arXiv subjects

Max Goldberg

Publications and source records attributed to Max Goldberg.

18 recordsLinked to original sources

arXiv2026

An Adolescent and Near-Resonant Planetary System Near the End of Photoevaporation

Young exoplanets provide vital insights into the early dynamical and atmospheric evolution of planetary systems. Many multi-planet systems younger than 100 Myr exhibit mean-motion resonances, likely established through convergent disk migration. Over time, however, these resonant chains are often disrupted, mirroring the Nice model proposed for the Solar System. We present a detailed characterization of the ~200-Myr-old TOI-2076 system, which contains four sub-Neptune planets between 1.4 and 3.5 Earth radii. We demonstrate that its planets are near but not locked in mean-motion resonances, making the system dynamically fragile. The four planets have comparable core masses but display a monotonic increase in hydrogen and helium (H/He) envelope mass fractions (stripped-1%-5%-5%) with decreasing stellar insolation. This trend is consistent with atmospheric mass-loss due to photoevaporation, which predicts that the envelopes of irradiated planets either erode completely or stabilize at a residual level of ~1% by mass within the first few hundred million years, with more distant, less-irradiated planets retaining most of primordial envelopes. Additionally, previous detections of metastable helium outflows rule out a pure water-world scenario for TOI-2076 planets. Our finding provides direct observational evidence that the dynamical and atmospheric reshaping of compact planetary systems begin early, offering an empirical anchor for models of their long-term evolution.

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arXiv2026

Two-source terrestrial planet formation with a sweeping secular resonance

The models that most successfully reproduce the orbital architecture of the Solar System terrestrial planets start from a narrow annulus of material that grows into embryos and then planets. However, it is not clear how this ring model can be made consistent with the chemical structure of the inner Solar System, which shows a reduced-to-oxidized gradient from Mercury to Mars and a parallel gradient in the asteroid belt. We propose that there were two primary reservoirs in the early inner Solar System: a narrow, refractory enriched ring inside of 1 au, and a less massive, extended planetesimal disk outside of 1 au with oxidation states ranging from enstatite chondrites to ordinary chondrites. We show through a suite of N-body simulations that an inwardly sweeping secular resonance, caused by aerodynamic drag and perturbations from a mean-motion resonant Jupiter and Saturn, gathers the outer planetesimal disk into a narrow ring that migrates radially, forms Mars, and contributes oxidized material to proto-Earth. Remaining unaccreted planetesimals can be implanted into the asteroid belt as the parent bodies of aubrites and non-carbonaceous iron meteorites, while the most reduced material is not implanted and thus unsampled in the meteorite collection. This model explains the oxidation and isotopic gradients within the inner Solar System within the context of a low-viscosity, magnetic wind-driven disk.

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arXiv2026

TOI-4495: A Pair of Aligned, Near-Resonant Sub-Neptunes that Likely Experienced Overstable Migration

We report the discovery of a sub-Neptune and a Neptune-like planet ($R_b = 2.48^{+0.14}_{-0.10}\,R_\oplus$, $R_c = 4.03^{+0.23}_{-0.15}\,R_\oplus$) orbiting the F-type star TOI-4495. The planets have orbital periods of 2.567 days and 5.185 days, lying close to a 2:1 mean-motion resonance (MMR). Our photodynamical analysis of the TESS light curves constrains the planetary masses to $M_b = 7.7 \pm 1.4\,M_\oplus$ and $M_c = 23.2 \pm 4.7\,M_\oplus$. The measured masses and radii indicate the presence of volatile-rich gaseous envelopes on both planets. The Rossiter-McLaughlin effect and the Doppler shadow of TOI-4495 c reveal a well-aligned orbit with a projected stellar obliquity of $\lambda = -2.3^{+8.3}_{-7.8}\,\mathrm{deg}$. Combined with the low mutual inclination constrained by the photodynamical analysis ($\Delta I < 8.7\,\mathrm{deg}$), the planetary orbits are likely coplanar and aligned with the host star's spin axis. We show that the planets are near, but not in, the 2:1 MMR, with a circulating resonant angle. We also find substantial free eccentricity for the inner planet, TOI-4495 b ($e_b = 0.078^{+0.020}_{-0.013}$). Given the observed proximity to the 2:1 resonance and the more massive outer planet, TOI-4495 b and c are particularly susceptible to resonant overstability, which can convert resonantly excited eccentricity into free eccentricity. However, additional mechanisms (e.g., planetesimal scattering) may be required to further excite the eccentricity by $\sim 4\%$. To prevent tidal damping from reducing the eccentricity below the observed level over the star's lifetime (1.9 Gyr), the reduced tidal quality factor of TOI-4495 b must be $Q' \gtrsim 10^5$, consistent with the presence of a thick envelope on the planet.

