# Elliptic orbit

In astrodynamics or celestial mechanics an elliptic orbit is a Kepler orbit with the eccentricity less than 1; this includes the special case of a circular orbit, with eccentricity equal to zero. 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 orbitMolniya orbit and tundra orbit.

## Velocity

Under standard assumptions the orbital speed ($v,$) of a body traveling along elliptic orbit can be computed from the Vis-viva equationas:

$v=sqrt{muleft({2over{r}}-{1over{a}} ight)}$

where:

• $mu,$ is the standard gravitational parameter,
• $r,$ is the distance between the orbiting bodies.
• $a,!$ is the length of the semi-major axis.

The velocity equation for a hyperbolic trajectory has either + ${1over{a}}$, or it is the same with the convention that in that case a is negative.

## Orbital period

Under standard assumptions the orbital period ($P,!$) of a body traveling along an elliptic orbit can be computed as:

$P=2pisqrt{a^3over{mu}}$

where:

Conclusions:

• The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis ($a,!$),
• 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, specific orbital energy ($epsilon,$) of elliptic orbit is negative and the orbital energy conservation equation (the Vis-viva equation) for this orbit can take the form:

${v^2over{2}}-{muover{r}}=-{muover{2a}}=epsilon<0$

where:

Conclusions:

• For a given semi-major axis the specific orbital energy is independent of the eccentricity.

Using the virial theorem we find:

• the time-average of the specific potential energy is equal to 2ε
• the time-average of r−1 is a−1
• the time-average of the specific kinetic energy is equal to –ε

## Flight path angle

The flight path angle is the angle between the orbiting body's velocity vector (= the vector tangent to the instantaneous orbit) and the local horizontal. Under standard assumptions the flight path angle $phi$ satisfies the equation:

$h, = r, v, cos phi$

where:

• $h,$ is the specific relative angular momentum of the orbit,
• $v,$ is orbital speed of orbiting body,
• $r,$ is radial distance of orbiting body from central body,
• $phi ,$ is the flight path angle

## Equation of motion

See orbit equation

## Orbital parameters

The state of an orbiting body at any given time is defined by the orbiting body's position and velocity with respect to the central body, which can be represented by the three-dimensional Cartesian coordinates (position of the orbiting body represented by x, y, and z) and the similar Cartesian components of the orbiting body's velocity. This set of six variables, together with time, are called the orbital state vectors. Given the masses of the two bodies they determine the full orbit. The two most general cases with these 6 degrees of freedom are the elliptic and the hyperbolic orbit. Special cases with less degrees of freedom are the circular and parabolic orbit.

Because at least six variables are absolutely required to completely represent an elliptic orbit with this set of parameters, then six variables are required to represent an orbit with any set of parameters. Another set of six parameters that are commonly used are the orbital elements.

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