Meteorology - radiative equilibrium timescale

Sojourner01

1. The problem statement, all variables and given/known data

Derive the radiative timescale for an atmosphere:

[tex]
\tau_{E} = \frac{c_{p} p_{0}}{4 g \sigma T^{3}_{E}}
[/tex]

2. Relevant equations

As above

3. The attempt at a solution

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

And the incoming solar flux on the object, [tex](1 - \sigma) S[/tex], is equal to an instantaneous rate of change of heat, [tex]\frac{dQ}{dt}[/tex]. 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 [tex](T + \Delta T)^{4}[/tex] as [tex]T^{4}(1 + \frac{4 \Delta T}{T})[/tex] using the first order binomial expansion.

Silly method, but there we go.