Orbital period

For the music album, see Orbital Period (album).

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.

Varieties of orbital periods

Orbital period is an approximated term, and can mean any of several periods, each of which is used in the fields of astronomy and astrophysics:

Small body orbiting a central body

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.

Inversely, for calculating the distance where a body has to orbit in order to pulse a given orbital period:

a = \sqrt[3]{\frac{GMT^2}{4\pi^2}}

where:

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 its center of mass.

Orbital period as a function of central body's density

When a very small body is in a circular orbit barely above the surface of a sphere of any radius and mean density ρ (in kg/m3), the above equation simplifies to (since M = \rho V = \rho {\frac {4}{3}} \pi a^3):

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)[1] we get:

T = 1.41 hours

and for a body made of water (ρ≈1000 kg/m3)[2]

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

In celestial mechanics, when both orbiting bodies' masses have to be taken into account, the orbital period T\, can be calculated as follows:[3]

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).

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

Synodic period

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}

Examples of sidereal and synodic periods

Table of synodic periods in the Solar System, relative to Earth:

Object Sidereal period (yr) Synodic period (yr) Synodic period (d)
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   (27.32 days)   0.0809   29.5306
99942 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

In the case of a planet's moon, the synodic period usually means the Sun-synodic period, namely, 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.

Binary stars

Binary starOrbital period
AM Canum Venaticorum 17.146 minutes
Beta Lyrae AB 12.9075 days
Alpha Centauri AB 79.91 years
Proxima Centauri - Alpha Centauri AB 500,000 years or more

See also

Notes

  1. Density of the Earth, wolframalpha.com
  2. Density of water, wolframalpha.com
  3. Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.

External links

Look up synodic in Wiktionary, the free dictionary.
This article is issued from Wikipedia - version of the Wednesday, April 20, 2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.