learn.xhelio.ai · a book

Finding Planets

Every detection method, from Herschel's telescope to tensor cores

On the night of 13 March 1781 William Herschel, working through faint stars in Gemini from Bath with a reflector of his own construction, noticed that one of them grew when he raised the magnification. In 2026 Kevin Napier and colleagues, computing on NVIDIA H200 GPUs on Harvard’s Cannon cluster, summed 1,118 Hubble images along more than a million trial orbits and reported two faint bodies that no single image shows. Between those two moments lie the photographic plate and the blink comparator, the perturbation calculation that found Neptune on paper, the CCD, the kd-tree, the matched filter, the Doppler spectrograph, the transit survey and the tensor core. This book is about all of them: every way that planets and planetary bodies, in the Solar System and around other stars, have been found or might be found.

Why this book exists

The methods of planet detection grew up in separate communities. Asteroid hunters, Kuiper-belt surveyors, exoplanet teams, pulsar astronomers, gravitational-wave analysts, radar engineers and space-debris trackers each built their own pipelines, wrote in their own journals and, again and again, rediscovered the same mathematics. The likelihood ratio, the trials factor, the template bank, the hierarchical search, the decision of when to throw a hypothesis away: these appear under different names in every field, often with different mistakes.

That separation matters most to one kind of reader: someone who wants to build a better search. A physicist or a GPU engineer who asks “could this be done ten times faster on modern hardware, without losing sensitivity?” needs to know what the search actually computes, what limits it, what has already been tried, and what the published speed-ups were measured against. That information exists, but it is spread across thousands of papers, and the claims in them are of very uneven strength. This book gathers it in one place, explains each method well enough to reproduce it, and keeps a running account, chapter by chapter, of which limits are set by physics and which by engineering. The answer is often “not here”, and the book says so.

What it covers

The book has forty-two chapters in nine parts.

Part I, Foundations, sets out the statistics shared by every search, from the matched filter and the two likelihood images that make stacking exact to trials factors, sequential tests and injection–recovery (Chapter 1), and the physics of how bright an undiscovered body is in reflected and thermal light (Chapter 2).

Part II, A history of discovery, follows the Solar System’s census from Uranus and Neptune through Pluto, the first Kuiper-belt objects and the slow, distant movers found since 2010 (Chapters 3–6).

Part III, Planets found by their pull, covers ephemerides and perturbation searches, then the case for Planet Nine and the optical hunt for it (Chapters 7–9).

Part IV, The survey pipeline, walks a modern survey from pixels to detections, streaks, linking, orbit determination and precovery, near-Earth objects, Rubin’s alert stream, and the simulators that turn discoveries into populations (Chapters 10–16).

Part V, Below the noise, is the book’s longest part. It covers shift-and-stack and KBMOD, synthetic and digital tracking, deep fields and moon searches, orbit-space stacking, trees and catalogues, machine learning, and searches from space (Chapters 17–23).

Part VI, Other light, other shadows, turns to thermal and millimetre searches, occultations, radar and gravity probes, comets, moons and binaries, and citizen science (Chapters 24–28).

Part VII, Exoplanets, gives each method its own chapter: radial velocity, transits, timing, microlensing, direct imaging, astrometry and spectroscopy, and a census of how more than six thousand planets were found (Chapters 29–35).

Part VIII, Algorithms and compute, borrows from pulsar and radio-burst searches, gravitational waves, radar and debris tracking, then treats the geometry of search, fast transforms and optimal coadds, GPUs, and the data and compute budgets of discovery (Chapters 36–41).

Part IX, The frontier, is a single chapter, Where the room is. It collects the closing sections of all the others into one map of physical and engineering limits, audits the large speed-up claims in the literature, and sets out a research programme.

A timeline of about three hundred dated milestones and an account of how this book was made complete it.

Who it is for

The intended reader is a physicist, a machine-learning researcher or a GPU engineer: clever and patient, but not yet an expert in this subfield. The book assumes linear algebra, probability and some calculus. It does not assume any astronomy beyond what a physics degree supplies; terms such as point-spread function, ecliptic or parallax are defined where they first appear. Astronomers will find the history and the methods familiar, and may find the cross-field comparisons and the compute accounting less so.

