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Kick Velocities and Mass Function of Free-Floating Planets from Dynamical Ejection in Hierarchical Three-Body Systems

This paper uses direct N-body simulations and a semi-analytic framework to demonstrate that while the mass of ejected free-floating planets has negligible impact on their velocities, the mass and eccentricity of the giant perturber critically determine the ejection timescale and the high-velocity tail of the distribution, thereby shaping the observable microlensing signatures for upcoming missions like Roman and Euclid.

Original authors: Hugh Kramer, Stefano Profumo

Published 2026-08-05

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hugh Kramer, Stefano Profumo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Cosmic Drift: Why Some Planets Go Rogue

Imagine our galaxy, the Milky Way, not as a static city of stars, but as a bustling, chaotic dance floor. Most stars have a partner or a family of planets orbiting them, moving in neat, predictable circles. But sometimes, a dancer gets pushed too hard, loses their grip, and flies off the floor entirely. These are "free-floating planets" (FFPs), also known as rogue planets. They are worlds the size of Earth or Jupiter that wander through the dark void of space, untethered to any sun. Astronomers care deeply about them because they are like cosmic fossils. If we can figure out how fast they are flying and how heavy they are, we can work backward to understand the violent history of the solar systems they were kicked out of. It's like finding a broken toy in a field and trying to guess how hard the kid threw it and what kind of game they were playing.

The main way we find these invisible wanderers is through a trick called "gravitational microlensing." Think of a rogue planet passing in front of a distant star. The planet's gravity acts like a magnifying glass, briefly bending the star's light and making it sparkle. The speed of this sparkle tells us how heavy the planet is, while the way the light shifts can hint at how fast the planet is moving. But to know exactly what these numbers mean, we need to understand the "launch mechanism." How does a planet get kicked out in the first place? Is it a gentle nudge, or a violent slam? This is the puzzle scientists are trying to solve.

The Great Galactic Slingshot

In this study, two researchers, Hugh Kramer and Stefano Profumo, decided to play a high-stakes game of cosmic billiards using a supercomputer. They wanted to simulate the most common scenario for a planet getting kicked out: a "hierarchical three-body system." Picture a massive star (like our Sun), a giant, heavy planet (like Jupiter) orbiting it, and a smaller, lighter planet (like Earth or a smaller Jupiter) orbiting closer in.

The story they tell is one of chaos and gravity. The heavy giant planet acts like a bully, constantly tugging on the smaller planet. Over time, this tugging gets the smaller planet's orbit messy and wobbly. Eventually, the two planets get close enough to have a "slingshot encounter." It's like the smaller planet is a tennis ball and the giant planet is a speeding truck; if the ball hits the truck just right, it gets launched forward at incredible speed. The researchers ran thousands of these simulations, changing the weight of the planets, how far apart they started, and how squiggly their orbits were, to see what kind of "kick" the smaller planet would get.

Here is what they found, and it's full of surprises:

1. The Weight Doesn't Matter (Much)
You might think that a heavy planet would be harder to kick than a light one. But the simulations showed that for planets ranging from tiny rocky worlds up to the size of Saturn, it barely matters how heavy they are. Whether the planet is a pebble or a boulder, the giant bully kicks it with the same force. The speed of the launch depends almost entirely on the giant planet, not the one being kicked. This confirms that for these smaller worlds, they act like "test particles"—tiny specks that just go wherever the big guy pushes them.

2. The Big Guy Sets the Pace
The size of the giant bully is the real boss. If the giant planet is heavier, the kick is harder, and the ejection happens much faster. The researchers found that if you make the giant planet 75 times heavier, the time it takes to kick the smaller planet out drops by a factor of 100. It's like a heavier mace swinging faster and harder, clearing the smaller object out of the way almost instantly.

3. The Shape of the Orbit is the Secret Sauce
This is where things get wild. The researchers tested what happens if the planets have "squiggly" (eccentric) orbits instead of perfect circles.

The Small Planet's Squiggle: If the little planet has a squiggly orbit, it can get kicked a bit faster, reaching speeds up to about 23 km/s. It's like the little planet is already running fast before the giant hits it, so the launch is extra energetic.
The Giant's Squiggle: But if the giant planet has a squiggly orbit, the results are explosive. When the giant swings in close to the star (its closest point, or pericenter), it moves incredibly fast. If the little planet gets caught in this high-speed zone, it can be launched at speeds nearing 80 km/s. That's fast enough to escape the galaxy's neighborhood entirely! The giant's squiggle acts like a turbo-boost, turning a normal kick into a super-highway launch.

