On the Detectability of Compact Multi-Planet Systems via Gravitational Microlensing

Stephen R. Kane

AI summary · arXiv: 2611.05535v1 · astro-ph.EP

On the Detectability of Compact Multi-Planet Systems via Gravitational Microlensing

Stephen R. Kane

Original paper Submitted to arXiv on 03 Nov. 2026 18 pages, 6 figures, 2 tables, accepted for publication in PASP License: NONEXCLUSIVE-DISTRIB 1.0

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The essentials

The study finds that TRAPPIST-1-like compact systems are effectively undetectable by Roman microlensing, leaving transit surveys to probe their demographics.

Question

Can the combined microlensing signal of several close-in terrestrial planets be detected even when individual planet signals are too weak? How does this sensitivity affect interpretations of Galactic planet demographics?

Method

The author numerically calculated binary-lens signals for the seven TRAPPIST-1 planets and the full eight-body system, using a 0.09-solar-mass lens at 4 kpc and a bulge source at 8 kpc. Signals were measured as residuals after fitting a single-lens model and assessed against Roman’s 15-minute cadence and a Δχ² detection threshold of 160; the analysis also considered finite-source effects, mass and orbit scaling, and stability constraints.

Findings

Result In the fiducial geometry, all seven planets lie at projected separations of s=0.0095–0.051, well inside the conventional lensing zone.
See the passage in the paper
All seven planets fall at $s=0.0095$–$0.051$, placing them 12–63 times closer to the host than the inner edge of the conventional lensing zone at $s=0.6$
Result After single-lens fitting, the strongest individual planetary residual is 6.6×10⁻⁶, and finite-source averaging reduces it further.
See the passage in the paper
After the single-lens model is fitted, the largest single-planet residual is that of planet g ($s=0.039$, $q=4.42\times 10^{-5}$) at $u_{0}=0.001$, $6.6\times 10^{-6}$, or $0.007$ mmag
Result Even in the maximally coherent configuration, the combined seven-planet signal remains far below Roman’s detection threshold.
See the passage in the paper
for the maximally coherent configuration, more than seven orders of magnitude short of the detection criterion of $\Delta\chi^{2}=160$.
Result Detectability by increasing planet masses conflicts with the system’s Hill-stability guarantee: the threshold is reached at a mass factor of about 315, while the guarantee is lost at about 6.8.
See the passage in the paper
The minimum spacing reaches the analytic two-planet Hill stability limit of $2\sqrt{3}\approx 3.46$ [18] at $f\approx 6.8$, a factor of about 46 below the $f\approx 315$ needed for detection at 1 mmag
Authors’ interpretation The authors argue that microlensing non-detections should not be treated as evidence that compact terrestrial systems are uncommon.
See the passage in the paper
The non-detection of compact terrestrial systems in current or future microlensing surveys should not be interpreted as evidence against their prevalence.

Limitations

The study uses one fiducial lensing geometry rather than integrating detectability over the Roman event-rate-weighted distribution of lens and source distances.
See the passage in the paper
First, we evaluate a single fiducial lensing geometry ($x=0.5$), bounding other lens distances only approximately (Section VI.1), rather than integrating detectability over the distribution of lens and source distances weighted by the Roman event rate
Events are treated as static configurations despite the planets completing between one and roughly fifteen orbits during an Einstein crossing.
See the passage in the paper
Second, we treat each event as a static configuration, whereas the TRAPPIST-1 planets complete between one and roughly fifteen orbits during a $22$-day Einstein crossing, so a full treatment would follow the orbital motion through the event.

Why it matters

The authors argue that compact systems occupy a structural microlensing blind spot because their small separations suppress observable signals, while masses high enough to compensate undermine the stability guarantee. They propose combining microlensing surveys of wider, colder planets with transit surveys of short-period planets to improve the Galactic census.

Written by AI from the paper. Passages are checked word for word against the text we analysed; the rest may contain errors.

