Research

Orbital Dynamics and Disk Hydrodynamics

From young planetary systems to the environments around supermassive black holes, gravity and hydrodynamics shape astrophysical systems across vastly different scales. I study how the two interact to produce the systems we observe, combining analytical theory, N-body integrations, and hydrodynamic simulations.

Figures and movies are from my own work unless noted. Click a figure to enlarge it, or a movie to pause it.

Dynamics of planetary systems

The planets of our Solar System move on nearly circular, well-spaced orbits, but many systems around other stars are far more crowded. Packed too tightly, planets tug on one another until their orbits cross: some collide and merge, while others are flung out of the system altogether. What survives this chaos sets the shape of the planetary systems we see today.

I study both what triggers these instabilities and what they leave behind. Using long-term N-body integrations, I showed how the precise spacing between planetary orbits influences the stability and evolution of tightly packed systems. Hydrodynamic simulations of planetary encounters then allowed me to quantify how fluid effects influence whether an instability ends in a merger or an ejection, as well as the properties of the collision remnants. These mergers, in turn, shape the spins and obliquities of the resulting planets. Together, these studies connect dynamical instability to the orbital and spin properties of observed exoplanets.

Instability timescale of planetary systems versus orbital spacing in Hill radii, for four planet masses — **When do packed systems go unstable?** How long a planetary system survives, versus the initial spacing of its orbits in Hill radii, for four planet masses. From Li, Rodet & Lai (2024).

Smoothed-particle hydrodynamics snapshots of three close encounters between giant planets: a one-shot merger, a two-step merger with tidal disruption, and a bypass — **Three outcomes of a giant-planet encounter.** From smoothed-particle hydrodynamics simulations, with time running left to right. Top: a near head-on collision merges the planets at once. Middle: after a gentle encounter, the planets swing back for a second encounter that tidally disrupts the smaller one. Bottom: a wide encounter where the planets pass each other and both survive. From Li, Lai, Anderson & Pu (2021).

More recently, I have asked how planets escape the resonant chains they assemble into during disk-driven migration. Such chains can stay stable far longer than observations of young systems suggest. With Christopher O’Connor and Fred Rasio, I showed that repeated flybys of planetesimals from an outer reservoir can break otherwise stable chains. An injected mass of only ≲ 0.1 M_⊕ is enough to trigger instability within ~100 Myr. This is one of the first quantitative mechanisms for breaking resonant chains on the timescales the observations imply.

Probability of breaking a resonant chain versus accumulated planetesimal perturbation, and the total planetesimal mass required versus single planetesimal mass — **Breaking resonant chains with planetesimal flybys.** Left: the probability that a chain is disrupted, versus the accumulated perturbation from flybys, for independent and recurring flybys. Right: the total planetesimal mass needed for each outcome, versus the mass of a single planetesimal. From Li, O’Connor & Rasio (2026).

Protoplanetary disks: eccentricity, rings, and warps

Planets are born inside rotating disks of gas and dust around young stars. These disks are not always neat and round: they can be stretched into ovals, or bent and twisted like a warped vinyl record. Telescopes such as ALMA now image them in remarkable detail, revealing rings, spirals, and warps. I work out which of these shapes a disk can make on its own, and how those shapes in turn steer the planets forming inside it.

Rings in protoplanetary disks are often read as the signature of an unseen planet, but they need not be. I developed the first theory of eccentric disk modes that includes both radiative cooling and disk self-gravity, and found an eccentric-mode instability that amplifies global disk eccentricity. In hydrodynamic simulations, this instability produces rings and spirals with no embedded planet and no external perturbation. Some of the substructures seen by ALMA may therefore come from the disk itself.

