1. MMC
  2. All Volumes and Issues
  3. Volume 10, Number 1, 2023
  4. The generalized polytropic model for the Sun-like stars

The generalized polytropic model for the Sun-like stars

MMC.
2023;
Volume 10, Number 1
: pp. 1–9
https://doi.org/10.23939/mmc2023.01.001
Received: July 14, 2022
Revised: September 06, 2022
Accepted: September 28, 2022

Mathematical Modeling and Computing, Vol. 10, No. 1, pp. 1–9 (2023)

Authors:
  1. M. Vavrukh
  2. D. Dzikovskyi
1
Ivan Franko National University of Lviv
2
Ivan Franko National University of Lviv

The Eddington method based on simultaneous consideration of gas and light pressures with a homogeneous сhemical composition of stellar matter was generalized for the case of model with a spatially inhomogeneous chemical composition.  As a result, it was obtained the equation of state, which is expressed by a generalized polytrope with index $n=3$.  As an example, it was solved the equilibrium equation for the Sun both using the standard polytropic equation of state and generalized polytrope.  The coordinate dependence of the Sun characteristics was calculated within two models.  Obtained results are compared with the results of numerical calculations for the Sun based on the system of Schwarzschild equations for the standard model.  It was shown that the standard polytropic model is applicable only for the Sun of zero-age.  The Sun characteristics calculated with help of generalized equation of state are close to the results of numerical calculations based on Schwarzschild equations.  It was concluded that the standard polytropic model is applicable for the stars of zero-age main sequence, and the generalized model – for the stars of finite age, in which thermonuclear reactions have already created a significant spatially inhomogeneity of chemical composition inside of the core.

polytropic stars
spatially inhomogeneous chemical composition
mechanical equilibrium equation
  1. Guenther D. B., Demarque P., Kim Y.-C., Pinsonneault M. H.  Standard Solar Model.  The Astrophysical Journal.  387, 372–393 (1992).
  2. Milne E. A.  The equilibrium of a rotating star.  Monthly Notices of the Royal Astronomical Society.  83 (3), 118–147 (1923).
  3. Chandrasekhar S.  The Equilibrium of Distorted Polytropes. I. The Rotational Problem.  Monthly Notices of the Royal Astronomical Society.  93 (5), 390–406 (1933).
  4. James R. A.  The Structure and Stability of Rotating Gas Masses.  Astrophysical Journal.  140, 552–582 (1964).
  5. Monaghan J. J., Roxburgh I. W.  The Structure of Rapidly Rotating Polytropes.  Monthly Notices of the Royal Astronomical Society.  131 (1), 13–22 (1965).
  6. Kopal Z.  Bemerkung zur Theorie der rotierenden Polytropen.  Zeitschrift für Astrophysik.  14, 135–138 (1937).
  7. Caimmi R.  Emden–Chandrasekhar Axisymmetric Solid–Body Rotating Polytropes. I: Exact Solutions for the Special Cases $N=0$, $1$ and $5$.  Astrophysics and Space Science. 71, 415-457 (1980).
  8. Williams P. S.  Analytical Solutions for the Rotating Polytrope $N=1$.  Astrophysics and Space Science.  143, 349–358 (1988).
  9. Vavrukh M. V., Tyshko N. L., Dzikovskyi D. V., Stelmakh O. M.  The self-consistent description of stellar equilibrium with axial rotation.  Mathematical Modeling and Computing.  6 (2), 153–172 (2019).
  10. Vavrukh M. V., Tyshko N. L., Dzikovskyi D. V.  New approach in the theory of stellar equilibrium with axial rotation.  Journal of Physical Studies.  24 (3), 3902 (2020).
  11. Vavrukh M. V., Dzikovskyi D. V.  Exact solution for the rotating polytropes with index unity, its approximations and some applications.  Contributions of the Astronomical Observatory Skalnaté Pleso.  50 (4), 748–771 (2020).
  12. Kong D., Zhang K., Schubert G.  An exact solution for arbitrarily rotating gaseous polytropes with index unity.  Monthly Notices of the Royal Astronomical Society.  448 (1), 456–463 (2015).
  13. Knopik J., Mach P., Odrzywołek A.  The shape of a rapidly rotating polytrope with index unity.  Monthly Notices of the Royal Astronomical Society.  467 (4), 4965–4969 (2017).
  14. Schwarzschild M.  Structure and Evolution of the Stars.  Princeton University Press, Princeton (1958).
  15. Eddington A. S.  The Internal Constitution of the Stars.  Cambridge University Press, Cambridge (1988).
  16. Vavrukh M. V., Smerechynskyi S. V., Tyshko N. L.  The inverse problem of the theory of degenerate dwarfs.  Astronomy Reports.  55, 505–524 (2011).
  17. Gibson E. G.  The Quiet Sun.  Washington, D.C. (1973).
  18. Chandrasekhar S.  An Introduction to the Study of Stellar Structure.  University of Chicago Press, Chicago (1939).
  19. Sears R. L.  Helium Content and Neutrino Fluxes in Solar Models.  Astrophysical Journal.  140, 477–484 (1964).
  20. Lamers H. J. G. L. M., Levesque E. M.  Understanding Stellar Evolution. IOP Publishing, Bristol, UK (2017).