Abstract
The evolutionary hydrodynamic model for the formation and growth of loose dust aggregates in the aerodisperse medium of a laminar disk, which was originally comprised of the gas and solid (sub)micrometer particles, is considered as applied to the problem of the formation of planetesimals in the Solar protoplanetary cloud. The model takes into account the fractal properties of dust clusters. It is shown that the clusters partly merge in the process of cluster-cluster coagulation, giving rise to the formation of large fractal aggregates that are the basic structure-forming elements of loose protoplanetesimals arising as a result of physicochemical and hydrodynamic processes similar to the processes of growth of the fractal clusters. Earlier, the modeling was conventionally performed in an “ordinary” continuous medium without considering the multifractional structure of the dust component of the protoplanetary cloud and the fractal nature of the dust clusters being formed during its evolution. Instead, we propose to consider a complex of loose dust aggregates as a special type of continuous medium, namely, the fractal medium for which there exist points and regions that are not filled with its particles. We suggest performing the hydrodynamic modeling of this medium, which has a noninteger mass dimensionality, in a fractional integral model (its differential form) that takes the fractality into account using fractional integrals whose order is determined by a fractal dimensionality of the disk medium.
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References
Adams, F.C. and Watkins, R., Vortices in Circumstellar Disks, Astrophys. J., 1995, vol. 451, pp. 314–327.
Barge, P. and Sommeria, J., Did Planet Formation Begin Inside Persistent Gaseous Vortices?, Astron. Astrophys., 1995, vol. 295, pp. L1–L4.
Bertini, I., Gutierrez, P.J., and Sabolo, W., The Influence of the Monomer Shape in the First Stage of Dust Growth in the Protoplanetary Disk, Astron. Astrophys., 2009, vol. 504, p. 625.
Blum, J., Wurm, G., Kempf, S., and Henning, Th., The Brownian Motion of Dust in the Solar Nebula: An Experimental Approach to the Problem of Pre-Planetary Dust Aggregation, Icarus, 1996, vol. 124, pp. 441–451.
Blum, J., Grain Growth and Coagulation, in ASP Conf. Ser. Astrophysics of Dust, Witt, A.N., Clayton, G.C., and Draine, B.T., Eds., 2004, San Francisco: ASP, vol. 309, p. 369.
Blum, J. and Wurm, G., The Growth Mechanisms of Macroscopic Bodies in Protoplanetary Disks, Annu. Rev. Astron. Astrophys., 2008, vol. 46, pp. 21–56.
Brandenburg, A. and Hodgson, L.S., Turbulence Effects in Planetesimal Formation, Astron. Astrophys., 1998, vol. 330, pp. 1169–1174.
Chapman, S. and Cowling, T.G., The Mathematical Theory of Non-Uniform Gases, Cambridge: Cambridge University Press, 1953; Moscow: Inostannaya literatura, 1960.
Chavanis, P.-H., Trapping of Dust by Coherent Vortices in the Solar Nebula, Astron. Astrophys., 2000, vol. 356, pp. 1089–1111. http://arxiv.org/abs/astro-ph/9912087
Chen, Z.-Y., Meakin, P., and Deutch, J.M., Comment on “Hydrodinamic Behavior of Fractal Aggregates”, Phys. Rev. Lett., 1987, no. 18, p. 2121.
Dominik, C. and Tielens, A.G.G.M., The Physics of Dust Coagulation and the Structure of Dust Aggregates in Space, Astrophys. J., 1997, vol. 480, pp. 647–673.
Dubrulle, B., Differential Rotation as a Source of Angular Momentum Transfer in the Solar Nebula, Icarus, 1993, vol. 106, pp. 59–76.
Dubrulle, B., Morfill, G., and Sterzik, M., The Dust Subdisk in the Protoplanetary Nebula, Icarus, 1995, vol. 114, pp. 237–246.
Emets, E.P., Novoselova, A.E., and Poluektov, P.P., In-situ Determination of Fractal Dimensionality of Aerosol Particles, Usp. Fiz. Nauk, 1994, vol. 164, no. 9, pp. 959–966.
Goldrich, P. and Ward, W.R., The Formation of Planetesimals, Astrophys. J., 1973, vol. 183, no. 3, pp. 1051–1061.
