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Man Hoi Lee

Publications and source records attributed to Man Hoi Lee.

At least 19 recordsLinked to original sources
arXiv2026

Constraining the orbit of the retrograde planet in the $ν$ Octantis system

The $ν$ Octantis system is composed of a compact stellar binary and a retrograde S-type planet orbiting near the edge of stability. Here, we investigate the dynamically viable architectures of this system by exploring the planetary orbital elements with a Monte Carlo method. To handle the large number of free parameters, we employed a novel approach in which all relevant orbital parameters were varied simultaneously, which we refer to as corner stability maps. Furthermore, since the osculating orbital period of the planet undergoes large oscillations, we combined these maps with frequency analysis to identify the stable configurations that remain consistent with the observational data. We find that the retrograde planetary orbit most likely lies in the region of influence of the high-order 28/$-$11 mean-motion resonance, where its long-term stability is enhanced by secular apsidal alignment.

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arXiv2026

Dynamical analysis of the 5:1 mean-motion resonance of the HD 202206 system

The HD 202206 system, which features two substellar companions with a 5:1 period ratio around a solar-type star, offers a rare opportunity to study high-order mean-motion resonances and provides new insights into the formation and evolution of substellar companions in extrasolar planetary systems. We revisited the HD 202206 system, aiming to conduct a more comprehensive analysis of orbital inclinations, companion masses, resonance dynamics, and potential formation mechanisms. Our analysis of the astrometric jitter around the best fit in the Gaia DR3 catalog and the proper motion anomalies between Hipparcos and Gaia places robust constraints on the orbital inclinations. We performed a new dynamical fit to all available radial velocity data from the CORALIE and HARPS spectrographs. We assessed the dynamical configuration and long-term stability of the system using samples generated by nested sampling. Our analysis shows that the orbital inclinations are strongly constrained to about $51^\circ$. Consequently, the best-fit dynamical solution, assuming a coplanar and inclined ($i = 51^\circ$) configuration, yields an inner brown dwarf of $21.56 M_\mathrm{J}$ and an outer giant planet of $3.12 M_\mathrm{J}$. The corresponding orbital periods are 256.26 days and 1298.87 days, with eccentricities of 0.426 and 0.180. An investigation of the five relevant resonance angles shows that only one is librating with a large amplitude. Our stability analysis confirms that the system is dynamically stable. Our results provide new constraints on the masses and inclination of the system, challenging earlier claims of a face-on configuration and shedding light on the formation and evolution of substellar companions in extrasolar planetary systems.

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arXiv2026

Coupling dynamical accretion and chemical differentiation: A unified framework for the diversity of Earth and Mars

The physical and geochemical differences between Earth and Mars provide fundamental constraints on terrestrial planet formation, yet a self-consistent framework linking dynamical and chemical aspects remains elusive. Here we present an integrated modeling framework that couples high-resolution N-body simulations with impact-driven metal-silicate equilibration to track the dynamical accretion history and chemical differentiation for Earth and Mars. Using a narrow ring planetesimal accretion scenario, we show that Earth and Mars analogs naturally sample systematically different solid reservoirs within the protoplanetary disk. Earth analogs preferentially accrete reduced material around the planetesimal ring center, whereas Mars analogs acquire a larger fraction of oxidized material exterior to the ring. This leads to diverse bulk redox states, with composition further modified by impact-dependent pressure-temperature equilibration conditions during core formation. As a result, Earth analogs experience deeper equilibration and more efficient transfer of iron into the core, producing mantles with low iron oxide contents and larger core mass fractions. In contrast, Mars analogs equilibrate at shallower conditions, retain more iron in their mantles, and develop smaller cores. Our results demonstrate that the dynamical and geochemical differences between Earth and Mars emerge from the coupled effects of accretion pathways, the disk's radial redox structure, and impact-controlled differentiation rather than from any single process. Our unified framework physically explains the geochemical diversity of terrestrial planets and offers a potential pathway to interpret compositions of rocky planets in exoplanetary systems.