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arXiv2025

Close-in compact super-Earth systems emerging from resonant chains: slow destabilization by unseen remnants of formation

Planet formation simulations consistently predict compact systems of numerous small planets in chains of mean motion resonances formed by planet-disk interaction, but transiting planet surveys have found most systems to be non-resonant and somewhat dynamically excited. A scenario in which nearly all of the primordial resonant chains undergo dynamical instabilities and collisions has previously been found to closely match many features of the observed planet sample. However, existing models have not been tested against new observations that show a steep decline in the resonant fraction as a function of stellar age on a timescale of ~100 Myr. We construct a simplified model incorporating Type I migration, growth from embryos, and N-body integrations continued to 500 Myr and use it to generate a synthetic planet population. Nearly all systems exit the disk phase in a resonant configuration but begin slowly diffusing away from the center of the resonance. Dynamical instabilities can arise on timescales of tens or hundreds of Myr, especially when systems formed in disks with a convergent migration trap. In this case, a secondary chain of smaller planets that remained at their birth location eventually breaks, destabilizing the inner resonant chain. We also show that the instability statistics are well modeled by a Weibull distribution, and use this to extrapolate our population to Gyr ages. The close match of our modeled systems to the observed population implies that the high resonance fraction predicted by this class of models is in fact consistent with the data, and the previously-reported overabundance of resonant systems was a consequence of comparing simulations of early evolution to mature Gyr-old systems. This result also suggests that instabilities triggered by disk dissipation or other very early mechanisms are unlikely to be consistent with observed young systems.

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arXiv2025

Unexpected Near-Resonant and Metastable States of Young Multi-Planet Systems

Recent observations suggest that the incidence of near-resonant planets declines as planetary systems age, making young planetary systems key signposts of early dynamical evolution. Here we investigate the dynamical states of three of the youngest multi-transiting planetary systems: AU Mic (3-planet, $\sim$20-Myr-old), V1298 Tau (4-planet, $\sim$23-Myr-old), and TOI-2076 (4-planet, $\sim$200-Myr-old). We find that most planet pairs in these systems lie near resonance with circulating rather than librating resonant angles. As a result, they are more susceptible to dynamical chaos than systems that are either securely locked in resonance or far removed from it. Even modest eccentricities of 0.04 to 0.08 may drive them to instability on timescales of tens to hundreds of Myr. Moreover, the observed orbital architectures are vulnerable to eccentricity excitation through mechanisms such as divergent resonance crossing triggered by planetesimal scattering. The observed near-resonant state may represent a transitional phase between a librating resonant chains and a mature non-resonant planetary system. Finally, we briefly discuss mechanisms that could give rise to the observed near-resonant configurations, including overstable libration, disk turbulence, and receding disk inner edge.