Gor’kavyi, N.N., Fridman, A.M., Fizika planetnykh kolets: Nebesnaya mekhanika sploshnoi sredy (Physics of Planetary Rings: Celestial Mechanics of Continuous Medium), Moscow: Nauka, 1994.
Heng, K. and Kenyon, S.J., Vortices as Nurseries for Planetesimal Formation in Protoplanetary Discs, Mon. Notic. Roy. Astron. Soc., 2010, vol. 408, no. 3, pp. 1476–1493.
Johansen, A., Klahr, H., and Henning, T., Gravoturbulent Formation of Planetesimals, Astrophys. J., 2006, vol. 636, pp. 1121–1134.
Johansen, A., Oishi, J.S., MacLow M.M., et al., Rapid Planetesimal Formation in Turbulent Circumstellar Disks, Nature, 2007a, vol. 448, pp. 1022–1025.
Johansen, A. and Youdin, A., Protoplanetary Disk Turbulence Driven by the Streaming Instability: Nonlinear Saturation and Particle Concentration, Astrophys. J., 2007b, vol. 662, pp. 627–641.
Johansen, A., Youdin, A., and Klahr, H., Zonal Flows and Long-Lived Axisymmetric Pressure Bumps in Magnetorotational Turbulence, Astrophys. J., 2009, vol. 697, pp. 1269–1289.
Jullien, R.M., Transparency Effects in Cluster-Cluster Aggregation with Linear Trajectories, J. Phys. A, 1984, vol. 17, pp. L771–L776.
Jullien, R., A New Model of Cluster Aggregation, J. Phys. A, 1986, vol. 19, no. 11, pp. 2129–2136.
Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Fractional Differential Equations: An Emergent Field in Applied and Mathematical Sciences, in Factorization, Singular Operators and Related Problems, Samko, S., Lebre, A., and Dos Santos, A.F., Eds., London: Kluwer Acad. Publ., 2003, pp. 151–173.
Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J., Theory and Appllications of Fractional Differential Equations, Amsterdam: Elsevier, 2006.
Klahr, H. and Bodenheimer, P., Turbulence in Accretion Disks: Vorticity Generation and Angular Momentum Transport via the Global Baroclinic Instability, Astrophys. J., 2003, vol. 582, pp. 869–892.
Klahr, H. and Bodenheimer, P., Formation of Giant Planets by Concurrent Accretion of Solids and Gas inside an Anticyclonic Vortex, Astrophys. J., 2006, vol. 639, pp. 432–440.
Kolb, M., Botet, R., and Jullien, R., Scaling of Kinetically Growing Clusters, Phys. Rev. Lett., 1983, vol. 51, no. 13, pp. 1123–1126.
Kolesnichenko, A.V., Synergetic Mechanism of the Development of Coherent Structures in the Continual Theory of Developed Turbulence, Solar Syst. Res., 2004, vol. 38, no. 5, pp. 351–371.
Kolesnichenko, A.V., The Role of Noise-Induced Nonequilibrium Phase Transitions in the Structurization of Hydrodynamic Turbulence, Solar Syst. Res., 2005, vol. 39, no. 3, pp. 214–230.
Kolesnichenko, A.V. and Marov, M.Ya., Turbulentnost’ i samoorganizatsiya. Problemy modelirovaniya kosmicheskikh i prirodnykh sred (Turbulence and SelfOrganization. Simulation Problems for Space and Nature Mediums), Moscow: BINOM. Laboratoriya znanii, 2009.
Kolesnichenko, A.V., On the Simulation of Helical Turbulence in an Astrophysical Nonmagnetic Disk, Solar Syst. Res., 2011, vol. 45, no. 3, pp. 246–263.
Kulak, M.I., Fraktal’naya mekhanika materialov (Fractal Mechanics for Materials), Minsk: Vysheishaya shkola, 2002.
Mandelbrot, B.B., The Fractal Geometry of Nature, New York: Freeman, 1982.
Marov, M.Ya., Kolesnichenko, A.V., Makalkin, A.B., et al., From Protosolar Cloud to Planetary System. Early Evolution Model of Gas-Dust Disk, in Problemy zarozhdeniya i evolyutsii biosfery (Problems of Biosphere Origin and Evolution), Galimov, E.M., Ed., Moscow: URSS, 2008, pp. 223–273.
Matthews, L.S., Land, V., and Hyde, T.W., Charging and Coagulation of Dust in Protoplanetary Plasma Environments, Astrophys. J., 2012, vol. 744, no. 1.