AI-generated keywords: gravitational microlensing TRAPPIST-1 Roman Space Telescope planetary system detectability dynamical stability

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Comprehensive Summary
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Original abstract

The paper asks whether several close-in terrestrial planets could produce a detectable collective gravitational-microlensing signal, even when no individual planet produces a detectable signal on its own. It examines TRAPPIST-1—a system of seven planets orbiting an ultracool dwarf about 0.09 times the Sun’s mass—as a case study, focusing on the Roman Space Telescope’s Galactic Bulge survey. The question matters for interpreting microlensing surveys: a lack of detected compact planetary systems might reflect limited sensitivity rather than an actual scarcity of such systems in the Galaxy. The authors model TRAPPIST-1 as a lens in front of a Galactic bulge source 8 kiloparsecs away, with the lens halfway to the source, at 4 kiloparsecs. This geometry gives an Einstein radius of about 1.21 astronomical units. The planets’ orbital distances, approximately 0.012 to 0.062 AU, therefore correspond to projected separations of about 0.010 to 0.051 Einstein radii—well inside the usual microlensing-sensitive range of 0.6 to 1.6 Einstein radii. For each planet, the authors solve the binary-lens equation, calculate magnification changes along source trajectories, and compare those changes with a best-fitting single-lens light curve. This fitting is essential because it absorbs the apparent shift in the lens’s center of mass, which is not an observable planetary signal. They also calculate the full eight-body lens signal rather than simply adding the seven individual signals. Detectability is assessed using Roman’s 15-minute cadence, a nominal fractional photometric error of 1 percent, and a detection threshold of Δχ² > 160. The largest residual from an individual planet occurs for planet g during an exceptionally high-magnification event. Even then, its peak fractional deviation is only 6.6 parts per million, far below Roman’s nominal single-epoch precision of about 1 percent. The authors find that the residual after fitting scales approximately with the planet-to-star mass ratio multiplied by the cube of the projected separation. Although Roman’s cadence would provide several measurements across the perturbation, the combined signal from all seven planets remains negligible. Even in a deliberately favorable configuration, with all planets aligned to maximize coherent addition, Δχ² is about 0.00001 at 1 percent photometric precision. At an optimistic precision of 1 millimagnitude, it is about 0.001—still far below the detection threshold. Finite-source effects further weaken the signal. The tiny central caustics are four to five orders of magnitude smaller than the angular size of a typical main-sequence bulge source, so averaging across the source’s disk suppresses the perturbation. The paper also considers what would need to change for the system to become detectable. At 1 millimagnitude precision, every planet’s mass would need to increase by about a factor of 315; at 10 millimagnitudes, the required increase would be about 1,900. Yet the tightest adjacent planet pair loses guaranteed Hill stability when the masses increase by only about a factor of 6.8. This does not mean a resonant system would immediately break apart, but it shows that the masses required for detection are in serious tension with the system’s dynamical stability. Expanding the orbits could also strengthen the signal, but detection would require increases of roughly seven to fifteen times, moving the system out of the compact regime. The authors conclude that, for the representative geometry studied, Roman microlensing is effectively insensitive to TRAPPIST-1-like systems. Their absence from microlensing detections therefore cannot constrain how common such systems are.

Can Microlensing Reveal a Compact System of Seven Planets?

Gravitational microlensing can reveal planets by measuring how their gravity bends and magnifies light from a distant star. But what if a planetary system is so compact that none of its planets makes a noticeable signal alone? Could several planets combine to create a detectable effect? In “On the Detectability of Compact Multi-Planet Systems via Gravitational Microlensing,” Stephen R. Kane investigates that question using TRAPPIST-1 as a case study.

The result has consequences for how astronomers interpret microlensing surveys. If a survey does not detect compact planetary systems, that may not mean they are rare. It could simply mean that the systems’ gravitational signals are too faint to distinguish from ordinary stellar lensing.

A compact system in a microlensing survey

TRAPPIST-1 is an ultracool dwarf with about 0.09 times the Sun’s mass, orbited by seven planets. Kane considers how the system might appear in the Roman Space Telescope’s Galactic Bulge survey, which will monitor stars toward the densely populated center of the Milky Way.

