nu$ at one frequency. Comparing with catalogue fluxes requires the catalogue’s reference spectrum and a colour correction; the IRAS explanatory supplement warns that its broad-band convention can require corrections as large as 50% in extreme cases (IRAS Explanatory Supplement).

Sun-warmed asteroids

For asteroids the standard tool is Alan Harris’s near-Earth asteroid thermal model (NEATM). The dayside temperature falls from a subsolar value

where $A$ is the Bond albedo, $S_\odot$ the solar irradiance at 1 au, $r$ the heliocentric distance in au, $\sigma$ the Stefan–Boltzmann constant, $\zeta$ the angle from the subsolar point, and $\eta$ a “beaming parameter” that absorbs roughness, thermal inertia, rotation and geometry. It is fitted or assumed, not a measurable material constant. Models with finite thermal inertia instead solve heat diffusion into the surface, which matters for fast rotators and high phase angles.

The reason infrared surveys are prized for asteroid sizes is the next relation. Visible absolute magnitude $H$ and visible geometric albedo $p_V$ fix the diameter:

An optical survey measures $H$ and so leaves $D$ and $p_V$ degenerate: a dark large body and a bright small one look the same. Thermal flux depends mainly on emitting area, so a thermal measurement plus $H$ gives both $D$ and $p_V$ (Mainzer et al. 2023). Two traps await implementers. Geometric albedo $p_V$ is not Bond albedo $A$; the latter sets absorbed energy and is related to $p_V$ through a phase integral. And infrared selection is less sensitive to albedo, not independent of rotation, thermal inertia or viewing geometry.

Self-luminous planets

A distant giant planet is a different object. Its effective temperature combines internal heat and absorbed sunlight:

assuming the absorbed sunlight is redistributed over the sphere. Esther Linder and Christoph Mordasini’s nominal model of a 10-Earth-mass planet at 700 au has $T_{\rm eff}=47$ K and a radius of $3.7R_\oplus$, while its equilibrium temperature is only about 10 K (Linder & Mordasini 2016). Internal heat dominates by a factor of about $(47/10)^4\approx500$ in emitted power.

Wien’s law, $\lambda_{\max}T\simeq2898\ \mu{\rm m\,K}$ for the peak of $B_\lambda$, places a 47 K planet’s emission peak near 62 μm. This is why far-infrared surveys at 60–160 μm are natural for cold planets, while Sun-warmed asteroids at a few au favour the mid-infrared. (The peak of $B_\nu$ falls at a different wavelength; quoted peak wavelengths must state which convention they use.)

Why a 3.4 micron search can work at all

A 47 K blackbody emits almost nothing at 3.4 μm. Yet the most extensive planet search in this chapter used exactly that band. The reason is atmospheric chemistry. Jonathan Fortney, Mark Marley, Gregory Laughlin, Nadine Nettelmann, Caroline Morley and colleagues modelled the atmospheres and thermal evolution of possible distant planets (Fortney et al. 2016). In a cold atmosphere methane can condense and leave the upper layers, removing strong molecular opacity. Hydrogen collision-induced absorption and the remaining bands leave windows through which radiation escapes from much warmer, deeper layers. Their models predict 3–5 μm emission up to roughly twenty orders of magnitude above a blackbody of the same effective temperature. This is spectral redistribution, not extra energy.

The catch is that the prediction depends sharply on chemistry. At 622 au, with Neptune’s radius and $T_{\rm eff}=50$ K, changing the methane abundance parameter from $10^{-1}$ to $10^{-6}$ moves the W1 (3.4 μm) magnitude from 28.0 to 16.1, while W2 (4.6 μm) barely changes, from 17.6 to 17.2. Their evolution models also overpredict Uranus’s internal luminosity under adiabatic assumptions, and the authors treat predicted internal temperatures with caution.

So a W1 search tests the bright, methane-depleted end of the model space, not every planet of a given mass. The figure makes the point: of the tabulated models, only the warmest and most depleted comes near the depth that WISE actually reached.

Motion: parallax and drift

Every thermal search eventually relies on motion. At distance $d$ in au, annual parallax has semimajor amplitude

and observations from opposite sides of Earth’s orbit can be up to $2p$ apart. For a circular orbit, Kepler’s third law gives period $P=d^{3/2}$ years and angular drift

At 700 au, $p\approx295''=4.9'$, and the orbital drift is $1296000/700^{1.5}\approx70''$ per year, or $27.3'$ over 23.4 years, the baseline between IRAS and AKARI. Both numbers match the tabulation of Sedgwick and Serjeant (Sedgwick & Serjeant 2022). A two-epoch separation is a vector sum of orbital drift and the difference in parallax, not a scalar distance indicator; arbitrary inclinations and eccentricities need projected orbit calculations.

Detection statistics

The pixel model and matched filter are those of Chapter 1. With exposures $\mathbf y_i=\mathbf b_i+f_i\mathbf P_i[\mathbf x_i(\theta)]+\mathbf n_i$, where $\mathbf b_i$ is static sky, $\mathbf P_i$ the (possibly trailed) PSF and $\theta$ the orbit, and with background-subtracted data of covariance $C$ and template $\mathbf s_\theta$,

Two infrared-specific points matter. Noise includes zodiacal photon noise, read noise, stellar and extragalactic confusion, Galactic cirrus, persistence and subtraction residuals, so a Gaussian covariance is a local approximation, not a model of the false-candidate tail. And smearing is a sensitivity loss even when flux is conserved: for a normalised pixel PSF the effective number of noise pixels is $n_{\rm eff}=1/\sum_j P_j^2$, and background-limited flux error grows as $\sqrt{n_{\rm eff}}$ (Meisner et al. 2017).