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arXiv2025

Terrestrial Planet Formation from Two Source Reservoirs

This work describes new dynamical simulations of terrestrial planet formation. The simulations started at the protoplanetary disk stage, when planetesimals formed and accreted into protoplanets, and continued past the late stage of giant impacts. We explored the effect of different parameters, such as the initial radial distribution of planetesimals and Type-I migration of protoplanets, on the final results. In each case, a thousand simulations were completed to characterize the stochastic nature of the accretion process. In the model best able to satisfy various constraints, Mercury, Venus, and Earth accreted from planetesimals that formed early near the silicate sublimation line near 0.5 au and migrated by disk torques. For Venus and Earth to end up at 0.7-1 au, Type-I migration had to be directed outward, for example as the magnetically driven winds reduced the surface gas density in the inner part of the disk. Mercury was left behind near the original ring location. We suggest that Mars and multiple Mars-sized protoplanets grew from a distinct outer source of planetesimals at 1.5-2 au. While many migrated inwards to accrete onto the proto-Earth, our Mars was the lone survivor. This model explains: (1) the masses and orbits of the terrestrial planets, (2) the chemical composition of the Earth, where ~70% and ~30% come from reduced inner-ring and more-oxidized outer-ring materials, and (3) the isotopic differences of the Earth and Mars. It suggests that the Moon-forming impactor Theia plausibly shared a similar isotopic composition and accretion history with that of the proto-Earth.

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arXiv2024

Nutation-orbit resonances: The origin of the chaotic rotation of Hyperion and the barrel instability

While numerous planetary and asteroid satellites show evidence for non-trivial rotation states, none are as emblematic as Hyperion, which has long been held as the most striking example of chaotic spin-orbit evolution in the Solar System. Nevertheless, an analytically tractable theory of the full 3D spin-orbit dynamics of Hyperion has not been developed. We derive the Hamiltonian for a spinning axisymmetric satellite in the gravitational potential of a planet without assuming planar or principal axis rotation and without averaging over the spin period. Using this model, we demonstrate the emergence of resonances between the nutation and orbital frequencies that act as the primary drivers of the spin dynamics. This analysis reveals that, contrary to long-held belief, Hyperion is not tumbling chaotically. Instead, it lies near or in a nutation-orbit resonance that is first-order in eccentricity, allowing it to rotate quasi-regularly. The most reliable observations are consistent with either nonchaotic motion or chaos that is orders of magnitude smaller than originally claimed. A separate phenomenon, the so-called barrel instability, is shown to be related to a different set of nutation-orbit resonances that generalize the planar spin-orbit resonances. Finally, we show that changes in spin states over long timescales are best understood by considering chaotic diffusion of quasi-conserved quantities.

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arXiv2024

The Prevalence of Resonance Among Young, Close-in Planets

Multiple planets undergoing disk migration may be captured into a chain of mean-motion resonances with the innermost planet parked near the disk's inner edge. Subsequent dynamical evolution may disrupt these resonances, leading to the non-resonant configurations typically observed among {\it Kepler} planets that are Gyrs old. In this scenario, resonant configurations are expected to be more common in younger systems. This prediction can now be tested, thanks to recent discoveries of young planets, particularly those in stellar clusters, by NASA's {\it TESS} mission. We divided the known planetary systems into three age groups: young ($<$100-Myr-old), adolescent (0.1-1-Gyr-old), and mature ($>1$-Gyr-old). The fraction of neighboring planet pairs having period ratios within a few percent of a first-order commensurability (e.g.~4:3, 3:2, or 2:1) is 70$\pm$15\% for young pairs, 24$\pm$8\% for adolescent pairs, and 15$\pm$2\% for mature pairs. The fraction of systems with at least one nearly commensurable pair (either first or second-order) is 86$\pm13$\% among young systems, 38$\pm12$\% for adolescent systems, and 23$\pm3$\% for mature systems. First-order commensurabilities prevail across all age groups, with an admixture of second-order commensurabilities. Commensurabilities are more common in systems with high planet multiplicity and low mutual inclinations. Observed period ratios often deviate from perfect commensurability by $\sim$1\% even among young planets, too large to be explained by resonant repulsion with equilibrium eccentricity tides. We also find that super-Earths in the radius gap ($1.5-1.9R_\oplus$) are less likely to be near-resonant (11.9$\pm2.0\%$) compared to Earth-sized planets ($R_p<1R_\oplus$; 25.3$\pm4.4\%$) or mini-Neptunes ($1.9R_\oplus \leq R_p<2.5R_\oplus$; 14.4$\pm1.8\%$).

