Skip to main content
SpringerLink
Log in

Jeans Instability of a Protoplanetary Circular Disk Taking into Account the Magnetic Field and Radiation in Nonextensive Tsallis Kinetics

  • Published:
Solar System Research Aims and scope Submit manuscript

Abstract—

Within the framework of Tsallis nonextensive statistics, the criteria for the Jeans gravitational instability are derived for a self-gravitating protoplanetary disk, whose substance consists of a mixture of a conducting ideal q-gas and modified radiation of a photon gas. The instability criteria are derived from the corresponding dispersion relations written for both neutral disk matter and magnetized plasma with modified blackbody radiation. The thermodynamics of a photon gas are constructed based on the nonextensive Tsallis quantum entropy, which depends on the deformation parameter. It is shown that blackbody q-radiation can stabilize the state of a nonextensive medium for a purely gaseous disk, and for an electrically conducting disk, the Jeans instability criterion is modified by the magnetic field and radiation pressure only in the transverse propagation mode of the disturbance wave.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. In the cited work, the kinetic theory was based on the Bhatnager–Gross–Krook (BGK) collision operator, which was generalized to an arbitrary parameter value q.

  2. Bose created statistical mechanics for photon gas; Einstein developed it to describe massive particles.

  3. Integration over dV often comes down to replacing dV by the total volume V of a homogeneous photon gas.

  4. The calculation of integrals of this kind was carried out in (Kolesnichenko, 2020 \(J_{q}^{{(n)}} = - \frac{{15\Gamma (1 - q)}}{{{{\pi }^{4}}{{{(q - 1)}}^{n}}}}\sum\limits_{k = 0}^n {\left\{ {\frac{{(n - k)n!}}{{k!(n - k + 1)!}}\frac{{\Gamma \left[ {(1 - q)(n - k)} \right]}}{{\Gamma \left[ {(1 - q)(n - k + 1)} \right]}}} \right\}} .\)

  5. In what follows, we will omit the index q for a number of hydrodynamic and thermodynamic variables.

  6. The particular importance of the relationship (1 – β) for the theory of stellar structure was first pointed out by Eddington. In a famous passage from his book, The Internal Structure of Stars, Eddington linked this relationship to a “happening of the stars.”

  7. When studying the disturbed states of self-gravitating cosmic material, one often has to deal with a variety of sound waves.

  8. It should be noted that the linearized momentum equation requires that the velocity u be parallel to the wave vector ±k (see Landau and Lifshitz, 1964). Consequently, the velocities of liquid particles associated with adiabatic sound waves are parallel to the direction of wave propagation.

  9. It is known that the problem of stability of a self-gravitating two-dimensional gas cloud, in principle, cannot be described within the framework of the two-dimensional approximation, since it is certainly highly unstable (see, for example, Fridman and Khoperskov, 2011). However, in the presence of a strong external gravitational field with a cylindrical geometry and with a generatrix along the axis of rotation of the cloud, it is possible to ensure its stability in the case when the angular velocity of rotation is sufficiently high. In this case, the structure of the preplanetary cloud along the axis of rotation will be determined exclusively by its self-gravity. Of course, this case is artificial, since in real astrophysical systems such cylindrical fields, even if they occur, are without embedded disks. At the same time, the consideration of such a self-gravitating gas disk embedded in a cylinder is of certain mathematical interest, since only in this case it is possible to single out the effects to which self-gravity leads in its pure form. Such models were considered in most of the classical works on astrophysical disks (see, for example, Goldreich and Lynden-Bell, 1965; Hunter, 1972; Toomre, 1964).

REFERENCES

  1. Abe, S. and Okamoto, Y., Nonextensive Statistical Mechanics and Its Applications, Lecture Notes in Physics, Berlin, New York: Springer, 2001. Anchrordoqui, L.A. and Torres, D.F., Non-extensivity effects and the highest energy cosmic ray affair, Phys. Lett. A, vol. 283, pp. 319–322.

