The star's wobble, the planet's lines, and other signs

In 1963 Peter van de Kamp of Swarthmore’s Sproul Observatory announced that Barnard’s star, one of the Sun’s nearest neighbours, wobbled. He had measured its position on 2,413 photographic plates, comprising 8,260 exposures on 609 nights between 1916 and 1962, taken with the observatory’s 24-inch refractor. After removing the star’s large proper motion and parallax he found an orbit with a 24-year period and a semimajor axis of $0.0245\pm0.002$ arcseconds, and inferred a companion of 1.6 Jupiter masses (van de Kamp 1963). It would have been the first planet found around another star.

It was not there. Ten years later John Hershey remeasured another star field from the same telescope and found that all its stars shifted together at about 1949, when the objective lens was worked on and the emulsion changed, and again around 1957, when the objective was adjusted (Hershey 1973). George Gatewood and Heinrich Eichhorn measured 241 plates from two other observatories and did not recover the orbit (Gatewood & Eichhorn 1973). The lesson was not that old astrometry was bad. It was that an instrument can produce coherent, slowly changing distortions that survive averaging and look exactly like the long periods one is searching for.

This chapter covers the methods that find planets without imaging them directly or watching them transit. Astrometry measures the star’s reflex motion across the sky. High-resolution cross-correlation spectroscopy (HRCCS) measures the planet’s own spectrum by following its lines as they Doppler-shift around the orbit. A collection of rarer signatures follows: kinks in the velocity field of a protoplanetary disk, reflected-light phase curves, polarisation, coherent radio bursts, the debris of planets polluting white dwarfs, and the search for moons and rings. Two of these, astrometry and HRCCS, are matched filters along trajectories, close relatives of the stacking methods in Part V. Most of the others are inverse problems where the hard part is excluding everything else that could produce the same data.

Astrometry: the star’s wobble

The signal

A star of mass $M_\star$ and a planet of mass $m$ on a relative orbit of semimajor axis $a$ orbit their common barycentre. The star’s barycentric orbit has semimajor axis $a_\star=a\,m/(M_\star+m)$, and at distance $d$ its angular size is $\alpha_\star[{\rm arcsec}]=a_\star[{\rm au}]/d[{\rm pc}]$. For $m\ll M_\star$,

(Lammers & Winn 2025). A Jupiter analogue at 10 pc moves its star by about 0.50 mas; an Earth analogue by about 0.30 µas (illustrative values from the formula). Because $a\propto P^{2/3}$ at fixed stellar mass, the signal grows with period, until the period exceeds the observing baseline and the mission sees only a fragment of the ellipse. That is the opposite bias to radial velocities and transits, which favour short periods. Astrometry also measures the orbit’s inclination, so for a dark companion it yields a true mass rather than the radial-velocity $m\sin i$ of Chapter 29.

The instrument, however, does not follow the star. For an unresolved pair it follows the photocentre, the light-weighted mean position. With mass fraction $\mathcal B=m/(M_\star+m)$ and light fraction $\beta=F_2/(F_1+F_2)$ in the instrument’s band, the photocentre orbit has semimajor axis

with $\varpi$ the parallax in the same angular unit (Gaia Collaboration, Arenou et al. 2023). A dark planet has $\beta\simeq0$ and the photocentre follows the star. A pair of nearly equal stars has $\mathcal B\simeq\beta$ and the photocentre barely moves, however large the true orbits are. In between, a stellar binary with an unequal light ratio can produce a photocentre orbit of exactly the size expected from a planet. This is a physical degeneracy, not a statistical fluctuation, and it is the main source of false positives in Gaia’s planetary candidates.

The measurement model

Gaia scans the sky and measures one coordinate at a time, along the scan direction. Let $x=\Delta\alpha\cos\delta$ point east and $y=\Delta\delta$ north. For scan angle $\psi_i$ measured from north through east, the along-scan measurement is

where the barycentric motion is $x_0+\mu_x\tau_i$, $y_0+\mu_y\tau_i$ and $f_{\varpi,i}$ is the known parallax factor. The single-star model has five parameters: two positions, two proper motions and parallax. A Keplerian companion adds, after solving $E_i-e\sin E_i=2\pi(t_i-T_0)/P$ and forming $X_i=\cos E_i-e$, $Y_i=\sqrt{1-e^2}\sin E_i$,

where the Thiele–Innes constants $A,B,F,G$ combine $\alpha_{\rm ph}$, $i$, $\Omega$ and $\omega$ (Holl et al. 2023). The decisive fact for computation is that at fixed $(P,e,T_0)$ the model is linear in nine parameters, the five astrometric ones and the four Thiele–Innes constants. A search therefore scans a three-dimensional nonlinear space and solves a small weighted least-squares problem at every point. This is the same variable-projection structure as a radial-velocity Keplerian periodogram.

Linear nuisance parameters also explain which orbits are hard to see. For a known covariance $C$ and null design matrix $H$, whiten with $\widetilde H=C^{-1/2}H$ and define the projector $R_H=I-\widetilde H(\widetilde H^T\widetilde H)^{-1}\widetilde H^T$. An orbital signal $\mathbf s$ has squared significance approximately

(a standard linear-regression result). Whatever part of the orbit resembles position, proper motion or parallax is projected away. When $P$ is much longer than the baseline, the first terms of the orbit’s Taylor expansion are absorbed into position and proper motion, leaving only curvature; near a one-year period, some orientations and phases mimic parallax. These are information limits. No optimiser can recover what the projection removed.

Two cautionary tales

Hipparcos brought space astrometry and one-dimensional intermediate measurements. Combining a radial-velocity orbit with those measurements seemed an economical route to true masses, and early reductions found a remarkable excess of nearly face-on orbits, implying large masses. Dimitri Pourbaix and Frédéric Arenou showed in 2001 that the excess was largely produced by the fitting: forcing an orbit into data with little orbital information pushes the inclination toward face-on, because an optimiser always returns an inclination, whether or not the data measured one (Pourbaix & Arenou 2001). A narrow conditional solution is not a detected orbit.

