Density distrubution and solar lifetime of the sun

[zeion]

Homework Statement

Kelvin could only see the photosphere (the glowing outer surface) of the sun, so didn't know how the mass was distributed inside
it. How would the potential energy of the sun change if it was a hollow shell? (No numbers needed; just indicate whether it
would be more or less than in the uniform density case). Similarly, what if the outer part is just fluff, so that most of the mass is
in the center? Which of these two cases makes a bigger difference to our calculation for the lifetime of the sun?

Homework Equations

Potential energy = GM^2 / r

There is a factor in this equation that depends on the density distribution within the
object (e.g. 0.6 for a uniform sphere) - but for rough calculations we can ignore that.

The Attempt at a Solution

I don't understand how this factor of density distribution fits into the equation.
But I'm assuming that if the sun was hollow it would significantly lower it's solar lifetime since the photosphere would have to radiate inside the shell as well thus depleting its potential energy quicker.

 

[zeion]

No one? :(

 

[Blueshift5]

On the other hand, if the outer part is just fluff, the potential energy would increase due to the concentration of mass in the center, potentially prolonging the solar lifetime. However, the difference between these two cases would depend on the actual density distribution within the sun, as the denser the center is, the more significant the change in potential energy will be. Therefore, it is difficult to determine which case would have a bigger impact on the calculation for the solar lifetime. Further research and analysis would be needed to accurately determine the effects of different density distributions on the potential energy and solar lifetime of the sun.