Spherical star in a hydrostatic equilibrium

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Discussion Overview

The discussion revolves around deriving an equation of motion for a perturbation in the context of a star in hydrostatic equilibrium. Participants are examining the momentum equation and its application to perturbations, with references to specific exercises and solutions.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant seeks guidance on deriving an equation of motion for perturbations, expressing uncertainty about their own solution.
  • Another participant clarifies the distinction between "d" and "delta," emphasizing the need to derive the momentum equation for delta variables before substituting them into the equations.
  • A participant recalls a previous attempt at the problem, suggesting not to follow the provided hints and referencing their own approach.
  • Another participant expresses concern that a previous solution is incorrect due to a misunderstanding of the relationship between density and mass, advocating for a different approach involving linearization of perturbations.
  • A later reply acknowledges the contributions of others without providing further clarification or resolution.

Areas of Agreement / Disagreement

Participants exhibit disagreement regarding the correct approach to deriving the equation of motion, with multiple competing views on how to handle perturbations and the relationships between variables.

Contextual Notes

There are unresolved mathematical steps and assumptions regarding the treatment of perturbations and the definitions of variables involved in the momentum equation.

ted1986
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Hello again,

I've got a question about a star in a hydrostatic equilibrium.
How do I derive an equation of motion for a pertubation in the full momentum equation? I'm attaching my solution (my_solution.jpg) , but I'm not quite sure about it.

The full exercise is attached as astro_problem.jpg.

Thank you.
 

Attachments

  • astro_problem.jpg
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  • my_solution.jpg
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It's not that simple, the meaning of "d" and "delta" are different. The meaning of "d" is "a change as you change location", but the meaning of "delta" is "perturbed from the original equation." So before you substitute and delta expressions, you first have to find the momentum equation that applies to the delta variables. When you're all done, you'll still have d/dr kinds of things, but they will apply to the delta variables, not the P and rho by themselves.
 
I think I did this once. I even thought it was my idea. Don't use their hints, see attachment. The "del" works like del f = f' del r.
 

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  • dada.jpg
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Helios said:
I think I did this once. I even thought it was my idea. Don't use their hints, see attachment. The "del" works like del f = f' del r.



OK, I tried to solve the exercise as you said (P=K*rho^\gamma), but the equaion I've got seems to be to complicated... (my derivation is attached - star_my_sol2.jpg)

Perhaps the derivation needed to solve it is less complicated?

Thank you.
 

Attachments

  • star_my_sol2.jpg
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Sorry, but that's way off because the way you related density and mass. Your equation would only work for mean density, were M the total mass. Their relationship is instead differential.
The knack here is to apply the variation ( perturb ) and then factor out ( linearize ) the del-r out. Since the variation is arbitrary, the parenthetical stuff must equal zero ( the derived equation ).
I don't get the hints they gave.
 

Attachments

  • dada02.jpg
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Thank you for your efforts :)
 

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