On the Modeling Aggregation of Dust Fractal Clusters in the Protoplanetary Laminar Disc

1. On the Modeling Aggregation of Dust Fractal Clusters in the Protoplanetary Laminar Disc A.V. Kolesnichenko*, M.Ya. Marov***M.V. Keldysh Institute of Applied Mathematics, RAS, Moscow**V.I.Vernadsky Institute of Geochemistry and Analytical Chemistry, RAS, Moscow The Third Moscow Solar System Symposium Space Research Institute, Moscow, Russia October 8-12, 2012

2. The Background The authors’ study in the field of stellar-planetary cosmogony is rooted in their research of: • multicomponent turbulent gases; • heterogenic mechanics; • physical - chemical kinetics; • non-equilibrium thermodynamics; • magnetohydrodynamics; • chaotic and self-organization processes; • coagulation processes; • mechanisms of hydrodynamic instability. Based on the results derived from these research methods of mathematical modeling were developed in application to disc formation and evolution involving Solar system origin.

3. What do we really know • Primary dust particles may combine in the collisional processes under velocities < 1 m/s because mostly of Van-der-Waals ‘ force or hydrogen bond. • Amount of monomers (primary fine-dyspersated particles) is fast exhausted as a result of sticking process in the big dust aggregates. • At low speeds of collisions the cloud of dust particles evolves in fractal clusters with quasi monodispersible allocation to the dimensions. • Fractal structure is not preserved when energy of the collisions grows which results in porous structures set up. • Radial drift separates solid particles from the carrying gas phase, particles migrating inside the protoplanetary disc influencing mass distribution and chemical processes in it. • Radial drift of small (< cm) particles can be partially compensated by the turbulent diffusion. • Vertical mixing and small replenishment of monomers thanks to accretion processes are necessary for an explanation of observable disk structure.

4. Problems demanding the answer • What maximum size of dust formations in the protoplanetary cloud is achievable? • What are the velocity constraints and dust composition/properties to make collisional combinations possible? • What main physical parameters (for example, a velocity, angle of attack, fractality, porosity, material, form, mass, etc.), defining outcomes of collisions? • If the disc is turbulent indeed, what is coherent structures (turbulent curls, rings, etc.) time life to afford concentrations of large size solid aggregates? • How radial drift affects the disc structure and how dust formations behave in dense regions of subdisk? • How fragmentation of relatively small dust formation occur and do they acquire fluffy/fractal structures?

5. The Goal • The goal of the study is to model hydrodynamic and coagulation processes in the accretion gas-dust protoplanetary disc based on the new developed approach of dust clusters set up and follow on evolution. • In contrast to the classic models of protoplanetary cloud based on the continuous mechanics approach when fractions of dusty medium and its fractal nature were not distinguished, our model deals with the set of disperse (fluffy) dust aggregates as a specific kind of fractal continuous medium where there are hollow regions not fulfilled with particles. • The point specifically emphasized is that fluffy structure of clusters significantly facilitates probability of integration in the collisional processes because of larger geometrical cross section and patterns of motion in the gas medium in terms of friction force change dependence. • Hydrodynamic modeling of such a medium having non-integer mass dimensionality can be performed in the framework of differential mode of the fractional-integral model with the use of fractional integrals of the order corresponding to fractal dimensionalityof the disc medium.

7. Образование пылевых сгущений  0.1-0.5 млн.лет Образование диска  0.5 -1 млн.лет Time sequence of the protoplanetary accretion disc evolution including dense dust subdisk formation caused by particles sedimentation towards equatorial plane and gravity instability development when condition of the critical density is fulfilled followed by the primary dust clusters and planetesimals set up. Аккреция газа и пыли через диск на Солнце Аккумуляция сгущений, образование зародышей Рост частиц, оседаниек средней плоскости диска, дрейф к Солнцу  1-10 млн.лет  0.4-0.9 млн.лет

8. Scenario of the Disc Evolution • Simultaneous protostar and gas-dust formation from a turbulent molecular cloud and continuing accretion of gas and dust onto disc and protostar (~ 5 mil. yrs.). • Viscous turbulent disc dissipation at the T-Tauristage (~ 5-10 mil. yrs.). • Grows of dust particles from submicron to decimeter size in due course of mutual collisions incorporating electrostatic charges influence. • Dust subdisk formation and its density increase up to critical (~ 0.1-0.5 mil. yrs.). • Subdisc fragmentation into dust clusters due to gravity instability and asteroid-size bodies (10-100 km) formation (~ 0.5-1 mil. yrs.). Basically, the key problem is: What is the physical mechanism of small particles integration in dust clusters giving birth to larger size formations and eventually to planetesimals of 0.1-1 km across? Is it porous rather than firm particles?

9. Problems of Meter Solid Structures Formation • Collisions resulting in destructions rather than integration. • Fast exhausting of sufficient matter supply owing to radial drift towards protostar and follow up evaporation of small particles. • Concurrency of gravitational and brownian coagulation of dust monomers. • Concurrency of the radial and vertical motions of disc particles at the accretion stage. • Efficiency of evaporation depending on opacity of the inner disc regions and dissipation of the turbulent energy.

