For other meanings of the term "orbit", see orbit (disambiguation)

In astrodynamics or celestial mechanics a circular orbit is an elliptic orbit with the eccentricity equal to 0. It is an example of a rotation around a fixed axis: this axis is the line through the center of mass perpendicular to the plane of motion.

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Circular acceleration

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Transverse acceleration (perpendicular to velocity) causes change in direction. If it is constant in magnitude and changing in direction with the velocity, we get a circular motion. For this centripetal acceleration we have

$\backslash mathbf\{a\}\; =\; -\; \backslash frac\{v^2\}\{r\}\; \backslash frac\{\backslash mathbf\{r\}\}\{r\}\; =\; -\; \backslash omega^2\; \backslash mathbf\{r\}$

where:

• $v\backslash ,$ is orbital velocity of orbiting body,
• $r\backslash ,$ is radius of the circle
• $\backslash omega\; \backslash$ is frequency, measured in radian per second.

Velocity

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

$v=\backslash sqrt\{\backslash mu\backslash over\{r\}\}$

where:

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

Conclusion:

• Velocity is constant along the path.

Orbital period

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

$T=\{2\backslash pi\backslash over\{\backslash sqrt\{\backslash mu\}\}\}r^\{3\backslash over\{2\}\}$

where:

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

Conclusions:

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

Energy

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

where:

• $v\backslash ,$ is orbital velocity of orbiting body,
• $r\backslash ,$ is radius of orbit equal to radial distance of orbiting body from central body,
• $\backslash mu\backslash ,$ 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

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Under standard assumptions, the orbital equation becomes:

$r=$

where:

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

Delta-v to reach a circular orbit

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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

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• Orbit
• Low Earth orbit
• Intermediate circular orbit
• Geostationary orbit
• Areostationary orbit
• Two-body problem

Astrodynamics | Orbits

Orbita circolare | 円軌道