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arXiv2024

Breaking Giant Chains: Early-Stage Instabilities in Long-Period Giant Planet Systems

Orbital evolution is a critical process that sculpts planetary systems, particularly during their early stages where planet-disk interactions are expected to lead to the formation of resonant chains. Despite the theoretically expected prominence of such configurations, they are scarcely observed among long-period giant exoplanets. This disparity suggests an evolutionary sequence wherein giant planet systems originate in compact multiresonant configurations, but subsequently become unstable, eventually relaxing to wider orbits--a phenomenon mirrored in our own solar system's early history. In this work, we present a suite of N-body simulations that model the instability-driven evolution of giant planet systems, originating from resonant initial conditions, through phases of disk dispersal and beyond. By comparing the period ratio and normalized angular momentum distributions of our synthetic aggregate of systems with the observational census of long-period Jovian planets, we derive constraints on the expected rate of orbital migration, the efficiency of gas-driven eccentricity damping, and typical initial multiplicity. Our findings reveal a distinct inclination toward densely packed initial conditions, weak damping, and high giant planet multiplicities. Furthermore, our models indicate that resonant chain origins do not facilitate the formation of Hot Jupiters via the coplanar high-eccentricity pathway at rates high enough to explain their observed prevalence.

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arXiv2024

Chaotic tides as a solution to the Hyperion problem

The dynamics of the outer regular satellites of Saturn are driven primarily by the outward migration of Titan, but several independent constraints on Titan's migration are difficult to reconcile with the current resonant orbit of the small satellite Hyperion. We argue that Hyperion's rapid irregular tumbling greatly increases tidal dissipation with a steep dependence on orbital eccentricity. Resonant excitation from a migrating Titan is then balanced by damping in a feedback mechanism that maintains Hyperion's eccentricity without fine-tuning. The inferred tidal parameters of Hyperion are most consistent with rapid Titan migration enabled by a resonance lock with an internal mode of Saturn, but a scenario with only equilibrium dissipation in Saturn is also possible.

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arXiv2023

A $5M_\text{Jup}$ Non-Transiting Coplanar Circumbinary Planet Around Kepler-1660AB

Over a dozen transiting circumbinary planets have been discovered around eclipsing binaries. Transit detections are biased towards aligned planet and binary orbits, and indeed all of the known planets have mutual inclinations less than $4.5^{\circ}$. One path to discovering circumbinary planets with misaligned orbits is through eclipse timing variations (ETVs) of non-transiting planets. Borkovits et al. (2016) discovered ETVs on the 18.6 d binary Kepler-1660AB, indicative of a third body on a $\approx 236$ d period, with a misaligned orbit and a potentially planetary mass. Getley et al. (2017) agreed with the planetary hypothesis, arguing for a $7.7M_{\rm Jup}$ circumbinary planet on an orbit that is highly misaligned by $120^{\circ}$ with respect to the binary. In this paper, we obtain the first radial velocities of the binary. We combine these with an analysis of not only the ETVs but also the eclipse depth variations. We confirm the existence of a $239.5$ d circumbinary planet, but with a lower mass of $4.87M_{\rm Jup}$ and a coplanar orbit. The misaligned orbits proposed by previous authors are definitively ruled out by a lack of eclipse depth variations. Kepler-1660ABb is the first confirmed circumbinary planet found using ETVs around a main sequence binary.

