I don't get the difference between the two terms named "Eccentricity" and "Ellipticity", especially, in astronomy. I understand eccentricity as a measure of the curvature of a orbit but what is ellipticity?
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2 Answers
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Both ellipticity $f$ (also called flattening) and eccentricity $e$ are measures of how elongated an ellipse is, based on the semi-major axis $a$ and the semi-minor axis $b$ (figure from wikipedia).
They are defined respectively as $$f=\frac{a-b}{a}$$ and $$e=\sqrt{1-\frac{b^2}{a^2}}$$
For a circle, $a=b$, which implies that $f=e=0$. In modern orbital mechanics, $e$ is used rather than $f$. This is due to convention as $e$ is one of the six orbital elements Kepler defined.
Notes:
We can extend the definition of eccentricity to parabolic and hyperbolic orbits: $$e=\sqrt{1+\frac{2\epsilon h^2}{\mu^2}}$$ where $h$ is the angular momentum of the orbiting body relative to the more massive body, $\epsilon$ is the specific orbital energy, and $\mu$ is the standard gravitational parameter. Then $e=1$ for a parabolic orbit, and $e>1$ for a hyperbolic orbit.
While $e$ is conventionally used to describe orbits, $f$ is still used in astronomy as a factor expressing the oblateness of nearly circular bodies.
Here is a table and plot that illustrates the difference in values by fixing the semi-major axis $a=1$ and varying the semi-minor axis $b$ between 0 and 1. Eccentricity and ellipticity are only equal when the orbit is either a circle or a line segment (when the orbit has no tangential component of velocity, in free fall towards the more massive body).
answered Apr 27, 2021 at 16:36
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2$\begingroup$ Eccentricity is not used just because of convention. Orbital mechanics are more concerned with distances relative to the focus of the ellipse than to its center (since that's where the orbited body is) and the eccentricity is more closely related to that. $\endgroup$– AetolCommented Apr 28, 2021 at 19:49
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$\begingroup$ @Aetol Fair point, but Kepler's orbital elements aren't a unique way to describe elliptical 3D orbits. They became convention because Kepler was the first one to figure it out. $\endgroup$– Connor Garcia ♦Commented Apr 28, 2021 at 22:02
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3$\begingroup$ Looking at that table gives another reason why eccentricity is preferred: astronomy deals with a whole lot of almost-circular orbits. Eccentricity provides far finer distinction of those orbits than ellipticity does. $\endgroup$– MarkCommented Apr 29, 2021 at 1:59
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Ellipses have a "long radius" called the "semi-major-axis" which is the length from the centre to the ellipse measured along the long axis. And a "semi-minor-axis" which is measured along the short axis. Call the semi-major-axis "a" and the semi-minor-axis "b".
Ellipses also have foci: which is where the central body, eg the sun, is found. The distance from the centre to a focus is called "c"
The eccentricity is a measure of how far the foci of the ellipse is from the centre. Mathematically it is $e = c/a$.
The ellipticity is the measure of how flattened the ellipse is $f = (a-b)/a$
They are related as $e^2 = 2f-f^2$ So if you know one value, you can calculate the other.
In orbit calculations, eccentricity is more convenient, for example, Kepler's equation $M= E-e\sin E$ is easily expressed with eccentricity, not with ellipicity.
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3$\begingroup$ Perhaps a nicer way to state the relationship is that $1-e^2 = (1-f)^2$. $\endgroup$ Commented Apr 28, 2021 at 17:45