Heat transfer in the planet is by conduction only in the lithosphere, which extends from r = b to r = a. For 0 ≤ r ≤ b heat transfer is by convection, which maintains the temperature gradient dT/dr constant at the adiabatic value −Γ. The surface temperature is T0. To solve for T (r), you need to assume that T and the heat flux are continuous at r = b.
Problem 4.16 It is assumed that a constant density planetary body of radius a has a core of radius b. There is uniform heat production in the core but no heat production outside the core. Determine the temperature at the center of the body in terms of a, b, k, T0 (the surface temperature), and q0 (the surface heat flow).
The Moon is a relatively small planetary body so it is a good approximation to assume that its density is constant. If we also assume that the Moon is in a steady-state thermal balance and that the mean heat production is the same
as the value we derived for the Earth’s mantle, that is, H = 7.38× 10−12 W kg−1, we can determine the surface heat flow on the Moon using Equation (4–43). With ρ = 3300 kg m−3 and a = 1738 km we find that q0 = 14.1 mW m−2. The mean of two lunar heat flow measurements on Apollos 15 and 17 is qs = 18 mW m−2. This approximate agreement suggests that the mean lunar abundances of the radioactive isotopes are near those of the Earth.
The difference may be partially attributable to the cooling of the Moon. Assuming that the conduction solution is applicable and that the Moon has a uniform distribution of radioactivity, the maximum temperature at the center of the Moon can be obtained from Equation (4–42) with the result Tmax = 3904 K, assuming k = 3.3 W m−1 K−1 and that the surface temperature is T0 = 250 K. This conduction solution indicates that a substantial fraction of the interior of the Moon is totally melted. However, the limited seismic results from the Apollo missions suggest that a sizable liquid core in the Moon is unlikely. Thus, either the conductive solution is not valid or the radioactive isotopes are not distributed uniformly throughout the Moon. There should be some upward concentration of radioactive isotopes in the relatively thick lunar highland crust (60 km) by analogy with the upward concentration of radioactive isotopes in the Earth’s continental crust. Problem 4.17 Determine the steady-state conduction temperature profile for a spherical model of the Moon in which all the radioactivity is confined to an outer shell whose radii are b and a (a is the lunar radius). In the outer shell H is uniform.