In 2009 Steven Pravdo and Stuart Shaklan announced a planet around the ultracool dwarf VB 10 from Palomar astrometry: $6.4^{+2.6}_{-3.1}\,M_{\rm J}$ on a 0.744-year orbit, with a single-star-model probability of $2\times10^{-8}$ and a periodogram false-alarm probability of $3\times10^{-5}$ (Pravdo & Shaklan 2009). Guillem Anglada-Escudé and collaborators obtained radial velocities with MIKE on Magellan and ESPaDOnS on CFHT and refitted the astrometry with proper motion, parallax and orbit together. The astrometry alone recovered the 270-day signal but preferred another near 50 days, and the velocities undermined the companion (Anglada-Escudé et al. 2010). Impressive nominal significance under one reduction says little about the physical claim.

Accelerations: Hipparcos and Gaia together

A planet with a period of decades produces no closed ellipse in a three-year mission, but it bends the star’s path. Timothy Brandt’s Hipparcos–Gaia Catalog of Accelerations (HGCA) cross-calibrates the two missions so that the positional difference over their 25-year separation gives a precise long-term proper motion $\boldsymbol\mu_{HG}=(\mathbf x_G-\mathbf x_H)/(t_G-t_H)$. Comparing it with Gaia’s own short-baseline proper motion gives the anomaly $\Delta\boldsymbol\mu_G=\boldsymbol\mu_G-\boldsymbol\mu_{HG}$ (Brandt 2018). The EDR3 edition matches 115,346 stars, calibrates local frame rotations, mixes the two Hipparcos reductions 60/40, and inflates Gaia uncertainties by a factor of 1.37; its colour and magnitude frame terms reach about 50 µas per year, large enough to matter for planets (Brandt 2021). Pierre Kervella and collaborators run a parallel programme and report a median precision in tangential-velocity anomaly of 26 cm s$^{-1}$ per parsec of distance, with 37,515 Hipparcos stars showing an anomaly above S/N 3, mostly due to stellar companions (Kervella et al. 2022).

The detection statistic is a two-dimensional chi-square, $Q=\Delta\boldsymbol\mu^TC_{\Delta\mu}^{-1}\Delta\boldsymbol\mu$, whose ideal tail is $e^{-Q/2}$. A Mahalanobis radius of 3 ($Q=9$) therefore has a tail probability of 1.1%, not the 0.27% of a one-dimensional “three sigma”. A proper-motion anomaly converts to a velocity as $\Delta v_t[{\rm m\,s^{-1}}]=4740.47\,\Delta\mu[{\rm mas\,yr^{-1}}]/\varpi[{\rm mas}]$, and for a dark companion on a circular face-on orbit the star’s speed is $v_\star\simeq m\sqrt{G/(aM_\star)}$. An isolated anomaly therefore constrains only the combination $m/\sqrt a$. As an illustration, Jupiter moves the Sun at about 12.5 m s$^{-1}$; seen from 10 pc that is an anomaly of about 0.26 mas per year, against Kervella’s median precision of about 2.6 m s$^{-1}$ at that distance (my arithmetic, ignoring projection and the smoothing by the mission windows, which is substantial).

The productive use of accelerations has been to choose targets. Thayne Currie and collaborators imaged the accelerating star HIP 99770 with Subaru’s SCExAO/CHARIS and found a companion near 17 au with a prior-dependent dynamical mass of 13.9–16.1 Jupiter masses (Currie et al. 2023). HIP 54515 b, found the same way in 2025 with five imaging epochs at 0.145–0.192 arcsec, has a dynamical mass of $17.7^{+7.6}_{-4.9}\,M_{\rm J}$, straddling the conventional planet–brown-dwarf boundary (Currie et al. 2025). The acceleration was the clue; the imaging and the joint fit made the discovery. The interferometric discoveries of Chapter 33 use the same logic with Gaia orbits.

Gaia DR3: the first orbit catalogue

Gaia DR3, based on 34 months of data, contains the first population-scale catalogue of astrometric orbits. Its dedicated exoplanet pipeline, developed by Berry Holl, Alessandro Sozzetti, Johannes Sahlmann and colleagues, used two independent search methods that both exploit the linear Thiele–Innes structure: a differential-evolution MCMC over period, eccentricity, periastron time and jitter, solving the linear part by QR, and a genetic algorithm with Fourier initialisation and explicit alias mutations. Each was allowed up to 60 seconds per source. They ran on 2,457,530 sources and on a targeted list of 19,845; after aggressive filtering, 1,162 solutions were published, of which 198 carried a “validated” label meaning agreement with existing information (Holl et al. 2023). The filters were specific: for the stochastic sources, more than 36 field-of-view transits, parallax agreement within 5% and an orbital semimajor-axis significance above 20; for the targeted list, heterogeneous cuts including a BIC improvement above 30 over both five- and seven-parameter models. Only 17 solutions fall below $20\,M_{\rm J}$ in the paper’s illustrative pseudo-mass, which assumes a solar-mass host. These are not 1,162 planets. Formal covariances were reconstructed from Jacobians rather than exported from the often asymmetric MCMC chains, so competing aliases and high-eccentricity tails are lost from the summary; repeating the model comparison needs the epoch data. Gaia’s wider multiplicity analysis selected 72 planetary candidates under assumptions about light and mass (Gaia Collaboration, Arenou et al. 2023).

