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Concept
Orbital period
Summary
The orbital period (also revolution period) is the amount of time a given astronomical object takes to complete one orbit around another object. In astronomy, it usually applies to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars. It may also refer to the time it takes a satellite orbiting a planet or moon to complete one orbit.
For celestial objects in general, the orbital period is determined by a 360° revolution of one body around its primary, e.g. Earth around the Sun.
Periods in astronomy are expressed in units of time, usually hours, days, or years.
According to Kepler's Third Law, the orbital period T of two point masses orbiting each other in a circular or elliptic orbit is:
where:
a is the orbit's semi-major axis
G is the gravitational constant,
M is the mass of the more massive body.
For all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity.
Inversely, for calculating the distance where a body has to orbit in order to have a given orbital period T:
For instance, for completing an orbit every 24 hours around a mass of 100 kg, a small body has to orbit at a distance of 1.08 meters from the central body's center of mass.
In the special case of perfectly circular orbits, the semimajor axis a is equal to the radius of the orbit, and the orbital velocity is constant and equal to
where:
r is the circular orbit's radius in meters,
This corresponds to times (≈ 0.707 times) the escape velocity.
For a perfect sphere of uniform density, it is possible to rewrite the first equation without measuring the mass as:
where:
r is the sphere's radius
a is the orbit's semi-major axis in metres,
G is the gravitational constant,
ρ is the density of the sphere in kilograms per cubic metre.
For instance, a small body in circular orbit 10.5 cm above the surface of a sphere of tungsten half a metre in radius would travel at slightly more than 1 mm/s, completing an orbit every hour. If the same sphere were made of lead the small body would need to orbit just 6.
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