# Earth-Sun distance equation

I was studying solar geometry and I read the next equation

$$r=r_{0}\left(1+0,017\sin\left[\frac{2\pi(d_{n}-93)}{365}\right]\right)$$

where $r_{0}$ is the average distance between the Earth and the Sun and 0.017 is the eccentricity of the earth orbit, $d_{n}$ is the days.

My question is how to proof the equation and why the argument of the sine. I suppose that is for properties of the ellipse (the orbit of the Earth is a ellipse) but doesn't work.

• What does the equation describe? The distance between the Sun and the Earth? – Neal Jan 4 '13 at 23:43
• Neal: Yes, DonAntonio: excuse me but you could explain more about that please – kEoz Jan 4 '13 at 23:51
• What's $d_n$ here? – Patrick Li Jan 5 '13 at 0:11
• days, for example dn=1 is the 1 january – kEoz Jan 5 '13 at 0:15
• – c.p. Jan 5 '13 at 0:21

The distance as a function of the angle from perihelion is $$r=\frac{a\left(1-e^2\right)}{1+e\cos(\theta)}$$ where $a$ is the semi-major axis, $e$ is the eccentricity of the orbit, and $\theta$ is the angle from perihelion.
The relation of $\theta$ to $t$, the time since perihelion passage, is $$\frac{2\pi t}{P}=\phi-e\sin(\phi)$$ and $$\tan\left(\frac\theta2\right)=\sqrt{\frac{1+e}{1-e}}\tan\left(\frac\phi2\right)$$ where $P$ is the period of revolution ($\phi$ is called the eccentric anomaly).
The formula cited in the question is a first-order deferent approximation assuming that the anomalistic year is 365 days and the earth passes perihelion when $d_n=1.75$.