In this problem you will estimate the amount of energy released per second by the radioactive decay of 1326Al inside Vesta during its lifetime.
(a) Vesta has a radius of 250 km and a density of 2900 kg m−3. Assuming spherical symmetry, estimate the asteroid's mass.
(b) Assume for the moment that the asteroid is composed entirely of silicon atoms. Estimate the total number of atoms inside Vesta. The mass of one silicon atom is approximately 28 u.
(c) The mass of 1326Al is 25.986892 u and the mass of 1226Mg is 25.982594 u. How much energy is released in the decay of one aluminum atom? Express your answer in joules.
(d) The ratio of 1326Al to all aluminum atoms formed in a supernova is about 5×10−5, and aluminum constitutes approximately 8680 ppm (parts per million) of the atoms in a chondritic meteorite. Assuming that these values apply to Vesta, estimate the number of 1326Al atoms originally present in the asteroid.
(e) Find an expression for the amount of energy released per second in the decay of 1326Al within Vesta as a function of time, and plot your results over the first 5×107 years on semilog graph paper. You may find Eq. (15.9) useful.
(f) How much time was required after the formation of Vesta before energy production due to the radioactive decay of 1326Al dropped to 1×1013 W, comparable to the current rate of energy output from the asteroid? See Problem 22.7.