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arXiv2022

Dynamics and Origins of the Near-Resonant Kepler Planets

Short-period super-Earths and mini-Neptunes encircle more than $\sim50\%$ of Sun-like stars and are relatively amenable to direct observational characterization. Despite this, environments in which these planets accrete are difficult to probe directly. Nevertheless, pairs of planets that are close to orbital resonances provide a unique window into the inner regions of protoplanetary disks, as they preserve the conditions of their formation, as well as the early evolution of their orbital architectures. In this work, we present a novel approach toward quantifying transit timing variations within multi-planetary systems and examine the near-resonant dynamics of over 100 planet pairs detected by \textit{Kepler}. Using an integrable model for first-order resonances, we find a clear transition from libration to circulation of the resonant angle at a period ratio of $\approx 0.6\%$ wide of exact resonance. The orbital properties of these systems indicate that they systematically lie far away from the resonant forced equilibrium. Cumulatively our modeling indicates that while orbital architectures shaped by strong disk damping or tidal dissipation are inconsistent with observations, a scenario where stochastic stirring by turbulent eddies augments the dissipative effects of protoplanetary disks reproduces several features of the data.

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arXiv2022

TOI-1136 is a Young, Coplanar, Aligned Planetary System in a Pristine Resonant Chain

Convergent disk migration has long been suspected to be responsible for forming planetary systems with a chain of mean-motion resonances (MMR). Dynamical evolution over time could disrupt the delicate resonant configuration. We present TOI-1136, a 700-Myr-old G star hosting at least 6 transiting planets between $\sim$2 and 5 $R_\oplus$. The orbital period ratios deviate from exact commensurability by only $10^{-4}$, smaller than the $\sim$\,$10^{-2}$ deviations seen in typical Kepler near-resonant systems. A transit-timing analysis measured the masses of the planets (3-8$M_\oplus$) and demonstrated that the planets in TOI-1136 are in true resonances with librating resonant angles. Based on a Rossiter-McLaughlin measurement of planet d, the star's rotation appears to be aligned with the planetary orbital planes. The well-aligned planetary system and the lack of detected binary companion together suggest that TOI-1136's resonant chain formed in an isolated, quiescent disk with no stellar fly-by, disk warp, or significant axial asymmetry. With period ratios near 3:2, 2:1, 3:2, 7:5, and 3:2, TOI-1136 is the first known resonant chain involving a second-order MMR (7:5) between two first-order MMR. The formation of the delicate 7:5 resonance places strong constraints on the system's migration history. Short-scale (starting from $\sim$0.1 AU) Type-I migration with an inner disk edge is most consistent with the formation of TOI-1136. A low disk surface density ($\Sigma_{\rm 1AU}\lesssim10^3$g~cm$^{-2}$; lower than the minimum-mass solar nebula) and the resultant slower migration rate likely facilitated the formation of the 7:5 second-order MMR. TOI-1136's deep resonance suggests that it has not undergone much resonant repulsion during its 700-Myr lifetime. One can rule out rapid tidal dissipation within a rocky planet b or obliquity tides within the largest planets d and f.

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arXiv2022

A Criterion for the Stability of Planets in Chains of Resonances

Uncovering the formation process that reproduces the distinct properties of compact super-Earth exoplanet systems is a major goal of planet formation theory. The most successful model argues that non-resonant systems begin as resonant chains of planets that later experience a dynamical instability. However, both the boundary of stability in resonant chains and the mechanism of the instability itself are poorly understood. Previous work postulated that a secondary resonance between the fastest libration frequency and a difference in synodic frequencies destabilizes the system. Here, we use that hypothesis to produce a simple and general criterion for resonant chain stability that depends only on planet orbital periods and masses. We show that the criterion accurately predicts the maximum mass of planets in synthetic resonant chains up to six planets. More complicated resonant chains produced in population synthesis simulations are found to be less stable than expected, although our criterion remains useful and superior to machine learning models.

