### 1 Star and Planet Formation spring 2014: Problem Set 3 1. Thermal

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Star and Planet Formation spring 2014: Problem Set 3
1. Thermal History of Early Solar System Bodies. Adapted from Dr. Geoff Blake
and TA Michael Busch fall 2009 course on The Formation and Evolution of Planetary
Systems at Caltech.
Consider a growing spherical body of mass M , radius R and uniform density ρ.
(a) Assuming a constant specific heat capacity Cp = 107 erg g−1 K−1 and given that there
is no energy loss from the system, derive an equation for the change in temperature of
the body due to accretion alone. What does this equation predict for the temperature
of the Moon and the Earth? Why is this an upper limit for the temperature?
(b) In reality, only a fraction of the energy due to accretion is trapped as heat in the
growing body. Assume that the efficiency of this trapping, , is approximately 2.5%.
Recalculate the temperatures for the Moon and the Earth, taking this into account.
The largest asteroid in the Solar System, Ceres, has a radius of 487 km. Assuming
a uniform density, what temperature would you expect for Ceres upon accretion? If
silicates begin to melt at a temperature of 1500 K, would Ceres be differentiated?
Would the Moon and/or the Earth be differentiated?
(c) Chondritic meteorites are the only direct samples we have of relatively unaltered rocks
from the early history of the Solar System. The parent bodies of these meteorites
were at least partially molten. In 1955, Harold Urey suggested that the decay of
the short-lived isotope 26 Al could produce sufficient heat. The decay of 26 Al occurs
either via β + decay or electron capture, both modes resulting in the stable nuclide
26 Mg. The half-life of 26 Al is 7.2×105 yr. If a meteorite contained some 26 Al initially,
what percentage of it remains in the meteorite today, 4.5 gigayears later?
(d) Below is a table of published data on the isotopic composition of minerals in chondrules from the Allende meteorite. Plot 26 Mg/24 Mg versus 27 Al/24 Mg and make
a linear least squares fit to the data. Calculate the initial 26 Al/27 Al and initial
26 Mg/24 Mg abundances of the chondrules.
Sample
27 Al/24 Mg
26 Mg/24 Mg
Anorthite-1
Anorthite-2
Melilite
Spinel
Fassaite
245
128
9.1
2.5
2.0
0.1517
0.1468
0.1404
0.1398
0.1398
(e) Assume that the energy lost to the by-products of decay is negligible. Use the initial
isotopic ratio of 26 Al/27 Al of the chondrules computed in the previous sub-question
and that the energy per decay of 26 Al, γ, is 3.3MeV. Also use that the average
abundance of Al in a chondrite is 0.868% by mass, while assuming that the only
isotopes of Al present are 26 Al and 27 Al. Write an expression for the rate of energy
release per gram of meteorite due to the 26 Al decay as a function of time and plot
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(f) Finally, let us do a slightly more realistic temperature calculation including radiogenic heating. Assume that the asteroid is chemically and isotopically homogeneous
with a density of 3.7 g cm−3 , and that the outward energy flux as a function of radius
is given by F = −kdT /dr, i.e. it is directly proportional to the radial temperature
gradient where k = 3.25 J s−1 m−1 K−1 is the thermal conductivity. Write an ordinary differential equation that relates the temperature to radius (in one dimension).
Assume that the outer boundary condition is the equilibrium temperature for a perfect blackbody radiating away to free space. Use the initial heat production (energy
per mass per unit time) calculated above to estimate the temperature at the center
of an asteroid as a function of its radius. Plot your results. If the melting point at
the center of the asteroid is 1500 K, what is the smallest radius at which the asteroid
melts at the center? Estimate the temperature versus radius for an asteroid given a 1
million year and a 10 million year intervals for 26 Al decay (that is, assume a window
between the initial injection of live 26 Al and the bulk accretion of 1 Myr as one test
case and then a delay of 10 Myr as a second situation). For these cases, what is the
minimum radius of an asteroid that melts at the center?
The evidence is that planetesimals somewhat smaller than 100 km experienced partial melting. What does this tell you about the timescale for the assembly of such
bodies in the Solar Nebula?
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