IGNOU PHE 15 2014 - What is meant by a polytropic change ? Startingwith the equations of

Question

What is meant by a polytropic change ? Starting
with the equations of hydrostatic equilibrium
and mass continuity, obtain Lane — Emden
equation for a polytropic index 'n'.

26 Jun 2018

Answer :

Word Count : 457

A polytropic change in astronomy and astrophysics refers to a process in which a gas undergoes a specific type of thermodynamic transformation, known as a polytropic process. A polytropic process is characterized by the equation \(P V^n = \text{constant}\), where \(P\) is the pressure, \(V\) is the volume, and \(n\) is the polytropic index. This equation is an extension of the more general form of the ideal gas law and is particularly useful in studying certain physical conditions in celestial bodies, such as stars.

To derive the Lane-Emden equation for a polytropic process, we begin with the equations of hydrostatic equilibrium and mass continuity. These equations describe the balance between the gravitational force trying to collapse a celestial object and the pressure force resisting the collapse.

1. Hydrostatic Equilibrium Equation:
The hydrostatic equilibrium equation is

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