How Does Hydrostatic Equilibrium Apply to a Polytropic Star?

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SUMMARY

This discussion focuses on calculating hydrostatic equilibrium for a spherically symmetric star using a polytropic equation of state. The key equation referenced is the Lane-Emden equation, which relates pressure and gravity in stellar structures. The participants emphasize the importance of selecting a specific polytropic index (n) and clarify that pressure and gravity are only equal at the star's surface, with pressure remaining non-zero at the center despite zero gravity. Understanding these concepts is crucial for solving problems related to stellar structure.

PREREQUISITES
  • Understanding of hydrostatic equilibrium in astrophysics
  • Familiarity with polytropic equations of state
  • Knowledge of the Lane-Emden equation
  • Basic concepts of gravity and pressure in stellar contexts
NEXT STEPS
  • Study the Lane-Emden equation in detail
  • Research different polytropic indices and their implications
  • Explore hydrostatic equilibrium conditions in stellar models
  • Learn about the relationship between pressure and gravity in astrophysical contexts
USEFUL FOR

Astronomy students, astrophysicists, and anyone studying stellar structure and hydrostatic equilibrium will benefit from this discussion.

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Homework Statement



I am working on a problem in which I need to calculate the hydrostatic equilibrium for a spherically symmetric star described by a polytropic equation of state.

There are two boundary conditions.



Homework Equations



All I know is gravity and pressure are going to be equal. This subject is [academically] new to me, so I don't really know where to start. Are there any other variables I need to consider? How would the lane-emden equation relate to this?



The Attempt at a Solution

 
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Wikipedia has the equation of state http://en.wikipedia.org/wiki/Lane–Emden_equation

This topic is new to me. But it seems you have to choose a polytropic specific index n. Solutions are for a few such indeces are given in the article.

Pressure and gravity are only necessarily equal at the surface. For instance, consider that at the center gravity will be zero but the pressure will definitely be non-zero.

That's all I can add. Good hunting.
 

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