Echoes and pulls: radar, gravity and the exotic

On 22 and 23 September 2000 the Goldstone antenna in California sent a radio carrier toward a near-Earth asteroid called 2000 DP107 and listened for the echo. The asteroid had been found by the LINEAR survey in February, and on 19 September it had passed within 0.048 au of Earth. Its echo spectrum was a broad hump, as expected for a spinning rock several hundred metres across. Riding on the hump was a narrow spike whose frequency changed from one observation to the next. A spike that narrow had to come from something small or slowly rotating, and a spike that wandered in frequency had to be moving relative to the rest. The next day range-resolved data showed two components up to 2 km apart. Goldstone had no more time to give, so the team moved to Arecibo and followed the companion from 30 September to 9 October. The result, published in 2002, was the first well-characterised radar image of a binary near-Earth asteroid, with a mutual orbit of period $1.755\pm0.007$ days and semimajor axis $2620\pm160$ m, and therefore a total mass of $(4.6\pm0.5)\times10^{11}$ kg (Margot et al. 2002).

That discovery says a great deal about the methods in this chapter. Nobody had seen the moon in reflected sunlight. It was found by a technique that controls its own illumination, measures distance and line-of-sight velocity instead of angles, and resolves a body far smaller than its own beam. Yet the same technique is useless for finding a new planet at 500 au, and the reason is a single exponent. The second half of the chapter turns to methods that do not need light reflected from the body at all: the bending of starlight by its gravity, the drift it induces in spacecraft and in the clocks of distant pulsars, the way it nudges comets, and, if the body happens to be a primordial black hole, the radiation it might emit. These are the most speculative methods in the book. Some are real searches with real null results, and many are forecasts. Telling them apart is the main skill this chapter tries to teach.

Talking to asteroids: a short history of planetary radar

The first astronomical radar echo came from the Moon in 1946, when the US Army Signal Corps’ Project Diana showed that a body in space could be interrogated with a transmitted signal (Virkki et al. 2023). The two instruments that dominated the field for sixty years opened soon after: Goldstone in 1958 and Arecibo in 1963.

An early triumph shows what radar is good at. Before 1965 Mercury was generally thought to rotate synchronously, once per 88-day orbit. Pettengill and Dyce used Arecibo to measure how far the echo was spread in frequency. A rotating planet’s limbs move toward and away from the observer, so the edges of the spectrum record the spin. The edges were wider than synchronous rotation allowed, and they gave a period of $59\pm5$ days, consistent with the now familiar three rotations per two orbits (Nolan et al. 2026). Radar discovered a fact about a known planet, by finding the faint edges of a spectrum rather than maximising its peak.

The first asteroid echoes came from (1566) Icarus in June 1968, when it passed within about 0.04 au. Richard Goldstein observed it at Goldstone and Pettengill’s group at Haystack, independently. (The Haystack paper appeared in 1969, hence the occasional wrong date.) Two decades later radar became a way to see shapes. On 22 August 1989 Arecibo imaged (4769) Castalia, then known as 1989 PB, with pixels of 2 microseconds in delay and 0.95 Hz in Doppler, about 300 m by 150 m at that geometry. Each lobe covered only about two range pixels, but the bifurcation was unmistakable. Steven Ostro’s group ran the observations, and R. Scott Hudson worked out how to recover a three-dimensional shape from the full sequence of radar projections (Hudson 1993).

The late-1990s Gregorian upgrade transformed Arecibo. The transmitter went from 0.5 MW to 1 MW, and gain and receiver performance away from zenith improved greatly. The combined gain was about twentyfold in sensitivity integrated over a target’s track. From 1997 to 2020 Arecibo observed 889 near-Earth asteroids and comets, against about 40 in the previous three decades (Nolan et al. 2026). Those are radar-observed bodies, every one of them found first by optical surveys. The binary discoveries came out of this program. Margot and colleagues found five binaries among about 50 radar-observed near-Earth asteroids and estimated that about 16% of those larger than 200 m are binary, a selected-sample estimate rather than a population census. In February 2008 Arecibo found two companions around (153591) 2001 SN263, the first triple system recognised in near-Earth space, with components about 2.8, 0.75 and 0.5 km across.

Radar also became the bridge between optical discovery and spacecraft. The pre-encounter shape model of Bennu, the target of OSIRIS-REx, combined Arecibo and Goldstone data from 1999 and 2005, with range sampling as fine as 7.5 m, and optical lightcurves (Nolan et al. 2013). It got the spinning-top shape right. It could not predict the boulder field the spacecraft found, because a smooth model rendered at high resolution is not the same as high-resolution information.

Arecibo collapsed on 1 December 2020, removing the most sensitive planetary radar in the world. Goldstone continues, often with the 100 m Green Bank Telescope (GBT) receiving. In the south, transmissions from Canberra received at Parkes began in 2015, and by 2025 the Southern Hemisphere Asteroid Research Program had characterised more than 50 near-Earth asteroids, including the close approach of 2024 MK on 29 June 2024, received with the Ceduna 30 m dish and small geodetic antennas (White et al. 2025a; White et al. 2025b). Distant work also continues: a 2026 study analysed 33 epochs of Europa echoes from 2011 to 2024 using Goldstone and the GBT (Xie et al. 2026). JPL’s public radar history index runs to entry NEA-1166, dated 24 June 2026 (JPL radar history).

The physics of an echo

The radar equation and the fourth power

Let a transmitter of power $P_t$ and antenna gain $G_t$ illuminate a target at distance $R_t$. The power per unit area arriving at the target is $P_tG_t/(4\pi R_t^2)$. The target intercepts and re-radiates as if it were an isotropic scatterer of area $\sigma$, the radar cross-section, so the power per unit area back at a receiver a distance $R_r$ away is that times $\sigma/(4\pi R_r^2)$. A receiving antenna of effective area $A_e = G_r\lambda^2/(4\pi)$ collects

with $\lambda$ the wavelength and $G = 4\pi A_e/\lambda^2$ for each antenna (Naidu et al. 2016). For a monostatic radar, which transmits and receives with one dish, $R_t = R_r = R$ and $P_r \propto R^{-4}$. For bistatic work with two stations on Earth the two distances are nearly equal for any asteroid, so the penalty is the same. The two inverse squares have a plain meaning: the outgoing beam spreads, and so does the echo.

The cross-section is not a geometric area. Write $\sigma = \hat\sigma A_{\rm proj}$, where $A_{\rm proj}$ is the projected area and $\hat\sigma$ is the radar albedo, a number that depends on direction, wavelength and polarisation.