IRAS: confirmation streams and rejected sources

IRAS was a joint US–Netherlands–UK project, begun in 1975, that operated from January to November 1983. It surveyed more than 96% of the sky in bands centred at 12, 25, 60 and 100 μm, with point-source limits outside confused regions of about 0.5 Jy in the first three bands and 1.5 Jy at 100 μm, and resolution from about half an arcminute to two arcminutes (IRAS Explanatory Supplement).

Its source catalogue was built on confirmation at successive timescales. Michael Rowan-Robinson, who searched for Planet X while preparing the IRAS Point Source Catalog in 1983, describes “seconds confirmation” across detector rows, “hours confirmation” across successive satellite orbits, “weeks confirmation” on repeat coverage and “months confirmation” after about half a year. A single hours-confirmed detection is an HCON. A moving body could appear as a confirmed source at some visits and as a separate rejected source at others, depending on when its positions stopped agreeing (Rowan-Robinson 2021). The lesson has outlived IRAS: a carefully purified catalogue of stationary sources can discard exactly the objects a moving-source search wants.

Rowan-Robinson’s own 1983 search found Comet Bowell, not a planet. He explains the notorious contemporary press story of an IRAS “Planet X” as a misunderstanding of a team briefing. In 2021, prompted by the Planet Nine hypothesis, he returned to the archive and searched confirmed and rejected detections together, including pairs and triplets of single HCONs separated by 2–35 arcminutes for lower-mass bodies at 200–400 au. He reported one candidate consistent with $225\pm15$ au and a model-dependent mass of 3–5 Earth masses, explicitly calling for dynamical tests and recovery. It has not been confirmed.

The search logic is worth stating because it recurs. Select unidentified sources with plausible far-infrared emission. Check which survey passes detected them; require absence in the passes where a moving body should have left; look in the reject file at the predicted offsets. Re-extract scan profiles with IRSA’s SCANPI tool rather than trusting a coadded map, which has already mixed all the hypotheses about when the source was where. “Not detected” means something only if an adequate observation actually occurred, so the coverage map is part of the data.

WISE and NEOWISE: one observatory, several experiments

The mission

The Wide-field Infrared Survey Explorer (WISE), led by Edward Wright, was a 40-cm telescope launched in December 2009 with bands W1–W4 at 3.4, 4.6, 12 and 22 μm (Wright et al. 2010). As its cryogen ran out, W4 and then W3 were lost; the spacecraft continued in W1 and W2, hibernated, and resumed survey observations in December 2013 as NEOWISE-Reactivation. It stopped observing on 1 August 2024 and was decommissioned on 8 August. The final release of 14 November 2024 contains 26,888,166 two-band exposure sets, 199,762,421,143 source detections and 1,632,566 confirmed detections of 44,597 distinct moving bodies during reactivation, which are observations and observed objects, not discoveries (NEOWISE Final Data Release).

From WISE’s single spacecraft came at least four different discovery experiments, which accept different motions and should not be confused.

Asteroids from single exposures

Amy Mainzer and colleagues built NEOWISE, the small-body enhancement, around the WISE Moving Object Processing System (WMOPS). By the end of its first year it had detected about 157,000 minor planets and discovered about 33,000 new ones (Mainzer et al. 2011). WMOPS starts from about 3–4 days of scans, removes low-reliability sources, rejects stationary pairs that move less than two arcseconds between adjacent overlapping exposures, builds moving pairs and chains them by position and velocity consistency, and keeps tracklets with at least five independent detections. About 80% of initial tracklets matched known bodies; the remaining 20% were checked by eye before going to the Minor Planet Center.

for processing window:
    detections = calibrate_extract_and_flag(single_exposures)
    movers = remove_stationary_matches(reject_low_quality(detections))
    pairs = find_kinematically_allowed_pairs(movers)
    tracklets = chain_pairs_with_consistent_position_and_velocity(pairs)
    keep tracklets with >= 5 independent detections
    associate known objects; inspect the rest
    report validated astrometry to the MPC; request follow-up

The stationary-source cut sets a distance horizon. Two arcseconds in about 90 minutes corresponds to a parallactic distance of roughly 70 au in the authors’ geometry. Slower, more distant bodies are removed as stars. At the other end, the fastest motion actually detected in the initial pipeline was $3.3^\circ$ per day. Neither limit means the images contain no slower or faster objects; they are properties of the pipeline. The pipeline mechanics and later operations are covered in Chapter 23.

A distant companion from parallax

Kevin Luhman’s 2014 search looked for something quite different: the annual parallactic displacement of a distant solar companion between WISE epochs. It reached $W2=14.5$ and found no companion (Luhman 2014). It also catalogued 762 newly identified high-proper-motion objects, which are stars and brown dwarfs, not planets.

unWISE: coadd, difference, detect, link

The most completely specified large thermal planet search is Aaron Meisner and collaborators’ use of time-resolved WISE coadds. Dustin Lang’s unWISE processing (2014) had already rebuilt WISE coadds at native resolution, instead of the deliberately PSF-smoothed official atlas (Lang 2014). Lang reported that a representative W1 tile of about $3\times10^8$ input pixels took 630 seconds and 2.3 GB on one CPU core, and a north-ecliptic-pole tile touching about $10^{10}$ input pixels took about twelve hours and 100 GB. Meisner, Benjamin Bromley and colleagues sliced this machinery by time. Their first search (2017) reached 90% completeness at $W1=16.66$ over about two thousand square degrees (Meisner et al. 2017); the expansion in 2018 searched more than three-quarters of the sky (Meisner et al. 2018).