Follow-up has been sobering. Gudmundur Stefánsson and collaborators observed 28 northern K and M dwarf candidates spectroscopically with HPF, NEID and FIES. Twenty-one turned out to be double-lined stellar binaries. Two were confirmed: Gaia-4b, a planet of $11.8\pm0.7\,M_{\rm J}$ with $P=571.3\pm1.4$ days, and Gaia-5b, a brown dwarf of $20.9\pm0.5\,M_{\rm J}$. For the original 72-candidate list they quoted a confirmed contamination of at least $23/72=32\%$ and up to $60/72=83\%$ if unresolved cases are false (Stefánsson et al. 2025). Three of the 72 solutions were also withdrawn after a software bug, a failure mode different from both noise and binaries. Holl et al. had estimated 5–10% spurious or incorrect solutions from internal checks; the numbers have different denominators and definitions and are not in contradiction. Gaia-4b is the first secure planet selected from Gaia astrometric orbits and independently confirmed.

Searching and fitting

A scalar Lomb–Scargle periodogram run on along-scan residuals ignores the changing scan geometry. Jean-Baptiste Delisle and Damien Ségransan’s scan-aware periodogram adds, at each angular frequency $\nu$, four columns $\cos\psi_i\cos\nu t_i$, $\sin\psi_i\cos\nu t_i$, $\cos\psi_i\sin\nu t_i$ and $\sin\psi_i\sin\nu t_i$, which fit the projected fundamental of any sky ellipse, and refits the null terms simultaneously. They derive an analytic maximum-peak false-alarm probability by the Rice method, and the Fourier coefficients initialise eccentric fits (Delisle & Ségransan 2022). Direct evaluation costs $O(MNp^2)$ for $M$ frequencies, $N$ measurements and design width $p$, and the grid should resolve phase over the baseline $T$, roughly $\Delta f\lesssim1/T$ before oversampling.

fit the null model H (position, proper motion, parallax) to all scans
for each frequency in a recorded search range:
    build the four scan-projected sinusoid columns S
    solve weighted least squares for [H, S]; record the chi-square improvement
keep several separated peaks and their aliases
estimate the maximum-peak false-alarm probability for this cadence
initialise eccentric Keplerian fits from the Fourier coefficients
calibrate the whole search with null simulations and injections

A Gaia field crossing gives eight or nine CCD measurements, and those share systematics; treating every CCD sample as independent with too small an error floor overstates significance. The DR3 pipeline added a typical 0.08 mas calibration term and a fitted jitter for this reason (Holl et al. 2023).

For full inference, Brandt’s orvara fits radial velocities, relative astrometry from imaging, and absolute astrometry together. It marginalises radial-velocity zero points, parallax and barycentric proper motion analytically, and uses htof to replay each trial orbit through the actual Hipparcos and Gaia sampling, refitting the catalogue model to predict what the catalogue would have measured (Brandt et al. 2021). Its timing breakdown is a useful Amdahl’s-law exercise for anyone planning a GPU port.

On one 2.2 GHz Xeon core, one iteration for the Gl 758 system took 85 µs, of which 44 went to solving Kepler’s equation. Running on five processors gave only about a twofold speedup. William Thompson’s differentiable Julia framework Octofitter took a different route, Hamiltonian sampling with gradients; in one 27-epoch relative-astrometry benchmark it converged in 11 minutes against 98 for orvara and 923 for orbitize! (Thompson et al. 2023). Sampler geometry can matter more than raw likelihood throughput. At population scale, Kareem El-Badry and collaborators’ gaiamock ran a DR3-like model cascade on 46 million simulated unresolved binaries in about 80 hours on one 64-core node, roughly 5,000 CPU-hours, dominated by the 1% of sources that reached the nonlinear orbit stage; they estimate 50,000 CPU-hours for a DR4 equivalent. The same simulations reproduce spurious orbital periods near 91, 31.5 and 182 days from marginally resolved binaries whose blended images change with scan angle (El-Badry et al. 2024).

Radio and ground-based astrometry

The VLBA measures positions of radio-emitting stars against distant quasars. Salvador Curiel and collaborators reported a candidate around the M9 dwarf TVLM 513-46546 with a $221\pm5$ day period and 0.35–0.42 Jupiter masses, but its fitted reflex amplitude was only $0.17\pm0.10$ mas, 1.7 sigma, with a combined periodogram false-alarm probability of 5.4% (Curiel et al. 2020). For the binary GJ 896AB they combined VLBA positions with decades of relative astrometry and found a planet of $2.26\pm0.57\,M_{\rm J}$ on a $284.39\pm1.47$ day orbit tilted about 148 degrees to the binary’s, a retrograde architecture (Curiel et al. 2022). The observable is the radio centroid, which may not coincide with the stellar centre when magnetospheric emission changes.

On 7 October 2026 Sahlmann and collaborators posted PALTA-1 b, a candidate around an early-L dwarf from 28 VLT/FORS2 epochs over 8.4 years, with a photocentre orbit of best-fit period $8.8^{+8.4}_{-1.1}$ years and a companion of 0.54–0.77 Jupiter masses, conditional on the host’s poorly known mass. Field distortion was fitted with polynomials of degree two to seven, because motion across a static, imperfectly known distortion can itself become an apparent orbit; image elongation tests excluded an equal-brightness binary under the adopted models (Sahlmann et al. 2026). It remains a candidate. Barnard’s star, sixty years on, is the reason for that care.

What Gaia DR4 will bring

ESA lists 2 December 2026 for Gaia DR4, based on 66 months of observations and including epoch astrometry (ESA Gaia release schedule). At the time of writing it has not been released, so its planet yield is a forecast.