Detection is limited by thermal noise in the receiver. With system temperature $T_{\rm sys}$, echo bandwidth $B$ and integration time $T$, the standard deviation of the noise power estimate is $k_BT_{\rm sys}\sqrt{B/T}$, so the signal-to-noise ratio is

This $\rho$ is total echo power divided by the noise in the power estimate, not a voltage-amplitude SNR. Now let the target be a sphere of diameter $D$ rotating with period $P$. Its cross-section scales as $D^2$. The echo bandwidth is set by the line-of-sight speed of its limbs, $\pi D/P$, and so scales as $D/P$. Substituting,

This is the scaling given by Ostro et al. (2002) and Naidu et al., for fixed wavelength, geometry, albedo and instrument. A slowly rotating body is easier to detect because its echo is packed into fewer frequency bins.

The consequences are severe, and worth deriving explicitly. At fixed SNR, $T \propto R^8$. Doubling the range costs $2^8 = 256$ times the integration time. Doubling transmitter power extends the range reachable in a fixed time by only $2^{1/4} = 1.19$, and a tenfold improvement in power-sensitivity buys $10^{1/4} = 1.78$ in range. A tenfold faster processing pipeline buys nothing in range at all, since the limit is photons, not arithmetic.

Delay, Doppler, and what a radar image means

Radar measures two quantities: when the echo arrives and how its frequency is shifted. For a monostatic radar at range $R$ and range rate $\dot R$,

with the sign chosen so an approaching object has positive Doppler. Resolution in delay is set by the transmitted bandwidth, $\Delta R \simeq c/(2B_{\rm tx})$, and resolution in frequency by the coherent integration time, $\Delta f \simeq 1/T_{\rm coh}$. A phase-coded waveform with a 50 ns chip (the shortest element of the code) gives about 7.5 m in range, which was Arecibo’s finest mode. The Goldstone DSS-14 and DSS-13 antennas offered 25 ns and 12.5 ns chips, 3.75 m and 1.875 m (Nolan et al. 2026; Naidu et al. 2016). These are modes, not guarantees, and oversampling gives more pixels without more information. For bistatic radar replace $2R$ by the path length $L = R_t + R_r$, so $\tau = L/c$ and $f_D = -\dot L/\lambda$, with transmitter, target and receiver each evaluated at the correct retarded time.

Subtract the motion of the centre of mass and look at a point $\boldsymbol x$ on the surface of a body spinning with angular velocity $\boldsymbol\omega$. If $\hat{\boldsymbol n}$ points from the asteroid toward the observer,

Delay measures depth along the line of sight, and Doppler measures line-of-sight velocity due to rotation. Neither is an angle on the sky. That is why a radar can image a 500 m body with a beam arcminutes across.

For a sphere of diameter $D$ seen at subradar latitude $\delta$ (the latitude of the point nearest the radar), the two limbs move at $\pm(\pi D/P)\cos\delta$ along the line of sight, so the total echo bandwidth is

A narrow spectrum therefore means a small body, a slow rotator, or a nearly pole-on view, and if only one of $D$ and $P$ is known the other is bounded rather than measured (Virkki et al. 2023). This is exactly how DP107’s moon announced itself: its power sat in a few frequency bins, so it stood out above the broad echo of the much larger primary.

Delay–Doppler coordinates are not a projection the eye understands. A surface of constant delay is a plane perpendicular to the line of sight, a surface of constant Doppler is a plane parallel to both the line of sight and the spin axis, and the two planes meet in a line that crosses the visible hemisphere at two points, one north and one south of the apparent equator. Both points land in the same pixel. This north–south ambiguity was the central problem Hudson’s method addressed, and no amount of arithmetic precision removes it (Hudson 1993). Only changes of viewing aspect over a rotation, or over an encounter, break it. An equatorial view maximises Doppler resolution but leaves the ambiguity at its worst, and a pole-on view removes the ambiguity along with the Doppler resolution.

Speckle, looks, and thresholds

A rough rotating asteroid is not a fixed point reflector. Its echo is the sum of contributions from many unresolved facets with random phases, so the echo power in each pixel fluctuates even with no receiver noise. This is speckle. For that reason radar images are built by averaging the power of many independent “looks” rather than by coherently integrating one long voltage stream.

The noise statistics follow. Under the noise-only hypothesis a single-look power, normalised to unit mean, is exponentially distributed. The sum $Q$ of $L$ independent looks is Gamma-distributed,

and the average $\bar I = Q/L$ has mean 1 and variance $1/L$, which is the familiar square-root gain of averaging (Ostro et al. 2002). Individual power images are therefore not additive Gaussian observations, though they approach that at large $L$. In the strong-echo regime the target’s own speckle adds self-noise, so the relative uncertainty does not vanish when the receiver gets colder.

A detection threshold follows from the Gamma survival function. With $M_{\rm eff}$ effectively independent trials, the global false-alarm probability is approximately $M_{\rm eff}$ times the single-cell probability, and for a single look the threshold is $q \simeq \ln(M_{\rm eff}/\alpha)$ for global false-alarm probability $\alpha$. For example, $10^6$ cells and $\alpha = 10^{-3}$ give a threshold of $\ln 10^9 \approx 20.7$ times the mean noise power. This is a statistical derivation, not any observatory’s rule. Because radar targets are known optical objects with tiny search windows, the trials problem of Chapter 1 matters much less, and Naidu et al.’s criterion of SNR 30 per track is a scheduling threshold, not a false-alarm rate. The trials problem returns when one searches an archive for faint moons.

From echo to image: the processing chain

Targeting and acquisition

Every radar observation starts with an optical orbit. The pipeline retrieves the orbit and its covariance, computes pointing and light-time predictions, forecasts the echo SNR, finds the windows when the target is above the horizon and within the antenna’s declination limits, and requests transmitter time. Forecasts need a size and spin, which for most targets are unknown. Naidu et al. assumed an optical albedo of 0.18 to turn absolute magnitude into size, and spin periods of 2.1 hours above 140 m diameter and 0.5 hours below (Naidu et al. 2016).

Their assumed facility parameters give a sense of the instruments:

(The Arecibo review’s nominal specification is 1 MW, the 900 kW being a forecasting assumption.) Normalised to one round-trip light time, Naidu et al. give relative SNRs of 1 for DSS-14, 15 for Arecibo, and 2.3 for DSS-14 transmitting to the GBT. Over a full track the bistatic pair averages about three times the DSS-14 monostatic SNR, because a monostatic radar alternates transmitting and receiving and so integrates for at most half the track. Arecibo could follow most targets for less than about three hours.

Acquisition proceeds from coarse to fine. Naidu et al. preferred three-sigma pointing uncertainties below 20 arcseconds, and when the orbit is worse than that, more optical astrometry is often a better use of time than a radar raster. A continuous-wave (CW) observation then pins down the Doppler, coarse coded ranging pins down the delay, and only then is the finest imaging mode attempted.

