Ph 441/541
Problem Set 1
Due: Friday, February 3, 2012
1. Hydrostatic Equilibrium:
Consider a sphere of mass
M
and radius
R
. Calculate the gravitational potential energy of the
sphere assuming (a) a density which is independent of the distance from the center and (b) a
density which increases towards the cener according to
ρ
(
r
) =
ρ
c
(1
−
r/R
)
.
In both cases (a) and (b), also write down the average internal pressure needed for hydrostatic
equilibrium and determine how the pressure within the sphere depends on the distance from the
center.
2. Virial Theorem (Ph 441 students only):
As the Sun evolved towards the main sequence, it contracted due to gravity while remaining close
to hydrostatic equilibrium and its internal temperature changed from about 30,000 K, Phillips
Eq. (1.23), to about 6
×
10
6
K, Eq. (1.31). This stage of evolution is called the KelvinHelmholtz
stage. Find the total energy radiated during this contraction. Assume that the luminosity during
this contraction is comparable to the present luminosity of the Sun and estimate the time taken
to reach the main sequence.
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 Spring '12
 PRYOR
 Energy, Kinetic Energy, Mass, Potential Energy, Static Equilibrium, hydrostatic equilibrium

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