Tidal locking results in the Moon rotating about its axis in about the same time it takes to orbit Earth. Except for libration
effects, this results in the Moon keeping the same face turned toward
Earth, as seen in the left figure. (The Moon is shown in polar view, and
is not drawn to scale.) If the Moon were not rotating at all, it would
alternately show its near and far sides to Earth, while moving around
Earth in orbit, as shown in the right figure.
A side view of the Pluto-Charon system. Pluto and Charon are tidally locked to each other. Charon is massive enough that the barycenter of Pluto's system lies outside of Pluto; thus Pluto and Charon are sometimes considered to be a binary system.
Tidal locking (also called gravitational locking, captured rotation and spin-orbit locking), in the most well-known case, occurs when an orbiting astronomical body always has the same face toward the object it is orbiting. This is known as synchronous rotation:
the tidally locked body takes just as long to rotate around its own
axis as it does to revolve around its partner. For example, the same
side of the Moon always faces the Earth, although there is some variability because the Moon's orbit is not perfectly circular. Usually, only the satellite is tidally locked to the larger body.
However, if both the difference in mass between the two bodies and the
distance between them are relatively small, each may be tidally locked
to the other; this is the case for Pluto and Charon.
The effect arises between two bodies when their gravitational
interaction slows a body's rotation until it becomes tidally locked.
Over many millions of years, the interaction forces changes to their
orbits and rotation rates as a result of energy exchange and heat dissipation.
When one of the bodies reaches a state where there is no longer any net
change in its rotation rate over the course of a complete orbit, it is
said to be tidally locked.
The object tends to stay in this state when leaving it would require
adding energy back into the system. The object's orbit may migrate over
time so as to undo the tidal lock, for example, if a giant planet
perturbs the object.
Not every case of tidal locking involves synchronous rotation.
With Mercury, for example, this tidally locked planet completes three
rotations for every two revolutions around the Sun, a 3:2 spin-orbit
resonance. In the special case where an orbit is nearly circular and the
body's rotation axis is not significantly tilted, such as the Moon,
tidal locking results in the same hemisphere of the revolving object
constantly facing its partner.
However, in this case the exact same portion of the body does not always
face the partner on all orbits. There can be some shifting due to variations in the locked body's orbital velocity and the inclination of its rotation axis.
Mechanism
If
the tidal bulges on a body (green) are misaligned with the major axis
(red), the tidal forces (blue) exert a net torque on that body that
twists the body toward the direction of realignment
Consider a pair of co-orbiting objects, A and B. The change in rotation rate necessary to tidally lock body B to the larger body A is caused by the torque applied by A's gravity on bulges it has induced on B by tidal forces.
The gravitational force from object A upon B will vary with
distance, being greatest at the nearest surface to A and least at the
most distant. This creates a gravitational gradient across object B that will distort its equilibrium
shape slightly. The body of object B will become elongated along the
axis oriented toward A, and conversely, slightly reduced in dimension in
directions orthogonal
to this axis. The elongated distortions are known as tidal bulges. (For
the solid Earth, these bulges can reach displacements of up to around
0.4 metres (1.3 ft).)
When B is not yet tidally locked, the bulges travel over its surface
due to orbital motions, with one of the two "high" tidal bulges
traveling close to the point where body A is overhead. For large
astronomical bodies that are nearly spherical due to self-gravitation, the tidal distortion produces a slightly prolate spheroid, i.e. an axially symmetric ellipsoid that is elongated along its major axis. Smaller bodies also experience distortion, but this distortion is less regular.
The material of B exerts resistance to this periodic reshaping
caused by the tidal force. In effect, some time is required to reshape B
to the gravitational equilibrium shape, by which time the forming
bulges have already been carried some distance away from the A–B axis by
B's rotation. Seen from a vantage point in space, the points of maximum
bulge extension are displaced from the axis oriented toward A. If B's
rotation period is shorter than its orbital period, the bulges are
carried forward of the axis oriented toward A in the direction of
rotation, whereas if B's rotation period is longer, the bulges instead
lag behind.
Because the bulges are now displaced from the A–B axis, A's
gravitational pull on the mass in them exerts a torque on B. The torque
on the A-facing bulge acts to bring B's rotation in line with its
orbital period, whereas the "back" bulge, which faces away from A, acts
in the opposite sense. However, the bulge on the A-facing side is closer
to A than the back bulge by a distance of approximately B's diameter,
and so experiences a slightly stronger gravitational force and torque.
The net resulting torque from both bulges, then, is always in the
direction that acts to synchronize B's rotation with its orbital period,
leading eventually to tidal locking.
Orbital changes
If
rotational frequency is larger than orbital frequency, a small torque
counteracting the rotation arises, eventually locking the frequencies
(situation depicted in green)
The angular momentum of the whole A–B system is conserved in this process, so that when B slows down and loses rotational angular momentum, its orbital
angular momentum is boosted by a similar amount (there are also some
smaller effects on A's rotation). This results in a raising of B's orbit
about A in tandem with its rotational slowdown. For the other case
where B starts off rotating too slowly, tidal locking both speeds up its
rotation, and lowers its orbit.
