arXiv subjects

Scott Tremaine

Publications and source records attributed to Scott Tremaine.

At least 37 records · Page 2Linked to original sources

arXiv2022

The origin and fate of the Gaia phase-space snail

The Gaia snail is a spiral feature in the distribution of solar-neighbourhood stars in position and velocity normal to the Galactic midplane. The snail probably arises from phase mixing of gravitational disturbances that perturbed the disc in the distant past. The most common hypothesis is that the strongest disturbance resulted from a passage of the Sagittarius dwarf galaxy close to the solar neighbourhood. In this paper we investigate the alternative hypothesis that the snail is created by many small disturbances rather than one large one, that is, by Gaussian noise in the gravitational potential. Probably most of this noise is due to substructures in the dark-matter halo. We show that this hypothesis naturally reproduces most of the properties of the snail. In particular it predicts correctly, with no free parameters, that the apparent age of the snail will be $ \sim 0.5$ Gyr. An important ingredient of this model is that any snail-like feature in the solar neighbourhood, whatever its cause, is erased by scattering from giant molecular clouds or other small-scale structure on a time-scale $\lesssim 1$ Gyr.

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arXiv2022

Vertical motion in the Galactic disc: unwinding the Snail

The distribution of stars in the Milky Way disc shows a spiral structure--the Snail--in the space of velocity and position normal to the Galactic mid-plane. The Snail appears as straight lines in the vertical frequency--vertical phase plane when effects from sample selection are removed. Their slope has the dimension of inverse time, with the simplest interpretation being the inverse age of the Snail. Here, we devise and fit a simple model in which the spiral starts as a lopsided perturbation from steady state, that winds up into the present-day morphology. The winding occurs because the vertical frequency decreases with vertical action. We use data from stars in Gaia EDR3 that have measured radial velocities, pruned by simple distance and photometric selection functions. We divide the data into boxels of dynamical invariants (radial action, angular momentum); our model fits the data well in many of the boxels. The model parameters have physical interpretations: one, $A$, is a perturbation amplitude, and one, $t$, is interpretable in the simplest models as the time since the event that caused the Snail. We find trends relating the strength and age to angular momentum: (i) the amplitude $A$ is small at low angular momentum ($<1\,600\mathrm{\,kpc\,km\,s}^{-1}$ or guiding-centre radius $< 7.3\,$kpc), and over a factor of three larger, with strong variations, in the outer disc; (ii) there is no single well-defined perturbation time, with $t$ varying between 0.2 and 0.6 Gyr. Residuals between the data and the model display systematic trends, implying that the data call for more complex models.

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arXiv2022

Secular Dynamics around a Supermassive Black Hole via Multipole Expansion

In galactic nuclei, the gravitational potential is dominated by the central supermassive black hole, so stars follow quasi-Keplerian orbits. These orbits are distorted by gravitational forces from other stars, leading to long-term orbital relaxation. The direct numerical study of these processes is challenging because the fast orbital motion imposed by the central black hole requires very small timesteps. An alternative approach, pioneered by Gauss, is to use the secular approximation of smearing out the $N$ stars over their Keplerian orbits, using $K$ nodes along each orbit. In this study we propose three novel improvements to this method. First, we re-formulate the discretisation of the rates of change of the variables describing the orbital states to ensure that all conservation laws are exactly satisfied. Second, we replace the pairwise sum over nodes by a multipole expansion up to order $\ell_{\mathrm{max}}$, reducing the overall computational costs from $O(N^2K^2)$ to $O(NK\ell_{\mathrm{max}}^2)$. Finally, we show that the averaged dynamical system is equivalent to $2N$ interacting unit spin vectors and provide two time integrators: a second-order symplectic scheme and a fourth-order Lie-group Runge-Kutta method, both of which are straightforward to generalize to higher order. These new simulations recover the diffusion coefficients of stellar eccentricities obtained through analytical calculations of the secular dynamics.

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arXiv2021

A Criterion for the Onset of Chaos in Compact, Eccentric Multiplanet Systems

We derive a semi-analytic criterion for the presence of chaos in compact, eccentric multiplanet systems. Beyond a minimum semimajor-axis separation, below which the dynamics are chaotic at all eccentricities, we show that (i) the onset of chaos is determined by the overlap of two-body mean motion resonances (MMRs), like it is in two-planet systems; (ii) secular evolution causes the MMR widths to expand and contract adiabatically, so that the chaotic boundary is established where MMRs overlap at their greatest width. For closely spaced two-planet systems, a near-symmetry strongly suppresses this secular modulation, explaining why the chaotic boundaries for two-planet systems are qualitatively different from cases with more than two planets. We use these results to derive an improved angular-momentum-deficit (AMD) stability criterion, i.e., the critical system AMD below which stability should be guaranteed. This introduces an additional factor to the expression from Laskar and Petit (2017) that is exponential in the interplanetary separations, which corrects the AMD threshold toward lower eccentricities by a factor of several for tightly packed configurations. We make routines for evaluating the chaotic boundary available to the community through the open-source SPOCK package.

