Brian K. Pickett

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks

We present a series of high-resolution, three-dimensional hydrodynamics simulations of a gravitationally unstable solar nebula model. The influences of both azimuthal grid resolution and the treatment of thermal processes on the origin and evolution of gravitational instabilities are investigated. In the first set of simulations, we vary the azimuthal resolution for a locally isothermal simulation, doubling and quadrupling the resolution used in a previous study; the largest number of grid points is (256, 256, 64) in cylindrical coordinates (r, , z). At this resolution, the disk breaks apart into a dozen short-lived condensations. Although our previous calculations underresolved the number and growth rate of clumps in the disk, the overall qualitative, but fundamental, conclusion remains: fragmentation under the locally isothermal condition in numerical simulations does not in itself lead to the survival of clumps to become gaseous giant protoplanets. Since local isothermality represents an extreme assumption about thermal processes in the disk, we also present several extended simulations in which heating from an artificial viscosity scheme and cooling from a simple volumetric cooling function are applied to two different models of the solar nebula. The models are differentiated primarily by disk temperature: a high-Q model generated directly by our self-consistent field equilibrium code and a low-Q model generated by cooling the high-Q model in a two-dimensional version of our hydrodynamics code. Here, high-Q and low-Q refer to the minimum values of the Toomre stability parameter Q in each disk, Qmin = 1.8 and 0.9, respectively. Previous simulations, by ourselves as well as others, have focused on initial states that are already gravitationally unstable, i.e., models similar to the low-Q model. This paper presents for the first time the numerical evolution of an essentially stable initial equilibrium state (the high-Q model) to a severely unstable one by cooling. The additional heating and cooling are applied to each model over the outer half of the disk or the entire disk. The models are subject to the rapid growth of a four-armed spiral instability; the subsequent evolution of the models depends on the thermal behavior of the disk. The cooling function tends to overwhelm the heating included in our artificial viscosity prescription, and as a result the spiral structure strengthens. The spiral disturbances transport mass at prodigious rates during the early nonlinear stages of development and significantly alter the disks vertical surface. Although dense condensations of material can appear, their character depends on the extent of the volumetric cooling in the disk. In the simulation of the high-Q model with heating and cooling applied throughout the disk, thin, dense rings form at radii ranging from 1 to 3 AU and steadily increase in mass; later companion formation may occur in these rings as cooling drives them toward instability. When heating and cooling are applied only over the outer radial half of the disk, however, a succession of single condensations appears near 5 AU. Each clump has roughly the mass of Saturn, and some survive a complete orbit. Since the clumps form near the artificial boundary in the treatment of the disk gas physics, the production of a clump in this case is a numerical artifact. Nevertheless, radially abrupt transitions in disk gas characteristics, for example, in opacity, might mimic the artificial boundary effects in our simulations and favor the production of stable companions in actual protostellar and protoplanetary disks. The ultimate survival of condensations as eventual stellar or substellar companions to the central star is still largely an open question.

Analysis of modern tides and implications for ancient tidalites

Abstract Recently, stacked successions of ancient tidal rhythmites have been found to preserve long records of tidal cycles. These include semidaily, daily, semimonthly, monthly, semiannual, annual and multiyear periods. Though such deposits reveal much about ancient tidal dynamics, the tidal signatures within the rhythmites can be masked or modified by basinal or nontidal effects. This paper discusses the results of an analysis of data from several different modern tidal stations. We show how actual tidal data can be abstracted to a form similar to what might ideally be preserved in the rock record, and then power spectra are calculated to yield estimates of the astronomical periods which can be compared to the current values. In this study, data from four modern tidal stations, ranging from diurnal to semidiurnal, are analyzed as both time- and event-series. A series of tests, which involve selective removal of parts of the tidal signal, are made using the modern tidal-station data. These tests were performed in order to determine to what extent the tidal signal can be degraded and still be recognized. Finally, we discuss some implications of the similarities of the modern and ancient tidal data and suggest how ancient data may be used to constrain basinal paleography and make inferences regarding ancient lunar orbital geometries.

Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. II. Torques, Bars, and Mode Saturation with Applications to Protostars and Fizzlers

Dynamic nonaxisymmetric instabilities in rapidly rotating stars and protostars have a range of potential applications in astrophysics, including implications for binary formation during protostellar cloud collapse and for the possibility of aborted collapse to neutron star densities at late stages of stellar evolution (fizzlers). We have recently presented detailed linear analyses for polytropes of the most dynamically unstable global modes, the barlike modes. These produce bar distortions in the regions near the rotation axis but have trailing spiral arms toward the equator. In this paper, we use our linear eigenfunctions to predict the early nonlinear behavior of the dynamic instability and compare these quasi-linear predictions with several fully nonlinear hydrodynamics simulations. The comparisons demonstrate that the nonlinear saturation of the barlike instability is due to the self-interaction gravitational torques between the growing central bar and the spiral arms, where angular momentum is transferred outward from bar to arms. We also find a previously unsuspected resonance condition that accurately predicts the mass of the bar regions in our own simulations and in those published by other researchers. The quasi-linear theory makes other accurate predictions about consequences of instability, including properties of possible end-state bars and increases in central density, which can be large under some conditions. We discuss in some detail the application of our results to binary formation during protostellar collapse and to the formation of massive rotating black holes.

Nonaxisymmetric Dynamic Instabilities of Rotating Polytropes. I. The Kelvin Modes

We study the dynamic instabilities of rotating polytropes in the linear regime using an approximate Lagrangian technique and a more precise Eulerian scheme. We consider nonaxisymmetric modes with azimuthal dependence proportional to exp (im), where m is an integer and is the azimuthal angle, for polytropes with a wide range of compressibilities and angular momentum distributions. We determine stability limits for the m = 2-4 modes and find the eigenvalue and eigenfunction of the most unstable m-mode for given equilibrium models. To the extent that we have explored parameter space, we find that the onset of instability is not very sensitive to the compressibility or angular momentum distribution of the polytrope when the models are parameterized by T/| W |. Here T is the rotational kinetic energy, and W is the gravitational energy of the polytrope. The m = 2, 3, and 4 modes become unstable at T/| W | ≈ 0.26-0.28, 0.29-0.32, and 0.32-0.35, respectively, limits consistent with those of the Maclaurin spheroids to within ±0.015 in T/| W |. The only exception to this occurs for the most compressible polytrope we test and then only for m = 4, where instability sets in at T/| W | ≈ 0.37-0.39. The eigenfunctions for the fastest growing low m-modes are similar to those of the Maclaurin spheroid eigenfunctions in that they do not show large vertical motions, are only weakly dependent on z, and increase strongly in amplitude as the equatorial radius of the spheroid is approached. The polytrope eigenfunctions are, however, qualitatively different from the Maclaurin eigenfunctions in one respect: they develop strong spiral arms. The spiral arms are stronger for more compressible polytropes and for polytropes whose angular momentum distributions deviate significantly from those of the Maclaurin spheroids. Nevertheless, our approximate Lagrangian method, which explicitly assumes nonspiral Maclaurin-like trial functions, yields reasonable estimates for the pattern periods and e-folding times of unstable m = 2 modes even for highly compressible and strongly differentially rotating polytropes. Comparisons for m = 2 between the linear analyses in this paper and nonlinear hydrodynamic simulations give excellent quantitative agreement in eigenfunctions, pattern speeds, and e-folding times for the dynamically unstable modes.

