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The Athena MHD code has been extended to integrates the motion of particles coupled with the gas via aerodynamic drag, in order to study the dynamics of gas and solids in protoplanetary disks and the formation of planetesimals. Our particle-gas hybrid scheme is based on a second order predictor-corrector method. Careful treatment of the momentum feedback on the gas guarantees exact conservation. The hybrid scheme is stable and convergent in most regimes relevant to protoplanetary disks. We describe a semi-implicit integrator generalized from the leap-frog approach. In the absence of drag force, it preserves the geometric properties of a particle orbit. We also present a fully-implicit integrator that is unconditionally stable for all regimes of particle-gas coupling. Using our hybrid code, we study the numerical convergence of the non-linear saturated state of the streaming instability. We find that gas flow properties are well converged with modest grid resolution (128 cells per pressure length eta r for dimensionless stopping time tau_s=0.1), and equal number of particles and grid cells. On the other hand, particle clumping properties converge only at higher resolutions, and finer resolution leads to stronger clumping before convergence is reached. Finally, we find that measurement of particle transport properties resulted from the streaming instability may be subject to error of about 20%.
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In a standard theory of the formation of the planets in our Solar System, terrestrial planets and cores of gas giants are formed through accretion of kilometer-sized objects (planetesimals) in a protoplanetary disk. Gravitational $N$-body simulations of a disk system made up of numerous planetesimals are the most direct way to study the accretion process. However, the use of $N$-body simulations has been limited to idealized models (e.g. perfect accretion) and/or narrow spatial ranges in the radial direction, due to the limited number of simulation runs and particles available. We have developed new $N$-body simulation code equipped with a particle-particle particle-tree (${rm P^3T}$) scheme for studying the planetary system formation process: GPLUM. For each particle, GPLUM uses the fourth-order Hermite scheme to calculate gravitational interactions with particles within cut-off radii and the Barnes-Hut tree scheme for particles outside the cut-off radii. In existing implementations, ${rm P^3T}$ schemes use the same cut-off radius for all particles, making a simulation become slower when the mass range of the planetesimal population becomes wider. We have solved this problem by allowing each particle to have an appropriate cut-off radius depending on its mass, its distance from the central star, and the local velocity dispersion of planetesimals. In addition to achieving a significant speed-up, we have also improved the scalability of the code to reach a good strong-scaling performance up to 1024 cores in the case of $N=10^6$. GPLUM is freely available from https://github.com/YotaIshigaki/GPLUM with MIT license.
Observations suggest that protoplanetary disks have moderate accretion rates onto the central young star, especially at early stages (e.g. HL Tau), indicating moderate disk turbulence. However, recent ALMA observations suggest that dust is highly settled, implying weak turbulence. Motivated by such tension, we carry out 3D stratified local simulations of self-gravitating disks, focusing on settling of dust particles in actively accreting disks. We find that gravitationally unstable disks can have moderately high accretion rates while maintaining a relatively thin dust disk for two reasons. First, accretion stress from the self-gravitating spirals (self-gravity stress) can be stronger than the stress from turbulence (Reynolds stress) by a factor of 5-20. Second, the strong gravity from the gas to the dust decreases the dust scale height by another factor of $sim 2$. Furthermore, the turbulence is slightly anisotropic, producing a larger Reynolds stress than the vertical dust diffusion coefficient. Thus, gravitoturbulent disks have unusually high vertical Schmidt numbers ($Sc_z$) if we scale the total accretion stress with the vertical diffusion coefficient (e.g. $Sc_zsim$ 10-100). The reduction of the dust scale height by the gas gravity, should also operate in gravitationally stable disks ($Q>$1). Gravitational forces between particles become more relevant for the concentration of intermediate dust sizes, forming dense clouds of dust. After comparing with HL Tau observations, our results suggest that self-gravity and gravity among different disk components could be crucial for solving the conflict between the protoplanetary disk accretion and dust settling, at least at the early stages.
The Streaming Instability (SI) is a mechanism to concentrate solids in protoplanetary disks. Nonlinear particle clumping from the SI can trigger gravitational collapse into planetesimals. To better understand the numerical robustness of the SI, we perform a suite of vertically-stratified 3D simulations with fixed physical parameters known to produce strong clumping. We vary the numerical implementation, namely the computational domain size and the vertical boundary conditions (vBCs), comparing newly-implemented outflow vBCs to the previously-used periodic and reflecting vBCs. We find strong particle clumping by the SI is mostly independent of the vBCs. However, peak particle densities are higher in larger simulation domains due to a larger particle mass reservoir. We report SI-triggered zonal flows, i.e., azimuthally-banded radial variations of gas pressure. These structures have low amplitudes, insufficient to halt particle radial drift, confirming that particle trapping in gas pressure maxima is not the mechanism of the SI. We find that outflow vBCs produce artificially large gas outflow rates at vertical boundaries. However, the outflow vBCs reduce artificial reflections at vertical boundaries, allowing more particle sedimentation, and showing less temporal variation and better convergence with box size. The radial spacing of dense particle filaments is $sim0.15$ gas scale heights ($H$) for all vBCs, which sets the feeding zone for planetesimal growth in self-gravitating simulations. Our results validate the use of the outflow vBCs in SI simulations, even with vertical boundaries close ($leq 0.4H$) to the disk midplane. Overall, our study demonstrates the numerical robustness of nonlinear particle clumping by the SI.
We address the question of convergence of evolving interacting particle systems as the number of particles tends to infinity. We consider two types of particles, called positive and negative. Same-sign particles repel each other, and opposite-sign particles attract each other. The interaction potential is the same for all particles, up to the sign, and has a logarithmic singularity at zero. The central example of such systems is that of dislocations in crystals. Because of the singularity in the interaction potential, the discrete evolution leads to blow-up in finite time. We remedy this situation by regularising the interaction potential at a length-scale $delta_n>0$, which converges to zero as the number of particles $n$ tends to infinity. We establish two main results. The first one is an evolutionary convergence result showing that the empirical measures of the positive and of the negative particles converge to a solution of a set of coupled PDEs which describe the evolution of their continuum densities. In the setting of dislocations these PDEs are known as the Groma-Balogh equations. In the proof we rely on the theory of $lambda$-convex gradient flows, a priori estimates for the Groma-Balogh equations and Orlicz spaces. The proof require $delta_n$ to converge to zero sufficiently slowly. The second result is a counterexample, demonstrating that if $delta_n$ converges to zero sufficiently fast, then the limits of the empirical measures of the positive and the negative dislocations do not satisfy the Groma-Balogh equations. These results show how the validity of the Groma-Balogh equations as the limit of many-particle systems depends in a subtle way on the scale at which the singularity of the potential is regularised.
X-ray spectroscopy is key to address the theme of The Hot Universe, the still poorly understood astrophysical processes driving the cosmological evolution of the baryonic hot gas traceable through its electromagnetic radiation. Two future X-ray observatories: the JAXA-led XRISM (due to launch in the early 2020s), and the ESA Cosmic Vision L-class mission Athena (early 2030s) will provide breakthroughs in our understanding of how and when large-scale hot gas structures formed in the Universe, and in tracking their evolution from the formation epoch to the present day.
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