I'm dealing with a confusing problem related to radiative transfer in atmospheres. In this problem, the solar flux on a planet it being modeled by: $S(t)=S_0+\sum_{n=1}^{\infty}S_n\,e^{in\omega t}$, and temperature modeled by $T(t)=T_0+\sum_{n=1}^{\infty}T_n\,e^{in\omega t-\phi}$

My question is a math one, not a thermodynamics one. By using the relation $\sigma\,T^4(t)=(1-A)\,S(t)$ (I think), I'm supposed to be able to find the amplitude of the fourier coefficient $T_n$ to be $T_n=\frac{\frac{1}{4}(S_n/S_0)T_0}{\sqrt{1+tan^2(\phi)}}$.

Working backwards, I've reduced this to $\frac{T_n}{T_0}=\frac{1}{4}\frac{S_n}{S_0}cos(\phi)$, but I'm not sure what else I can do.

Potentially useful relation: $tan(\phi)=n\omega\tau,\ \tau=C_PP_0/4\sigma T_0^3g$.

  • $\begingroup$ What is $A$ and $\sigma$ in your question? $\endgroup$ – Jacky Chong Apr 18 '18 at 2:59
  • $\begingroup$ Sorry, $A$ is albedo (unitless) and $\sigma$ is the Stefan–Boltzmann constant. I'm not how they disappear from the solution. $\endgroup$ – Spuds Apr 18 '18 at 3:02
  • $\begingroup$ Do you have an expression for $A$? $\endgroup$ – Jacky Chong Apr 18 '18 at 3:04
  • $\begingroup$ Not in this context, it's just a constant. $\endgroup$ – Spuds Apr 18 '18 at 3:13
  • $\begingroup$ Could you provide some reference to the problem? As it is I don't know how you could get rid of $\sigma$ and $A$. $\endgroup$ – Jacky Chong Apr 18 '18 at 3:52

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