Continuous-wave spectroscopy

The simplest observation transmits an almost pure carrier with the predicted centre-of-mass Doppler removed. The receiver records two polarisations, downconverts and digitises, corrects residual phase drift, cuts the stream into coherent intervals, Fourier-transforms each, squares, and averages the spectra. Excess power above the off-echo noise floor, integrated over plausible echo widths, is the detection. CW gives radial velocity and bandwidth but no absolute range.

For $N$ complex samples split into transforms of length $F$ the work is $O(N\log F)$, with $O(F)$ memory per channel when streamed. Scanning many echo widths is cheap with prefix sums over the averaged spectrum, though the scan needs a trials correction, and a careless interference mask can erase a real narrow secondary. The 2026 Europa analysis is a reproducible example, from polyphase downsampling of the voltages to 62.5 kHz through 10 Hz spectra to gain and system-temperature calibration (Xie et al. 2026). Calibration is part of the detection statistic, not decoration added after it.

Phase-coded ranging and delay–Doppler imaging

To measure delay with fine resolution while still transmitting a long, energetic signal, the transmitter flips the carrier phase between two states according to a pseudorandom code. Correlating the echo with a delayed copy of the code compresses the long transmission into a narrow delay response. A Fourier transform along slow time within each delay bin then resolves Doppler. Chip duration sets delay resolution, and the code’s repetition length sets the delay ambiguity. The DP107 campaign of 2008 used codes of 8,191 and 65,535 chips with 0.2 microsecond chips, giving 30 m range bins, and used longer coherent intervals for the slowly rotating secondary than for the broad primary (Naidu et al. 2015).

A reference processing plan, synthesised from published methods rather than taken from any observatory’s code, is:

for each receive interval and polarisation:
    read complex voltages and metadata
    remove predicted centre-of-mass phase and delay drift
    flag corrupted samples, keep their exposure weights
    correlate with the transmitted code        -> delay streams
    for each coherent block:
        FFT each delay stream along time        -> Doppler bins
        form calibrated power and noise estimates
    register look powers using the updated ephemeris
    average looks, keeping look counts and masks
compare CW centroid and coded delay with the orbit prediction
refit the orbit, update waveform and window, repeat as SNR allows

Direct correlation of $N$ samples with a $K$-chip code costs $O(NK)$, and overlap-save FFT correlation approaches $O(N\log K)$. A grid of $N_\tau$ delays and $N_f$ slow-time samples costs $O(N_\tau N_f\log N_f)$ per look for the Doppler FFTs, and accumulating $L$ looks costs $O(LN_\tau N_f)$. These are asymptotic estimates. Pulse compression is not free sensitivity beyond the transmitted energy. It only lets a long waveform deliver the resolution of a short one. Code sidelobes, transmitter bandwidth, spectral leakage and fractional-chip delays shape the real point-spread function.

Motion compensation and multistatic reception

Before integrating, the predicted motion must be removed. If the residual delay drifts by a substantial fraction of a range bin the target smears in range, and if the residual frequency drifts by a fraction of a bin the power spreads in Doppler. The residual phase is modelled as a polynomial,

and the data multiplied by $e^{-i\phi(t)}$. A bank over uncertain coefficients is the same object as the acceleration search in pulsar astronomy (Chapter 36) and the phase templates of continuous gravitational-wave searches (Chapter 37). The difference is that a radar target comes with an accurate optical orbit, so the huge blind bank collapses to one prediction plus a small local residual search. Keeping that prior is the biggest computational saving in the field.

Data movement is a concrete constraint. Margot’s Portable Fast Sampler records two polarisations of complex samples at 20 MHz with two bits per real component, which is 20 MB/s, or about 72 GB per hour of reception, and four-level quantisation set correctly retains 88.12% correlator efficiency (Margot 2021). The Australian program adapted JIVE’s spacecraft-tracking software SDtracker, which takes recorded voltages and phase polynomials and integrates coherently and incoherently (White et al. 2025a). For 2024 MK a 20 kW transmitter near 7159.4 MHz was received at Ceduna, Katherine, Yarragadee and ATCA. Ceduna’s integrated spectrum over a 136-minute scan peaked at 62 sigma in the dominant circular polarisation and eight sigma in the other (White et al. 2025b). Those are peak spectral significances, not per-track SNRs, and cross-correlating stations to get differential arrival times is proposed future work, not a demonstrated asteroid VLBI image.

Finding moons and weighing them

A satellite first appears as a narrow spectral feature moving relative to the broad primary, or as a separate island in a delay–Doppler image. One thresholded frame is not enough. What distinguishes a moon from interference, code sidelobes or a bright patch on the primary is persistence and orbital motion across epochs, and a contact binary, two lobes touching, is not a moon at all.

The natural test fits a mutual orbit. Given orbit parameters $\theta$, predict the separation $\boldsymbol r(t)$ and relative velocity $\boldsymbol v(t)$ and project them into the radar coordinates,

with $r_{\rm los}$ positive away from the observer. Fitting measured offsets works for a bright moon. For a faint one, the companion could be stacked along each trial orbit, using a Gaussian weak-echo statistic over images $k$ and pixels $p$,

where $h$ is the predicted secondary response including masks and exposure, $\mu$ and $s$ the background mean and standard deviation. This is a proposed detector from the research dossier, not a published search, and low-look images need the Gamma likelihood instead. It is the same idea as orbit-space stacking in optical surveys (Chapter 20). Searching $H$ orbits across $E$ epochs with $A$ pixels per template costs $O(HEA)$ naively.

Once the orbit is known, Kepler’s third law weighs the system:

Check it with DP107: $a = 2620$ m and $P_{\rm orb} = 1.755$ days $=1.516\times10^5$ s give $4\pi^2(2620)^3/(6.674\times10^{-11}\times(1.516\times10^5)^2) \approx 4.6\times10^{11}$ kg, the published value. Add shape volumes and the bulk density follows. The 2008 campaign, with 335 Arecibo and 534 Goldstone images at 30 m resolution, gave a primary 863 m across and a density of $1381\pm244$ kg m$^{-3}$ (Naidu et al. 2015). This is why finding a moon matters so much: it turns a shape into a mass. One subtlety is that the brightest pixel is not the centre of mass, and the DP107 analysis alternated shape and orbit fits to locate the component centres. Radar-discovered companions and their population statistics are taken up in Chapter 27. By the 2023 review, radar had observed 69 multi-body systems, including confirmations of systems found by other means (Virkki et al. 2023).

Shape and spin inversion

Hudson’s method is forward modelling. For shape and spin parameters $\theta$, predict each power image,

where $V$ handles visibility and shadowing, $q$ is the scattering law as a function of incidence angle $\iota$, and $H_\tau,H_f$ are the instrument responses in delay and Doppler. Fit all images, often jointly with optical lightcurves, with weighted residuals plus regularisation. The usual progression is an ellipsoid, then spherical harmonics, then a triangular-facet model, so that fine structure is not given freedom before the coarse shape is pinned down.

