arXiv subjects

Yoram Lithwick

Publications and source records attributed to Yoram Lithwick.

At least 19 recordsLinked to original sources

arXiv2026

Two-stage disruption of resonant chains

TESS is enabling the discovery of transiting planets around young stars. These observations suggest that most close-in planets were born in chains of mean-motion resonances that break on a characteristic timescale of order 100 Myr. This observation is surprising because the same dissipative forces that capture planets into resonance render their orbits long-term stable. We explore a two-stage disruption scenario for resonant chains of super-Earths. First, the chains have their (free) eccentricities excited by some mechanism. We show that any such mechanism that seeds eccentricities of a few percent sets in motion a second stage of dynamical instability on a $\sim$100 Myr timescale. A possible stage-one mechanism is the accretion of a handful of Mercury-sized bodies totaling a few percent of the planetary system mass, which excites the requisite eccentricities and triggers a stage two that reproduces the observed decline in the incidence of resonance. Impacts from such bodies can also explain why some young systems have period ratios narrow of commensurability. We sketch how these impactors may have grown out of debris left over from an earlier epoch of planet formation. We also identify two new trends in the observational data: a decline in multiplicity on the same timescale as the decline in the incidence of resonance, and an increase in the occupation of resonances with multiplicity.

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arXiv2025

Eccentric Disks With Self-Gravity

Can a disk orbiting a central body be eccentric, when the disk feels its own self-gravity and is pressureless? Contradictory answers appear in the literature. We show that such a disk can be eccentric, but only if it has a sharply truncated edge: the surface density $\Sigma$ must vanish at the edge, and the $\Sigma$ profile must be sufficiently steep at the point where it vanishes. If either requirement is violated, an eccentric disturbance leaks out of the bulk of the disk into the low density edge region, and cannot return. An edge where $\Sigma$ asymptotes to zero but never vanishes, as is often assumed for astrophysical disks, is insufficiently sharp. Similar results were shown by Hunter & Toomre (1969) for galactic warps. We demonstrate these results in three ways: by solving the eigenvalue equation for the eccentricity profile; by solving the initial value problem; and by analyzing a new and simple dispersion relation that is valid for any wavenumber, unlike WKB. As a byproduct, we show that softening the self-gravitational potential is not needed to model a flat disk, and we develop a softening-free algorithm to model the disk's Laplace-Lagrange-like equations. The algorithm is easy to implement and is more accurate than softening-based methods at a given resolution by many orders of magnitude.

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arXiv2025

Enceladus's Limit Cycle

Enceladus exhibits some remarkable phenomena, including water geysers spraying through surface cracks, a global ice shell that is librating atop an ocean, a large luminosity, and rapid outward orbital migration. Here we model the coupled evolution of Enceladus's orbit and interior structure. We find that Enceladus is driven into a periodic state: a limit cycle. Enceladus's observed phenomena emerge from the model, and the predicted values for the orbital eccentricity, libration amplitude, shell thickness, and luminosity agree with observations. A single limit cycle lasts around ten million years, and has three distinct stages: (1) freezing, (2) melting, and (3) resonant libration. Enceladus is currently in the freezing stage, meaning that its ice shell is getting thicker. That pressurizes the ocean, which in turn cracks the shell and pushes water up through the cracks. In this stage the orbital eccentricity increases, as Saturn pushes Enceladus deeper into resonance with Dione. Once the eccentricity is sufficiently high, tidal heating begins to melt the shell, which is the second stage of the cycle. In the third stage the shell remains close to 3km thick. At that thickness the shell's natural libration frequency is resonant with the orbital frequency. The shell's librations are consequently driven to large amplitude, for millions of years. Most of the tidal heating of Enceladus occurs during this stage, and the observed luminosity is a relic from the last episode of resonant libration.

