Editing Orbital eccentricity (section)

==Definition==
In a [[two-body problem]] with inverse-square-law force, every [[orbit]] is a [[Kepler orbit]]. The [[eccentricity (mathematics)|eccentricity]] of this Kepler orbit is a [[non-negative number]] that defines its shape.

The eccentricity may take the following values:
* [[Circular orbit]]: {{math|''e'' {{=}} 0}}
* [[Elliptic orbit]]: {{math|0 < ''e'' < 1}}
* [[Parabolic trajectory]]: {{math|''e'' {{=}} 1}}
* [[Hyperbolic trajectory]]: {{math|''e'' > 1}}

The eccentricity {{mvar|e}} is given by<ref>{{cite book| last=Abraham |first=Ralph | year=2008 | title=Foundations of Mechanics | edition=2nd | publisher=AMS Chelsea Pub./[[American Mathematical Society]] | others=Marsden, Jerrold E. | isbn=978-0-8218-4438-0 | location=Providence, RI | oclc=191847156}}</ref>

<math display="block">e = \sqrt{1 + \frac{\ 2\ E\ L^2\ }{\ m_\text{rdc}\ \alpha^2\ }}</math>

where {{math|''E''}} is the total [[orbital energy]], {{math|''L''}} is the [[angular momentum]], {{math|''m''<sub>rdc</sub>}} is the [[reduced mass]], and <math>\alpha</math> the coefficient of the inverse-square law [[central force]] such as in the theory of [[gravity]] or [[electrostatics]] in [[classical physics]]:
<math display="block">F = \frac{\alpha}{r^2}</math>
(<math>\alpha</math> is negative for an attractive force, positive for a repulsive one; related to the [[Kepler problem]])

or in the case of a gravitational force:<ref name="BateEtAl">{{cite book|url=https://books.google.com/books?id=UEC9DwAAQBAJ | title=Fundamentals of Astrodynamics | last1=Bate|first1=Roger R. | last2=Mueller | first2=Donald D. | last3=White|first3=Jerry E. | last4=Saylor|first4=William W. | publisher=[[Courier Dover]] | date=2020 | access-date=4 March 2022 | isbn=978-0-486-49704-4}}</ref>{{rp|p=24}}
<math display="block">e = \sqrt{1 + \frac{2 \varepsilon h^{2}}{\mu^2}}</math>

where {{math|''ε''}} is the [[specific orbital energy]] (total energy divided by the reduced mass), {{math|''μ''}} the [[standard gravitational parameter]] based on the total mass, and {{math|''h''}} the [[specific relative angular momentum]] ([[angular momentum]] divided by the reduced mass).<ref name="BateEtAl"/>{{rp|pp=12–17}}

For values of {{mvar|e}} from {{math|0}} to just under {{math|1}} the orbit's shape is an increasingly elongated (or flatter) ellipse; for values of {{mvar|e}} just over {{math|1}} to infinity the orbit is a [[hyperbola]] branch making a total turn of {{nobr|{{math|&hairsp; 2 [[Inverse trigonometric functions|arccsc]](''e'')}} ,}} decreasing from 180 to 0 degrees. Here, the total turn is analogous to [[Winding number#Turning number|turning number]], but for open curves (an angle covered by velocity vector). The [[limit (mathematics)|limit]] case between an ellipse and a hyperbola, when {{mvar|e}} equals {{math|1}}, is parabola.

Radial trajectories are classified as elliptic, parabolic, or hyperbolic based on the energy of the orbit, not the eccentricity. Radial orbits have zero angular momentum and hence eccentricity equal to one. Keeping the energy constant and reducing the angular momentum, elliptic, parabolic, and hyperbolic orbits each tend to the corresponding type of radial trajectory while {{mvar|e}} tends to {{math|1}} (or in the parabolic case, remains {{math|1}}).

For a repulsive force only the hyperbolic trajectory, including the radial version, is applicable.

For elliptical orbits, a simple proof shows that <math>\ \arcsin(e)\ </math> gives the projection angle of a perfect circle to an [[ellipse]] of eccentricity {{mvar|e}}.  For example, to view the eccentricity of the planet Mercury ({{math|''e'' {{=}} 0.2056}}), one must simply calculate the [[inverse trigonometric functions|inverse sine]] to find the projection angle of 11.86&nbsp;degrees. Then, tilting any circular object by that angle, the apparent ellipse of that object projected to the viewer's eye will be of the same eccentricity.