  2. Boghosian, B.M., Navier–Storts equations for generalized thermostatistics, Bras. J. Phys., 1999, vol. 29, no. 1, pp. 91–107.

    ADS  Google Scholar 

  3. Bonnor, W.B., Jeans’ formula for gravitational instability, Mon. Not. R. Astron. Soc., 1957, vol. 117, no. 1, pp. 104–117.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Büyükkilic, F. and Demirhan, D., A fractal approach to entropy and distribution functions, Phys. Lett. A, 1993, vol. 181, pp. 24–28.

    Article  ADS  MathSciNet  Google Scholar 

  5. Büyükkilic, F. and Demirhan, D., A unified grand canonical description of the nonextensive thermostatistics of the quantum gases: Fractal and fractional approach, Eur. Phys. J. B, 2000, vol. 14, pp. 705–711.

    Article  ADS  Google Scholar 

  6. Cadez, V.M., Applicability problem of Jeans criterion to a stationary self-gravitating cloud, Astron. Astrophys., 1990, vol. 235, pp. 242–244.

    ADS  Google Scholar 

  7. Cadez, V.M., Instabilities in stratified magnetized stellar atmospheres, Publ. Astron. Obs. Belgrade, 2010, vol. 90, pp. 121–124.

    ADS  Google Scholar 

  8. Camenzind, M., Demole, F., and Straumann, N., The stability of radiation-pressure-dominated accretion discs, Astron. Astrophys., 1986, vol. 158, pp. 212–216.

    ADS  Google Scholar 

  9. Chamati, H., Djankova, A.T., and Tonchev, N.S., On the application of nonextensive statistical mechanics to the black-body radiation, Phys. A, 2006, vol. 360, pp. 297–303.

    Article  Google Scholar 

  10. Chandrasekhar, S., An Introduction to the Study of Stellar Structure, New York: Dover, 1939.

    MATH  Google Scholar 

  11. Chandrasekhar, S. and Fermi, E., Problems of gravitational stability in the presence of a magnetic field, Astrophys. J., 1953, vol. 118, pp. 116–141.

    Article  ADS  MathSciNet  Google Scholar 

  12. Curado, E.M.F. and Tsallis, C., Generalized statistical mechanics: connection with thermodynamics, J. Phys. A, 1991, vol. 24, pp. L69–72.

    Article  ADS  MathSciNet  Google Scholar 

  13. Daroczy, Z., Generalized information function, Inform. Control, 1970, vol. 16, pp. 36–51.

    Article  MathSciNet  MATH  Google Scholar 

  14. Dhiman, J.S. and Dadwal, R., On the Jeans criterion of a stratified heat conducting gaseous medium in the presence of non-uniform rotation and magnetic field, J. Astrophys. Astron., 2012, vol. 33, no. 4, pp. 363–373.

    Article  ADS  Google Scholar 

  15. Eddington, A.S., The Internal Constitution of the Stars, Cambridge: Cambridge Univ. Press, 1988.

    Book  MATH  Google Scholar 

  16. Fridman, A.M. and Khoperskov, A.V., Fizika galakticheskikh diskov (Physics of Galactic Disks), Moscow: Fizmatlit, 2011.

  17. Fridman, A.M. and Polyachenko, V.L., Physics of Gravitating System, New York: Springer-Verlag, 1984, vols. 1, 2.

    Book  MATH  Google Scholar 

  18. Fridman, A.M. and Polyachenko, V.L., Physics of Gravitating Systems I: Equilibrium and Stability, New York: Springer Science & Business Media, 2012.

    Google Scholar 

  19. Gell-Mann, M. and Tsallis, C., Nonextensive Entropy – Interdisciplinary Applications, Oxford: Oxford Univ. Press, 2004.

    Book  MATH  Google Scholar 

  20. Grigolini, P., Tsallis, C., and West, B.J., Classical and quantum complexity and nonextensive thermodynamics, Chaos, Solitons Fractals, 2002, vol. 13, no. 3, pp. 367–370.