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arXiv2026

Capture and Stability of Resonant Planet Pairs in Turbulent Disk

We present a theoretical framework for the resonance capture and stability of two-planet systems in turbulent disks. By incorporating stochastic forcing (parameterized by $κ$) alongside laminar angular momentum and eccentricity damping timescales ($τ_{\rm m}, τ_{e}$), we derive an analytical criterion for the general $j:j-1$ mean motion resonances, and validate it through N-body simulations. The outcome is mapped in $κ$-$τ_{\rm m}/τ_{e}$ parameter space, revealing two distinct regimes: resonance trapping and turbulence-induced disruption -- which occurs either directly cross or via temporary capture followed by escape through turbulent diffusion. Crucially, our analysis identifies turbulence as a universal destabilizer. It amplifies the intrinsic overstability mechanism: In laminar disks, escape requires $τ_{\rm m}/τ_{e}$ to drop below a critical limit due to excessive eccentricity excitation. We demonstrate that turbulent diffusion lowers this limit, demanding stronger damping (larger $τ_{\rm m}/τ_{e}$) for stability. Thus, greater turbulence promotes escape, and sufficiently strong diffusion precludes resonance retention irrespective of eccentricity damping.

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arXiv2025

Dynamical Instability of Multi-planet Systems and Free-floating Planets

The ejection of planets by the instability of planetary systems is a potential source of free-floating planets. We numerically simulate multi-planet systems to study the evolution process, the properties of surviving systems, and the statistics of ejected planets. For systems with only super-Earth planets, we find that the time (in units of the orbital period $P_{1}$ of the innermost planet) for the system to lose the first planet by collision or ejection increases with the semimajor axis of the innermost planet. In contrast, the time (in units of $P_{1}$) for the first close encounter between two planets is identical. These two timescales also depend differently on the orbital spacing between the planets. Most systems with only super-Earths do not have planets ejected. In systems with super-Earths and a cold Jupiter, we discover that a cold Jupiter significantly increases the probability of ejection of the super-Earths by close encounters. Of 38\% of ejected super-Earths, most velocities relative to their parent stars are smaller than $6\ \mathrm{km\ s^{-1}}$. We conservatively estimate that more than 86\% of the surviving two-planet systems in the super-Earths plus cold Jupiter sample are long-term stable by using empirical criteria. Most super-Earths in the remaining two-planet systems are on highly elliptical but stable orbits and have migrated inwards compared with their initial states.

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arXiv2024

Giant planet formation in the solar system

The formation history of Jupiter has been of interest due to its ability to shape the solar system's history. Yet little attention has been paid to the formation and growth of Saturn and the other giant planets. Here, we explore the implications of the simplest disc and pebble accretion model with steady-state accretion on the formation of giant planets in the solar system through N-body simulations. We conducted a statistical survey of different disc parameters and initial conditions of the protoplanetary disc to establish which combination best reproduces the present outer solar system. We examined the effect of the initial planetesimal disc mass, the number of planetesimals and their size-frequency distribution slope, pebble accretion prescription, and sticking efficiency on the likelihood of forming gas giants and their orbital distribution. The results reveal that the accretion sticking efficiency is the most sensitive parameter for controlling the final masses and number of giant planets. We have been unable to replicate the formation of all three types of giant planets in the solar system in a single simulation. The probability distribution of the final location of the giant planets is approximately constant in $\log r$, suggesting there is a slight preference for formation closer to the Sun but no preference for more massive planets to form closer. The eccentricity distribution has a higher mean for more massive planets, indicating that systems with more massive planets are more violent. The formation timescales of the cores of the gas giants are distinct, suggesting that they formed sequentially.

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arXiv2024

Scattering of Giant Planets and Implications for the Origin of the Hierarchical and Eccentric Two-planet System GJ 1148

The GJ 1148 system has two Saturn-mass planets orbiting around an M dwarf star on hierarchical and eccentric orbits, with orbital period ratio of 13 and eccentricities of both planets of 0.375. The inner planet is in the regime of eccentric warm Jupiters. We perform numerical experiments to study the planet-planet scattering scenario for the origin of this orbital architecture. We consider a third planet of $0.1 M_J$ (Jupiter's mass) in the initial GJ 1148 system with initial orbital separations of 3.5, 4, and 4.5 mutual Hill radii and initial semimajor axis of the innermost planet in the range of 0.10-0.50 au. The majority of scattering results in planet-planet collisions, followed by planet ejections, and planet-star close approaches. Among them, only planet ejections produce eccentric and widely separated two-planet systems, with some having similar orbital properties to the GJ 1148 system. We also examine the effects of general relativistic apsidal precession and a higher mass of $0.227 M_J$ for the third planet. The simulation results suggest that the GJ 1148 system may have lost a giant planet. We also perform simulations of the general problem of the origin of warm Jupiters by planet-planet scattering. As in the GJ 1148 simulations, a nontrivial number of stable two-planet systems are produced by ejection, which disagrees with the result from a previous study showing that two-planet systems arise exclusively through planet-planet collisions.

