### Synodic 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 about 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):

• The sidereal period is the temporal cycle that it takes an object to make a full orbit, relative to the stars. This is the orbital period in an inertial (non-rotating) frame of reference.
• The synodic period is the temporal interval that it takes for an object to reappear at the same point in relation to two or more other objects, e.g. when the Moon relative to the Sun as observed from Earth returns to the same illumination phase. The synodic period is the time that elapses between two successive conjunctions with the Sun–Earth line in the same linear order. The synodic period differs from the sidereal period due to the Earth's orbiting around the Sun.
• The draconitic period, or draconic period, is the time that elapses between two passages of the object through its ascending node, the point of its orbit where it crosses the ecliptic from the southern to the northern hemisphere. This period differs from the sidereal period because both the orbital plane of the object and the plane of the ecliptic precess with respect to the fixed stars, so their intersection, the line of nodes, also precesses with respect to the fixed stars. Although the plane of the ecliptic is often held fixed at the position it occupied at a specific epoch, the orbital plane of the object still precesses causing the draconitic period to differ from the sidereal period.
• The anomalistic period is the time that elapses between two passages of an object at its periapsis (in the case of the planets in the solar system, called the perihelion), the point of its closest approach to the attracting body. It differs from the sidereal period because the object's semimajor axis typically advances slowly.
• Also, the Earth's tropical period (or simply its "year") is the time that elapses between two alignments of its axis of rotation with the Sun, also viewed as two passages of the object at right ascension zero. One Earth year has a slightly shorter interval than the solar orbit (sidereal period) because the inclined axis and equatorial plane slowly precesses (rotates in sidereal terms), realigning before orbit completes with an interval equal to the inverse of the precession cycle (about 25,770 years).

## Relation between the sidereal and synodic periods

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

 Sidereal period (yr) Synodic period (yr) Synodic period (d) Solar surface 0.069 (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 84.32 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. 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.

## Calculation

### 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:

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

where:

• $a\,$ is the orbit's semi-major axis, in meters
• $\mu = GM \,$ is the standard gravitational parameter, typically in $m^3/s^2$
• $G \,$ 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.

### 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 radius R and average density ρ in g/cm3, the above equation simplifies to:

$T = \sqrt\left\{ \frac \left\{3\pi\right\}\left\{G \rho\right\} \right\}$

So, for the Earth as central body (or for any other spherically symmetric body with the same average density) we get:

$T = 1.4$ hours

and for a body made of water

$T = 3.3$ 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:

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

where:

• $a\,$ is the sum of the semi-major axes of the ellipses in which the centers of the bodies move, or equivalently, the semi-major axis of the ellipse in which one body moves, in the frame of reference with the other body at the origin (which is equal to their constant separation for circular orbits),
• $M_1+M_2\,$ is the sum of the masses of the two bodies,
• $G\,$ is the gravitational constant.

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.

## Escape velocities at altitude

orbit center-to-center
distance
altitude above
the Earth's surface
speed Orbital period specific orbital energy
Earth's surface (for comparison) 6,400 km 0 km 7.89 km/s (17,650 mph) 85 minutes −62.6 MJ/kg
Low Earth orbit 6,600 to 8,400 km 200 to 2,000 km circular 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 orbit 6,900 to 46,300 km 500 to 39,900 km 10.0 to 1.5 km/s (22,370 mph to 3,335 mph) respectively 11 h 58 min −4.7 MJ/kg
Geostationary 42,000 km 35,786 km 3.1 km/s (6,935 mph) 23 h 56 min −4.6 MJ/kg
Orbit of the Moon 363,000 to 406,000 km 357,000 to 399,000 km 1.08 to 0.97 km/s (2,416 to 2,170 mph) respectively 27.3 days −0.5 MJ/kg

## Binary stars

Binary star Orbital 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