So I'm writing a paper for a math class on Kepler's equation, and I've ran into a snag on deriving the equation. I've been mostly following the book Solving Kepler's Equations by Peter Colwell. I think the problem may be fairly simple, but I have searched and searched and couldn't find anything.
So here's the problem. given the following image
Let $a$ and $b$ be the lengths of the semi-major axis and the semi minor axis, respectively, of the ellipse with focus $S$. The author says that the ratio $b/a = PR/QR$. It seems simple but I can't figure out why that is. I'm trying find the justification here.
With the above relationship being shown, it is used to directly show that $rsinv = bsinE$, which is pretty straight forward. I just can't figure out why the equality between the two ratios is true. Any help on this would be greatly appreciated.
I found a mention of the fact here: http://science.larouchepac.com/kepler/newastronomy/
But that source didn't give any justification either.
Best Answer
That outer circle is clearly the auxiliary circle of that ellipse, so the $\angle QCR$ is obviously the eccentric angle of point $P$. In the right triangle $\triangle QCR$, use simple trig to get $QR=a \cdot \sin\angle QCR$. From the parametrization of the ellipse $(a\cos\theta,b\sin\theta)$ it follows that the ordinate of point $P$ is $PR=b\sin\angle QCR$.
From there it is not a big leap to conclude that $\dfrac{PR}{QR}=\dfrac{b}{a}$