Michael Perryman and collaborators’ 2014 forecast predicted $21{,}000\pm6{,}000$ high-mass planets from a five-year mission and $70{,}000\pm20{,}000$ from ten years (Perryman et al. 2014). Caleb Lammers and Joshua Winn’s 2025 update, using demonstrated astrometric performance (a 54 µas floor per field crossing for bright stars) and updated occurrence rates, predicts $7{,}500\pm2{,}100$ and $120{,}000\pm22{,}000$, of which $1{,}900\pm540$ and $38{,}000\pm7{,}300$ would have well-constrained masses and periods. Their uncertainty is dominated by planet occurrence; their simulated binary impostors approach 50% of the planets for M dwarfs inside 1 au (Lammers & Winn 2025). Multi-planet systems add another trap: an eccentric single orbit and a 2:1 resonant pair have nearly the same astrometric signature, with apparent eccentricity $e_{\rm eff}\simeq2^{1/3}m_i/m_o$ for inner and outer masses $m_i$ and $m_o$ (Yahalomi et al. 2026), and one-planet fits near resonance can bias masses by up to 100% in simulation (Greenfield & Owen 2026). The NASA Exoplanet Archive currently credits astrometry with six confirmed discoveries. DR4 is the moment that number is expected to change by orders of magnitude.

High-resolution cross-correlation: following the planet’s lines

Two experiments on CRIRES

On 7 August 2009 Ignas Snellen, Remco de Kok, Ernst de Mooij and Simon Albrecht observed the transiting hot Jupiter HD 209458 b with the CRIRES spectrograph and adaptive optics on the VLT. Over about five hours, covering the three-hour transit and baselines on both sides, they took 51 spectra between 2291 and 2349 nm at resolving power $R=100{,}000$. Their model predicted 56 strong carbon monoxide lines in the planet’s transmission spectrum. During the transit the planet’s orbital velocity along the line of sight changed by about 30 km s$^{-1}$, so its absorption lines moved while those of Earth’s atmosphere and the star stayed almost still. Each spectrum contained the signal at about unity S/N. Cross-correlating with a CO template and aligning in the planet’s frame gave a combined $5.6\sigma$ detection, a planetary orbital velocity of $140\pm10$ km s$^{-1}$, and with the stellar velocity and the known inclination, masses of $1.00\pm0.22\,M_\odot$ for the star and $0.64\pm0.09\,M_{\rm J}$ for the planet (Snellen et al. 2010).

These were detections of planetary light from known planets, not discoveries of new ones. Their importance for this book is the method: an invisible signal recovered by summing evidence along a trajectory.

The signal model

After normalisation and removal of telluric and stellar features, write the residual time series as $\mathbf d=\alpha\,\mathbf s(\vartheta)+\boldsymbol\epsilon$, with model $\mathbf s=\mathcal P\,\mathcal D(\vartheta)\,\mathbf m(\theta)$. Here $\mathbf m$ is the planet spectrum for atmospheric parameters $\theta$, $\mathcal D$ applies Doppler shifts, broadening and phase weights for orbital parameters $\vartheta$, and $\mathcal P$ is the effect of the reduction itself, which must be applied to the model as it was to the data (Brogi & Line 2019; Gibson et al. 2022). For known covariance the matched filter is again

with fitted amplitude $\widehat\alpha=A/B$. If lines of equal strength contribute independently, S/N grows as $\sqrt{N_{\rm lines}}$, and equal exposures add another $\sqrt{N_t}$. For $\tau$ Boo, $\sqrt{452}\simeq21$ from exposures and a roughly fivefold gain from lines explain why a signal invisible in any spectrum reached $6\sigma$. “Number of lines” is an effective quantity: telluric overlap, blends, weak lines and template errors reduce it. Removing a perfect model of the stellar spectrum does not remove its photon noise, a point William Sparks and Holland Ford made in their 2002 proposal for coronagraphic integral-field spectroscopy (Sparks & Ford 2002).

For a circular orbit with phase $\phi(t)=(t-T_0)/P$,

where $\gamma$ is the systemic velocity and $v_{\rm obs}$ the observer’s barycentric correction. Measuring both the star’s amplitude $K_\star$ and the planet’s $K_p$ makes the system a double-lined binary: $M_p/M_\star=K_\star/K_p$ and $(M_\star+M_p)\sin^3i=P(1-e^2)^{3/2}(K_\star+K_p)^3/(2\pi G)$. For a non-transiting planet, an independent stellar mass is still needed for the inclination. At $R=100{,}000$ one resolution element is $c/R\simeq3.0$ km s$^{-1}$, but combining many lines measures the velocity far more precisely.

Spectral shift-and-stack

The HRCCS algorithm has two compressions. First, for each exposure $t$ and order $o$, correlate the residual spectrum with the template at each trial velocity:

This aligns many lines within one spectrum. Second, for each orbital hypothesis $\eta=(K_p,\gamma,\ldots)$, sum along its velocity trajectory:

with $a_t$ the planet’s relative visibility at that phase. This aligns many exposures. It is precisely the shift-and-stack of Chapter 17 with wavelength in place of sky position: each exposure is turned into a pair of matched-filter maps, now functions of velocity instead of position, and an orbit selects one sample from each map. Summing individually squared correlations instead would turn a common signed amplitude test into a weaker power test.

reduce spectra -> flux, uncertainty, masks, barycentric times and phases
fit the stationary nuisance (tellurics, star) with a fixed, documented rule
for each atmospheric template theta:
    m = broaden_and_normalise(spectrum(theta))
    for each exposure t, order o, trial velocity v:
        A[t,o,v], B[t,o,v] = weighted dot(residual, shift(m, v)), weighted norm
    for each orbit (Kp, gamma, ...):
        score = sum of A along v_p(t) / sqrt(sum of B along v_p(t))
inject planets before detrending and repeat every choice in null tests

Brogi and collaborators’ $\tau$ Boo search correlated from $-150$ to $+150$ km s$^{-1}$ in 1.5 km s$^{-1}$ steps, 201 velocities, interpolated the correlations into trial planetary frames and combined three nights. For eccentric orbits the trajectory becomes $K_p[\cos(\nu+\omega_p)+e\cos\omega_p]+\gamma$ with the planet’s own argument of periastron, which differs from the star’s by $180^\circ$; using the stellar value is a sign error that an injected spectrum would expose, as it would a wrong sign in the barycentric correction.

