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Meteorology - radiative equilibrium timescale

  1. May 7, 2007 #1
    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.
     
    Last edited: May 7, 2007
  2. jcsd
  3. May 9, 2007 #2
    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.
     
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