Locking of the larger body
The tidal locking effect is also experienced by the larger body A,
but at a slower rate because B's gravitational effect is weaker due to
B's smaller mass. For example, Earth's rotation is gradually being
slowed by the Moon, by an amount that becomes noticeable over geological
time as revealed in the fossil record.
Current estimations are that this (together with the tidal influence of
the Sun) has helped lengthen the Earth day from about 6 hours to the
current 24 hours (over ≈ 4½ billion years). Currently, atomic clocks show that Earth's day lengthens, on average, by about 15 microseconds every year. Given enough time, this would create a mutual tidal locking between Earth and the Moon. The length of the Earth's day would increase and the length of a lunar month would also increase. The Earth's sidereal day would eventually have the same length as the Moon's orbital period,
about 47 times the length of the Earth's day at present. However, Earth
is not expected to become tidally locked to the Moon before the Sun
becomes a red giant and engulfs Earth and the Moon.
For bodies of similar size the effect may be of comparable size
for both, and both may become tidally locked to each other on a much
shorter timescale. An example is the dwarf planetPluto and its satellite Charon. They have already reached a state where Charon is visible from only one hemisphere of Pluto and vice versa.
Eccentric orbits
A widely spread misapprehension is that a tidally locked body
permanently turns one side to its host.
— Heller et al. (2011)
For orbits that do not have an eccentricity close to zero, the rotation rate tends to become locked with the orbital speed when the body is at periapsis,
which is the point of strongest tidal interaction between the two
objects. If the orbiting object has a companion, this third body can
cause the rotation rate of the parent object to vary in an oscillatory
manner. This interaction can also drive an increase in orbital
eccentricity of the orbiting object around the primary – an effect known
as eccentricity pumping.
In some cases where the orbit is eccentric and the tidal effect is relatively weak, the smaller body may end up in a so-called spin–orbit resonance,
rather than being tidally locked. Here, the ratio of the rotation
period of a body to its own orbital period is some simple fraction
different from 1:1. A well known case is the rotation of Mercury, which is locked to its own orbit around the Sun in a 3:2 resonance.
Many exoplanets (especially the close-in ones) are expected to be
in spin–orbit resonances higher than 1:1. A Mercury-like terrestrial
planet can, for example, become captured in a 3:2, 2:1, or 5:2
spin–orbit resonance, with the probability of each being dependent on
the orbital eccentricity.
Occurrence
Moons
Due to tidal locking, the inhabitants of the central body will never be able to see the satellite's green area.
Most major moons in the Solar System − the gravitationally rounded satellites − are tidally locked with their primaries, because they orbit very closely and tidal force increases rapidly (as a cubic function) with decreasing distance. Notable exceptions are the irregular outer satellites of the gas giants, which orbit much farther away than the large well-known moons.
Pluto and Charon
are an extreme example of a tidal lock. Charon is a relatively large
moon in comparison to its primary and also has a very close orbit. This results in Pluto and Charon being mutually tidally locked. Pluto's other moons are not tidally locked; Styx, Nix, Kerberos, and Hydra all rotate chaotically due to the influence of Charon.
The tidal locking situation for asteroid moons is largely unknown, but closely orbiting binaries are expected to be tidally locked, as well as contact binaries.
Earth's Moon
Because Earth's Moon is 1:1 tidally locked, only one side is visible from Earth.
Earth's Moon's rotation and orbital periods are tidally locked with
each other, so no matter when the Moon is observed from Earth the same
hemisphere of the Moon is always seen. The far side of the Moon was not seen until 1959, when photographs of most of the far side were transmitted from the Soviet spacecraft Luna 3.
When the Earth is observed from the moon, the Earth does not
appear to translate across the sky but appears to remain in the same
place, rotating on its own axis.
Despite the Moon's rotational and orbital periods being exactly
locked, about 59% of the Moon's total surface may be seen with repeated
observations from Earth due to the phenomena of libration and parallax. Librations are primarily caused by the Moon's varying orbital speed due to the eccentricity
of its orbit: this allows up to about 6° more along its perimeter to be
seen from Earth. Parallax is a geometric effect: at the surface of
Earth we are offset from the line through the centers of Earth and Moon,
and because of this we can observe a bit (about 1°) more around the
side of the Moon when it is on our local horizon.
Planets
It was thought for some time that Mercury
was in synchronous rotation with the Sun. This was because whenever
Mercury was best placed for observation, the same side faced inward.
Radar observations in 1965 demonstrated instead that Mercury has a 3:2
spin–orbit resonance, rotating three times for every two revolutions
around the Sun, which results in the same positioning at those
observation points. Modeling has demonstrated that Mercury was captured
into the 3:2 spin–orbit state very early in its history, within 20 (and
more likely even 10) million years after its formation.
Venus's
583.92-day interval between successive close approaches to Earth is
equal to 5.001444 Venusian solar days, making approximately the same
face visible from Earth at each close approach. Whether this
relationship arose by chance or is the result of some kind of tidal
locking with Earth is unknown.
Proxima Centauri b, the "Earth-like planet" discovered in 2016 that orbits around the star Proxima Centauri is tidally locked, either in synchronized rotation, or otherwise expresses a 3:2 spin–orbit resonance like that of Mercury.