The SHAPE package historically minimised one parameter at a time with bracketing and Brent searches. Greenberg & Margot (2015) replaced this with a square-root information filter (SRIF): linearise, triangularise the weighted Jacobian with Householder transformations, and solve for a simultaneous correction with step-length control. With $m$ data values and $p$ parameters, dense QR costs $O(mp^2)$ and finite-difference Jacobians cost $p$ forward models per update. Their synthetic tests used 121 shape parameters and 2.4 million data values, a useful benchmark scale. On real data for asteroid 2000 ET70:

The harmonic stage ran about 2.2 times faster to the same fit, while the ellipsoid stage was slower but reached a better fit. No hardware configuration is given, so this is not a GPU benchmark. A public CUDA port of SHAPE exists, with GPU kernels for visibility rendering, delay–Doppler projection and $\chi^2$ reductions, still using one-parameter-at-a-time minimisation and not benchmarked in the sources here (CUDA SHAPE). The ADAM package fits range–Doppler images jointly with lightcurves, adaptive-optics images and occultations (ADAM), which matters because other data types constrain exactly the directions radar alone leaves ambiguous.

Landmark radar results

Sources: Nolan et al. 2026, Virkki et al. 2023, and the papers cited in the rows’ text above.

What limits radar

The binding limits are physical: $R^{-4}$, illumination, and geometry. A receiver cannot recover bandwidth that was never transmitted, and no regularisation adds observations of terrain that was never illuminated. Scattering is not composition. With $\mu_c = \sigma_{\rm SC}/\sigma_{\rm OC}$ the ratio of same-sense to opposite-sense circular polarisation echo, Europa’s 2026 values are OC and SC albedos of $0.92\pm0.11$ and $1.35\pm0.13$ and $\mu_c = 1.44\pm0.12$. A polarisation ratio above one reflects directional coherent backscatter, not a violation of energy conservation, and the inferred 32 m bound on penetration depth is model-dependent (Xie et al. 2026).

The most striking limit, though, is operational. In 2015, 430 near-Earth asteroids were potentially detectable by Arecibo above SNR 30 per track, but only 253 were discovered and announced before their windows closed, and 77 were observed.

Naidu et al. computed that a uniform two-day reduction in discovery-reporting delay would have made 17 more asteroids available to Arecibo that year, raising 253 to about 270, with no change in sensitivity. Radar’s yield is limited by how fast optical surveys find targets and by who staffs the transmitter, as much as by antennas. NRAO’s proposed GBT transmitter has so far produced 700 W pilot experiments near 13.7 GHz, against a proposed 500 kW, and the sources here do not show the full system commissioned by October 2026 (NRAO 2024; Virkki et al. 2023).

Finding a mass without its light

A black hole the size of a fist

In September 2019 Jakub Scholtz and James Unwin posted a short paper with a provocative title: “What if Planet 9 is a Primordial Black Hole?” (Scholtz & Unwin 2020). The orbital clustering of distant trans-Neptunian objects that motivates Planet Nine (Chapter 8) requires a mass of several Earth masses, not a planet. Separately, the OGLE microlensing survey had reported six ultrashort events, 0.1 to 0.3 days long, which they argued are compatible with a Galactic population of compact lenses of about 0.5 to 20 Earth masses. If some of those lenses are primordial black holes (PBHs), formed in the early universe rather than from stars, the Sun could have captured one at roughly the rate it might capture a free-floating planet. The comparison is conditional on adopted densities and velocities. It is not a probability that Planet Nine is a black hole.

The physical scales are startling. A black hole’s horizon radius and temperature are

and for five Earth masses $r_s \simeq 4.5$ cm and $T_H\simeq 0.004$ K. Their paper printed the silhouette at actual size. Such a body reflects nothing, emits essentially nothing, and would be invisible to every optical search in Chapter 9. Far outside the horizon, though, its gravity is identical to a planet’s. The rest of this chapter is about what that gravity, and possibly some radiation, could reveal.

What gravity can and cannot see

Put a body of mass $M$ at $\mathbf R = R\hat{\mathbf n}$. Its acceleration on a test body at $\mathbf r$, expanded for $r\ll R$, is

The first term accelerates everything in the inner Solar System equally. By the equivalence principle, the relative positions of the Sun, Earth and a spacecraft do not feel it at all. What they feel is the tidal term $\mathbf T(\mathbf r_1-\mathbf r_2)$, smaller by a factor of order $r/R$. To see the common acceleration one needs an external reference, such as distant pulsars. To see the full nonlinear field, a sensor must pass close to the body. Each method below measures a different combination of $M$ and $R$, and none of them distinguishes a planet from a black hole by its distant field alone.

Lensing by a nearby dark mass

Einstein radius and induced parallax

A mass bends light passing near it. For a lens at distance $D_L$ and a source at $D_S$, the angular Einstein radius is

where $\pi_{\rm rel}$ is the relative parallax of lens and source and $\kappa = 8.144$ mas per solar mass when $\pi_{\rm rel}$ is in mas (Gaudi & Bloom 2005). For a source at angular separation $\beta$ from a dark point lens, with $u = \beta/\theta_E$, the unresolved centroid shifts by $\delta\boldsymbol\theta = \theta_E^2\boldsymbol\beta/(\beta^2+2\theta_E^2)$ and the brightness is magnified by $A(u) = (u^2+2)/(u\sqrt{u^2+4})$. At large separation the centroid shift falls only as $1/u$ while $A-1 \simeq 2/u^4$. Astrometry therefore senses a lens over a far wider area than photometry, the reason to look at star positions rather than brightnesses. General microlensing theory is developed in Chapter 32.

A worked example, with numbers calculated from these formulas: a five-Earth-mass lens at 500 au has an annual parallax of about 413 arcseconds, so $\theta_E \simeq 7.1$ mas, which is about 2,580 km at that distance. The largest centroid excursion of a background star is $\theta_E/(2\sqrt2) \simeq 2.5$ mas. At a projected relative speed of 30 km/s, an Einstein radius is crossed in about 86 seconds. A large peak excursion can thus come and go between two visits of a sparse survey.

Gaudi and Bloom’s 2005 insight was to stop waiting for one spectacular event. A nearby lens traces a large annual ellipse on the sky because of Earth’s motion, and as it does so it drags every background star it passes in a small, correlated pattern. Summing these weak, coherent distortions over many stars is the lensing equivalent of shift-and-stack. They called it induced parallax and forecast it for the then-planned Gaia mission.

