# Temp Gradient derivation at Surface, earth as infinite half-space

## Homework Statement

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.)

## Homework Equations

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?

peace

Redbelly98
Staff Emeritus