    Article  ADS  MATH  Google Scholar 

  21. Goldreich, P. and Lynden-Bell, D.I., Gravitational stability of uniformly rotating disks, Mon. Not. R. Astron. Soc., 1965, vol. 130, pp. 97–124.

    Article  ADS  Google Scholar 

  22. Goldreich, P. and Ward, W.R., The formation of planetesimals, Astrophys. J., 1973, vol. 183, pp. 1051–1062.

    Article  ADS  Google Scholar 

  23. Gor’kavyi, N.N. and Fridman, A.M., Fizika planetnykh kolets (Physics of Planetary Rings), Moscow: Nauka, 1994.

  24. Gough, D.O., Heliophysics gleaned from seismology, Progress in Solar/Stellar Physics with Helio- and Asteroseismology, Proc. 65th Fujihara Seminar, Astron. Soc. Pacific Conf. Ser., 2011, vol. 462, pp. 429–454. https://arxiv.org/abs/1210.1114.

    ADS  Google Scholar 

  25. Gough, D.O. and Hindman, B., Helioseismic detection of deep meridional flow, J. Astrophys., 2010, vol. 714, no. 1, pp. 960–970.

    Article  ADS  Google Scholar 

  26. Havrda, J. and Charvat, F., Quantification method of classification processes, Kybernetika, 1967, vol. 3, pp. 30–35.

    MathSciNet  MATH  Google Scholar 

  27. Herrmann, H.J., Barbosa, M., and Curado, E.M.F., Trends and perspectives in extensive and non-extensive statistical mechanics, Phys. A, 2004, vol. 344, nos. 3–4, pp. v–vi.

    Article  Google Scholar 

  28. Hunter, C., Self-gravitating gaseous disks, Ann. Rev. Fluid Mech., 1972, vol. 4, pp. 219–242.

    Article  ADS  MATH  Google Scholar 

  29. Jaynes, E.T., Information theory and statistical mechanics, Statistical Physics 3. Lectures from Brandeis Summer Institute 1962, New York: W.A. Benjamin, 1963.

    Google Scholar 

  30. Jeans, J.H., The stability of a spherical nebula 199, Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 1902, vol. 199, pp. 1–53.

    ADS  MATH  Google Scholar 

  31. Jeans, J.H., Astronomy and Cosmogony, Cambridge: Cambridge Univ. Press, 2009.

    Book  MATH  Google Scholar 

  32. Joshi, H. and Pensia, R.K., Effect of rotation on Jeans instability of magnetized radiative quantum plasma, Phys. Plasmas, 2017, vol. 24, id. 032113.

  33. Kaniadakis, G. and Lissia, M., News and expectations in thermostatistics, Phys. A: Stat. Mech. Its Appl., 2004, vol. 340, no. 1, pp. xv–xix.

    Article  MathSciNet  Google Scholar 

  34. Kaniadakis, G., Lissia, M., and Rapisarda, A., Non extensive thermodynamics and physical applications, Phys. A, 2002, vol. 305, nos. 1–2, pp. xv–xvii.

    MathSciNet  Google Scholar 

  35. Kaniadakis, G., Carbone, A., and Lissia, M., News, expectations and trends in statistical physics, Phys. A: Stat. Mech. Its Appl., 2006, vol. 365, no. 1, p. xi.

    Article  Google Scholar 

  36. Kaothekar, S. and Chhajlani, R.K., Jeans instability of self gravitating partially ionized Hall plasma with radiative heat loss functions and porosity, AIP Conf. Proc., 2013, pp. 1288–1289.

  37. Khoperskov, A.V. and Khrapov, S.S., Instability of sound waves in a thin gaseous disk, Astron. Lett., 1995, vol. 21, pp. 347–351.

    ADS  Google Scholar 

  38. Kolesnichenko, A.V., On the simulation of helical turbulence in an astrophysical nonmagnetic disk, Sol. Syst. Res., 2011, vol. 45, no. 3, pp. 246–263.