One form of hypothetical tidal locked exoplanets are eyeball planets, that in turn are divided into "hot" and "cold" eyeball planets.
Stars
Close binary stars throughout the universe are expected to be tidally locked with each other, and extrasolar planets
that have been found to orbit their primaries extremely closely are
also thought to be tidally locked to them. An unusual example, confirmed
by MOST, may be Tau Boötis, a star that is probably tidally locked by its planet Tau Boötis b. If so, the tidal locking is almost certainly mutual.
However, since stars are gaseous bodies that can rotate with a
different rate at different latitudes, the tidal lock is with Tau
Boötis's magnetic field.
Timescale
An estimate of the time for a body to become tidally locked can be obtained using the following formula:
and are generally very poorly known except for the Moon, which has . For a really rough estimate it is common to take (perhaps conservatively, giving overestimated locking times), and
where
is the density of the satellite
is the surface gravity of the satellite
is the rigidity of the satellite. This can be roughly taken as 3×1010 N·m−2 for rocky objects and 4×109 N·m−2 for icy ones.
Even knowing the size and density of the satellite leaves many parameters that must be estimated (especially ω, Q, and μ),
so that any calculated locking times obtained are expected to be
inaccurate, even to factors of ten. Further, during the tidal locking
phase the semi-major axis may have been significantly different from that observed nowadays due to subsequent tidal acceleration, and the locking time is extremely sensitive to this value.
Because the uncertainty is so high, the above formulas can be
simplified to give a somewhat less cumbersome one. By assuming that the
satellite is spherical, ,
and it is sensible to guess one revolution every 12 hours in the
initial non-locked state (most asteroids have rotational periods between
about 2 hours and about 2 days)
with masses in kilograms, distances in meters, and in newtons per meter squared; can be roughly taken as 3×1010 N·m−2 for rocky objects and 4×109 N·m−2 for icy ones.
There is an extremely strong dependence on semi-major axis .
For the locking of a primary body to its satellite as in the case
of Pluto, the satellite and primary body parameters can be swapped.
One conclusion is that, other things being equal (such as and ), a large moon will lock faster than a smaller moon at the same orbital distance from the planet because grows as the cube of the satellite radius . A possible example of this is in the Saturn system, where Hyperion is not tidally locked, whereas the larger Iapetus,
which orbits at a greater distance, is. However, this is not clear cut
because Hyperion also experiences strong driving from the nearby Titan, which forces its rotation to be chaotic.
The above formulae for the timescale of locking may be off by
orders of magnitude, because they ignore the frequency dependence of .
More importantly, they may be inapplicable to viscous binaries (double
stars, or double asteroids that are rubble), because the spin–orbit
dynamics of such bodies is defined mainly by their viscosity, not
rigidity.
The
most successful detection methods of exoplanets (transits and radial
velocities) suffer from a clear observational bias favoring the
detection of planets near the star; thus, 85% of the exoplanets detected
are inside the tidal locking zone, which makes it difficult to estimate
the true incidence of this phenomenon. Tau Boötis is known to be locked to the close-orbiting giant planetTau Boötis b.
Bodies likely to be locked
Solar System
Based
on comparison between the likely time needed to lock a body to its
primary, and the time it has been in its present orbit (comparable with
the age of the Solar System for most planetary moons), a number of moons
are thought to be locked. However their rotations are not known or not
known enough. These are:
Historically, the apparent motions of the planets were described by European and Arabic philosophers using the idea of celestial spheres.
This model posited the existence of perfect moving spheres or rings to
which the stars and planets were attached. It assumed the heavens were
fixed apart from the motion of the spheres, and was developed without
any understanding of gravity. After the planets' motions were more
accurately measured, theoretical mechanisms such as deferent and epicycles
were added. Although the model was capable of reasonably accurately
predicting the planets' positions in the sky, more and more epicycles
were required as the measurements became more accurate, hence the model
became increasingly unwieldy. Originally geocentric, it was modified by Copernicus
to place the Sun at the centre to help simplify the model. The model
was further challenged during the 16th century, as comets were observed
traversing the spheres.
The basis for the modern understanding of orbits was first formulated by Johannes Kepler whose results are summarised in his three laws of planetary motion. First, he found that the orbits of the planets in our Solar System are elliptical, not circular (or epicyclic), as had previously been believed, and that the Sun is not located at the center of the orbits, but rather at one focus.
Second, he found that the orbital speed of each planet is not constant,
as had previously been thought, but rather that the speed depends on
the planet's distance from the Sun. Third, Kepler found a universal
relationship between the orbital properties of all the planets orbiting
the Sun. For the planets, the cubes of their distances from the Sun are
proportional to the squares of their orbital periods. Jupiter and Venus,
for example, are respectively about 5.2 and 0.723 AU
distant from the Sun, their orbital periods respectively about 11.86
and 0.615 years. The proportionality is seen by the fact that the ratio
for Jupiter, 5.23/11.862, is practically equal to that for Venus, 0.7233/0.6152, in accord with the relationship. Idealised orbits meeting these rules are known as Kepler orbits.
