The well known results of the theory of classical gaseous polytropes are presented in the framework of an integral approach where the standard Lane-Emden differential equation for a spherically symmetric gravitating mass is examined via its equivalent in the form of a nonlinear integral Volterra equation of the 2nd kind. It is shown that the inverse Laplace transform of the Lane-Emden equation for polytropes with an index of n=5 (Schuster model) is a recurrence relation for Bessel functions of the first kind. The invariance of the nonlinear integral Volterra equation with respect to homological transformations is shown, as well as the possibility of obtaining singular solutions under certain conditions. It is also shown that for whole integral and half integral polytrope indices, this equation is equivalent to a multidimensional integral equation, and finding the expansion of the Emden function in a power series of the dimensionless distance ξ from the center of the polytrope is equivalent to finding the Neumann series and the iterated nuclei in the Fredholm theory. Approximations of the Emden functions in closed form and their applicability to various astrophysical objects will be presented and discussed in the second part of this paper. Polytropes of other geometries and dimensionalities are not considered here.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. R. Emden, Gaskugeln, Leipzig (1907).
S. Chandrasekhar, An Introduction to the Study of Stellar Structure, University of Chicago Press (1939).
A. C. Eddington, The Internal Constitution of the Stars, Cambridge University Press (1930).
G. Horedt, Polytropes. Applications in Physics and Astrophysics (2004).
W. J. Chung and L. A. Nelson, Bose-Einstein condensate & degenerate Fermi cored dark matter, https://arxiv.org/pdf/1801.00565.pdf.
G. P. Horedt, Astron. Astrophys. 172, 359 (1987).
M. V. Medvedev and G. Rybicki, Astrophys. J. 555, 863 (2001).
M. Milgrom, Phys. Rev. D 103, 044043 (2021).
G. A. Saiyan, Com. BAO 68, 509 (2021).
V. V. Ivanov, Astrophysics of Stars [in Russian], St. Petersburg Univ. (2006).
P. M. Krasnov, Integral Equations [in Russian], Nauka, Moscow (1975).
G. A. Saiyan, Candidate’s Dissertation, Academy of Sciences of the Republic of Armenia (1997).
U. M. Shaudt, Ann. Henri Poincare 1, 945 (2000).
M. V. Vavrukh and D. V. Dzikovskyi, Mathematical Modeling and Computing 8, 338 (2021).
G. S. Saakyan, Space, Time, and Gravitation, Izd-vo. Erevanskogo Universiteta (1985).
G. Adomian, Solving Frontier Problems in Physics. The Decomposition Method, Kluwer Academic Publishers (1993).
G. Bateman and A. Erdelyi, Higher Transcendental Functions II [Russian translation], Nauka, Moscow (1974).
S. Srivastava, Astrophys. J. 136, 680 (1962).
H. Goenner and P. Havas, J. Math. Phys. 41, 7029 (2000).
P. Mach, J. Math. Phys. 53, 062503 (2012).
J. M. Huntley and W. C. Saslaw, Astrophys. J. 199, 325 (1975).
M. V. Mirkin and A. J. Bard, J. Electroanal. Chem. 323, l-27 (1992).
M. V. Mirkin and A. J. Bard, J. Electroanal. Chem. 323, 29-51 (1992).
V. G. Gurzadyan and A. G. Kechek, Preprint, Lebedev Institute AN SSSR, No. 180, (Moscow) (1979).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Astrofizika, Vol. 66, No. 1, pp. 151-166 (February 2023).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Saiyan, G.A. Theory of Classical Gaseous Polytropes in an Integral Representation. I. Some General Results. Astrophysics 66, 140–156 (2023). https://doi.org/10.1007/s10511-023-09776-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10511-023-09776-0