1 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 this rate of radiogenic heating. 2 (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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