Simulated disk developing concentric rings over 9000 orbits, beside ALMA DSHARP images of real disks and synthetic images from the simulations — **Rings without planets.** Left: gas density in a simulated disk undergoing the eccentric-mode instability, at 0, 1500, 3500, and 9000 orbits. Right: ALMA images of real disks from the DSHARP survey (top), beside synthetic dust images made from the simulations with RADMC-3D (bottom). Simulation from Li, Dempsey, Li & Li (2021); observations from Huang et al. (2018).

Ring formation. Surface density in the orbital plane (left) and unwrapped in radius and azimuth (right): the eccentric mode grows, saturates, and leaves the disk organised into several long-lived concentric rings. A faster version of this movie is also available.

Structure of the eccentric mode. Top: perturbations in surface density and in the radial and azimuthal velocities. Bottom: the radial profiles of the eccentricity and its argument, showing the mode precessing coherently.

Disk eccentricity can also pass to planets. As an eccentric disk disperses, a planet can sweep through a secular resonance where its apsidal precession matches the disk’s, and crossing it strongly excites the planet’s eccentricity. The result is an eccentric planet whose orbit stays well aligned with the stellar spin, a configuration that planet–planet scattering struggles to explain.

Planet eccentricity jumping from near zero to about 0.36 as the disk loses mass, at the point where the disk and planet precession rates match — **Eccentricity excitation at the secular resonance.** The disk mass M_d decreases from left to right as the disk disperses. Right: the disk’s precession rate ω_d (black) falls through the planet’s precession rate ω_p,d (orange). Left: at that crossing (dashed line), the planet’s eccentricity E_p (orange) jumps from nearly zero to about 0.36, even though the disk’s eccentricity E_d (blue) is only about 0.05. From Li & Lai (2023).

Disks are not always flat. When a companion star, an inclined planet, or a spinning black hole pulls on different parts of a disk in different directions, the disk can bend into a warp, with its orientation changing with distance from the centre. Warps appear across astrophysics: from protoplanetary disks, to in X-ray binaries, and to the accretion disks around the supermassive black holes. One of the examples is in the GW Orionis system (shown below), where the disk appears to have been torn into a misaligned inner ring and a warped outer disk.

ALMA and VLT/SPHERE images of the GW Orionis disk, showing a bright inner ring misaligned with the outer rings and shadows cast on the outer disk — **A warped, torn disk: GW Orionis.** ALMA images of thermal dust emission at 1.3 mm (A, B) show a bright, eccentric inner ring (R3) inside fainter outer rings; scattered-light images at 1.6 and 0.8 μm (C, D) show arcs and shadows on the outer disk, cast by the misaligned inner ring. From Kraus et al. (2020), *Science* 369, 1233; see also ESO’s press release.

Artist’s impression. As the camera moves around the system, you can see a misaligned ring that has detached from the outer disk. The ring casts two rays of shadow onto the disk beyond it, forming the dark features S3 and S4 in the images on the left. The outer disk itself is strongly warped, and the warp acts like a curtain, blocking starlight from reaching S2. Credit: L. Calçada/ESO, S. Kraus et al., University of Exeter.

My current work is on the nonlinear dynamics of warped disks: when does a warped disk stay smooth, and when does it break into misaligned pieces? With Yoram Lithwick, I combine analytical theory, one-dimensional models, and large three-dimensional Athena++ simulations. We find that a radial change in disk inclination can concentrate into a strong local warp. That warp drives extreme vertical compression and expansion (“bouncing”) of the gas and, in some circumstances, breaks the disk.

Gas bouncing vertically in a nonlinear warp, and a 3D simulation of a warped disk breaking into misaligned pieces — **Nonlinear warps and disk breaking.** Left: a strong warp squeezes the gas vertically two times per orbit, and lets it expand again in between, so the gas “bounces”. Right: a three-dimensional simulation in which an initially smooth warp steepens and the disk breaks into misaligned pieces. From Li & Lithwick (in prep.).

A strongly warped disk that holds together. Gas density in a vertical slice through a 3D Athena++ simulation, with the inner and outer edges of the disk held at different inclinations (β_out = 0.3). The warp steepens, but the disk stays connected.