The lines traced out by orbits dominated by the gravity of a central source are conic sections: the shapes of the curves of intersection between a plane and a cone. Parabolic (1) and hyperbolic (3) orbits are escape orbits, whereas elliptical and circular orbits (2) are captive.
This image shows the four trajectory categories with the gravitational potential well
of the central mass's field of potential energy shown in black and the
height of the kinetic energy of the moving body shown in red extending
above that, correlating to changes in speed as distance changes
according to Kepler's laws.
Isaac Newton demonstrated that Kepler's laws were derivable from his theory of gravitation and that, in general, the orbits of bodies subject to gravity were conic sections
(this assumes that the force of gravity propagates instantaneously).
Newton showed that, for a pair of bodies, the orbits' sizes are in
inverse proportion to their masses, and that those bodies orbit their common center of mass.
Where one body is much more massive than the other (as is the case of
an artificial satellite orbiting a planet), it is a convenient
approximation to take the center of mass as coinciding with the center
of the more massive body.
Advances in Newtonian mechanics were then used to explore
variations from the simple assumptions behind Kepler orbits, such as the
perturbations due to other bodies, or the impact of spheroidal rather
than spherical bodies. Lagrange (1736–1813) developed a new approach to Newtonian mechanics emphasizing energy more than force, and made progress on the three body problem, discovering the Lagrangian points. In a dramatic vindication of classical mechanics, in 1846 Urbain Le Verrier was able to predict the position of Neptune based on unexplained perturbations in the orbit of Uranus.
Albert Einstein (1879-1955) in his 1916 paper The Foundation of the General Theory of Relativity explained that gravity was due to curvature of space-time and removed Newton's assumption that changes propagate instantaneously. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in understanding orbits. In relativity theory,
orbits follow geodesic trajectories which are usually approximated very
well by the Newtonian predictions (except where there are very strong
gravity fields and very high speeds) but the differences are measurable.
Essentially all the experimental evidence that can distinguish between
the theories agrees with relativity theory to within experimental
measurement accuracy. The original vindication of general relativity is
that it was able to account for the remaining unexplained amount in precession of Mercury's perihelion
first noted by Le Verrier. However, Newton's solution is still used
for most short term purposes since it is significantly easier to use and
sufficiently accurate.
Owing to mutual gravitational perturbations, the eccentricities of the planetary orbits vary over time. Mercury, the smallest planet in the Solar System, has the most eccentric orbit. At the present epoch, Mars has the next largest eccentricity while the smallest orbital eccentricities are seen with Venus and Neptune.
As two objects orbit each other, the periapsis is that point at which the two objects are closest to each other and the apoapsis is that point at which they are the farthest. (More specific terms are used for specific bodies. For example, perigee and apogee are the lowest and highest parts of an orbit around Earth, while perihelion and aphelion are the closest and farthest points of an orbit around the Sun.)
In the case of planets orbiting a star, the mass of the star and
all its satellites are calculated to be at a single point called the
barycenter. The paths of all the star's satellites are elliptical orbits
about that barycenter. Each satellite in that system will have its own
elliptical orbit with the barycenter at one focal point of that ellipse.
At any point along its orbit, any satellite will have a certain value
of kinetic and potential energy with respect to the barycenter, and that
energy is a constant value at every point along its orbit. As a result,
as a planet approaches periapsis, the planet will increase in speed as its potential energy decreases; as a planet approaches apoapsis, its velocity will decrease as its potential energy increases.
Understanding orbits
There are a few common ways of understanding orbits:
A force, such as gravity, pulls an object into a curved path as it attempts to fly off in a straight line.
As the object is pulled toward the massive body, it falls toward that body. However, if it has enough tangential velocity
it will not fall into the body but will instead continue to follow the
curved trajectory caused by that body indefinitely. The object is then
said to be orbiting the body.
As an illustration of an orbit around a planet, the Newton's cannonball model may prove useful (see image below). This is a 'thought experiment',
in which a cannon on top of a tall mountain is able to fire a
cannonball horizontally at any chosen muzzle speed. The effects of air
friction on the cannonball are ignored (or perhaps the mountain is high
enough that the cannon is above the Earth's atmosphere, which is the
same thing).
Conic
sections describe the possible orbits (yellow) of small objects around
the Earth. A projection of these orbits onto the gravitational potential
(blue) of the Earth makes it possible to determine the orbital energy
at each point in space.
If the cannon fires its ball with a low initial speed, the trajectory
of the ball curves downward and hits the ground (A). As the firing
speed is increased, the cannonball hits the ground farther (B) away from
the cannon, because while the ball is still falling towards the ground,
the ground is increasingly curving away from it (see first point,
above). All these motions are actually "orbits" in a technical sense –
they are describing a portion of an elliptical path around the center of
gravity – but the orbits are interrupted by striking the Earth.
If the cannonball is fired with sufficient speed, the ground
curves away from the ball at least as much as the ball falls – so the
ball never strikes the ground. It is now in what could be called a
non-interrupted, or circumnavigating, orbit. For any specific
combination of height above the center of gravity and mass of the
planet, there is one specific firing speed (unaffected by the mass of
the ball, which is assumed to be very small relative to the Earth's
mass) that produces a circular orbit, as shown in (C).
