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

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Dynamically ejected free-floating planets leave their birth systems with a two-part velocity distribution: a Hill-scale bulk at a few km/s and an eccentricity-driven slingshot tail reaching about 80 km/s.

desk verdict Solid N-body survey with a genuinely new slingshot envelope; the high-velocity tail rests on an unquantified quality cut and fixed apsidal phase, but the central results hold and it deserves refereeing. read the letter →

arxiv 2608.02945 v1 pith:XJP3VTP4 submitted 2026-08-03 astro-ph.EP astro-ph.GAhep-ph

Hugh Kramer , Stefano Profumo This is my paper · ORCID
classification astro-ph.EPastro-ph.GAhep-ph
keywords free-floatingplanetsdynamicalejectionhierarchicalthree-bodyproblemgravitationalslingshotHillradiusTisserandparametermassfunctionmicrolensing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper uses direct simulations of a star, a giant planet, and a lighter inner planet to claim that the free-floating planets produced by dynamical ejection have a two-part velocity distribution. The typical asymptotic speed is set by the Hill velocity, the orbital speed at the edge of the giant's gravitational sphere of influence, so the median kick scales with the cube root of the giant's mass and is only a few km/s for a Jupiter analog. The high-velocity tail is bounded instead by an eccentricity-dependent gravitational-slingshot ceiling: an eccentric inner planet can be kicked up to about 23 km/s, while a highly eccentric giant can produce rare kicks near 80–86 km/s. The ejected mass has almost no effect over four orders of magnitude, so the free-floating planet mass function should directly trace the underlying planet occurrence rate. These claims matter because they tie the velocities and masses of rogue planets to the architectures of the systems they came from, with testable consequences for microlensing surveys.

What carries the argument

The argument is carried by three linked analytical objects. The gravitational slingshot energy exchange $\Delta\epsilon = \mathbf{v}_J\cdot(\mathbf{u}_{\rm out}-\mathbf{u}_{\rm in})$ between the planet and the giant sets the eccentricity-dependent envelope $v_{\infty,\rm max}$; the Hill velocity $v_H = \Omega_J R_H$, the orbital speed at the edge of the giant's gravitational sphere of influence, sets the typical encounter scale and therefore the median kick; and the Tisserand parameter $T_J = 1/x + 2\sqrt{x(1-e_p^2)}$ organises the encounter speed through $u^2/v_J^2 = 3 - T_J$. For an eccentric giant, the same slingshot budget is evaluated with the pericentre speed replacing the circular speed, which is what opens the 80–86 km/s tail. These expressions predict the ceiling of the kick distribution as a function of system architecture rather than merely fitting it.

What would settle it

A rerun of the eccentric-giant grids with the energy-error cut relaxed and with the argument of pericentre randomised would settle whether the 80–86 km/s maxima and the fraction of ejections above 50 km/s survive; if they vanish or shift strongly, the high-velocity tail is an artifact of excluded runs or of unrandomised apsidal orientation.

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Extended reading notes

Core claim

The central claim is that, in the restricted hierarchical three-body problem, the giant perturber controls both the scale and the ceiling of ejection velocities. The median asymptotic speed follows $\langle v_\infty\rangle \sim \mathrm{few}\times v_H \propto M_J^{1/3}$, where $v_H$ is the Hill velocity, while the upper envelope of the kick distribution follows the slingshot formula $v_{\infty,\rm max}^2/v_J^2 \approx 4u/v_J - 1/x$, with $u$ the relative encounter speed and $x=a_p/a_J$. Simulations reproduce this envelope: the maximum kick grows from about 7.5 km/s for circular orbits to about 23 km/s for an inner planet at $e_p=0.9$, and to 80–86 km/s when the giant itself is eccentric, because the slingshot energy must then be evaluated at the giant's pericentre speed. The ejected body's mass leaves all statistics unchanged from $10^{-5}$ to $10^{-1}$ Jupiter masses, confirming the test-particle limit; the direct corollary is that the field free-floating planet mass function, up to a nearly constant factor, is the planet occurrence rate.

Load-bearing premise

The load-bearing premise is that runs with fractional energy error above $10^{-6}$ can safely be discarded; the paper does not state how many runs this removes or whether those are exactly the strongest slingshot encounters, which would bias the quoted tail maxima.

