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 central mass being orbited at a focal point of the ellipse, as described by Kepler\'s laws of planetary motion . Current understanding of the mechanics of orbital motion is based on
CONTENTS * 1 History * 2 Planetary orbits * 2.1 Understanding orbits * 3 Newton\'s laws of motion * 3.1 Newton\'s law of gravitation and laws of motion for two-body problems * 3.2 Defining gravitational potential energy * 3.3 Orbital energies and orbit shapes * 3.4 Kepler\'s laws * 3.5 Limitations of Newton\'s law of gravitation * 3.6 Approaches to many-body problems * 4 Newtonian analysis of orbital motion
* 5 Relativistic orbital motion
* 6 Orbital planes
* 7
* 9 Orbital perturbations * 9.1 Radial, prograde and transverse perturbations
* 9.2
* 10 Strange orbits
* 11 Astrodynamics
* 12
HISTORY Part of a series on SPACEFLIGHT HISTORY *
APPLICATIONS *
SPACECRAFT *
*
*
LAUNCH *
DESTINATIONS * Sub-orbital
* Orbital
*
SPACE AGENCIES * ESA
* CSA
* RKA
*
PRIVATE SPACEFLIGHT *
* v * t * e 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
The basis for the modern understanding of orbits was first formulated
by
Advances in
PLANETARY ORBITS Within a planetary system , planets, dwarf planets , asteroids and other minor planets , comets , and space debris orbit the system's 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 or within 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 ,
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 will be above the Earth's
atmosphere, which comes to the same thing). Newton\'s cannonball
, an illustration of how objects can "fall" in a curve 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
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
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: * 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 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
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. This may be the case with most comets if they come from outside the solar system. 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
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
LIMITATIONS OF NEWTON\'S LAW OF GRAVITATION Note that while bound orbits of a point mass or a spherical body with
a
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 (See also
The
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. F 2 = G m 1 m 2 r 2 {displaystyle F_{2}=-{frac {Gm_{1}m_{2}}{r^{2}}}} 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. F 2 = m 2 A 2 {displaystyle F_{2}=m_{2}A_{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: G m 1 m 2 r 2 = m 2 A 2 {displaystyle -{frac {Gm_{1}m_{2}}{r^{2}}}=m_{2}A_{2}} Solving for the acceleration, A2: A 2 = F 2 m 2 = 1 m 2 G m 1 m 2 r 2 = r 2 {displaystyle A_{2}={frac {F_{2}}{m_{2}}}=-{frac {1}{m_{2}}}{frac {Gm_{1}m_{2}}{r^{2}}}=-{frac {mu }{r^{2}}}} where {displaystyle mu ,} is the standard gravitational parameter , in this case G m 1 {displaystyle Gm_{1}} . 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 A = F / m = k r . {displaystyle A=F/m=-kr.} Due to the way vectors add, the component of the force in the x {displaystyle {hat {mathbf {x}}}} or in the y {displaystyle {hat {mathbf {y}}}} directions are also proportionate to the respective components of the distances, r x = A x = k r x {displaystyle r''_{x}=A_{x}=-kr_{x}} . Hence, the entire analysis can be done separately in these dimensions. This results in the harmonic parabolic equations x = A cos ( t ) {displaystyle x=Acos(t)} and y = B sin ( t ) {displaystyle y=Bsin(t)} of the ellipse. In contrast, with the decreasing relationship A = / r 2 {displaystyle A=mu /r^{2}} , the dimensions cannot be separated. The location of the orbiting object at the current time t
{displaystyle t} is located in the plane using
We use r {displaystyle {dot {r}}} and {displaystyle {dot {theta }}} 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 t {displaystyle t} from that at time t + t {displaystyle t+delta t} and dividing by t {displaystyle delta t} . The result is also a vector. Because our basis vector r {displaystyle {hat {mathbf {r}}}} moves as the object orbits, we start by differentiating it. From time t {displaystyle t} to t + t {displaystyle t+delta t} , the vector r {displaystyle {hat {mathbf {r}}}} keeps its beginning at the origin and rotates from angle {displaystyle theta } to + t {displaystyle theta +{dot {theta }} delta t} which moves its head a distance t {displaystyle {dot {theta }} delta t} in the perpendicular direction {displaystyle {hat {boldsymbol {theta }}}} giving a derivative of {displaystyle {dot {theta }}{hat {boldsymbol {theta }}}} . r = cos ( ) x + sin ( ) y {displaystyle {hat {mathbf {r}}}=cos(theta ){hat {mathbf {x}}}+sin(theta ){hat {mathbf {y}}}} r t = r = sin ( ) x + cos ( ) y = {displaystyle {frac {delta {hat {mathbf {r}}}}{delta t}}={dot {mathbf {r}}}=-sin(theta ){dot {theta }}{hat {mathbf {x}}}+cos(theta ){dot {theta }}{hat {mathbf {y}}}={dot {theta }}{hat {boldsymbol {theta }}}} = sin ( ) x + cos ( ) y {displaystyle {hat {boldsymbol {theta }}}=-sin(theta ){hat {mathbf {x}}}+cos(theta ){hat {mathbf {y}}}} t = = cos ( ) x sin ( ) y = r {displaystyle {frac {delta {hat {boldsymbol {theta }}}}{delta t}}={dot {boldsymbol {theta }}}=-cos(theta ){dot {theta }}{hat {mathbf {x}}}-sin(theta ){dot {theta }}{hat {mathbf {y}}}=-{dot {theta }}{hat {mathbf {r}}}} We can now find the velocity and acceleration of our orbiting object. O = r r {displaystyle {hat {mathbf {O}}}=r{hat {mathbf {r}}}} O = r t r + r r t = r r + r {displaystyle {dot {mathbf {O}}}={frac {delta r}{delta t}}{hat {mathbf {r}}}+r{frac {delta {hat {mathbf {r}}}}{delta t}}={dot {r}}{hat {mathbf {r}}}+r} O = + {displaystyle {ddot {mathbf {O}}}=+} = r + {displaystyle ={hat {mathbf {r}}}+{hat {boldsymbol {theta }}}} The coefficients of r {displaystyle {hat {mathbf {r}}}} and {displaystyle {hat {boldsymbol {theta }}}} give the accelerations in the radial and transverse directions. As said, Newton gives this first due to gravity is / r 2 {displaystyle -mu /r^{2}} and the second is zero. r r 2 = r 2 {displaystyle {ddot {r}}-r{dot {theta }}^{2}=-{frac {mu }{r^{2}}}} (1) r + 2 r = 0 {displaystyle r{ddot {theta }}+2{dot {r}}{dot {theta }}=0} (2) Equation (2) can be rearranged using integration by parts. r + 2 r = 1 r d d t ( r 2 ) = 0 {displaystyle r{ddot {theta }}+2{dot {r}}{dot {theta }}={frac {1}{r}}{frac {d}{dt}}left(r^{2}{dot {theta }}right)=0} We can multiply through by r {displaystyle r} because it is not zero unless the orbiting object crashes. Then having the derivative be zero gives that the function is a constant. r 2 = h {displaystyle r^{2}{dot {theta }}=h} (3) which is actually the theoretical proof of Kepler\'s second law (A
line joining a planet and the
In order to get an equation for the orbit from equation (1), we need
to eliminate time. (See also
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: u ( ) = h 2 A cos ( 0 ) {displaystyle u(theta )={frac {mu }{h^{2}}}-Acos(theta -theta _{0})} 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 e h 2 A / {displaystyle eequiv h^{2}A/mu } be the eccentricity , letting a h 2 / ( ( 1 e 2 ) ) {displaystyle aequiv h^{2}/(mu (1-e^{2}))} be the semi-major axis. Finally, letting 0 0 {displaystyle theta _{0}equiv 0} so the long axis of the ellipse is along the positive x coordinate. r ( ) = a ( 1 e 2 ) 1 e cos {displaystyle r(theta )={frac {a(1-e^{2})}{1-ecos theta }}} RELATIVISTIC ORBITAL MOTION The above classical (Newtonian ) analysis of orbital mechanics assumes that the more subtle effects of general relativity , such as frame dragging and gravitational time dilation are negligible. Relativistic effects cease to be negligible when near very massive bodies (as with the precession of Mercury\'s orbit about the Sun), or when extreme precision is needed (as with calculations of the orbital elements and time signal references for GPS satellites. ) ORBITAL PLANES Main article:
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:
The orbital period is simply how long an orbiting body takes to complete one orbit. SPECIFYING ORBITS Main article:
Six parameters are required to specify a
The traditionally used set of orbital elements is called the set of
*
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 Main article:
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.
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 Main article: n-body problem The effects of other gravitating bodies can be significant. For
example, the orbit of the
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 Main article:
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.
EARTH ORBITS Main article:
* Low
SCALING IN GRAVITY The gravitational constant G has been calculated as: * (6.6742 ± 0.001) × 10−11 (kg/m3)−1s−2. 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. These properties are illustrated in the formula (derived from the formula for the orbital period ) G T 2 = 3 ( a r ) 3 , {displaystyle GT^{2}sigma =3pi left({frac {a}{r}}right)^{3},} for an elliptical orbit with semi-major axis a, of a small body around a spherical body with radius r and average density σ, 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. SEE ALSO * Astronomy portal
*
*
REFERENCES * ^ orbit (astronomy) – Britannica Online Encyclopedia
* ^ The Space Place :: What\'s a Barycenter
* ^ Kuhn, The Copernican Revolution, pp. 238, 246–252
* ^ Encyclopædia Britannica, 1968, vol. 2, p. 645
* ^ M Caspar, Kepler (1959, Abelard-Schuman), at pp.131–140; A
Koyré, The Astronomical Revolution: Copernicus, Kepler, Borelli
(1973, Methuen), pp. 277–279
* ^ Jones, Andrew. "Kepler\'s Laws of Planetary Motion". about.com
. Retrieved 2008-06-01.
* ^ See pages 6 to 8 in Newton\'s "Treatise of the System of the
World" (written 1685, translated into English 1728, see Newton\'s
\'Principia\' – A preliminary version ), for the original version of
this 'cannonball' thought-experiment.
* ^ Fitzpatrick, Richard (2006-02-02). "Planetary orbits".
Classical Mechanics – an introductory course. The University of
Texas at Austin. Archived from the original on 3 March 2001. Retrieved
2009-01-14.
* ^ Pogge, Richard W.; "Real-World Relativity: The GPS Navigation
System". Retrieved 25 January 2008.
* ^ Iorio, L. (2011). "Perturbed stellar motions around the
rotating black hole in Sgr A* for a generic orientation of its spin
axis".
FURTHER READING * Abell; Morrison & Wolff (1987). Exploration of the Universe (fifth ed.). Saunders College Publishing. * Linton, Christopher (2004). From Eudoxus to Einstein. Cambridge: University Press. ISBN 0-521-82750-7 * Swetz, Frank; et al. (1997). Learn from the Masters!. Mathematical Association of America. ISBN 0-88385-703-0 * 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. EXTERNAL LINKS Look up ORBIT in Wiktionary, |