Skip to main content
Log in

Jeans Instability of a Protoplanetary Gas Cloud with Radiation in Nonextensive Tsallis Kinetics

  • Published:
Solar System Research Aims and scope Submit manuscript

Abstract

In the framework of Tsallis statistics, we study the effect of medium nonextensivity on the Jeans gravitational instability criterion for a self-gravitating protoplanetary cloud, the substance of which consists of a mixture of perfect gas and blackbody radiation. Generalized Jeans instability criteria have been found from the corresponding dispersion relations obtained both for a uniform cloud with radiation and for a rotating protoplanetary cloud. An integral generalized Chandrasekhar stability criterion for a gravitating spherical cloud has also been obtained. The presented results are analyzed for various values of deformation parameters \(q,\) dimensionality \(D\) of the velocity space and coefficient \(\beta \), characterizing the fraction of radiation in the total pressure of the system. It is shown that radiation stabilizes the substance of nonextensive protoplanetary clouds, and for rotating clouds, the Jeans instability criterion is modified by the Coriolis force only in the transverse modes of perturbation wave propagation.

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

Access this article

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. Numerous journal articles, collections and monographs are devoted to surveys of studies within the framework of nonextensive Tsallis statistics. In addition, there is a constantly updated full bibliography (Nonextensive statistical mechanics and thermodynamics: Bibliography/ http://tsallis.cat.cbpf.br/biblio.htm), which today includes more than 5600 references.

  2. In the cited paper, the kinetic theory was based on the Bhatnagar-Gross-Krook collision integral (BGK), which was generalized for an arbitrary value of parameter \(q\).

  3. In what follows, index “q” for some hydrodynamic and thermodynamic variables will be omitted.

  4. Eddington first pointed out the special importance of quantity \((1 - \beta )\) for the theory of stellar structure. In a famous passage from his book “The Internal Structure of the Stars”, Eddington associated this quantity with the “happening of the stars.”

  5. When studying perturbed states of self-gravitating cosmic matter, one often has to deal with some form of sound waves.

  6. It should be noted that the linearized equation of momentum requires that the velocity u be parallel to the wave vector \( \pm {\mathbf{k}}\) (see Landau, Lifshitz, 1976). Consequently, the velocities of the fluid particles associated with adiabatic sound waves are parallel to the direction of wave propagation.

  7. In particular, it follows from (60) that for a star with a mass equal to the solar mass and with an average molecular weight equal to unity, the radiation pressure in the center of the star cannot exceed three percent of the total pressure, i.e., \(1 - \beta_* \cong 0.03\) (Chandrasekhar, 1985).

  8. It is known that the problem of stability of a self-gravitating two-dimensional gas cloud cannot be described, in principle, in the framework of the two-dimensional approximation, since it is a priori very unstable (see, for example, Fridman and Khoperskov, 2011). However, when the angular velocity of rotation is sufficiently high, in the presence of a strong external gravitational field with cylindrical geometry and with a generatrix along the axis of rotation of the cloud, it is possible to ensure its stability. In this case, the structure of the protoplanetary cloud along the axis of rotation will be determined solely by its self-gravity. It is clear that this case is artificial, since such cylindrical fields, if they occur in real astrophysical systems, are without embedded disks. At the same time, the analysis of such a self-gravitating thick gas disk embedded in the cylinder is of certain theoretical interest, since only in this case one can allocate the effects arising under the action of pure gravity. It is precisely such models were studied in most classical works on astrophysical disks (see, for example, Goldreich and Lynden-Bell, 1965; Hunter, 1972; Toomre, 1964).

REFERENCES

  1. Boghosian, B.M., Navier-Stokes equations for generalized thermostatistics, Braz. J. Phys., 1999, vol. 29, no. 1, pp. 91–107.

    Article  ADS  Google Scholar 

  2. Bonnor, W.B., Jeans’ formula for gravitational instability, Mon. Not. R. Astron. Soc., 1957, vol. 117, no. 1, pp. 104–117. https://doi.org/10.1093/mnras/117.1.104

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Cadez, V.M., Instabilities in stratified magnetized stellar atmospheres, Publ. Astron. Ops. Beogradu, 2010, vol. 90, pp. 121–124.

    ADS  Google Scholar 

  5. 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 

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

    MATH  Google Scholar 

  7. Chandrasekhar, S., On Stars, Their Evolution and Their Stability: Nobel Lecture, Stockholm, 1983.

  8. 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 

  9. Curado, E.M.F. and Tsallis, C., Generalized statistical mechanics: connection with thermodynamics, J. Phys. A, 1991, vol. 24, pp. L69–L72.

    Article  ADS  MathSciNet  Google Scholar 

  10. 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 

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

    Book  MATH  Google Scholar 

  12. Fridman, A.M. and Gorkavyi, N.N., Physics of Planetary Rings, New York: Springer-Verlag, 1999.

    Book  MATH  Google Scholar 

  13. Fridman, A.M. and Khoperskov, A.V., Physics of Galaxies: Observation and Investigation of Galact, New Delhi: Viva Books, 2014.

    Google Scholar 

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

  15. Fridman, A.M. and Polyachenko, V.L., Physics of Gravitating Systems I: Equilibrium and Stability, New York: Springer-Verlag, 2012.

    Google Scholar 

  16. 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 

  17. Gough, D.O., Heliophysics gleaned from seismology, Proc. 61st Fujihara Seminar “Progress in Solar/Stellar Physics with Helio- and Asteroseismology,” ASP Conference Series vol. 462, San Francisco: Astron. Soc. Pac., 2011, pp. 429–454.