Their fit to the forecast is easy to reproduce. At a threshold signal-to-noise $S$, with coefficients $C_f = (290, 490, 750)\,M_\oplus$ and break distances $D_b = (4470, 1550, 780)$ au for searches covering 10%, 50% and 100% of the sky,

At 500 au and $S = 5$ this gives about 14.5, 24.5 and 37.5 Earth masses. The headline that Gaia could detect a Jupiter at 2,000 au is consistent with this, but a five-Earth-mass Planet Nine is below the forecast at every sky fraction.

Several qualifications apply. The forecast used prelaunch noise and cadence, about a billion usable stars over five years, and a signal-energy SNR in which each star’s mean deflection was subtracted. That is not a full search likelihood with every source’s parallax, proper motion and binarity fitted.

Photometric events are rarer still. An annual path sweeps a strip $2\theta_E$ wide and $2\pi\pi_L$ long, about 37 square arcseconds per year for the example lens. At stellar densities of $10^{-5}$, $10^{-3}$ and $0.1$ stars per square arcsecond, that is about 0.0004, 0.04 and 4 geometric crossings per year, for continuous monitoring of one hypothetical lens. These are upper-bound estimates before cadence and sky coverage.

A finite body complicates things. A planet of three Earth radii at 500 au is about seven Einstein radii across (an estimate), so near alignment it simply occults the background star rather than forming two images (Philippov & Chobanu 2016). A 4.5 cm black hole does not. In principle, a lensing event without an occultation could tell the two apart, a rare case where geometry, not gravity, discriminates. Occultations by solid bodies are the subject of Chapter 25.

Searching a catalogue

The data from a scanning astrometric mission are along-scan positions, one projection per transit, not two-dimensional positions. For source $s$ at epoch $i$ the datum is $d_{si} = \hat{\mathbf q}_{si}\cdot\mathbf x_s(t_i)+\epsilon_{si}$, where

combines the star’s own position, proper motion $\boldsymbol\mu_s$, parallax $\pi_s$ along the parallax curve $\mathbf p_s$, and the lensing term from a lens with parameters $\boldsymbol\theta_L$. A search loop, proposed in the dossier rather than published, looks like this:

partition the stellar catalogue into overlapping sky tiles
for each coarse lens position, distance and allowed velocity:
    build the observer-dependent lens trajectory
    query sources inside a conservative lensing tube
    evaluate along-scan lens displacements at the actual epochs
    project out source astrometry and shared calibration terms
    accumulate numerator h^T P d and normalisation h^T P h
    keep candidates above a calibrated coarse threshold
refine survivors with the exact lens equation
compare with binary-source, calibration and null models
calibrate global significance and completeness with injections

Naively, $K$ lens hypotheses and $N_{\rm obs}$ stellar observations cost $O(KN_{\rm obs})$. Spatial indexing replaces $N_{\rm obs}$ by the few observations inside each lens’s tube, and source projections can be precomputed once if the covariance is fixed. No paper reports the number of independent lens hypotheses or the compute needed.

There is one real implementation of a related search. Mondino et al. (2024) applied matched filters to Gaia DR3 proper-motion fields of Magellanic Cloud stars and stacked foreground–background star pairs to constrain dark matter subhalos of $10^7$ to $10^9$ solar masses, with public code (lens-velocity). It is a different mass regime and its kernels assume a static lens, so a Solar-System version must carry the full annual trajectory. But it is proof that the template machinery runs on real Gaia data.

Probes: fast fleets and slow cruisers

Witten’s swarm

On 29 April 2020 Edward Witten posted “Searching for a Black Hole in the Outer Solar System” (Witten 2020). Breakthrough Starshot is a concept for laser-driven miniature spacecraft. Witten suggested repurposing it at a much lower speed, about $0.001c$, as a gravitational survey: launch many probes in different directions and see which one is pulled.

Take a probe on a straight path $\mathbf r_0(t) = (vt, b, 0)$ past a mass at impact parameter $b$, closest approach at $t=0$. To first order the along-track acceleration and velocity perturbations are

and integrating once more,

If the probe transmits timestamps, this displacement shifts their arrival by $\delta\tau = \delta x/c$. With $GM/v^2c$ for five Earth masses at $0.001c$,

This is a change in the probe’s trajectory, not the Shapiro delay of light passing the mass. In Witten’s illustrative design, 1,000 probes cover half the sky toward a body at 500 au, so the nearest passes within roughly 40 au. At $0.001c$ a probe reaches 500 au in about eight years. Over two years around closest approach the timing shifts by about $1.7\times10^{-4}$ s, which demands clocks good to better than $10^{-5}$ s on a gram-scale craft hundreds of au away.

The “typical spacing” in such estimates hides a probability. For $N$ randomly aimed probes covering solid angle $\Omega$ at distance $R$, the chance that at least one passes within $b$ is approximately $1 - \exp[-N\pi b^2/(\Omega R^2)]$ (an estimate for a small cap, derived in the dossier). For 1,000 probes over half the sky ($\Omega = 2\pi$) at 500 au, this is about 96% for $b = 40$ au but only about 55% for $b = 20$ au, by our calculation. Mission completeness is a probability contour, not one impact parameter.

Deflection instead of timing

A day later Scott Lawrence and Zeeve Rogoszinski proposed measuring the sideways kick instead (Lawrence & Rogoszinski 2020). The transverse impulse is $|\Delta v_\perp| = 2GM/(bv)$, which turns the probe through an asymptotic angle

Earth-based very-long-baseline interferometry (VLBI) could track the probe’s angular position, so no precision clock would be needed on board. Two caveats matter. The angle is not instantaneous. In their impulse approximation the apparent angle after a time $t$ is $\alpha_\infty t/(T+t)$ with $T = R/v$, so it reaches half its asymptotic value only after another full flight time. And their comparison threshold of about $10^{-9}$ rad applies to bright VLBI sources, not to a gram-scale transmitter hundreds of au away.

The interstellar medium pushes back

On 3 May 2020 Thiem Hoang and Abraham Loeb pointed out that space beyond the heliosphere is not empty (Hoang & Loeb 2020). Gas drag on a probe scales as $\rho_gv^2A$, while the gravitational pull is $GM/b^2$. For a compact cube of density $\rho$ and mass $m$, the area scales as $(m/\rho)^{2/3}$, and the mass needed for gravity to exceed the mean drag is

with $n_H$ the hydrogen number density. The sixth power of speed is the problem. At $0.001c$ a gram-scale cube suffices against the mean drag. At $0.01c$ it must weigh about 1.6 tonnes.