4. The "Kick" vs. The "Drift"
The paper makes a crucial distinction between two speeds. The "kick speed" is how fast the planet is moving right when it leaves the system. But as it climbs away from the star's gravity, it slows down, like a ball thrown upward. The "asymptotic velocity" is how fast it's going once it's far away in deep space. The researchers found that even though the kick can be fast, the final drift speed for most planets is only a few kilometers per second. This means that most rogue planets are moving at roughly the same speed as the stars they were kicked from. They are kinematically invisible; you can't tell them apart from the crowd just by looking at how fast they move. Only the rare, extreme cases (the ones kicked by the super-squiggly giants) stand out as speedsters.

5. What This Means for the Galaxy
Because the kick speed doesn't depend much on the planet's mass, the number of rogue planets we see should match the number of planets that exist in the first place. If there are lots of tiny Earth-sized planets, there should be lots of tiny rogue Earths. The researchers suggest that the "mass function" (the list of how many planets of each size exist) of rogue planets is just a mirror of the planets still orbiting stars.

They also looked at how this affects what telescopes like the upcoming Roman space telescope will see. If the universe is full of tiny, ejected planets, we should see a lot of very short "sparkles" (microlensing events) that last less than a day. The speed of the planet helps determine how long that sparkle lasts.

The Bottom Line
This paper didn't just guess; they ran massive computer simulations to prove these points. They showed that the "kick" a planet gets is mostly about the giant planet's mass and how squiggly its orbit is, not the size of the planet being kicked. While most rogue planets will be wandering quietly at normal speeds, a rare few, launched by the most eccentric giants, will be zooming through the galaxy at breakneck speeds. It's a reminder that in the cosmic dance, the biggest, wildest dancers are the ones who send the others flying.

Technical Summary: Kick Velocities and Mass Function of Free-Floating Planets from Dynamical Ejection in Hierarchical Three-Body Systems

Problem Statement
Free-floating planets (FFPs), or rogue planets, are sub-stellar objects unbound to any host star. Their velocity and mass distributions serve as kinematic fossils of their formation and ejection history. While gravitational microlensing has detected a population of FFPs ranging from Earth-mass to Jupiter-mass, the dynamical mechanisms governing their ejection velocities and the resulting mass function remain incompletely characterized. Specifically, there is a need to map the ejection outcomes of hierarchical three-body systems (a host star, a massive giant perturber, and a lighter inner planet) to the observable properties of the Galactic FFP population, including their asymptotic velocities ( $v_\infty$ ) and the timescales of ejection ( $t_{ejec}$ ).

Methodology
The authors present a suite of direct $N$ -body simulations using the rebound integration package with the IAS15 integrator (15th-order adaptive timestepping) to ensure high accuracy during close encounters. The study focuses on the restricted three-body problem, treating the inner planet as a test particle relative to a Solar-mass host star ( $M_\star = 1 M_\odot$ ) and a massive giant perturber.

The simulation grid is controlled and systematic:

Fixed Parameters: Host mass ( $1 M_\odot$ ), giant planet semi-major axis ( $a_J = 5.2$ AU, a Jupiter analog), and initial inclinations ( $0^\circ$ – $3^\circ$ ).
Varied Parameters:
- Ejected planet mass ( $m_p$ ): $10^{-5}$ to $10^{-1} M_{Jupiter}$ .
- Giant planet mass ( $M_J$ ): $1$ to $75 M_{Jupiter}$ .
- Ejected planet semi-major axis ( $a_p$ ): $0.7$ to $1.5 a_J$ .
- Eccentricities of both planets ( $e_p$ and $e_J$ ): $0.0$ to $0.9$.
Realizations: Each configuration is simulated with $N=10,000$ independent realizations using shared random seeds for mean anomalies to isolate parameter effects.
Metrics: The study measures the asymptotic ejection velocity ( $v_\infty$ ), the finite-radius kick velocity ( $v_{kick}$ ) at the moment of ejection confirmation, and the ejection timescale ( $t_{ejec}$ ). Ejection is defined by specific energy positivity, distance from the host exceeding $10 \times$ the system's maximum semi-major axis, and outward radial velocity persisting for ten orbital periods.

Analytical Framework
The numerical results are interpreted through a semi-analytic framework combining:

Energy Budget: The energy increment required for ejection is derived from the difference between the bound orbital energy and the asymptotic kinetic energy.
Gravitational Slingshot: The maximum kick velocity is bounded by the slingshot mechanism in the giant's rest frame, dependent on the encounter speed $u$ and the giant's orbital velocity.
Hill-Scale Scattering: The median ejection speed is governed by typical encounters within the giant's Hill sphere, scaling as $v_H \propto M_J^{1/3}$ .
Tisserand Parameter: Used as a conserved diagnostic to relate orbital elements to encounter speeds.
Secular Evolution: Timescales for instability are linked to secular pumping and mean-motion resonance (MMR) overlap.