The geometry also explains the cross-shaped peaks in published $K_p$–$v_{\rm sys}$ maps. A wrong trial line $(K',\gamma')$ intersects the true line at $s^\ast=-(\gamma'-\gamma)/(K'-K)$ and collects part of the signal whenever $s^\ast$ falls inside the observed range of $s_t$. The set of such trial points is a fan of lines through the true point, with slopes set by the observed phases: two arms of an X. For $\tau$ Boo’s phases 0.37–0.63, $s_t$ runs from about $+0.73$ to $-0.73$. Daniel Sebastian, Amaury Triaud and Brogi’s Saltire fits this predictable shape instead of a one-dimensional cut, which measures the centre more robustly (Sebastian et al. 2024).

Removing the stationary spectra

Before correlation, the telluric and stellar spectra must be removed, and the planet must survive. Options include fitting a telluric radiative-transfer model, regressing on airmass, principal-component analysis, and SYSREM, which iteratively subtracts weighted rank-one matrices $u_tw_j$ and reduces to PCA under uniform weights. Too few components leave strong stationary residuals; too many fit away the moving planet. A moving spectrum is not immune to PCA: several stationary components can approximate a diagonal trail, and removing all 79 available components in one simulation destroyed the signal (Savel et al. 2025).

The model must be filtered the same way. Re-running the whole detrending for every trial atmosphere is faithful but expensive. Neale Gibson and collaborators showed in 2022 that one can keep the fitted temporal basis $U$ and project it out of each model column with $\mathcal P=I-U(U^TWU)^{-1}U^TW$, at cost $O(ELr)$ per order for rank $r$, $E$ exposures and $L$ pixels, assuming a weak planet does not change the nuisance subspace (Gibson et al. 2022). They also found slightly more than a twofold forward-model speedup from FP32 instead of FP64.

Correlation values are not likelihoods. Brogi and Michael Line’s 2019 likelihood, for a spectrum of $N$ channels with data variance $s_f^2$, model variance $s_g^2$ and cross-covariance $R$, profiles an unknown common variance to give

up to constants (Brogi & Line 2019). Unlike a normalised correlation coefficient, it keeps the model’s amplitude and the sign of the correlation, so absorption and emission are no longer equivalent. Independent log-likelihoods add over exposures, orders, nights and instruments; arbitrary normalised correlation peaks do not.

Trials and robustness

A map’s off-trail scatter is a convenient scale, not a tail probability. Neighbouring velocities are correlated over the template’s autocorrelation width, and neighbouring $K_p$ values share almost all their spectra, so counting map pixels as independent trials overcorrects while ignoring the search undercorrects. Worse, analysts choose molecules, order masks, detrending ranks and filter widths. Samuel Cabot and collaborators found apparent signals of $4\sigma$ or more at incorrect velocities in nearly 30% of the optimisation cases they tested (Cabot et al. 2019). Connor Cheverall and collaborators recommend tuning on the difference between injected and uninjected correlations, which reduces the tendency to pick settings that amplify favourable noise (Cheverall et al. 2023).

Velocities are not settled either. A 2026 CRIRES+ study of the non-transiting 51 Pegasi b found $K_p=102.8^{+8.3}_{-9.1}$ km s$^{-1}$ and $M_p=0.61^{+0.06}_{-0.05}\,M_{\rm J}$, against earlier values around 133 km s$^{-1}$; its authors present it as an alternative orbital solution requiring further observations (Cont et al. 2026). Short phase coverage lets $K_p$ trade against systemic velocity and ephemeris errors, and winds and changing line-forming regions can move the apparent centroid off the centre-of-mass track.

Molecule maps

When the planet is spatially resolved, an integral-field spectrograph supplies $d(x,y,\lambda)$, and spatial separation does the work that Doppler motion does for unresolved planets, so much lower resolution suffices. In 2018 Jens Hoeijmakers and colleagues reanalysed archival 2014 SINFONI observations of $\beta$ Pictoris at $R\sim5000$. They divided each spaxel’s spectrum by a master stellar spectrum, removed the smooth chromatic scaling, cleaned residuals with SYSREM, and correlated with templates from $-2500$ to $+2500$ km s$^{-1}$ in 10 km s$^{-1}$ steps. CO and water recovered the known planet at S/N 14.5 and 17.0, and a full atmospheric template at 25.0 (Hoeijmakers et al. 2018). VLT/ERIS observations of AF Lep b at $R\sim11{,}000$ gave water, CO and full-template S/N of 9.3, 9.7 and 20.7, and a planet–star radial velocity of $7.8\pm1.7$ km s$^{-1}$ that resolved the sign ambiguity of its astrometric orbit (Hayoz et al. 2025). Ignas Snellen and collaborators had quantified why combining high-dispersion spectroscopy with high-contrast imaging works: when stellar photon noise dominates, suppressing the starlight by a factor $K$ improves S/N by $\sqrt K$, until other noise terms take over (Snellen et al. 2015).

In September 2026 R. Kravtchenko and collaborators applied molecular mapping to JWST integral-field data of $\epsilon$ Indi Ab, the cold giant imaged by MIRI (Chapter 33). They report ammonia maps with S/N up to 15.6 and full-template S/N of 15.2 with NIRSpec (Kravtchenko et al. 2026). They also chose, for each cube, the high-pass cutoff that maximised S/N from 200 trial values, so these are target-optimised map statistics rather than calibrated blind-search significances. The figure shows where the planet’s line structure carries information, which is not where its flux is greatest.