10. Example of Fractal Aggregate

11. The Refined Compression Model

13. Number of Monomers Entering Cluster

14. Initial Cluster Forms and Collisions Outcomes

16. Fractal Dust Clusters Generation • We assume that at the early stage of the gas-dust disc evolution monomers embedded in the gas phase coalesced in the collision events provided relative velocities were ∼ 10 cm/s. • Collisions result in both mechanical and chemical couplings and dust clusters with mono-disperse size distribution and fluffy structure are formed. • Such cluster structures have fractional fractal dimension • As clusters grow forming large fractal clusters mechanism of particle-cluster interaction changes to cluster-cluster interaction giving rise to fractal structures of larger mass dimension. • Water ice clusters following this scenario may grow under much higher velocities (up to ∼50-60 м/с) and their mass dimension achieves ∼2.5while compression rate depends on collision energy. Therefore, large compressed dust clusters may form in the collisions of fluffy fractal clusters.

18. The Baseline • Evolutionary hydrodynamic model of the formation and growth of fluffy dust aggregates (clusters) in disperse medium of the protoplanetary laminar disc is developed. • Basically, the model proceeds from the idea of fractal structure of the primary dust clusters composed originally of the gas and submicron dust particles, which eventually results in planetesimals set up. • The disc medium is considered as thermodynamic heterogeneous complex consisted of two interacting subsystems: gas phase of the solar composition (gas continuum) and polydisperse fractal dust phase. • Polydisperse phase is addressed as multi-rate heterogenic medium composed of dust fractal aggregates and pristine condensed monomers. These subsystems are assumed to fill up simultaneously every volume of the Euclid phase space. • An original approach to the modeling of hydrodynamic and coagulation processes in such a complex is suggested. It is shown that the process of cluster-cluster coagulation and their partial integration gives rise to the progressive aggregates growing. • Dust aggregates of different scales and their internal structure influencing the follow on formation of the intermediate fluffy proto-planetesimals are specially addressed, the latter appearing as the result of the combined physical-chemical and hydrodynamic processes similar to the processes of fractal clusters grow.

19. The Model: Basic Assumptions

22. Momentum Balance Equations for the Gas and Dust Disc Fractions

23. Modes of Fractal Clusters Interaction • Cluster size and physical properties depend on motion patterns of primary monomers before collision and coalescence capacity. • Two mechanisms of grow of clusters possessing fractal structure are possible, both depending on number density of monomers in the unit volume: either due to monomer attachment to cluster or cluster-cluster aggregation. • Attachment of unit nucleus in moving in the straight direction corresponds to the kinetic regime ; combination of numerous monomers in diffusion motion corresponds to the diffusion or hydrodynamic regime.

25. Generalized Set of the Equations of Motion The following set of the equations of motion underlies the new approach to the modeling of protoplanetary disc evolution.

26. Kinetics of Clusters Formation in the Fractal Medium

27. Coagulation Cores (Kernels) Two groups of models of cluster formations in the disc fractal medium were considered: adhering monomers to cluster; two clusters association. Monomer-cluster coagulation occurs in the rarefied aero-disperse medium due to drag of monomers when colliding with a cluster or due to diffuse sticking of monomer to a cluster surface.

30. Protoplanetary Disc Stationary Model

31. Generalized Smoluchowski Equation in the Cylindrical Coordinate System

32. Imitative Model of ClustersFormation • In a basis of research of processes of coagulation of dust particles and the formation of fractal clusterslying generalized Smoluchowski equation. • To solve the generalized Smoluchowski equation are applied the scheme of numerical modeling of Monte-Carlo using the method of variable weight factors. • Essence of imitating model in weighted schemas consists that the considered system of dust condensations in volume V is replacedbysystem from comparative small number of “model particles”. • Field of flows breaks into a series of the cells which dimensions are small in comparison with characteristic scales of change of hydrodynamic parameter's of medium. The width on time ∆t is small in comparison with mean time between collisions. • Random process is constructed as sequence of collisions of "model particles", played out according to the scheme of Monte- Carlo. • Algorithm "sustains" in cells constant number of “model particles” irrespective of intensity of the collisions, both uniform, and on variable meshes. Algorithm ensures functioning with adaptable grids and calculations 2D and 3D of flows.

33. Comparison of the Distribution Function (by Mass) of Compact and Fractal Dust Clusters

39. Пример расчёта изменения массовой плотности пылевой фазы субдиска. Плотность увеличивается в процессе вертикального и радиального сжатия субдиска. Широкая диагональная полоса - критическая плотность, при которой субдиск становится гравитационно-неустойчивым и распадается на пылевые сгущения. Кривые 1 - 6 соответствуют моментам времени: 0, 1103, 5103, 2104, 5104 и 1105 лет от начала образования субдиска (начальный радиус rd0 = 100 а.е., диаметры частиц d = 10 сми d = 1 см). Распад субдиска на пылевые сгущения Критическая плотность Условие гравитационной неустойчивости субдиска t~ 5104 лет t~ 6104 лет

40. Future Priorities • Research role in evolution of a protoplanetary cloud of relative velocities of collisions of dust aggregates in the areas of a disk “overloaded mass” (for example, in a subdisk, in large-scale rotational formations, in stagnant areas). • Problem of origin of turbulence in a disk at very high values of a Reynolds number. • Modeling formation of dust fractal aggregates in turbulent aerodispersible disk medium. • Research optical properties of dust clusters including of fractal aggregates. • Development evolutionary macroscopic model of the disk considering fractality of dust aggregates and their collisions with arbitrary velocities. • Building of global evolutionary model of a protoplanetary disk including of turbulization mediums, electromagnetic effects, a fragmentation of dust particles, their erect and radial drift, and also turbulent agitating.

41. Thanks for attention