Expert: James Gort - 12/30/2005

Question
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Followup To
Question -
Hi James,
How can I quantify the degree of ellipticity of various oval shapes or orbits?  Can I measure the widest point and the narrowest point and divide one into the other?  Does the result then need to be squared or square rooted to get a linear measure of the degree of ellipticity?  Thanks.

Regards,
Usuff
Answer -
Hello Usuff,

Yes, you can measure the widest point (call this length 2a) and the narrowest point (call this length 2b).  The "degree of ellipticity" - called the eccentricity "e" - is defined as:

    e = sqrt(1 - (b squared) / (a squared) )

The length of the major axis is 2a, and the length of the minor axis is 2b.  Be sure you use the radius (a or b) and not the major or minor axis (2a or 2b).

The eccentricity "e" can only vary betwee 0 and 1.  If e = 0, it's a circle.  If e = 1, it's a parabola.

Hope that helps.

Prof. James Gort
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Professor James,

Thanks, my astronomy studies are coming back to me.  Is there another way of measuring eccentricity?  The scale 0 to 1 seems so small, yet when you look at the following three orbital paths at :-

http://www.usuff.com/Eccentricity.html

which are traces of pendulum swings, the inner most path seems hugely more eccentric than the outer most path, yet e is 0.995 and 0.298 respectively, which doesn't seem to reflect the perception of the difference. Another example, the middle orbit has an e of 0.863, only slightly less than the inner most orbit, yet visually there seems a huge difference between them.   

Would e be the only scale of this kind to quantify how ellpitical an orbit is?  Thanks.

Regards,
Usuff  

Answer
Hi Usuff,

Yes, that's the only scale I'm aware of to quantify the eccentricity of an ellipse.  But you've hit on a very important characteristic of that function.  As the eccentricity gets more pronounced, there is less of a "difference" between eccentricity values.  To see how the function behaves, make a graph of b/a versus e (where e = sqrt(1 - (b squared) / (a squared) )).  You'll see that when b/a is close to 1, e will change quickly.  But when b/a is a small number, e doesn't change as fast.  It's fascinating to see how functions behave as the variables change.

Cheers,

Prof. James Gort