Annie C. Mejia

THE THERMAL REGULATION OF GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS. III. SIMULATIONS WITH RADIATIVE COOLING AND REALISTIC OPACITIES

This paper presents a fully three-dimensional radiative hydrodymanics simulation with realistic opacities for a gravitationally unstable 0.07 Mdisk around a 0.5 Mstar. We address the following aspects of disk evolution: the strengthofgravitationalinstabilitiesunderrealisticcooling,masstransportinthediskthatarisesfromGIs,comparisons betweenthegravitationalandReynoldsstressesmeasuredinthediskandthoseexpectedinan � -disk,andcomparisons between the SED derived for the disk and SEDs derived from observationally determined parameters. The mass transport in this disk is dominated by global modes, and the cooling times are too long to permit fragmentation for all radii. Moreover, our results suggest a plausible explanation for the FU Ori outburst phenomenon. Subject headingg accretion, accretion disks — convection — hydrodynamics — instabilities — solar system: formation

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks. II. Extended Simulations with Varied Cooling Rates

In order to investigate mass transport and planet formation through gravitational instabilities (GIs), we have extended our three-dimensional hydrodynamic simulations of protoplanetary disks from a previous paper. Our goal is to determine the asymptotic behavior of GIs and how it is affected by different constant cooling times. Initially, Rdisk = 40 AU, Mdisk = 0.07 M☉, M* = 0.5 M☉, and Qmin = 1.5. Sustained cooling, with tcool = 2 ORPs (outer rotation periods; 1 ORP ≈ 250 yr), drives the disk to instability in about 4 ORPs. This calculation is followed for 23.5 ORPs. After 12 ORPs, the disk settles into a quasi-steady state with sustained nonlinear instabilities, an average Q = 1.44 over the outer disk, a well-defined power law Σ(r), and a roughly steady ≈ 5 × 10-7 M☉ yr-1. The transport is driven by global low-order spiral modes. We restart the calculation at 11.2 ORPs with tcool = 1 and ORPs. The latter case is also run at high azimuthal resolution. We find that shorter cooling times lead to increased -values, denser and thinner spiral structures, and more violent dynamic behavior. The asymptotic total internal energy and the azimuthally averaged Q(r) are insensitive to tcool. Fragmentation occurs only in the high-resolution tcool = ORP case; however, none of the fragments survive for even a quarter of an orbit. Ringlike density enhancements appear and grow near the boundary between GI-active and GI-inactive regions. We discuss the possible implications of these rings for gas giant planet formation.

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.

THE EFFECTS OF METALLICITY AND GRAIN SIZE ON GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS

Observational studies show that the probability of finding gas giant planets around a star increases with the stars metallicity. Our latest simulations of disks undergoing gravitational instabilities (GIs) with realistic radiative cooling indicate that protoplanetary disks with lower metallicity generally cool faster and thus show stronger overall GI activity. More importantly, the global cooling times in our simulations are too long for disk fragmentation to occur, and the disks do not fragment into dense protoplanetary clumps. Our results suggest that direct gas giant planet formation via disk instabilities is unlikely to be the mechanism that produced most observed planets. Nevertheless, GIs may still play an important role in a hybrid scenario, compatible with the observed metallicity trend, where structure created by GIs accelerates planet formation by core accretion.

THE THERMAL REGULATION OF GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS. IV. SIMULATIONS WITH ENVELOPE IRRADIATION

It is generally thought that protoplanetary disks embedded in envelopes are more massive and thus more suscep- tible to gravitational instabilities (GIs) than exposed disks. We present three-dimensional radiative hydrodynamic simulationsof protoplanetarydiskswiththepresenceof envelopeirradiation.Foradiskwitharadiusof 40AUanda mass of 0.07 Maround a young star of 0.5 M� , envelope irradiation tends to weaken and even suppress GIs as the irradiating flux is increased. The global mass transport induced by GIs is dominated by lower order modes, and irradiation preferentially suppresses higher order modes. As a result, gravitational torques and mass inflow rates are actually increased bymild irradiation. None of the simulations producedense clumps or rapid cooling byconvection, arguing against direct formation of giant planets by disk instability, at least in irradiated disks. However, dense gas rings andradial mass concentrationsare produced,and these mightbeconduciveto accelerated planetary coreforma- tion. Preliminary results from a simulation of a massive embedded disk with physical characteristics similar to one of thedisksintheembeddedsourceL1551IRS5indicatealongradiativecoolingtimeandnofragmentation.TheGIsin this disk are dominated by global two- and three-armed modes. Subject headingg accretion,accretiondisks — hydrodynamics — instabilities — planetarysystems:formation — planetary systems: protoplanetary disks

Protostellar Disk Instabilities and the Formation of Substellar Companions

Recent numerical simulations of self-gravitating protostellar disks have suggested that gravitational instabilities can lead to the production of substellar companions. In these simulations, the disk is typically assumed to be locally isothermal; i.e., the initial, axisymmetric temperature in the disk remains everywhere unchanged. Such an idealized condition implies extremely efficient cooling for outwardly moving parcels of gas. While we have seen disk disruption in our own locally isothermal simulations of a small, massive protostellar disk, no long-lived companions formed as a result of the instabilities. Instead, thermal and tidal effects and the complex interactions of the disk material prevented permanent condensations from forming, despite the vigorous growth of spiral instabilities. In order to compare our results more directly with those of other authors, we here present three-dimensional evolutions of an older, larger, but less massive protostellar disk. We show that potentially long-lived condensations form only for the extreme of local isothermality, and then only when severe restrictions are placed on the natural tendency of the protostellar disk to expand in response to gravitational instabilities. A more realistic adiabatic evolution leads to vertical and radial expansion of the disk but no clump formation. We conclude that isothermal disk calculations cannot demonstrate companion formation by disk fragmentation but only suggest it at best. It will be necessary in future numerical work on this problem to treat the disk thermodynamics more realistically.

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.

Erratum: “The Effects of Metallicity and Grain Size on Gravitational Instabilities in Protoplanetary Disks” (ApJ 636, L149 [2006])

We have found that the total cumulative radiative energy losses shown in Figure 2 of the above-mentioned Letter were computed for only half the disk. This caused the final global cooling times tcool in the eighth column of the original Table 1 to be too large by a factor of 2. Proper values of tcool are given in the revised Table 1 below. To be more consistent with what the Letter states, we now use instantaneous values for both the total internal energy and the final total net cooling rates to compute final tcool’s, instead of averaging the loss rates over an interval of time at the end of the calculations before dividing, as was done in the Letter. We also take this opportunity to make a few other inconsequential corrections to the fourth column of the table. In addition to the changes to Table 1, the approximate initial tcool relation in the fourth paragraph of § 3.2 becomes to within tens t ∼ Z/Z cool , of percent. Despite the corrections, our conclusions in the Letter remain unchanged. Most importantly, the final tcool’s vary with metallicity and are still too long for disk fragmentation to occur with our equation of state over the range of Z examined. We regret any inconvenience our errors may have caused.