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arXiv2022

Architectures of Compact Super-Earth Systems Shaped by Instabilities

Compact non-resonant systems of sub-Jovian planets are the most common outcome of the planet formation process. Despite exhibiting broad overall diversity, these planets also display dramatic signatures of intra-system uniformity in their masses, radii, and orbital spacings. Although the details of their formation and early evolution are poorly known, sub-Jovian planets are expected to emerge from their natal nebulae as multi-resonant chains, owing to planet-disk interactions. Within the context of this scenario, the architectures of observed exoplanet systems can be broadly replicated if resonances are disrupted through post-nebular dynamical instabilities. Here, we generate an ad-hoc sample of resonant chains and use a suite of N-body simulations to show that instabilities can not only reproduce the observed period ratio distribution, but that the resulting collisions also modify the mass uniformity in a way that is consistent with the data. Furthermore, we demonstrate that primordial mass uniformity, motivated by the sample of resonant chains coupled with dynamical sculpting, naturally generates uniformity in orbital period spacing similar to what is observed. Finally, we find that almost all collisions lead to perfect mergers, but some form of post-instability damping is likely needed to fully account for the present-day dynamically cold architectures of sub-Jovian exoplanets.

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arXiv2021

A Tidal Origin for a 3-body Resonance in Kepler-221

Over the course of the last two decades, traditional models of planet formation have been repeatedly challenged by the emerging census of extrasolar planets. Key among them is the orbital architecture problem: while standard models of orbital migration predict resonant orbits for short-period objects, most planets do not appear to lie in orbital resonances. Here we show that the four-planet system Kepler-221, not previously recognized to have active orbital resonances, has a three-body commensurability relation unique within the Kepler sample. Using a suite of numerical experiments as well as a perturbative analysis, we demonstrate that this system likely began as a resonant chain and proceeded to undergo large-scale divergence away from resonance, under the action of tidal dissipation. Our results further indicate that obliquity tides, driven by a secular spin-orbit resonance and mutual inclination, are an excellent candidate for driving this orbital divergence, and that the high tidal luminosity may also explain the anomalous size of planet b, which lies within the Fulton radius gap.

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arXiv2018

Prospects for Refining Kepler TTV Masses using TESS Observations

In this paper we investigate systems previously identified to exhibit transit timing variations (TTVs) in Kepler data, with the goal of predicting the expected improvements to the mass and eccentricity constraints that will arise from combining Kepler data with future data from the TESS mission. We advocate for the use of the Kullback-Leibler (KL) divergence as a means to quantify improvements in the measured constraints. Compared to the original Kepler data, the TESS data will have a lower signal-to-noise ratio, rendering some of the planetary transits undetectable, and lowering the accuracy with which the transit mid-time can be estimated. Despite these difficulties, out of the 55 systems (containing 143 planets) investigated, we predict that the collection of short-cadence data by TESS will be of significant value (i.e. it will improve the mass uncertainty such that the KL divergence is > 0.1) for approximately 6 - 14 planets during the nominal mission, with the range primarily driven by the uncertain precision with which transit mid-times will be recovered from TESS data. In an extended mission this would increase to a total of approximately 12 - 25 planets.

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arXiv2018

Dynamical Evolution of Planetary Systems

Planetary systems can evolve dynamically even after the planets themselves have fully formed, and there is circumstantial evidence that most planetary systems become unstable after the disappearance of the gaseous protoplanetary disk. Theories of planet formation predict that chains of mean motion resonances are the natural outcome of disk-driven planet migration, leading to the pile up of super-Earths resonant chains close to the inner edge of the disk and the formation of fragile chains for distant giant planets. Observations of young systems suggest that they are more often locked in these chains than older ones, which are instead mostly non-resonant. The instabilities thought responsible for this trend can arise intrinsically if the original systems are too closely packed, or be due to external perturbations such as tides, planetesimal scattering, or torques from distant stellar companions. The Solar System was not exceptional in this sense, as the outer giants saw the disruption of a resonant chain; meanwhile, the inner system was likely built through a series of giant impacts between closely packed planetary embryos. Thus, the orbital distributions of planetary systems that is observed today, both solar and extrasolar, can be different from those emerging from formation and assembly processes within the disk, and it is important to consider possible long-term dynamics to connect the two.

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