Density of a Star: Calculating Mass & Average Density

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Homework Help Overview

The discussion revolves around calculating the mass and average density of a star, given a density function that varies with radial distance from the center to the surface. The original poster presents a formula for density and attempts to relate mass and volume using a constant density approach, which raises questions about the validity of this method in a non-uniform density context.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants discuss the need to integrate to account for the varying density throughout the star's volume, rather than applying a constant density formula. There is a suggestion to consider the volume of a shell and the use of calculus to derive the mass function.

Discussion Status

The discussion is active, with participants providing guidance on the necessity of integration and the correct approach to calculating mass in the context of a non-uniform density. There is acknowledgment of the original poster's misunderstanding, and some participants are exploring the implications of the density function on the calculations.

Contextual Notes

There is a mention of a specific textbook reference, which may indicate a shared resource among participants, but the exact average density value is not provided in the posts.

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Homework Statement


For a star of mass M and radius R, the density increases from the centre to the surface as a function of radial distance r, according to

[tex]\rho = \rho_{c}[1-(\frac{r}{R})^2][/tex]


where is the central density constant.

a) Find M(r).
b) Derive the relation between M and R and show that the average density of the star is .


i know that D=M/V
so M = D.V ; i simply substiuted the Given Density and the volume by 4Pi/3r^3
but somethin is wrong!


Homework Equations





The Attempt at a Solution

 
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D=M/V only when the density is constant over the entire volume, or when you're trying to compute an average density. In this case, the density is not constant over the entire volume, so you can't just figure out M from M=D*V. You need to account for the fact that D is different in different parts of the volume. How can you do that? (Hint: calculus is required)

Also, I can't see some parts of your post. What is the average density supposed to be?
 
right! i have to integrate over dr... and i should take the volume of the shell . the thank you for your help : )
 
For part a, you need the integral

[tex] m(r)=\int_0^r 4\pi r^2\rho(r)\,dr[/tex]

Then the total mass in part b, [itex]M[/itex], comes from using [itex]r=R[/itex] in the result of part a.

This is from Dina Prialnik's Stellar Astrophysics textbook, right?
 

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