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arXiv2020

Thermal equilibrium of an ideal gas in a free-floating box

The equilibrium and fluctuations of an ideal gas in a rigid container are studied by every student of statistical mechanics. Here we study the less well-known case when the box is floating freely; in particular we determine the fluctuations of the box in velocity and position due to interactions with the gas it contains. This system is a toy model for the fluctuations in velocity and position of a black hole surrounded by stars at the center of a galaxy. These fluctuations may be observable in nearby galaxies.

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arXiv2020

Relaxation in a Fuzzy Dark Matter Halo. II. Self-consistent kinetic equations

Fuzzy dark matter (FDM) is composed of ultra-light bosons having a de Broglie wavelength that is comparable to the size of the stellar component of galaxies at typical galactic velocities. FDM behaves like cold dark matter on large scales. However, on the scale of the de Broglie wavelength, an FDM halo exhibits density fluctuations that lead to relaxation, a process similar to the two-body relaxation that occurs in classical gravitational N-body systems and is described by the Fokker-Planck equation. We derive the FDM analog of that kinetic equation, and solve it to find the evolution of the velocity distribution in a spatially homogeneous FDM halo. We also determine the dielectric function and the dispersion relation for linear waves in an FDM halo.

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arXiv2020

Peas in a Pod? Radius correlations in Kepler multi-planet systems

We address the claim of Weiss et al. (2018) that the radii of adjacent planets in Kepler multi-planet systems are correlated. We explore two simple toy models---in the first the radii of the planets are chosen at random from a single universal distribution, and in the second we postulate several types of system with distinct radius distributions. We show that an apparent correlation between the radii of adjacent planets similar to the one reported by Weiss et al. (2018) can arise in both models. In addition the second model fits the radius and signal-to-noise distribution of the observed planets. We also comment on the validity of a commonly used correction that is used to estimate intrinsic planet occurrence rates, based on weighting planets by the inverse of their detectability.

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arXiv2020

Resonant capture in quadruple stellar systems

Some quadruple star systems in the hierarchical 2+2 configuration exhibit orbit-orbit resonances between the two compact binaries. We show that the most important resonances occur at period ratios of 1:1, 3:2 and 2:1. We describe the conditions required for capture and show that they can be satisfied at the 3:2 and 2:1 resonances in binaries that migrate significantly in semimajor axis after circularization, probably through magnetic braking or gravitational radiation.

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arXiv2020

Order-disorder phase transition in black-hole star clusters -- III. A mono-energetic cluster

Supermassive black holes at the centres of galaxies are often surrounded by dense star clusters. For a wide range of cluster properties and orbital radii the resonant relaxation times in these clusters are much shorter than the Hubble time. Since resonant relaxation conserves semimajor axes, these clusters should be in the maximum-entropy state consistent with the given semimajor axis distribution. We determine these maximum-entropy equilibria in a simplified model in which all of the stars have the same semimajor axes. We find that the cluster exhibits a phase transition from a disordered, spherical, high-temperature equilibrium to an ordered low-temperature equilibrium in which the stellar orbits have a preferred orientation or line of apsides. Here `temperature' is a measure of the non-Keplerian or self-gravitational energy of the cluster; in the spherical state, temperature is a function of the rms eccentricity of the stars. We explore a simple two-parameter model of black-hole star clusters -- the two parameters are semimajor axis and black-hole mass --- and find that clusters are susceptible to the lopsided phase transition over a range of ~100 in semimajor axis, mostly for black-hole masses less than $10^{7.5}$ solar masses.

