Lane-Emden Eqn 5: Solving for Polytropic Index

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In summary, the Lane-Emden equation is a mathematical formula used in astrophysics to model the structure and behavior of self-gravitating, spherically symmetric, polytropic gas spheres. The polytropic index in the equation represents the compressibility of the gas and can be solved using various numerical and analytical techniques. Its applications in astrophysics include modeling the structure and evolution of stars, studying stellar dynamics, and formation of galaxies. However, the equation has limitations such as assuming certain conditions and only being valid for polytropic gases.
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Apashanka das
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Can any please suggest me about how to solve the lane emden equation for polytrophic index 5 ,
Any portal or journal
 
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1. What is the Lane-Emden equation?

The Lane-Emden equation is a mathematical formula that describes the structure and behavior of a self-gravitating, spherically symmetric, polytropic gas sphere. It is commonly used in astrophysics to model the structure of stars, as well as other celestial bodies.

2. What is the physical significance of the polytropic index in the Lane-Emden equation?

The polytropic index, denoted by n, represents the relationship between pressure and density in a polytropic gas. It is a measure of the compressibility of the gas, and it determines the overall behavior and stability of the system.

3. How is the Lane-Emden equation solved for the polytropic index?

The Lane-Emden equation can be solved using various numerical and analytical techniques, such as the shooting method, the finite difference method, and the Frobenius method. Each method involves making certain assumptions and approximations to arrive at a solution for the polytropic index.

4. What are the applications of the Lane-Emden equation in astrophysics?

The Lane-Emden equation is used to model the structure and evolution of stars, as well as other celestial objects such as white dwarfs, neutron stars, and black holes. It is also used in the study of stellar dynamics and the formation of galaxies.

5. What are the limitations of the Lane-Emden equation?

The Lane-Emden equation assumes a spherically symmetric, non-rotating, and non-magnetic gas sphere. It also does not take into account the effects of radiation pressure and nuclear reactions, which are important in the study of stars. Additionally, it is only valid for polytropic gases, which may not accurately represent the composition of certain celestial bodies.

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