Orbital period

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The orbital period is the time taken for a given object to make one complete orbit around another object.

When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.

There are several kinds of orbital periods for objects around the Sun (or other celestial objects):

Relation between the sidereal and synodic periods[edit]

Table of synodic periods in the Solar System, relative to Earth:[citation needed]

   Sidereal period (yr)Synodic period (yr)Synodic period (d)
Solar surface      0.069[1] (25.3 days)  0.074  27.3
Mercury      0.240846 (87.9691 days)  0.317  115.88
Venus      0.615 (225 days)  1.599  583.9
Earth      1 (365.25636 solar days)    —    —
Moon      0.0748    0.0809  29.5306
Apophis (near-Earth asteroid)      0.886  7.769  2,837.6
Mars      1.881  2.135  779.9
4 Vesta      3.629  1.380  504.0
1 Ceres      4.600  1.278  466.7
10 Hygiea      5.557  1.219  445.4
Jupiter      11.86  1.092  398.9
Saturn      29.46  1.035  378.1
Uranus      84.01  1.012  369.7
Neptune      164.8  1.006  367.5
134340 Pluto      248.1  1.004  366.7
136199 Eris      557  1.002  365.9
90377 Sedna      12050  1.00001  365.1[citation needed]

In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to complete its illumination phases, completing the solar phases for an astronomer on the planet's surface —the Earth's motion does not determine this value for other planets because an Earth observer is not orbited by the moons in question. For example, Deimos's synodic period is 1.2648 days, 0.18% longer than Deimos's sidereal period of 1.2624 d.[citation needed]

Calculation[edit]

Small body orbiting a central body[edit]

According to Kepler's Third Law, the orbital period T\, (in seconds) of two bodies orbiting each other in a circular or elliptic orbit is:

T = 2\pi\sqrt{a^3/\mu}

where:

For all ellipses with a given semi-major axis the orbital period is the same, regardless of eccentricity.

Orbital period as a function of central body's density[edit]

When a very small body is in a circular orbit barely above the surface of a sphere of radius R and mean density ρ (in kg/m3), the above equation simplifies to (since a≈R and M=ρV):[citation needed]

T = \sqrt{ \frac {3\pi}{G \rho} }

So, for the Earth as the central body (or any other spherically symmetric body with the same mean density, about 5515 kg/m3)[2] we get:

T = 1.41 hours

and for a body made of water (ρ≈1000 kg/m3)[3]

T = 3.30 hours

Thus, as an alternative for using a very small number like G, the strength of universal gravity can be described using some reference material, like water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time if we have a unit of mass, a unit of length and a unit of density.

Two bodies orbiting each other[edit]

In celestial mechanics, when both orbiting bodies' masses have to be taken into account, the orbital period T\, can be calculated as follows:[4]

T= 2\pi\sqrt{\frac{a^3}{G \left(M_1 + M_2\right)}}

where:

Note that the orbital period is independent of size: for a scale model it would be the same, when densities are the same (see also Orbit#Scaling in gravity).[citation needed]

In a parabolic or hyperbolic trajectory, the motion is not periodic, and the duration of the full trajectory is infinite.[citation needed]

Synodic period[edit]

When two bodies orbit a third body in different orbits, and thus different orbital periods, their respective, synodic period can be found. If the orbital periods of the two bodies around the third are called P_1 and P_2, so that P_1 < P_2, their synodic period is given by

\frac{1}{P_{syn}}=\frac{1}{P_1}-\frac{1}{P_2}

Tangential velocities at altitude[edit]

orbitcenter-to-center
distance
altitude above
the Earth's surface
speedOrbital periodspecific orbital energy
Earth's surface (for comparison)6,400 km0 km7.89 km/s (17,650 mph)85 minutes−62.6 MJ/kg
Low Earth orbit6,600 to 8,400 km200 to 2,000 kmcircular orbit: 7.8 to 6.9 km/s (17,450 mph to 15,430 mph) respectively
elliptic orbit: 8.2 to 6.5 km/s respectively
89 to 128 min−29.8 MJ/kg
Molniya orbit6,900 to 46,300 km500 to 39,900 km10.0 to 1.5 km/s (22,370 mph to 3,335 mph) respectively11 h 58 min−4.7 MJ/kg
Geostationary42,000 km35,786 km3.1 km/s (6,935 mph)23 h 56 min−4.6 MJ/kg
Orbit of the Moon363,000 to 406,000 km357,000 to 399,000 km1.08 to 0.97 km/s (2,416 to 2,170 mph) respectively27.3 days−0.5 MJ/kg

Binary stars[edit]

Binary starOrbital period
AM Canum Venaticorum17.146 minutes
Beta Lyrae AB12.9075 days
Alpha Centauri AB79.91 years
Proxima Centauri - Alpha Centauri AB500,000 years or more

See also[edit]

Notes[edit]

  1. ^ The motion of the solar surface is not purely gravitational and therefore does not follow Kepler's laws of motion
  2. ^ Density of the Earth, wolframalpha.com 
  3. ^ Density of water, wolframalpha.com 
  4. ^ Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.

External links[edit]