Meteorology - radiative equilibrium timescale

[Sojourner01]

Homework Statement

Derive the radiative timescale for an atmosphere:

$$\tau_{E} = \frac{c_{p} p_{0}}{4 g \sigma T^{3}_{E}}$$

Homework Equations

As above

The Attempt at a Solution

I've gathered that the difference between the radiative power of an object,
$$\sigma (T + \Delta T)^{4}$$

And the incoming solar flux on the object, $$(1 - \sigma) S$$, is equal to an instantaneous rate of change of heat, $$\frac{dQ}{dt}$$. I don't know how to proceed from here; my derivation of the answer doesn't appear to conform to the one above.

edit: oh for crying out loud, I hate TeX. It never does what I want it to, and I have the 'how to program tex' thread open here in front of me. You can see what I was trying to achieve.

 

[Sojourner01]

Well, I've sorted it out in case anyone is interested. Turns out you can factor out the bracket $$(T + \Delta T)^{4}$$ as $$T^{4}(1 + \frac{4 \Delta T}{T})$$ using the first order binomial expansion.

Silly method, but there we go.