WISE’s observing geometry helps. The spacecraft looks roughly at right angles to the Sun, so at low ecliptic latitude a distant body’s parallax is near an extremum during a visit, and its within-visit motion is far smaller than its maximum annual parallax speed. Meisner et al. derived the visit length and within-visit parallactic trail,

where $\beta$ is ecliptic latitude. At 600 au and $|\beta|=15^\circ$ the trail is about $1.6''$, comfortably smaller than the $2.75''$ coadd pixels. At high latitude, longer visits and larger reflex motion both increase smearing, which they calibrated with simulated WISE profiles.

Image stages. The expanded analysis processed about six million W1 framesets (intensity, uncertainty and mask), about 40 TB and 18 terapixels, into 129,483 epochal coadds of $2048\times2048$ pixels at $2.75''$. A typical sky position had nine coadd epochs, each from about twelve exposures. For each epoch a reference image is built from epochs more than five months away. Astrometry is corrected with Source Extractor and SCAMP, images warped with SWarp, and the reference PSF-matched and subtracted with HOTPANTS. Positive residuals are detected, and stationary counterparts, bright-star artefacts, persistence latents and kernel failures rejected. The temporal slicing is the method: a full-depth stack would smear a body that moves between visits, and a body that moves too little between reference and search epochs subtracts itself.

Linking stages. For each transient, find later transients reachable by a bound body beyond the minimum distance. Form triplets and fit a parallax-plus-linear-drift model,

where $\mathbf q(t)$ is the projected observer position, keeping $\chi^2<30$. Extend survivors to quadruplets with a permissive $\chi^2<150$ prescreen, fit them with Gary Bernstein and Bharat Khushalani’s orbfit, accept $\chi^2<40$, merge compatible groups sharing detections, and require at least five transients for a candidate. The model is linear in its parameters, so it is cheap to fit before the nonlinear orbit.

The burden. The filtered catalogue held 5,603,879 transients. The prescreen enumerated more than 92 billion triplets and 275 million quadruplets, followed by more than eight million orbfit calls. It returned 2,735 candidate linkages: 2,560 quintuplets, 163 sextuplets, eleven septuplets and one octuplet. Of these, 737 were known high-proper-motion sources. Inspection rejected all the rest; the octuplet was a white dwarf. For a seed with $k$ plausible later neighbours, naive triplet and quadruplet enumeration grows as $O(k^2)$ and $O(k^3)$, and the paper notes the steep power-law growth of tuple counts with transient surface density. Masking a small crowded region can make a disproportionate difference to the compute.

The selection function. The team injected 300,000 artificial sources into difference images, measured recovery against magnitude and depth, and simulated the chance of detecting at least five epochs given masks and cadence. High-latitude completeness was 90% at $W1=16.70$ and 50% at 16.86, with the spatial 90% limit ranging from 15.97 to 16.93 across the sky. The searched footprint was 31,480 square degrees, 76.3% of the sky. For the favourable W1-bright models, low ecliptic latitudes were tested over about 250–800 au; near $|\beta|=55^\circ$ the range narrowed to about 650–850 au. The formal self-subtraction ceiling was 2,250 au. Their forecast that W2 processing could reach $W2\approx15.9$ and test the brightest W2 models to about 1,700 au remains a forecast. The expanded search acknowledges NASA high-end computing and NERSC but reports no end-to-end CPU-hour total.

The same time-resolved WISE images feed the Backyard Worlds citizen-science project, whose volunteers spot moving brown dwarfs by eye (Kuchner et al. 2017); that story is in Chapter 28.

The far infrared: IRAS meets AKARI

The AKARI satellite’s cooled 68.5-cm telescope surveyed the sky in 2006–2007, and its Far-Infrared Surveyor bands at 65, 90, 140 and 160 μm sample the expected emission peak of a planet at tens of kelvin. Paired with IRAS, it offers a 23.4-year baseline. Three searches have exploited it, each loosening the catalogue’s confirmation rules further.

Sedgwick and Serjeant, 2022. Chris Sedgwick and Stephen Serjeant matched unidentified or rejected IRAS sources to displaced AKARI sources, interpreting the separation with roughly circular Keplerian motion in distance bins from 8,000 au inward to 700 au (Sedgwick & Serjeant 2022). They excluded stationary matches and high-cirrus areas, compared colours, remeasured IRAS fluxes with SCANPI, applied a catalogue-wide $1/0.7$ correction to AKARI 90 μm fluxes for an inter-survey inconsistency, and fitted planetary blackbody spectra with a $\chi^2$ over bands. Their handling of upper limits was nonstandard: limits were treated as measurements with errors of half their values. A rigorous treatment would use a censored likelihood, $\Phi[(F_{\rm lim}-F_{\rm model})/\sigma]$, for each non-detection. Spectral screening left 535 candidates, mostly suggestive of Neptune-mass or smaller bodies near the inner distance boundary. Image inspection found every one associated with or near Galactic cirrus, and none convincing enough for follow-up.

Phan et al., 2025. AKARI’s bright-source catalogue requires detection in at least three-quarters of the scans through a position. A distant mover seen on one day and displaced by the next half-year visit fails that rule. Terry Long Phan, Tomotsugu Goto, Issei Yamamura and colleagues used instead the Monthly Unconfirmed Source List (MUSL), which admits sources with two or more detections without monthly persistence (Phan et al. 2025). They paired IRAS and AKARI sources separated by 42–69.6 arcminutes, which in their geometry corresponds to 500–700 au, for planets of 7–17 Earth masses, with working limits of 0.2 Jy at IRAS 60 μm and 0.21 Jy at AKARI 90 μm. Detectability was far from uniform: the 7-Earth-mass model was detectable by IRAS only over about 500–520 au.

