1.3. The Binding Energy of the Sun The gravitational potential energy of a shell of mass dm, thickness dr, resting on a sphere of radius r is -GMspheredm dU where Msphere is the mass of the sphere. i) For a shell of constant density p, draw a diagram and explain the form of the expression dm 4Tr pdr ii) Write down an expression for Msphere in terms of p and r, and show that 16 m²Gp°r*dr 3 iii) Integrate this quantity to show that the total gravitational binding energy of a sphere of uniform density p, and radius R is 3GM? U = 5R Calculate U for the Sun. Given the Sun's luminosity, give an order of magnitude estimate of how long it would take for the Sun to convert all its binding energy into radiation (this is known as the Kelvin-Helmholtz timescale) Checkpoint 4: What did you get for the Kelvin-Helmholtz timescale? Is it larger or smaller than Checkpoint 2? How does it compare with the lifetime of the Sun?
1.3. The Binding Energy of the Sun The gravitational potential energy of a shell of mass dm, thickness dr, resting on a sphere of radius r is -GMspheredm dU where Msphere is the mass of the sphere. i) For a shell of constant density p, draw a diagram and explain the form of the expression dm 4Tr pdr ii) Write down an expression for Msphere in terms of p and r, and show that 16 m²Gp°r*dr 3 iii) Integrate this quantity to show that the total gravitational binding energy of a sphere of uniform density p, and radius R is 3GM? U = 5R Calculate U for the Sun. Given the Sun's luminosity, give an order of magnitude estimate of how long it would take for the Sun to convert all its binding energy into radiation (this is known as the Kelvin-Helmholtz timescale) Checkpoint 4: What did you get for the Kelvin-Helmholtz timescale? Is it larger or smaller than Checkpoint 2? How does it compare with the lifetime of the Sun?
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Transcribed Image Text:1.3. The Binding Energy of the Sun
The gravitational potential energy of a shell of mass dm, thickness dr, resting on a sphere of
radius r is
-GMapheredm
dU =
where Msphere is the mass of the sphere.
i) For a shell of constant density p, draw a diagram and explain the form of the expression
dm =
4Tr? pdr
ii) Write down an expression for Msphere in terms of p and r, and show that
16
nGp°r*dr
dU
iii) Integrate this quantity to show that the total gravitational binding energy of a sphere of uniform
density p, and radius R is
3GM?
U =
5R
Calculate U for the Sun. Given the Sun's luminosity, give an order of magnitude estimate of how
long it would take for the Sun to convert all its binding energy into radiation (this is known as the
Kelvin-Helmholtz timescale)
Checkpoint 4: What did you get for the Kelvin-Helmholtz timescale? Is it larger or smaller
than Checkpoint 2? How does it compare with the lifetime of the Sun?
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