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arXiv2024

Can the giant planets of the Solar System form via pebble accretion in a smooth protoplanetary disc?

Prevailing $N$-body planet formation models typically start with lunar-mass embryos and show a general trend of rapid migration of massive planetary cores to the inner Solar System in the absence of a migration trap. This setup cannot capture the evolution from a planetesimal to embryo, which is crucial to the final architecture of the system. We aim to model planet formation with planet migration starting with planetesimals of $\sim10^{-6}$ -- $10^{-4}M_\oplus$ and reproduce the giant planets of the Solar System. We simulated a population of 1,000 -- 5,000 planetesimals in a smooth protoplanetary disc, which was evolved under the effects of their mutual gravity, pebble accretion, gas accretion, and planet migration, employing the parallelized $N$-body code SyMBAp. We find that the dynamical interactions among growing planetesimals are vigorous and can halt pebble accretion for excited bodies. While a set of results without planet migration produces one to two gas giants and one to two ice giants beyond 6 au, massive planetary cores readily move to the inner Solar System once planet migration is in effect. Dynamical heating is important in a planetesimal disc and the reduced pebble encounter time should be considered in similar models. Planet migration remains a challenge to form cold giant planets in a smooth protoplanetary disc, which suggests an alternative mechanism is required to stop them at wide orbits.

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arXiv2024

Resonant Chains and the Convergent Migration of Planets in Protoplanetary Disks

An increasing number of compact planetary systems with multiple planets in a resonant chain have been detected. The resonant chain must be maintained by convergent migration of the planets due to planet-disk interactions if it is formed before the dispersal of the protoplanetary gas disk. For type I migration in an adiabatic disk, we show that an analytic criterion for convergent migration can be developed by requiring that any part of the resonant chain should be convergently migrating toward the remaining part. The criterion depends primarily on the logarithmic gradients $α$ and $β$ of the surface density and temperature profiles of the disk, respectively, and it is independent of the absolute values of the surface density and temperature. The analytic criterion is applied to the Kepler-60, Kepler-80, Kepler-223, TOI-178, and TRAPPIST-1 systems. Due to the variation of planetary masses within the resonant chains, we find that convergent migration typically requires rather extreme values of $(α, β)$ that have little or no overlap with common disk models. Finally, we show that there is an empirical relationship between the distance of the innermost planet from the central star and the stellar mass for the observed resonant chain systems, which supports the idea that the resonant chains are formed and maintained by stalling the migration of the innermost planet near the inner edge of the disk truncated by the magnetic fields of the protostar.

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arXiv2023

Parallelization of the Symplectic Massive Body Algorithm (SyMBA) $N$-body Code

Direct $N$-body simulations of a large number of particles, especially in the study of planetesimal dynamics and planet formation, have been computationally challenging even with modern machines. This work presents the combination of fully parallelized $N^2/2$ interactions and the incorporation of the GENGA code's close encounter pair grouping strategy to enable MIMD parallelization of the Symplectic Massive Body Algorithm (SyMBA) with OpenMP on multi-core CPUs in shared-memory environment. SyMBAp (SyMBA parallelized) preserves the symplectic nature of SyMBA and shows good scalability, with a speedup of 30.8 times with 56 cores in a simulation with 5,000 fully interactive particles.

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arXiv2023

TOI-2525 b and c: A pair of massive warm giant planets with a strong transit timing variations revealed by TESS

