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Perhaps I am misunderstanding elementary math, but....

How is it possible to know the eccentricity of an ellipse (e) but not the semimajor axis (a)?

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    $\begingroup$ Eccentricity is defined as a ratio. $\endgroup$
    – AJN
    Jan 1 at 3:42

1 Answer 1

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The two parameters are independent; any combination of non-negative semi-major axis and eccentricity between 0 and 1 results in a valid ellipse.

Semi-major axis for an ellipse is half the longest dimension of the ellipse. For orbital mechanics it is an important parameter that's related to specific orbital energy and orbital period.

Take an ellipse, and scale it up or down in size, keeping the same shape. All the resulting ellipses will have the same eccentricity, but different semi-major axis length.

Ellipses sharing a primary focus with eccentricity $e= 0.5$ and varying semi-major axis $(a)$
Six ellipses of eccentricity 0.5 and semi-major axes ranging from 0 to 1
$$a\in\{0,0.2,0.4,0.6,0.8,1\}$$

Eccentricity is the ratio between the distance from a focus to the center of the ellipse (the linear eccentricity) and the length of the semi-major axis. For ellipses, it ranges between 0 and 1.

Go back to your original ellipse, and flatten or stretch it in one direction, perpendicular to its longest dimension (along its minor axis), but keep the major axis the same. Stretch it until the minor axis is the same length as its major axis, and your ellipse will be a circle with eccentricity 0. Flatten it until it's a line, and your ellipse will be a degenerate linear ellipse with eccentricity 1. At all points during the transformation, you'll have an ellipse with the same semi-major axis as the original, but a unique eccentricity.

Ellipses sharing a primary focus with semi-major axis $a=1$ and varying eccentricity $(e)$
Ellipses of semi-major axis 1 and eccentricities ranging from 0 to 1
$$e \in\{0, 0.2,0.4,0.6,0.8,1\}$$

Knowing just the eccentricity gives you information about the shape of the ellipse. Knowing just the semi-major axis gives you information about the size of the ellipse. To reproduce a congruent ellipse, neither parameter is sufficient alone.

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    $\begingroup$ I love the plots! Well done. $\endgroup$
    – Ryan C
    Jan 1 at 17:16

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