2. Consider a metal sphere of radius R and heat capacity C, initially at a temperature To which is much hotter than the background temperature.
a) Derive an analytical result for the temperature of such a sphere as a function of time. Clearly state any simplifying assumptions.
b) Early models to calculate the age of the Earth presumed that the Earth had the heat capacity of iron3 and started out at a temperature of T ∼ 4000 K4. Using these assumptions, how long would it take Earth to cool to its current surface temperature of Tsurf ≃ 280K? Make a reasonable speculation about why this estimate fails so spectacularly.
c) For a sufficiently hot object, obeying the same assumptions as part (a) and the initial problem statement, what is the time scale of cooling to a background temperature T1, independent of the initial temperature?
I am only asking for help with part a)
dU/dt = σA(Tbackground4 - Tsphere4)
C = dU/dT
dT/dt = (σA/C)(Tbackground4 - Tsphere4)
The Attempt at a Solution
This appears to be a separable differential equation, but not an easy one to solve. It is fairly straight forward to non-dimensionalize this equation and by assuming Tsphere/Tbackground >> 1.
But this leads to results which seem like they can't possibly be correct;
Tsphere = Tbackground2(σA/3tC)1/3
I suppose there should also be an integration constant as well, but ignoring that for now;
I can't see how this result could be used to solve part b), attempting to do so results in a much too small time.
This equation is also undefined at infinity, possibly a result of the assumption I mentioned above?
I would appreciate it if someone could suggest a coherent approach to this problem as I have been struggling with it for a while now