Newton's Law of Cooling, age of Earth, weak math skills I'm curious about a problem concerning the age of the earth, but I don't have the math skills to think properly about it. I've found the solution to Newton's Law of Cooling, and I can handle that much:
$$T(t) = T_s + (T_0 - T_s)e^{-kt}$$
The part I'm stuck on is that I don't want to know the temperature of the whole earth. I want to know the temperature at the bottom of a deep mineshaft, and under two different conditions: (1) with Lord Kelvin's mistaken assumption that the interior of the earth is static and (2) with our current knowledge that the mantle convects.
I'm thinking that the way to approach condition (1) is to treat Earth as though it were made up of infinitely many concentric spheres, all infinitely thin, and then just graph the temperature at the bottom of the mineshaft. I'm too weak in calculus to figure out how to deal with the infinitely many spheres.
I'm thinking that the way to approach condition (2) is to treat the entire mantle as a single sphere that has a definite thickness and even temperature, among all the other infinitely thin spheres both inside it and out.
So I think I have the qualitative approach right, but I just don't have the math skills to consummate the deal. Can anyone show me the functions needed to graph the two cases over time?
 A: This is a rather difficult problem to solve, but one can think about it qualitatively, at least. As it turns out, it is just as hard to find the temperature distribution at one point of the earth as it is everywhere. 
Assume the earth is spherical and all the fields spherically symmetric: you'll still have two independent variables: distance from the center of the earth r and time t. The dependent variables are the temperature $T(r,t)$ and the phase of a slice at $r$ (liquid / solid) 
The physical processes that can occur are 
(1) heat conduction along r 
(2) convection, which is a toughie to simplify to 1D without introducing sources / sinks
(3) Phase change from liquid to solid as the earth cools, absorbing the heat of phase 
transformation 
(4) A nonlinear radiative boundary condition at the end r=Re (this is too important to 
ignore especially in the early stages of the earth)
(5) A source term if heat generation from radiation is included
(6) Shrinkage / expansion due to cooling
The initial conditions are the phase and initial temperature distribution of the earth at t=0. 
You'll need to make a bunch of assumptions and need data for the (average) thermal conductivity $K$, or $K(T)$, specific heat capacity $c_p(T)$, latent heat of melting, the radiative constant $\sigma$, coefficient of thermal expansion $\alpha$ and some equivalent heat transfer law for the convective term and the densities $\rho(T)$. 
You can formulate the above as a regular IBVP (initial-boundary value problem). Your solution will be T(r=r_{mantle}, t=4.5x10^9 years) :)
Of course, you could simplify things further. Ignore convection, for one thing. Or you could try to assume that the earth has been around for long enough that a steady-state approximation is good enough.
In any case, using Newton's law will not give you a very meaningful answer, I'm afraid.
Edited to add: The solid / liquid interface position $r_i$ will also arise as a result of the calculation. Of course, you will need different sets of properties for the different phases.
