Circular orbit

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0.

Velocity

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

<math>v=\sqrt{\mu\over{r}}<math>

where:

• <math>r\,<math> is radius of orbit equal to radial distance of orbiting body from central body,
• <math>\mu\,<math> is standard gravitational parameter.

Conclusion:

• Velocity is constant along the path.

Orbital period

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

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

where:

• <math>r\,<math> is orbit radius equal to radial distance of orbiting body from central body,
• <math>\mu\,<math> is standard gravitational parameter.

Conclusions:

• The orbital period is the same as that for an elliptic orbit with the semi-major axis (<math>a\,\!<math>) equal to orbit radius.

Energy

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

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

where:

• <math>v\,<math> is orbital velocity of orbiting body,
• <math>r\,<math> is radius of orbit equal to radial distance of orbiting body from central body,
• <math>\mu\,<math> is standard gravitational parameter.

The virial theorem applies even without taking a time-average:

• the potential energy of the system is equal to twice the total energy
• the kinetic energy of the system is equal to minus the total energy

Thus the escape velocity from any distance is √2 times the speed in a circular orbit at that distance: the kinetic energy is twice as much, hence the total energy is zero.

Equation of motion

Under standard assumptions, the orbital equation becomes:

<math>r={{h^2}\over{\mu}}<math>

where:

• <math>r\,<math> is radial distance of orbiting body from central body,
• <math>h\,<math> is specific angular momentum of the orbiting body,
• <math>\mu\,<math> is standard gravitational parameter.

Delta-v to reach a circular orbit

Maneuvering into a large circular orbit, e.g. a geostationary orbit, requires a larger delta-v than an escape orbit, although the latter implies getting arbitrarily far away and having more energy than needed for the orbital speed of the circular orbit. It is also a matter of maneuvering into the orbit. See also Hohmann transfer orbit.

See also

• Orbit
• Geostationary orbit
• Areostationary orbit