- #1
Irid
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Homework Statement
Given the density function [tex]\rho = \rho(r)[/tex] calculate the pressure at the center of a star.
Homework Equations
[tex]F = \frac{GMm}{r^2}[/tex]
[tex]P = \frac{\Delta F}{\Delta A}[/tex]
The Attempt at a Solution
Choose some radius [tex]r[/tex]. Then the gravitational attraction there is
[tex]\Delta F = \frac{GM(r) \Delta m}{r^2}[/tex]
and the resulting pressure is
[tex]P = \frac{\Delta F}{\Delta A} = \frac{GM(r)}{r^2} \frac{\Delta m}{\Delta A}[/tex].
We can interpret [tex]\Delta m[/tex] as the total mass above radius [tex]r[/tex] and [tex]A[/tex] as the area of the sphere at that radius. Then
[tex]P = \frac{GM(r) [M_0-M(r)]}{4\pi r^4}[/tex].
Near the center
[tex]\frac{M(0)}{4\pi r^3} \approx \rho_c/3[/tex]
and so
[tex]P(0) \approx \frac{G\rho _c M_0}{3r} \rightarrow \infty[/tex].
Where's my error?