Chen et al., 2025. Amos Chen and the same group went further, to the AKARI Single Scan Detection List, and searched a predicted region ($30^\circ<{\rm RA}<50^\circ$, $-20^\circ<{\rm Dec}<20^\circ$) for sources repeated within a day but without six-month persistence (Chen et al. 2025). They estimated an expected 0.029 chance overlaps of false single-scan sources in the region and about 0.014 astrophysical transients, mostly supernovae. Those are conditional calculations under a uniform, independent false-source model, not total false-alarm probabilities.

Phan’s candidate pairs IRAS F02211−4844 with AKARI-MUSL J0220440−491247, 47.46 arcminutes apart, with listed fluxes of 0.24 Jy at 60 μm, 0.09 Jy at 65, 0.27 Jy at 90 and 0.52 Jy at 100. The AKARI source is not a clear point source in the coadded image; the authors assess it with scan-level detection-probability maps and call the low 65 μm flux possibly unreliable. Chen’s two candidates, at $(42.676^\circ,-15.021^\circ)$ on 28 July 2007 and $(45.297^\circ,-16.711^\circ)$ on 30 July 2006, each have two detections within a day. They are two separate candidates, not two epochs of one body, and no displaced detection consistent with the expected half-year parallax was found for either.

These are candidates, not planets. “Absent at the old position, present somewhere else” does not determine a unique orbit; two widely separated positions leave a large recovery region and real association ambiguity. The hardest alternatives are instrumental: cosmic rays tied to the South Atlantic Anomaly, scan-edge effects and cirrus. The appropriate next step is independent scan-level reanalysis and a recovery search over a deliberately broad orbital posterior. No confirmation was found in the sources examined for this book.

Deep and narrow: Spitzer and JWST

Narrow, deep infrared observations taken for other purposes are also full of asteroids. Erin Ryan and colleagues mined the Spitzer MIPSGAL and Taurus surveys at 24 μm by subtracting visits separated by 3–7 hours (MIPSGAL, 29.4 deg²) and 5–12 hours (Taurus, 53.12 deg²), photometering positive and negative residuals, and rejecting opposite-sign pairs within 1.5 pixels as imperfectly subtracted stars (Ryan et al. 2015). Known-object ephemerides identified 1,865 asteroids, whose fluxes went into NEATM fits; the adopted 90% completeness was 2 mJy at 24 μm. This was characterisation of known bodies, not discovery.

The sharpest recent result came from exoplanet data. Artem Burdanov, Julien de Wit and colleagues reprocessed 93.5 hours of JWST MIRI time-series observations of the TRAPPIST-1 system, taken in 11 visits during 2022–2023 to study its planets (Burdanov et al. 2024). They found 8 known asteroids and 138 previously unknown asteroid detections, which under their radiometric interpretation reach main-belt bodies about ten metres across.

The method is thermal synthetic tracking, the infrared version of Chapter 18. A wrapper around the Tycho Tracker software searched all position angles and speeds of 0.001–1.200 arcseconds per minute, using overlapping groups of 100 exposures for faster objects and whole visits of up to about 1,000 exposures for slower ones. For linear motion the core operation is

with brute-force cost $O(N_{\rm pix}N_tN_v)$. Because the velocity spacing must keep end-to-end mismatch below the PSF width, a two-dimensional velocity bank grows roughly as the square of the time baseline at fixed velocity range. Injection tests with 100 frames, a $4\times11$ grid of sources and eleven flux levels gave recovery of at least 80% down to 1.5 μJy and $50\pm2$% at 1.1 μJy. The faintest detections, near 0.5 μJy, are not the 50% level. Timestamp perturbations and split-sequence checks helped separate noise peaks from real sources; no calibrated all-bank false-alarm probability or GPU-hour budget is published.

Detection did not determine size directly. Short arcs leave distance ambiguous, so the authors used known asteroids near the field in speed and position-angle space as a prior on distance and phase angle, and thermal radiometry then gave sizes. The 138 detections are isolated tracklets awaiting linkage, not 138 orbits. The associated arXiv preprint (2502.01744) retains an earlier count of 139; the published article controls.

A September 2026 JWST result marks the boundary of this chapter: Eduardo et al. found 27 trans-Neptunian objects in NIRCam images near 1.5 μm, reaching a 40% detection threshold at $m_{F150W2}=28.8$ over 0.05 square degrees (Eduardo et al. 2026). Those detections are reflected sunlight, and the method belongs with the deep stacks of Chapter 19.

NEO Surveyor: thermal discovery as mission design

NEO Surveyor, described in Chapter 23 and Chapter 14, is the first space telescope designed around thermal asteroid discovery. Its passively cooled 50-cm telescope at Sun–Earth L1 images at 4–5.2 μm (NC1) and 6–10 μm (NC2), with required five-sigma sensitivities of 65–120 μJy and 110–280 μJy depending on zodiacal background (Mainzer et al. 2023). Its simulator predicts reflected plus thermal band fluxes for every synthetic object, requires at least four detections above signal-to-noise five and motion between 0.008 and 8 degrees per day, and applies assumed 99% tracklet and linking efficiencies (Masiero et al. 2023). The requirement concerns potentially hazardous asteroids larger than 140 m. NASA’s mission page gives a launch no earlier than September 2027 (NASA).

