Problem Ge/Ay133 Set #7 Due February 26th , 2009 (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 coreinstability model of Jovian planet formation and also about comets, asteroids and the dierentiation of planetesimals and/or oligarchs. (a) Show that the gravitational potential energy of a spherical body of uniform density is given by 3GM2 . 5R
Assuming a constant and Cp , 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 specic heat of C p = 107 erg/g/K, and use densities of = 3.3, 5.5 g cm3 for the Moon and Earth. (b) In reality, only a fraction of the accretional energy is trapped as heat in the growing body. Lets assume this eciency of trapping, , 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 a uniform density of = 3 g cm3 , what temperature would you expect for Ceres shortly after accretion? If silicates begin to melt at a temperature of 1500 K, would Ceres be dierentiated? The Moon? The Earth? (c) Now for a little (nuclear) chemistry... Chondritic meteorites are the only direct samples we have of relatively unaltered rocks from the early history of the solar system. A couple of lines of evidence suggests that the parent bodies of these meteorites were at least partially molten: igneous textures and the separation of silicates from metallic phases. The accretional energy analysis above suggests that these bodies should not be dierentiated, and there is also insucient energy from the long lived radionuclides that supply much of the internal heat of the present day earth ( 40 K, 235/238 U, and 232 Th). In 1955, Harold Urey suggested that the beta decay of the short-lived isotope 26 Al could produce sucient heat. The reaction is:
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Al 26 Mg
,
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with a half-life of 750,000 years. If a meteorite contained some 26 Al remains in the meteorite today, 4.5 AE later?
Al initially, what percent of the
(d) Below is a table of published data on the isotopic composition of minerals in chondrule from the Allende meteorite: Anorthite Phase Anorthite Melilite Spinel Fassaite
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Formula CaAl2 Si2 O8 CaAl2 Si2 O8 Ca2 (Mg,Al,Si)3 O7 MgAl2 O4 Ca(Mg,Al,Ti)(Si,Al)2 O6
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Al/24 Mg 245 128 9.1 2.5 2.0
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Mg/24 Mg 0.1517 0.1468 0.1404 0.1398 0.1398
Al, 26 Mg, and 24 Mg are stable isotopes. Neither 27 Al nor 24 Mg are decay products. Plot Mg/24 Mg versus 27 Al/24 Mg and draw a least-squares line through the data points. Calculate the initial 26 Al/27 Al and initial 26 Mg/24 Mg of the chondrule.
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(e) Using this initial isotopic ratio (of 26 Al/27 Al) of the chondrule, write an expression for the rate of energy release per gram of meteorite due to the 26 Al decay as a function of time. The energy released per decay of an 26 Al atom is 3.3 MeV. The average abundance of Al in a chondrite is 0.868% by mass and the only isotopes of Al initially present are 26 Al and 27 Al. Plot your results. (f) Finally, lets do a slightly more realistic temperature calculation including radioactive heating. Assume that the asteroid is chemically and isotopically homogeneous with a density of 3.7 g/cm 3 , and that the outward energy ux as a function of radius is F = k(dT /dr) (that is, is proportional to the radial temperature gradient), where the thermal conductivity k equals 3.25 J/smK. First, write an ordinary dierential equation that relates the temperature to radius (in one dimension). Assume the outer boundary condition is the equilibrium temperature for a perfect blackbody radiating away to free ...