Modification in the framework of nonadditive Tsallis statistics of the gravitational instability criterions of astrophysical disks

A. V. Kolesnichenko

Matematičeskoe modelirovanie, Tome 28 (2016) no. 3, pp. 96-118.

Résumé

Unlike classical studies in which the gravitational instability criterions for astrophysical disks are derived in the framework of traditional kinetics or hydrodynamics, we propose to totality of fluffy dust clusters of various astrophysical objects, in particular, protoplanetary subdisks, as a special type of continuous medium, i.e., fractal medium for which there are points and areas not filled with its components. Within the deformed Tsallis statistics formalism, which is intended to describe the behavior of anomalous systems with strong gravitational interaction and fractal nature of phase space, we derive, on the basis of the modified hydrodynamic equations of Navier–Stokes (the so-called equations $q$-hydrodynamics) the linearized equations of oscillations a solid-state rotating disk and the conclusion of the dispersing equation in VKB-approach is given. Considering the linearization of the $q$-hydrodynamics equations for viscosity solid-state rotating clouds we investigate the instability of an infinitely homogeneous medium to obtain a simplified version of the modified gravitational instability Jeans and Toomre criterions for an astrophysical disk with fractal structure.

Détail

Keywords: entropy Tsallis, nonadditive statistics, astrophysical disks, gravitational noninstability
Mots-clés : dispersion equation.

Bibliographie

[1] A. V. Kolesnichenko, M. Ja. Marov, “Modeling of aggregation of fractal dust clusters in laminar protoplanetary disk”, Solar. Syst. Res., 47:2 (2013), 80–98 | DOI

[2] A. Toomre, “On the gravitational stability of a disk of stars”, Astrophys. J., 139 (1964), 1217–1238 | DOI

[3] V. S. Safronov, Jevoljucija doplanetnogo oblaka i obrazovanie Zemli i planet, Nauka, M., 1969, 244 pp.

[4] P. Goldrich, W. R. Ward, “The formation of planetesimals”, Astrophys. J., 183:3 (1973), 1051–1061 | DOI

[5] T. Nakamoto, Y. Nakagawa, “Formation, early evolution, and gravitational stability of protoplanetary disks”, Astrophys. J., 421 (1994), 640–651 | DOI

[6] O. M. Guilera, G. C. de El'{\i}a, A. Brunini, P.J. Santamar'{\i}a, The role of planetesimal fragmentation on giant planet formation, 2014, 15 pp., arXiv: 1401.7738v1 [astro-ph.EP]

[7] N. J. Turner, S. Fromang, C. Gammie, H. Klahr, G. Lesur, M. Wardle, X-N. Bai, Transport and Accretion in Planet-Forming Disks, 2014, 24 pp., arXiv: 1401.7306v1 [astro-ph.EP]

[8] C. Dominik, J. Blum, J. Cuzzi, G. Wurm, “Growth of dust as the initial step toward planet formation”, Protostars and Planets V, Arizona Press, AZ, 2007

[9] S. Wolf, F. Malbet, R. Alexander, J.-Ph. Berger, M. Creech-Eakman, G. Duchene, A. Dutrey, C. Mordasini, E. Pantin, F. Pont, J.-U. Pott, E. Tatulli, L. Testi, Circumstellar disks and planets Science cases for nextgeneration optical/infrared long base line Interferometers, 2012, 83 pp., arXiv: 1203.6271v1 [astro-ph.IM]

[10] J. A. S. Lima, R. Silva, J. Santos, “Jeans' gravitational instability and nonextensive kinetic theory”, Astronomy and Astrophysics, 396 (2002), 309–313 | DOI | Zbl

[11] M. Sakagami, A. Taruya, “Self-gravitating stellar systems and nonextensive thermostatistics”, Continuum Mechanics and Thermodynamics, 16:3 (2004), 279–292 | DOI | MR | Zbl

[12] K. Nobuyoshi, K. Shigeo, K. Takahiro, “Nonequilibrium process of self-gravitating N-body systems and quasi-equilibrium structure using normalized q-expectation values for Tsallis' generalized entropy”, J. of Physics: Conference Series, 201:1 (2010), 012009

[13] D. B. de Freitas, J. R. de Medeiros, “Nonextensivity in the solar neighborhood”, Europhysics Letters, 97:1 (2012), 19001 | DOI

[14] A. I. Olemskoj, Sinergetika slozhnyh sistem: Fenomenologija i statisticheskaja teorija, KRASAND, M., 2009, 384 pp.

