Density distrubution and solar lifetime of the sun

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SUMMARY

The discussion focuses on the implications of the sun's internal density distribution on its potential energy and solar lifetime. It establishes that if the sun were a hollow shell, its potential energy would be less than that of a uniform density case, leading to a shorter solar lifetime. Conversely, if most of the mass were concentrated in the center, this configuration would significantly impact calculations regarding the sun's longevity. The key takeaway is that the density distribution directly influences the potential energy and, consequently, the solar lifetime.

PREREQUISITES
  • Understanding of gravitational potential energy, specifically the equation GM^2 / r.
  • Familiarity with concepts of density distribution in astrophysics.
  • Basic knowledge of the structure of the sun, including the photosphere and core.
  • Awareness of how mass distribution affects stellar lifetimes.
NEXT STEPS
  • Research the implications of density distribution on stellar evolution.
  • Study the gravitational potential energy in non-uniform spheres.
  • Learn about the solar structure and its layers, focusing on mass distribution.
  • Explore models of stellar lifetimes and how they relate to mass and density.
USEFUL FOR

Astronomy students, astrophysicists, and anyone interested in understanding the factors influencing stellar lifetimes and the internal structure of stars.

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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.
 
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