    Article  ADS  Google Scholar 

  39. Kolesnichenko, A.V., Modification of the criteria for gravitational instability of astrophysical disks with a fractal phase space structure within Tsallis statistics, Math. Montisnigri, 2015, vol. 32, pp. 93–118.

    Google Scholar 

  40. Kolesnichenko, A.V., Konstruirovanie kontinual’nykh modelei turbulentnykh kosmicheskikh sred. Problemy matematicheskogo modelirovaniya astrofizicheskikh akkretsionnykh diskov (Construction of Continual Models of Turbulent Space Media. Problems of Mathematical Modeling of Astrophysical Accretion Disks), Saarbrücken, Germany: LAMBERT Academic Publishing, 2016a.

  41. Kolesnichenko, A.V., Modification of criteria for gravitational instability of astrophysical disks within non-additive Tsallis statistics, Mat. Model., 2016b, vol. 28, no. 3, pp. 96–118.

    MathSciNet  MATH  Google Scholar 

  42. Kolesnichenko, A.V., Nekotorye problemy konstruirovaniya kosmicheskikh sploshnykh sred. Modelirovanie akkretsionnykh protoplanetnykh diskov (Some Problems of Constructing Space Continuous Media. Modeling of Accretion Protoplanetary Disks), Moscow: Inst. Prikl. Mat. im. Keldysha, 2017.

  43. Kolesnichenko, A.V., To the construction of non-additive thermodynamics of complex systems based on the Kurado–Tsallis statistics, Preprints of Keldysh Inst. of Appl. Math., Russ. Acad. Sci., Moscow, 2018, no. 25.

  44. Kolesnichenko, A.V., Statisticheskaya mekhanika i termodinamika Tsallisa neadditivnykh sistem. Vvedenie v teoriyu i prilozheniya (Statistical Mechanics and Tsallis Thermodynamics of Non-Additive Systems. Introduction to Theory and Applications), Sinergetika: ot proshlogo k budushchemu (Synergetics: From the Past to the Future), no. 87, Moscow: Lenand, 2019.

  45. Kolesnichenko, A.V., Thermodynamics of the Bose gas and blackbody radiation in non-extensive Tsallis statistics, Sol. Syst. Res., 2020a, vol. 54, no. 5, pp. 420–431.

    Article  ADS  Google Scholar 

  46. Kolesnichenko, A.V., Jeans instability of a protoplanetary gas cloud with radiation in nonextensive Tsallis kinetics, Sol. Syst. Res., 2020b, vol. 54, no. 2, pp. 137–149.

    Article  ADS  Google Scholar 

  47. Kolesnichenko, A.V. and Chetverushkin, B.N., Kinetic derivation of a quasi-hydrodynamic system of equations on the base of nonextensive statistics, RJNAMM (Russ. J. Numer. Anal. Math. Model.), 2013, vol. 28, no. 6, pp. 547–576.

    Article  MATH  Google Scholar 

  48. Kolesnichenko, A.V. and Marov, M.Ya., Thermodynamic model of MHD turbulence and some of its applications to accretion disks, Sol. Syst. Res., 2008, vol. 42, no. 3, pp. 226–255.

    Article  ADS  Google Scholar 

  49. Kolesnichenko, A.V. and Marov, M.Ya., Modeling of aggregation of fractal dust clusters in a laminar protoplanetary disk, Sol. Syst. Res., 2013, vol. 47, no. 2, pp. 80–98.

    Article  ADS  Google Scholar 

  50. Kolesnichenko, A.V. and Marov, M.Ya., Modification of the Jeans instability criterion for fractal-structure astrophysical objects in the framework of nonextensive statistics, Sol. Syst. Res., 2014, vol. 48, no. 5, pp. 354–365.