[15] P. Bak, How nature works: The science of self-organized criticality, Springer-Verlag, New York, 1996, 276 pp. | MR | Zbl

[16] C. Tsallis, “Possible generalization of Boltzmann–Gibbs statistics”, J. Stat. Phys., 52 (1988), 479–487 | DOI | MR | Zbl

[17] E. M. F. Curado, C. Tsallis, “Generalized statistical mechanics: connection with thermodynamics”, J. Phys. A, 24 (1991), L69–L72 | DOI | MR

[18] C. Tsallis, R. S. Mendes, A. R. Plastino, “The role of constraints within generalized Nonextensive statistics”, Physica A, 261 (1998), 534–554 | DOI | MR

[19] S. J. Weidenschilling, “Dust to planetesimals: Settling and coagulation in the solar nebula”, Icarus, 44 (1980), 172–189 | DOI

[20] Y. Nakagawa, K. Nakazawa, C. Hayashi, “Growth and sedimentation of dust grains in the primordial solar nebula”, Icarus, 45 (1981), 517–528 | DOI

[21] Y. Nakagawa, C. Hayashi, K. Nakazawa, “Accumulation of planetesimals in the solar nebula”, Icarus, 54 (1983), 361–376 | DOI

[22] Y. Nakagawa, M. Sekiya, C. Hayashi, “Settling and growth of dust particles in a laminar phase of a low-mass Solar nebula”, Icarus, 67 (1986), 375–390 | DOI

[23] J. Blum, “Grain growth and coagulation”, Astrophysics of Dust, ASP Conf. Ser., 309, eds. A. N. Witt, G. C. Clayton, B. T. Draine, ASP, San Francisco, 2004, 369

[24] C. W. Orme, M. Spaans, A. G. G. M. Tielens, “Dust coagulation in protoplanetary disks: porosity matters”, Astron. Astrophys, 461 (2007), 215–236 | DOI

[25] T. Suyama, K. Wada, H. Tanaka, “Numerical simulation of density evolution of dust aggregates in protoplanetary disks. I: Headon collisions”, Astroph. J., 684 (2008), 1310–1322 | DOI

[26] T. Suyama, K. Wada, H. Tanaka, S. Okuzumi, Geometrical cross sections of dust aggregates and a compression model for aggregate collisions, 2012, 28 pp., arXiv: 1205.1894v1 [astro-ph.EP]

[27] K. Wada, H. Tanaka, T. Suyama, H. Kimura, T. Yamamoto, “Simulation of dust aggregate collisions. ii. compression and disruption of three-dimensional aggregates in head-on collisions”, Astrophys. J., 677 (2008), 1296–1308 | DOI

[28] K. Wada, H. Tanaka, T. Suyama, H. Kimura, T. Yamamoto, “Collisional growth conditions for dust aggregates”, Astrophys. J., 702 (2009), 1490–1501 | DOI

[29] S. Okuzumi, H. Tanaka, M.-A. Sakagami, “Numerical modeling of the coagulation and porosity evolution of dust aggregates”, ApJ., 707 (2009), 1247–1264 | DOI

[30] S. Okuzumi, H. Tanaka, T. Takeuchu, M.-A. Sakagami, “Electrostatic barrier against dust growth in protoplanetary disks. 1: Classifying the evolution of size distribution”, ApJ., 731 (2011), 95 | DOI

