Accretion and Collapse

  • D. Lynden-Bell
Part of the NATO ASI Series book series (NSSB, volume 156)

Abstract

There is a three-line proof that all specific heats are positive. In a canonical ensemble of systems at equilibrium, the average energy of a system is given by the sum over all energy levels i of the system
$$ < E > \, = \,\mathop \sum \limits_i \,{E_i}\,{e^{ - \beta {\rm E}}}\,/\,\sum {e^{ - \beta {\rm E}}}i$$
where ß= (kT)-1. Evidently \( {C_{V}} = d < E > /dT = - k{\beta ^{2}}\frac{d}{{d\beta }} < E > \) and so \({\beta ^{ - 2}}{C_V} = k[{\sum\limits_i {{E_i}} ^{{2_{{e^{ - \beta {E_{{i_{{/_{\sum\limits_i {{e^{\beta {E_{{i_ - }}}}}} {{(\Sigma {E_i}{e^{ - \beta {E_{i/\Sigma {e^{ - \beta {E_i}}}}}}})}^2}] = k( < {E^2} > - < E{ > ^2}) = k < {{(E - < E > )}^2} > }}}}}}}}}}}} \) which is clearly positive.

Keywords

Entropy Vortex Convection Enthalpy Stratification 

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References

  1. Antonov, V.A., 1962, Vest Leningrad. gos Univ. 7 ,135.Google Scholar
  2. Bardeen, J., 1970, Nature 226, 64.ADSCrossRefGoogle Scholar
  3. Begelman, M.C. & Meyer, D.L., 1982, Astrophys. J. 253, 873.ADSCrossRefGoogle Scholar
  4. Bettwieser, E. & Sugimoto, D., 1984, Mon. Not. R. astr. Soc., 208, 493.ADSGoogle Scholar
  5. Blandford, R.D. & Rees, M.J., 1974, Mon. Not. R. astr. Soc, 169, 395.ADSGoogle Scholar
  6. Bondi, H., 1952, Mon. Not. R. astr. Soc., 112, 195.MathSciNetADSGoogle Scholar
  7. Cohn, H., 1980, Astrophys. J., 242, 765; also IAU Symp. 113, Reidel.ADSCrossRefGoogle Scholar
  8. Fillmore, J.A. & Goldreich, P., 1984, Ap. J. 281, 1, 9.MathSciNetADSCrossRefGoogle Scholar
  9. Gilham, S., 1981, Mon. Not. R. astr. Soc., 195, 755.ADSMATHGoogle Scholar
  10. Hachisu, I., Nakada, Y., Nomoto, K. & Sugimoto, D., 1978, Prog. Theor. Phys. 60, 393.ADSCrossRefGoogle Scholar
  11. Heggie, D.C., 1985, Goodman, J. & Hut, P., eds IAU Symp. 113, Reidel.Google Scholar
  12. Henon, M., 1961, Ann. Astrophys. 24, 369.MathSciNetADSGoogle Scholar
  13. Hoyle, F. & Lyttleton, R.A., 1939, Proc. Camb. Phil. Soc. 35, 405.ADSCrossRefGoogle Scholar
  14. Hunt, R., 1971, Mon. Not. R. astr. Soc. 154, 152.Google Scholar
  15. Inagaki, S. & Lynden-Bell, D., 1983, Mon. Not. R. astr. Soc. 205, 913.ADSMATHGoogle Scholar
  16. Katz, J. & Lynden-Bell, D., 1978, Mon. Not. R. astr. Soc. 184, 708.ADSGoogle Scholar
  17. Landau, C.D. & Lifshitz, E.M., 1966, Classical Theory of Fields, MIR, Moscow.Google Scholar
  18. Lynden-Bell, D. & Wood, R., 1968, Mon. Not. R. astr. Soc. 138, 495.ADSGoogle Scholar
  19. Lynden-Bell, D., 1969, Nature 223, 690.ADSCrossRefGoogle Scholar
  20. Lynden-Bell, D. & Pringle, J.E., 1974, Mon. Not. R. astr. Soc. 168, 603.ADSGoogle Scholar
  21. Lynden-Bell, D. & Lynden-Bell, R.M., 1977, Mon. Not. R. astr. Soc. 181, 405ADSGoogle Scholar
  22. Lynden-Bell, D., 1978, Physica Scripta 17, 185•ADSCrossRefGoogle Scholar
  23. Lynden-Bell, D. & Eggleton, P.P., 1980, Mon. Not. R. astr. Soc. 191, 483.MathSciNetADSGoogle Scholar
  24. Michel, F.C., 1972, Astrophys. & Sp. Sc. 15, 153.ADSCrossRefGoogle Scholar
  25. Penston, M.V., 1969, Mon. Not. R. astr. Soc. 144, 425.ADSGoogle Scholar
  26. Tolman, R.C., 1934, Relativity Thermodynamics & Cosmology.Google Scholar

Copyright information

© Plenum Press, New York 1987

Authors and Affiliations

  • D. Lynden-Bell
    • 1
  1. 1.Clare College and Institute of AstronomyThe ObservatoriesCambridgeUK

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