# Finding amplitude of Fourier coefficient

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

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