Three ways to read it

The chapters are written to stand alone, with cross-references wherever a concept belongs to another chapter. Three paths through them suit different readers.

The historian. Start with the discoveries themselves: Uranus to Neptune, Planet X and Pluto, The Kuiper belt opens and The modern outer Solar System. Continue with The case for Planet Nine and the search for it, near-Earth objects and interstellar visitors, shift-and-stack, deep fields and hidden moons and citizen science, then read A census of exoplanet discovery and the timeline. Return to Chapter 1 when the statistics start to matter.

The algorithm or GPU designer. Read Signals and noise first; everything else depends on it. Then the moving-object core: shift-and-stack and KBMOD, orbit-space stacking, trees and catalogues and linking. Then Part VIII in order: pulsars and radio bursts, gravitational waves, radar and debris, the geometry of search, fast transforms and optimal coadds, GPUs and data and compute. Finish with Where the room is. Machine learning and transits are good side trips: one shows the price of learned shortcuts, the other a field where GPU search has already been done well.

The exoplanet reader. Read Signals and noise and How bright is a hidden world, then Part VII in order: radial velocity, transits, timing, microlensing, direct imaging, astrometry, spectroscopy and other signatures and the census. Add the geometry of search for template banks and fast transforms and optimal coadds for image combination, and end with Where the room is.

Conventions

One notation. Chapter 1 fixes the symbols used throughout: the images $D_i$ and variances $V_i$ , the PSF $P_i$ , the two likelihood images $\psi_i=(D_i/V_i)\star P_i$ and $\phi_i=(1/V_i)\star P_i^2$ , the signed numerator $A$ and information $B$ summed along a trajectory, the significance $\nu=A/\sqrt B$ , the threshold $u$ and the number of templates $M$ . Chapters that borrow a field’s own notation say so.

Measurement, model and estimate. The book distinguishes three kinds of number. A measured number comes from a stated experiment on stated hardware or data, with its source. A model or projection comes from a calculation, such as a roofline model of a GPU kernel or a forecast of a survey’s yield, and is labelled as such. An estimate is the author’s own arithmetic from published figures, marked “(estimate)”. Speed-ups are always quoted with their comparator, and a speed-up at changed sensitivity is called a trade, not a speed-up.

Physical versus engineering limits. Every chapter ends with a section called “Where the room is”, which asks what a new method could change and separates the limits set by photons, geometry and missing information from those set by memory traffic, bank design, thresholds and calibration. The final chapter collects them.

The project’s own work. A few chapters report design studies and GPU measurements made for this book, including a tensor-core benchmark on an NVIDIA A100 in Chapter 40. These are unpublished, have not been independently reproduced, and are always labelled as the project’s own.

Citations. Sources are cited inline by author and year, linked to the arXiv abstract where one exists and otherwise to a DOI or the original page. Each chapter ends with its reference list, and a combined reference index covers the whole book. Figures are drawn from sourced data, and each caption names its source. Boxes mark key ideas, historical vignettes, algorithmic opportunities and contested claims.

The date. The book describes the field as of 16 October 2026. Several of the most important results it discusses appeared in 2025 and 2026, some only as preprints, and some software was changing as the book was written. Where a claim rests on a preprint, a repository’s documentation or a single unreplicated benchmark, the text says so. The book was researched and drafted with the help of AI systems, as described in How this book was made. Its numbers were taken from sources the research workers and writers opened, but errors are possible, and anyone reusing a figure or a citation should check it against the original.

Part I

Foundations

1. Signals and noise: the statistics of finding what is barely there
The mathematics shared by every planet search: the likelihood ratio and matched filter, the two likelihood images ψ and φ that make stacking exact, square-root integration and its 1.25 log₁₀ N magnitude gain, trials factors, sequential tests, injection–recovery completeness and the Cramér–Rao bound. The information in the data and the threshold set what a search can find. Computation decides how close a search gets to that limit.
2. How bright is a hidden world?
The physics that decides whether an unseen body can be detected at all: reflected sunlight falling as the inverse fourth power of distance, thermal emission that falls only as the inverse square when a planet keeps its own heat, and motion whose parallax encodes distance. Together they give a regime map of which method can find what, and where. It has branches that depend on albedo, temperature and survey depth, not one clean boundary.

