Orbital period

Last updated

The orbital period is the time a given astronomical object takes to complete one orbit around another object, and applies in astronomy usually to planets or asteroids orbiting the Sun, moons orbiting planets, exoplanets orbiting other stars, or binary stars.

Contents

For objects in the Solar System, this is often referred to as the sidereal period, determined by a 360° revolution of one celestial body around another, e.g. the Earth orbiting the Sun. The term sidereal denotes that the object returns to the same position relative to the fixed stars projected in the sky. When describing orbits of binary stars, the orbital period is usually referred to as just the period. For example, Jupiter has a sidereal period of 11.86 years while the main binary star Alpha Centauri AB has a period of about 79.91 years.

Another important orbital period definition can refer to the repeated cycles for celestial bodies as observed from the Earth's surface. An example is the so-called synodic period, applying to the elapsed time where planets return to the same kind of phenomena or location, such as when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

Periods in astronomy are conveniently expressed in various units of time, often in hours, days, or years. They can be also defined under different specific astronomical definitions that are mostly caused by small complex external gravitational influences by other celestial objects. Such variations also include the true placement of the centre of gravity between two astronomical bodies (barycenter), perturbations by other planets or bodies, orbital resonance, general relativity, etc. Most are investigated by detailed complex astronomical theories using celestial mechanics using precise positional observations of celestial objects via astrometry.

There are many periods related to the orbits of objects, each of which are often used in the various fields of astronomy and astrophysics. Examples of some of the common ones include the following:

Small body orbiting a central body

The semi-major axis (a) and semi-minor axis (b) of an ellipse Ellipse semi-major and minor axes.svg
The semi-major axis (a) and semi-minor axis (b) of an ellipse

According to Kepler's Third Law, the orbital periodT (in seconds) of two point masses orbiting each other in a circular or elliptic orbit is: [2]

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 have a given orbital period:

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 the central body's center of mass.

In the special case of perfectly circular orbits, the orbital velocity is constant and equal (in m/s) to

where:

This corresponds to 1√2 times (≈ 0.707 times) the escape velocity.

Effect of central body's density

For a perfect sphere of uniform density, it is possible to rewrite the first equation without measuring the mass as:

where:

For instance, a small body in circular orbit 10.5 cm above the surface of a sphere of tungsten half a meter 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 6.7 mm above the surface for sustaining the same orbital period.

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 =  = 4/3πa3ρ)

Thus the orbital period in low orbit depends only on the density of the central body, regardless of its size.

So, for the Earth as the central body (or any other spherically symmetric body with the same mean density, about 5,515 kg/m3, [3] e.g. Mercury with 5,427 kg/m3 and Venus with 5,243 kg/m3) we get:

T = 1.41 hours

and for a body made of water (ρ  1,000 kg/m3) [4] , respectively bodies with a similar density, e.g. Saturn's moons Iapetus with 1,088 kg/m3 and Tethys with 984 kg/m3 we get:

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 periodT can be calculated as follows: [5]

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.

Synodic period

One of the observable characteristics of two bodies which orbit a third body in different orbits, and thus have different orbital periods, is their synodic period, which is the time between conjunctions.

An example of this related period description is the repeated cycles for celestial bodies as observed from the Earth's surface, the so-called synodic period, applying to the elapsed time where planets return to the same kind of phenomena or location. For example, when any planet returns between its consecutive observed conjunctions with or oppositions to the Sun. For example, Jupiter has a synodic period of 398.8 days from Earth; thus, Jupiter's opposition occurs once roughly every 13 months.