TOI-2525 is a K-type star with an estimated mass of M = 0.849$_{-0.033}^{+0.024}$ M$_\odot$ and radius of R = 0.785$_{-0.007}^{+0.007}$ R$_\odot$ observed by the TESS mission in 22 sectors (within sectors 1 and 39). The TESS light curves yield significant transit events of two companions, which show strong transit timing variations (TTVs) with a semi-amplitude of a $\sim$6 hours. We performed TTV dynamical, and photo-dynamical light curve analysis of the TESS data, combined with radial velocity (RV) measurements from FEROS and PFS, and we confirmed the planetary nature of these companions. The TOI-2525 system consists of a transiting pair of planets comparable to Neptune and Jupiter with estimated dynamical masses of $m_{\rm b}$ = 0.088$_{-0.004}^{+0.005}$ M$_{\rm Jup.}$, and $m_{\rm c}$ = 0.709$_{-0.033}^{+0.034}$ M$_{\rm Jup.}$, radius of $r_b$ = 0.88$_{-0.02}^{+0.02}$ R$_{\rm Jup.}$ and $r_c$ = 0.98$_{-0.02}^{+0.02}$ R$_{\rm Jup.}$, and with orbital periods of $P_{\rm b}$ = 23.288$_{-0.002}^{+0.001}$ days and $P_{\rm c}$ = 49.260$_{-0.001}^{+0.001}$ days for the inner and the outer planet, respectively. The period ratio is close to the 2:1 period commensurability, but the dynamical simulations of the system suggest that it is outside the mean motion resonance (MMR) dynamical configuration. TOI-2525 b is among the lowest density Neptune-mass planets known to date, with an estimated median density of $ρ_{\rm b}$ = 0.174$_{-0.015}^{+0.016}$ g\,cm$^{-3}$. The TOI-2525 system is very similar to the other K-dwarf systems discovered by TESS, TOI-2202 and TOI-216, which are composed of almost identical K-dwarf primary and two warm giant planets near the 2:1 MMR.

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arXiv2022

A new third planet and the dynamical architecture of the HD33142 planetary system

Based on recently-taken and archival HARPS, FEROS and HIRES radial velocities (RVs), we present evidence for a new planet orbiting the first ascent red giant star HD33142 (with an improved mass estimate of 1.52$\pm$0.03 M$_\odot$), already known to host two planets. We confirm the Jovian mass planets HD33142 b and c with periods of $P_{\rm b}$ = 330.0$_{-0.4}^{+0.4}$ d and $P_{\rm c}$ = 810.2$_{-4.2}^{+3.8}$ d and minimum dynamical masses of $m_{\rm b}\sin{i}$ = 1.26$_{-0.05}^{+0.05}$ M$_{\rm Jup}$ and $m_{\rm c}\sin{i}$ = 0.89$_{-0.05}^{+0.06}$ M$_{\rm Jup}$. Furthermore, our periodogram analysis of the precise RVs shows strong evidence for a short-period Doppler signal in the residuals of a two-planet Keplerian fit, which we interpret as a third, Saturn-mass planet with $m_\mathrm{d}\sin{i}$ = 0.20$_{-0.03}^{+0.02}$ M$_{\rm Jup}$ on a close-in orbit with an orbital period of $P_{\rm d}$ =89.9$_{-0.1}^{+0.1}$ d. We study the dynamical behavior of the three-planet system configurations with an N-body integration scheme, finding it long-term stable with the planets alternating between low and moderate eccentricities episodes. We also performed N-body simulations, including stellar evolution and second-order dynamical effects such as planet-stellar tides and stellar mass-loss on the way to the white dwarf phase. We find that planets HD33142 b, c and d are likely to be engulfed near the tip of the red giant branch phase due to tidal migration. These results make the HD33142 system an essential benchmark for the planet population statistics of the multiple-planet systems found around evolved stars.

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arXiv2022

Dynamical architecture of the HD 107148 system

We present an independent Doppler validation and dynamical orbital analysis of the two-planet system HD 107148, which was recently announced in Rosenthal et al. (2021). Our detailed analyses are based on literature HIRES data and newly obtained HARPS and CARMENES radial velocity (RV) measurements as part of our survey in search for additional planets around single planet systems. We perform a periodogram analysis of the available HIRES and HARPS precise RVs and stellar activity indicators. We do not find any apparent correlation between the RV measurements and the stellar activity indicators, thus linking the two strong periodicities to a moderately compact multiple-planet system. We carry out orbital fitting analysis by testing various one- and two-planet orbital configurations and studying the posterior probability distribution of the fitted parameters. Our results solidify the existence of a Saturn-mass planet (HD 107148 b, discovered first) with a period $P_b\sim77.2$ d, and a second, eccentric ($e_c \sim$ 0.4), Neptune-mass exoplanet (HD 107148 c), with an orbital period of $P_c\sim18.3$ d. Finally, we investigate the two-planet system's long-term stability and overall orbital dynamics with the posterior distribution of our preferred orbital configuration. Our N-body stability simulations show that the system is long-term stable and exhibits large secular osculations in eccentricity but in no particular mean-motion resonance configuration. The HD 107148 system, consisting of a Solar-type main sequence star with two giant planets in a rare configuration, features a common proper motion white dwarf companion and is, therefore, a valuable target for understanding the formation and evolution of planetary systems.

