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