Part II

A history of discovery

3. Uranus to Neptune: the eye, the chart and the equation
How the planets, asteroids and moons found between 1781 and 1900 were detected: Herschel's magnification test, Piazzi's repeated transits and Gauss's recovery orbit, Le Verrier's inverse perturbation problem and the star chart that confirmed it, the failure of Vulcan, and Max Wolf's photographic trails. Every stage of a modern survey pipeline is already here, carried out by hand.
4. Planet X and Pluto: blink comparators, Schmidt plates and the first CCDs
From Lowell's hunt for Planet X to Spacewatch's real-time CCD software, 1906–1990: how Tombaugh actually found Pluto, why he photographed at opposition and took three plates, how Palomar–Leiden turned asteroid discovery into statistics, how Chiron, Charon and the IRAS 'mystery sources' were found or explained, and how Standish showed that the residuals that had motivated Planet X were errors in the known-planet model.
5. The Kuiper belt opens: CCDs, moving-object software and the bright end
From 1992 QB1 to the Palomar distant-body survey of 2010: how CCD imaging found the Kuiper belt, how wide-field mosaics and automated triplet linking turned discovery into characterised surveys, how shift-and-stack reached fainter, and how Sedna, Eris and Makemake were found at the bright end. Eris was in the discovery images and was first rejected by a motion threshold, which makes it the chapter's central lesson: a software cut can be a detection limit.
6. The slowest movers: the outer Solar System, 2010–2026
How OSSOS, the Dark Energy Survey, the Sheppard–Trujillo search, Subaru, DEEP, New Horizons and early Rubin mapped the far outer Solar System, and why the physics of distant bodies (light falling as r⁻⁴, motion that is mostly our own parallax) makes distance quick to measure but orbits slow to secure.

Part III

Planets found by their pull

7. Weighing the dark: ephemerides, perturbations and the planets they exclude
How planets are found, and ruled out, by their pull: from Adams's and Le Verrier's inversion of Uranus's wanderings to relativistic ephemerides fitted to spacecraft ranging and pulsar timing. The central lesson is that the refit, not the raw perturbation, decides what can be detected.
8. Six orbits pointing one way: the case for Planet Nine
Planet Nine is inferred from the orbits of a dozen or so distant bodies, not from any measured pull on a known planet. This chapter covers the clustering evidence and the fair arguments over survey bias, how the orbit is inferred by N-body simulation and emulation, the predicted sky path, and the competing explanations and other proposed hidden planets. The main lesson is that the evidence is a statistical claim about a selected sample, and it stands or falls with the selection function.
9. Searching the swath: the hunt for Planet Nine
Every direct search for Planet Nine so far, from dedicated Subaru and DECam fields to the archival sweeps of ZTF, DES and Pan-STARRS, and what each one excludes. The chapter explains why a slow, looping, sparsely observed planet defeats ordinary asteroid pipelines, how a null result becomes an exclusion of a reference population, where the surviving 22% lies, and what Rubin and image stacking could do next.