If the orbital periods of the two bodies around the third are called T1 and T2, so that T1 < T2, their synodic period is given by: [6]

Examples of sidereal and synodic periods

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

ObjectSidereal period
(yr)
Synodic period
(yr)(d) [7]
Mercury 0.240846 (87.9691 days)0.317115.88
Venus 0.615 (225 days)1.599583.9
Earth 1 (365.25636 solar days)
Mars 1.8812.135779.9
Jupiter 11.861.092398.9
Saturn 29.461.035378.1
Uranus 84.011.012369.7
Neptune 164.81.006367.5
134340 Pluto 248.11.004366.7
Moon 0.0748 (27.32 days)0.080929.5306
99942 Apophis (near-Earth asteroid)0.8867.7692,837.6
4 Vesta 3.6291.380504.0
1 Ceres 4.6001.278466.7
10 Hygiea 5.5571.219445.4
2060 Chiron 50.421.020372.6
50000 Quaoar 287.51.003366.5
136199 Eris 5571.002365.9
90377 Sedna 120501.0001365.3 [ citation needed ]

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.[ citation needed ]

Synodic periods relative to other planets

The concept of synodic period does not just apply to the Earth, but also to other planets as well, and the formula for computation is the same as the one given above. Here is a table which lists the synodic periods of some planets relative to each other:

Orbital period (years)
Relative toMarsJupiterSaturnChironUranusNeptunePlutoQuaoarEris
Sol 1.88111.8629.4650.4284.01164.8248.1287.5557.0
Mars 2.2362.0091.9541.9241.9031.8951.8931.887
Jupiter 19.8515.5113.8112.7812.4612.3712.12
Saturn 70.8745.3735.8733.4332.8231.11
2060 Chiron 126.172.6563.2861.1455.44
Uranus 171.4127.0118.798.93
Neptune 490.8386.1234.0
134340 Pluto 1810.4447.4
50000 Quaoar 594.2

Binary stars

Binary star Orbital period
AM Canum Venaticorum 17.146 minutes
Beta Lyrae AB12.9075 days
Alpha Centauri AB79.91 years
Proxima CentauriAlpha Centauri AB500,000 years or more

See also

Notes

  1. Oliver Montenbruck, Eberhard Gill (2000). Satellite Orbits: Models, Methods, and Applications. Springer Science & Business Media. p. 50. ISBN   978-3-540-67280-7.
  2. Bate, Mueller & White (1971), p. 33.
  3. Density of the Earth, wolframalpha.com
  4. Density of water, wolframalpha.com
  5. Bradley W. Carroll, Dale A. Ostlie. An introduction to modern astrophysics. 2nd edition. Pearson 2007.
  6. Hannu Karttunen; et al. (2016). Fundamental Astronomy (6th ed.). Springer. p. 145. ISBN   9783662530450 . Retrieved December 7, 2018.
  7. "Questions and Answers - Sten's Space Blog". www.astronomycafe.net.

Bibliography

Related Research Articles

Orbit Orbital gravitationally curved path of an object around a point in outer space; circular or elliptical path of one object around another object

In physics, an orbit is the gravitationally curved trajectory of an object, such as the trajectory of a planet around a star or a natural satellite around a planet. Normally, orbit refers to a regularly repeating trajectory, although it may also refer to a non-repeating trajectory. To a close approximation, planets and satellites follow elliptic orbits, with the center of mass being orbited at a focal point of the ellipse, as described by Kepler's laws of planetary motion.

Tidal acceleration Natural phenomenon due to which tidal locking occurs

Tidal acceleration is an effect of the tidal forces between an orbiting natural satellite, and the primary planet that it orbits. The acceleration causes a gradual recession of a satellite in a prograde orbit away from the primary, and a corresponding slowdown of the primary's rotation. The process eventually leads to tidal locking, usually of the smaller first, and later the larger body. The Earth–Moon system is the best-studied case.

Escape velocity Concept in celestial mechanics

In physics, escape velocity is the minimum speed needed for a free, non-propelled object to escape from the gravitational influence of a massive body, that is, to achieve an infinite distance from it. Escape velocity is a function of the mass of the body and distance to the center of mass of the body.

Roche limit astronomical concept

In celestial mechanics, the Roche limit, also called Roche radius, is the distance within which a celestial body, held together only by its own force of gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction. Inside the Roche limit, orbiting material disperses and forms rings, whereas outside the limit material tends to coalesce. The term is named after Édouard Roche, who was the French astronomer who first calculated this theoretical limit in 1848.

Free fall Motion of a body where its weight is the only force acting upon it

In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it.

An equatorial bulge is a difference between the equatorial and polar diameters of a planet, due to the centrifugal force exerted by the rotation about the body's axis. A rotating body tends to form an oblate spheroid rather than a sphere.

