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1. The problem statement, all variables and given/known data

A pressure of [itex] \textrm{2} \cdot \textrm{10}^{9} [/itex] Pa on a time scale of [itex] \textrm{10}^{9} [/itex] yr, holds half way to the center of the "critical" objects.

Estimate the diameter using two densities: [itex] \rho = 3 \textrm{g}/ \textrm{cm} ^{3} [/itex] and [itex] \rho = 1 \textrm{g}/ \textrm{cm} ^{3} [/itex].

2. Relevant equations

I was told to use the equation for hydrostatic equilibrium

[tex] \frac{dp_}{dr} = -\frac{GM_{r}\rho}{{r}^{2}} [/tex]

where

[tex] M_{r} = \frac{4\pi {r}^{3}\rho}{3} [/tex]

3. The attempt at a solution

I chose2ras the radius for the entire object, thenris the radius half way.

I then inserted [itex]M_{r}[/itex] into the equation for hydrostatic equilibrium and integrated the new equation

[tex]p = \int_{2r}^r -\frac{{4\pi G} \rho^{2}}{3}r^\prime\, dr^\prime = \frac{{4\pi G} \rho^{2}}{3} ((2r)^{2}-r^{2}) = 2 \pi G \rho^{2} r^{2}[/tex]

Then I just inserted all the constants and solved the equation forr.

[tex]\rho = 3 \textrm{g}/ \textrm{cm} ^{3} \Rightarrow r \approx 728,183 km \Rightarrow d=4r \approx 3000km [/tex]

and

[tex]\rho = 1 \textrm{g}/ \textrm{cm} ^{3} \Rightarrow r \approx 2184,550 km \Rightarrow d=4r \approx 8740 km [/tex]

I was told that I should get a diameter between 500-600 km. Obviously I'm no where near that, so I really need help with this because I have to present a solution on October 18th.

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# Homework Help: Diameter of critical object

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