Part IV

The survey pipeline

10. From pixels to detections
How a survey turns raw exposures into a list of candidate moving sources: calibration and astrometry, image subtraction from Alard–Lupton to ZOGY and SFFT, matched-filter detection, real–bogus classification and false-alarm accounting. The thread through all of it is that a detection statistic is only as good as its denominator, and that masks, templates and interpolation quietly change that denominator.
11. Streaks
When a body moves fast enough to trail during one exposure, its photons form a line. This chapter covers how surveys find such lines: thresholded Hough voting, matched-filter template banks, the Radon transform and the fast Radon transform that shares partial sums across lines, and the neural classifiers and detectors that now screen millions of candidates a month. The physical cost of trailing is fixed; how much of the remaining signal a detector keeps is not.
12. Linking: which dots belong together
A survey sees millions of anonymous detections a night. Linking decides which of them are the same moving body. This chapter counts why the problem explodes, then follows the answers: kd-tree MOPS, the heliocentric transformations of HelioLinC and THOR, and the Dark Energy Survey's multi-year searches. The lasting lesson is that the big gains came from changing the coordinates so that wrong associations are easy to reject, not from trying combinations faster.
13. Orbits, identities and the archive
A linked track becomes a body only when an orbit fits it, and a short arc admits a whole family of orbits. This chapter follows orbit determination from Gauss to admissible regions, statistical ranging and the Bernstein–Khushalani method, then the three ways an orbit reaches into the past: attribution of new tracklets, identification of separate orbits, and precovery in archival images. The main lesson: a short-arc orbit is a probability distribution whose shape is set as much by assumptions as by data, and an old image can shrink it more than any new one.
14. Twenty hours' notice: near-Earth objects and interstellar visitors
How the near-Earth object network turns a moving smudge into a designation, an orbit and, rarely, an impact warning, from Spacewatch and LINEAR to ATLAS, ZTF and NEOWISE, the NEO Confirmation Page, digest2, Scout and Sentry-II. The same machinery found 'Oumuamua, Borisov and 3I/ATLAS. Discovery is a property of the whole network, and most of its losses are not in the pixels.
15. Rubin, the alert stream and the discoverer's toolkit
How the Rubin Observatory's Solar System Processing turns difference-image detections into tracklets, HelioLinC3D linkages and MPC submissions, what its first data actually showed, how community alert brokers like Fink and ALeRCE handle moving objects, and which services and libraries a discoverer uses to go from a candidate to a recognised object. Detection, association, orbit and discovery are four different states, and the software should keep them apart.
16. The missing denominator: survey simulators and debiasing
A catalogue of discoveries is a population multiplied by a selection function. This chapter shows how CFEPS, OSSOS, DES, DEEP, JWST and Rubin's Sorcha measure that selection with injected objects and forward simulation, how population models are tested against what a survey would have seen, why apparent orbital clustering can be manufactured by the telescope, and where faster simulation is still needed.

Part V

Below the noise: stacking

17. Shift-and-stack: summing along a guess
How astronomers learned to find bodies too faint to see in any single image by adding exposures along hypothetical trajectories, from the pencil-beam Kuiper-belt searches of the 1990s and Hubble's 10¹⁴-trajectory ACS search to KBMOD, which turns the method into two precomputed images and a GPU gather. The main lesson is that the matched filter is simple, and almost everything hard lies in the size of the search, the artifacts and the bookkeeping.
18. Synthetic and digital tracking: stacking at the speed of near-Earth objects
Shift-and-stack rebuilt for fast movers: second-scale exposures from low-noise cameras, tens of thousands of trial velocities, and GPUs that keep up with the sky. The chapter follows the two lineages that made it work, Shao and Zhai's synthetic tracking for near-Earth objects and Heinze, Metchev and Trollo's large-format digital tracking for faint main-belt asteroids, and shows that their limits are set as much by read noise, curvature, artifacts and memory traffic as by photons.
19. Deep fields and hidden moons
The surveys that made stacking pay: DEEP on DECam, Subaru's search toward New Horizons, the Hubble hunt that found Arrokoth, the fifteen-day HST restack and the JWST deep field, and the searches that took the known moons of the giant planets from dozens to hundreds. The lesson throughout is that a stack is only one stage of an experiment whose cadence, subtraction, vetting, confirmation and selection function decide what it can claim.
20. Stacking along orbits
How to add up the light of a body that is invisible in every exposure when its path across days or years is a curve: the coordinates that straighten orbits, the metric g = JᵀηJ that says how densely trial orbits must be placed, why their number grows as a high power of the baseline, and what heliostack and multi-year digital tracking actually achieved with that machinery.
21. Reusing the sums: trees and catalogues
Two ways to make sub-threshold stacking cheaper: share partial sums across trajectories (Grid Tree, FaXT, exact and interpolating trees, pruning hierarchies), or compress each image to a list of faint peaks and stack the list (Stetzler's catalogue Hough search). Both work, and both make you pay in sensitivity unless the accounting is done carefully.
22. What the networks learn: machine learning at the noise floor
Machine learning has transformed the vetting of moving-object candidates and is starting to propose them directly from image sequences. It has not yet replaced coherent matched-filter integration at the faint limit, and the published comparisons show why: every learned stage that sees fewer photons, or a different simulated sky, pays for it in depth.
23. From space: the survey inside every spacecraft
TESS, Kepler/K2, Gaia, Euclid, NEOWISE, Hubble, Roman, CSST, NEO Surveyor and New Horizons all record moving Solar System bodies, most of them by accident. Each spacecraft applies its own observation operator to the same moving source, and the decisive question is usually not how fast the final search runs but what information the spacecraft kept.

