The self-consistent description of stellar equilibrium with axial rotation

New method for the description of mechanical equilibrium of stellar structure with axial rotation was proposed.  The self-consistent calculation is based on simultaneous use of differential and integral forms of mechanical equilibrium equation, which allows us to correctly determine the integration constants.  In the frame of polytropic model with indexes n=0 and 1 were first obtained the analytical solutions, for n=2 and 3 numerically.  The geometrical parameters of stellar surface as well as mass, volume and moment of inertia  were calculated as the functions of angular velocity.  It was found the maximal value of angular velocity in which the stability is disturbed.  Obtained results improve the results of E. Milne, S. Chandrasekhar and R. James, obtained with help of the approximate numerical integration of mechanical equilibrium  equation.

polytropic stars
heterogeneous ellipsoids
axial rotation
mechanical equilibrium equation
stability of stars
  1. Lane H. J.  On the Theoretical Temperature of the Sun under the Hypothesis of a gaseous mass Maintining Its Volume by Its Internal Heat and Depending on the Law of Gases Known to Terrestrial Experiment.  Am. J. Sci. 50, 57--74 (1870).
  2. Emden K.  Gaskugeln: Anwendungen der mechanischen Wärmetheorie.  Leipzig, Berlin (1907).
  3. Fowler R. H.  Emden's equation: The solutions of Emden's and similar differential equations.  Monthly Notices of the Royal Astronomical Society. 91 (1), 63--91 (1930).
  4. Ritter A.  Untersuchungen über die Höhe der Atmosphäre und die Constitution gasförmiger Weltkörper.  Ann. Phys. 241, 543--558 (1878).
  5. Kelvin W. Mathematical and Physical Papers. 3, 255 (1911).
  6. Eddington A. S.  The Internal Constitution of the Stars.  Cambridge, Cambridge University Press (1926).
  7. Milne E. A.  The equilibrium of a rotating star.  Monthly Notices of the Royal Astronomical Society. 83, 118--147 (1923).
  8. Chandrasekhar S.  The Equilibrium of Distorted Polytropes.   I. The Rotational Problem.  Monthly Notices of the Royal Astronomical Society. 33, 390--406 (1933).
  9. James R. A. The Structure and Stability of Rotating Cas Masses.  Astrophysical Journal. 140, 552--582 (1964).
  10. Vavrukh M. V., Smerechynskyi S. V., Tyshko N. L.  The microscopic parameters and the macroscopic characteristics of real degenerate dwarfs.  Journal of Physical Studies. 14 (4), 4901 (2010).
  11. Shapiro S. L., Teukolsky S. A.  Black Holes, White Dwarfs and Neutron Stars.  Cornell University, Ithaca, New York (1983).
  12. Chandrasekhar S.  An Introduction to the Study of Stellar Structure.  Chicago, University of Chicago Press (1939).
  13. Lyttleton R. A.  The Stability of Rotating Liquid Masses.  Cambridge, Cambridge University Press (1953).
  14. Abramowitz M., Stegun I. A.  Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. Government Printing Office Washington (1972).