As the firing speed is increased beyond this, non-interrupted
elliptic orbits are produced; one is shown in (D). If the initial firing
is above the surface of the Earth as shown, there will also be
non-interrupted elliptical orbits at slower firing speed; these will
come closest to the Earth at the point half an orbit beyond, and
directly opposite the firing point, below the circular orbit.
At a specific horizontal firing speed called escape velocity, dependent on the mass of the planet, an open orbit (E) is achieved that has a parabolic path. At even greater speeds the object will follow a range of hyperbolic trajectories.
In a practical sense, both of these trajectory types mean the object is
"breaking free" of the planet's gravity, and "going off into space"
never to return.
The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:
It is worth noting that orbital rockets are launched vertically at
first to lift the rocket above the atmosphere (which causes frictional
drag), and then slowly pitch over and finish firing the rocket engine
parallel to the atmosphere to achieve orbit speed.
Once in orbit, their speed keeps them in orbit above the
atmosphere. If e.g., an elliptical orbit dips into dense air, the
object will lose speed and re-enter (i.e. fall). Occasionally a space
craft will intentionally intercept the atmosphere, in an act commonly
referred to as an aerobraking maneuver.
Newton's laws of motion
Newton's law of gravitation and laws of motion for two-body problems
In most situations relativistic effects can be neglected, and Newton's laws
give a sufficiently accurate description of motion. The acceleration of
a body is equal to the sum of the forces acting on it, divided by its
mass, and the gravitational force acting on a body is proportional to
the product of the masses of the two attracting bodies and decreases
inversely with the square of the distance between them. To this
Newtonian approximation, for a system of two-point masses or spherical
bodies, only influenced by their mutual gravitation (called a two-body problem),
their trajectories can be exactly calculated. If the heavier body is
much more massive than the smaller, as in the case of a satellite or
small moon orbiting a planet or for the Earth orbiting the Sun, it is
accurate enough and convenient to describe the motion in terms of a coordinate system
that is centered on the heavier body, and we say that the lighter body
is in orbit around the heavier. For the case where the masses of two
bodies are comparable, an exact Newtonian solution is still sufficient
and can be had by placing the coordinate system at the center of mass of
the system.
Defining gravitational potential energy
Energy is associated with gravitational fields. A stationary body far from another can do external work if it is pulled towards it, and therefore has gravitational potential energy.
Since work is required to separate two bodies against the pull of
gravity, their gravitational potential energy increases as they are
separated, and decreases as they approach one another. For point masses
the gravitational energy decreases to zero as they approach zero
separation. It is convenient and conventional to assign the potential
energy as having zero value when they are an infinite distance apart,
and hence it has a negative value (since it decreases from zero) for
smaller finite distances.
Orbital energies and orbit shapes
When only two gravitational bodies interact, their orbits follow a conic section. The orbit can be open (implying the object never returns) or closed (returning). Which it is depends on the total energy (kinetic + potential energy) of the system. In the case of an open orbit, the speed at any position of the orbit is at least the escape velocity
for that position, in the case of a closed orbit, the speed is always
less than the escape velocity. Since the kinetic energy is never
negative, if the common convention is adopted of taking the potential
energy as zero at infinite separation, the bound orbits will have
negative total energy, the parabolic trajectories zero total energy, and
hyperbolic orbits positive total energy.
An open orbit will have a parabolic shape if it has velocity of
exactly the escape velocity at that point in its trajectory, and it will
have the shape of a hyperbola
when its velocity is greater than the escape velocity. When bodies with
escape velocity or greater approach each other, they will briefly curve
around each other at the time of their closest approach, and then
separate, forever.
All closed orbits have the shape of an ellipse.
A circular orbit is a special case, wherein the foci of the ellipse
coincide. The point where the orbiting body is closest to Earth is
called the perigee,
and is called the periapsis (less properly, "perifocus" or
"pericentron") when the orbit is about a body other than Earth. The
point where the satellite is farthest from Earth is called the apogee, apoapsis, or sometimes apifocus or apocentron. A line drawn from periapsis to apoapsis is the line-of-apsides. This is the major axis of the ellipse, the line through its longest part.
Kepler's laws
Bodies
following closed orbits repeat their paths with a certain time called
the period. This motion is described by the empirical laws of Kepler,
which can be mathematically derived from Newton's laws. These can be
formulated as follows:
The orbit of a planet around the Sun is an ellipse, with the Sun in one of the focal points of that ellipse. [This focal point is actually the barycenter
of the Sun-planet system; for simplicity this explanation assumes the
Sun's mass is infinitely larger than that planet's.] The planet's orbit
lies in a plane, called the orbital plane.
The point on the orbit closest to the attracting body is the periapsis.
The point farthest from the attracting body is called the apoapsis.
There are also specific terms for orbits about particular bodies; things
orbiting the Sun have a perihelion and aphelion, things orbiting the Earth have a perigee and apogee, and things orbiting the Moon have a perilune and apolune (or periselene and aposelene respectively). An orbit around any star, not just the Sun, has a periastron and an apastron.