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arXiv2021

A pair of warm giant planets near the 2:1 mean motion resonance around the K-dwarf star TOI-2202

TOI-2202 b is a transiting warm Jovian-mass planet with an orbital period of P=11.91 days identified from the Full Frame Images data of five different sectors of the TESS mission. Ten TESS transits of TOI-2202 b combined with three follow-up light curves obtained with the CHAT robotic telescope show strong transit timing variations (TTVs) with an amplitude of about 1.2 hours. Radial velocity follow-up with FEROS, HARPS and PFS confirms the planetary nature of the transiting candidate (a$_{\rm b}$ = 0.096 $\pm$ 0.002 au, m$_{\rm b}$ = 0.98 $\pm$ 0.06 M$_{\rm Jup}$), and dynamical analysis of RVs, transit data, and TTVs points to an outer Saturn-mass companion (a$_{\rm c}$ = 0.155 $\pm$ 0.003 au, m$_{\rm c}$= $0.37 \pm 0.10$ M$_{\rm Jup}$) near the 2:1 mean motion resonance. Our stellar modeling indicates that TOI-2202 is an early K-type star with a mass of 0.82 M$_\odot$, a radius of 0.79 R$_\odot$, and solar-like metallicity. The TOI-2202 system is very interesting because of the two warm Jovian-mass planets near the 2:1 MMR, which is a rare configuration, and their formation and dynamical evolution are still not well understood.

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arXiv2021

Precise radial velocities of giant stars XV. Mysterious nearly periodic radial velocity variations in the eccentric binary $ε$ Cygni

Using the Hamilton Echelle Spectrograph at Lick Observatory, we have obtained precise radial velocities (RVs) of a sample of 373 G- and K-giant stars over more than 12 years, leading to the discovery of several single and multiple planetary systems. The RVs of the long-period (~53 years) spectroscopic binary $ε$ Cyg (HIP 102488) are found to exhibit additional regular variations with a much shorter period (~291 days). We intend to improve the orbital solution of the $ε$ Cyg system and attempt to identify the cause of the nearly periodic shorter period variations, which might be due to an additional substellar companion. We used precise RV measurements of the K-giant star $ε$ Cyg from Lick Observatory, in combination with a large set of RVs collected more recently with the SONG telescope, as well as archival data sets. Our Keplerian model to the RVs characterizes the orbit of the spectroscopic binary to higher precision than achieved previously, resulting in a semi-major axis of $a = 15.8 \mathrm{AU}$, an eccentricity of $e = 0.93$, and a minimum mass of the secondary of $m \sin i = 0.265 M_\odot$. Additional short-period RV variations closely resemble the signal of a Jupiter-mass planet orbiting the evolved primary component with a period of $291 \mathrm{d}$, but the period and amplitude of the putative orbit change strongly over time. Furthermore, in our stability analysis of the system, no stable orbits could be found in a large region around the best fit. Both of these findings deem a planetary cause of the RV variations unlikely. Most of the investigated alternative scenarios, such as an hierarchical triple or stellar spots, also fail to explain the observed variability convincingly. Due to its very eccentric binary orbit, it seems possible, however, that $ε$ Cyg could be an extreme example of a heartbeat system.

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arXiv2020

A Numerical Method for Determining the Elements of Circumbinary Orbits and Its Application to Circumbinary Planets and the Satellites of Pluto-Charon

