# Meteorology - radiative equilibrium timescale

by Sojourner01
 P: 372 1. The problem statement, all variables and given/known data Derive the radiative timescale for an atmosphere: $$\tau_{E} = \frac{c_{p} p_{0}}{4 g \sigma T^{3}_{E}}$$ 2. Relevant equations As above 3. 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.
 P: 372 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.