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arXiv2025

Enceladus's Tidal Heating

Saturn raises a time-dependent tide on its small moon Enceladus, due to the eccentricity of the orbit. As shown in a companion paper (Goldreich et al.), the resulting tidal heating drives Enceladus into a limit cycle, in which its eccentricity and shell thickness vary in tandem, on a timescale of ~ 10 Myr. The limit cycle explains a variety of observed phenomena on Enceladus, including its large luminosity and cracked ice shell. Here we derive the tidal heating rate needed for that study, starting from a simple first-principles derivation of Enceladus's tidal response. Enceladus is comprised of three layers: a rocky core, an outer ice shell, and an ocean sandwiched in between. Tides force the shell to librate and distort, which generates heat. We calculate the libration amplitude and tidal heating rate by minimizing the sum of elastic and gravitational energies. The final expressions are analytic, and account for the finite hardness of the shell, and for resonant libration. Although we specialize to Enceladus, our approach may be extended to other bodies that have a similar three layer structure, such as Europa and Titan.

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arXiv2024

Extreme scale height variations and nozzle shocks in warped disks

Accretion disks around both stellar-mass and supermassive black holes are likely often warped. Whenever a disk is warped, its scale height varies with azimuth. Sufficiently strong warps cause extreme compressions of the scale height, which fluid parcels "bounce" off of twice per orbit to high latitudes. In this paper, we study the dynamics of such strong warps using two methods: (i) the nearly analytic "ring theory" of Fairbairn & Ogilvie (2021a), which we generalize to the Kerr metric; and (ii) 3D general-relativistic hydrodynamic simulations of tori ("rings") around black holes, using the H-AMR code. We initialize a ring with a warp and study the subsequent evolution on tens of orbital periods. The simulations agree excellently with the ring theory until the warp amplitude, $\psi$, reaches a critical value $\psi_{\rm c}$. When $\psi>\psi_{\rm c}$, the rings enter the bouncing regime. We analytically derive (and numerically validate) that $\psi_{\rm c}\approx (r/r_{\rm g})^{-1/2}$ in the non-Keplerian regime, where $r_{\rm g}=GM/c^2$ is the gravitational radius and $M$ is the mass of the central object. Whenever the scale height bounces, the vertical velocity becomes supersonic, which leads to a "nozzle shock" as the gas collides at the scale height minima. Nozzle shocks damp the warp within $\approx10-20$ orbits in the simulations; but, that damping is not captured by the ring theory. Nozzle shock dissipation leads to inflow timescales that are 1-2 orders of magnitude shorter than unwarped $\alpha$ disks which may result in rapid variability, such as in changing-look active galactic nuclei or in the soft state of X-ray binaries. We also propose that steady disks with strong enough warps may self-regulate to have amplitudes near $\psi_{\rm c}$.

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arXiv2024

Repelling Planet pairs by Ping-pong Scattering

The Kepler mission reveals a peculiar trough-peak feature in the orbital spacing of close-in planets near mean-motion resonances: a deficit and an excess that are a couple percent to the narrow, respectively wide, of the resonances. This feature has received two main classes of explanations, one involving eccentricity damping, the other scattering with small bodies. Here, we point out a few issues with the damping scenario, and study the scattering scenario in more detail. We elucidate why scattering small bodies tends to repel two planets. As the small bodies random-walk in energy and angular momentum space, they tend to absorb, fractionally, more energy than angular momentum. This, which we call "ping-pong repulsion", transports angular momentum from the inner to the outer planet and pushes the two planets apart. Such a process, even if ubiquitous, leaves identifiable marks only near first-order resonances: diverging pairs jump across the resonance quickly and produce the MMR asymmetry. To explain the observed positions of the trough-peaks, a total scattering mass of order a few percent of the planet masses is required. Moreover, if this mass is dominated by a handful of Mercury-sized bodies, one can also explain the planet eccentricities as inferred from transit-time-variations. Lastly, we suggest how these conditions may have naturally arisen during the late stages of planet formation.