Key Results

Test-Particle Limit: The mass of the ejected planet ( $m_p$ ) has negligible impact on ejection statistics over four orders of magnitude ( $10^{-5}$ to $10^{-1} M_{Jupiter}$ ). The ejection fraction varies by less than 2.3%, and mean velocities remain stable. This confirms that for $m_p/M_J \lesssim 10^{-2}$ , the dynamics are dominated by the giant perturber, and the ejected body behaves as a test particle.
Giant Planet Mass Scaling: The giant planet mass ( $M_J$ $M_{J}$ ) sets the ejection scale and time.
- Mean ejection velocity scales as $\langle v_\infty \rangle \propto M_J^{0.42}$ .
- Ejection timescale scales as $\langle t_{ejec} \rangle \propto M_J^{-1.09}$ , consistent with the secular prediction $t_{sec} \propto M_J^{-1}$ .
- The ejection fraction is non-monotonic, peaking near $10 M_{Jupiter}$ and declining at very low and very high masses due to secular timescale limitations and Hill sphere capture, respectively.
Semi-Major Axis Dependence: For configurations within the resonance-overlap zone ( $|a_p - a_J|/a_J \lesssim 0.40$ for a $10 M_{Jupiter}$ perturber), ejection times are flat ( $\sim 1.9 \times 10^4$ yr), governed by the fast chaotic overlap timescale rather than secular evolution. Mean velocities peak near $a_p = 0.9 a_J$ .
Eccentricity Effects (Two-Speed Response):
- Inner Planet Eccentricity ( $e_p$ ): Primarily affects the high-velocity tail. Increasing $e_p$ from 0 to 0.9 raises the maximum kick velocity by a factor of $\sim 3$ (from $\sim 7.8$ to $\sim 23$ km/s) but only increases the mean velocity by a factor of $\sim 1.7$ .
- Giant Planet Eccentricity ( $e_J$ ): Has a substantially stronger effect on the tail. Increasing $e_J$ from 0 to 0.9 raises the maximum kick velocity by a factor of $\sim 10$ (from $\sim 7.8$ to $\sim 80$ km/s). This is driven by the nonlinear growth of the giant's pericentre speed ( $v_{peri} \propto \sqrt{(1+e_J)/(1-e_J)}$ ), which allows for rare, high-energy slingshot encounters.
Analytic Upper Envelope: The authors derive an analytic formula for the maximum ejection speed based on the slingshot mechanism and Tisserand parameter. This envelope accurately predicts the ceiling of the simulated velocity distributions, particularly for $e_p \gtrsim 0.5$ .

Significance and Implications
The paper claims that these results provide a direct link between the architecture of planetary systems and the observable properties of the Galactic FFP population:

Mass Function: Since the ejection fraction is independent of planet mass, the present-day FFP mass function directly traces the underlying planet occurrence rate ( $dN_{FFP}/dm_p \approx \bar{f}_{eject} \cdot dN_{planet}/dm_p$ ). This implies that if the occurrence rate rises toward lower masses (as suggested by Kepler data), the sub-Earth FFP population will be numerically dominant.
Microlensing Signatures: The transformed microlensing timescale distribution ( $dN/d\log t_E$ ) is predicted to be flat in $\log t_E$ if the planet occurrence rate is flat, or enhanced at short timescales if the occurrence rate rises toward lower masses. This offers a testable prediction for the Roman Wide Field Survey and Euclid.
Kinematics: The bulk of ejected planets have asymptotic velocities of only a few km/s, making them kinematically indistinguishable from their parent stellar population (changing the 3D velocity dispersion by $\sim 1\%$ ). However, systems with highly eccentric giant perturbers ( $e_J \gtrsim 0.5$ ) can produce a high-velocity tail ( $v_\infty \sim 80$ km/s), potentially creating proper-motion outliers detectable by Gaia or Roman.
Ejection Timescales: Most ejections from massive or eccentric systems occur rapidly ( $10^3$ – $10^4$ yr), often during the pre-main-sequence phase. Consequently, the spatial distribution of the general FFP population is effectively indistinguishable from the field, with only young, recently ejected objects retaining a memory of their birth environment.

Limitations
The authors note that the restriction to exactly three bodies is the primary limitation, as real systems may involve multiple planets leading to cascading instabilities. Additionally, the simulations do not include protoplanetary discs, stellar mass loss, or a full range of inclinations (specifically the Kozai-Lidov mechanism), which could alter ejection statistics. Future work is identified as necessary to incorporate these factors into a full population synthesis model.

The Cosmic Drift: Why Some Planets Go Rogue

The Great Galactic Slingshot

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