# As orbits get more "out of round" in shape, does eccentricity increase or decrease?

Mar 23, 2016

Increase

#### Explanation:

A circular orbit has $\text{eccentricity} = 0$.

An elliptical orbit has $0 < \text{eccentricity} < 1$.

A parabolic orbit has $\text{eccentricity} = 1$.

A hyperbolic orbit has $\text{eccentricity} > 1$.

Mar 23, 2016

We can see the orbit getting more out of shape with increase in eccentricity, by fixing semi-major axis a and increasig the eccentricity e

#### Explanation:

The semi-minor axis b = asqrt(1-e^2 = a function of e..

Now, b' = $- \frac{e}{\sqrt{1 - {e}^{2}}} < 0$.

So, b is a decreasing function of e. We can see the orbit shrinking towards the fixed major axis, as e increases.

As $e \to 1 - , b \to 0$. This is an interesting degenerate case for the ellipse, in becoming a line segment. Note that the focus S (a star for the orbit) $\to$ the end A of the major axis. The perihelion $\to 0$.

If at all this happens, after billions of years, for the closer-to-star orbiter (like Mercury in solar system), It would be the orbiter's apocalypse...

I think that I could possibly make any reader of my answer to ponder over this degenerate case. I like to add that, likewise, the limit of a hyperbola, with fixed a and eccentricity decreasing $\to$ 1+, could be reviewed. .