  18. 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 

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

    Article  ADS  MATH  Google Scholar 

  20. Jeans, J.H., The stability of a spherical nebula, Philos. Trans. R. Soc., A, 1902, vol. 199, pp. 1–53.

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

    Book  MATH  Google Scholar 

  22. Joshi, H. and Pensia, R.K., Effect of rotation on Jeans instability of magnetized radiative quantum plasma, Phys. Plasmas, 2017, vol. 24, pp. 032113-1–032113-8.

  23. 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, vol. 1536, no. 1, pp. 1288–1289.

    Article  ADS  Google Scholar 

  24. Khoperskov, A.V. and Khrapov, S.S., Instability of sound waves in a thin gas disk, Pis’ma Astron. Zh., 1995, vol. 21, pp. 388–393.

    ADS  Google Scholar 

  25. Kochin, N.E., Vektornoe ischislenie i nachala tenzornogo ischisleniya (Vector Calculus and the Beginnings of Tensor Calculus), Moscow: Akad. Nauk SSSR, 1961.

  26. Kolesnichenko, A.V., Modification in framework of Tsallis, statistics of gravitational instability criterions of astrophysical disks with fractal structure of phase space, Math. Montisnigri, 2015, vol. 32, pp. 93–118.

    MathSciNet  Google Scholar 

  27. Kolesnichenko, A.V., Modification in the framework of nonadditive Tsallis, statistics of the gravitational instability criterions of astrophysical disks, Matem.Model., 2016, vol. 28, no. 3, pp. 96–118.

    MATH  Google Scholar 

  28. Kolesnichenko, A.V., The construction of non-additive thermodynamics of complex systems based on the Curado-Tsallis, statistics, Preprint of Keldysh Inst. of Applied Mathematics, Russ. Acad. Sci., Moscow, 2018, no. 25.

  29. Kolesnichenko, A.V., Statisticheskaya mekhanika i termodinamika Tsallisa neadditivnykh sistem. Vvedenie v teoriyu i prilozheniya (Tsallis’s Statistical Mechanics and Thermodynamics: Theory and Application), Sinergetika: ot proshlogo k budushchemu no. 87, Moscow: Lenand, 2019.

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

    MathSciNet  MATH  Google Scholar 

  31. 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 

  32. 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 

  33. 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 

  34. Kolesnichenko, A.V. and Marov, M.Ya., Rényi thermodynamics as a mandatory basis to model the evolution of a protoplanetary gas–dust disk with a fractal structure, Sol. Syst. Res., 2019, vol. 53, no. 6, pp. 443–461.

    Article  ADS  Google Scholar 

  35. Kumar, V., Sutar, D.L., Pensia, R.K., and Sharma, S., Effect of fine dust particles and finite electron inertia of rotating magnetized plasma, AIP Conf. Proc., 2018, vol. 1953, no. 1, pp. 060036-1–060036-4.

  36. Landau, L.D. and Lifshitz, E.M., Course of Theoretical Physics, Part 1, Vol. 5: Statistical Physics, Oxford: Butterworth-Heinemann, 1980.

  37. 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 

  38. 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 

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

    Article  ADS  Google Scholar 

  40. Masood, W., Salimullah, M., and Shah, H.A., A quantum hydrodynamic model for multicomponent quantum magnetoplasma with Jeans term, Phys. Lett. A, 2008, vol. 372, no. 45, pp. 6757–6760.

    Article  ADS  MATH  Google Scholar 

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

    ADS  Google Scholar 

  42. Nonextensive statistical mechanics and thermodynamics: bibliography. http://tsallis.cat.cbpf.br/biblio.htm.

  43. Oliveira, D.S. and Galvao, R.M.O., Transport equations in magnetized plasmas for non-Maxwellian distribution functions, Phys. Plasmas, 2018, vol. 25, pp. 102308-1–102308-13.

  44. Owen, J.M. and Villumsen, J., Baryons, 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 

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

    Article  Google Scholar 

  46. 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, AIP Conf. Proc., 2018, vol. 1953, no. 1, pp. 060044-1–060044-4.

  47. Radwan, A.E., Variable streams self-gravitating instability of radiating rotating gas cloud, Appl. Math. Comput., 2004, vol. 148, pp. 331–339.

    MathSciNet  MATH  Google Scholar 

  48. Safronov, V.S., Evolyutsiya doplanetarnogo oblaka i obrazovanie Zemli i planet (The Evolution of the Pre-Planetary Cloud and the Formation of the Earth and Planets), Moscow: Nauka, 1969.

  49. 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 

  50. 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 

  51. Shukla, P.K. and Stenflo, L., Jeans instability in a self-gravitating dusty plasma, Proc. R. Soc. A, 2006, vol. 462, pp. 403–407.

    Article  ADS  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  53. 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: Stat., Nonlinear, Soft Matter Phys., 2004, vol. 69, pp. 066403–066405.

    Article  Google Scholar 

  54. 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 

  55. Tsallis, C., Nonextensive statistics: theoretical, experimental and computational evidences and connections, Braz. J. Phys., 1999, vol. 29, no. 1, pp. 1–35.

    Article  ADS  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  Google Scholar 

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

    Article  ADS  Google Scholar 

  59. 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 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Kolesnichenko.

Additional information

Translated by G. Dedkov

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 Gas Cloud with Radiation in Nonextensive Tsallis Kinetics. Sol Syst Res 54, 137–149 (2020). https://doi.org/10.1134/S0038094620020045

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0038094620020045

Keywords:

Navigation