Eccentricity

Eccentricity, simply put, is the measure of how much a conic section deviates from a perfect circle.

• The eccentricity of a circle is 0.
• The eccentricity of an ellipse is a real number between 0 and 1 given by the formula.
• The eccentricity of a parabola is always 1.
• And the eccentricity of a hyperbola is a real number greater than 1 given by the forumla.

The term "first eccentricity" is sometimes used for eccentricity to distinguish it from second eccentricity and third eccentricity, which is defined for both ellipses. Below is a chart that shows how to calculate the first, second and third eccentricities of conics, along with linear eccentricity (defined for ellipses and hyperbolae). Linear eccentricity is defined as the distance between the center and a focal point of an ellipse or hyperbola.

Conic	Formula	1st eccentricity	Linear eccentricity	2nd eccentricity	3rd eccentricity
Circle	$x^2 + y^2 = r^2$	$0$	$0$	$0$	$0$
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	$\sqrt{1-\frac{b^2}{a^2}}$	$\sqrt{a^2-b^2}$	$\sqrt{\frac{b^2}{a^2}-1}$	$\frac{\sqrt{a^2-b^2}}{\sqrt{a^2+b^2}}$
Parabola	$x^2=ay^2$	$1$	n/a	n/a	n/a
Hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	$\sqrt{1+\frac{b^2}{a^2}}$	$\sqrt{a^2+b^2}$	$\sqrt{-\frac{b^2}{a^2}-1}$	$\frac {\sqrt{a^2+b^2}}{\sqrt{a^2-b^2}}$

Orbital eccentricity

Eccentricity is important to astrodynamics in the form of orbital eccentricity, which is the measure of how much an orbit or trajectory deviates from a perfect circle. Circular orbits have an eccentricity of 0, elliptic orbits have an eccentricity between 0 and 1, parabolic trajectories have an eccentricity of 1, and hyperbolic trajectories have an eccentricity greater than 1.

The eccentricity can be given with the formula

$\sqrt{1-\frac{2EL^2}{m_{\rm red}\alpha^2}}$

where $E$ is the total orbital energy, $L$ is the angular momentum, $m_{\rm red}$ is the reduced mass and $\alpha$ is the coefficient of the inverse-square law central force. Below is a table of the eccentricities of well-known objects.

Object	Semi-major axis (AU)	Orbital eccentricity
Mercury	0.387098	0.205630
Venus	0.723332	0.006772
Earth	1.000000	0.016709
The Moon	0.002570	0.054900
Mars	1.523679	0.093400
Vesta	2.361790	0.088740
Ceres	2.767500	0.075823
Jupiter	5.204400	0.048900
Io	0.002819	0.004100
Europa	0.004485	0.009000
Ganymede	0.007155	0.001300
Callisto	0.012585	0.007400
Saturn	9.582600	0.056500
Titan	0.008168	0.028800
Halley's Comet	17.83400	0.967140
Uranus	19.21840	0.046381
Neptune	30.11039	0.009456
Pluto	39.48000	0.248800
Haumea	43.21800	0.191260
Makemake	45.71500	0.155860
Eris	67.78100	0.440680
Comet Hale-Bopp	186.0000	0.995086
Sedna	506.8000	0.854910
Comet ISON	?	0.999995
ʻOumuamua	-1.279800	1.199510