The millimetre: CMB telescopes as planet finders

The proposal

In February 2016, weeks after the renewed dynamical case for Planet Nine, Nicolas Cowan, Gil Holder and Nathan Kaib asked whether cosmologists already had the right instruments (Cowan, Holder & Kaib 2016). Ground-based CMB telescopes repeatedly map thousands of square degrees with large arrays of millimetre detectors and arcminute beams. Using Uranus and Neptune as empirical brackets on internal heat, they estimated that a Neptune-radius planet at 700 au and 40 K would appear as a source of about 30 mJy at 300 GHz (1 mm) and 8 mJy at 150 GHz, with an annual parallax of about 5 arcminutes, much larger than the beam. The difficulty, they noted, would be distinguishing it from the roughly 4,000 foreground asteroids brighter than 30 mJy at 1 mm, which are known and can be removed. They suggested maps every few months, difference imaging and, for a fainter planet, matched filtering through the data cube.

Eric Baxter, Bhuvnesh Jain, Cullen Blake, Gary Bernstein, Mark Devlin and Gil Holder extended the idea in 2018 to a cross-survey strategy (Baxter et al. 2018). A microwave detection with measurable parallax gives a position and distance for targeted optical follow-up. Even a microwave source without measurable parallax, if its spectrum is thermal, can seed a restricted digital-tracking search in an optical archive. Dusty galaxies and active nuclei make spectral selection impure, so this narrows the search rather than classifying the source.

Signal

On the Rayleigh–Jeans tail ($h\nu\ll kT$) the flux becomes

so the signal scales as $TR^2\nu^2\Delta^{-2}$, linearly in temperature rather than as $T^4$. Measurements entirely on this tail constrain roughly $\epsilon R^2T$, not radius and temperature separately; higher-frequency data and a distance are needed to break the degeneracy.

The search’s two reference planets come from Fortney’s evolution grid, with band fluxes calculated by ACT (Naess et al. 2021):

All fluxes are at 500 au. Other compositions in the grid give 4.1–8.9 mJy at 150 GHz for 5 Earth masses and 6.6–15 mJy for 10. Those are model variations, not error bars. ACT also estimated that the planet’s blocking of the CMB behind it would change the contrast by only 2.6%, 1.4% and 0.6% in its three bands, negligible beside the model uncertainty.

For small bodies the temperature law is different. A passive asteroid heated by the Sun has $T\propto r^{-1/2}$, so its millimetre flux scales as roughly $R^2r^{-1/2}\Delta^{-2}$. Applying the planet’s constant-temperature law to asteroids is wrong.

Motion

The searched trajectories combine a large parallax ellipse with slow heliocentric drift. ACT gives the parallax semimajor axis and maximum parallax speed as

and, for an orbit of semimajor axis $a$ and eccentricity $e$, the heliocentric angular speed

The parallax-induced speed is roughly twenty times the intrinsic drift at 500 au, so it sets the smearing within a map. The search model has five parameters: two for sky position, two for linear drift, and distance, with flux fitted analytically. It is not a general six-element orbit search.

Mapmaking and the matched filter

CMB telescopes do not take exposures. They scan the sky with detector arrays and record time streams, and maps are estimated afterwards. Write the time-ordered data as $d=Pm+n$, where $P$ projects a beam-convolved sky map $m$ into detector samples and the noise $n$ has covariance $N$, dominated by correlated atmospheric emission. The maximum-likelihood map and its covariance are

For a point source at a trial position with response vector $a$ (beam, spectrum, pixel window and unit conversion) and map noise covariance $U$, the matched-filter quantities are

ACT computes a numerator map $\rho$ and a normalisation map $\kappa$ for every short time interval. Keeping both is essential: summing signal-to-noise maps or unweighted flux maps gives the wrong statistic when depth varies. Along a trajectory $\lambda=(r,v_x,v_y)$ the stacked quantities are

where $\mathcal T_i$ maps a reference position into its observed location at epoch $i$. “Coherent” here means consistent positional alignment, not preservation of electromagnetic phase.

ACT’s search for Planet Nine

The data and the bank

Sigurd Naess led the Atacama Cosmology Telescope collaboration’s analysis, published in 2021 (Naess et al. 2021). ACT is a six-metre telescope on Cerro Toco in Chile. The search used 150 GHz data from 2013–2019, 98 GHz from 2015–2019 and 229 GHz from 2017–2019, with beam widths (FWHM) of 1.40, 2.05 and 0.98 arcminutes, over about 18,000 square degrees including low ecliptic latitudes that the main South Pole Telescope fields do not cover. The raw data amounted to seven years and 140 TB.

Observations were split into three-day chunks and mapped at 0.5-arcminute pixels. The mean static sky was subtracted first, bright sources were handled, and numerator and inverse-variance maps were built, then downsampled to one-arcminute pixels; matching before downsampling reduces aliasing. Time-stream samples within 0.8 degrees of the major planets and within three arcminutes of listed bright asteroids were cut; bright variable sources and dusty Galactic regions were masked. Those masks are part of the selection function. The result was 3,834 numerator/weight map pairs occupying 1.9 TB, combined across simultaneous arrays and bands into 787 pairs occupying 220 GB, all in equivalent 150 GHz flux under an assumed 40 K blackbody spectrum.

The motion bank was set by the then-favoured dynamical range for Planet Nine. At each of 14 distances,

it searched an annulus in $(v_x,v_y)$ with velocity spacing 0.1 arcminutes per year, 25,837 cells in all. (The paper gives the annulus as about 1.6–2.3 arcminutes per year at 500 au in the text and 1.50–2.26 in a footnote describing the executed search; a literal reproduction should follow the code.) Because parallax is linear in inverse distance $u=1/r$, spacing nodes roughly uniformly in $u$ gives roughly uniform parallax mismatch, which the node list approximately does.