As the planet moves in its orbit, the line from the Sun to planet sweeps a constant area of the orbital plane
for a given period of time, regardless of which part of its orbit the
planet traces during that period of time. This means that the planet
moves faster near its perihelion than near its aphelion,
because at the smaller distance it needs to trace a greater arc to
cover the same area. This law is usually stated as "equal areas in equal
time."
For a given orbit, the ratio of the cube of its semi-major axis to the square of its period is constant.
Limitations of Newton's law of gravitation
Note that while bound orbits of a point mass or a spherical body with a Newtonian gravitational field are closed ellipses,
which repeat the same path exactly and indefinitely, any non-spherical
or non-Newtonian effects (such as caused by the slight oblateness of the
Earth, or by relativistic effects, thereby changing the gravitational field's behavior with distance) will cause the orbit's shape to depart from the closed ellipses characteristic of Newtonian two-body motion. The two-body solutions were published by Newton in Principia in 1687. In 1912, Karl Fritiof Sundman developed a converging infinite series that solves the three-body problem; however, it converges too slowly to be of much use. Except for special cases like the Lagrangian points, no method is known to solve the equations of motion for a system with four or more bodies.
Approaches to many-body problems
Rather
than an exact closed form solution, orbits with many bodies can be
approximated with arbitrarily high accuracy. These approximations take
two forms:
One form takes the pure elliptic motion as a basis, and adds perturbation
terms to account for the gravitational influence of multiple bodies.
This is convenient for calculating the positions of astronomical bodies.
The equations of motion of the moons, planets and other bodies are
known with great accuracy, and are used to generate tables for celestial navigation. Still, there are secular phenomena that have to be dealt with by post-Newtonian methods.
The differential equation
form is used for scientific or mission-planning purposes. According to
Newton's laws, the sum of all the forces acting on a body will equal the
mass of the body times its acceleration (F = ma). Therefore
accelerations can be expressed in terms of positions. The perturbation
terms are much easier to describe in this form. Predicting subsequent
positions and velocities from initial values of position and velocity
corresponds to solving an initial value problem.
Numerical methods calculate the positions and velocities of the objects
a short time in the future, then repeat the calculation ad nauseam.
However, tiny arithmetic errors from the limited accuracy of a
computer's math are cumulative, which limits the accuracy of this
approach.
Differential simulations with large numbers of objects perform the
calculations in a hierarchical pairwise fashion between centers of mass.
Using this scheme, galaxies, star clusters and other large assemblages
of objects have been simulated.
Newtonian analysis of orbital motion
The Earth follows an ellipse round the sun.
But unlike the ellipse followed by a pendulum or an object attached to a
spring, the sun is at a focal point of the ellipse and not at its
centre.
The following derivation applies to such an elliptical orbit.
We start only with the Newtonian
law of gravitation stating that the gravitational acceleration towards
the central body is related to the inverse of the square of the distance
between them, namely
eq 1.
where F2 is the force acting on the mass m2 caused by the gravitational attraction mass m1 has for m2, G is the universal gravitational constant, and r is the distance between the two masses centers.
From Newton's Second Law, the summation of the forces acting on m2 related to that bodies acceleration:
eq 2.
where A2 is the acceleration of m2 caused by the force of gravitational attraction F2 of m1 acting on m2.
Combining Eq 1 and 2:
Solving for the acceleration, A2:
where is the standard gravitational parameter, in this case . It is understood that the system being described is m2, hence the subscripts can be dropped.
We assume that the central body is massive enough that it can be
considered to be stationary and we ignore the more subtle effects of general relativity.
When a pendulum or an object attached to a spring swings in an ellipse,
the inward acceleration/force is proportional to the distance
Due to the way vectors add, the component of the force in the or in the directions are also proportionate to the respective
components of the distances, . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations and of the ellipse. In contrast, with the decreasing relationship , the dimensions cannot be separated.
The location of the orbiting object at the current time is located in the plane using
Vector calculus in polar coordinates both with the standard Euclidean basis and with the polar basis
with the origin coinciding with the center of force.
Let be the distance between the object and the center and
be the angle it has rotated.
Let and be the standard Euclidean bases and let and be the radial and transverse polar
basis with the first being the unit vector pointing from the central
body to the current location of the orbiting object and the second being
the orthogonal unit vector pointing in the direction that the orbiting
object would travel if orbiting in a counter clockwise circle. Then the
vector to the orbiting object is
We use and
to denote the standard derivatives of how this distance and angle
change over time. We take the derivative of a vector to see how it
changes over time by subtracting its location at time
from that at time and dividing by . The result is also a vector. Because our basis vector moves as the object orbits, we start by differentiating it.
From time to ,
the vector keeps its beginning at the origin and rotates from
angle to which moves its head a distance in the perpendicular direction giving a derivative of .
We can now find the velocity and acceleration of our orbiting object.
The coefficients of
and give the accelerations in the radial and transverse directions.
As said, Newton gives this first due to gravity is and the second is zero.
(1)
(2)
Equation (2) can be rearranged using integration by parts.
We can multiply through by because it is not zero unless the orbiting object crashes.
Then having the derivative be zero gives that the function is a constant.
