Question

4. Consider a planet of radius R in which the density decreases linearly from center to edge, vanishing at the edge: This should look familiar: we calculated its mass (MToR/3) and moment of inertia (1 = 4MR2/15) in lecture. Its contents are unit mass Л and thermal conductivity taking its surface temperature to be T, and its central temperature to be finite. characterized by radioactive heating power per KT. Derive a formula for its internal temperature T(r), . Consider a simple differentiated spherical planet with mass M, radius R, uniform-density core with density Pe and radius Re 2R/3, and a uniform-density mantle and crust with density m2pe/5 for r - Rc to R. You have already shown that the mass M and core density Pe are related by pe 133M/104T R3, and that its moment of inertia is I-0.33MR2, so that its internal structure may be much like Earth's a) The core is also characterized by the radioactive heating rate and thermal conductivity Ke. Find a formula for the heat flux fr(Re) at the core-mantle boundary (b) The planet 's mantle has thermalconductivity Kom and the same radioactive heating power per unit mass as the core. The planet's surface temperature is T, . Use this boundary condition, and the boundary condition found in part a, to show that the temperature in the planet's mantle is given by 6km

Answer 1