Orbit
In physics, an orbit is the gravitationally curved path of an object around a point in space, for example the orbit of a planet around the center of a star system, such as the Solar System.[1][2] Orbits of planets are typically elliptical.
Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics. For ease of calculation, relativity is commonly approximated by the force-based theory of universal gravitation based on Kepler's laws of planetary motion.[3]
Contents
History
Historically, the apparent motions of the planets were first understood geometrically (and without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions.[4] Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were in fact (at least approximately) elliptical motions.[5]In the geocentric model of the solar system, the celestial spheres model was originally used to explain the apparent motion of the planets in the sky in terms of perfect spheres or rings, but after the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' position in the sky, more and more epicycles were required over time, and the model became more and more unwieldy.
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.[6] 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.
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, if the force of gravity propagated instantaneously. Newton showed that, for a pair of bodies, the orbits' sizes are in inverse proportion to their masses, and that the bodies revolve about their common center of mass. Where one body is much more massive than the other, it is a convenient approximation to take the center of mass as coinciding with the center of the more massive body.
Albert Einstein was able to show that gravity was due to curvature of space-time, and thus he was able to remove Newton's assumption that changes propagate instantaneously. In relativity theory, orbits follow geodesic trajectories which approximate very well to the Newtonian predictions. However there are differences that can be used to determine which theory describes reality more accurately. Essentially all experimental evidence that can distinguish between the theories agrees with relativity theory to within experimental measurement accuracy, but the differences from Newtonian mechanics are usually very small (except where there are very strong gravity fields and very high speeds). The first calculation of the relativistic distortion came from the speed of Mercury's orbit and the strength of the solar gravity field because these are enough to cause Mercury's orbital elements to change.
However, Newton's solution is still used for most short term purposes since it is significantly easier to use.
Planetary orbits
Within a planetary system, planets, dwarf planets, asteroids (a.k.a. minor planets), comets, and space debris orbit the barycenter in elliptical orbits. A comet in a parabolic or hyperbolic orbit about a barycenter is not gravitationally bound to the star and therefore is not considered part of the star's planetary system. Bodies which are gravitationally bound to one of the planets in a planetary system, either natural or artificial satellites, follow orbits about a barycenter near that planet.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 in 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 from each other. (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 elliptical orbit, the center of mass of the orbiting-orbited system is at one focus of both orbits, with nothing present at the other focus. As a planet approaches periapsis, the planet will increase in speed, or velocity. As a planet approaches apoapsis, its velocity will decrease.
Understanding orbits
There are a few common ways of understanding orbits:- As the object moves sideways, it falls toward the central body. However, it moves so quickly that the central body will curve away beneath it.
- A force, such as gravity, pulls the object into a curved path as it attempts to fly off in a straight line.
- As the object moves sideways (tangentially), it falls toward the central body. However, it has enough tangential velocity to miss the orbited object, and will continue falling indefinitely. This understanding is particularly useful for mathematical analysis, because the object's motion can be described as the sum of the three one-dimensional coordinates oscillating around a gravitational center.
If the cannon fires its ball with a low initial velocity, the trajectory of the ball curves downward and hits the ground (A). As the firing velocity 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 velocity, 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 velocity (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 velocity is increased beyond this, 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 elliptical orbits at slower velocities; these will come closest to the Earth at the point half an orbit beyond, and directly opposite, the firing point.
At a specific velocity called escape velocity, again dependent on the firing height and mass of the planet, an open orbit such as (E) results – a parabolic trajectory. At even faster velocities 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".
The velocity relationship of two moving objects with mass can thus be considered in four practical classes, with subtypes:
- No orbit
- Suborbital trajectories
- Range of interrupted elliptical paths
- Orbital trajectories (or simply "orbits")
- Range of elliptical paths with closest point opposite firing point
- Circular path
- Range of elliptical paths with closest point at firing point
- Open (or escape) trajectories
- Parabolic paths
- Hyperbolic paths
Then, their orbits keep them above the atmosphere. If e.g., an elliptical orbit dips into dense air, the object will lose speed and re-enter (i.e. fall).
Newton's laws of motion
This section is too long to read comfortably, and needs subsections. (December 2013) |
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 without limit as they approach zero separation, and it is convenient and conventional to take the potential energy as zero when they are an infinite distance apart, and then negative (since it decreases from zero) for smaller finite distances.
With two bodies, an orbit is a conic section. The orbit can be open (so the object never returns) or closed (returning), depending 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, always less. 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 have negative total energy, parabolic trajectories have zero total energy, and hyperbolic orbits have positive total energy.
An open orbit has the shape of a hyperbola (when the velocity is greater than the escape velocity), or a parabola (when the velocity is exactly the escape velocity). The bodies approach each other for a while, curve around each other around the time of their closest approach, and then separate again forever. This may be the case with some comets if they come from outside the solar system.
A closed orbit has the shape of an ellipse. In the special case that the orbiting body is always the same distance from the center, it is also the shape of a circle. Otherwise, the point where the orbiting body is closest to Earth is the perigee, called periapsis (less properly, "perifocus" or "pericentron") when the orbit is around a body other than Earth. The point where the satellite is farthest from Earth is called 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.
Orbiting bodies in closed orbits repeat their paths after a constant period of time. 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 the 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 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 around 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 around its orbit during a fixed amount of time, the line from the Sun to planet sweeps a constant area of the orbital plane, 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.
Instead, 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 will equal the mass 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 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. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.
Analysis of orbital motion
- (See also Kepler orbit, orbit equation and Kepler's first law.)
To analyze the motion of a body moving under the influence of a force which is always directed towards a fixed point, it is convenient to use polar coordinates with the origin coinciding with the center of force. In such coordinates the radial and transverse components of the acceleration are, respectively:
The equation of the orbit described by the particle is thus:
Orbital planes
Main article: Orbital plane (astronomy)
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
Main article: Orbital period
The orbital period is simply how long an orbiting body takes to complete one orbit.Specifying orbits
Main article: Orbital trajectory
See also: Keplerian elements
Six parameters are required to specify a Keplerian orbit
about a body. For example, the 3 numbers which describe the body's
initial position, and the 3 values which describe its velocity will
describe a unique orbit that can be calculated forwards (or backwards).
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:
- Inclination (i)
- Longitude of the ascending node (Ω)
- Argument of periapsis (ω)
- Eccentricity (e)
- Semimajor axis (a)
- Mean anomaly at epoch (M0).
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
Main article: 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. See statite for one such proposed use.
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.
Multiple gravitating bodies
Main article: n-body problem
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.Astrodynamics
Main article: Orbital mechanics
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).Earth orbits
Main article: List of orbits
This article should include a summary of List of orbits. See Wikipedia:Summary style for information on how to incorporate it into this article's main text. (December 2013) |
Scaling in gravity
The gravitational constant G has been calculated as:- (6.6742 ± 0.001) × 10−11 (kg/m3)−1s−2.
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.
These properties are illustrated in the formula (derived from the formula for the orbital period)
Further reading
- Andrea Milani and Giovanni F. Gronchi. Theory of Orbit Determination (Cambridge University Press; 378 pages; 2010). Discusses new algorithms for determining the orbits of both natural and artificial celestial bodies.
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