Other ways to use the same data

Christopher Watson and collaborators adapted Doppler tomography from binary stars: instead of backprojecting along each sinusoid, solve for a velocity-space intensity map whose forward projection fits all spectra, with maximum-entropy regularisation. On the original $\tau$ Boo data it gave $K_p=109.6\pm2.2$ km s$^{-1}$, though regularised maps have non-Gaussian noise, so peak over scatter is not a significance (Watson et al. 2019). Emily Garvin and collaborators fed whole correlation-versus-velocity vectors to a one-dimensional CNN; on synthetic planets injected into real SINFONI noise, completeness at false-discovery rates up to 5% rose from 0.7% to 55.5% (Garvin et al. 2024), an injection-experiment result, not a 77-fold gain in flux limit. Nikolai Piskunov and collaborators’ TSD instead treats transmission spectroscopy as a regularised inverse problem for the stellar, telluric and planetary spectra, running 300 iterations over 114 phases and 36,000 wavelengths in 25 minutes on an Apple M2 Ultra CPU (Piskunov et al. 2026).

Where the time goes

For $E$ exposures, $O$ orders, $L$ pixels per order and $V$ velocities, direct correlation costs $O(EOLV)$ per template and trajectory summation $O(EOQ)$ for $Q$ orbits. Take the IGRINS time series of WASP-77 Ab reanalysed in POSEIDON, 79 exposures and 54 orders, and assume 2048 pixels per order and 401 velocities: one template’s correlations are about $3.5\times10^9$ multiply-accumulates, while a $201\times201$ $(K_p,\gamma)$ map needs only about $3.2\times10^6$ lookups (illustrative estimates). The trajectory sum, the part that looks like stacking, is negligible. Retrievals are another regime: each of hundreds of thousands of likelihood calls rebuilds a radiative-transfer model, broadens it, shifts it and filters it. POSEIDON’s typical emission retrieval took about seven hours on 24 CPUs with 12 parameters and 800 live points (Wang et al. 2025). Arjun Savel and collaborators cut one retrieval from eight days on 24 CPUs plus an A100 to 15 hours on one CPU plus the A100, by moving to JAX, changing interpolation and switching samplers, or to five hours with local sampling only (Savel et al. 2025). That is a whole-workflow change of about 13 times, not a kernel benchmark.

Other signatures

Disk kinematics

Gaps and spirals in protoplanetary disks have many possible causes. The decisive step was to measure gas velocities. In May 2018 two independent teams analysed archival ALMA observations of HD 163296. Christophe Pinte, Daniel Price, François Ménard and colleagues found a localised kink in the CO isovelocity contours, on one side of the disk only and in two CO transitions, and compared it with three-dimensional simulations; a planet of about two Jupiter masses at about 260 au best reproduced its morphology among their 1, 2, 3 and 5 Jupiter-mass models (Pinte et al. 2018). Richard Teague, Jaehan Bae, Edwin Bergin, Tilman Birnstiel and Daniel Foreman-Mackey measured the rotation curve in annuli and found pressure-gradient signatures consistent with Jupiter-mass planets at 83 and 137 au (Teague et al. 2018). These were different signatures at different radii. The NASA Exoplanet Archive lists one planet discovered by disk kinematics.

Radial force balance in a thin disk gives the rotation speed relative to Keplerian,

for emission height $z$, sound speed $c_s$ and gas pressure $P$ (Pinte et al. 2023). A gap has opposite pressure slopes at its edges, giving a structured residual, but an error in the assumed emission height imitates part of it. Teague’s method is itself a stack: in each annulus a trial rotation velocity predicts every pixel’s Doppler shift; shifting the spectra into a common frame and combining them sharpens the line most when the velocity is right. It is shift-and-stack along a velocity field, with beam-correlated pixels that must not be counted as independent spectra.

By 2025 the exoALMA Large Program, observing fifteen disks at about 0.15 arcsec and 100 m s$^{-1}$ resolution, had turned this into a survey: kinematic structures in thirteen disks, wake-compatible signatures in six (Teague et al. 2025; Pinte et al. 2025). A companion benchmark of four grid hydrodynamics codes, an SPH code and two radiative-transfer codes found perturbation amplitudes differing by at most about 20% in its test setup (Bae et al. 2025). Mass estimates from gap depths, such as Kanagawa’s $\Sigma_{\rm gap}/\Sigma_0=1/(1+0.04K)$ with $K=(M_p/M_\star)^2h^{-5}\alpha^{-1}$, inherit uncertainty in disk thickness $h$ and viscosity $\alpha$: at fixed depth $M_p\propto h^{5/2}\alpha^{1/2}$ (Kanagawa et al. 2015).

Phase curves

A close-in companion modulates its star’s brightness through relativistic beaming, tidal (ellipsoidal) distortion and reflection, even without an eclipse. In a circular orbit,

with $A_{\rm beam}\simeq4\alpha_{\rm beam}K_\star/c$ and $A_{\rm ellip}\simeq\alpha_{\rm ellip}(M_p/M_\star)(R_\star/a)^3\sin^2i$. Simchon Faigler and Tsevi Mazeh’s BEER algorithm (2011) scans trial periods, fits the first two harmonics, and checks that amplitudes and phases are physically consistent; a bootstrap test on 10,118 Kepler stars gave no false detections above its threshold (Faigler & Mazeh 2011). Sarah Millholland and Gregory Laughlin searched 142,630 Kepler FGK stars for reflected-light phase curves, with injection-trained logistic regression, and produced sixty non-transiting hot-Jupiter candidates for radial-velocity follow-up (Millholland & Laughlin 2017). At fixed period the harmonic fit is linear, so a period scan is cheap; the difficulty is that spots, pulsations and ellipsoidal binaries also produce near-sinusoids. The archive credits nine planets to orbital brightness modulation.

Polarimetry

Starlight is nearly unpolarised; light scattered by a planet’s atmosphere is polarised. For an unresolved system the Stokes parameters are $q=q_0+\epsilon p(\alpha)\cos2\chi$ and $u=u_0+\epsilon p(\alpha)\sin2\chi$, with $\epsilon$ the planet/star flux ratio, $p(\alpha)$ the planet’s polarisation at phase angle $\alpha$ and $\chi$ the polarisation angle. Fits should use the signed $q,u$, since $\sqrt{q^2+u^2}$ is biased upward by noise. Svetlana Berdyugina and collaborators claimed a blue-band polarisation peak of about $2\times10^{-4}$ from HD 189733 b using a 60-cm telescope in 2006–2007 (Berdyugina et al. 2008), a historical claim on a known planet that needs replication. Polarimetry removes unpolarised starlight from the mean signal but not its photon noise, and a tiny instrumental leakage from intensity into polarisation can exceed the planet’s signal. The deepest polarimetric planet search so far, ZIMPOL on $\epsilon$ Eridani, is an orbit-stacking search described in Chapter 33.

Radio emission from star–planet interaction

A planet orbiting inside its star’s sub-Alfvénic region can, like Io at Jupiter, drive currents that make the star emit coherent radio bursts at the electron cyclotron frequency $\nu_{ce}=eB/(2\pi m_e)\simeq2.80\,B[{\rm G}]$ MHz. J. Sebastian Pineda and Jackie Villadsen observed YZ Ceti with the VLA at 2–4 GHz for about 26 hours over five epochs and found coherent bursts reaching $620\pm80$ µJy. The frequency implies a field of roughly 700–1,400 gauss at the emission site (my arithmetic from the formula), a stellar field, not the planet’s. The bursts did not recur at exactly the same orbital phase of the innermost known planet, and a stochastic-burst hypothesis gave a 5.1% probability for the observed coincidence. They describe a candidate interaction, not a radio discovery (Pineda & Villadsen 2023). Self-calibration lowered their local image noise from 120 to 25 µJy, a reminder that calibration often matters more than search. A natural statistical model, not the one they fitted, treats detected bursts at times $t_j$ as a point process observed through an exposure window $W(t)$, with rate $\lambda(t)=\lambda_\star+\lambda_pg[\phi_p(t)]$ split into stellar and planet-phased parts:

The integral is the important term: counting bursts per orbital-phase bin without correcting for how long each phase was observed can manufacture modulation.

White dwarfs

A white dwarf is small, so a planet or fragment can block a large fraction of its light, and its atmosphere should be pure hydrogen or helium because heavy elements sink, so any metals were accreted recently. Andrew Vanderburg and collaborators found dips with periods of 4.5–4.9 hours around WD 1145+017 in K2 data; ground-based follow-up showed dips of up to 40%, some lasting only about five minutes and changing in depth and phase, together with infrared excess and photospheric metals: a disintegrating minor planet (Vanderburg et al. 2015). WD 1856+534 b, found in TESS data in 2020, dims its star by up to 56% for about eight minutes every 1.4 days; Spitzer limits excluded masses above $13.8\,M_{\rm J}$ (Vanderburg et al. 2020). Boris Gänsicke and collaborators interpreted hydrogen, oxygen and sulfur emission around WD J0914+1914 as gas from an evaporating giant planet (Gänsicke et al. 2019). In October 2026 Jamie Williams, Gänsicke and colleagues reported a white dwarf, HS 0209+0832, polluted by trans-iron elements and interpreted it as a second-generation planet candidate, with a 4.4-day photometric signal (Williams et al. 2026). Pollution is found in roughly a quarter to a half of white dwarfs. It securely establishes accreted planetary material but rarely identifies a surviving planet’s orbit.

Moons, rings and the limits of one event

A moon makes its own small transit, displaces the planet around their barycentre so that transit times shift by about $\Delta t_p\sim(a_m/v_{\rm tr})\,M_m/(M_p+M_m)$, and can produce mutual events. Alex Teachey and David Kipping’s 2018 Hubble transit of Kepler-1625 b arrived about 77.8 minutes early, with a moon-like dip (Teachey & Kipping 2018). Kipping and collaborators’ survey of 70 cool giants yielded Kepler-1708 b-i, a 4.8-sigma candidate of about 2.6 Earth radii; planet-only injections produced apparent moons in two of 200 tests (Kipping et al. 2022). René Heller and Michael Hippke’s reanalysis found false-positive rates of 10.9% and 1.6% for the two systems and judged large moons unlikely (Heller & Hippke 2024); Kipping and collaborators replied that the reanalysis missed deeper likelihood maxima (Kipping et al. 2025). The candidates remain unresolved; reduction choices and sampler convergence decided the disagreement.

Hippke and Heller’s Pandora code made photodynamical moon fits fast: about 13,900 complete model-and-likelihood evaluations per second on one laptop core, and 225,000 per second with a CUDA prototype on a GTX 1080, about twenty times its CPU comparison. Nested sampling then limited proposals to about 50,000 per second (Hippke & Heller 2022). In July 2026 Kevin Hoy, Alice Zurlo and collaborators reported a roughly 170-day signal in CRIRES+ velocities of the brown dwarf CD−35 2722 B, a satellite of minimum mass about $0.9\,M_{\rm J}$ around a host of about $37\,M_{\rm J}$, a hierarchical substellar system rather than a moon of an ordinary planet (Hoy et al. 2026).

Rings and spectroastrometry complete the list. The 56-day eclipse of J1407 in 2007, deeper than 95%, was modelled with 37 rings extending to about 0.6 au (Kenworthy & Mamajek 2015); a 2024 ALMA observation did not recover a previously reported millimetre counterpart (Klaassen et al. 2025). For fixed projected geometry, splitting the ring into $R$ radial bins makes the light curve linear in their transmissions, $\mathbf F=A\mathbf T$ with $0\le T_j\le1$, where $A_{ij}$ is the fraction of starlight behind ring $j$ at time $i$. That part is a bounded convex problem; the geometry and transverse speed remain nonlinear, and a one-dimensional chord does not uniquely determine a two-dimensional ring system, so regularisation can manufacture persuasive gaps from noise (van Dam & Kenworthy 2024). Eric Agol and collaborators proposed finding moons through the wavelength dependence of the photocentre of a planet–moon pair: in a band where one body is dark, the centroid moves toward the other (Agol et al. 2015). For a pair with flux ratio $f_\lambda=F_2/F_1$ and separation $\mathbf s$, the centroid moves between two bands $a$ and $b$ by

a shift that can be measured far below the angular resolution because photon-limited centroiding improves as $\lambda/(D\sqrt{N_\gamma})$. Their fiducial simulations, a 12-metre telescope with 20% throughput and day-long exposures on an Earth–Moon analogue at 1.34 pc, are idealised model systems, not instrument limits. A broad band admits more photons but can average away the very colour difference that moves the centroid. For a luminous moon of mass ratio $q=M_m/M_p$ and flux ratio $f=F_m/F_p$, the photocentre sits at $(f-q)/[(1+f)(1+q)]$ of the separation from the barycentre: the same cancellation as in stellar astrometry when light ratio equals mass ratio. Astrometric limits have already been set on moons of a real planet: GRAVITY and SPHERE astrometry of $\beta$ Pictoris b excludes satellites above about 65, 50 and 30 Earth masses near periods of 300, 700 and 1,100 days (Macias et al. 2025).

Landmark results

Limitations and open problems

Impostors dominate the false positives. In astrometry the main failure is not a Gaussian noise peak but a stellar binary with an unlucky light ratio, a scan-dependent blend, a software error or a single orbit fitted to two planets. One number called “false-alarm probability” cannot summarise all four, and DR3 follow-up shows spectroscopy is part of the detection algorithm.

Long periods and annual periods are poorly conditioned. Partial arcs constrain combinations of mass, period and inclination; annual orbits overlap parallax. New baseline, radial-velocity acceleration or a resolved image is new information; re-optimising the same data is not.

HRCCS detects structured contrast, not flux. Isothermal layers, clouds, rotation, high mean molecular weight and saturated telluric bands remove line information physically. There is no universally best resolution: for fixed detector resources higher resolution narrows the wavelength range or field, and for self-luminous planets some cases favour roughly $R=8000$–20,000 (Bidot et al. 2024).

Significance is heterogeneous. Map S/N, Bayes factors, classifier scores, single-period false-alarm probabilities and empirical false-positive rates answer different questions. The rarer signatures in this chapter have the least calibrated thresholds, and two hundred injections cannot pin down a tail probability of $10^{-3}$ precisely.

Signal reality is not cause. A disk kink may come from a warp or infall; a radio burst may be purely stellar; a transit anomaly may be a spot. Pipelines should report these alternatives rather than one “planet probability”.

Where the room is

Astrometry: batch the small problems. The likelihood is tiny, so per-call overhead dominates; orvara’s five-processor speedup of two shows how easily communication swamps it. The natural GPU workload is population-scale: thousands of stars or orbital proposals at once, data resident on the device, Kepler solves fused with accumulation of $A^TWA$, $A^TWy$ and $y^TWy$, and a Schur complement against the precomputed null factorisation for the added orbital columns. The scan-aware periodogram is a set of weighted sinusoidal sums that non-uniform FFTs could accelerate, provided the denominator and cross-terms are transformed too. Positions must be formed as tangent-plane offsets with centred times; subtracting large absolute angles in low precision destroys microarcsecond residuals. A tenfold end-to-end gain at matched recovery and contamination is a reasonable research target against a strong CPU baseline, not an established result. And the saved time should go to the hard null: injecting synthetic stars, blended binaries and multi-planet systems through the full DR4 selection cascade, and ranking candidates under planet and luminous-binary models so that spectroscopic follow-up is spent where it discriminates. That converts compute into telescope time saved.

HRCCS: transform the right part. In log-wavelength the Doppler shift is a translation, so dense correlations can use FFTs, with the template norm $B_t(v)$ computed under the same masks. That identity is old; the gain is in reusing data transforms across many templates. The trajectory sum is a Radon transform in $(s_t,v)$, and Fourier-slice, NUFFT or tree algorithms of the kind used for dedispersion (Chapter 36) apply, but it is already cheap for small maps. The real cost is retrieval: radiative transfer, opacity interpolation, broadening and model filtering for every likelihood call, which is where GPU batching, mixed precision with careful accumulation, and Gibson’s reusable projector pay off. Template banks should be laid out by mismatch, $1-\rho\simeq\langle(\delta\gamma+s_t\delta K+2\pi K\cos2\pi\phi_t\,\delta\phi)^2\rangle/(2w^2)$ for velocity width $w$, rather than by square grids, which oversample some directions and miss others near narrow phase intervals (Chapter 38).

Exomoons and disks: parallelise the sampler, not only the likelihood. Pandora’s CUDA likelihood was twenty times faster, yet the sampler capped throughput. Batched proposals, independent replicated runs and separating rare mutual-event branches from ordinary transits would let the hardware work. For disks, beam-convolved derivative templates of localised perturbations could screen locations cheaply before full hydrodynamics, which FARGO3D and Idefix already run on GPUs.

Hard limits. No computation recovers line information absent from an isothermal atmosphere, photons absorbed by Earth’s atmosphere, a photocentre orbit cancelled by $\mathcal B=\beta$, or a moon that transits only once a decade. Those need new observations: another epoch, another wavelength, a longer astrometric baseline. The engineering limits, repeated detrending, unbatched likelihoods, uncalibrated thresholds, are where new algorithms can move the field; Chapter 42 compares them with the other methods in this book.

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