Rotating ellipse

An ellipse $E$ is rotating about its center with a constant angular speed $\omega$ in the counter clockwise direction. However, during its rotation, its eccentricity, $e$ , is varying with time so as to satisfy the following criteria.

1. The foci of the ellipse $E$ trace out another fixed, concentric ellipse $E'$ whose eccentricity is $\displaystyle \frac{1}{2}$ .

2. The foci of the ellipse $E'$ always lie on the ellipse $E$ .

Evaluate the root mean square eccentricity of the ellipse $E$ i.e., $\displaystyle \sqrt{< e^2 > }$ .

Details and Assumptions:

• Both the ellipses are concentric as mentioned before. Moreover, the length of semi-major axis of both the ellipses is the same equal to $a = 5$ .

• The conditions are satisfied at all instants of time and hence the eccentricity is continuous function of time except when their axes are parallel.

• The average has to be taken over one complete rotation.

• The root mean square of continuous function $p(x)$ defined as $\displaystyle [ a , b ] \to \mathbb{R}$ is given as $\displaystyle \sqrt{< (p(x))^2 > } = \sqrt{\frac{\int \limits_a^b (p(x))^2 \text{ d}x }{\int \limits_a^b \text{ d}x }}$