Orbital mechanics Field of classical mechanics concerned with the motion of spacecraft

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Orbital speed speed at which it orbits around the barycenter of a system, usually around a more massive body. It can be used to refer to either the mean orbital speed, i.e. the average speed as it completes an orbit, or the speed at a particular point in its orbit

In gravitationally bound systems, the orbital speed of an astronomical body or object is the speed at which it orbits around either the barycenter or, if one object is much more massive than the other bodies in the system, its speed relative to the center of mass of the most massive body.

Schwarzschild radius distance from a massive body where the escape velocity equals the speed of light

The Schwarzschild radius is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.

Gaussian gravitational constant

The Gaussian gravitational constant is a parameter used in the orbital mechanics of the solar system. It relates the orbital period to the orbit's semi-major axis and the mass of the orbiting body in Solar masses.

Hill sphere The region in which an astronomical body dominates the attraction of satellites

The Hill sphere or Roche sphere of an astronomical body is the region in which it dominates the attraction of satellites. The outer shell of that region constitutes a zero-velocity surface. To be retained by a planet, a moon must have an orbit that lies within the planet's Hill sphere. That moon would, in turn, have a Hill sphere of its own. Any object within that distance would tend to become a satellite of the moon, rather than of the planet itself. One simple view of the extent of the Solar System is the Hill sphere of the Sun with respect to local stars and the galactic nucleus.

Sun-synchronous orbit type of geocentric orbit

A Sun-synchronous orbit is a nearly polar orbit around a planet, in which the satellite passes over any given point of the planet's surface at the same local mean solar time. More technically, it is an orbit arranged so that it precesses through one complete revolution each year, so it always maintains the same relationship with the Sun.

Hyperbolic trajectory trajectory of any object around a central body with more than enough speed to escape the central objects gravitational pull

In astrodynamics or celestial mechanics, a hyperbolic trajectory is the trajectory of any object around a central body with more than enough speed to escape the central object's gravitational pull. The name derives from the fact that according to Newtonian theory such an orbit has the shape of a hyperbola. In more technical terms this can be expressed by the condition that the orbital eccentricity is greater than one.

Elliptic orbit Keplers orbit with an eccentricity of less than 1

In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0. In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1. In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1.

In orbital mechanics, mean motion is the angular speed required for a body to complete one orbit, assuming constant speed in a circular orbit which completes in the same time as the variable speed, elliptical orbit of the actual body. The concept applies equally well to a small body revolving about a large, massive primary body or to two relatively same-sized bodies revolving about a common center of mass. While nominally a mean, and theoretically so in the case of two-body motion, in practice the mean motion is not typically an average over time for the orbits of real bodies, which only approximate the two-body assumption. It is rather the instantaneous value which satisfies the above conditions as calculated from the current gravitational and geometric circumstances of the body's constantly-changing, perturbed orbit.

The free-fall time is the characteristic time that would take a body to collapse under its own gravitational attraction, if no other forces existed to oppose the collapse. As such, it plays a fundamental role in setting the timescale for a wide variety of astrophysical processes—from star formation to helioseismology to supernovae—in which gravity plays a dominant role.

A gravity train is a theoretical means of transportation for purposes of commuting between two points on the surface of a sphere, by following a straight tunnel connecting the two points through the interior of the sphere.

For the majority of numbered asteroids, almost nothing is known apart from a few physical parameters and orbital elements and some physical characteristics are often only estimated. The physical data is determined by making certain standard assumptions.

"Clearing the neighbourhood around its orbit" is one of three necessary criteria for a celestial body to be considered a planet in the Solar System, according to the definition adopted in 2006 by the International Astronomical Union (IAU). In 2015, a proposal was made to extend this definition to exoplanets.

The gravitational two-body problem concerns the motion of two point particles that interact only with each other, due to gravity. This means that influences from any third body are neglected. For approximate results that is often suitable. It also means that the two bodies stay clear of each other, that is, the two do not collide, and one body does not pass through the other's atmosphere. Even if they do, the theory still holds for the part of the orbit where they don't. Apart from these considerations a spherically symmetric body can be approximated by a point mass.