Derivation of the sun constant formula (using the elliptical orbit of the earth around the sun) In a book, I've to read for a class at my university, they gave a formula that looks like:
$$\text{G}_\text{sc}\left(\text{n}\right)=1367\cdot\left(1+0.03\cos\left(\frac{360}{365}\cdot\text{n}\right)\right)\space\space\space\space\space\space\space\space\space\left[\text{W}/\text{m}^2\right]\tag1$$
Where $\text{n}$ is a day in a year.
That formula is called the sun constant formula, but I do not know where it comes from. 

Question: Can someone help me derive this formula?


My work:
I think that the energy flux is given by:
$$\text{G}=\mathcal{k}\cdot\text{T}^4\cdot\left(\frac{\text{R}}{\text{D}}\right)^2\tag2$$
Where $\mathcal{k}$ is the Boltzmann constant, $\text{T}$ is the surface temprature of the sun, $\text{R}$ is the radius of the sun and $\text{D}$ is the distance from the sun to the earth.
Now, for $\text{D}$ we know that is changes over a year because the earth makes a elliptical orbit around the sun.
 A: Let us put a few things together. By Kepler's first law the distance of the Earth from the Sun is given by
$$ D(\theta) = \frac{p}{1+\varepsilon\cos\theta} $$
with $\varepsilon$ being the eccentricity of Earth's orbit, $p$ being the semi-latus rectum and $\theta$ being the angle perihelion-Sun-Earth. The way $\theta$ changes over time depends on Kepler's second law, giving that between day $0$ (Earth at the perihelion) and day $A$ (assuming the year has exactly $365$ days) the identity
$$ \frac{A}{365}\int_{0}^{2\pi}\frac{1}{2}\left(\frac{p}{1+\varepsilon\cos\theta}\right)^2\,d\theta = \int_{0}^{\theta_A}\frac{1}{2}\left(\frac{p}{1+\varepsilon\cos\theta}\right)^2\,d\theta$$
holds. That gives Kepler's equation: in order to find $\theta$ at a given time of the year, we have to solve a trascendental equation (through Newton's method, for instance). However, since the orbit of the Earth has a low eccentricity, we may assume that the Earth has a constant angular speed with respect to the Sun, leading to $\theta_A = \frac{A}{365}\cdot 360^\circ$. The energy emitted by the Sun is proportional to $T^4$ by Boltzmann's law, hence the flux of such energy through a sphere with radius $D$ is proportional to $\frac{1}{D^2}$, i.e. to
$$ \left(1+\varepsilon \cos\theta\right)^2 = 1+ 2\varepsilon\cos\theta + \varepsilon^2 \cos^2\theta $$
where the $\varepsilon^2\cos^2\theta$ term can be neglected by the same reason as before: $\varepsilon$ is pretty small.