Planets and satellites orbiting a binary system exist in the solar system and extrasolar planetary systems. Their orbits can be significantly different from Keplerian orbits, if they are close to the binary and the secondary-to-primary mass ratio is high. A proper description of a circumbinary orbit is in terms of the free eccentricity $e_{\rm free}$ at the epicyclic frequency $κ_0$, forced eccentricity $e_{\rm forced}$ at the mean motion $n_0$, and oscillations at higher frequencies forced by the non-axisymmetric components of the binary's potential. We show that accurate numerical values for the amplitudes and frequencies of these terms can be extracted from numerical orbit integrations by applying fast Fourier transformation (FFT) to the cylindrical distance between the circumbinary object and the center of mass of the binary as a function of time. We apply this method to three Kepler circumbinary planets and the satellites of Pluto-Charon. For the satellite Styx of Pluto-Charon, the FFT results for $κ_0$ and $e_{\rm free}$ differ significantly from the first-order analytic value and the value reported by Showalter & Hamilton (2015), respectively. We show that the deviation in $κ_0$ is likely due to the effect of the 3:1 mean-motion resonance and discuss the implications of the lower value for $e_{\rm free}$.

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arXiv2019

Precise radial velocities of giant stars. XI. Two brown dwarfs in 6:1 mean motion resonance around the K giant star $ν$ Ophiuchi

We present radial-velocity (RV) measurements for the K giant $ν$ Oph (= HIP88048, HD163917, HR6698), which reveal two brown dwarf companions with a period ratio close to 6:1. For our orbital analysis we use 150 precise RV measurements taken at Lick Observatory between 2000 and 2011, and we combine them with RV data for this star available in the literature. Using a stellar mass of $M = 2.7\,M_\odot$ for $ν$ Oph and applying a self-consistent N-body model we estimate the minimum dynamical companion masses to be $m_1\sin i \approx 22.2\,M_{\mathrm{Jup}}$ and $m_2\sin i \approx 24.7\,M_{\mathrm{Jup}}$, with orbital periods $P_1 \approx 530$ d and $P_2 \approx 3185$ d. We study a large set of potential orbital configurations for this system, employing a bootstrap analysis and a systematic $χ_ν^2$ grid-search coupled with our dynamical fitting model, and we examine their long-term stability. We find that the system is indeed locked in a 6:1 mean motion resonance (MMR), with $Δω$ and all six resonance angles $θ_{1}, \ldots, θ_{6}$ librating around 0$^\circ$. We also test a large set of coplanar inclined configurations, and we find that the system will remain in a stable resonance for most of these configurations. The $ν$ Oph system is important for probing planetary formation and evolution scenarios. It seems very likely that the two brown dwarf companions of $ν$ Oph formed like planets in a circumstellar disk around the star and have been trapped in a MMR by smooth migration capture.

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arXiv2019

Two Jovian planets around the giant star HD202696. A growing population of packed massive planetary pairs around massive stars?

We present evidence for a new two-planet system around the giant star HD202696 (= HIP105056, BD+26 4118). The discovery is based on public HIRES radial velocity measurements taken at Keck Observatory between July 2007 and September 2014. We estimate a stellar mass of 1.91$^{+0.09}_{-0.14}M_\odot$ for HD202696, which is located close to the base of the red giant branch. A two-planet self-consistent dynamical modeling MCMC scheme of the radial velocity data followed by a long-term stability test suggests planetary orbital periods of $P_{\rm b}$ = 517.8$_{-3.9}^{+8.9}$ days and $P_{\rm c}$ = 946.6$_{-20.9}^{+20.7}$ days, eccentricities of $e_{\rm b}$ = 0.011$_{-0.011}^{+0.078}$ and $e_{\rm c}$ = 0.028$_{-0.012}^{+0.065}$ , and minimum dynamical masses of $m_{\rm b}$ = 2.00$_{-0.10}^{+0.22}$\,$M_{\mathrm{Jup}}$ and $m_{\rm c}$ = 1.86$_{-0.23}^{+0.18}$,$M_{\mathrm{Jup}}$, respectively. Our stable MCMC samples are consistent with orbital configurations predominantly in a mean period ratio of 11:6 and its close-by high order mean-motion commensurabilities with low eccentricities. For the majority of the stable configurations we find an aligned or anti-aligned apsidal libration (i.e.\ $Δω$ librating around 0$^\circ$ or 180$^\circ$), suggesting that the HD202696 system is likely dominated by secular perturbations near the high-order 11:6 mean-motion resonance. The HD202696 system is yet another Jovian mass pair around an intermediate mass star with a period ratio below the 2:1 mean motion resonance. Therefore, the HD202696 system is an important discovery, which may shed light on the primordial disk-planet properties needed for giant planets to break the strong 2:1 mean motion resonance and settle in more compact orbits.

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