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arXiv2023

Irradiated Disks May Settle into Staircases

Much of a protoplanetary disk is thermally controlled by irradiation from the central star. Such a disk, long thought to have a smoothly flaring shape, is unstable to the so-called 'irradiation instability'. But what's the outcome of such an instability? In particular, is it possible that such a disk settles into a shape that is immune to the instability? We combine Athena++ with a simplified thermal treatment to show that passively heated disks settle into a 'staircase' shape. Here, the disk is punctuated by bright rings and dark gaps, with the bright rings intercepting the lion's share of stellar illumination, and the dark gaps hidden in their shadows. The optical surface of such a disk (height at which starlight is absorbed) resembles a staircase. Although our simulations do not have realistic radiative transfer, we use the RADMC3d code to show that this steady state is in good thermal equilibrium. It is possible that realistic disks reach such a state via ways not captured by our simulations. In contrast to our results here, two previous studies have claimed that irradiated disks stay smooth. We show here that they err on different issues. The staircase state, if confirmed by more sophisticated radiative hydrodynamic simulations, has a range of implications for disk evolution and planet formation.

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arXiv2021

The Criterion for Chaos in Three-Planet Systems

We establish the criterion for chaos in three-planet systems, for systems similar to those discovered by the Kepler spacecraft. Our main results are as follows: (i) The simplest criterion, which is based on overlapping mean motion resonances MMR's), only agrees with numerical simulations at a very crude level. (ii) Much greater accuracy is attained by considering neighboring MMR's that do not overlap. We work out the width of the chaotic zones around each of the neighbors, and also provide simple approximate expressions for the widths. (iii) Even greater accuracy is provided by the overlap of three-body resonances (3BR's), which accounts for fine-grained structure seen in maps from N-body simulations, and also predicts the Lyapunov times. Previous studies conflict on whether overlap of MMR's or of 3BR's drive interplanetary chaos. We show that both do, and in fact they are merely different ways of looking at the same effect. (iv) We compare both criteria with high-resolution maps of chaos from N-body simulations, and show that they agree at a high level of detail.

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arXiv2021

The Irradiation Instability of Protoplanetary Disks

The temperature in most parts of a protoplanetary disk is determined by irradiation from the central star. Numerical experiments of Watanabe \& Lin (2008) suggested that such disks, also called `passive disks', suffer from a thermal instability. Here, we use analytical and numerical tools to elucidate the nature of this instability. We find that it is related to the flaring of the optical surface, the layer at which starlight is intercepted by the disk. Whenever a disk annulus is perturbed thermally and acquires a larger scale height, disk flaring becomes steeper in the inner part, and flatter in the outer part. Starlight now shines more overhead for the inner part and so can penetrate into deeper layers; conversely, it is absorbed more shallowly in the outer part. These geometric changes allow the annulus to intercept more starlight, and the perturbation grows. We call this the irradiation instability. It requires only ingredients that are known to exist in realistic disks, and operates best in parts that are both optically thick and geometrically thin (inside 30AU, but can extend to further reaches when, e.g., dust settling is considered). An unstable disk develops travelling thermal waves that reach order-unity in amplitude. In thermal radiation, such a disk should appear as a series of bright rings interleaved with dark shadowed gaps, while in scattered light it resembles a moving staircase. Depending on the gas and dust responses, this instability could lead to a wide range of consequences, such as {\w ALMA rings and gaps,} dust traps, vertical circulation, vortices and turbulence.

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arXiv2021

Outward Migration of Super-Jupiters

Recent simulations show that giant planets of about one Jupiter mass migrate inward at a rate that differs from the Type II prediction. Here we show that at higher masses, planets migrate outward. Our result differs from previous ones because of our longer simulation times, lower viscosity, and our boundary conditions that allow the disk to reach viscous steady state. We show that, for planets on circular orbits, the transition from inward to outward migration coincides with the known transition from circular to eccentric disks that occurs for planets more massive than a few Jupiters. In an eccentric disk, the torque on the outer disk weakens due to two effects: the planet launches weaker waves, and those waves travel further before damping. As a result, the torque on the inner disk dominates, and the planet pushes itself outward. Our results suggest that the many super-Jupiters observed by direct-imaging at large distances from the star may have gotten there by outward migration.

