Circular_orbit

Two bodies with a slight difference in mass orbiting around a common barycenter with circular orbits.

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.

Contents 1 Circular acceleration 2 Velocity 3 Orbital period 4 Energy 5 Equation of motion 6 Delta-v to reach a circular orbit 7 See also

Circular acceleration

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 angular frequency, measured in radian per second.

Velocity

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

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

where:

• $r\backslash ,$ is radius of orbit equal to radial distance of orbiting body from central body,
• $\backslash mu=GM\backslash ,$ is the standard gravitational parameter, which is the product of the gravitational constant $G$ and the central mass $M$.

Conclusion:

• Velocity is constant along the path.

Orbital period

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

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

where:

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

Energy

Under standard assumptions, specific orbital energy ($\backslash epsilon\backslash ,$) is negative for a closed orbit and the orbital energy conservation equation (the Vis-viva equation) can take 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 boundary case is $\backslash epsilon\backslash ,=0$ which corresponds to escape from the primary (parabolic orbit), with $v=\backslash sqrt\{2\}v\_c=\backslash sqrt\{2\backslash mu\backslash over\{r\}\}$.

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:

$r=\{\{h^2\}\backslash over\{\backslash mu\}\}$

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

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
• Elliptic orbit
• List of orbits
• Two-body problem

v • d • e Orbits
Orbital parameters Classical orbital elements $i\backslash ,\backslash !$     Inclination $\backslash Omega\backslash ,\backslash !$   Longitude of the ascending node $e\backslash ,\backslash !$    Eccentricity    $\backslash omega\backslash ,\backslash !$    Argument of periapsis    $a\backslash ,\backslash !$     Semi-major axis $M\_o\backslash ,\backslash !$   Mean anomaly at epoch Other parameters  $\backslash nu\backslash ,$  True anomaly  $b\backslash ,$   Semi-minor axis  $\backslash epsilon\backslash ,$   Linear eccentricity $E\backslash ,$  Eccentric anomaly $L\backslash ,$  Mean longitude  $l\backslash ,$    True longitude $T\backslash ,$  Orbital period
Types	General Box · Circular · Non-inclined · Elliptic (Highly elliptical) · Graveyard · Hyperbolic trajectory · Inclined · Osculating · Parabolic trajectory · Capture · Escape · Semi-synchronous · Subsynchronous · Synchronous · Parking Geocentric Geosynchronous · Geostationary · Sun-synchronous · Low Earth · Medium Earth · Molniya · Near equatorial · Moon · Polar · Tundra Other Areosynchronous · Areostationary · Halo · Lissajous · Lunar · Heliocentric · Heliosynchronous
Maneuvers	Bi-elliptic transfer · Geostationary transfer · Gravity assist · Hohmann transfer · Inclination change · Phasing · Rendezvous
Related topics	Apsis · Celestial coordinate system · Delta-v budget · Epoch · Ephemeris · Equatorial coordinate system · Gravity turn · Ground track · Interplanetary Transport Network · Kepler\'s laws of planetary motion · Lagrangian point · n-body problem · Oberth effect · Orbit equation · Orbital state vectors · Perturbation · Retrograde and direct motion · Specific orbital energy · Specific relative angular momentum
List of orbits

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia