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Concept
Orbite elliptique
Résumé
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.
In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter. Also the relative position of one body with respect to the other follows an elliptic orbit.
Examples of elliptic orbits include Hohmann transfer orbits, Molniya orbits, and tundra orbits.
Under standard assumptions, no other forces acting except two spherically symmetrical bodies m1 and m2, the orbital speed () of one body traveling along an elliptic orbit can be computed from the vis-viva equation as:
where:
is the standard gravitational parameter, G(m1+m2), often expressed as GM when one body is much larger than the other.
is the distance between the orbiting body and center of mass.
is the length of the semi-major axis.
The velocity equation for a hyperbolic trajectory has either + , or it is the same with the convention that in that case a is negative.
Under standard assumptions the orbital period() of a body travelling along an elliptic orbit can be computed as:
where:
is the standard gravitational parameter.
is the length of the semi-major axis.
Conclusions:
The orbital period is equal to that for a circular orbit with the orbital radius equal to the semi-major axis (),
For a given semi-major axis the orbital period does not depend on the eccentricity (See also: Kepler's third law).
Under standard assumptions, the specific orbital energy () of an elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:
where:
is the orbital speed of the orbiting body,
is the distance of the orbiting body from the central body,
is the length of the semi-major axis,
is the standard gravitational parameter.
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