Editorial extensions

If this is right

  • The bulk of dynamically ejected planets is kinematically indistinguishable from thin-disc stars: a few km/s asymptotic speed changes the roughly 47 km/s stellar velocity dispersion by about one percent.
  • A highly eccentric giant with $e_J\simeq0.9$ creates a rare but genuine population of fast escapees at 50–86 km/s, so a free-floating planet detected with such a velocity implies a violent, eccentric birth environment.
  • Because the ejection fraction is almost flat in ejected mass, the microlensing timescale distribution directly mirrors the planet occurrence rate; a flat occurrence rate predicts a flat distribution in $\log t_E$, while a rising occurrence rate boosts short-timescale events.
  • Ejection timescales fall roughly as $M_J^{-1}$, so massive or eccentric perturbers expel planets during or shortly after the pre-main-sequence phase of the host star.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the energy-error cuts remove the closest encounters disproportionately, the quoted tail maxima are lower limits; the paper's own quality control may be hiding the strongest kicks rather than inventing them.
  • Convolving the simulated exceedance fractions with a thermal eccentricity prior, which weights high-$e_J$ orbits more heavily than the paper's uniform scan, would likely raise the predicted fraction of fast free-floating planets.
  • The slingshot envelope can be used in reverse: a microlensing parallax measurement of a single event with transverse velocity above the circular-giant envelope would point to an eccentric giant or a non-coplanar encounter rather than a new physics channel.
  • The three-body restriction is the most fragile link in the population estimate, because real multi-planet systems may eject several planets in a cascade, changing the assumption of one ejected planet per system.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The manuscript studies the dynamical ejection of a light planet from a hierarchical three-body system (Solar-mass star, a giant-planet perturber, and a lighter planet in the restricted regime) using a large suite of direct N-body simulations with the rebound/IAS15 integrator. The authors scan the ejected-planet mass, giant-planet mass, light-planet semi-major axis, and both eccentricities, with 10,000 realizations per configuration, and measure the asymptotic ejection velocity v_infinity, the finite-radius kick velocity v_kick, the ejection timescale, and the ejection fraction. The central claims are: (i) the ejected mass has essentially no effect over four orders of magnitude (test-particle limit), so the FFP mass function directly traces the planet occurrence rate; (ii) the median ejection speed scales with the Hill velocity, v_infinity ∼ few × v_H ∝ M_J^{1/3}, while the ejection time scales as t_ejec ∝ M_J^{-1}; (iii) the high-velocity tail is controlled by an eccentricity-dependent slingshot envelope (Eq. 8), with maximum v_kick reaching ~23 km/s for e_p = 0.9 and ~80-86 km/s for e_J = 0.9 via a pericentre-enhanced slingshot; and (iv) the bulk of ejected planets has asymptotic speeds of only a few km/s, producing a negligible change in the Galactic FFP velocity dispersion relative to the stellar baseline, while the mass function translates into a directly observable microlensing timescale distribution.

Significance. If the results are correct, the paper provides the most systematic controlled mapping to date of ejection kinematics in the hierarchical restricted three-body problem, and it offers a physically motivated, analytic upper envelope for the kick-velocity distribution that can serve as a prior in microlensing-parallax analyses. The manuscript is transparent in several important ways: it carefully distinguishes the finite-radius v_kick from the asymptotic v_infinity (Section 2.6), states the apsidal-orientation limitation explicitly (Section 6.2), identifies the three-body restriction as the chief limitation (Section 6.5), uses a high-order adaptive integrator, and makes the simulation outputs and summary statistics publicly available on Zenodo. The two-component structure of the ejection-velocity distribution (Hill-scale-controlled median, slingshot-controlled tail) is a valuable organizing principle. The analytic framework is derived from first principles rather than fitted to the data, and the simulations are used as a test of the envelope, not as a calibration set.

major comments (3)
  1. [Section 3.1; Sections 4.4, 4.5, 5.2]
  2. [Sections 3.2 and 4.2]
  3. [Section 4.2, Equation (23)]
minor comments (4)
  1. [Sections 4.1, 5.3, and Conclusions]
  2. [Figure 9 and Section 4.4]
  3. [Equations (26)-(28), Section 4.5]
  4. [Section 4.4 and Section 6.5]

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the slingshot/Hill framework is derived from first principles, the simulations measure the outputs, and the envelope is applied as a ceiling rather than fitted.

full rationale

The derivation chain is self-contained. Equation (8) is built from the vis-viva encounter speed (Eq. 7), the elastic-slingshot energy exchange scaling (Eq. 4), and the binding-energy budget (Eq. 2); no constant is fitted to the simulation maxima, and the paper explicitly treats the formula as an upper envelope that the measured vkick tail must lie below (Figs. 4, 5, and 9). The median estimate of Equation (12) is an order-of-magnitude Hill-scale scaling with an unspecified constant, but it is not fed into the simulations, and the measured power-law index α(v∞) ≈ 0.42 is honestly reported as steeper than the 1/3 Hill prediction. The M_J^{-1} ejection-time scaling is derived from secular theory (Eq. 18) and the measured index −1.09 is compared with it rather than imposed. The mass-function tracing statement (Eq. 35) uses the measured near-flat ejection fraction, so it is an output of the simulation campaign rather than an assumed relation. The pericentre-enhanced envelope (Eq. 28) is introduced as a ceiling with an explicit discussion that the measured 80–86 km/s maxima lie below the strict maximal choice u_eff ∼ v_peri, not as a fitted curve. The only self-citation is the Zenodo data-release entry (Kramer & Profumo, 2026), which is not load-bearing for any result. Section 6.5 and Section 6.2 flag the three-body restriction, the unrandomized argument of pericentre, and the absence of gas as limitations; Section 3.1's unquantified ΔE/E0 > 10^-6 cut is a possible selection effect on the high-velocity tail and therefore a correctness risk, but it is independent of the predicted quantities and is not a circularity. No equation reduces by construction to an input or fitted parameter, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central claims rest on the restricted three-body approximation (verified internally), standard secular and resonance-overlap results from the literature, and the paper's own ejection detection criteria. The main non-standard choices are the energy-error exclusion threshold and the apsidal-orientation defaults, which are documented but not fully characterized.

free parameters (2)
  • Energy-error exclusion threshold = ΔE/E0 > 1e-6
    Runs exceeding this fractional energy error are excluded (Section 3.1). The excluded fraction is not stated; if violent slingshots correlate with larger energy errors, the high-velocity tail and ejection fractions could be biased.
  • Apsidal orientation = rebound defaults (not randomized)
    Arguments of pericentre are left at rebound defaults (Section 6.2), so maximum kick velocities sample only a subset of apsidal-orientation phase space, affecting the tail maxima.
assumptions (6)
  • domain assumption The inner planet can be treated as a test particle (mp << MJ), so its gravity does not perturb the star or giant (Section 2.1).
    Confirmed numerically in Section 4.1 over mp from 1e-5 to 1e-1 M_Jup; the restricted three-body approximation is used throughout the analytic framework.
  • domain assumption Initial inclinations are drawn uniformly from 0 to 3 degrees, so the Kozai-Lidov mechanism is not activated (Sections 2.5 and 3.2).
    The grid is nearly coplanar; high-inclination ejection channels are explicitly out of scope.
  • standard math Secular timescale scaling t_sec ∝ (M*/MJ) P_J^2 / P_p (Equation 18) from Laplace-Lagrange theory.
    Used to interpret the measured ejection-time scaling α(tejec) ≈ -1.09; a standard result cited in the paper.
  • standard math Resonance-overlap criterion Δa/aJ ≲ 1.5 (MJ/M*)^{2/7} (Equation 20).
    Used to interpret the flat ejection-time profile across ap/aJ; from Wisdom 1980 and Duncan et al. 1989.
  • standard math Tisserand parameter conservation during encounters (Equation 15).
    Used to organize encounter kinematics; standard in the restricted three-body problem (Murray and Dermott).
  • ad hoc to paper Ejection criteria: distance > 10 times maximum semi-major axis, positive energy, outward radial velocity, persisting for 10 orbital periods (Section 3.3).
    The specific detection radius and confirmation window are choices by the authors; they define the measured vkick and t_ejec.