Mean drag can in principle be fitted as a nuisance. Fluctuating drag and magnetic forces on a charged probe are harder, and Hoang and Loeb’s required masses scale as $b^6v^6$ against density fluctuations and $b^3v^{9/2}$ against magnetic ones. A deployed light sail has far more area per gram than a cube, so it couples to the medium much more strongly, and its mass must not be substituted into the cube formula. The paper also contains editorial inconsistencies in its fluctuation expressions, and the physical force equations should be trusted over the printed normalisations. A later trajectory study by Köprücü & Tekin (2024), integrating with MATLAB’s ode45 at tolerances of $10^{-13}$, found radiation pressure dominant at lower probe speeds and drag at higher ones for their sail, with relativistic corrections negligible.

The speed trade is the whole story of the fleet idea. Signal scales as $v^{-2}$, flight time as $v^{-1}$, and the drag-mass requirement as $v^6$:

Flight time is a simple $R/v$ estimate. The angle follows Lawrence and Rogoszinski’s normalisation and the mass Hoang and Loeb’s compact-cube model with $n_H = 1$ cm$^{-3}$. The two impact parameters differ, deliberately, because they come from different papers.

A conventional spacecraft on its way to Uranus

A slower route uses a spacecraft that would fly anyway. Bucko, Soyuer & Zwick (2023) simulated Doppler tracking during a nine-year Jupiter-to-Uranus cruise, with a fiducial perturber of 6.3 Earth masses at 460 au. Its acceleration on the inner Solar System is $5.3\times10^{-13}$ m/s$^2$, which they motivate as about 20 km of displacement over ten years. They integrated the spacecraft under the Sun, giant planets and the hypothetical mass, fitted nine parameters (five gravitational parameters for the Sun and giant planets, plus the target’s $GM$, distance, right ascension and declination) with the emcee ensemble sampler, and found a one-sigma sky area of

where $\sigma_A$ is the Allan deviation of the Ka-band link and $T_{\rm res}$ the sampling interval. A threefold noise improvement would shrink it to about 0.01 square degree.

This is a forecast with a simplified Solar System and walkers started near the injected solution, so it demonstrates local recovery, not a blind search. The equivalence principle adds a sharper caution. The 20 km figure is an absolute displacement under $GM/R^2$, but Earth feels almost the same acceleration, and the Earth–spacecraft range responds only to the much smaller tidal difference. A full ranging analysis would be needed to confirm the localisation numbers (Chapter 7 treats ephemerides and perturbations).

Clocks in the sky: pulsar timing

The barycentre dipole

Pulsar timing corrects every arrival time to the Solar System barycentre, computed from a planetary ephemeris. An unmodelled mass shifts the true barycentre by $\delta\mathbf r_{\rm SSB} \simeq (M/M_\odot)\mathbf R(t)$, and pulsar $p$ in direction $\hat{\mathbf n}_p$ sees a residual

the same trajectory projected differently onto each pulsar: a dipole pattern on the sky. Guo, Lee & Caballero (2018) built a Bayesian search with seven Keplerian parameters and joint timing noise. Simulating five pulsars over ten years, they could exclude unknown masses above about $10^{-9}M_\odot$ within about 5 au.

That result does not extrapolate to 500 au, and the reason is instructive. A five-Earth-mass body at 500 au displaces the barycentre by a raw light-time of about 3.7 s, an enormous number by timing standards. But its orbital period is about 11,200 years. Over a few decades, the trajectory is $\mathbf R_0 + \mathbf V_0t + \tfrac12\mathbf a_0t^2 + \cdots$, and every pulsar’s fitted phase, spin frequency and spin-down absorb the constant, linear and quadratic terms exactly. Only the tiny non-polynomial remainder survives. Guo et al. find mass uncertainties growing as $a^{7/2}$ at long periods. The raw amplitude is a mirage, and this is the $\mathbf h^T\mathbf P\mathbf h$ lesson in its starkest form.

Binary pulsars as accelerometers

A different clock sidesteps spin-down. The orbital period of a binary pulsar changes because of gravitational-wave emission, which can be calculated, and because of any acceleration of the binary relative to us. Subtracting the gravitational-wave term and the Shklovskii term from proper motion $\mu_i$ at distance $d_i$ gives the line-of-sight acceleration,

After removing the Galactic differential acceleration, each residual is $-\hat{\mathbf n}_i\cdot\mathbf a_{\rm loc}$ plus noise, and a weighted least-squares solve gives the three components of the Sun’s own acceleration. Wang et al. (2026), a September 2026 preprint, did this with 26 binary pulsars and $10^6$ Monte Carlo realisations over three Galactic potential models. The acceleration is consistent with zero, with null probabilities of about 0.20. Their conservative 95% upper bounds on $a/c$ are 0.198, 0.262 and 0.693 μas/yr over 50%, 75% and 100% of sky directions.

Converting the median bound gives about $9.1\times10^{-12}$ m/s$^2$, and with $M = aD^2/G$ the mass limits at 500 au are about 128, 169 and 448 Earth masses (estimates from the published bounds). So the first real three-dimensional clock-based acceleration measurement does not exclude a several-Earth-mass Planet Nine. It constrains heavier or closer dark companions, and its arithmetic is a three-dimensional solve: compute is not the constraint here.

Encounters are a separate question. Dey et al. (2025) searched the NANOGrav 15-year data for short hyperbolic encounters of free-floating objects with individual pulsars, fitting a localised acceleration history. Among 68 pulsars they found no significant event, with Bayes factors at most about one, and set a combined number-density limit on Jupiter-mass objects of $6.03\times10^5$ pc$^{-3}$, far above any expected population. That is a method demonstration and a null result, not a search for an Earth-term Planet Nine. Timing methods for exoplanets are in Chapter 31.

Gravitational waves are not the relevant channel. A bound Sun–Planet-Nine pair radiates at roughly $6\times10^{-12}$ Hz (an estimate), and Earth sits deep in its near zone, where the signal is simply acceleration and tide. A PBH flying past Earth would instead give the observer a velocity kick, a smoothed change of slope in timing residuals. Trivedi & Loeb (2025) give $\Delta v/c \simeq 1.5\times10^{-15}$ for a $10^{23}$ g PBH passing at 1 au, but the expected number of such encounters in 15 years is about $3\times10^{-5}$ even if all dark matter were in such objects (a dossier estimate from their own density), so their population arguments are not observational limits.

Light from the dark: gamma rays, flares and Hawking radiation

If Planet Nine were a PBH, it might not be completely dark. Scholtz and Unwin noted that a PBH formed in the early universe should carry a dense dark-matter halo, with density roughly $\rho\propto r^{-9/4}$, truncated by the Sun’s tide at about 8 au in their example. If dark matter annihilates, the halo glows in gamma rays at a rate set by $\int\rho^2\langle\sigma v\rangle/m_\chi^2\,dV$. Their highly model-dependent example gives of order $10^{20}$ annihilations per second, and they warned that a source at 500 au moves several degrees over the Fermi-LAT mission, so fixed-source catalogue thresholds do not apply. A proper search would use an unbinned Poisson likelihood over photon directions, energies and times along a moving track, a sparse, highly parallel problem nobody has published for this target.