    Article  ADS  Google Scholar 

  51. Kolesnichenko, A.V. and Marov, M.Ya., Modification of the Jeans and Toomre instability criteria for astrophysical fractal objects within nonextensive statistics, Sol. Syst. Res., 2016, vol. 50, no. 4, pp. 251–261.

    Article  ADS  Google Scholar 

  52. Kolesnichenko, A.V. and Marov, M.Ya., Streaming instability in the gas-dust medium of the protoplanetary disc and the formation of fractal dust clusters, Sol. Syst. Res., 2019, vol. 53, no. 3, pp. 181–198.

    Article  ADS  Google Scholar 

  53. Kumar, V., Sutar, D.L., Pensia, R.K., and Sharma, S., Effect of fine dust particles and finite electron inertia of rotating magnetized plasma, 2nd Int. Conf. Condensed Matter and Appl. Phys. (ICC 2017), AIP Conf. Proc., 2018, vol. 1953, id. 060036.

  54. Landau, L.D. and Lifshitz, E.M., Statisticheskaya fizika (Statistical Physics), Mosow: Nauka, 1964.

  55. Leubner, M.P., Nonextensive theory of dark matter and gas density profiles, Astrophys. J., 2005, vol. 632, pp. L1–L4.

    Article  ADS  Google Scholar 

  56. Lima, J.A.S., Silva, R., Jr., and Santos, J., Plasma oscillations and nonextensive statistics, Phys. Rev. E, 2000, vol. 61, no. 3, pp. 3260–3263.

    Article  ADS  Google Scholar 

  57. Lima, J.A.S., Silva, R., and Santos, J., Jeans’ gravitational instability and nonextensive kinetic theory, Astron. Astrophys., 2002, vol. 396, pp. 309–313.

    Article  ADS  MATH  Google Scholar 

  58. Low, C. and Lynden-Bell, D., The minimum Jeans mass or when fragmentation must stop, Mon. Not. R. Astron. Soc., 1976, vol. 176, no. 2, pp. 367–390.

    Article  ADS  Google Scholar 

  59. Ma, P., Zheng, Y., and Qi, G., The nonextensive Bose-Einstein condensation and photon gas with parameter transformation, Eur. Phys. J. Plus, 2019, vol. 134, art. no. 502.

    Article  Google Scholar 

  60. Mace, R.L., Verheest, Frank., and Hellberg, M.A., Jeans stability of dusty space plasmas, Phys. Lett. A, 1998, vol. 237, pp. 146–151.

    Article  ADS  Google Scholar 

  61. Makalkin, A.B. and Ziglina, I.N., Gravitational instability in the dust layer of a protoplanetary disk with interaction between the layer and the surrounding gas, Sol. Syst. Res., 2018, vol. 52, no. 6, pp. 518–533.

    Article  ADS  Google Scholar 

  62. Makalkin, A.B., Ziglina, I.N., and Artyushkova, M.E., Topical problems in the theory of planet formation: Formation of planetesimals, Izv., Phys. Solid Earth, 2019, vol. 55, pp. 87–101.

    Article  Google Scholar 

  63. Marov, M.Ya. and Kolesnichenko, A.V., Turbulence and Self-Organization. Modeling Astrophysical Objects, New York: Springer Science & Business Media, 2013.

    Book  Google Scholar 

  64. Martinez, S., Nicolas, F., Pennini, F., and Plastino, A., Tsallis’ entropy maximization procedure revisited, Phys. A, 2000, vol. 286, pp. 489–502.

    Article  MathSciNet  MATH  Google Scholar 

  65. Masood, W., Salimullah, M., and Shah, H.A., A quantum hydrodynamic model for multicomponent quantum magnetoplasma with Jeans term, Phys. Lett. A, 2008, vol. 45, pp. 6757–6760.