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arXiv2020

Long-Lived Eccentric Modes in Circumbinary Disks

Hydrodynamical simulations show that circumbinary disks become eccentric, even when the binary is circular. Here we demonstrate that, in steady state, the disk's eccentricity behaves as a long-lived free mode trapped by turning points that naturally arise from a continuously truncated density profile. Consequently, both the disk's precession rate and eccentricity profile may be calculated via the simple linear theory for perturbed pressure-supported disks. By formulating and solving the linear theory we find that (i) surprisingly, the precession rate is roughly determined by the binary's quadrupole, even when the quadrupole is very weak relative to pressure; (ii) the eccentricity profile is largest near the inner edge of the disk, and falls exponentially outwards; and (iii) the results from linear theory indeed agree with what is found in simulations. Understanding the development of eccentric modes in circumbinary disks is a crucial first step for understanding the long term (secular) exchange of eccentricity, angular momentum and mass between the binary and the gas. Potential applications include the search for a characteristic kinematic signature in disks around candidate binaries and precession-induced modulation of accretion over long timescales.

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arXiv2020

Inner Boundary Condition in Quasi-Lagrangian Simulations of Accretion Disks

In simulations of viscously evolving accretion disks, the inner boundary condition is particularly important. If treated incorrectly, it induces incorrect behavior very quickly, because the viscous time is shortest near the inner boundary. Recent work has determined the correct inner boundary in Eulerian simulations. But in quasi-Lagrangian simulations (e.g., SPH, moving mesh, and mesh-less), where the inner boundary is modeled by removing mass within a finite zone, the inner density profile typically becomes anomalously depleted. Here we show how the boundary condition should be applied in such codes, via a simple modification of the usual approach: when one removes mass, one must speed up the remaining material so that the disk's angular momentum is unchanged. We show with both 1D and 2D moving-mesh (AREPO) simulations that this scheme works as desired in viscously evolving disks. It produces no spurious density depletions and is independent of the mass removal rate, provided that the disk is adequately resolved and that the mass removal rate is not so extreme as to trigger instabilities. This "torque-free" mass removal technique permits the use of quasi-Lagrangian codes to simulate viscously evolving disks, while including a variety of additional effects. As an example, we apply our scheme to a 2D simulation of an accretion disk perturbed by a very massive planet, in which the disk is evolved to viscous steady state.

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arXiv2020

Convection with Misaligned Gravity and Rotation: Simulations and Rotating Mixing Length Theory

We present numerical simulations, using two complementary setups, of rotating Boussinesq thermal convection in a three-dimensional Cartesian geometry with misaligned gravity and rotation vectors. This model represents a small region at a non-polar latitude in the convection zone of a star or planet. We investigate the effects of rotation on the bulk properties of convection at different latitudes, focusing on determining the relation between the heat flux and temperature gradient. We show that our results may be interpreted using rotating mixing length theory (RMLT). The simplest version of RMLT (due to Stevenson) considers the single mode that transports the most heat. This works reasonably well in explaining our results, but there is a systematic departure from these predictions (up to approximately $30\%$ in the temperature gradient) at mid-latitudes. We develop a more detailed treatment of RMLT that includes the transport afforded by multiple modes, and we show that this accounts for most of the systematic differences. We also show that convectively-generated zonal flows and meridional circulations are produced in our simulations, and that their properties depend strongly on the dimensions of the box. These flows also affect the heat transport, contributing to departures from RMLT at some latitudes. However, we find the theoretical predictions of the multi-mode theory for the mid-layer temperature gradient, the root-mean-square (RMS) vertical velocity, the RMS temperature fluctuation, and the spatial spectrum of the heat transport at different latitudes, are all in reasonably good agreement with our numerical results when zonal flows are small.

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arXiv2020

Pileups and Migration Rates for Planets in Low Mass Disks

We investigate how planets interact with viscous accretion disks, in the limit that the disk is sufficiently low mass that the planet migrates more slowly than the disk material. In that case, the disk's surface density profile is determined by the disk being in viscous steady state (VSS), while overflowing the planet's orbit. We compute the VSS profiles with 2D hydrodynamical simulations, and show that disk material piles up behind the planet, with the planet effectively acting as a leaky dam. Previous 2D hydrodynamical simulations missed the pileup effect because of incorrect boundary conditions, while previous 1D models greatly overpredicted the pileup due to the neglect of non-local deposition. Our simulations quantify the magnitude of the pileup for a variety of planet masses and disk viscosities. We also calculate theoretically the magnitude of the pileup for moderately deep gaps, showing good agreement with simulations. For very deep gaps, current theory is inadequate, and we show why and what must be understood better. The pileup is important for two reasons. First, it is observable in directly imaged protoplanetary disks, and hence can be used to diagnose the mass of a planet that causes it or the viscosity within the disk. And second, it determines the planet's migration rate. Our simulations determine a new Type-II migration rate (valid for low mass disks), and show how it connects continuously with the well-verified Type-I rate.