For the modeled geometry, the background source lies 8 kiloparsecs away, while TRAPPIST-1 is placed halfway between Earth and the source, at 4 kiloparsecs. This gives the lens a projected Einstein radius of about 1.21 astronomical units. The Einstein radius is a useful scale for microlensing: planets near it generally produce the strongest detectable features.

TRAPPIST-1’s planets orbit much closer in, at distances of roughly 0.012 to 0.062 AU. In the modeled system, that corresponds to projected separations of just 0.010 to 0.051 Einstein radii—far inside the range of about 0.6 to 1.6 Einstein radii where planetary microlensing signals are usually most apparent.

Testing the planetary signals

Kane calculates the lensing effect of each planet and compares it with the light curve expected from a single star acting as the lens. That comparison matters because a planet can shift the apparent center of mass of the lens system. Fitting a single-lens model absorbs this shift, preventing it from being mistaken for a genuine planetary signal.

The analysis also evaluates the full eight-body system—the host star plus all seven planets—rather than assuming the planets’ separate effects can simply be added together. Detectability is assessed using Roman’s 15-minute observing cadence, a nominal photometric error of 1 percent, and a threshold of Δχ² greater than 160. In practical terms, the threshold asks whether the planetary model improves the fit enough to stand out from measurement noise.

Why even seven planets are not enough

The strongest signal from an individual planet comes from planet g during an exceptionally high-magnification event. Even in that favorable case, the peak deviation is only 6.6 parts per million. That is far below Roman’s nominal single-measurement precision of about 1 percent.

The study finds that, after the single-lens fit is accounted for, the residual signal scales approximately with the planet-to-star mass ratio multiplied by the cube of the planet’s projected separation. The planets’ small masses and tight orbits therefore work together to make their signals faint.

Roman’s frequent observations would capture multiple measurements during a microlensing perturbation, but cadence cannot compensate for a signal that is too weak. Even when Kane deliberately places all seven planets in a configuration intended to make their effects add coherently, the combined signal remains negligible:

At 1 percent photometric precision, the modeled Δχ² is about 0.00001.
At an optimistic precision of 1 millimagnitude, the modeled Δχ² is about 0.001.
Both values are far below the detection threshold of 160.

The planets’ tiny central caustics—the regions of a lensing pattern where magnification changes most sharply—make matters worse. They are four to five orders of magnitude smaller than the apparent size of a typical main-sequence star in the Galactic bulge. Because the source star has a measurable disk rather than being a point of light, its light averages over the small caustics, further smoothing out the signal.

Could a heavier or wider system be detected?

The paper also asks how much the system would need to change for Roman to see it. At 1 millimagnitude precision, each planet’s mass would need to rise by about a factor of 315. At 10 millimagnitudes, the required increase would be about 1,900.

Those mass increases are in serious tension with the system’s dynamical stability. The closest neighboring pair would lose guaranteed Hill stability at a mass increase of only about 6.8. That does not mean a resonant system would immediately fall apart, but it highlights how far the masses needed for detection lie beyond a stable configuration.

Wider orbits could strengthen the microlensing signal too, but the planets would need to move roughly seven to fifteen times farther from their star. Such an arrangement would no longer be a compact system like TRAPPIST-1.

What the result means for planet searches

For the representative geometry Kane studies, Roman microlensing is effectively insensitive to a TRAPPIST-1-like system. The planets’ combined gravity does not produce a detectable signal, even under an intentionally favorable alignment and with optimistic photometric precision.

That conclusion is not evidence that compact multi-planet systems are uncommon. Rather, it shows that microlensing surveys may be unable to find systems of this kind in the first place. Their absence from a survey’s detections cannot, on its own, tell us how common they are across the Galaxy.

Created on 05 Nov. 2026

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On the Detectability of Compact Multi-Planet Systems via Gravitational Microlensing

The essentials

Question

Method

Findings

Limitations

Why it matters

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Definitions

Can Microlensing Reveal a Compact System of Seven Planets?

A compact system in a microlensing survey

Testing the planetary signals

Why even seven planets are not enough

Could a heavier or wider system be detected?

What the result means for planet searches

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