The Role of the Cooling Prescription for Disk Fragmentation: Numerical Convergence & Critical Cooling Parameter in Self-Gravitating Disks

Protoplanetary disks fragment due to gravitational instability when there is enough mass for self-gravitation, described by the Toomre parameter, and when heat can be lost at a rate comparable to the local dynamical timescale, described by t_c=beta Omega^-1. Simulations of self-gravitating disks show that the cooling parameter has a rough critical value at beta_crit=3. When below beta_crit, gas overdensities will contract under their own gravity and fragment into bound objects while otherwise maintaining a steady state of gravitoturbulence. However, previous studies of the critical cooling parameter have found dependence on simulation resolution, indicating that the simulation of self-gravitating protoplanetary disks is not so straightforward. In particular, the simplicity of the cooling timescale t_c prevents fragments from being disrupted by pressure support as temperatures rise. We alter the cooling law so that the cooling timescale is dependent on local surface density fluctuations, a means of incorporating optical depth effects into the local cooling of an object. For lower resolution simulations, this results in a lower critical cooling parameter and a disk more stable to gravitational stresses suggesting the formation of large gas giants planets in large, cool disks is generally suppressed by more realistic cooling. At our highest resolution however, the model becomes unstable to fragmentation for cooling timescales up to beta = 10.Comment: 10 pages, 6 figures. Accepted for publication in Ap

Filling in the Gaps: Can Gravitationally Unstable Discs Form the Seeds of Gas Giant Planets?

Circumstellar discs likely have a short window when they are self-gravitating and prone to the effects of disc instability, but during this time the seeds of planet formation can be sown. It has long been argued that disc fragmentation can form large gas giant planets at wide orbital separations, but its place in the planet formation paradigm is hindered by a tendency to form especially large gas giants or brown dwarfs. We instead suggest that planet formation can occur early in massive discs, through the gravitational collapse of dust which can form the seeds of giant planets. This is different from the usual picture of self-gravitating discs, in which planet formation is considered through the gravitational collapse of the gas disc into a gas giant precursor. It is familiar in the sense that the core is formed first, and gas is accreted thereafter, as is the case in the core accretion scenario. However, by forming a $\sim 1 M_{\oplus}$ seed from the gravitational collapse of dust within a self-gravitating disc there exists the potential to overcome traditional growth barriers and form a planet within a few times $10^5$ years. The accretion of pebbles is most efficient with centimetre-sized dust, but the accretion of millimetre sizes can also result in formation within a Myr. Thus, if dust can grow to these sizes, planetary seeds formed within very young, massive discs could drastically reduce the timescale of planet formation and potentially explain the observed ring and gap structures in young discs.Comment: MNRAS accepted. 15 pages, 12 figure

The fragmentation criteria in local vertically stratified self-gravitating disk simulations

Massive circumstellar disks are prone to gravitational instabilities, which trigger the formation of spiral arms that can fragment into bound clumps under the right conditions. Two dimensional simulations of self-gravitating disks are useful starting points for studying fragmentation, allowing for high-resolution simulations of thin disks. However, convergence issues can arise in 2D from various sources. One of these sources is the 2D approximation of self-gravity, which exaggerates the effect of self-gravity on small scales when the potential is not smoothed to account for the assumed vertical extent of the disk. This effect is enhanced by increased resolution, resulting in fragmentation at longer cooling timescales $\beta$. If true, it suggests that the 3D simulations of disk fragmentation may not have the same convergence problem and could be used to examine the nature of fragmentation without smoothing self-gravity on scales similar to the disk scale height. To that end, we have carried out local 3D self-gravitating disk simulations with simple $\beta$ cooling with fixed background irradiation to determine if 3D is necessary to properly describe disk fragmentation. Above a resolution of $\sim 40$ grid cells per scale height, we find that our simulations converge with respect to the cooling timescale. This result converges in agreement with analytic expectations which place a fragmentation boundary at $\beta_\mathrm{crit} = 3$.Comment: 11 pages, 9 figures. Accepted for publication in Ap

Formation Criteria and Initial Constraints on Objects Formed in Gravitationally Unstable Disks

Early protoplanetary disks are cool and massive and thus subject to gravitational instabilites and fragmentation of the disk into dense clumps of gas. These fragments are massive enough to become gas giant planets and brown dwarfs in the distant regions of the disks beyond 50 au where traditional planet formation scenarios have trouble creating planetary cores fast enough to explain directly observed planets. I used high-resolution three-dimensional hydrodynamic simulations to model the collapse of self-gravitating disks to constrain the formation location of these fragments and characterize their initial gas and particle masses to compare to directly observed planets and brown dwarfs. I find the traditional cooling criterion, which constrains the formation location to the outer disk, is converged in these simulations and overall masses are consistent with massive gas giants bordering on brown dwarfs. The concentration of solid material in these fragments leads to an increase of the overall metallicity of the fragment and a solid core several tens of Earth masses. To model fragmentation with full disk simulations, I have also implemented a multigrid self-gravity solver in the PLUTO code which uses adaptive mesh refinement to resolve both the disk and fragments

Elective affinities of the Protestant ethic : Weber and the chemistry of capitalism

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