Elliptic orbit - Definition and Overview

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.

Velocity

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

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

where:

• <math>\mu\,<math> is standard gravitational parameter,
• <math>r\,<math> is radial distance of orbiting body from central body,
• <math>a\,\!<math> is length of semi-major axis.

Conclusion:

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

Orbital period

Under standard assumptions the orbital period (<math>T\,\!<math>) of a body traveling along elliptic orbit can be computed as:

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

where:

• <math>\mu\,<math> is standard gravitational parameter,
• <math>a\,\!<math> is length of semi-major axis.

Conclusions:

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

Energy

Under standard assumptions, specific orbital energy (<math>\epsilon\,<math>) of elliptic orbit is negative and the orbital energy conservation equation for this orbit takes form:

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

where:

• <math>v\,<math> is orbital velocity of orbiting body,
• <math>r\,<math> is radial distance of orbiting body from central body,
• <math>a\,<math> is length of semi-major axis,
• <math>\mu\,<math> is standard gravitational parameter.

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

Flight path angle

Equation of motion

See orbit equation.

Orbital parameters

Solar system

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.

See also

• Orbit
• Circular orbit
• Parabolic trajectory
• Hyperbolic trajectory
• Orbital equation
• Characteristic energy

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