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arXiv2019

Long-Lived Eccentricities in Accretion Disks

Accretion disks can be eccentric: they support $m=1$ modes that are global and slowly precessing. But whether the modes remain trapped in the disk---and hence are long-lived---depends on conditions at the outer edge of the disk. Here we show that in disks with realistic boundaries, in which the surface density drops rapidly beyond a given radius, eccentric modes are trapped and hence long-lived. We focus on pressure-only disks around a central mass, and show how this result can be understood with the help of a simple second-order WKB theory. We show that the longest lived mode is the zero-node mode in which all of the disk's elliptical streamlines are aligned, and that this mode decays coherently on the viscous timescale of the disk. Hence such a mode, once excited, will live for the lifetime of the disk. It may be responsible for asymmetries seen in recent images of protoplanetary disks.

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arXiv2019

Memoirs of a giant planet

Saturn is ringing weakly. Exquisite data from the Cassini mission reveal the presence of f-mode oscillations as they excite density waves in Saturn's rings. These oscillations have displacement amplitudes of order a metre on Saturn's surface. We propose that they result from large impacts in the past. Experiencing little dissipation inside Saturn on account of its weak luminosity, f-modes may live virtually forever; but the very ring waves that reveal their existence also remove energy from them, in 10^4 to 10^7 yrs for the observed f-modes (spherical degree 2-10). We find that the largest impacts that arrive during these times excite the modes to their current levels, with the exception of the few lowest degree modes. To explain the latter, either a fortuitously large impact in the recent past, or a new source of stochastic excitation, is needed. We extend this scenario to Jupiter which has no substantial rings. With an exceedingly long memory of past bombardments, Jovian f-modes and p-modes can acquire much higher amplitudes, possibly explaining past reports of radial-velocity detections, and detectable by the Juno spacecraft.

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arXiv2018

Eccentric Modes in Disks With Pressure and Self-Gravity

Accretion disks around stars, or other central massive bodies, can support long-lived, slowly precessing $m=1$ disturbances in which the fluid motion is nearly Keplerian with non-zero eccentricity. We study such `slow modes' in disks that are subject to both pressure and self-gravity forces. We derive a second-order WKB dispersion relation that describes the dynamics quite accurately, and show that the apparently complicated nature of the various modes can be understood in a simple way with the help of a graphical method. We also solve the linearized fluid equations numerically, and show that the results agree with the theory. We find that when self-gravity is weak ($Q\gtrsim 1/h$, where $Q$ is Toomre's parameter, and $h$ is the disk aspect ratio) the modes are pressure dominated. But when self-gravity is strong ($1<Q\lesssim 1/h$), two kinds of gravity-dominated modes appear: one is an aligned elliptical pattern and the other is a one-armed spiral. In the context of protoplanetary disks, we suggest that if the radial eccentricity profile can be measured, it could be used to determine the total disk mass.

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arXiv2018

A Criterion for the Onset of Chaos in Systems of Two Eccentric Planets

We derive a criterion for the onset of chaos in systems consisting of two massive, eccentric, coplanar planets. Given the planets' masses and separation, the criterion predicts the critical eccentricity above which chaos is triggered. Chaos occurs where mean motion resonances overlap, as in Wisdom (1980)'s pioneering work. But whereas Wisdom considered only nearly circular planets, and hence examined only first order resonances, we extend his results to arbitrarily eccentric planets (up to crossing orbits) by examining resonances of all orders. We thereby arrive at a simple expression for the critical eccentricity. We do this first for a test particle in the presence of a planet, and then generalize to the case of two massive planets, based on a new approximation to the Hamiltonian (Hadden, in prep). We then confirm our results with detailed numerical simulations. Finally, we explore the extent to which chaotic two-planet systems eventually result in planetary collisions.

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