Elliptic orbit

Specific energy of an elliptical orbit is negative. An orbit with an eccentricity of 0 is a circular orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit and tundra orbit.

v\,

) of a body traveling along elliptic orbit can be computed as:

v=\sqrt{\mu\left({2\over{r}}-{1\over{a}}\right)}

where:

Conclusion:

Velocity does not depend on eccentricity but is determined by length of semi-major axis ( $a\,\!$ ),
Velocity equation is similar to that for hyperbolic trajectory with the difference that for the latter, ${1\over{2a}}$ is positive.

T\,\!

) of a body traveling along elliptic orbit can be computed as:

T={2\pi\over{\sqrt{\mu}}}a^{3\over{2}}

where:

Conclusions:

The orbital period is equal to that for a circular orbit with the orbit radius equal to the semi-major axis ( $a\,\!$ ),
The orbital period does not depend on the eccentricity (See also: Kepler's third law).

\epsilon\,

) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form:

{v^2\over{2}}-{\mu\over{r}}=-{\mu\over{2a}}=\epsilon<0

where:

Conclusions:

Specific energy for elliptic orbits is independent of eccentricity and is determined only by semi-major axis of the ellipse.

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

In the solar system planets, asteroids, comets and space debris have elliptical orbits around the Sun.

Moons have an elliptic orbit around their planet.

Many artificial satellites have various elliptic orbits around the Earth.

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