A question asks for me to show that the minimum pressure of a star is given by:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]P_{min}=\frac{GM^2}{8\pi R^4}[/tex]

where M and R are the mass and radius of the star.

My answer goes like this:

Gravitational force towards center = [tex]\frac{GM(r)\delta M(r)}{r^2}[/tex]

Pressure force outwards = [tex](P(r)-P(r+\delta r))\delta A = \delta P (r)\delta A[/tex]

[tex]\delta M(r) = \delta A\delta r \rho (r)[/tex]

In equilibrium Gravitational up + Pressure down = 0

:. [tex]\frac{GM(r) \delta A \delta r \rho (r)}{r^2} + \delta P (r)\deltaA = 0[/tex]

[tex]\delta P(r) = - \frac{GM(r) \rho (r) \delta r}{r^2}[/tex]

as [tex]\rho (r) = \frac{M(r)}{\frac{4}{3}\pi r^3}[/tex]

:. [tex]\delta P(r) = \frac{-GM(r)M(r) \delta r}{\frac{4}{3} \pi r^5}[/tex]

after reducing [tex]\delta[/tex] to zero and integrating, I end up with:

[tex]P(r) = \frac{3G(M(r))^2}{16\pi r^4}[/tex]

The minimum pressure will be when at r=R, so the variables will check out, but I still have incorrect constants. Can you spot anywhere I've gone wrong, or an incorrect assumption in this derivation?

Thanks guys

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# Homework Help: Deriving the hydrostatic equilibrium equation.

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