The paper budgets four contributions to the broadening of the stacked beam: velocity discretisation 0.29 arcminutes, the finite three-day map at 300 au 0.40, distance discretisation 0.31, and one-arcminute pixels 0.68. Their quadrature sum is about 0.90 arcminutes, broadening the beams by 9%, 19% and 34% in the three bands. Half-arcminute pixels would have cut these to 5%, 11% and 21% at four times the CPU and memory cost.

The kernel

The search tiles the sky into ten-degree squares with a one-degree halo on each side and, for each distance, loops over velocities, interpolating and accumulating the displaced $\rho_i$ and $\kappa_i$ maps.

for tile in sky_tiles_with_halo:
    load epoch maps rho[i], kappa[i] covering tile
    for r in distance_nodes:
        best = empty maps
        for (vx, vy) in velocity_annulus(r):
            num, wt = zeros
            for i in valid_epochs:
                xy = observed_positions(tile, r, vx, vy, epoch[i])
                num += bilinear(rho[i], xy)
                wt  += bilinear(kappa[i], xy)
            update best with num / sqrt(wt), keep (vx, vy, num, wt)
        save best for this distance

Displacement varies slowly across a tile, so the implementation computes it once per eight-by-eight pixel block and reuses it, eliminating most of the trigonometry. It keeps the best velocity at each position separately for each distance, because the bias and background statistics depend on distance. A specialised C kernel with AVX fused multiply-adds and OpenMP was 480 times faster than a straightforward NumPy/SciPy version. The code is public: the driver planet9.py in the tenki repository and the kernel displace_map_blocks_avx_omp in enlib (tenki; enlib). A comment in the kernel notes that OpenMP gained only 23% on the developer’s laptop and suggests memory bandwidth is the bottleneck, an informal observation rather than a benchmark. No GPU version exists in these routines, and the paper reports no total CPU-hours.

Some rough arithmetic gives the scale. With $P$ pixels, $E$ epochs and $K$ templates, direct work is $O(PEK)$ interpolations. Taking 18,000 square degrees at one-arcminute pixels ($6.5\times10^7$), 787 epochs and 25,837 templates gives about $1.3\times10^{15}$ pixel–epoch–template updates. Each bank pass logically reads the 220 GB of maps, so all 25,837 passes traverse about 5.7 PB; at a sustained 1 TB/s that alone would take about 1.6 hours. These are estimates that ignore masks, halos and actual epoch coverage, not ACT timings. They show why memory traffic, not arithmetic, should be measured.

Self-subtraction and significance

Subtracting the mean sky partly subtracts the planet, because its smeared track is in the mean and overlaps each instantaneous position. ACT measured the surviving fraction $b(r,\mathbf x)$ with noiseless injections across position, distance and velocity. Typical values run around 0.55–0.70 at 300 au and 0.15–0.475 at 2,000 au, with strong position dependence. Debiasing sets $\rho\rightarrow b\rho$ and $\kappa\rightarrow b^2\kappa$: the flux estimate is divided by $b$, the significance is unchanged, and the information removed by subtraction is not restored.

The maximum over velocities is not Gaussian even if each fixed-template statistic is. ACT counts about thirty million independent beams and a naive template product near $10^{12}$, and treating that product as independent trials with one global threshold would both overcount correlations and penalise distances unequally. Instead it estimates the local mean and scatter of $q_{\max}$ in 0.5-by-4-degree blocks and maps each distance’s high tail onto Gaussian quantiles:

The affine tail fit works empirically; it is not a theorem. Peaks above $z=3.5$ form a deliberately permissive candidate list, and the whole search is repeated with the sign of the data reversed to give an empirical background sharing the same edge and artefact structure.

The result

About 38,000 raw candidates emerged, roughly 3,500 “Planet Nine-like” and 35,000 “general” (the two differ by an inclination cut). Inspecting the strongest hundred in each class removed three and seventeen artefacts. The strongest Planet Nine-like candidate had $z=5.17$ and the strongest general one $z=5.65$, and the positive and sign-reversed populations were statistically consistent: no significant detection. Inspecting candidates did turn up real millimetre transients, which became a separate ACT paper.

Noisy injections run through subtraction and the full detection procedure are recovered 95% of the time when their strength corresponds to $z=5.3$, so the flux limit is

where $\xi$ is the empirical map from raw signal-to-noise to $z$. The 5.3 is a completeness calibration, not a false-alarm threshold. The model-independent result is a 95% limit of about 4–12 mJy at 150 GHz over 300–600 au depending on sky position, and about 5–15 mJy at 1,500 au. Translated through the nominal planet models, a 5 Earth-mass planet would have been detected out to 325–625 au and a 10 Earth-mass planet out to 425–775 au. Overall the search excludes roughly 17% and 9% of the then-favoured parameter space for those masses. ACT’s coverage was shallow near the expected aphelion direction, where the planet would spend most of its time.

Known asteroids at millimetre wavelengths

Known-orbit stacking removes the blind template dimension and turns CMB surveys into radiometers for asteroids. P. M. Chichura, A. Foster, C. Patel, N. Ossa-Jaen and South Pole Telescope collaborators subtracted each SPTpol field’s mean map, obtained positions from JPL Horizons, coadded residual stamps at the predicted locations of about one hundred observations of two roughly 270-square-degree fields over a month, and matched-filtered the stack, using map-derived effective times because a raster-scanned map is not an instantaneous exposure (Chichura et al. 2022). At 150 GHz they measured Bamberga at $30.6\pm2.7$ mJy, Egeria at $44.5\pm4.3$ mJy and Kalliope at $16.5\pm2.7$ mJy, with significances 11.2, 10.4 and 6.1. Kalliope’s effective emissivity of $0.64\pm0.11$ shows why unit-emissivity predictions fail for rocky bodies.

John Orlowski-Scherer and ACT colleagues scaled this up with 29,175 “depth-1” maps, each pixel timestamped to the roughly four-minute array crossing (Orlowski-Scherer et al. 2024). To remove distance and phase effects they modelled $F_i=F_0W_i$ with

where $\alpha_i$ is the phase angle in degrees. Of about 1,200 bright asteroids examined, 170 exceed five sigma in at least one band, 70 in all bands, and 222 under the combined-band criterion. Relative to thermal models fitted to WISE data, the weighted sample shows flux deficits of $-4\pm2$%, $-24\pm2$% and $-22\pm3$% in the three bands. Any forecast of blind small-body detections at these wavelengths needs a realistic emissivity model.

Things that move, flash, or orbit Earth

Temporal searches are the control channel. Nathan Whitehorn, Thibaut Natoli and SPT colleagues used 6,000 observing hours over 100 square degrees of SPTpol data in 2012–2013 to search for day-to-month transients at about the 10 mJy level; their single candidate had a probability of 0.01 and was not confirmed (Whitehorn et al. 2016). S. Guns and colleagues found fifteen transient events in a 1,500-square-degree SPT-3G field during March–November 2020: thirteen flares from eight stars and two longer events probably extragalactic (Guns et al. 2021). Their likelihood uses Gaussian flare templates with a $2\ln(w/\Delta T)$ term that penalises the many possible placements of short flares, and a threshold of TS > 100, about 9.7 sigma under their calibrated null. A planet recurs along a common motion model; a flare appears once at a fixed position. But a distant planet moves little during a short flare, so a strong transient can contaminate a sparse moving stack. Time-split diagnostics are needed.

Artificial satellites are now an observed foreground too. Foster et al. detected thermal millimetre emission from low-Earth-orbit satellites on timescales of tens of milliseconds, with predicted orbital-element positions sometimes off by several arcminutes (Foster et al. 2025). Their timescales are far shorter than a planet’s motion, so time-stream information separates them. Compressed maps can lose that information.

What comes next

The Simons Observatory’s updated forecast for its enhanced Large Aperture Telescope covers about 25,000 square degrees over 2025–2034 and keeps Planet Nine in scope, with a nominal reach of about 500–900 au for 5 Earth masses, about ten thousand asteroid detections under an extrapolated flux distribution, and at least five-sigma detections of Quaoar, Makemake, Haumea and Pluto (Simons Observatory Collaboration 2025). All are forecasts. SPT-3G+, planned to start in 2029, would survey 800 square degrees for six years to about 0.7 μK-arcmin in each of its 90 and 150 GHz bands; its small footprint emphasises depth rather than the low-ecliptic coverage a planet search wants (Natoli et al. 2026). CMB-S4, often quoted in older forecasts with a 0.4 mJy (five-sigma) 150 GHz point-source sensitivity, is no longer coming: its website reports that the US Department of Energy and National Science Foundation withdrew support on 9 July 2025 and that the project is shutting down (CMB-S4).

Landmark results

Limitations

Confusion and cirrus. The IRAS supplement already warned that Galactic cirrus can produce well-confirmed apparent point sources and corrupt real fluxes, and Sedgwick and Serjeant’s entire final list proves it. Longer integration suppresses detector noise but does not separate a compact cirrus knot from a planet in a coarse beam. Only consistent measurements of the same scene across epochs can.

Selection rules erase planets. Stationary-source rejection removes distant bodies from WMOPS; confirmation requirements remove movers from AKARI’s bright catalogue; reference subtraction suppresses slow sources in unWISE and ACT; all-epoch coadds dilute displaced ones. Each rule has a sound origin, and relaxing it raises contamination, so it must come with a better likelihood and validation.

A mass is not a flux. Radius, envelope, methane depletion, clouds, internal entropy and heat transport all matter. Fortney’s twelve-magnitude W1 range and Luhman’s threefold model dependence are the clearest examples. The absence of a W1 source cannot rule out every dynamically plausible Planet Nine. ACT’s flux limits are more portable than its mass and distance translations.

Sparse cadence. WISE samples mostly opposite phases of the annual parallax, with a long gap in the middle of the mission. IRAS–AKARI has a long baseline but only two epochs. A deep CMB map with observations clustered in one season samples little of the parallax ellipse. More detectors do not guarantee a better planet search.

Correlated artefacts. WISE latents recur at predictable offsets from bright stars and move with scan direction. AKARI cosmic rays cluster near the South Atlantic Anomaly. Asymmetric bright-source residuals and positive-only flares break the symmetry that sign-reversed searches assume. A candidate with several detections can still be a trajectory made by the instrument.

Short-arc sizes. JWST’s thermal fluxes are exquisitely sensitive, but size depends on distance and temperature, and the distance prior borrowed from nearby known asteroids may fail for an unusual orbit.

