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Abstract
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This article proposes an analytical solution for the Emden–Chandrasekhar equation (ECE) that model
polytropic stellar structure in astrophysics. The mathematical approach is based on the Hybrid
Analytical and Numerical (HAN). The most remarkable feature of this problem is that the ECE has
no known explicit analytic solution despite all this time and effort. The existing explicit solutions
were valid when describing the star’s core, where the temperature is approximately constant. The
HAN-method can provide an explicit analytical solution for the ECE. Unlike other solutions that only
provide an exact solution near the center of the star, the HAN-method can obtain an explicit analytical
solution for all points of the star. The density, mass, and pressure functions of the star due to the
solution of the ECE are constructed. In addition, the dimensionless constant α, which we have named
‘‘the fundamental constant of the stellar structures,’’ has appeared in most of the equations of the
theory of stellar structures. This constant is calculated as α = 0.1719381834. The average temperature
of the Sun is obtained at 1.0305257597×107 K due to the temperature function considering the Sun as
an isothermal gas sphere. The mass, density, and pressure of the Sun at different radii from the center
of the Sun to its surface are shown in a Table. The validity of the HAN-method solution is proven
compared to other solutions that have been published before. These values are consistent with the
values obtained from experimental models to describe the dynamics of the Sun.
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