Siraj & Loeb (2020) proposed a different signature. Small icy bodies falling toward a PBH would be heated, sublimated, disrupted and accreted, producing flashes that peak in the optical and ultraviolet. Their accretion timescale is about 0.5 s at $10^6$ cm for ten Earth masses. Assuming LSST’s 9.6 square degree field, a one-third duty cycle and 30 s exposures, a survey intercepts about $10^{-4}$ of all brief flashes from a point source, and their model predicts at least a few detected flares per year. That conclusion depends on extrapolated small-body populations and accretion physics. A search would also have to keep single-detection transients that standard asteroid pipelines, which require two detections, deliberately discard (Chapter 15).

Finally, a black hole radiates Hawking photons. For a planetary-mass hole at 0.004 K they are hopeless. Arbey & Auffinger (2021) computed the signal for a spacecraft flyby with the public BlackHawk code and found an optimistic signal-to-CMB ratio of about $10^{-10}$. This is a physical signal-to-background problem, not a computing one.

A modern comet analysis would simulate the Oort cloud with and without a companion, including Galactic tides, stellar encounters, planetary scattering, fading and survey selection, and compare full distributions rather than one band. Tides and passing stars make a structured null, not an isotropic one. A 2025 study by Arani integrated orbits over 500 Myr with REBOUND and proposed a recurring stellar perturber, a simulation-based proposal rather than a discovery. With $N_c$ massless comets, $N_b$ massive bodies and $N_s$ steps the cost is $O(N_cN_bN_s)$, and the comets are independent, which suits GPUs.

What has been observed and what is forecast

No source reports a confirmed Planet Nine or a Solar-System black hole. The three real-data analyses are a Gaia search in a far heavier mass regime, a NANOGrav search for encounters near pulsars, and a binary-pulsar acceleration limit that is not yet sensitive to five Earth masses at 500 au.

Limitations and open problems

Existence and identity are separate questions. A gravitational detection finds mass, whatever it is made of. Identifying a PBH needs a discriminator, such as a lensing event without occultation, or radiation that depends on extra physics: a surviving dark-matter halo, an impactor supply, or Hawking emission that is far too faint. A non-detection of light can exclude a luminous model without excluding the mass.

The null hypothesis is structured. Binary source stars, correlated astrometric calibration, Galactic tides, comet selection, pulsar red noise, ephemeris errors and unmodelled spacecraft forces all vary slowly, as the signal does, so a flexible model will often find a nonzero best-fit mass. Local SNRs ignore search trials, and Bayes factors depend on priors.

Sparse sampling loses information for good. A 90-second lensing peak can fall between visits, a brief flash is diluted in a 30 s exposure, and a long orbital arc is absorbed into a pulsar’s spin-down fit. Faster template evaluation cannot restore epochs never recorded.

Proposal papers rarely release searches. Most give analytic signals but no pipeline, injection set, false-alarm curve or compute profile, and none of the opened sources gives CPU or GPU hours. The Gaia DR3 analysis is the exception, in a different regime.

Where the room is

The two halves of this chapter have very different answers to the question of what a new algorithm could change.

Radar is limited by physics and operations, not arithmetic. The $R^{-4}$ law, illumination by a narrow beam, collecting area and transmitter power set what can be detected, and no compute budget changes them. A tenfold faster pipeline is not a tenfold more sensitive radar. The processing chain (FFT channelisation, code correlation, phase correction, power accumulation) is regular, batched and already transform-based, so GPU work here is sound engineering with no hidden asymptotic gain. The one published operational gain came from latency: cutting discovery-reporting delay by two days would have given Arecibo 17 more targets in 2015 (Naidu et al. 2016). Faster optical linking (Chapter 12) and automated scheduling convert existing sensitivity into observations.

Three algorithmic directions do look open. First, stacking faint moons along trial orbits across a full image sequence, after subtracting the primary, could find companions below the threshold of any single image. That is genuine new-body discovery within radar archives, and it borrows directly from orbit-space stacking (Chapter 20), pulsar acceleration searches and radar track-before-detect (Chapter 37). It needs realistic primary residuals and off-track controls, and the archives of raw voltages and non-detections are incomplete. Second, shape inversion could move from one-parameter-at-a-time fits and finite-difference Jacobians to a GPU renderer with automatic derivatives, sparse elimination of per-image parameters, and stable QR solves. The SRIF result shows that changing the optimiser alone gave about a factor of two at the same fit. An order of magnitude end to end is a plausible research hypothesis, not a result, and visibility boundaries make the derivatives non-smooth. Third, coherent multistatic processing of stations like those that observed 2024 MK is explicitly left as future work, and it needs calibration more than FLOPs.

For the gravitational probes, the room is in coherent catalogue searches. An astrometric lens search is a sparse template problem: millions of stars, many trial trajectories, and a likelihood that is linear in mass. Tiling stars and epochs, reusing each source’s nuisance projection, batching many trajectories per tile and accumulating $\mathbf h^T\mathbf P\mathbf d$ and $\mathbf h^T\mathbf P\mathbf h$ is a natural GPU kernel. An order-of-magnitude throughput gain is a plausible engineering target, though no paper has demonstrated it, and the right metric is injected-lens completeness at fixed global false-alarm rate. Template placement uses the same metric machinery as Chapter 38. Fourier or tree methods can help locally, since a weak lens field is proportional to $\boldsymbol\beta/\beta^2$, but irregular epochs, scan directions and annual parallax deform it, so this is not a simple image shift-and-stack (Chapter 39).

Two smaller openings are worth naming. Comet ensembles are embarrassingly parallel, and the hard part is preserving rare loss-cone crossings over billions of years, not integrating faster. Exploiting linearity beats brute force: when mass enters linearly, profile it analytically, and build variational equations once around a reference orbit rather than re-integrating every trial.

The hard limits are physical: the equivalence principle cancels common acceleration in local ranging, distant monopole fields cannot tell a planet from a black hole, star alignments and photons are finite, probes take years to fly, and timing models absorb short arcs. Scheduling, redundant integrations, nuisance fitting and memory traffic are engineering limits, and only those yield to better algorithms. Chapter 42 compares these opportunities with those elsewhere in the book.