(3)
which is actually the theoretical proof of Kepler's second law (A line joining a planet and the Sun sweeps out equal areas during equal intervals of time). The constant of integration, h, is the angular momentum per unit mass.
In order to get an equation for the orbit from equation (1), we need to eliminate time.
In polar coordinates, this would express the distance of the orbiting object from the center as a function of its angle . However, it is easier to
introduce the auxiliary variable and to express as a function of . Derivatives of with respect to time may be rewritten as derivatives of with respect to angle.
(reworking (3))
Plugging these into (1) gives
So for the gravitational force – or, more generally, for any inverse square force law – the right hand side of the equation becomes a constant and the equation is seen to be the harmonic equation (up to a shift of origin of the dependent variable). The solution is:
where A and θ0 are arbitrary constants.
This resulting equation of the orbit of the object is that of an ellipse in Polar form relative to one of the focal points. This is put into a more standard form by
letting be the eccentricity,
letting be the semi-major axis.
Finally, letting so the long axis of the ellipse is along the positive x coordinate.
The analysis so far has been two dimensional; it turns out that an unperturbed
orbit is two-dimensional in a plane fixed in space, and thus the
extension to three dimensions requires simply rotating the
two-dimensional plane into the required angle relative to the poles of
the planetary body involved.
The rotation to do this in three dimensions requires three
numbers to uniquely determine; traditionally these are expressed as
three angles.
Orbital period
The orbital period is simply how long an orbiting body takes to complete one orbit.
Specifying orbits
Six parameters are required to specify a Keplerian orbit
about a body. For example, the three numbers that specify the body's
initial position, and the three values that specify its velocity will
define a unique orbit that can be calculated forwards (or backwards) in
time. However, traditionally the parameters used are slightly different.
The traditionally used set of orbital elements is called the set of Keplerian elements, after Johannes Kepler and his laws. The Keplerian elements are six:
In principle once the orbital elements are known for a body, its
position can be calculated forward and backwards indefinitely in time.
However, in practice, orbits are affected or perturbed,
by other forces than simple gravity from an assumed point source (see
the next section), and thus the orbital elements change over time.
Orbital perturbations
An
orbital perturbation is when a force or impulse which is much smaller
than the overall force or average impulse of the main gravitating body
and which is external to the two orbiting bodies causes an acceleration,
which changes the parameters of the orbit over time.
Radial, prograde and transverse perturbations
A small radial impulse given to a body in orbit changes the eccentricity, but not the orbital period (to first order). A prograde or retrograde impulse (i.e. an impulse applied along the orbital motion) changes both the eccentricity and the orbital period. Notably, a prograde impulse at periapsis raises the altitude at apoapsis,
and vice versa, and a retrograde impulse does the opposite. A
transverse impulse (out of the orbital plane) causes rotation of the orbital plane without changing the period or eccentricity. In all instances, a closed orbit will still intersect the perturbation point.
Orbital decay
If an orbit is about a planetary body with significant atmosphere, its orbit can decay because of drag. Particularly at each periapsis,
the object experiences atmospheric drag, losing energy. Each time, the
orbit grows less eccentric (more circular) because the object loses
kinetic energy precisely when that energy is at its maximum. This is
similar to the effect of slowing a pendulum at its lowest point; the
highest point of the pendulum's swing becomes lower. With each
successive slowing more of the orbit's path is affected by the
atmosphere and the effect becomes more pronounced. Eventually, the
effect becomes so great that the maximum kinetic energy is not enough to
return the orbit above the limits of the atmospheric drag effect. When
this happens the body will rapidly spiral down and intersect the central
body.
The bounds of an atmosphere vary wildly. During a solar maximum, the Earth's atmosphere causes drag up to a hundred kilometres higher than during a solar minimum.
Some satellites with long conductive tethers can also experience orbital decay because of electromagnetic drag from the Earth's magnetic field.
As the wire cuts the magnetic field it acts as a generator, moving
electrons from one end to the other. The orbital energy is converted to
heat in the wire.
Orbits can be artificially influenced through the use of rocket
engines which change the kinetic energy of the body at some point in its
path. This is the conversion of chemical or electrical energy to
kinetic energy. In this way changes in the orbit shape or orientation
can be facilitated.
Another method of artificially influencing an orbit is through the use of solar sails or magnetic sails.
These forms of propulsion require no propellant or energy input other
than that of the Sun, and so can be used indefinitely.
Orbital decay can occur due to tidal forces for objects below the synchronous orbit for the body they're orbiting. The gravity of the orbiting object raises tidal bulges
in the primary, and since below the synchronous orbit the orbiting
object is moving faster than the body's surface the bulges lag a short
angle behind it. The gravity of the bulges is slightly off of the
primary-satellite axis and thus has a component along the satellite's
motion. The near bulge slows the object more than the far bulge speeds
it up, and as a result the orbit decays. Conversely, the gravity of the
satellite on the bulges applies torque
on the primary and speeds up its rotation. Artificial satellites are
too small to have an appreciable tidal effect on the planets they orbit,
but several moons in the Solar System are undergoing orbital decay by
this mechanism. Mars' innermost moon Phobos is a prime example, and is expected to either impact Mars' surface or break up into a ring within 50 million years.