Nonaxisymmetric secular instabilities driven by star/disk coupling

We determine conditions for the onset of nonaxisymmetric secular instabilities in polytropes with a wide range of angular momentum distributions using Lagrangian techniques, and then calculate the growth rate of such instabilities when driven by the coupling of the perturbed star to a circumstellar disk. We use Langrangian displacement vectors with azimuthal dependence proportional to exp (im phi), where m is an integer and phi is the azimuthal coordinate. The onset of secular instability in terms of the quantity T/absolute value of W, the ratio of rotational kinetic energy to gravitational potential energy, is affected by both the compressibility and the angular momentum distribution of the polytrope. The largest effects occur when the angular momentum distribution is varied. For polytropic index n = 3/2, the onset of secular instability for the m = 2 mode (the bar mode), as determined by its neutral point, shifts from T/absolute value of W = 0.141 to 0.093, while the m = 5 mode neutral point shifts from T/absolute value of W = 0.088 to 0.031 over the range of angular momentum distributions we consider. The smallest critical T/absolute value of W-values occur for the angular momentum distributions which are the most peaked toward the equator. For the angular momentum distribution of a Maclaurin spheroid, as the polytropic index n is increased from 3/2 to 5/2, the neutral point for m = 2 shifts from T/absolute value of W = 0.141 to 0.144 and the netural point for m = 5 shifts from T/absolute value of W = 0.069 to 0.078. The netural points for m = 2 and 5 for the Maclaurin sequence (n = 0) are 0.137 and 0.0629, respectively. As the angular momentum distribution becomes more peaked toward the equatorial radius of the polytropes, the critical T/absolute value of W-values generally become less sensitive to the compressibility of the polytrope. Star/disk coupling can drive the secular instability in systems where the star is surrounded by a massive disk and, if the instability can grow to moderate amplitude, then the coupling can transport significant amounts of angular momentum from the star into the circumstellar disk. We find, for the particular case of rotating protostars during the accretion phase, that the instability growth time can be shorter than the accretion time. Further, if the instability can grow to amplitudes on the order of several percent, the star/disk coupling can remove angular momentum from the forming star faster than it is added by accretion.

Gravitational Instabilities in the Disks of Massive Protostars as an Explanation for Linear Distributions of Methanol Masers

Evidence suggests that some masers associated with massive protostars may originate in the outer regions of large disks, at radii of hundreds to thousands of AU from the central mass. This is particularly true for methanol (CH3OH), for which linear distributions of masers are found with disklike kinematics. In three-dimensional hydrodynamics simulations we have made to study the effects of gravitational instabilities in the outer parts of disks around young low-mass stars, the nonlinear development of the instabilities leads to a complex of intersecting spiral shocks, clumps, and arclets within the disk and to significant time-dependent, nonaxisymmetric distortions of the disk surface. A rescaling of our disk simulations to the case of a massive protostar shows that conditions in the disturbed outer disk seem conducive to the appearance of masers if it is viewed edge-on.

Three-Dimensional Hydrodynamics of Protostellar Disks: The Effects of Thermal Energetics

We use a second-order three-dimensional hydrodynamics code with self-gravity to investigate the role that thermal energetics plays in the development of nonaxisymmetric instabilities in protostellar disks. The initial axisymmetric equilibrium state is a continuous, fluid star/disk system, in which the star, the disk, and the star/disk boundary are resolved in 3D in our hydrodynamics code. An adiabatic evolution is compared to two previous simulations of the same model in which either local isentropy or local isothermality is maintained in the disk throughout the calculation. In all three cases, the model is highly unstable to multiple low-order nonaxisymmetric disturbances which induce significant mass and angular momentum transport in a few dynamical times. The star and star/disk boundary are dominated by three- and four-armed disturbances, whereas the disk is susceptible to a two-armed spiral. These disturbances saturate at moderate nonlinear amplitude in the adiabatic and isentropic evolutions; the same instabilities in the isothermal evolution lead to the disruption of the disk and concentration of material into several dense, thin arcs and arclets. Figures, tables and animations are included in the CD-ROM supplement.

Nonaxisymmetric Dynamic Instabilities of Star/Disk Systems

Dynamic nonaxisymmetric instabilities of star/disk systems are studied in the linear regime. We consider perturbations with azimuthal dependence ∝ exp(±imo) where m = 1 – 4, and o is the azimuthal angle. The eigenvalues and eigenfunctions for the most dynamically unstable m- modes are found by evolving initially random infinitesimal perturbations away from equilibrium in the linear regime. For star-like objects, dynamic instability sets in for T/|W| > 0.27, similar to well-known case of the uniformly rotating, incompressible fluid Maclaurin spheroids. Here T is the bulk kinetic energy of the object and W is the gravitational energy of the object. For star/disk-like objects, dynamic instability sets in significantly earlier, at T/|W| ∼ 0.20. For both star-like and and star/disk-like objects, dynamic instability sets in via an m = 2, i.e., two-armed or bar-like mode. All models tested were stable to m = 1 perturbations.