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Pith. "Pith review of Kick Velocities and Mass Function of Free-Floating Planets from Dynamical Ejection in Hierarchical Three-Body Systems." pith.science (2026). https://pith.science/paper/XJP3VTP4

@misc{pith2026260802945,
  author       = {Pith},
  title        = {Pith review of: Kick Velocities and Mass Function of Free-Floating Planets from Dynamical Ejection in Hierarchical Three-Body Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJP3VTP4}},
  note         = {Machine review of arXiv:2608.02945}
}
abstract

Free-floating planets (FFPs), also known as rogue planets, are sub-stellar objects that travel through the Galaxy unbound to any host star. Their velocity distribution carries information about the dynamical channel that released them from their birth systems, while their mass distribution encodes the underlying abundance of planets available for ejection. We present a suite of direct $N$-body simulations of hierarchical three-body systems consisting of a Solar-mass host star, a massive giant perturber, and a lighter planet treated in the restricted three-body regime. We vary the mass of the ejected planet, the mass of the giant perturber, the light planet's semi-major axis, and the eccentricities of both planets, measuring the asymptotic ejection velocity $v_\infty$ and the finite-radius ejection speed $v_{\rm kick}$ at the first accepted ejection output. The ejected body's mass has little effect on the outcome over four orders of magnitude, confirming the test-particle limit. By contrast, the giant-planet mass sets the ejection scale and time, following the secular scaling $t_{\rm ejec}\propto M_J^{-1}$. The eccentricities mainly affect the high-velocity tail rather than the median: a higher light-planet eccentricity extends the kick ceiling to $\sim23$ km/s, while a highly eccentric giant can yield rare kicks near $80$ km/s via pericentre-enhanced slingshot encounters. We interpret these trends with a semi-analytic framework based on Hill-scale scattering, the Tisserand parameter, and an eccentricity-dependent upper envelope for slingshot energy exchange, and discuss how the ejection velocities map onto the Galactic FFP velocity dispersion and how the mass function sets the microlensing timescale distribution relevant for Roman and Euclid.

Figures

Figures reproduced from arXiv: 2608.02945 by the authors.

Figure 1
Figure 1. Mean ejection velocity ⟨v∞⟩ (left), mean ejection time ⟨tejec⟩ (centre), and mean kick velocity ⟨vkick⟩ (right) as functions of the mass of the ejected planet mp, spanning four orders of magnitude from 10−5 to 10−1 MJupiter. The heavy planet is held at MJ = 10 MJupiter and the light planet’s initial semi-major axis at ap = 0.9 aJ . The ejection fraction rises from 96.54% at 10−1 MJupiter to 98.21% at 10−5 MJupiter. … view at source ↗
Figure 2
Figure 2. Mean ejection velocity ⟨v∞⟩ (left), mean ejection time ⟨tejec⟩ (centre), and mean kick velocity ⟨vkick⟩ (right) as functions of the giant-planet mass MJ on a log-log scale. The light planet is fixed at mp = 10−3 MJupiter and ap = 0.9 aJ . All three statistics increase or decrease monotonically across 1–75 MJupiter: ⟨v∞⟩ rises from 1.15 to 7.07 km s−1 (power-law index ≈ 0.42); ⟨tejec⟩ falls from 3.84 × 105 to 3.51 × … view at source ↗
Figure 3
Figure 3. Distributions of ejection velocity v∞ (left), ejection time tejec (centre), and kick velocity vkick (right) for varying initial semi-major axis of the ejected planet ap, expressed as a fraction of aJ and spanning from 0.7 aJ (purple) to 1.5 aJ (orange). Each coloured curve corresponds to a single value of ap/aJ , as identified in the legend. Ejection times are nearly flat at ≈ 1.9 × 104 yr for ap ∈ [0.8, 1.2], becau… view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Distributions of ejection velocity v∞ (left), ejection time tejec (centre), and kick velocity vkick (right) as a function of the initial eccentricity ep of the ejected planet, for the inner configuration ap = 0.8 aJ . Each coloured curve corresponds to a single value o…
Figure 5
Figure 5. Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Distributions of ejection velocity v∞ (left), ejection time tejec (centre), and kick velocity vkick (right) as a function of the giant-planet eccentricity eJ , for the inner configuration ap = 0.8 aJ with ep = 0. Each coloured curve corresponds to a single value of eJ …
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Transformed lens-timescale distribution dN/d(log tE) implied by two illustrative planet mass functions. The solid curve is flat in log mp, dN/d log mp = const, and the dashed curve rises toward lower masses as m−0.5 p ; both are normalised to unity at one Jupiter mass.…
Figure 9
Figure 9. Figure 9: Analytic upper envelope of the ejection speed v∞,max (Equation 8, solid curves) as a function of the initial eccentricity ep of the ejected planet, for the inner (x = ap/aJ = 0.8, purple) and outer (x = 1.2, amber) configurations with a Jupiter-analog perturber on a ci…

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