Orbits can decay via the emission of gravitational waves.
This mechanism is extremely weak for most stellar objects, only
becoming significant in cases where there is a combination of extreme
mass and extreme acceleration, such as with black holes or neutron stars that are orbiting each other closely.
Oblateness
The
standard analysis of orbiting bodies assumes that all bodies consist of
uniform spheres, or more generally, concentric shells each of uniform
density. It can be shown that such bodies are gravitationally equivalent
to point sources.
However, in the real world, many bodies rotate, and this introduces oblateness and distorts the gravity field, and gives a quadrupole
moment to the gravitational field which is significant at distances
comparable to the radius of the body. In the general case, the
gravitational potential of a rotating body such as, e.g., a planet is
usually expanded in multipoles accounting for the departures of it from
spherical symmetry. From the point of view of satellite dynamics, of
particular relevance are the so-called even zonal harmonic coefficients,
or even zonals, since they induce secular orbital perturbations which
are cumulative over time spans longer than the orbital period.
They do depend on the orientation of the body's symmetry axis in the
space, affecting, in general, the whole orbit, with the exception of the
semimajor axis.
Multiple gravitating bodies
The effects of other gravitating bodies can be significant. For example, the orbit of the Moon
cannot be accurately described without allowing for the action of the
Sun's gravity as well as the Earth's. One approximate result is that
bodies will usually have reasonably stable orbits around a heavier
planet or moon, in spite of these perturbations, provided they are
orbiting well within the heavier body's Hill sphere.
When there are more than two gravitating bodies it is referred to as an n-body problem. Most n-body problems have no closed form solution, although some special cases have been formulated.
Light radiation and stellar wind
For smaller bodies particularly, light and stellar wind can cause significant perturbations to the attitude and direction of motion of the body, and over time can be significant. Of the planetary bodies, the motion of asteroids is particularly affected over large periods when the asteroids are rotating relative to the Sun.
Strange orbits
Mathematicians
have discovered that it is possible in principle to have multiple
bodies in non-elliptical orbits that repeat periodically, although most
such orbits are not stable regarding small perturbations in mass,
position, or velocity. However, some special stable cases have been
identified, including a planar figure-eight orbit occupied by three moving bodies.
Further studies have discovered that nonplanar orbits are also
possible, including one involving 12 masses moving in 4 roughly
circular, interlocking orbits topologically equivalent to the edges of a cuboctahedron.
Finding such orbits naturally occurring in the universe is
thought to be extremely unlikely, because of the improbability of the
required conditions occurring by chance.
Astrodynamics
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 Newton's law of universal gravitation.
It is a core discipline within space mission design and control.
Celestial mechanics treats more broadly the orbital dynamics of systems
under the influence of gravity, including spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbit plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers. General relativity
is a more exact theory than Newton's laws for calculating orbits, and
is sometimes necessary for greater accuracy or in high-gravity
situations (such as orbits close to the Sun).
Both geosynchronous orbit (GSO) and geostationary orbit (GEO) are orbits around Earth matching Earth's sidereal rotation period. All geosynchronous and geostationary orbits have a semi-major axis of 42,164 km (26,199 mi).
All geostationary orbits are also geosynchronous, but not all
geosynchronous orbits are geostationary. A geostationary orbit stays
exactly above the equator, whereas a geosynchronous orbit may swing
north and south to cover more of the Earth's surface. Both complete one
full orbit of Earth per sidereal day (relative to the stars, not the
Sun).
Thus the constant has dimension density−1 time−2. This corresponds to the following properties.
Scaling of distances (including sizes of bodies, while keeping the densities the same) gives similar
orbits without scaling the time: if for example distances are halved,
masses are divided by 8, gravitational forces by 16 and gravitational
accelerations by 2. Hence velocities are halved and orbital periods
remain the same. Similarly, when an object is dropped from a tower, the
time it takes to fall to the ground remains the same with a scale model
of the tower on a scale model of the Earth.
Scaling of distances while keeping the masses the same (in the
case of point masses, or by reducing the densities) gives similar
orbits; if distances are multiplied by 4, gravitational forces and
accelerations are divided by 16, velocities are halved and orbital
periods are multiplied by 8.
When all densities are multiplied by 4, orbits are the same;
gravitational forces are multiplied by 16 and accelerations by 4,
velocities are doubled and orbital periods are halved.
When all densities are multiplied by 4, and all sizes are halved,
orbits are similar; masses are divided by 2, gravitational forces are
the same, gravitational accelerations are doubled. Hence velocities are
the same and orbital periods are halved.
In all these cases of scaling. if densities are multiplied by 4,
times are halved; if velocities are doubled, forces are multiplied by
16.
for an elliptical orbit with semi-major axisa, of a small body around a spherical body with radius r and average density ρ, where T is the orbital period. See also Kepler's Third Law.
Patents
The application of certain orbits or orbital maneuvers to specific useful purposes have been the subject of patents.
Tidal locking
Some bodies are tidally locked with other bodies, meaning that one
side of the celestial body is permanently facing its host object. This
is the case for Earth-Moon and Pluto-Charon system.