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arXiv2019

Order-disorder phase transition in black-hole star clusters. II. A scale-free cluster

The supermassive black holes found at the centers of galaxies are often surrounded by dense star clusters. The ages of these clusters are generally longer than the resonant-relaxation time and shorter than the two-body relaxation time over a wide range of radii. We explore the thermodynamic equilibria of such clusters using a simple self-similar model. We find that the cluster exhibits a phase transition between a high-temperature spherical equilibrium and a low-temperature equilibrium in which the stars are on high-eccentricity orbits with nearly the same orientation. In the absence of relativistic precession, the spherical equilibrium is metastable below the critical temperature and the phase transition is first-order. When relativistic effects are important, the spherical equilibrium is linearly unstable below the critical temperature and the phase transition is continuous. A similar phase transition has recently been found in a model cluster composed of stars with a single semimajor axis. The presence of the same phenomenon in two quite different cluster models suggests that lopsided equilibria may form naturally in a wide variety of black-hole star clusters.

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arXiv2019

An order-disorder phase transition in black-hole star clusters

The centers of most galaxies contain massive black holes surrounded by dense star clusters. The structure of these clusters determines the rate and properties of observable transient events, such as flares from tidally disrupted stars and gravitational-wave signals from stars spiraling into the black hole. Most estimates of these rates enforce spherical symmetry on the cluster. Here we show that, in the course of generic evolutionary processes, a star cluster surrounding a black hole can undergo a robust phase transition from a spherical thermal equilibrium to a lopsided equilibrium, in which most stars are on high-eccentricity orbits with aligned orientations. The rate of transient events is expected to be much higher in the ordered phase. Better models of cluster formation and evolution are needed to determine whether clusters should be found in the ordered or disordered phase.

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arXiv2019

Relaxation in a Fuzzy Dark Matter Halo

Dark matter may be composed of light bosons, ${m_b \sim 10^{-22}\, \mathrm{eV}}$, with a de Broglie wavelength $λ\sim 1 \,\mathrm{kpc}$ in typical galactic potentials. Such `fuzzy' dark matter (FDM) behaves like cold dark matter (CDM) on much larger scales than the de Broglie wavelength, but may resolve some of the challenges faced by CDM in explaining the properties of galaxies on small scales ($\lesssim 10\,\mathrm{kpc}$). Because of its wave nature, FDM exhibits stochastic density fluctuations on the scale of the de Broglie wavelength that never damp. The gravitational field from these fluctuations scatters stars and black holes, causing their orbits to diffuse through phase space. We show that this relaxation process can be analyzed quantitatively with the same tools used to analyze classical two-body relaxation in an $N$-body system, and can be described by treating the FDM fluctuations as quasiparticles, with effective mass $\sim 10^7 M_\odot {(1\,\mathrm{kpc}/r)}^2{(10^{-22}\,\mathrm{eV}/m_b)}^3$ in a galaxy with a constant circular speed of $200\,\mathrm{kms}$. This novel relaxation mechanism may stall the inspiral of supermassive black holes or globular clusters due to dynamical friction at radii of a few hundred pc, and can heat and expand the central regions of galaxies. These processes can be used to constrain the mass of the light bosons that might comprise FDM.

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arXiv2017

Measuring the mass distribution in stellar systems

One of the fundamental tasks of dynamical astronomy is to infer the distribution of mass in a stellar system from a snapshot of the positions and velocities of its stars. The usual approach to this task (e.g., Schwarzschild's method) involves fitting parametrized forms of the gravitational potential and the phase-space distribution to the data. We review the practical and conceptual difficulties with this approach and describe a novel statistical method for determining the mass distribution that does not require determining the phase-space distribution of the stars. We show that this new estimator out-performs other distribution-free estimators for the harmonic and Kepler potentials.

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arXiv2017

Producing Distant Planets by Mutual Scattering of Planetary Embryos

It is likely that multiple bodies with masses between those of Mars and Earth ("planetary embryos") formed in the outer planetesimal disk of the solar system. Some of these were likely scattered by the giant planets into orbits with semi-major axes of hundreds of AU. Mutual torques between these embryos may lift the perihelia of some of them beyond the orbit of Neptune, where they are no longer perturbed by the giant planets so their semi-major axes are frozen in place. We conduct N-body simulations of this process, and its effect on smaller planetesimals in the region of the giant planets and the Kuiper belt. We find that (i) there is a significant possibility that one sub-Earth mass embryo, or possibly more, is still present in the outer solar system; (ii) the orbit of the surviving embryo(s) typically has perihelion of 40--70 AU, semi-major axis less than 200 AU, and inclination less than 30 degrees; (iii) it is likely that any surviving embryos could be detected by current or planned optical surveys or have a significant effect on solar-system ephemerides; (iv) whether or not an embryo has survived to the present day, their dynamical influence earlier in the history of the solar system can explain the properties of the detached disk (defined in this paper as containing objects with perihelia > 38 AU and semi-major axes between 80 and 500 AU).