Part VI

Other light, other shadows

24. Heat: finding planets by their own glow
Asteroids and cold distant planets can be found by the heat they emit, from IRAS in 1983 through WISE, AKARI, JWST and NEO Surveyor in the infrared to cosmic-microwave-background telescopes at millimetre wavelengths. The distance-squared advantage of self-luminous planets is real but conditional, and every thermal search is in the end a motion search against a confused, static sky.
25. Shadows on the Earth: occultations
A body too faint to see can still block a star. This chapter treats both kinds of occultation search: predicted events that map the silhouettes, rings, moons and atmospheres of known bodies, and blind high-cadence surveys that try to count kilometre-sized Kuiper belt objects from sub-second diffraction dips. The physics is Fresnel diffraction, the hard part is the extreme tail of the noise.
26. Echoes and pulls: radar, gravity and the exotic
Planetary radar measures distance and spin directly and has discovered moons of near-Earth asteroids, but its fourth-power distance penalty rules out blind searches. Gravity, lensing, clocks and spacecraft can sense a dark mass whatever it is made of, and this chapter shows what each method actually measures, what has been observed, and which numbers are still forecasts.
27. Comets, moons and binaries: when a moving point is not a point
How dust comae, faint tails, unseen satellites and blended pairs are found around bodies that are already known: the physics of each signal, the statistics that separate it from a single point source, and why every method has its own blind spot. The main lesson is that the extra information is usually already in the pixels, and that calibration, not arithmetic, limits how much of it is recovered.
28. Many eyes on the archive
How volunteers and skilled amateurs find comets, asteroids, activity and cold brown dwarfs in public data, the statistics that turn many noisy votes into a candidate list, and why the real bottleneck is the human queue rather than the classifier.

Part VII

Exoplanets

29. The star that moves: radial velocity
How a star's reflex motion is measured to a few tens of centimetres per second, and how planets are searched for in the resulting velocities. Instruments are no longer the main limit. The star's own surface is, and separating the two has become an inference problem in which the useful algorithmic room lies in what information is kept, not in how fast a periodogram runs.
30. Shadows on the star: transits
How a transit survey turns light curves into planets: detrending that is part of the detector, a matched-filter search over periodic combs (BLS, TLS, Kepler's TPS, fast folding, GPU codes) with its grid and false-alarm statistics, and the vetting and statistical validation that decide what a dip is. The search is a trajectory search, algebraically the same as shift-and-stack, and its hard problems are noise models and selection calibration rather than arithmetic.
31. Timing: planets that make clocks run late
How planets are found from the arrival times of pulses, transits, eclipses and stellar oscillations: light-travel delays, dynamical transit-timing variations and the impostors that mimic both. The central computational problem is inversion, not detection: many architectures fit the same times, and proving uniqueness can cost ten thousand CPU-hours per system.
32. Microlensing: planets that bend starlight
How a planet is found by the brief distortion its gravity adds to a background star's lensing brightening: the physics of caustics, the search pipelines from EventFinder to AnomalyFinder, the solvers from ray shooting to GPU contour integration, and why the hard computational problem is not fitting the best model but proving that no other model fits.
33. Direct imaging: subtracting a star to see its planets
How coronagraphs, angular and spectral diversity, PSF subtraction (ADI, LOCI, KLIP, PACO, TRAP), forward models and honest small-sample statistics pull planets out from under stellar speckles; how K-Stacker and PACOME stack epochs along Keplerian orbits exactly as Solar-System searches stack along heliocentric ones; and what machine learning and interferometry add. The main lesson is that detection power comes from diversity and calibration, and that on modern hardware the orbit search itself should be cheap.
34. The star's wobble, the planet's lines, and other signs
Astrometry finds planets in the changing position of their stars, from Barnard's-star plates to Gaia; high-resolution cross-correlation spectroscopy finds them by summing molecular lines along Doppler trajectories, which is shift-and-stack in velocity space; and disk kinematics, phase curves, polarimetry, radio emission, white-dwarf debris and exomoon searches add rarer, more model-dependent signatures. The common lesson: the matched filter is easy, while calibration, impostors and the definition of a detection are hard.
35. A census of exoplanet discovery
How 6,451 accepted exoplanets were found, method by method and false claim by false claim, and how a catalogue shaped by instruments, algorithms and follow-up is turned into occurrence rates. The main lesson: the count is the easy part, and the denominator, measured by injection, recovery and reliability, is where both the science and the compute lie.