[31] A. Kataoka, H. Tanaka, S. Okuzumi, K. Wada, “Static compression of porous dust aggregates”, Protostars and Planets VI, Heidelberg, 2013, Poster #2B0929, arXiv: 1307.7984v2 [astro-ph.EP]

[32] A. Kataoka, H. Tanaka, S. Okuzumi, K. Wada, “Fluffy dust forms icy planetesimals by static compression”, Astronomy Astrophysics, 557 (2013), id.L4, 4 pp., arXiv: 1307.7984v2 [astro-ph.EP]

[33] V. E. Tarasov, Fractional Dynamics. Applications of fractional calculus to dynamics of particles, fields and media, Springer, Berlin, 2010, 567 pp. | MR | Zbl

[34] A. V. Kolesnichenko, “On construction of the entropy transport model based on the formalism of nonextensive statistics”, Math. Models and Comp. Simulations, 6:6 (2014), 587–597 | DOI | MR

[35] V. E. Tarasov, “Fractional hydrodynamic equations for fractal media”, Ann. Phys., 318 (2005), 286–307 | DOI | MR | Zbl

[36] A. I. Olemskoi, “Axiomatic theory of self-organizing system”, Physics A, 310 (2002), 223–233 | DOI | Zbl

[37] B. M. Boghosian, “Thermodynamic description of the relaxation of two-dimensional turbulence using Tsallis statistics”, Phys. Rev. E, 53 (1996), 47–54 | DOI

[38] T. Arimitsu, N. Arimitsu, “Analysis of turbulence by statistics based on generalized Entropies”, Physica A, 295 (2001), 177 | DOI | MR | Zbl

[39] B. M. Boghosian, “Navier–Stokes Equations for Generalized Thermostatistics”, Bras. J. Phys., 29:1 (1999), 91–107

[40] N. N. Bogoljubov, Problems of Dynamic Theory in Statistical Physics, Technical Information Service, Oak Ridge, TN, 1960

[41] A. G. Morozov, “Dissipativnye jeffekty v gazovyh podsistemakh ploskikh galaktik”, Astron. Zhurn., E.59 (1982), 864–869 | Zbl

[42] A. G. Morozov, “Lokal'nyi kriterii ustojchivosti gazovykh podsistem ploskikh galaktik”, Astron. Zhurn., E.62 (1985), 209–217

[43] A. M. Fridman, A. V. Hoperskov, Fizika galakticheskikh diskov, Fizmatlit, M., 2011, 640 pp.

[44] A. M. Fridman, O. V. Khoruzhii, V. A. Minin, E. V. Polyachenko, V. L. Polyachenko, O. K. Sil'chenko, A. V. Zasov, V. L. Afanasiev, S. N. Dodonov, A. V. Moiseev, J. Boulesteix, J. Knape, “New Structures in Galactic Disks: Predictions and Discoveries”, Galaxy Disks and Disk Galaxies, ASP Conf. Series, 230, Astronomical Society of the Pacific, 2001, 187–198

[45] A. M. Fridman, V. L. Polyachenko, Physics of gravitating system, v. 1, Springer-Verlag, N. Y., 1984, 468 pp. ; v. 2, 358 pp. | MR

[46] A. M. Fridman, A. A. Boyarchuk, D. V. Bisikalo, O. A. Kuznetsov, Yu. M. Torgashin, A. A. Kilpio, “From hydrodynamical instability to turbulent viscosity in accretion disks”, Plasmas in the laboratory and in the universe: New insights and new challenges, AIP Conference Proceedings, 703, 2004, 250–259 | DOI

[47] A. M. Fridman, N. N. Gor'kavyi, Physics of Planetary Rings. Celestial Mechanics of Continuous Media, Springer, New York, 1999, 431 pp. | Zbl

[48] P. J. E. Peebles, Principles of Physical Cosmology, Princeton U. Press, Princeton, 1993, 718 pp. | MR