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# Elliptic orbit

In astrodynamics or celestial mechanics a elliptic orbit is an orbit with the eccentricity greater than 0 and less than 1.

Specific energy of an elliptical orbit is negative.

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## Velocity

Under standard assumptions the orbital velocity ($v\,$) of a body traveling along elliptic orbit can be computed as:

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

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 one ${1\over{2a}}$ is positive.

## Orbital period

Under standard assumptions the orbital period ($T\,\!$) 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).

## Energy

Under standard assumptions, specific orbital energy ($\epsilon\,$) 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:

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

## Equation of motion

See orbit equation.

## Orbital parameters

Last updated: 06-02-2005 01:34:38
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