Meakin, P., Formation of Fractal Clusters and Networks by Irreversible Diffusion-Limited Aggregation, Phys. Rev. Lett., 1983, vol. 51, no. 13, pp. 1119–1122.
Meakin, P., Effects of Cluster Trajectories on Cluster-Cluster Aggregation: A Comparison of Linear and Brownian Trajectories in Twoand Three-Dimensional Simulations, Phys. Rev., 1984, vol. 29, no. 2, pp. 997–999.
Meakin, P., Donn, B., and Mulholland, G., Collisions between Point Masses and Fractal Aggregates, Langmuir, 1989, vol. 5, pp. 510–518.
Mikhailov, E.F. and Vlasenko, S.S., Fractal Structures Formation in Gas Phase, Usp. Fiz. Nauk, 1995, vol. 165, no. 3, pp. 263–283.
Mizuno, H., Markiewicz, W.J., and Völk, H.J., Grain Growth in Turbulent Protoplanetary Accretion Disks, Astron. Astrophys., 1988, vol. 195, pp. 183–192.
Mizuno, H., Grain Growth in the Turbulent Accretion Disk Solar Nebula, Icarus, 1989, vol. 80, pp. 189–201.
Monin, A.S. and Yaglom, A.M., Statisticheskaya gidromekhanika (Statistical Hydromechanics), St. Petersburg: Gidrometeoizdat, 1996, vol. 2.
Nakagawa, Y., Nakazawa, K., and Hayashi, C., Growth and Sedimentation of Dust Grains in the Primordial Solar Nebula, Icarus, 1981, vol. 45, pp. 517–528.
Nakagawa, Y., Hayashi, C., and Nakazawa, K., Accumulation of Planetesimals in the Solar Nebula, Icarus, 1983, vol. 54, pp. 361–376.
Nakagawa, Y., Sekiya, M., and Hayashi, C., Settling and Growth of Dust Particles in a Laminar Phase of a Low-Mass Solar Nebula, Icarus, 1986, vol. 67, pp. 375–390.
Nakamoto, T. and Nakagawa, Y., Formation, Early Evolution, and Gravitational Stability of Protoplanetary Disks, Astrophys. J., 1994, vol. 421, pp. 640–651.
Nigmatulin, R.I., Dinamika mnogofaznykh sred (Dynamics of Multiphase Mediums), Moscow: Nauka, 1987, part 1.
Okuzumi, S., Electric Charging of Dust Aggregates and Its Effect on Dust Coagulation in Protoplanetary Disks, Astrophys. J., 2009a, vol. 698, pp. 1122–1135.
Okuzumi, S., Tanaka, H., and Sakagami, M.-A., Numerical Modeling of the Coagulation and Porosity Evolution of Dust Aggregates, Astrophys. J., 2009b, vol. 707, pp. 1247–1264.
Okuzumi, S., Tanaka, H., Takeuchu, T., and Sakagami, M.-A., Electrostatic Barrier against Dust Growth in Protoplanetary Disks. 1. Classifying the Evolution of Size Distribution, Astrophys. J., 2011, vol. 731, p. 95.
Ormel, C.W., Spaans, M., and Tielens, A.G.G.M., Dust Coagulation in Protoplanetary Disks: Porosity Matters, Astron. Astrophys., 2007, vol. 461, pp. 215–236.
Ossenkopf, V., Dust Coagulation in Dense Molecular Clouds: The Formation of Fluffy Aggregates, Astron. Astrophys., 1993, vol. 280, pp. 617–646.
Perry, J., Kimery, J., Matthews, L.S., and Hyde, T.W., Effects of Monomer Shape on the Formation of Fractal Aggregates under a Power Law Distribution, Proc. 43rd Lanuar and Planetary Sci. Conf., Woodlands, 2012.
Piskunov, V.N., Dinamika aerozolei (Aerosols Dynamics), Moscow: Fizmatlit, 2010.
Roy, N. and Ray, A.K., Fractal Features in Accretion Discs, Mon. Notic. Roy. Astron. Soc., 2009, vol. 397, no. 3, pp. 1374–1385.
Safronov, V.S., On Gravity Instability in Flat Rotating Systems with Axial Symmetry, Dokl. Akad. Nauk SSSR, 1960, vol. 130, no. 1, pp. 53–56.