Where the room is

The thermal and millimetre searches are a useful case study for Chapter 42 because their limits separate so cleanly. Shot noise at fixed aperture, wavelength and integration time is a physical limit; no algorithm beats the optimal likelihood by trying more templates. A planet’s unknown internal luminosity cannot be computed away. A JWST field cannot become an all-sky survey, and two positions cannot determine an arbitrary orbit. Confusion is partly physical (finite resolution mixes sources) and partly inferential (better multi-epoch scene models can separate them). Trailing, coarse grids, discarded exposures, biased catalogue rules and redundant template computation are engineering losses. A benchmark should report each separately: flux sensitivity on the true trajectory, mismatch loss from the bank, veto and association loss, false-candidate workload, and end-to-end compute.

Pixels before catalogues in WISE. The unWISE search detected sources in each visit and then enumerated 92 billion triplets to link them. A distance-aware alternative would precompute whitened numerator and normalisation images per visit, remove the observer’s reflex motion at each trial inverse distance, and sum likelihoods across visits without requiring a detection in each, the HelioLinC idea of Chapter 12 carried into pixels and the orbit-space stacking of Chapter 20 applied to WISE. Where catalogues are kept, suppressing duplicate and physically impossible combinations before the nonlinear fit, with position–velocity–inverse-distance indexing and batched small least-squares fits, may matter more than accelerating each fit. The final NEOWISE archive, with ten more years of data than the expanded search used, is the obvious testbed, but its source table of almost 200 billion rows should be staged by sky tile with per-epoch image statistics, not scanned globally.

A GPU ACT kernel. ACT’s inner loop is two weighted bilinear gathers per epoch per template, a sparse, bandwidth-bound workload. A GPU block can own an output patch and a small batch of nearby velocities, load each input patch once for several templates, keep accumulators in registers or shared memory, and fuse interpolation, accumulation, division and the running maximum, never writing a full stack per template. A twelve-degree padded tile at one-arcminute sampling is about $720^2$ pixels; 787 numerator/weight pairs in 4-byte floats occupy about 3.3 GB, so a tile fits on one accelerator while the survey streams from the host. A dense matrix formulation would multiply mostly zeros, so tensor cores help only if a genuine low-rank or compressed basis exists. Because the CPU code is already 480 times faster than NumPy, an order-of-magnitude gain over it is a research target, not an expectation, and a fair comparison must include all distance banks, preprocessing, and the sign-reversed and injection reruns.

Reuse across inverse distance and velocity. In a small tile the observed position is approximately $\mathbf x_i=\mathbf x+\mathbf v\tau_i+u\mathbf p_i$ with $u=1/r$ and $\mathbf p_i$ the known projected observer displacement. At fixed $u$, deparallax every map once, then search the remaining linear drift, applying the same warp to the weights. For idealised shift-invariant maps the numerator stack has Fourier transform

which links the problem to non-uniform FFTs and generalised Radon transforms (Chapter 39). Irregular cadence, varying footprints and the position- and trajectory-dependent denominator mean this does not drop in directly; a transform that mishandles $\kappa$ changes the statistic. With only 25,837 templates, asymptotic complexity is not the argument. Measured injection recovery is.

Banks get much bigger. A template metric based on beam overlap, $\mathcal O\simeq\langle\exp[-|\delta\mathbf x_i|^2/(4\sigma_b^2)]\rangle_w$ for Gaussian beam width $\sigma_b$, would allocate templates where the data are deep and the cadence informative, rather than on one worst-case grid (Chapter 38). It will be needed. At fixed allowed drift error the two-dimensional velocity bank grows as $T^2$ with baseline $T$, and epochs grow as $T$, so naive work grows roughly as $T^3$. Shrinking the beam by a factor $f$ at fixed footprint multiplies position sampling by $f^2$, velocity sampling by $f^2$ and inverse-distance sampling by $f$: about $f^5$ more work. These are scaling estimates under fixed cadence and velocity range, but they mean that sharper, longer future surveys make reuse and hierarchy necessary rather than optional. Any hierarchy must accumulate likelihood before pruning, because a source detectable in the full stack can be below threshold in every segment.

Model the static sky inside the likelihood. Mean-sky subtraction followed by a debiasing factor is an approximation. Formally, if $A$ maps an unknown static sky into all observations, profiling it out gives the precision operator

and the moving-source statistic becomes $a_\lambda^TWy/\sqrt{a_\lambda^TWa_\lambda}$. This expresses self-subtraction and correlated noise exactly. It is expensive at full resolution, but the static component has repeated structure that local, low-rank or operator methods could exploit. It also exposes a hard limit. If the motion is much smaller than the beam, the time-varying part of the signal is the displacement times the beam gradient. In a parallax-dominated, internally heated case the displacement scales as $r^{-1}$ and the flux as $r^{-2}$, so the variability signal falls roughly as $r^{-3}$. The simple $r^{-2}$ thermal advantage overstates the reach of any search that must separate the planet from an unknown static sky. Sub-beam motion is not undetectable, because centroid precision improves with signal-to-noise, but better pointing, independent background estimates and well-spread temporal coverage are what help, not more arithmetic.

Joint far-infrared scene models. For IRAS and AKARI, the better test compares a moving point source plus a common diffuse sky against sky alone, fitting all scans jointly with each scan’s beam and calibration and treating non-detections as censored data. That avoids equating flux differences between unlike catalogue products with motion. Learned cirrus or artefact models could supply priors or proposals, but completeness must come from moving sources injected into the real scans, through every veto and inspection step.

Benchmarks. The single most useful contribution may be infrastructure: a public, small but representative benchmark of timestamped filtered maps, weights, masks and injected trajectories for ACT-like data, and the equivalent for WISE visits, with a fixed recovery test. Without it, claims of much faster GPU stacking cannot be compared with ACT’s already optimised CPU code or with the unWISE linker. That is a tractable problem, and one this field has not yet solved.

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