Lord Kelvin used the heat flow at the surface of the Earth to argue that the Earth was 100 million years old withing a factor of 4 error.
a.) Reproduce his logic by deriving the temperature gradient at the surface of the Earth for a planet that is cooling by conduction. Model the planet as an infinite half space.
b.) for this part we will infer the age of the earth using kelvin's measurements. I feel fine doing this but am stuck on part a.)
for all eqns: T = temp, t = time, k = kappa (thermal diffusivity), x is a variable used to define erf function, and z = positive in downward direction towards T0 (i.e. towards center of earth)
T = T0 erf(z/(kt).5), t > 0
Where "erf" is the error function and defined as:
Erf(x) = (2 / (pi)^.5) * integral( e(-(x^2)) dx) from 0 to x
The Attempt at a Solution
First I set these boundary conditions:
At t = 0, T = T0 everywhere
At z = 0, T = T1
And as z goes to infinity, T = T0 everywhere
Next, by substituting z / [(kt)^.5] into erf i got:
T = T0 * (2 / (pi)^.5) * [integral of (e^ -(z/root(kt))2) d(-z/root(kt))] from 0 to [z/root(kt)]
Then i attempted to take dT / dz and got very confused with the calculus.
Any ideas on how to go about solving this problem?