Part VIII

Algorithms and compute

36. Summing along a curve: lessons from pulsars and radio bursts
Pulsar and fast-radio-burst searches solve the faint-moving-object problem in one dimension: integrate weak evidence along an unknown curve, over a vast bank of curves, without being fooled by the noise. This chapter explains their statistics, their trees and their pruning schemes, from Taylor's 1974 dedispersion tree and the fast folding algorithm to FDMT and LOKI, and maps each one onto the search for faint bodies in stacks of images.
37. Evidence before detection: lessons from gravitational waves, radar and space debris
Continuous gravitational-wave searches, radar track-before-detect and optical space-debris surveys all accumulate evidence along unknown trajectories before deciding that anything is there. This chapter explains how each field builds its template banks, splits coherent from semicoherent integration, organises hierarchies and pays for them, and maps every idea onto the search for faint moving bodies in astronomical images.
38. The geometry of search
How many trial trajectories a blind search must test, and how many of them it must finish. Mismatch metrics, lattice and random template banks, orbit libraries, and the sequential and hierarchical pruning that lets pulsar, gravitational-wave and planet searches decline most hypotheses without losing the faint ones. Geometry buys factors of a few; calibrated pruning can buy more, but only with an honest accounting of what it discards.
39. Fast transforms and optimal coadds
Every stacking search is a large set of weighted sums. This chapter derives what those sums must preserve (the proper coadd for a static sky, the score and information maps for a moving one), then surveys the fast algorithms that share work between sums: FFTs and NUFFTs, recursive Radon and X-ray transforms, FDMT-style trees, butterflies, multipoles and sketches. Each is exact or fast under a specific structural assumption, and the hard part is keeping the noise normalisation right.
40. GPUs: where the bytes go
How graphics processors accelerate the search for faint moving bodies, why a trajectory gather and a matrix multiplication sit on opposite sides of the roofline, what tensor cores and low precision do to a detection statistic, and an honest account of this project's first A100 measurement of tensor-core velocity stacking: 5.9 times faster than the benchmark's best gather, but probably less than twice as fast as a gather done well.
41. Data and the compute of discovery
Which survey archives can actually be searched for moving bodies and what an archive must preserve to make that possible, followed by a history of what discovery has cost, from Tombaugh's three days per plate pair to tens of millions of CPU-hours and a few hundred GPU-hours. The lesson is that resources, the search performed and validated yield have to be reported together, and that data movement, not arithmetic, increasingly sets the price.

Part IX

The frontier

42. Where the room is: photons, engineering and the next order of magnitude
The book's synthesis. Across every method in the book, which limits are physical (photon noise, confusion, missing information, the matched-filter ceiling) and which are engineering or algorithmic; which reported speed-ups kept the statistic; a ranked map of where an order of magnitude in compute at equal sensitivity may remain; the project's own conclusions with their evidence level; and a concrete research programme with kill criteria.