    Article  ADS  MATH  Google Scholar 

  66. Mather, J.C., Cheng, E.S., Cottingham, D.A., Eplee, R.E., Fixsen, D.J., Hewagama, T., Isaacman, R.B., Jensen, K.A., Meyer, S.S., Noerdlinger, P.D., Read, S.M., Rosen, L.P., Shafer, R.A., Wright, E.L., Bennett, C.L., Boggess, N.W., Hauser, M.G., Kelsall, T., Moseley, S.H., Silverberg, R.F., Smoot, G.F., Weiss, R., and Wilkinson, D.T., Measurement of the cosmic microwave background spectrum by the COBE FIRAS instrument, Astrophys. J., 1994, vol. 420, pp. 439–444.

    Article  ADS  Google Scholar 

  67. McKee, M.R., The radial-azimuthal stability of accretion disks around black holes, Astron. Astrophys., 1990, vol. 235, pp. 521–525.

    ADS  Google Scholar 

  68. Nonextensive statistical mechanics and thermodynamics: Bibliography. http://tsallis. cat.cbpf.br/biblio.htm.

  69. Owen, J.M., Villumsen, J., and Baryons, V., Dark matter, and the Jeans mass in simulations of cosmological structure formation, J. Astrophys., 1997, vol. 481, no. 1, pp. 1–21.

    Article  ADS  Google Scholar 

  70. Pandey, B.P. and Avinash, K., Jeans instability of a dusty plasma, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top., 1994, vol. 49, no. 6, pp. 5599–5606.

    Article  Google Scholar 

  71. Pensia, R.K., Sutar, D.L., and Sharma, S., Analysis of Jeans instability of optically thick quantum plasma under the effect of modified Ohms law, 2nd Int. Conf. Condensed Matter and Appl. Phys. (ICC 2017), AIP Conf. Proc., 2018, vol. 1953, id. 060044.

  72. Pessah, M.E., Torres, D.F., and Vucetich, H., Statistical mechanics and the description of the early universe. (I). Foundations for a slightly non-extensive cosmology, Phys. A: Stat. Mech., 2001, vol. 297, nos. 1–2, pp. 164–200.

    Article  MATH  Google Scholar 

  73. Plastino, A.R., Plastino, A., and Vucetich, H., A quantitative test of Gibbs’ statistical mechanics, Phys. Lett. A, 1995, vol. 207, pp. 42–46.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  74. Rovenchak, A., Ideal Bose-gas in nonadditive statistics, Low Temp. Phys., 2018, vol. 44, no. 10, pp. 1025–1031.

    Article  ADS  Google Scholar 

  75. Safronov, V.S., Evolyutsiya doplanetnogo oblaka i obrazovanie Zemli i planet (Evolution of the Protoplanetary Cloud and the Formation of the Earth and Planets), Moscow: Nauka, 1969.

  76. Sakagami, M. and Taruya, A., Self-gravitating stellar systems and non-extensive thermostatistics, Continuum Mech. Thermodyn., 2004, vol. 16, no. 3, pp. 279–292.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  77. Sistema, P.D. and Vucetich, H., Cosmology, oscillating physics, and oscillating biology, Phys. Rev. Lett., 1994, vol. 72, no. 4, pp. 454–457.

    Article  ADS  Google Scholar 

  78. Shakura, N.I. and Sunyaev, R.A., A theory of the instability of disk accretion onto black holes and the variability of binary X-ray sources, galactic nuclei and quasars, Mon. Not. R. Astron. Soc., 1976, vol. 175, pp. 613–632.

    Article  ADS  Google Scholar 

  79. Shukla, P.K. and Stenflo, L., Jeans instability in a self-gravitating dusty plasma, Proc. R. Soc. A: Mathematical, Physical and Engineering Sciences, 2006, pp. 403–407.

  80. Tirnakli, U., Büyükkiliç, F., and Demirhan, D., Generalized distribution functions and an alternative approach to generalized Planck radiation law, Phys. A: Stat. Mech. Its Appl., 1997, vol. 240, nos. 3–4, pp. 657–664.

    Article  ADS  Google Scholar 

  81. Toomre, A., On the gravitational stability of a disk of stars, Astrophys. J., 1964, vol. 139, pp. 1217–1238.

    Article  ADS  Google Scholar 

  82. Trigger, S.A., Ershkovich, A.I., van Heijst, G.J.F., and Schram, P.P.J.M., Kinetic theory of Jeans instability, Phys. Rev. E, 2004, vol. 69, pp. 066403–066405.

    Article  ADS  Google Scholar 

  83. Tsiklauri, D., Jeans instability of interstellar gas clouds in the background of weakly interacting massive particles, Astrophys. J., 1998, vol. 507, no. 1, pp. 226–228.

    Article  ADS  Google Scholar 

  84. Tsintsadze, N.L., Chaudhary, R., Shah, H.A., and Murtaza, G., Jeans instability in a magneto-radiative dusty plasma, J. Plasma Phys., 2008, vol. 74, no. 6, pp. 847–853.

    Article  ADS  Google Scholar 

  85. Tsallis, C., Possible generalization of Boltzmann-Gibbs-statistics, J. Stat. Phys., 1988, vol. 52, nos. 1–2, pp. 479–487.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  86. Tsallis, C., Nonextensive statistic: Theoretical, experimental and computational evidences and connections, Braz. J. Phys., 1999, vol. 29, no. 1, pp. 1–35.

    Article  ADS  Google Scholar 

  87. Tsallis, C., Introduction to Nonextensive Statistical Mechanics. Approaching a Complex World, New York: Springer, 2009.

    MATH  Google Scholar 

  88. Tsallis, C., Sa Barreto, F.C., and Loh, E.D., Generalization of the Planck radiation law and application to the cosmic microwave background radiation, Phys. Rev. E, 1995, vol. 52, no. 2, pp. 1448–1451.

    Article  ADS  Google Scholar 

  89. Tsallis, C., Mendes, R.S., and Plastino, A.R., The role of constraints within generalized nonextensive statistics, Phys. A, 1998, vol. 261, pp. 534–554.

    Article  Google Scholar 

  90. Wang, Q.A. and Le Méhauté, A., Nonextensive black-body distribution function and Einstein’s coefficients A and B, Phys. Lett. A, 1998, vol. 242, pp. 301–306.

    Article  ADS  Google Scholar 

  91. Wang, Q.A. and Nivanen, L., and Le Méhauté, A., Generalized blackbody distribution within the dilute gas approximation, Phys. A, 1998, vol. 260, pp. 490–498.

    Article  Google Scholar 

  92. Zaripov, R.G., Samoorganizatsiya i neobratimost’ v neekstensivnykh sistemakh (Self-Organization and Irreversibility in Non-Extensive Systems), Kazan: Fen, 2002.

  93. Zaripov, R.G., Elementary particle physics and field theory. Evolution of the difference information in the process of the Fermi and Bose gas self-organization for nonextensive systems, Russ. Phys. J., 2009, vol. 52, no. 4, pp. 329–336.

    Article  MATH  Google Scholar 

  94. Zaripov, R.G., Printsipy neekstensivnoi statisticheskoi mekhaniki i geometriya mer besporyadka i poryadka (Principles of Nonextensive Statistical Mechanics and Geometry of Measures of Disorder and Order), Kazan: Izd. Kazan. Gos. Tekhn. Univ., 2010.

Download references

Funding

The author is grateful to the Government of the Russian Federation and the Ministry of Higher Education and Science of the Russian Federation for support under the grant 075-15-2020-780 (N13.1902.21.0039).

Author information

Authors and Affiliations

  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Moscow, Russia

    A. V. Kolesnichenko

Authors
  1. A. V. Kolesnichenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. V. Kolesnichenko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kolesnichenko, A.V. Jeans Instability of a Protoplanetary Circular Disk Taking into Account the Magnetic Field and Radiation in Nonextensive Tsallis Kinetics. Sol Syst Res 55, 132–149 (2021). https://doi.org/10.1134/S0038094621020039

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0038094621020039

Keywords:

Search

Navigation