Disk breaking. The same set-up with a larger misalignment (β_out = 0.4): the warp concentrates in a narrow region, the density there drops, and the disk tears into two misaligned pieces. From Li & Lithwick (in prep.).

Related papers: Li & Lithwick · Li & Lai (2023) · Li, Dempsey, Li & Li (2021)

Black holes in AGN disks

At the centres of many galaxies, a supermassive black hole is fed by a vast, swirling disk of gas. Smaller black holes, a few to tens of times the mass of the Sun, can be caught up in that disk, where they drift close to one another and can pair up. When such a pair finally merges, it sends out ripples in spacetime (gravitational waves) that detectors such as LIGO pick up on Earth. I study how these pairs form, and whether the gas around them pushes them toward a merger or drives them apart.

These disks in active galactic nuclei (AGN) are a promising factory for gravitational-wave sources, but how initially unbound black holes actually become binaries there remains poorly understood. I ran the first global hydrodynamic simulations of black-hole encounters in AGN disks and identified a gas-assisted capture mechanism. When two black holes pass close by, the gaseous mini-disks around each of them collide, dissipate orbital energy, and can leave the pair bound. Binaries formed this way can be compact, eccentric, and retrograde. With a student I mentored, I then developed simplified dynamical-friction models of this capture.

Gas-assisted capture. Two black holes meet inside their Hill sphere (dashed circle). Each carries its own circum-single disk; the two disks collide at closest approach, drain orbital energy, and leave the pair bound.

The same process, another view. Here the frame follows one of the black holes. Again the circum-single disks collide as the pair passes close by, and the two black holes end up bound to each other.

Three black-hole encounters that end in bound binaries, each with its trajectory and the evolution of separation, semi-major axis, eccentricity, and angular momentum — **Gas-assisted capture.** Three encounters from ongoing hydrodynamic simulations in which initially unbound black holes become bound binaries, with different orbital orientations. Each trajectory is shown with the evolution of the binary’s separation, semi-major axis, eccentricity, and angular momentum.

A bound binary is only the first step toward a merger, because interactions with the surrounding gas decide whether it hardens or expands. In collaborative hydrodynamic studies (led by Ya-Ping Li), we showed that prograde binaries can expand under positive torques from their circum-single disks, while retrograde binaries tend to shrink. We also showed that the gas thermodynamics matters: heating the circum-single disks weakens those torques and can turn expansion into contraction.

Gas surface density around a binary black hole in an AGN disk, zooming from the whole disk to the binary's circum-single disks and their spiral arms — **Spirals in the mini-disks.** Gas surface density around a prograde black-hole binary in an AGN disk, zooming in from left to right: the binary opens a gap in the AGN disk; inside its Hill sphere (dashed white circle) gas circulates around the pair; and each black hole carries a circum-single disk with spiral arms. Arrows show the gas flow. Once these spirals are resolved, their positive torque makes prograde binaries expand. From Y.-P. Li et al. (2021).

Surface density around a binary with cold and with heated circum-single disks, and a map of which binaries expand or contract as the disks are heated — **Hot mini-disks drive contraction.** Left and middle: time-averaged gas surface density around the binary when its circum-single disks have the background temperature (T₀ = 0) and when they are heated (T₀ = 20); the hotter disks are fainter and their spirals weaker (the coloured outlines mark the regions used in the torque analysis). Right: whether the binary expands (blue ⊕) or contracts (red ⊖), as a function of the heating T₀ and how steeply it falls off with distance from each black hole (γ_CSD); the green band marks the conditions covered by simulations of isolated binaries. From Y.-P. Li et al. (2022).

Earlier work

As an undergraduate at the University of Toronto, I worked with Artur Izmaylov on quantum molecular dynamics. I studied geometric-phase effects in nonadiabatic transitions through a conical intersection, and which parameters of an N-dimensional linear vibronic coupling model suppress them.