Orbital Eccentricity

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The orbital eccentricity of an astronomical body is the amount by which its orbit deviates from a perfect circle, where 0 is perfectly circular, and 1.0 is a parabola, and no longer a closed orbit.
## Definition

In a two-body problem with inverse-square-law force, every orbit is a Kepler orbit. The eccentricity of this Kepler orbit is a positive number that defines its shape.

The eccentricity may take the following values:

The eccentricity <math>e</math> is given by

where*E* is the total orbital energy, <math>L</math> is the angular momentum, <math>m_text</math> is the reduced mass. and <math>alpha</math> the coefficient of the inverse-square law central force such as gravity or electrostatics in classical physics:

(<math>alpha</math> is negative for an attractive force, positive for a repulsive one) (see also Kepler problem).

or in the case of a gravitational force:

where <math>epsilon</math> is the specific orbital energy (total energy divided by the reduced mass), <math>mu</math> the standard gravitational parameter...

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The eccentricity may take the following values:

- circular orbit: <math>e=0,!</math>
- elliptic orbit: <math>0<e<1,!</math> (see Ellipse)
- parabolic trajectory: <math>e=1,!</math> (see Parabola)
- hyperbolic trajectory: <math>e>1,!</math> (see Hyperbola)

The eccentricity <math>e</math> is given by

- <math>

where

- <math>

(<math>alpha</math> is negative for an attractive force, positive for a repulsive one) (see also Kepler problem).

or in the case of a gravitational force:

- <math>

where <math>epsilon</math> is the specific orbital energy (total energy divided by the reduced mass), <math>mu</math> the standard gravitational parameter...

Read More

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