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arXiv2017

Hot Jupiters driven by high-eccentricity migration in globular clusters

Hot Jupiters (HJs) are short-period giant planets that are observed around ~ 1% of solar-type field stars. One possible formation scenario for HJs is high-eccentricity (high-e) migration, in which the planet forms at much larger radii, is excited to high eccentricity by some mechanism, and migrates to its current orbit due to tidal dissipation occurring near periapsis. We consider high-e migration in dense stellar systems such as the cores of globular clusters (GCs), in which encounters with passing stars can excite planets to the high eccentricities needed to initiate migration. We study this process via Monte Carlo simulations of encounters with a star+planet system including the effects of tidal dissipation, using an efficient regularized restricted three-body code. HJs are produced in our simulations over a significant range of the stellar number density n_*. Assuming the planet is initially on a low-eccentricity orbit with semimajor axis 1 au, for n_* < 1e3 pc^{-3} the encounter rate is too low to induce orbital migration, whereas for n_* > 1e6 pc^{-3} HJ formation is suppressed because the planet is more likely ejected from its host star, tidally disrupted, or transferred to a perturbing star. The fraction of planets that are converted to HJs peaks at ~ 2% for intermediate number densities of ~ 4e4 pc^{-3}. Warm Jupiters, giant planets with periods between 10 and 100 days, are produced in our simulations with an efficiency of up to ~ 0.5%. Our results suggest that HJs can form through high-e migration induced by stellar encounters in the centers of of dense GCs, but not in their outskirts where the densities are lower.

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arXiv2017

Collisional Fragmentation is Not a Barrier to Close-in Planet Formation

Collisional fragmentation is shown to not be a barrier to rocky planet formation at small distances from the host star. Simple analytic arguments demonstrate that rocky planet formation via collisions of homogeneous gravity-dominated bodies is possible down to distances of order the Roche radius ($r_\mathrm{Roche}$). Extensive N-body simulations with initial bodies ${\gtrsim}1700$ km that include plausible models for fragmentation and merging of gravity-dominated bodies confirm this conclusion and demonstrate that rocky planet formation is possible down to ${\sim}$1.1 $r_\mathrm{Roche}$. At smaller distances, tidal effects cause collisions to be too fragmenting to allow mass build-up to a final, dynamically stable planetary system. We argue that even differentiated bodies can accumulate to form planets at distances that are not much larger than $r_\mathrm{Roche}$.

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arXiv2017

Isotropic-Nematic Phase Transitions in Gravitational Systems

We examine dense self-gravitating stellar systems dominated by a central potential, such as nuclear star clusters hosting a central supermassive black hole. Different dynamical properties of these systems evolve on vastly different timescales. In particular, the orbital-plane orientations are typically driven into internal thermodynamic equilibrium by vector resonant relaxation before the orbital eccentricities or semimajor axes relax. We show that the statistical mechanics of such systems exhibit a striking resemblance to liquid crystals, with analogous ordered-nematic and disordered-isotropic phases. The ordered phase consists of bodies orbiting in a disk in both directions, with the disk thickness depending on temperature, while the disordered phase corresponds to a nearly isotropic distribution of the orbit normals. We show that below a critical value of the total angular momentum, the system undergoes a first-order phase transition between the ordered and disordered phases. At the critical point the phase transition becomes second-order while for higher angular momenta there is a smooth crossover. We also find metastable equilibria containing two identical disks with mutual inclinations between $90^{\circ}$ and $180^\circ$.

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arXiv2017

Galactic disc profiles and a universal angular momentum distribution from statistical physics

We show that the stellar surface-brightness profiles in disc galaxies---observed to be approximately exponential---can be explained if radial migration efficiently scrambles the individual stars' angular momenta while conserving the circularity of the orbits and the total mass and angular momentum. In this case the disc's distribution of specific angular momenta $j$ should be near a maximum-entropy state and therefore approximately exponential, $dN\propto\exp(-j/\langle j\rangle)dj$. This distribution translates to a surface-density profile that is generally not an exponential function of radius: $Σ(R)\propto\exp[-R/R_e(R)]/(RR_e(R))(1+d\log v_c(R)/d\log R)$, for a rotation curve $v_c(R)$ and $R_e(R)\equiv\langle j\rangle/v_c(R)$. We show that such a profile matches the observed surface-brightness profiles of disc-dominated galaxies as well as the empirical exponential profile. Disc galaxies that exhibit population gradients cannot have fully reached a maximum-entropy state but appear to be close enough that their surface-brightness profiles are well-fit by this idealized model.

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