Safronov, V.S., Evolyutsiya doplanetnogo oblaka i obrazovanie Zemli i planet (Evolution of Protoplanetary Cloud and the Earth and Planets Formation), Moscow: Nauka, 1969.
Samko, S.G., Kilbas, A.A., and Marichev, O.I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya (Integrals and Fractional Derivatives and Their Applications), Minsk: Nauka i tekhnika, 1987.
Smirnov, B.M., Fractal Clusters, Usp. Fiz. Nauk, 1986, vol. 149, no. 2, pp. 177–219.
Smirnov, B.M., Fizika fraktal’nykh kllasterov (Physics of Fractal Clusters), Moscow: Nauka, 1991.
Smirnov, B.M., Processes with Clusters and Small Particles Participation in Buffer Gas, Usp. Fiz. Nauk, 2011, vol. 181, no. 7, pp. 713–745.
Strichartz, R.S., Analysis on Fractals, Notic. Amer. Math. Soc., 1999, vol. 46, no. 10, pp. 1199–1208.
Sutherland, D.N., Comments on Vold’s Simulation of Floc Formation, J. Colloid Interface Sci., 1966, vol. 22, pp. 300–302.
Suyama, T., Wada, K., and Tanaka, H., Numerical Simulation of Density Evolution of Dust Aggregates in Protoplanetary Disks. I. Head-on Collisions, Astrophys. J., 2008, vol. 684, pp. 1310–1322.
Suyama, T., Wada, K., Tanaka, H., and Okuzumi, S., Geometrical Cross Sections of Dust Aggregates and a Compression Model for Aggregate Collisions, Astrophys. J., 2012, vol. 753, p. 115.
Tanga, P., Babiano, A., Dubrulle, B., and Provenzale, A., Forming Planetosimals in Vortices, Icarus, 1996, vol. 121, pp. 158–170.
Tarasov, V.E., Fractional Hydrodynamic Equations for Fractal Media, Ann. Phys., 2005, vol. 318, no. 2, pp. 286–307.
Tarasov, V.E., Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer. Higher Education Press, 2010.
Toomre, A., On the Gravitational Stability of a Disk of Stars, Astrophys. J., 1964, vol. 139, pp. 1217–1238.
Uchaikin, V.V., Metod drobnykh proizvodnykh (Fractional Derivatives Method), Ulyanovsk: Artishok, 2008.
Vityazev, A.V., Pechernikova, G.N., and Safronov, V.S., Planety zemnoi gruppy. Proiskhozhdenie i rannyaya evolyutsiya (Planets of the Earth Group. Origin and Early Evolution), Moscow: Nauka, 1990.
Vold, M.J., Computer Simulation of Floc Formation in a Colloidal Suspension, J. Colloid Interface Sci., 1963, vol. 18, pp. 684–695.
Wada, K., Tanaka, H., Suyama, T., et al., Simulation of Dust Aggregate Collisions. II. Compression and Disruption of Three-Dimensional Aggregates in Head-on Collisions, Astrophys. J., 2008, vol. 677, pp. 1296–1308.
Wada, K., Tanaka, H., Suyama, T., et al., Collisional Growth Conditions for Dust Aggregates, Astrophys. J., 2009, vol. 702, pp. 1490–1501.
Weidenschilling, S.J., Dust to Planetesimals: Settling and Coagulation in the Solar Nebula, Icarus, 1980, vol. 44, pp. 172–189.
Wiltzius, P., Hydrodinamic Behavior of Fractal Aggregates, Phys. Rev. Lett., 1987, vol. 58, no. 7, pp. 710–713.
Witten, T.A. and Sander, L.M., Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon, Phys. Rev. Lett., 1981, vol. 47, pp. 1400–1403.
Youdin, A.N. and Shu, F., Planetesimal Formation by Gravitational Instability, Astrophys. J., 2002, vol. 580, pp. 494–505.
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Original Russian Text © A.V. Kolesnichenko, M.Ya. Marov, 2013, published in Astronomicheskii Vestnik, 2013, Vol. 47, No. 2, pp. 92–111.
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Kolesnichenko, A.V., Marov, M.Y. Modeling of aggregation of fractal dust clusters in a laminar protoplanetary disk. Sol Syst Res 47, 80–98 (2013). https://doi.org/10.1134/S0038094613020056
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DOI: https://doi.org/10.1134/S0038094613020056