Suppose we have an elliptical orbit with the sun at one focus $F_1$ -- See diagram. Let $F_1$ be the origin and let $\theta$ be the the angle between $F_1P$ and $F_1B$. Find $\theta$ as a function of time $t$.
My attempt:
According to Kepler's 2nd law, $$\frac{dA}{dt}=\frac{1}{2}r^2\frac{d\theta}{dt}=k\tag1$$
where $k$ is a constant.
We know that $\displaystyle r=\frac{p}{1-\epsilon\cos\theta}$ where $p$ is the semi-latus rectum and $\epsilon$ is the eccentricity of the ellipse.
Substituting into (1), we get
$$\frac{1}{2}\left(\frac{p}{1-\epsilon\cos\theta}\right)^2\frac{d\theta}{dt}=k$$
We can now integrate to find $\theta$ in terms of $t$.
Is this correct? Is there is a simpler way to do this that I'm missing?