References

1. Arani, S. S. (2025). Solar Periodic Companion and Random Stellar Flybys: Dynamical Perturbations of Highly Eccentric Comets in the Oort Cloud. arXiv preprint. arXiv:2511.15121
2. Arbey, A., Auffinger, J. (2021). Detecting Planet 9 via Hawking radiation. arXiv preprint. arXiv:2006.02944
3. Bucko, J., Soyuer, D., Zwick, L. (2023). Prospects for localising Planet 9 with a future Uranus mission. MNRAS Letters. arXiv:2303.16830
4. Dey, L., Jennings, R. J., Taylor, J. D. et al. (NANOGrav) (2025). The NANOGrav 15-year Data Set: Search for Gravitational Scattering of Pulsars by Free-Floating Objects in Interstellar Space. arXiv preprint. arXiv:2507.19475
5. Engels, M. CUDA SHAPE source code. GitHub. github.com/MattEngels/CUDA-SHAPE-SOURCE-CODE
6. Gaudi, B. S., Bloom, J. S. (2005). Astrometric Microlensing Constraints on a Massive Body in the Outer Solar System with Gaia. ApJ 635, 711. arXiv:astro-ph/0506426
7. Greenberg, A. H., Margot, J.-L. (2015). Improved Algorithms for Radar-Based Reconstruction of Asteroid Shapes. AJ 150, 114. doi:10.1088/0004-6256/150/4/114
8. Guo, Y. J., Lee, K. J., Caballero, R. N. (2018). A dynamical approach in exploring the unknown mass in the Solar system using pulsar timing arrays. MNRAS 475, 3644. arXiv:1802.05452
9. Hoang, T., Loeb, A. (2020). Can Planet Nine Be Detected Gravitationally by a Sub-Relativistic Spacecraft? ApJL. arXiv:2005.01120
10. Hudson, R. S. (1993). Three-Dimensional Reconstruction of Asteroids from Radar Observations. Remote Sensing Reviews 8, 195. doi:10.1080/02757259309532195
11. JPL (2026). Asteroid Radar History (snapshot of 16 October 2026). NASA JPL. echo.jpl.nasa.gov/History
12. Köprücü, Ş. U., Tekin, B. (2024). A search for Planet Nine with small spacecraft: Three-body, post-Newtonian, non-gravitational, planetary and Kuiper Belt effects. Acta Astronautica. arXiv:2309.11863
13. Lawrence, S., Rogoszinski, Z. (2020). The Brute-Force Search for Planet Nine. arXiv preprint. arXiv:2004.14980
14. Margot, J.-L. (2021). A Data-Taking System for Planetary Radar Applications. Journal of Astronomical Instrumentation 10, 2150001. doi:10.1142/S225117172150001X
15. Margot, J.-L., Nolan, M. C., Benner, L. A. M., Ostro, S. J. et al. (2002). Binary Asteroids in the Near-Earth Object Population. Science 296, 1445. doi:10.1126/science.1072094
16. Matese, J. J., Whitmire, D. P. (2011). Persistent Evidence of a Jovian Mass Solar Companion in the Oort Cloud. Icarus 211, 926. arXiv:1004.4584
17. Mondino, C., Tsantilas, A., Taki, A.-M., Van Tilburg, K., Weiner, N. (2024). Astrometric weak lensing with Gaia DR3 and future catalogues: searches for dark matter substructure. MNRAS 531, 632. arXiv:2308.12330
18. Naidu, S. P., Benner, L. A. M., Margot, J.-L., Busch, M. W., Taylor, P. A. (2016). Capabilities of Earth-Based Radar Facilities for Near-Earth Asteroid Observations. AJ 152, 99. arXiv:1604.01080
19. Naidu, S. P., Margot, J.-L., Taylor, P. A., Nolan, M. C. et al. (2015). Radar Imaging and Characterization of the Binary Near-Earth Asteroid (185851) 2000 DP107. AJ 150, 54. doi:10.1088/0004-6256/150/2/54
20. Nolan, M. C., Carter, L. M., Rivera-Valentín, E. G. (2026). Planetary Radar at the Arecibo Observatory. Reviews of Modern Physics 98, 011003. arXiv:2604.00332
21. Nolan, M. C., Magri, C., Howell, E. S., Benner, L. A. M. et al. (2013). Shape model and surface properties of the OSIRIS-REx target Asteroid (101955) Bennu from radar and lightcurve observations. Icarus 226, 629. doi:10.1016/j.icarus.2013.05.028
22. NRAO (2024). Can Astronomers Use Radar to Spot a Cataclysmic Asteroid? NRAO news release. public.nrao.edu
23. Ostro, S. J., Hudson, R. S., Benner, L. A. M., Giorgini, J. D. et al. (2002). Asteroid Radar Astronomy. Asteroids III, 151–168. seti.ucla.edu (PDF)
24. Philippov, J. P., Chobanu, M. I. (2016). Nemesis, Tyche, Planet Nine Hypotheses. I. Can We Detect the Bodies Using Gravitational Lensing? PASA 33, e033. doi:10.1017/pasa.2016.28
25. Scholtz, J., Unwin, J. (2020). What if Planet 9 is a Primordial Black Hole? Physical Review Letters 125, 051103. arXiv:1909.11090
26. Siraj, A., Loeb, A. (2020). Searching for Black Holes in the Outer Solar System with LSST. ApJL. arXiv:2005.12280
27. Trivedi, O., Loeb, A. (2025). Novel Solar System Probes for Primordial Black Holes. Physics of the Dark Universe. arXiv:2511.04895
28. Van Tilburg, K. et al. lens-velocity analysis code. GitHub. github.com/kenvantilburg/lens-velocity
29. Viikinkoski, M. ADAM: All-Data Asteroid Modelling. GitHub. github.com/matvii/ADAM
30. Virkki, A. K., Neish, C. D., Rivera-Valentín, E. G., Bhiravarasu, S. S. et al. (2023). Planetary Radar—State-of-the-Art Review. Remote Sensing 15, 5605. doi:10.3390/rs15235605
31. Wang, Z.-L., Xia, Z.-Q., Zhang, B., Fan, Y.-Z. (2026). Binary-pulsar Timing: A 3D Solar-System Accelerometer and Unknown-Source Probe. arXiv preprint. arXiv:2609.29162
32. White, O. J., Molera Calvés, G., Horiuchi, S., Edwards, P. et al. (2025a). Development of Radar and Optical Tracking of Near-Earth Asteroids at the University of Tasmania. Remote Sensing 17, 352. arXiv:2502.02890
33. White, O., Molera Calvés, G., Horiuchi, S., Kruzins, E. et al. (2025b). Ground-Based Radar Tracking of Near-Earth Objects With VLBI Radio Telescopes: 2024 MK Test Case. URSI Radio Science Letters 7. arXiv:2509.15684
34. Witten, E. (2020). Searching for a Black Hole in the Outer Solar System. arXiv preprint. arXiv:2004.14192
35. Xie, T., Margot, J.-L., Padovan, S., Ghigo, F. et al. (2026). Radar observations of Europa in 2011–2024: New insights into radar scattering properties. arXiv preprint. arXiv:2608.17689