Ge/Ay133 – Problem Set #6
Revenge of the (Geo)Chemists
Due November 17
th
(1) This problem is to help you think about the thermal history of bodies that are assembled
in the early solar system.
Information of this sort is important when thinking about the core
instability model of Jovian planet formation and also about comets, asteroids and the differentiation
of planetesimals and/or oligarchs.
(a) Show that the gravitational potential energy of a spherical body of uniform density is given by

3
G
M
2
5R
.
Assuming a constant
ρ
and
C
p
, and given no energy loss from the system, use this fact to generate
an equation for the temperature of a body in terms of these constants. What does this relationship
predict for the temperature for the moon and for the earth due to accretional energy alone? Why
is this an upper limit to the temperature? For both, use a specific heat of
C
p
= 10
7
erg/g/K, and
use densities of
ρ
= 3.3, 5.5 g cm

3
for the Moon and Earth.
(b) In reality, only a fraction of the accretional energy is trapped as heat in the growing body. Let’s
assume this efficiency of trapping,
ǫ
, is something like 2.5%. Recalculate the equations from (a),
what temperatures do you derive? The largest asteroid, Ceres, has a radius of 487 km. Assuming
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 Fall '10
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 Solar System, Meteorite

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