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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's 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 orbit, Molniya orbit, and tundra orbit.

Velocity

Under standard assumptions the orbital speed () of a body traveling along an elliptic orbit can be computed from the vis-viva equation as:

where:

The velocity equation for a hyperbolic trajectory has either + , or it is the same with the convention that in that case a is negative.

Orbital period

Under standard assumptions the orbital period () of a body travelling along an elliptic orbit can be computed as:

where:

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).

Energy

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:

  • 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 orbit, Molniya orbit, and tundra orbit.

    Under standard assumptions the orbital speed () of a body traveling along an elliptic orbit can be computed from the vis-viva equation as:

    where:

    • is the standard gravitational parameter,
    • where:

      • is the standard gravitational parameter,
      • is the distance between the orbiting bodies.
      • hyperbolic trajectory has either + , or it is the same with the convention that in that case a is negative.

        Orbital period

        Under standard assumptions the orbital period () of a body travelling along an elliptic orbit can be computed as:

        where:

        • orbital period () of a body travelling along an elliptic orbit can be computed as: