Given area of sector and a starting angle from focus of an ellipse, finding angle needed to get area.

Question

Problem Background:

I'm trying to make a rough simulation of Kepler's second law (equal areas over equal time) and to do this I've divided the area of the ellipse into some number of pieces. I want to figure out how much I need to increase the angle by so that it sweeps an arc which forms a sector that has an area equal the slice area (which is (Area of Ellipse/n)

I am representing this ellipse with the polar equation for an ellipse from:

http://mathworld.wolfram.com/Ellipse.html

where e is the eccentricity of the ellipse.

This question:

How to calculate ellipse sector area *from a focus*

describes solving the area given the angle, how do I approach this if I am trying to solve for change in theta while knowing the area that I want to solve for?

Answer 1

I just wrote a new answer to the question you referenced. That post describes how you can get from true anomaly $\theta$ via eccentric anomaly $E$ to mean anomaly $M$ which is proportional to both area and time.

To find the change in true anomaly $\theta$ corresponding to a given change in area, you'd first turn the original angle $\theta_1$ into some mean anomaly $M_1$, then you'd add to that and try to reverse the process, i.e. compute $\theta_2$ from $M_2$. But the relation between $M$ and $E$ is Kepler's equation:

$$M=E-e\sin E$$

This is a transcendental equation. Wikipedia writes:

This equation does not have a closed-form solution for $E$ given $M$. It is usually solved by numerical methods, e.g. Newton-Raphson method.

The lemma on Kepler's equation has a whole section on this inverse problem, followed by one about numeric approximations. So I guess that would be the way to go.